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UniversiW M ioO Tlnns International 300 N. Zeeb Road Ann Arbor, Ml 48106
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St-Pierre, Normand Roger
MINIMUM COST REQUIREMENTS FROM A RESPONSE FUNCTION AND INCORPORATION OF UNCERTAINTY IN COMPOSITION OF FEEDS INTO CHANCE-CONSTRAINED PROGRAMMING MODELS OF LIVESTOCK RATIONS
The Ohio State University Ph.D. 1985
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University Microfilms International
MINIMUM COST REQUIREMENTS FROM A RESPONSE FUNCTION AND
INCORPORATION OF UNCERTAINTY IN COMPOSITION OF FEEDS INTO
CHANCE-CONSTRAINED PROGRAMMING MODELS OF LIVESTOCK RATIONS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Normand Roger St-Pierre, B.S., M.S.
*****
The Ohio State University
1985
Reading Committee: Approved By
W.R. Harvey
C.S . Thra en
F .R. Walker ,1. / ^ , H.R. Conrad -WiïHîV.lt J.'> Advis er Department of Dairy Science Copyright by Normand Roger St-Pierre 1985 "Truth is so large a target that nobody can wholly miss
hitting it, but at the same time, nobody can hit all of it
with one throw".
Aristotle (384-330 BC)
"Science not founded on exact experience and mathematics are either deception or madness - a banner of charlatans, blown full by the wind, after which the foolish rabble flocks".
Lenordo Da Vinci (1452-1519)
- ii - ACKNOWLEDGEMENT
The author wants to express his sincere appreciation to Dr.
W.R. Harvey for being such a devoted adviser during the time this research was conducted. His motto 'think' should always inspire me.
Many thanks to Drs. H.R. Conrad, F.R. Walker and C.S.
Thraen for their time in reviewing this manuscript. Dr.
Conrad has been and will always be a model of excellence to me. Dr. Walker was very instrumental in my formation in mathematical programming. Dr. Thraen's interest in joining expertise from our two respective departments has been a keypoint in the initiation of this research.
Special thanks to my parents, brothers and sisters for their constant support. My parents have always emphasized the importance of education. Now, I finally understand why.
And thanks to all my fellow graduate students, especial ly Paul Ferguson and Bill Weiss. Their friendship will be remembe red.
Dee, there is no words good enough for you. This dis sertation is a partial result of your patience and under- stending.
- iii - VITA
October 1, 1955 . Born - Montreal, Canada
19 78 ...... B.Sc., Université Laval, Quebec city, Canada
1980 ...... M.Sc., Université Laval, Quebec city, Canada
1979-1981 . . . . Animal nutritionist. Cooperative Federe de Quebec, Montreal, Canada
1981- ...... Research scientist, Agriculture- Canada, Lennoxville, Canada
PUBLICATIONS
St-Pierre, N.R. 1980. Effets du gel et du stade de maturi té sur la valeur alimentaire de l'ensilage de mais pour la vache laitiere. M.Sc. Thesis, Université Laval, Quebec city, Canada, april 1980.
St-Pierre, N.R., R. Bouchard, G.J. S t-Laurent, C . Vinet and G.L. Roy. 1983. Effects of stage of maturity and frost on nutritive value of corn silage for lactating dairy cows. J. Dairy Sci. 66 : 1466-1473.
St-Pierre, N.R., R. Bouchard, G.J. S t-Laurent, C . Vinet and G.L. Roy. Performance of dairy cows fed corn silage affec ted by frost and evaluation of corn silage energy values. J. Dairy Sci. (submitted, accepted).
St-Pierre, N.R., R. Bouchard, G.J. S t-Laurent and G.L Roy. Relationship between chemical composition of corn and In vitro dry matter digestibility. Can. J. Anim. Sci. (submitted).
- IV - St-Pierre, N.R., R. Bouchard and J.G. St-Laurent. 1980. Effect of frost on the nutritive value of corn silage fed to lactating dairy cows. J. Dairy Sci. 63(suppl) : 151.
St-Pierre, N.R., R. Bouchard and G.J. St-Laurent. 1980. Performance of dairy cows fed corn silage affected by frost and evaluation of corn silage energy values. J. Dairy Sci. 63(suppl): 151(abstract) .
St-Pierre, N.R., C.S. Thraen and W.R. Harvey. 1984. Energy-protein substitution in the growth of feeder calves; evaluation of alternative models. J. Anim Sci 59(supple- ment 1): 413 (abstract).
St-Pierre, N.R., C.S. Thraen and W.R. Harvey. Impact of energy-protein substitution on growth of feeder calves : a biological and economic analysis. Ohio Agric. Res. Bull, (submitted, accepted).
St-Pierre, N.R., C.S. Thraen and W.R. Harvey. Nutrient energy/protein substitution in feeder cattle; a study in applied production economics. Amer. J. Agric. Econ. (submi t ted).
FIELDS OF STUDY
Major Field: Management
Studies in Dairy Science and Management. Professors R.M. Porter, D.E. Pritchard, N.S. Fech- he imer.
Studies in Statistical Methods. Professors W.R. Harvey, D. Kikuchi, H.N. Nagaraja, J.S. Verducci, T.A. Willke, D.A. Wolfe.
Studies in Animal Breeding and Population Genetics. Professors W.R. Harvey, F.R. Allaire, M.E. Davies, K .M. Irvin.
Studies in Mathematics, Systems Simulation and Optimiza tion. Professors A. Cronheim, G . Papalios, J.P. Klein, F.R. Walker
Studies in Agricultural Economics. Professors D.D. Southgathe, T.T. Stout, C.S. Thraen
- V - TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... ill
VITAE ...... iv
LIST OF TABLES...... vili
LIST OF F I G U R E S ...... xii
Chapt er page
I. INTRODUCTION ...... 1
II. ENERGY-PROTEIN RESPONSE FUNCTIONS FOR GROWTH . . . 6
INTRODUCTION ...... 6 FROM THE PHYSIOLOGY TO A PRODUCTION FUNCTION ...... 8 METHODOLOGY AND D A T A ...... 13 A n i m a l s ...... 13 D i e t s ...... 13 Statistical methods andmodels ...... 16 Polynomial model...... 16 Non-linear model...... 18 Transcendental logarithmic production function...... 18 C obb-Douglas...... 20 STATISTICAL RESULTS ...... 20 INTERPRETATION ...... 32 P h y s i c a l ...... 32 E c o n o m i c ...... 42 CONCLUSION...... 57
III. UNCERTAINTY IN COMPOSITION OF FEEDSTUFFS .... 59
INTRODUCTION ...... 59 LITERATURE REVIEW ...... 63 Least-cost ration as a linear programming problem...... 63 The mo de l ...... 63
— vi — The assumptions...... 64 Consequences of uncertainty ...... 66 Mathematical programming under uncertainty...... 68 Stochastic programming ...... 68 Simple-chanca-constrained programming . 71 Joi nt-chance-cons trained programming . . 78 Least-cost and max-profit rations under uncertainty...... 83 MATERIAL AND M E T H O D S ...... 90 Estimation of Var-Cov matrices of f eeds tuf f s ...... 90 Distribution of nutrients within feeds tuf f s ...... 94 Solutions to the SCCP p r o b l e m ...... 95 Solutions to JCCP p r o b l e m s ...... 107 RESULTS AND DISCUSSION ...... 121 Composition of feedstuffs ...... 121 S.C.C.P. problem ...... 138 J.C.C.P. problem ...... 149 CONCLUSION...... 175
IV. CONCLUSION ...... 177
REFERENCES C I T E D ...... 180
APPENDIX...... 190
- vi i - LIST OF TABLES
Table Page
1. Body weight, daily gain, and energy and protein intake of a n i m a l s ...... 15
2. Estimated response functions. Partial regres sion coefficient estimates, standard errors and goodness of fit statistics...... 21
3. Marginal rate of technical substitution (MRTS) of protein for energy at the NRC and ARC ad justed recommended nutrient requirements for three rates of gain (body weight set at 200 k g ) ...... 36
4. Elasticity of factor substitution ( p ) for three levels of MRTS and three rates of gain (body weight set at 200 k g ) ...... 37
5. Elasticity of MRTS at three levels of MRTS and three rates of gain (body weight set at 200 k g ) ...... 39
6. Annual average U.S. prices of corn grain, soybean meal, their ratio of prices and estimated prices per unit of ME and CP, from 1949 to 1980 ...... 46
7. Points of minimum costs, for three rates of gain (body weight set at 200 kg)...... 47
8. Comparison of the mean recommended nutrient le vels generated by the POLY model, with the adjusted recommendations by the NRC and ARC . . . 48
9. Comparison of the amounts of corn and SBM required at the points of minimum costs for a gain of 0.5 kg/day at a body weight of 200 kg . . 50
- viii - 10. Average cost of being wrong for a 0.5 kg/day rate of gain with annual average U.S. prices for 1949 to 1980 at a body weight of 200 kg. . . 55
11. Average annual prices of feeds tuffs used in the feed formulation problem ...... 96
12. Approximate Rahman-Bender algorithm with A fixed (RBF) . Model for the premix feed...... 99
13. Iterated Taylor series expansion algorithm (ITS). Model for the premix feed...... 104
14. General nonlinear programming algorithm (NLP). Model for the premix feed...... 106
15. Industry standards model (IS) for the supple ment feed...... Ill
16. Rahman-Bender estimate of X with Bonferroni inequality (RBB). Model for the supplement f e e d ...... 113
17. Bonferroni inequality with equal weight on row's probability (BEW). Model for the sup plement f e e d ...... 114
18. Bonferroni inequality with equal weight on row's probability (BEW). Model for the total mixed ration ( T M R ) ...... 115
19. Bonferroni inequality with unequal weight on row's probability (BUW). Model for the sup plement f e e d ...... 116
20. Bonferroni inequality with unequal weight on row's probability (BUW). Model for the total mixed ration ( T M R ) ...... 117
21. Overall probability by multivariate integra tion (MI) algorithm. Model for the total mixed ration ( T M R ) ...... 119
22. Univariate statistics on feed composition. . . . 122
23. Solution vectors and related statistics for the SCCP problem solved under a conventional deterministic LP model ...... 139
- IX - 24. Solution vectors and related statistics for the SCCP problem (premix) solved with the RBF algorithm...... 141
25. Solution vectors and related statistics for the SCCP problem (premix) solved with the IRB algorithm...... 143
26. Solution vectors and related statistics for the SCCP problem (premix) solved with the ITS algorithm...... 145
27. Solution vectors and related statistics for the SCCP problem (premix) solved with the NLP algorithm...... 146
28. Cost of ingredients for the solution vectors of the SCCP problem (premix)...... 148
29. Solution vectors and related statistics for the JCCP problem (supplement feed) solved with the IS model...... 150
30. Solution vectors and related statistics for the JCCP problem (grain mix) solved with the IS m o d e l ...... 151
31. Solution vectors and related statistics for the JCCP problem (supplement feed) solved with the RBB m o d e l ...... 153
32. Solution vectors and related statistics for the JCCP problem (grain mix) solved with the RBB model...... 15 4
33. Solution vectors and related statistics for the JCCP problem (supplement feed) solved with the BEW m o d e l ...... 156
34. Solution vectors and related statistics for the JCCP problem (grain mix) solved with the BEW model...... 157
35. Solution vectors and related statistics for the JCCP problem (supplement feed) solved with the BUW m o d e l ...... 159
- X - 36. Solution vectors and related statistics for the JCCP problem (grain mix) solved with the BUW model...... 161
37. Average achieved rates of success found by Monte Carlo experiments for the solution vectors of the JCCP problem (supplement feed and grain m i x ) ...... 163
38. Solution vectors and related statistics for the JCCP problem (TMR) solved with the deterministic and BEW mo d e l s ...... 165
39. Solution vectors and related statistics for the JCCP problem (TMR) solved with the BUW mode 1 ...... 16 7
40. Solution vectors and related statistics for the JCCP problem (TMR) solved with the MI m o d e l ...... 169
41. Achieved global rates of success found by Monte Carlo experiments for the solution vectors of the TMR p r o b l e m ...... 171
42. Variance-covariance matrices of nutrient within feedstuffs...... 191
43. Correlation coefficients between nutrient within feedstuffs...... 231
- XI - LIST OF FIGURES
Figure page
1. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POLY m o d e l ...... 22
2. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the NLIN model. . 23
3. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the POWER model . 24
4. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the TRANSLOG model...... 25
5. Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 100 kg body weight for the POLY model...... 27
6. Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POLY model...... 28
7. Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the NLIN model...... 29
8. Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POWER m o d e l ...... 30
- xii - 9. Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the TRANSLOG model...... 31
10. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for a fixed factor m o d e l ...... 33
11. Pattern of residuals expected if the fixed factor is the 'true' model and a smooth function is f i t t e d ...... 34
12. Comparison of least-cost input combination for alternative production models...... 52
13. Comparison of alternative production models with equal input cost for energy and protein but lower gross revenues...... 54
14. Illustration of the first 2 stages in the golden search...... 102
15. Illustration of the effect of a random input when output is a nonlinear function of input level. . . 174
- xi±i - Chapter I
INTRODUCTION
Feeding cost is the biggest input cost in livestock produc tion. It is estimated that feeding cost represents 50% of the total cost of production in dairy and over 7U% in beef.
Therefore, overall economic efficiency of production is largely dependent on the economic efficiency of feeding.
This dissertation deals with feeding efficiency, or how best to equate nutritional inputs with nutritional outputs.
Here, nutritional outputs refer to nutrient requirements and will be examined first. Thereafter, we will focus our attention on nutritional inputs, or how feeds can be best combined to provide the required nutrients.
An important step in the development of efficient feed ing models is the assesment of nutrient requirements for the various classes of livestock. At present, animal sci entists have placed their best estimates of requirements in the form of tables, which differ substantially from one world authority to the other. The first problem encoun tered when trying to explain these differences is that the assumptions and models underlying the tables are often
- 1 - 2 poorly stated. A second problem is that requirements for each individual nutrient are expressed as a function of output and weight as in:
(1) = g(0 1 W)
where
N = requirements for nutrient i
g = some unspecified function
0 = level of output
W = body weight
This form does not allow the expression of interactions be tween classes of nutrients. In a recent review, Oldham
(1984) has reported numerous results showing that energy and protein interact. Now, if a function of the general form
(2) 0 = f (N^ Np I W) is taken instead of (1), then various mathematical forms
for the function f can be proposed, allowing for varying degrees of interaction (substitution) between inputs. Un
less the interaction is null, then model (2) implies that
minimum cost requirements must vary with the relative pric
es of nutrients, and cannot be put in tabulated forms.
Traditionaly, agricultural economists have by-passed the
problem by fitting a function of the form: (3) 0 = h(I^IW) i n ' where
h = some unspecified function
= input level of feed i.
Examples of this can be found in Epplin et al. (1980) and
Heady and Bhide (1983). Problems with this approach are numerous :
1. Different functions must be fit for each combination
of feedstuffs. Since the number of feeds tuffs is
large, an almost infinite number of functions have
to be fitted.
2. It fails to recognize the non-uniformity within each
feedstuff. This research will report results show
ing tremendous levels of variation in composition
within feedstuffs.
3. It does not recognize the almost linear rate of sub
stitution between feeds tuffs. Clear examples of
this are found in Epplin et al. (1980).
Chapter 2 reports results obtained when various func- tions of the type expressed by (2) were fitted. Average daily gain of beef calves was the dependent variable, while metabolizable energy and crude protein intake were the nu tritional inputs. Eventhough direct application of the re sults is not warranted, the methodology presented should be instrumental in the future development of reliable esti mates of nutrient requirements. 4
In chapter 3, the problem of optimal combination of feeds as nutritional inputs is reevaluated. The current standard method for solving such a problem is linear pro gramming (LP). However, there are some major assumptions in LP, one of which relates to deterministic parameters.
This assumption cannot hold in the feed formulation problem since feed composition is never known with certainty. Very little work has been reported on ways to improve over regu lar LP solutions. Van de Panne and Popp (1963) have shown a model for minimum-cost cattle feed under probabilistic protein constraints. However, the size of their model was ridiculously small and uncharacteristic of problems seen in practice. Nott and Comb (1967) simplified the problem of probabilistic constraints to the point where results are seriously biased. The same can be said for the. work re ported by Rahman and Bender (1971). Reyes et al. (1981) used a dynamic approach to ration formulation, setting the problem as a multistage LP model. Their results showed promises. Recently, the ration formulation problem has been presented as a mult ip le-goal programming problem (Reh- man and Romero, 1984). Appropriate weights to be put on each specific goal are still totally undetermined.
But what is wrong with using LP for ration formulation?
Could it be that one is desperately trying to find tiny faults in a method which appears to be so successful? The 5 answer resides in the assesment of success itself. This research will report clear evidences of substantial levels of variation in the composition of feedstuffs. This im plies that some parameters in the LP are stochastic.
Whether performances of an LP model are affected to a sig nificant extent by such a deviation from the assumption of deterministic parameters remains to be determined. Either the LP model will prove to be 'robust' enough so that devi ations from the assumption can be ignored, or its perform ances will be affected, requiring a different method to tackle the problem. In chapter 3, different methods are proposed for dealing with uncertainty in composition of feedstuffs. When applied to practical problems and com pared to regular LP solutions, some of these methods showed performances that should warrant their utilization, espe cially by the feed industry. Chapter II
ENERGY-PROTEIN RESPONSE FUNCTIONS FOR GROWTH
2.1 INTRODUCTION
Feed costs represent approximately 70% of the total costs of production for a feedlot enterprise. Production re searchers have dedicated much of their efforts toward ob taining more reliable estimates of animal requirements for various classes of nutrients. These requirements are used
typically as constraints in least-cost ration models. With this approach, the composition and the cost of the ration is dependent largely on the values used as requirements.
However, world authorities in animal nutrition, the Agri cultural Research Council (ARC), the Institut National de la Recherche Agronomique (INRA) and the National Research
Council (NRC), do not agree on the magnitude of these re quirements. It is unclear which set of requirements is
'best' under U.S.A. conditions. Least-cost rationing based on linear mathematical programming also assumes no substi tution between major classes of nutrients such as energy and protein, since the requirements are fixed tabular val ues for a given physiological state. However, both the NRC and the ARC have stated that protein requirements are de-
— 6 — 7
pendent on energy concentration, which Implies indirectly
that substitution does exist.
Failure to account explicitly for energy-protein substi
tution can lead to lost profit opportunities for livestock
feeding enterprises. Profit maximization in an environment
of changing relative prices for energy and protein necessi
tates shifts in the energy-proteln ratio. This requires
detailed knowledge of the underlying production technology
and the explicit tradeoffs between energy and protein in
the process of weight gain in livestock. Different func
tional forms can be chosen, some of which offer a response
surface that is more acceptable on the grounds of actual
biological theory, while some others give a more tractable
function in terms of economic optimum.
This work attempts to combine biological considerations with production economics theory in the study of nutrition
al requirements. The objectives are:
1. to show the applicability of the principle of ener-
gy-protein substitution in the growth of cattle,
2. to empirically substantiate the substitution princi
ple by directly estimating the weight gain produc
tion function,
3. to analyze the empirical results both from a biolo
gical and an economics point of view, including the
potential costs incured by a misspecification of the
production function form. 8
The first section reviews the physiology and the
problems associated with protein and energy requirements of
growing livestock. In section II, we report the results
obtained when different functional forms were fitted to re
late gain as a function of weight, energy and protein in
take. Section III calculates the cost of using misspeci-
fied functional forms as the basis of least-cost rationing.
We conclude on the basis of the sample data used and the best production function estimates, that substitution of
energy and protein is possible in the feeding of livestock.
However, the elasticity of substitution is small and there fore the NRC and ARC recommendations, which do not allow substitution of energy-protein, do not result in signifi cantly higher feed costs per unit of weight gain.
2.2 FROM THE PHYSIOLOGY TO A PRODUCTION FUNCTION
Energy concentration in the diet affects protein utiliza
tion (Garret, 1977; Journet and Verite, 1977; Orskov, 1977;
Preston, 1972 and 1980). Energy and protein interact be cause protein can be used by the animal as a source of en ergy. Protein is also a necessary nutrient for various es sential synthetic processes in the body. Amino acids, the components of protein, are required by all cells and tis sues in the body. Many secretions in the body, including enzymes, hormones, mucin, and milk require specific assort 9
ments of amino acids. Hence there exists a strict minimum
requirement for protein at a given physiological state,
which is essentially the quantity of protein needed to cov
er the essential functions. Above this minimum, protein
can be used as energy but with a different efficiency than
lipids and carbohydrates. This implies that within pre
scribed limits, protein and energy may be substituted for
each other with no detrimental effects on animal health or
rate of weight gain.
The National Research Council (NRC) estimates protein
requirements by a factorial method, which attempts to meas
ure separately the different components of the total re
quirements and to vary each component as changes in age and
functional activity of the animal require (Blaxter and
Mitchell, 1949). The problem with this approach is that it
does not take into account interactions between the compo
nents of requirements which are likely to exist, and the
influence of the non-protein fraction on nitrogen metabo lism.
The British Agricultural Research Council (ARC), and the
Institut National de la Recherche Agronomique (INRA) in
France, use a different approach to estimate protein re quirements. They calculate the amount of amino acids-N of microbial origin that could be retained in the body for tissue synthesis when the maximal rate of fermentation of a 10 particular energy input is achieved. This amount of amino acid N is compared with the total tissue needs for the par ticular energy input. If the amount of microbial amino acid N available to the tissues is greater than the tissue needs, the N requirement is the amount of degraded N needed by the rumen microorganisms. If the microbial amino acid N is less than the tissue needs, then the difference must be supplied by amino acids from undegraded dietary protein.
Many of the coefficients used by the ARC and INRA are based on limited scientific information. As a result, the ARC and the INRA requirements differ widely (Waldo and Glenn,
1980 and 1984). Even though this approach does consider some effects of energy on the protein requirements by in- puting energy into the rate of fermentation, it does not take into account the basic principle of substitution be tween energy and protein. If energy can be substituted for protein and vice versa, with no adverse effect on animal health and weight gain, then feeding requirements for pro tein and energy should vary with the relative prices of en ergy and protein. Recommendations which do not recognize this potential substitution can lead to economic ineffi ciency over time.
Research studies have generally not addressed the possi bility of substitution. There are some exceptions however.
The ARC recognizes this substitution principle, but it does 11
not incorporate it into its working model (ARC, 1980 pl59).
Protein-energy subtitution has been well demonstrated in
poultry (Crampton and Harris, 1969). In 1966, Winchester
and Harvey reported a study on the effects of protein and
energy intake on nitrogen retention and growth of cattle.
Their results showed substitution of these two nutrients in a beef cattle ration. Their research design, while ad
dressing the question of energy-protein substitution, was
not complete. Body weight should have been included in the model since maintenance requirements increase with body size. Their parameter estimates were biased since weight varied among experimental units (Snedecor and Cochran, 1980 p353-54). Also, these authors only considered a polynomial model, while other alternatives could be considered.
To empirically test the hypothesis that significant en- ergy-protein substitution can be achieved in the feeding of livestock, a production function relating weight gain per day to protein and energy intake per day can be statisti cally estimated. From this estimated function, an auxil iary function or isoquant can be derived which relates the level of energy and protein required to maintain a given rate of weight gain. The curvature of this isoquant func tion is indicative of the degree of substitution between the energy and protein inputs. 12
The process of fitting a production function to observed
data is not without problems. Discussions of the proper
ties of some common functional forms used in agriculture
response studies are given in Heady and Dillon (1961).
Cady and Laird (1969) have shown that an incorrect choice of functional form can be a source of bias. Fuller (1969) has stated that "in practical situation, the choice of functional form will rest upon theoreticalconsiderations, ease of estimation and acceptance by the data". In a biol ogical process, a function is desired that:
1 . is continuous
2. possesses continous first derivatives
3. if. easy to estimate
4. gives a good statistical fit
5. has reasonable biological implications
6. permits easy computation of economic optima.
The last two characteristics tend to conflict. It is not uncommon to find different functions that satisfy require ments 1 to 4, but result in significant differences in both biological and economic implications. Hall (1983) has re ported such findings when different functional forms were used to estimate yields of three crops in response to lime application. Bay and Schoney (1982) reported similar re sults when corn yields, in response to nitrogen and potas sium fertilization, were predicted by four different meth- 13
ods. Lerman and Bie (1975) have shown some drastic
economic consequences from the choice of model which re
lates feeding costs to the percentage crude protein in the
diet. Their work also s.iowed the importance of adjusting
the animal requirements and implicitly the characteristics
of the diet, to the economic environment.
2.3 METHODOLOGY AND DATA
2.3.1 Animals
Data used in this study wereobtained from studies of com
pensatory growth in cattle subsequently reported by Win
chester and Harvey (1966). Eighty-four purebred, crossbred
or grade cattle of the beef type breeds were used plus 8
crossbred dairy-beef type animals, which were dropped from our analysis since they represent a different type of ani mal probably possessing a different response function.
Details of the management of the experimental animals and methods of measuring body weight have been reported elsewhere (Winchester and Ellis, 1957; Winchester, Hiner and Scarborough, 1957).
2.3.2 Diets
Diets were composed basically of corn and corn by-products
(corncobs, cornstarch), a protein source (linseed meal), supplemental minerals and vitamins, and a variable amount 14 of alfalfa hay (Winchester and Harvey, 1966). Diets ranged in energy levels too low to maintain body weight to those as high in energy as the animal would consume (Table 1).
Protein content ranged from 5 to 25 % crude protein on a dry matter basis.
Digestible protein and energy of the diets were estimat ed on the basis of chemical analysis. The original data were converted to current units used by the NRC. Digesti ble protein was converted to crude protein, using the equa tion reported by Winchester and Harvey (1966). Digestible energy was converted to metabolizable energy with the equa tion of Vermorel (1978). 15
TABLE 1
Body weight, daily gain, and energy and protein intake of aniraals .
Variable Mea n S.D. Minimum Maximum
Crude Protein (kg/day) 0.457 0.257 0.100 1.363
Met. Energy (Mcal/day) 7.136 3.200 2.220 15.110
Body Weight (kg) 162. 1 52.51 57.60 313.0
Daily Gain (kg/day) 0.310 0.272 -0.101 1.014 16
2.3.3 Statistical methods and models
To Investigate the hypothesis of substitution between ener
gy and protein, it is necessary to estimate an explicit
production function from the implicit function defined by:
(4) F(G,E,P,W) = 0
where
G = Rate of weight gain per day
E = Energy intake in Meal per day
P = Protein intake in kg per day
W = Weight of the animal in kg.
However, the implicit function does not indicate what alge
braic form the true response function should take. Recom
mendations by the NRC imply a function of the form:
(5 ) G = mi n 1. I «1 «2 where
units of nutrient i per unit of growth.
This functional form implies no substitution between energy and protein. In contrast to this fixed factor production function, four functions were estimated which allowed vary
ing degrees of energy/protein substitution. These four
functional forms were selected based on the criteria stated
earlier.
2.3.3.1 Polynomial model 17
The first model considered was an additive third degree
polynomial mooel with interaction terms for all variables.
Multiple least squares regression procedures were used to
estimate the parameters for crude protein intake, metaboli
zable energy intake and body weight, with rate of gain as a
dependent variable. The initial model was the following:
2 2 2 G =b„+bE+bP+bW+ b EW + b E + b P + b_EP 0 1 2 3 4 5 6 7
(6) + bgPW + bgW^ + b^gEPW + b^^E^p + b^gE^y + b^^EP^
+ bi^P^W +
+ e
where
E = Metabolizable energy intake (Mcal/day)
P = Crude protein intake (kg/day)
W = (W^- W) = corrected body weight
W^= Body weight of the ith animal (kg)
W = Average body weight of the n animals (163.59 kg)
G = Average daily gain of the ith animal (kg/day)
e = Random error N(0,(7^)
The type IV sum of squares was used to test only the last term in the model (highest degree). When non-signifi cant (Pr>0.10), the term was dropped and the reduced model 18 was fit in order to test for the next term. This procedure has been outlined by Freund and Littell (1981), and used by
Harvey (1977). In this paper, the polynomial model will be denoted as POLY.
2.3.3.2 Non-linear model
From the estimates obtained on the POLY model it was hy pothesized that the gain response would be strictly in creasing with energy, strictly decreasing with weight and curvilinear of a second order in protein. The initial mod el was ;
b b b, b^ G = b„+bW +bE +bP + bP + bW E 0 1 3 5 6 7 (7) ^2 ^4 ^2 ^4 + bgW P + bgE P + b^^W E P + e
where ;
W = Body weight ikg/100)
E , P and G are as defined in (6).
The NLIN procedure of SAS was used since the model is non linear in the parameters. This model will be referred to as NLIN.
2.3.3.3 Transcendental logarithmic production function 19
Generally referred to as a TRANSLOG function, this form has both linear and quadratic terms in the natural log of an arbitrary number of inputs (Berndt and Christensen, 1973).
It is considered a general form that imposes no separabili ty restrictions a priori, and is economically tractable.
This functional form has gained considerable popularity since it was first proposed (Christensen et al., 1971).
The specific model considered was:
G '= b^ + b^lnP + b^lnE + b^lnW + b^(lnP)^ + b^(lnE)^
(8) 2 + bg(lnW) + bylnP X InE + bglnE X InW + bglnP X InW
+ e
where:
W = Body weight (kg)
G' = ln(G + 1.0)
G = Average daily gain (kg/day)
P and E are as defined in (6).
The dependent variable gain was transformed by adding 1 kg so that there would be no attempt to take the log of a non-positive number. This figure should be close to the daily weight loss of fasting cattle. 20
2.3.3.4 Cobb-Douglas
This form, often referred to as the POWER function, has no
plateau (strictly increasing or decreasing in all the in
puts) and has been included since it is used widely. The
specific model considered was:
bi b£ b^ (9) G'=b^P E W e
where :
G', P, E, W are as defined in (8).
2.4 STATISTICAL RESULTS
The partial regression coefficient estimates for the four response functions, with their standard errors, are given in table 2 . Four terms in POLY and two in NLIN were de leted as a result of the stepdown procedure. The four mod els fit the data well with R squares higher than 0.90 and standard error of estimates less than .085 kg/day.
The functions are plotted in three dimensions in figures
1 to 4 . In each of these plots, body weight was set at
200 kg. This value is the closest weight to the average of this experiment for which the NRC and ARC give recommended requirements. It is obvious that the functional form has a drastic effect on the response surface. These differences will be quantified in a later section. 21
TABLE 2
Estimated response functions. Partial regression coef ficient estimates, standard errors and goodness of fit statistics.
Parameter NLIN TRANSLOG
-coefficient eatimatee (standard errors)-
bo -6.577E-1 -5.793E-1 3.004EOO 1.921E00 (4.417E-1) (2.407E-1 ) ( 1 .93OE0O) (O.412E0O)
bl 9.771E-2 -2.675E-1 -9.386E-1 7.503E-2 (15.10E-2) ( 1 . 2 7 1 E - 1 ) (6.151E-1) ( 1 . 8 7 8 E - 2 )
b2 1 .O32E00 -1.D14E00 1 .B45EOO 4.594E-1 ( 1 . 155E00) (O.4O8EO0) (O.795E0Q) ( 0 . 3 6 3 E - 1 )
b3 4.036E-3 1.43IE-1 - 1 .868EOO -2.361E-1 (4.229E-3) (0.07IE-1 ) ( 1 .081EOÛ) (0.378E-1 )
b4 -2.479E-3 4.425E-1 -9.157E-3 (1.225E-3) (1.941E-1) (27.93E-3)
bS -8.374E-3 -5.438E-1 3.860E-1 (13.00E-2) (2.831E-1) (3 . 1 9 2 E - 1 )
b6 -7.970E00 -8.276E-1 5.159E-1 (1.765E00) (2.267E-1) (3 . 2 4 0 E - 1 )
b7 7.249E-1 3.268E-1 -1.482E-1 (2.902E-1) ( 1 . 5 3 7 E - 1 ) (1.952E-1)
b8 2.375E-3 - -8.477E-1 (J0.46E-3) (5 . 9 92E-1)
b9 4.449E-5 6.263E-1 4.411E-1 (2.236E-5) ( 3 . 1 2 6 E - 1 ) ( 2 . 8 2 0 E - 1 )
blO -4.171E-4 (19.34E-4)
bl 1 4.911E-2 ( 2 . 5 5 3 E - 2 )
bl2 2 . 538E-4 (1.151E-4)
bl3 5.648E-1 ( 2 . 3 4 1 E - 1 )
bl4 -6.452E-3 (1.696E-3)
bli -B.OOOE-6 (3.480E-6)
R2 0.95 0.93 0.92 0.91 Sji . X 0.068 0.079 0.085 0.084 d • f • 60 67 66 72
bo to bjj refer to specific parameters given by each unique
cod e 1 discussed in the t e x t . A given b ^ does not refer to
the Game variable for each of the 4 models. 22
0.5
0.0
0.5
ADC
(KC/OPTI
0.0 'O
‘o
Figure 1. Average daily gain (ADC) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POLY model 23
0.5
0.0
y 0.5
ADC
IMC/OATI
Figure 2. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the NLIN model 24
0,5
ADG
0.0
0.0
'C
c
Figure 3- Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the POWER model 25
RDG
tKC/Û«Tl
0.5
ADC
IKC/OATI
0.0
Figure 4. Average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake at 200 kg body weight for the TRANSLOG model 26
The functions are also shown in contour plots in figures
5 to 9. Sample data points are included in these figures.
The shape of the isoquants suggests some widely different elasticities of substitution across the four models. Fig ures 5 and 6 are from the POLY model but for two different body weight, 100 and 200 kg respectively. The area of eco nomic consideration increases with body weight, for all four models. This has some important implications that will be discussed later. 27
AOG .1 .<
1 0 -
ME
*-
3 .0 1.0
CP(kg/d»y)
Figure 5 . Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 100 kg body weight for the POLY model. Shown are the isoquants (----) , the ridgelines (<--■-•) 28
4 A D G 2
ME
.7 ,4 1.0 CP(UQId»y)
Figure 6 . Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POLY model. Shown are the isoquants (--- ) , the ridgelines (...•) , the adjusted NRC requirements (*)» the adjusted ARC requirements (o) and data points (*)• 29
Mf
(mctlldtyl
CP Iko/dtyl
Figure 7 . Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the NLIN model. Shown are the isoquants (--- ) , a ridgeline (■••■..) , the adjusted NRC requirements («), the adjusted ARC requirements (°) and data points (•)• 30
10
(McêHdêy)
3 .4 >
CP (kglday)
Figure 8 • Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the POWER model. Shown are the isoquants (----) , the adjusted NRC requirements («)» the adjusted ARC requirements (°) and data points (•)• 31
1 0 -
me
(Mcil/dêr]
,1 .* .«
CP (tgldtrl
Figure 9 . Contour plot of average daily gain (ADG) as a function of crude protein intake (CP) and metabolizable energy intake (ME) at 200 kg body weight for the TRANSLOG model. Shown are the isoquants (--- ) , the adjusted NRC requirements (*), the adjusted ARC requirements (°) and data points (•). 32
2.5 INTERPRETATION
2.5.1 Phys1ca I
Estimates from all four models suggest the existence of
substitution between energy and protein. If substitution
does not exist, the inputs would be complementary with a
fixed factor response surface. Under the NRC recommenda
tions, the implied production function is of the form given
by (5) which is depicted in figure 10 . If this model is
true, a plot of the residuals (observed - predicted values) versus the predicted weight gain from any of the four mod
els would show a definite pattern depicted in figure 11 .
This pattern never showed up in the analysis of the residu als. This fact combined with the good fit observed with any of the four models is interpreted as strong evidence of energy-protein substitution. The non-substitution model also assumes that gain reaches a plateau at a certain pro tein level, Po, which is a function of the energy level.
However, there is evidence in the literature that gain de creases when protein is fed at a higher level than Po
(ARC).
The NRC and ARC adjusted recommended requirements are shown in figure 6 to 9 . These values were adjusted from ones published in the original references since one set of requirements cannot lay on the isoquants of the four dif ferent models. The adjustment was made in such a way that ^ 3
^ •<=
^ 0. 0.8 0.6 0.8 C . P . iKC/onri 34
G. - G. I I
G.
Figure 11. Pattern of residuals expected if the fixed factor is the 'true' model and a smooth function is fitted. = observed gain, = predicted gain 35
the reported requirements are on the specific isoquant at
points with the same energy to protein ratio as the ones
from the original references. This ratio is assumed con
stant by the ARC at 30.8 Meal ME per kg CP. The same ratio
is consistently lower for the NRC requirements and varies
slightly with the rate gain level considered (23.3 at main
tenance and 21.3 at .5 kg/day).
Table 3 reports the computed marginal rate of technical
substitution (MRTS) of protein for energy at the predeter
mined adjusted NRC and ARC requirements. The MRTS increas
es with an increase in gain for the TRANSLOG and POWER mod
els. As a general rule, the MRTS increases in the
following model order: POLY, NLIN, TRANSLOG, POWER.
The variation in elasticity of factor substitution as
defined below, can also be used to examine the functions.
(10) p = % AE = = MRTS % Ap 3P [t]
Elasticities for three levels of MRTS and three rates of
gain are in table 4. The POWER function forces this elas
ticity to be constant at a ratio -bl/b2, which is equal to
-0.16. The POLY function yields values of P that are con sistently lower than with any other models. The NLIN is intermediate between the POLY and POWER functions. With respect to p, the TRANSLOG is generally close to the POWER function. 36
TABLE 3
Marginal rate of technical substitution (MRTS) of protein for energy at the NRC and ARC adjusted recommended nutrient requirements for three rates of gain. (Body weight set at 200 kg).
Gain (kg/day)
0 .0 0 .3 0.5
Model NRC ARC NRC ARC NRC ARC
POLY -14.55 -31.23 -13.58 -75.75 -19.33 -
NLIN -4.53 -6.29 -5.76 -9.57 -5.63 -10.99
TRANSLOG -9.35 -10.56 -3.99 -5 .08 -2.55 -3.46
POWER -3.79 -5.06 -3.60 -5.05 -3.47 -5.04 37
TABLE 4
Elasticity of factor substitution ( p ) for three levels of MRTS and three rates of gain. (Body weight set at 200 kg).
Gal n (kg/day)
0.0 0.3 0.5
MRTS MRTS MRTS
Model -4 —8 — 12 -4 —8 —12 -4 —8 —12
POLY -.23 -.40 -.55 -.22 -.40 -.56 -.22 — .42 -.60
NLIN -.19 -.20 -.14 -.21 -.30 -.32 -.21 -.32 -.36
TRANSLOG - - -.30 — .18 -.14 -.13 -.11 -.10 -.09
POWER -.16 — .16 — .16 -.16 -.16 -.16 -.16 —.16 -.16 38
A final way of comparing the functions is to look at the
elasticity of substitution as defined by:
(II) , . Z ' F . “ Ill & i n % aMRTS iW /mpp U1mppmp Pj,J U J Table 5 reports elasticities calculated at three levels of
MRTS and three rates of gain. With the POWER function, this elasticity is forced to be constant at 1.0. The POLY function gives values of e much lower than with the other models. The NLIN model shows elasticities intermediate be tween the POLY and POWER funtions. The TRANSLOG model gives elasticities that are always higher than unity.
These results suggest some large differences in the sensi tivity of the models' optima to changing economic situ ations. Results are not shown for different body weight, but as could be expected from the isoquant maps of figures
5 and 6, the elasticity of MRTS increases with body weight, i.e., 3e/3W > 0. One should evaluate the models in terms of their biological significance before examining more closely the problem of sensitivity.
As seen in figures 5 to 9, only the POLY model showed a true minimum crude protein requirement, which is the level of protein that is both an element of a given Isoquant and the ridgeline found by setting f ^ to 0, i.e.. 39
TABLE 5
Elasticity of MRTS at three levels of MRTS and three rates of gain. (Body weight set at 200 kg).
Gain (kg/day)
0.0 0.3 0.5
MRTS MRTS MRTS
-4 -8 -12 -4 -8 -12 -4 -8 -12
POLY 0.13 0.22 0.29 0.08 0.18 0.21 0.07 0.10 0.13
NLIN 0.51 1.15 2.35 0.20 0.55 0.74 0.20 0.34 0.67
TRLOG - - 1.81 1.43 1 .26 1.22 1.19 1 .14 1 .12
POWER 1.00 1 .00 1 .00 1 .00 1 .00 1 .00 1 .00 1 .00 1 .00 40
2 ( b2 + b7 E + b8 W + blO E W + bll E ) (12) P = ------2 (b6 + bl3 E + bl4 W)
This way, the POLY model yields an estimate of 132 g/day of
CP as an estimate of the true minimum protein requirements for maintenance of a 200 kg animal. Assuming a crude pro tein digestibility of 50%, this value is equivalent to 1.24 g DP/kg W*^^, where DP represents digestible protein and
Wthe metabolic body weight. Early in the century,
Arrasby (1922) estimated the DP maintenance requirements to be around 2.8 g DP/kg W H e pointed out however that maintenance requirements ranged between 0.97 and 4.45 g/kg
depending on the supply of non-nitrogenous materials.
More recently, Elliot (1963) showed that body weight of cattle can be maintained when fed a diet containing only
48% of Armsby's maintenance DP level, i.e., 1.7 g DP/kg . 75 W . Further results from the same laboratory have con firmed that the maintenance DP requirements of cattle are between 25 and 50% of Armsby's value, i.e., between 0.7 and
1.4 g DP/kg (Elliott and Topps, 1963 a and b; Elliott and Topps, 1964). Our estimate of 1.24 g DP/kg is well within that range. By extrapolation, all the other models assume that an animal could be maintained on a ni trogen-free diet, which is biological nonsense since prote in is needed for specific functions within the body. 41
Although the POLY and NLIN models do not a priori Impose
minimum energy requirements, results of the unconstrained
estimation of these models indicates the existence of such
a minimum. With the same approach as the one discussed
previously for protein, the following values of true mini
mum energy requirements for maintenance of a 200 kg animal
are obtained: 4.21 Meal ME/day and 4.32 Meal ME/day for the
POLY and NLIN model respectively. Reported per unit of
metabolic body weight, these are equivalent to 79.2 and
81.2 Kcal ME/kg W"^^. These values are lower than the 104
Kcal ME/kg reported by Moe et al. ( 1972). The latter value was determined at an intermediate level of protein
and does not represent a true minimum. If one takes the
energy requirements for maintenance from the POLY model at an intermediate level of protein (.180 kg CP/day, 5.02 Meal
ME /day), 95 Kcal ME/kg is obtained, which is a value much closer to the value reported by Moe et al. (1972).
It seems that only the POLY model generates extrapola tion that has some biological sense. But, as one can easi ly verify, this particular function is not easy to work with and the conditions for economic optima cannot be de rived analytically. The results shown in the paper were found by numerical methods and have been estimated by com puter approximation. In that respect, the tractability of the POWER function makes it appealing. Since within the 42 range of our experimental data, all four models yielded a very good fit, it seems advisable at this point not to dis card any model solely on their poor biological behavior when extrapolated.
2.5.2 Economic
Our interest was to find the minimum cost requirements, i.e., mi nimiz e
C = Pp P + Pg E
(13) subject to :
f(E,P|W) = Go
where
Go = desired rate of gain (kg/day)
Pg = price per unit of energy ($/Mcal ME)
= price per unit of protein ($/kg CP)
E, P and W as defined in (4)
The problem stated as a Lagrangian function is to minimize:
(14) L = Pg E + Pp P + A(f(E,P|w) - Go)
The necessary conditions are given by: 43
(15) 21 = P_ + Af = 0 3E ^ ^
(16) IL = P + Afp = 0 3?
(17) IL = f(E,p|w) - Go = 0 3X
The first two conditions can be combined:
(1 8) fp Pp I Pp \ and thus MRTS = - I I E E \ e J
These conditions can be easily solved for the POWER func tion, but are not as easily solved for the other models. A computer approximation was used in those cases and the point of minimum costs was found as the point on a given isoquant with the same slope (MRTS) as the isocost slope
(negative ratio of prices).
Prices of corn grain and soybean meal (SBM) were used to estimate Pp and Pe, the respective prices of protein and energy. Alternative sources of protein or energy that could be cheaper in some regions or at some time of the year were not considered. The possibility of using non protein nitrogen (urea for example) as a possible source of protein was also not considered. However, since it is cur rent practice to use corn - SBM in most feedlot operations, those restrictions should not affect the general conclu sions to a large extent. For a given set of prices of corn and SBM, Pp and Pe were found by solving the following set of corn and SBM equations respectively: 44
(19) 3,088.3 Pe + 89 Pp = 100
2,803.5 Pe + 440 Pp = 100 X Ratio
where
Pe = price per unit of ME ($/mcal)
Pp = price per unit of CP ($/kg)
Ratio = ratio of prices of SBM ($/ton)
to corn grain ($/ton)
The coefficients in the equation (e.g. 3,088.3) represent
standard composition per ton of corn and SBM. This proce
dure was used since both corn and SBM contain a certain
amount of energy and protein. Corn and SBM prices were
then converted into energy and protein prices. The right hand side coefficients could be replaced by the direct prices of corn and SBM, but historical prices would yield
estimates of Pp and Pe that contain inflation effects. The procedure used indexes all prices to a value of $100/ton for corn ($2.80/bushel) so that the resulting estimates of
Pp and Pe are only affected by the change in the ratio of prices of SBM to corn. Annual average U.S. prices for corn and SBM from 1949 to 1980 were taken from the United States
Department of Agriculture - Agicultural Statistics. The ratio of prices of SBM to corn grain varied considerably between 1945 and 1981, with a range of 1.03 to 4.08. Con 45 sequently, the Pp to Pe price ratio varied tremendously, with a range of 1.08 to 128.2 (Table 6). This indicates a significant variation in the optimum point due to fluctua tions of prices.
Minimum costs requirements for each of the four models and for each of the 32 sets of annual prices were computed at three rates of gain. Average minimum cost values, stan dard deviations and minimum and maximum values are given in table 7. On the average, the POLY model gave consistently higher crude protein requirements, followed closely by NLIN and then by the POWER and TRANSLOG. Similarly, the POLY model gave the lowest ME requirements while the three other models gave generally similar ME requirements. As expect ed, these requirements showed a substantial variation due to price fluctuations. The magnitude of this variation was dependent on the model under consideration, the POLY model having the lowest variation in both protein and energy. It is of interest to note how close the NRC requirements are to the mean requirements given by the POLY model (Table 8).
Considering that the POLY model gives the lowest variation in requirements (low e ), then the use of NRC require ments should result in a low loss of profit if the 'true' response surface is given by the POLY function.
The minimum cost requirements are converted to amounts of corn and SBM in table 9. Large differences in the 46
TABLE 6
Annual average U.S. prices for corn grain, soybean meal, their ratio of prices and estimated prices per unit of ME and CP, from 1949 to 1980.
Variable Mea n S.D. Minimum Maximum
Corn grain ^ 55.79 21.96 35.71 111.07 ($/ton) ^ Soybean Meal 99.58 53.95 47.45 228.99 ($/ton) Corn/SBM ratio 1.76 0.58 1.03 4 .08
ME price 2.81 0.51 0.76 3.46 (c/Mcal) CP price 26.2 17.7 3.75 97.5 (c/kg) CP/ME price ratio 12.7 21.8 1 .08 128.2
^ Seasons average prices. Ag Stats 19 66, table 39, p29 for 1949-1964 ^ Ag Stats 1982 , table 39, p30 for 1965-1980. 44% protein. bulk, Decatur, 111. Ag Stats 1956, table 188 , pl30 for 1949-1954 Ag Stats 1966, table 85, p60 for 1955-1964 Ag Stats 1982, table 80, p59 for 1965-1980. 47
TABLE 7
Points of minimum costs, for three rates of gain. (Body weight set at 200 kg).
Model Nutrient Mean S.D. Mi niraum Maximum
0 0 K g /aay——
POLY CP^ 0.221 0.027 0.138 0.267 ME 4 .538 0.530 4 .220 7.220 NLIN CP 0.133 0.092 0.001 0.332 ME 5.400 0.813 4.360 7 .140 TRANSLOG CP 0.202 0.088 0.010 0.292 ME 4.992 1.138 4.030 9.100 POWER CP 0.144 0.121 0.010 0.602 ME 5.312 0.694 3.990 7.800
o 4 rs — p U . J Kg/aay———
POLY CP 0.335 0.028 0.264 0.389 ME 6.817 0.399 6.500 8.650 NLIN CP 0.300 0.117 0.001 0.522 ME 8.583 1.482 7.280 14.980 TRANSLOG CP 0.277 0.335 0.009 1.578 ME 8.872 1.403 5.090 12.500 POWER CP 0.237 0.198 0.016 0.984 ME 8.683 1.138 6.520 12.780
a n = 0 • JQ Kg/cayn I A ^ 1 7 wm
POLY CP 0.440 0.024 0.389 0.491 ME 8.429 0.262 8 . 160 9.490 NLIN CP 0.400 0.132 0.001 0.632 ME 10.897 2.340 9.370 22.520 TRANSLOG CP 0.219 0.239 0.008 1.190 ME 11.445 1.074 8.870 14.660 POWER CP 0.302 0.235 0.021 1 .096 ME 11.349 1.475 8.680 16.700
^ CP = Crude protein intake (kg/day) ME = Metabolizable energy intake (Mcal/day) 48
TABLE B
Comparison of the mean recommended nutrient levels generated by the POLY model, with the adjusted recommendations by the NRC and ARC.
Model
POLY NRC ARC
Gain (kg/day) CP® ME CP ME CP ME
0.0 0.221 4.538 0.200 4.65 0.171 5.27
0.3 0.335 6.817 0.314 6.94 0.268 8.28
0.5 0.440 8.429 0.410 8.72 0.388 9.61
^ CP = Recommended crude protein intake (kg/day) ME = Recommended metabolizable energy intake (Mcal/day). 49
amounts of corn and SBM required are observed between mod
els. For each model, these amounts show appreciable varia
tions depending upon the relative corn and SBM prices.
A final analysis is provided in order to estimate the
possible savings by switching from fixed nutrient require
ments. Since the 'true' model is unknown, we adopted the procedure of Bay and Schoney (1982) originating from Havli- cek and Seagraves (1962). In this method, a true model is selected and the optimum input levels from each alternative model are used to predict levels of output from the 'true' model. Net returns are then calculated for each alterna tive model and the difference between these and net returns from the 'true' model represents the cost of being wrong.
The cost of being wrong, i.e., choosing the inappropriate production model upon which to base least-cost input deci sions can be measured as either of two components. First, the choice of a wrong model will result in a greater than least-cost combination of inputs for the expected level of gain per feeding period. Second, the actual weight gain will not be as high as expected, resulting in a lower mar ket weight at the termination of the feeding period. In the first case, the total revenue from the sale of live stock will be the same regardless of the correctness of the production model used to calculate the least-cost energy and protein ratio. However, the cost of the suboptimal in- 50
TABLE 9
Comparison of the amounts of corn and SBM required at the points of minimum costs for a gain of 0.5 kg/day at a body weight of 200 kg.
Model Ingredient Mea n Mi nimum Maximum
POLY Corn 2.23 2.00 2.78 SBM 0.55 0.32 0.71
NLIN Corn 3.31 2.12 7.29 SBM 0.24 0.00 1.01
TRANSLOG Corn 3.71 0.51 4.75 SBM 0.00 0.00 2.60
POWER Corn 3.67 0.67 5.41 SBM 0.00 0.00 2.35 51 put combination will be higher than necessary. This case is illustrated in figure 12 . In this graph, it is assumed that the only error is in the ratio of inputs and that the two different production models being used are evaluated at the same level of output per feeding period. Also, the case is represented for a production model with substantial substitution between inputs, denoted SS, and one with no implied substitution possibilities, labelled FF. The opti mal input combination is given as point A on the assumption that SS is the correct model. Point B is the recommenda tion corresponding to the incorrect model for the same ex pected weight gain. This would correspond to the case of comparing NRC recommended feeding rates with the POLY mod el, after adjusting the recommendation so the implied pro duction levels are the same. The total cost of the input combinations can be expressed as the quantity of either en ergy or protein multiplied by their respective prices. As long as the imputed total cost line for input point B is strictly to the right of the same cost line for A, we can conclude that the cost implied for B is greater than that for the correct model SS, point A.
The second case is illustrated in figure 13 . In this case we drop the assumption that the implied levels of weight gain are the same and allow the case where the iso quants can cross each other. The correct isoquant is la- 52
ENERGY
X”Y • Increased cost due to excessive feeding rates.
Y PROTEIN
Figure 12. Comparison of least-cost input combinations for alternative production models 53
beled as S'S' and the incorrect one is labeled F'F', We
can see that at the given price ratio for energy and prote in, the imputed cost of the wrong model will be lower than for the correct model, i.e., point D versus point C, re spectively. The cost of choosing the wrong model comes about not because of inflated feed cost with equivalent revenues, but because of lower gross revenues due to under feeding. This would correspond to the case where we are comparing two different models both of which imply some in put substitution. In the analysis which follows, we have accounted for either of these costs depending upon which is relevant.
Net price of beef was set at $60/cwt which represents
60% of the indexed price of corn grain per ton. This ratio of price represents the 1949-80 average ratio of prices of beef at Chicago (1949-65) and Omaha (1966-80) to the price of corn grain per ton.(l) This analysis was performed using annual prices and optimum points from 1949 to 1981, at three levels of gain. Results are summarized in table 10 for the highest rate of gain considered (0.5kg/day).
(1) Beef Steers ; Choice grade;
table 463, p386, Ag Stats 1957 for 1949-1956
table 467, p319, Ag Stats 1966 for 1957-1965
table 454, p306, Ag Stats 1981 for 1966-1980. 54
ENERC
X“Y « decline in gross revenue due CO underfeeding
-
PROTEINY
Figure 13. Comparison of alternative production models with equal input cost for energy and protein but lower gross revenues 55
TABLE 10
Average cost of being wrong for a 0.5 kg/day rate of gain with annual average U.S. prices for 1949 to 1980 at a body weight of 200 kg.
Alternative model
True model POLY NLIN POWER TRANSLOG NRC ARC
a POLY 0 15.44 47.39 64.54 0.54 2.44 - (16.92) (29.58) (33.54) (0.49) (1.11)
NLIN 12.93 0 8.12 14.87 2.17 1.74 (4.32) - (5.53) (7.42) (6.29) (4.41)
POWER 14.80 2.92 0 4.65 4.35 2.47 (4.87) (3.33) - (4.07) (7.17) (4.81)
TRANSLOG 18.38 4.25 1.69 0 5.93 3.50 (5.47) (1.82) (0.75) (7.91) (5.40)
^ Mean of 32 observations; standard deviation in parentheses . 56
The calculated profit loss incured by the fixed require ments models (NRC, ARC) vary depending on the true model chosen. The loss is only 0.5 cent/head/day if one uses the
NRC requirements, assuming that POLY is the true model.
This amounts to about $1.50 per head for a full feeding period (300 days). Even though the use of minimum costs requirements instead of fixed requirements can be accom plished at a low cost, it is not clear if these theoretical savings are of enough magnitude to be attractive in prac tice. However, the MRTS and its elasticity increase with body weight. Over a full growing period (200 to 450 kg) our estimates of average lost profit are conservative since the magnitude of loss increases over the feeding period.
The range of our data with respect to weight does not allow analysis for a heavier animal. It is also likely that po tential costs increase with weight and the timing of the feeding period might be an important factor to consider.
Until more evidence is presented on the magnitude of the energy-protein substitution, it appears that following the
NRC recommendations would be reasonably safe. 57
2.6 CONCLUSION
This research provided evidence of protein-energy substitu tion in the growth of cattle. Among the four models that were proposed, the polynomial model seems to be the one that is biologically most meaningful. However, direct ap plication of this function is not warranted. Data used were old (mid-fifties) and were not obtained on the cross bred type of animal the industry is using now. Animals on this experiment were in a study of compensatory growth, which can alter the nitrogen and energy metabolism. This leads to bias estimates when results are used with animals following a normal growth. One also cannot ignore that an imals used were much lighter than the average ones used by a feedlot operation (350 kg). The approach used was static in considering only a body weight of 200 kg. Since it is likely that the costs of being wrong increase with the weight of the animals, the possible savings by using mini mum requirements compared to fixed requirements should be larger than the ones reported here. Finally, the models studied involved few variables. A more general model could be developed that would consider other factors known to af fect rate of gain, such as the breed, frame size, sex, body condition, use of growth stimulants, environmental factors and estimated genetic potential. With such a model, and with the use of dynamic techniques, the analysis could be 58 taken further by determining levels of inputs needed for profit maximization. Chapter III
UNCERTAINTY IN COMPOSITION OF FEEDSTUFFS
3 .1 INTRODUCTION
Over the last few decades, ration formulation has evolved from mystic recipes to a scientific approach using modern mathematical tools. Development of high speed computers has allowed the coding of efficient optimization algor ithms, such as the revised simplex in the case of linear programming (LP). This procedure has been used successful ly to solve problems in civil and chemical engineering, medicine, computer science, agriculture and transportation
(Coffin, 1982; Feldman and Curry, 1982).
The exact origin of using LP for ration formulation can be debated but came early in the mathematical programming developments (Waugh, 1951; Hutton and Allison, 1957). The procedure is now widely used in this country. Efficient programs for least-cost rations are made available to farm ers through the State Cooperative Extension Services, like
Wisconsin (Schmidt and Howard, 1981), Virginia (Staling and
Jones, 1983), New York (Milligan et al., 1981), and Ohio
- 59 - 60
(Pritchard and Staubus, 1981). Design considerations of such a program have been clearly stated by Black and Hlubik
(1980). LP has been used also by California researchers in an attempt to develop maximum-profit rations (Bath, 1975).
Their program is now available to users by remote computer terminals (Bath and Bennett, 1980).
There are some major implicit assumptions in the basic linear programming model, including: divisibility, propor tionality, additivity (independence), no "either/or" activ ities, and deterministic parameters. Later in this paper, we shall explain and discuss the implications of these as sumptions . For many years, it has been assumed that ration formulation by LP was meeting these assumptions, mainly for « practical reasons since no other alternative was available.
The assumption of deterministic parameters is not realistic since feedstuff prices are not always perfectly known, nut rient requirements vary among animals within groups and feedstuff composition is never known with certainty. Under these conditions, the LP model should be altered so as to take into account the uncertainty of the parameters . This sort of model has been labeled as chance-constrained pro gramming (CCP), a member of the large family of stochastic programming (SP).
Dantzig (1955) is probably the first author that has considered the implications of uncertainty in a linear pro- 61 gramming model. The first basic theory on CCP is due to
Charnes and Cooper (1959 and 1963). Since then, the method has been applied successfully by Van de Panne and Popp
(1963) to a minimum-cost cattle feed under probabilistic protein constraints; Lohani and Saleemi (1982) on water- quality management; De, Acharya and Sahu (1982)on capital budgeting and Guildman (1983) on supply, storage and ser vice reliability decisions by gas distribution companies .
A stochastic approach to feed formulation could not only improve economic performance of livestock, but it could also improve the reliability of commercial feeds. Hanks
(1982) reports some alarming results on commercial feed analysis. Out of 3,259 samples analyzed by the feed con trol program in Indiana, the following proportions did not meet their implied feed analysis (tag values): 12.5% for crude protein, 11.2% for ether extract, 4.7% for crude fi ber, 15.4% for calcium, 5.7% for phosphorus and 17.3% for sodium chloride. The combined rate of failure is not re ported. Because correlation coefficients between nutrients are generally between -0.5 and 0.5, the global rate of failure has to be larger than 20%. As will be shown, det erministic LP can be held partly responsible for such a poor performance from the feed industry. 62
1. Outline models and solution procedures to incorpo
rate probabilistic constraints in a least-cost or
maximum-profit ration.
2. Evaluate the seriousness of assuming deterministic
parameters in terms of animal performances and eco
nomic efficiency.
This research will address these needs with the following specific objectives;
1. Estimate the variance-covariance matrix of nutrients
within each of the major feeds used in North Ameri
ca .
2. Determine the acceptability of assuming a multivari
ate normal distribution of nutrients within each
feed.
3. Test the efficiency and accuracy of proposed algor
ithms for solving simple-chance-constrained program-
mi ng .
4. Test efficiency and accuracy of proposed models for
solving joint-chance-constrained programming.
5. Determine the effect of certainty level of ration
composition on feeding costs. 63
3.2 LITERATURE REVIEW
Traditlonaly, least-cost rations have been formulated using a linear programming approach. As a first step, the linear programming model will be discussed with its assumptions and consequences. Then the theory of mathematical program ming under uncertainty will be reviewed with emphasis on chance-constrained programming. Finally, some results found in the literature on least-cost or maximum-profit ra- tinns under uncertainty will be reported.
3.2.1 Least-cost ration as a^ linear programming problem
3.2.1.1 The model
In the case of minimization (least-cost rations), linear programming involves selection of a set of structural vari ables and their level of activity that minimizes a linear objective function subject to a series of linear restric tions or constraints. In algebraic terms, the model is:
r Minimize Z = j^j^j j = 1, ..., n
(20) Subject to: ^ ^ij^j ^i 1=1» •••» m
X. k 0 1
The coefficients of the structural variables in the objec tive function, c ., are typically cost per unit for the j J feedstuff, . The b^ represent nutritional requirements, while a^. is the quantity of the i^^ nutrient per unit of the j^^ feed. 64
3.2.1.2 The assumptions.
The major assumptions in LP are: divisibility, proportion
ality, additivity, no "either/or" activities, and determin
istic parameters (Black and Hlubik, 1980).
Divisibility means that all decision variables (e.g.
quantity of feed i) can be divided into any fraction. In
farm decision models, this assumption rarely holds, as it
is very difficult, for example, to buy 2.5 tractors. But
for all practical purposes, this assumption holds in the
feed formulation problem.
Proportionality means that for any given decision vari
able, X., its contribution to cost is c.X. and its contri- J J J bution to the i^^ constraint is a..X.. If X. is doubled, ij J J so is its contribution to cost and to each of the con straints (i.e., if 1 kg of soybean meal contributes 490g of crude protein, then 2 kg contribute 980g) . However the nu
tritional value of some feedstuff is known to be greater when fed in small amounts than at higher amounts, which clearly violates the proportionality assumption. However, the non-proportional effect is generally small, and could be easily accomodated in a linear programming environment by using step-functions, unless it represents increasing returns to the input.
Additivity implies that the total cost is the sum of all the individual costs, which is clearly met by the feed for 65
mulation problem, and that the total contribution to the
i*"^ restriction is the sum of the contributions of the in
dividual feedstuff. This is to say that feedstuffs do not
interact with each other. The presence of associative ef
fects would violate this assumption (Byers et al., 1975).
But these effects are possibly due to an unaccounted time
effect (like differences in fermentation rates), which
could be considered in a dynamic approach, like multi-peri
od models.
"Either/or" activities cannot be handled directly by the
simplex algorithm, a very popular and efficient algorithm
to solve LP. For example, it is impossible to include a
constraint of the form "either include ingredient A in the
ration or ingredient B, but not both". However, "either/
or'' choices are one of the most easily solved mixed integer programming problems (Pfaffenberger and Walker, 1976).
Furthermore, multiple LP runs using simplex could give the
same results provided that the number of "either/or" vari
ables is limited.
Deterministic implies that the parameters (i.e., c ^, a ^^ and b ) are known with certainty. Nutrient values, a^^, are never known with certainty in practical problems. Fox
(1977) reported an estimated coefficient of variation for protein in Michigan corn silage in the order of 15%. Also, nutrient requirements b^: vary among animals within groups, due to biological variations. 6 6
3.2.1.3 Consequences of uncertainty
As stated previously, the regular LP model assumes certain
ty of the parameters. For a feed manufacturer, this as
sumption holds for the c^ and since the ingredients are
already purchased at the time the the ration is formulated
and the composition of the ration is perfectly known from
the guaranteed analysis. The situation is different on the
farm. The prices (cy) are not necessarily perfectly known at the time the ration is formulated. The represent the average requirementsof all similar animals (size) in a similar physiological state (milk production and composi tion). No doubt that there are some biological variations around these computed values. Finally, the nutrient val ues, aUj, are never known with certainty, and this is true both for the feed manufacturer and the farmer. This fact is obvious if the ingredients are not chemically analyzed for their nutrient composition. But even if they are, un certainty remains as the variation due to sampling and the analytical procedures are still present.
To evaluate the consequences of the deterministic ap proach, let us define the following:
(21) T^ = Ç j = 1» » "
where
T^ = amount of nutrient i
The expected amount of nutrient i in the ration is: 67 (22) E(T^) . t E(a..)X.
I.e.
(23) E(ip = 2 P..X.
Let us assume a symmetrical probability distribution of the
or . If we use the p^^'s (mean value of nutrient i in
feedstuff j) in a LP environment, then 50% of the time the ration will be under the desired specified level b ^ for
nutrient i. Similarly, if rations are formulated for aver
age requirement values (b^), the strict requirements of 50%
of the animals will not be met even if feedstuffs composi
tion are perfectly known. As a result, if one uses the mean composition of the ingredients and the mean require ments, then one should expect to meet the requirements only
50% of the time, assuming a symmetrical probability density function of the a and b .. Therefore, there is a need to 13 1 assess the form of the distribution of nutrients within
each feedstuff.
Consequences of uncertainty differ across the segments of the feed industry:
1. For the farmer, the variance in might be large
enough so that inadequate rations will be formulated
(i.e., by chance, some rations will be low enough in
a given nutrient to affect the animal performances).
Black et al. (1978) have demonstrated that it pays
to have a safety factor in a beef ration. However, 68
the economically optimal safety factors have yet to
be assessed with good confidence.
2. For the feed manufacturer, the law states that the
guaranteed analysis for some nutrients must be ex
pressed as a minimum concentration (e.g. crude pro
tein). Hence, the feed must meet the desired con
centration more than 50% of the time in order to
avoid costly lawsuits.
3.2.2 Mathematical programming under uncertainty
3.2.2.1 Stochastic programming
The general field of stochastic programming started in the mid-fifties from the expose of Dantzig (1955). Since then, there has been some very active research reported in the literature, which has been summarized by Vajda (1970) and
Sengupta (1973).
A stochastic problem can be classified into one of the three following categories, depending on the problem's characteristics and one's interests: 'wait-and-see',
'here-and-now', and chance-constrained (Pfaffenberger and
Walker, 1976). We start with a study of problems of the
'wait-and-see' type. Consider the program of:
Minimize z = c'l (24) Subject to: Ax ^ b 69
where c and x are n-vectors, b is an ra-vector, A is an mxn
matrix and x is to be determined. A natural question to be
asked is one about the probability distribution of the min
imum of c'x. This product of two vectors, or its minimum,
will be a random variable when the c-coefficients are ran
dom variables themselves. If we know that the set of opti
mal basic variables remain the same whatever the variations
of the coefficients, then the question could be answered by
methods which analyze the solution of a set of algebraic
linear equations when the coefficients vary. But as these
coefficients vary, the set of optimal basic variables might
vary as well. Then the probability of the minimum of c'x
not exceeding a value C is a sum of terms, each term being
the product of a probability that the set of optimal basic
variables is a given set, multiplied by the conditional
probability that in this case the minimal value of c'x does
not exceed C. A problem is of the 'wait-and-see' type when
one's interest is in finding the probability distribution
of the optimum when the c's are random.
Now let us modify the program to the following:
Minimize z = c'x (25) Subject to: Ax = b
and assume that b is not precisely known, but only its dis
tribution function. Then it is plainly impossible to de mand that X should be determined in a way that Ax equals 70
whatever value of b. The difference between Ax and b will
in effect be a random variable whose distribution function
depends on x. We can then argue that we understand that we have to pay a penalty for any discrepancy, and we might de
cide to minimize the sum of c x plus the expected value of
such possible penalties.
We can make various assumptions about the penalties to be paid. A general formulation reported by Vajda (1970) is as f ollow : Minimize c'x + E min d'y v (26) f Subject to: Ax + By = b
This is interpreted as saying that an x must be found
'here-and-now', before the actual values of the components of b are known; when they become known, then y must be found from the second stage program:
Minimi z e d'y (27) Subject to: By = b - Ax where all coefficients and x are known. Then the vector x must be determined in such a way that
^28) c'x+E min d 'y is minimized.
With the third type of problem, chance-constrained pro gramming, we do not demand that the constraints should al ways be satisfied, but rather that they be satisfied in a 71
proportion of cases or, in other words, with certain given probabilities. Thus we might want to:
Minimize c'x (29) Subject to: Pr (a^x i b^) i where the a, are given probabilities and the symbol Pr( ) stands for 'the probability of ( ) to hold'. The proce dures which we will discuss in the next sections consist in finding for such a probability constraint, an equivalent deterministic constraint, i.e., one which no longer con tains any probabilistic element.
Recent advances in stochastic programming (SP) can be found in Cunninghame-Green (1977), Bawa (1982) and Stancu-
Minasian (1982). Interesting applications of SP in opera tions research can be found in Itami (1974), Heyman et al.
(1982) and Sengupta (1981, 1982). Research is now being conducted on extensions of SP in dynamic programming
(Schal, 1979). Much controversy remains about the benefits of specific SP in decision analysis (Blau, 1974; Charnes and Cooper, 1975; Heilman, 1980).
3.2.2.2 Simple-chance-constrained programming
Chance-constrained programming (CCP) has been introduced in the previous section. The problem is repeated here for convenience. 72
Minimize z = c'x
(30) Subject to: Pr(a|x ^ b^) ^
X ^ 0
The magnitude of i allows for one more step in our classi
fication. In (30), if i = 1 then the problem is a simple-
chance-constrained program (SCCP); otherwise, it is a
joint-chance-constrained program (JCCP).
3.2.2.1 Origin
Elmaghraby (1959) is one of the first authors that reported work on SCCP, but somehow his approach has not proven use ful and has been abandoned by the scientific community.
The whole field was really initiated and structured under
the work of Charnes and Cooper (1959, 1962, 1963) who set
the basic theory of CCP, which has not really changed since. This theory was also outlined by Sinha (1963) in
California. During chat time, Madansky (1962) got inter ested in the numerical solution to the problem but without great success. The state of the art was summarized in the early seventies by Kirby (1970), Prekopa(1970) and Sengupta
(1970). But the best exposition to the topic can be found in Sposito (1975) whose work will be heavily used and par aphrased in the next section.
3.2.2.2 Models
The general SCPP model is: 73
Minimize Z = j i = 1...... n
Subject to: Pr{% a^jXj ^ b^} ^ i = 1, e{0,l} (31) j SijXj - i = 2, ...,m
Xj ^ 0
where e stands for 'element of'. Depending on which of the
parameters are random, different models can be outlined.
3.2.2.2.1 Constraint requirements random
This is to say that the b^ cannot be determined exactly but
comes from some distribution. The objective is to meet the
constraint for most parameter values. The constraint can be formulated as:
(32) Pr {? a^jXj - b^} - ot^ i e m, “i ^ {0,1} where relates to the rate of success and 1-a^ denotes
the allowable risk of failure. Standardizing, we get:
^ a X - E(b ) b - E(b ) (33) Pr {Z a^jXj ^ b^} = Pr J L1_J------L_ k - 1 ------
where E(b.) and o. are the mean and standard deviation of i b^ respectively.
If we assume that b^ = N(p^ , ) , then i i b. - E(b ) (34) — = N(0,1) ^i
Therefore, 74
X - E(b.) (35) Pr = a “i
We can note that this probability will increase if Z is “i increased. Hence,
? a X. - E(b.) b. - E(b.) (36) Pr -1 3______L- à — = a
if and only .if
(37) a Z
which is to say that
(38) ; a^j Xj ^ E(b^) +
Hence, the model
Min imize Z . I C.X.
Subject to : (39) j aijXj " + z..°b.
j 'ijXj ^ "i Xj ^ 0 is the deterministic equivalent of (31) under a random nor mally distributed b^^. The normality assumption is not es sential in this derivation. We only need to find a such that f,(Z ) =a., where f, denotes the cumulative dis- b 1 D tribution function of b. Charnes and Cooper (1964) have adapted the procedure to a nonparametric environment, i.e., using information from the median and the ranks, which 75
could be considered if the does not follow a tabulated
distribution.
3.2.2.2.2 Input-output coefficients random
Let us consider model (10) when the input-output coeffi cients a are random. Again, let us assume that each a i j i j is normally and independently distributed with mean p and variance o ^ . The independence of the a. . does not Hi- XJ cause any problem in the feed formulation problem since the protein in corn silage is independent from the protein in % alfalfa, for example. Let amount of nut rient i. Then
E(Rj^) . : E(a^j)Xj
■ 3 and V(E,) . : 3 # 3'
which reduces to
(41) V(R ) = ^ q 2 X? "ij ' due to the independence of the a . .. Then XJ
(42) ! a X = N ( ? X., ? 02 X? ) and ] iJ J J J 1 a^^ J
(43) ? XjE(a.j) - ^ a.jXj = N ( 0, x’Vx )
where 76
0
0 12 V =
0 0 in
So if we consider
(44) Pr { f a_, .X. - b_, } then
.X. - ? X.E(a. .) b, - Ç X.E(a. .) (45) Pr JLL_ \ n (x'Vx) (x'Vx)
if and only if
(46) "i - i (x'Vx)
Therefore, the following model
Minimize j Subject to: ? V ..X. - Z (x'Vx)^ ^ b . (4 7)
is a deterministic equivalent to (31) when the a^^ are in dependent random normal variâtes. In the present problem, the normal distribution assumption of the a^^ is of impor- 77 tance since our derivation rest upon the statistical theo rem stating that any linear combination of random variables that are independent and normally distributed has also a normal distribution (De Groot, 1975). Note also that (47) is a nonlinear model.
3.2.2.2.3 Input-output coefficients and constraint
requirements random.
This is a straight-forward extension of the two previous sections. Let b. = N(p_ ,o^ ) and 1 b . b . 1 X a. . = N(p , a t ) and b^ and a^^ independent. ij Then
(48) j *ijXj - b. = N ( ) where
(49) " L - b ■ j - E(b.) a nd 1.2 = : X? 0% + of (50) Ax-b j j a^j b^
Hence
(51)
Pr - a . 1 Ax-b 'Ax-b if and only if
(52) E(bi) - : X.E(a,.) » Ax-b 78
i.e., the stochastic constraint is replaced by its
deterministic equivalent:
Is ^ E(b^)
3.2.2.3 Recent developments
Millier (1967) has extended the theory to the case of 0-1
decision variables, the 'either/or' activities. Much has
been published on the decision rules for specific models
(Eisner et al., 1971) as well as for the general chance-
constrained programming model (Sengupta, 1969, 1970; Smith,
1973; Mukherje, 1980). Thus far, results are inconclusive
and the extension of CCP to decision problems under risk is
still debated (Hogan et al., 1981; Charnes and Cooper,
1983). Babievsk et al. (1980) derived the optimality con
ditions for problems with probabilistic constraints, but
the consequences of their findings are still unknown.
Computer structures have been successfully optimized by
CCP (Rao, 1980). More recently, Sniedovich (1983) has ex amined a special class of CCP which, like the work of Heil-
mann (1983), does not seem to be closely related to our proposed research.
3.2.2.3 Joint-chance-constrained programming
3 . 2 . 3 .1 Origin 79
Miller and Wagner (1965) are the first authors that report
ed some work on joint-chance-constrained programming
(JCCP), i.e., a problem when one constraint has the form:
Pr n (54) i j ij j J (n is the standard symbol for the intersection of sets)
with random input-output coefficients, constraint require
ments, or both, random. However, their work was limited to
the case of independent random right-hand-side (RHS) ele
ments. Balintfy (1970) extended the theory to the case of
dependent random RHS and showed an exact solution when k=2
and b^ bivariate normal. Bawa (1973) proposed two lower
bound approximations to the former constraint, based on
Bonferroni's inequality and Slepian's inequality. A major
contribution to the field was made by Jagannathan (1974)
when he fully relaxed the assumption of independence of the
random RHS elements, treated the case of a random input-
output coefficient matrix and determined the conditions for
convergence (convex programming) of the JCCP. His results
will be heavily used in the next section.
3 .2 . 3 . 2 Models
The general JCCP model is: (55) Minimize ^ c^X^
(56) Subject to: Pr ' rf 'i]
(57)
(53) ^ 0 j 80
In the case of a random RHS vector, (56) is expressed as:
k Pr n (59) i - a
where a ., c. and a are given constants, Ç ,...,Ç are con- ij J I k tinuous random variables with a known joint distribution F.
If are independent, then F = ttF ^ , where it denotes
product and F^ is the marginal distribution function of
Then constraint (59) has a deterministic equivalent of the
form: (60) F^( j a_jXj) \ a i = 1,..., k which can be recast in an additive form (Miller and Wagner,
1965) as:
? In F (y ) ^ In a (61) j 'ijXj - °
If we add the assumption that Ç. are normally distributed,
i.e., with a density function of the form:
(62) f ^(Ç.i I .cTp i exp - 1 Si - ( n)^o, 2 °i
then
- 1 (63) Fi(Ci) = ^ exp dç. (2n)^o. 81 which does not have any known analytical solution. There
exist good numerical approximations to integral (63) and
their use would not preclude the use of derivative-based
NLP algorithms, since = f^(S^) which is given by (62) and can be computed easily.
The case when the are not mutually independent is not as straightforward. Let us examine the special case when k = 2 and C ^, ?2 have a bivariate normal distribution func tion. Let
(64) g,(X) = ( j *ijXj ^i) 1 = 1,2
where = E(Ç^) ,2 = E((?i - y^) )
(Ci -Pi) /
Then (59) has the form
Pr (X) ^ e, , g^ (X) (65) 1 where variables E ^have standard bivariate normal dis tribution denoted by EXZ^.Zgir), and
"l %2 E ^ - 2re^E2 + e ^ FXZ^.Zgrr) = exp dci dE2 2 s h 2TT(l-rO 2 ( l - r 2 ) 0 0 (66) E(e^) = E^Eg) 0 E(ep = E(e2) = 1
£(£^£2) = r
It follows that 82
(67) ^ Z^, ^2 ^ ^2 ) FCZj^jZ^jT) and (65) can be rewritten as:
(68) F ( g^(x), g2(x);r ) ^ a
So, if we let g^j= au^/o^ and G(Z) = F(Z^r), the determin istic equivalent to (65) is
(69) ? SijXj - Zi = 0
G(Z) ^ c
However, in the process of casting this idea into nonlinear programming models, insurmountable computational difficul ties arise from the process of evaluating the function of multivariate normal probability integrals (Balintfy, 1970).
In fact, at present sufficiently fast numerical analytical procedures and tables are available only for the bivariate normal distribution (Gupta, 1963; National Bureau of Stan dards, 1959) and no experience exists in handling similar computation for k>2.
The extension to the case of random input-output coeffi cients should now be obvious. Variances and covariances are given constants when the RHS elements are random.
These statistics are nonlinear functions of the structural variables x when input-output coefficients are random, and T can be included easily in the model. If we let T .= a X. 1 j i j 1 then V(T^) reduces to
(70) V(T.) = E o2 1 a.j J 83 in the feed formulation problem due to the independence of the »^ij' jfj '» and
(71) Cov(T.,T.,) = J ^,XjX.,Cov(a.j,a^,j,) which reduces to
(72) Cov(T^,T.,) = j Cov(a^j,a^,j) xj in the feed formulation problem since Cov(a^^,a^,^ ,) = 0 for j=j' (i.e., no correlation between protein in corn si lage and fiber in alfalfa).
3.2.3 Least-cost and max-profit rations under uncertainty
Van de Panne and Popp (1963) are the first authors that have applied the theory of CCP to the feed formulation problem. Essentially, they have applied model (26) to a cattle feed problem with a probabilistic protein con straint. Since the resulting model is nonlinear in the constraints, they used Zoutendijk's method of feasible di rections to determine the optimum (Zoutendijk, 1959). But much of their discussion is on Zoutendijk's method and on the care that must be taken in each step of computation so that the feasible directions and optimal lengths of move ments are properly derived. Later, Van de Panne and Whin- ston (1966) simplified the method, but the non-availability of a computer code has precluded its use. During that period, Balintfy and Prekopa (1966) began their work on hu 84
man diets which resulted in marked progress in the field of
joint-chance-constraint programming (Balintfy, 1970).
As a consequence of the non-availability of an efficient
computer code for solving nonlinear programming, some re
searchers have proposed methods to linearize the nonlinear
constraints. Nott and Combs (1967) and Combs and Nott
(1967) suggested that a proportion of the nutrient's stan
dard deviation be subtracted from the nutrient's mean,
i.e., a! . = a..- Xo. .. The LP model becomes: ij iJ iJ
Minimize ? c .X . J J J Subject to: ? (a.. - Xo..)X, ^ b. J 1] 1] j i where X is the adjustment factor needed to obtain the de
sired probability of success. Black and Hlubik (1980) re
ported values of X required to obtain alternative probabil
ities of meeting the requirement for the case where the a^^
follow the normal distribution (i.e., Pr = 0.95 for
^ = 1.65). According to these authors, "this approximation
procedure is accurate when the desired rates are in the
0.50 to 0.80 range (0.83 < X < 0)". This statement is er
roneous as it can be demonstrated that the odds of success are much larger than stated. This is because the proce dure ignores that the variance of nutrient i in the final
solution is a function of the number of ingredients in the solution and their proportions. Due to its conservatism,
the method is safe but not accurate. However, it is easy 85
to implement since all equations are linear. Computer
software can be designed to store standard deviations in
addition to means in the nutrient data file. The model
would adjust automatically the mean according to the appro
priate adjustment factor and would place the adjusted mean
in the LP matrix.
Rahman and Bender (1971) proposed two linear approxima
tions of the variance equation. The first is based on a
Taylor series expansion of a . To see how this works, let E i T .= . a . .X .. Then 1 J iJ J Ç o2 X? (74) 1 a.. ] 1 The RHS of this equation can be expanded through a Taylor
series expansion at the point X^ = X°, j = l, 2,..., n. In
the neighborhood of this point, the terms of second and
higher order in the expansion can be ignored, to yield the
following:
(75) = j «ijXj where
J?,X?
Then the LP could be Iterated with 6.. and a re-estimated 13 i after each iteration. Unfortunately, Rahman and Bender
(1971) did not consider this alternative in their applica tion. Instead, they used the second approximation method which can be described as the following. Let 86 *
By squaring both sides, we obtain:
(78) - .2^ + Ç ; differing from equation (74) by the second term only, which being the sum of positive cross-products is positive.
Therefore, *2 ^ 2 (79) •k As a result, if a is approximated by a , the result i i would be biased, but in a known direction. The implication of this bias is that the actual probability of meeting the requirements would be more than the specified value. The deterministic equivalent to the probabilistic constraint has the form:
(80) ; a..X. - X Z o^.X. i b^ which is equivalent to (73), the Nott and Combs' method of downgrading the mean nutrient contained in each ingredient to allow for a safety margin. However, (80) allows A to be parametrized easily, i.e., by repetitive runs and post-op timization computation of o^ , A could be determined so i that the actual probability of meeting the requirements is close to the specified value. This procedure has not been used by the authors.
Rightfully, Chen (1973) argued that under the Rahman and
Bender's approach, the desired level of probability of sue- 87
cess is likely to deviate greatly from the true optimal
nonlinear programming result. The fact Is that although
parametric linear programming for X over an appropriate
range generates a ration that does satisfy the desired lev
el of probability of success, the mixture Is possibly no- noptlmal In terms of the original stochastic programming problem. Cost Is likely to be greater than the minimum at tainable for this ration because the solution thus obtained Z 2 minimizes ( j XjOj) rather than the true measurement of 2 2 total variance j (XjOj) , for any given ration cost and mean protein content restrictions.
Chen (1973) used a different and original approach to the problem. The author showed that when other things are equal, it cannot cost less to produce a mixture with a higher attainable probability of success. Therefore, a re vised stochastic programming problem can be specified to maximize the probability of success, such that the cost and other linear Inequality restrictions are satisfied. Then the model Is:
Maximize Pr ( ? a^^X^ à b )
Subject to: c^X^ é. k (81) Ax ^ b
X h 0
The deterministic equivalent Is: 88
Maximize ? a. ,X. - 8x ' Wx ( 8 - 0 to <”) ] 1 ] ] Subject to: c.X. 4 k J J (82) Ax \ b
X ^ 0 with k parametrized uniformly over a large range within which the desired level of success may exist. The problem is solved and the results at every basis change are record ed. The highest attainable q value (i.e., highest
7 0 0 ' 0 ( a X - b ) / (x Wx ) ) is computed, from which the im- j ij j i plied probability of success is determined. Then k is mod ified and the procedure is repeated until convergence. The whole procedure is rather cumbersome to say the least and an efficient computer code for solving quadratic program ming is needed. Such a code does exist (Cutler and Pass,
1971), but the model can only be applied to simple-chance- constrained programming.
Other than the limited work reported by Balintfy (1970), nothing has been done on the application of joint-chance- constrained programming to the feed formulation problem.
With respect to maximum-profit rations, much remains to be done. The California dairy model (Bath, 1975) yields results that are highly dependent on the user's arbitrary choice of response function. Kagan et al. (1982) designed an approach to maximize returns over feed costs with re spect to supplemental fat in turkey diets. However, we are 89 unable to obtain the same regression equations using the reported data, which makes their results questionable.
Also, the accuracy and simplicity of the model could be greatly improved by considering direct daily intake instead of feed efficiency ratios. None of these maximum-profit models considered the effect of uncertainty on the optimal solution. 90
3.3 MATERIAL AND METHODS
3.3.1 Es tlmatlon of Var-Cov matrices of feedstuff s
Two sources of data have been used in order to estimate the varlance-covarlance matrices of nutrients within each feedstuff. The first source was from the Research-Exten- tlon Analytical Laboratory In Wooster, Ohio. Feed analysis from all samples submitted to the lab In 1981, 1982 and
1983 were used, and a few records were deleted for obvious coding errors (I.e., dry matter higher than100%.) The analysis was performed on nearly 14,000 records from the following feedstuffs (number of Individual records In pa rentheses ) .
1. Legume hay - 1st cutting (357)
2. Legume hay - 2nd cutting (618)
3. Legume hay - 3rd cutting (257)
4. Legume-grass hay - 1st cutting (1274)
5. Legume-grass hay - 2nd cutting (1536)
6. Legume-grass hay - 3rd cutting (315)
7. Grass hay - 1stcutting (477)
8. Grass hay - 2nd cutting (118)
9. Grass hay - 3rd cutting (29)
10. Legume silage - 1st cutting (459)
11. Legume silage - 2nd cutting (237)
12. Legume silage - 3rd cutting (124)
13. Legume-grass silage - 1st cutting (1517) 91
14. Legume-grass silage - 2nd cutting (510)
15. Legume-grass silage - 3rd cutting (144)
16. Grass silage - 1st cutting (184)
17. Grass silage - 2nd cutting (23)
18. Grass silage - 3rd cutting (8)
19. Corn Silage - no NPN added (3024)
20. Corn silage - NPN added (1524)
21. Sorghum silage (55)
22. Small grain silage (283)
23. Dry shelled corn (118)
24. High moisture shelled corn (275)
25. Dry ground ear corn (62)
26. High moisture ground ear corn (373)
Records Included information on the following nutrients
(abbreviations and units in parentheses)
1. Dry matter (DM, %)
2. Crude protein (CP, %)
3. Acid detergent fiber (ADF, %)
4. Estimated net energy for lactation (NE(1), Mcal/kg)
5. Phosphorus (P, %)
6. Potassium (K, %)
7 . Calcium (Ca, %)
8. Magnesium (Mg, %)
9. Manganese (Mn. ppm)
10. Iron (Fe, ppm) 92
11. Copper (Cu, ppm)
12. Zinc (Zn, ppm)
Net energy for lactation has been estimated from the ADF
content using the equations developed at Penn State (Bath
et al., 1978). A recent summative equation derived by Con
rad et al.(1984) seems to have much better predictive prop
erties, but the lack of information on lignin, neutral de
tergent fiber, ash and ether extracts of these feedstuffs precludes its utilization in this research.
The second source of data was from a major feed manufac
turer in Eastern Canada. More than 1,400 recordsfrom 1982 and 1983 were available in the following feedstuffs (abbre viations and number of records in parentheses).
1. Barley grain (BRY, 47)
2. Brewer dried grains (BDG, 86)
3. Corn grain (CRN, 184)
4. Corn gluten feed (CGF, 66)
5. Corn gluten meal (CGM, 81)
6. Distillers dried grains (DDG, 98)
7. Feather meal (FTH, 67)
8. Meat and bone meal (MBM, 88)
9. Oat grain (OAT, 38)
10. Rapeseed meal (RPS, 255)
11. Soybean meal (SBM, 100)
12. Sunflower meal (SFM, 101) 93
13. Wheat grain (WHT, 88)
14. Wheat shorts (WSH, 162)
Records included information on (abbreviations and units in
parentheses):
1. Crude protein (CP, %)
2. Acid detergent fiber (ADF, %)
3. Ether extract (FAT, %)
4. Phosphorus (P, %)
5. Potassium (K, %)
6. Calcium (Ca, %)
7. Sodium chloride (Salt, %)
8. Ash (Ash, %)
Except for feather meal and meat and bone meal, net energy for lactation was again estimated from the ADF content by the Penn State equations (Bath et al., 1978). NE(1) in feather meal and meat and bone meal was predicted using the following respective equations:
(83) NE(1) = 0.0245 (.93CP + FAT + .92 (100-ASH-CP-l) + 0.75) - 0.12
(84) NE(1) = 0.0245 (.7CP + 0.1 + 0.70 (lOO-ASH-CP-0.5)) - 0.12
Both equations were derived from the general scheme pro posed by Conrad et al, (1984). 94
Univariate statistics were computed using the MEANS pro cedure of SAS. The CORR procedure was used to estimate the variance-covariance matrices of nutrients within each feedstuff.
3.3.2 Distribution of nutrients wi thin f eeds tuffs
Many of the procedures described in section 3.2 were assum ing a normal or multivariate normal distribution of nut rients within feedstuffs. Therefore it is of interest to test whether this assumption holds under the empirical data.
Necessary conditions for a multivariate normal distribu tion are that each variate must follow a univariate normal distribution, but these conditions are not sufficient.
However, results from this testing phase were conclusive enough so that no further testing was deemed necessary.
Tests of normality for each nutrient within each feeds tuff were performed by the UNIVARIATE procedure of SAS.
For sample size smaller than 51, the Shapiro-Wilks statis- ti was computed and tested (Shapiro and Wilks, 1965). For sample size greater than 50, the data were tested against a normal distribution with parameters equal to the sample mean and variance. The usual KoImogorov-Smirnov D statis tic was computed and tested against critical values (Ste phens, 1974). 95
3.3.3 Solutions to the SCCP problem
The deterministic equivalent to the stochastic model invol ving randomness in one single row (47) implies a search for the solution of a nonlinear programming problem. Algor ithms used to solve such problems are more complex and not as available as the revised simplex for linear programming problems. Currently the feed industry possesses powerful custom-made computer programs using the simplex algorithm to formulate rations as linear programming models. There fore, an algorithm based on a linear approximation to the nonlinear constraint would present a clear advantage as it could be used by the industry with minimal changes from their current procedures. Three such algorithms have been derived and compared to the exact solution given by a non linear programming algorithm.
Data on expected feed composition and their variances were taken from the results obtained by the procedures out lined in the previous section. Feed prices from 1970 to
1979 were taken monthly from Feedstuffs magazine and aver aged per year, resulting in 10 sets of prices. This period has been characterized by a great variation in relative prices of feedstuffs (St-Pierre et al., 1984). These annu al prices are reported in table 11 .
Algorithms were tested on a dairy premix. Such a feed is designed to be added to grains and protein sources in 96
TABLE 11
Average annual prices of feedstuffs used in the feed formulation problems.
Year
Feed* 123456789 10
BRY 36.07
BDG 54.31
CGF 54.22
CGM 151.52 144.76 151.56 294.51 258.02 238.09 271.26 279.24 257.53 302.26
CRN 53.21
DDG 70.34
FTH 146.40
MAB 118.99
MOL 38.85
OAT 71.40
SBM 94.54
WHT 54.11
FAT 169.60
DCL 114.63
LME 11.02
SLT 20.78
^ Abbreviations are as defined in the text.
^ From Feedstuffs magazine. 97
order to form a complete grain mix. Typically, it is manu
factured by a central mill which sells and distributes it
to local feed mills. Under those circumstances, SCCP is a reasonable model. The desired nutrient composition of the feed is :
Crude protein: 25% minimum
Fat : 1 % mi nimum
Crude fiber: 13% maximum
Calcium: 6.6%
Phosphorus: 3.1%
Salt: 12.7%
A micro premix was incorporated at the rate of 4%. The stochastic constraint was applied on crude protein. The other nutrients were considered in a deterministic fashion not because they do not present randomness, but because the magnitude of this randomness is low enough that it does not put the composition of the final product (grain mix) out of legal bounds. The probability level on the protein content was arbitrarily set at 95%. It is thought that such a con fidence level should satisfy regulating agencies. The al gorithms considered were the following:
(1) The approximate Rahman and Bender algorithm with X fixed (RBF).
This algorithm approximates the standard deviation of pro tein in the feed mix by using equation ( 7 7). Since the ap- 98
proxlraated a is larger than the true o, the actual prob
ability of meeting the desired level of protein has to be
more than the specified value. The specific model for the
dairy premix is given in table 12 . The 95% probability
level implied that A be set at -1.63. This linear program
ming problem was solved using the subroutine ZX4LP from the
IMSL library.
(2) Iterated Rahman - Bender (IRB).
In the RBF algorithm, A is set to a given value depending on the desired probability level. However, due to the con servatism of the method, the achieved probability is guar anteed to be larger than the desired probability. There fore, a second algorithm could be designed where o is linearized by a as in equation (77), but A parametrized so that the achieved probability equals the desired one.
For a probability of 95% , we know that -1.65 < A < 0.
If we let f(A) = ABS(achieved Pr - 0.95), then the problem is to find which value of \ yields the minimum of f(A).
This involves a unidimensional search on A. For efficiency reasons, the golden search (search by golden section) was chosen (Hiramelblau, 1972). This search is based on the splitting of a line into two segments known in ancient times as the 'golden section', i.e., the ratio of the whole line to the larger segment is the same as the ratio of the larger segment to the smaller. The fractions employed are: TABLE 12
Approximate Rahman-Bender algorithm with X fixed (RBF). Model for the premix feed
Ingredients ^
R{JU5° FAT BRY BDG CGF CGM CRN DCI. DDG FTH LME MAB MOL OAT SLT SBM WHT HPX RHS^
Protein 0.0 11.331 25.359 20.865 63.510 9. 128 0.0 27.534 85,260 0.0 50.795 3.200 12.079 0.0 49.499 15.225 0.0 -1.65 25
Fat 100.0 1.90 6.60 3.06 2.20 3.80 0.0 5.88 2.90 0.0 9.55 0.10 4.80 0.0 0.90 1.80 0.0 0.0 I
Fiber 0,0 s.o 13.2 8.7 2.0 2.6 0.0 4,6 1.4 0.0 2.2 0.0 10.8 0.0 3.4 2.6 0.0 0.0 13
Ca 0.0 0.101 0.300 0.330 0.150 0.036 17.860 0.100 0.394 34.000 10.602 0.750 0.159 0.0 0.260 0.095 0.0 0,0 6.6
P 0.0 0.340 0.510 0.740 0,450 0.310 21.100 0.400 0.252 0.080 5.217 0.080 0.392 0.0 0.630 0.404 0.0 0.0 3.1
Snlt 0,0 0.207 0.222 0.420 0.215 0.158 0.250 0.876 0.337 0.560 1,768 0.410 0.133 100.00 0.217 0.171 0.0 0.0 12.7
Kclasses 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 - U.04
Hnx 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 - 0.04
Quantity 1.0 I.O 1.0 1.0 1.0 1.0 1.0 1,0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 - 1.0
0 transfer 0.0 0.784 2.003 1.313 4.084 0.514 0.0 2.429 1.777 0.0 2.880 0.334 0.877 0.0 1.234 0.982 0.0 -1.0 - U.U
Abbreviations are as defined In the text
^ Right-hand side elenents
to VO 100
(85)
Let x° = -1.65 and = 0.0 and let = x ^ - x At the
stage, the next interval is computed as follows:
Determine: , ,
'Î ■ • '' ■'
Then,
1: If f(Y^) < fCYg);
ak+l = (Y% _ xk), xf+l = xk. and x^+l = Y%
2: If f(Yk) > f(Y^) ;
ak+l = (%k _ Yk), xk+1 = Yk, and x^+l =
3: If f (Y^) = f(Yk) ;
ak+1 = (yk _ %k+l = ^k, and x^+l = Y%
This procedure is illustrated in figure 14 for the first 2 stages. One should realize that the evaluation of f(x^) i nvolves:
1. setting \ = x^,
2. solving the resulting linear model with the Rahman-
Bender linearizatrion of a.
3. computing the real o of protein from the solution, •
4. computing the achieved probability for protein (Pa), 101
5. computing f(x^) = ABS (Fa - 0.95).
Therefore, if k is the number of phases required for con
vergence in the golden search, then 2xk linear programming
problems must be solved for, before the solution of the
original problem is found. Unless k is relatively small,
this algorithm may become extremely cumbersome. But in
practical applications, the algorithm might converge con
sistently to about the same value of A. If this is the
case, then the range of search for A could be shortened
with a dramatic improvement in the performance of the al
gorithm.
(3) Iterated Taylor series expansion (ITS).
A Taylor series is the polynomial expansion of a function
at a given point. If f(x,, x^, ..., x ) is a function of n 1 2 n variables, then the Taylor series expansion at the point
X. = x ° (j = 1...... n) is: J J
(87) F(Xj, x^, ..., x^) = f(x°, x°, ... x°)
2 + (x -x°) + 1 5 (x.-xO)Z ^ J j 9Xj X =x° 2 ^ J j 9x^ x. = xÇ j J j
Then, if the x^'s are sufficiently close to the x 9, the
difference, (Xj-x9), will converge to zero as it is raised
to second and higher powers and the series will converge
fast enough to justify an approximation with only the first 102
f (x)
-1.650 -1.023 -0.627 0
y O
Figure 14. Illustration of the first 2 stages in the golden search. Because f(Y^) < f(Y°) a1 — Yp — v-O a rtf'i pnrl = -vO Y ano X ; ^2" 103
y 2 2 ^ two terms. Recall that o =(.a,X.) . Using a Taylor se- ^ J J 3 ries expansion with only the first two terms, the equation
becomes :
C O yO . j j j X. a. X. = 7 % ^ - -
Then the LP could be iterated, with 6^ and consequently Op
re-estimated after each iteration. In the early applica
tion, it became evident that in many instances, the proce
dure was diverging or oscillating from the optimal solu
tion. The problem is inherent in the method itself, since
if the Xj 's are not sufficiently close to the Xj's, the se
ries may not converge fast enough or may even diverge.
Consequently, the algorithm was modified in order to limit
the step size from one iteration to the other , a procedure
similar to the method of approximation programming (MAP)
described by Himmelblau (1972) and Palacios-Gomez et al.
(1982). The model used for the dairy premix is reported in
table 13 . At each iteration step, the resulting linear programming model was solved using the subroutine ZX4LP of
the IMSL library.
4. General nonlinear programming algorithm (NLP).
A general nonlinear programming code, MINOS (Modular Incore
Nonlinear Optimization System) was used to solve the exact deterministic equivalent to the stochastic programming TABLE 13
Iterated Taylor series expansion algorithm (ITS). Model for the premix feed.
Ingredients ^
Rows BRY BDG CGF CGM CRN DCL DDG FTH LME MAB MOL OAT SLT SBM WHT HPX RHS^FAT
Protein 0.0 II.321 25.359 20.865 63.510 9. 128 0.0 27.534 85.260 0.0 50.795 3.200 12.079 0.0 49.499 15.225 0.0 -1.65 i 25
Fat 100.0 1.90 6.60 3.06 2.20 3.80 0.0 5.88 2.90 0.0 9.55 0.10 4.80 0.0 0.90 1.80 0.0 0.0 > 1
Fiber 0.0 5.0 13.2 8.7 2.0 2.6 0,0 4.6 1.4 0.0 2.2 0.0 10.8 0.0 3.4 2.6 0.0 0.0 1 13
Ca 0.0 0.101 0.300 0.330 0.150 0.036 17.860 0.100 0.394 34.000 10.602 0.750 0.159 0.0 0.260 0.095 0.0 0.0 ' 6.6
P 0.0 0.340 0.510 0.740 0.450 0.310 21.100 0.400 0.252 0.080 5.217 0.080 0.392 0.0 0.630 0.404 0.0 0.0 > 3.1
Salt 0.0 0.207 0.222 0.420 0.215 0. 158 0.250 0.876 0.337 0.560 1.768 0.410 0.133 100.00 0.217 0.171 0.0 0.0 > 12.7
Molasses 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 • Ü.04
Mdx 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 ■ 0.04
Quantity 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 - 1.0
n transfer 0.0 0.784 2.003 1.313 4.084 0.514 0.0 2.429 1.777 0.0 2.860 0.334 0.877 0.0 1.234 0.982 0.0 -1.0 - Ü.U
Max change 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 : 1-0^ " 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 : 1-0
" oio o!o o!o 0.0 o!o 0.0 oio oio o L 0:0 oio oio oio oio oio oio lio oio : .:u
Abbreviations are as defii>ed in the text
Right-hand side eletaenta
Listed RllS are for Iteration 0 only. At the Iteration, they are replaced byxj*+ O.OI*C, where X* * is the solution vector
at iteration 1-1 ».nd C • l/log(i+10).
O 105 problem. In its augmented version, this code can handle both nonlinear constraints and objective functions. Meth ods used are fully described by Murtagh and Saunders (1977,
1980). The dairy premix model is reported in table 14 . TABLE 14
General nonlinear programming algorithm (NLP). Model for the premix feed
Ingredients^
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Row i FAT BRY BDC CCF COM CRN nci. nuc Fin I. MI: M All MOI. (Ml SLT SBM WHT KPX K its *’
Pro :eln 0.0 11.321 25.359 20.865 63.510 9. 128 0.0 27,534 85.260 0.0 50,795 3.200 12.079 0.0 49.499 15.225 0.0 -1.65 ; 25
Fat 100.0 1.90 6.60 3,06 2.20 3.80 0.0 5.88 2.90 0.0 9.55 0.10 4.80 0.0 0.90 1.80 0.0 0.0 ; 1
Fib Mr 0.0 5.0 13.2 8.7 2.0 2.6 0.0 4.6 1.4 0.0 2.2 0.0 10.8 0.0 3.4 2.6 0.0 0.0 ; 13
Ca 0.0 0.101 0.300 0.330 0.150 0.036 17.860 0.100 0.394 34.000 10.602 0.750 0.159 0.0 0.260 0.095 0.0 0.0 ; 6.6
P 0.0 0.340 0.510 0.740 0,450 0.310 21.100 0.400 0.252 0,080 5.217 0.080 0.392 0.0 0.630 0.404 0.0 0.0 m 3. 1
Sal). 0.0 0.207 0.222 0.420 0.215 0. 158 0.250 0.876 0.337 0.560 1.768 0.410 0,133 100.00 0.217 0.171 0.0 0.0 > 12.7
Kol.Msses 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 .0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.04
Md x 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 - 0.04
Quai tlty 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 I.O 1.0 1.0 1,0 1.0 1.0 1.0 1.0 0.0 « .0
0.0 0.784 2.003 1.313 4.084 0.514 0.0 2.429 1,777 0.0 2.880 0.334 0.877 0.0 1.234 0.982 0.0
Abbreviations are as defined In the text.
Right-hand side elements.
F(l) : ( ): 0, •••» 17. Listed on the row are the j ’ ^ 'r
o O' 107
3.3.4 Solutions to JCCP problems
These problems are characterized by multiple stochastic
rows, i.e., two or more nutrients are imperfectly known.
Most feed formulation problems should fall into this cat
egory, since an array of nutrients must be considered in
order to formulate a least-cost balanced ration.
Data on expected feed composition and variance- covari
ance matrices of nutrients within feedstuffs were those ob
tained from the procedures described in section 3.3.1.
Feed prices were those reported in table 11.
The proposed algorithms were tested with three different
types of feed: a dairy supplement, a dairy grain mix and a
total mixed ration.
A supplement is a mixture of protein, vitamins and min
erals designed to be added to grains in order to form a
complete grain mix. It normally represents 25 to 30% of
the final grain mix. This type of product is sold to small
local feed mills or directly to dairy farmers mixing their own grain mix on the farm. The desired composition of the supplement was:
NE(1): 1.64 Mcal/kg minimum
Crude protein: 27% minimum
Calcium : 2.2%
Phos phorus: 1.7%
Salt: 3.33% 108
Feather meal and meat and bone meal were constrained at a
level lower than 15% of the mix. A micro premix was incor
porated at a rate of 1.5%.
A dairy grain mix is a mixture of protein, vitamins,
mineral sources and grains designed to be added to forages
in order to form a balanced ration. It normally represents
40 to 50% of the ration of a lactating dairy cow. The de
sired composition of the grain mix was:
NE(1): 1.64 Mcal/kg minimum
Crude protein: 16% minimum
Calcium: 0.7%
Phosphorus: 0.65%
Salt: 1.0%
Additional constraints were added to insure adequate palat-
ability and physical properties of the mix: maximum 5% of
feather and bone meal; 20% of corn gluten feed and brewers
dried grains and 40% wheat grain. Molasses and a micro
premix were forced at 5% and .5% respectively. It was
thought that the added number of constraints of this feed
as compared to the dairy supplement could significatly
change the performance of the proposed algorithms.
A total mixed dairy ration (TMR) is a mixture of prote
in, vitamins and mineral sources, grains and forages de
signed to be fed as the sole feed to lactating dairy cows.
The TMR was set for a theoretical cow weighing 650 kg and 109
producing 30 kg of milk at 3.5% fat. The maximum dry mat
ter intake was set at 19 kg and the corresponding nutrient
requirements were computed from tabulated values (NRC,
1978). Dry fat, a mixture of tallow blended with the non
nutritive carrier verxite (Jenkins and Palmquist, 1984) was
allowed in the ration as a high source of energy with very
little variability in its composition. A mineral and vita
min mix containing 18% calcium and 10% phosphorus was also
incorporated into the models. Prices for ingredients were
those suggested by the State Extention Office (Pritchard,
personnal communication) and are reported per metric ton of
dry matter: legume-grass hay - 2nd cutting (HAY), $95.28;
legume-grass silage - 1st cutting (HLG), $85.81; corn si
lage -no NPN added (CSL), $70.77; high moisture ground ear
corn (GEC), $128.62; soybean meal (SBM), $214.33; dry fat
(DPT), $400.00; mineral mix (MIN), $250.00.
Among the algorithms considered for solving JCCP prob
lems, some use the Rahman-Bender estimate of a variance as described previously. A well known statistical inequality,
the Bonferroni inequality was also used in order to get simplified algorithms. This inequality goes as follows: for any set of random events,
(89) Pr ZgàOg...... ZpàOp) i
Pr (Z^èap + Pr (ZgàOg) + ... + Pr ( Z ^ è a ^ ) - (P-1) 110
where z.» 1 = 1 , are random variables, are given
constants and P is the number of events in the set. The
algorithm considered were:
1: Industry standards (IS)
The way that feed manufacturers cope with uncertainty in
composition of feedstuffs varies among them. We defined arbitrarily the following scheme to be the one used for
comparison with the alternate algorithms.
Discount feedstuffs' energy level by 1 .65 times
their individual standard deviation for energy.
Discount feedstuffs' protein level by 1.65 times
their individual standard deviation for protein.
- Set the other nutrients as if they were known with
certainty and set their respective RHS to the desired
values .
The resulting model is reported in table 15 for the supple ment feed. In short, the model tries to adjust for uncer tainty of CP and NE(1) by incorporating Rahman-Bender esti mates. The remaining nutrients are considered deterministic and the RHS is set to the tag value.
2: Rahman-Bender estimate of with Bonferroni inequal ity (RBB) TABLE 15
Industry standards model (IS) for the supplement feed.
INIW.nlKNlS
2 3 4 5 6 7 8 9 10 11 12 13 14 B(tws HDC CCF COM CRN 1)1)0 HIT HAB MOL SBM WHT 1 A3 nci. i.w: SLT MPX RHS "2
Sig(l) 0.70 0.84 0.91 0.43 1. 12 0. 14 0.93 0.10 0.49 0.35 0.54 0.0 0.0 0.0 -10.0 0.0 0.0 0.0
Sig(2) 20,03 13.13 40.84 05. 14 24.29 17.77 28.80 3,34 12.34 9.82 0.0 0.0 0.0 0.0 0.0 -10.0 0.0 0.0
NE(1) t.47 1.72 l.Bl I. 78 1.86 1.48 1.51 1.23 1.81 5.35 0.0 0.0 0.0 0.0 -1 .95 0.0 0.0 1.64
CP 25. 359 20.865 63.510 9.128 27.534 85.260 50.795 3.200 49.499 15.225 0.0 0.0 0.0 0.0 0.0 -1.35 0.0 27.0
CA 0. 30 0.33 0. 15 0.036 0. 10 0. 394 10.602 0.75 0.26 0.095 0.0 17.86 34.0 0.0 0.0 0.0 0.0 2.2
P 0.51 0. 74 0.45 0.31 0.40 0.252 5.217 0.08 0.63 0.404 0.0 21. 10 0.08 0.0 0.0 0.0 0.0 1.7
SALT 0.222 0.420 0.215 0. 158 0.876 0.337 1.768 0.410 0.217 0.171 0.0 0.25 0.56 100.0 0,0 0.0 0.0 3.33
FAT 6.60 3.06 2.20 3.80 5.86 2.90 9.55 0.10 0.90 1.80 100.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0
HAB 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.15
HIT 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.15
MOL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05
MPX 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.015
Quantity 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0,0 0.0 1.0 1.0
Abbreviations as defined in the text. 112
Each nutrient's standard deviation was estimated using the
Rahman-Bender approximation given by equation (77). Over
all level of probability was insured by a Bonferroni ine
quality, each row's probability of success being set to the
same value. The resulting model is reported in Table 16
for the supplement. Since this is a linear model, it could
be solved by regular linear programming procedures (IBM'S
MPS package) .
3: Bonferroni inequality with equal weight on row's
probability (BEW).
This model computes the true o, while overall probability of success is insured by a Bonferroni inequality where each row is assigned an equal probability level. Models are re ported for the supplement (table 17) and the TMR (Table
18). Due to its nonlinear constraints (computation of
a 's), the model had to be solved by nonlinear programming techniques (MINOS).
4 : Bonferroni inequality with unequal weight on rows'
probability (BUW).
In this model, each row's probability of success level is individually computed, the sum of all being forced to meet a Bonferroni inequality (Table 19 and 20). Normal distri bution of nutrients in the final mix is assumed. TABLE 16
Rahman-Bender estimate of X with Bonferroni inequality (RBB) Model for the sup- plement feed.
ihi;kli)Ikhts
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Rows BDC CCF COM CRN DOC niT KAU MOL SBM WHT FAT DCL LKE SLT MPX RllS °2 %
Slg(I) 0.70 0.84 0.91 0.43 1.12 0.14 0.93 0.10 0.49 0.35 0.54 0.0 0.0 0.0 -10.0 0.0 0.0 0.0 0.0 0.0 0,0
SIb (2) 20.03 13, 13 40.84 05.14 24.29 17.77 28.80 3.34 12.34 9.82 0.0 0.0 0.0 0.0 0.0 -10.0 0.0 0.0 0.0 0.0 0.0
SlgC) 0.42 0.80 0.23 0. 18 0. 12 0.91 12.60 0.75 0.54 0.27 0.0 6.75 6.90 0.0 0.0 0.0 -10.0 0.0 0.0 o.u 0,0
5lR(4) 0.33 1.37 0.29 0.48 0.25 0.80 5.71 0.08 0.40 0.40 0.0 2.40 0.11 0.0 0.0 0.0 0.0 -10.0 0.0 o.u 0.0
Si?(5) 1.25 0.91 0.95 0.85 5.09 1.25 5.89 0,41 0.98 0.25 0.0 0.21 0.96 10.0 0.0 0.0 0.0 0.0 -10.0 0.0 0.0
NE(1) 1.47 1.72 1.61 1.78 1.86 1.48 1.51 1.23 1.81 5.35 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 0.0 0.0 1.64
CP 25.359 20.665 63.510 9.128 27.534 85.260 50.795 3.200 49.499 15.225 0.0 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 0.0 27.0
CA 0.30 0.33 0.15 0.036 0.10 0.394 10.602 0.75 0.26 0.095 0.0 17.86 34.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 1.76
P 0.51 0.74 0.45 0.31 0.40 0.252 5.217 0.08 0.63 0.404 0.0 21. 10 0.08 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 1.36
SALT 0.222 0.420 0.215 0. 158 0.876 0.337 1. 768 0.410 0.217 0.171 0.0 0.25 0.56 100.0 0.0 0.0 0.0 0.0 -2.33 0.0 2.66
FAT 6.60 3.06 2.20 3.80 5.88 2.90 9.55 0.10 0.90 1.80 100.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0,0 3.0
HAB 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0,15
FUT 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.15
MOL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.05
MPX 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.015
Quant: ty 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 I .0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0,0 1,0 1.0
Abbreviations are as listed In the text. TABLE 17
Bonferroni inequality with equal weight on row's probability (BEW). Model for the supplement feed.
INGREDIENTS ^
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Rows BDC CCF COM CRN DDC. FTIT MAD MOL SBM WHT FAT DCL LHE SLT MPX RHS *’l "2 "3 ^^4 "5
0.70 0.84 0.91 0.43 1.12 0.14 0.93 0.10 0.49 0.35 0.54 0.0 0.0 0.0
K(2) 20.03 13.13 40.84 05,14 24.29 17.77 28.80 3.34 12.34 9.82 0.0 0.0 0.0 "2J 0.0 F(3) 0.42 0.80 0.23 0. 18 0. 12 0.91 12.60 0.75 0.54 0.27 0.0 6.75 6.90 0.0
F(4) 0.33 1.37 0.29 0.48 0.25 0.80 5.71 0.08 0.40 0.40 0.0 2.40 0.11 0.0
F(5) 1.25 0.91 0.95 0.85 5.09 1.25 5.89 0.41 0.98 0.25 0.0 0.21 0.96 10.0 '5j
NE(1) 1.47 1.72 1.81 1.78 1.86 1.48 1.51 1,23 1,81 5,35 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 0.0 0.0 1.64
CP 25.359 20.865 63.510 9.128 27.534 85.260 50.795 3.200 49.499 15.225 0.0 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 0.0 27.0
CA 0.30 0.33 0.15 0.036 0.10 0.394 10.602 0.75 0.26 0.095 0.0 17.86 34.0 0.0 0.0 0.0 -2.33 0.0 0.0 0.0 1.76
P 0,51 0.74 0.45 0.31 0.40 0.252 5.217 0.08 0.63 0.404 0.0 21. 10 0.08 0.0 0.0 0.0 0.0 -2.33 0.0 0.0 1.36
SALT 0.222 0.420 0.215 0.158 0.876 0.337 1.768 0.410 0.217 0.171 0.0 0.25 0.56 100.0 0.0 0.0 0.0 0.0 -2.33 0.0 2.66
FAT 6.60 3.06 2.20 3.80 5.88 2.90 9.55 0. 10 0.90 1.80 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0.0 3.0
HAB 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.15
HIT 0.0 0.0 0.0 0.0 0.0 1 .0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 U. 15
MOL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05
MPX 0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 1.0 0.015
Ouantity 1.0 1.0 1 .0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0
Abbre.'latlonfl are as defined in the text.
*’ F(l) : ( r. sj - o - 0, ., li. Listed on the first five rows arc x 10 J ^ 115
TABLE 18
Bonferroni inequality with equal weight on row's probabi lity (BEW). Model for the total mixed ration (TMR).
INGREDIENTS ^
1 2 3 4 5 Rows HAY HLG CSL GEC SBM
F(l)* s^j: 9.4435 3.6740 2.2650 0.6236 0.512S
F(2) Sgj: 8.7593 10.304 1.7989 2.5418 1.9219
NE(1) 1.25 1 1.139 1.480 1.850 1.810
CP 15 . 894 14.590 7.875 9.445 54.999
ADF 36.369 40.244 28.239 9.576 4.599
CA 0.9 14 0.894 0.274 0.052 0 .289
P 0.298 0 .287 0.230 0.314 0 . 700
Quanti ty 1.0 1.0 1.0 1.0 1.0
^ F(i): ( j Xj )^ - = 0 ; i = 1, 2; j
2 2 Listed on the first two rows are s^^ x 10 c
^ X is set to -0.675, -0.841, -1.038, -1.281,
of 50, 60, 70 , 80.and 90% respectively.
Abbreviations are as reported in the text.
115
lENTS ^
4 5 6 GEC SBM DFT MIN RHS "2
1 0.6236 0,5129 0.0183
' 2.5418 1.9219 0.0
1.850 1.810 2.700 0.0 0.0 > 29 . 1
i 9.445 54.999 0.0 0.0 0.0 > 275 .5
) 9.576 4.599 0.0 0.0 0.0 0.0 > 399 .0
0.052 0.289 0.0 0.0 0.0 18,0 > 11.4
0.314 0.700 0.0 0.0 0.0 10.0 > 7.6
1. 0 1.0 1.0 0.0 0.0 1,0 = 19 .0
; i = 1, 2 ; j = 1 , .. ., 6. 2 . _2 . 2 are x 10 and .
-1.038, -1.281, -1.654 for overall probability pe ctively.
I in the text.
116
TABLE 19
Bonferroni inequality with unequal weight on row's proba bility (BUW). Model for the supplement feed.
INCREUIENTS
1 2 3 4 5 6 7 8 9 10 1 Rows BDC CCF CCM CRN DDC niT KAfi MOL SBM WHT FA*
: 1.47 I. 72 1.81 1.78 1.86 1.48 1.51 1.23 1.81 1.81 5. F(l)° ‘J : 0.70 0.84 0.91 0.43 1.12 0.14 0.93 0.10 0.49 0.35 0.
:25.359 20.865 63.510 9.128 27.534 85.260 50.795 3.200 49.499 15.225 0.( F(2) ‘J :20.03 13.13 40.84 5.14 24.29 17.77 26.80 3.34 12.34 9.82 0.(
: 0.30 0.33 0.15 0.036 0.10 0.394 10.602 0.75 0.26 0,095 0.( F(3) ^ : 0.42 0.80 0.23 0.18 0.12 0.91 12.60 0.75 0.54 0.27 0.(
: 0.51 0.74 0.45 0.31 0.40 0.252 5.217 0.08 0.63 0.404 0.( F(M “J : 0.33 1.37 0.29 0.48 0.25 0.80 5.71 0.08 0.40 0.40 0.{
: 0.222 0.420 0.215 0. 158 0.876 0.327 1.768 0.410 0,217 0. 171 0.( F(5) : 1.25 0.91 0.95 0.85 5.09 1.25 5.89 0.41 0.98 ' 0.25 0.(
S ump rob 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.
Fat 6.60 3.06 2.20 3.80 5.88 2.90 9.55 0.10 0.90 1.80 100
Mab 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.
Fht 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.
Kol 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.
Mpx 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.
Abbrevlaclortb are as defined In the text.
1 (- \ F(l) : du - Pr - Ü.I-1...... 5, j»l, U (2")
116
'ow * s pr ob a-
INCKEIHENTS
5 6 7 8 9 10 11 12 13 14
DDC niT MA8 MOL SBM UlIT FAT DCL LHE SLT If X KJIS '^l '■^2 "'3
1.86 1.48 1.51 1.23 1.81 1.81 5.35 0.0 0.0 0.0 1.64
1.12 0.14 0.93 0. 10 0.49 0.35 0.54 0.0 0.0 0.0
7.534 85.260 50. 795 3.200 49.499 15.225 0.0 0.0 0.0 0.0 27.0
4.29 17.77 28.80 3.34 12.34 9.82 0.0 0.0 0.0 0.0
0.10 0.394 10.602 0.75 0.26 0.095 0.0 17.86 34.0 0.0 1.76
0.12 0.91 12.60 0.75 0.54 0.27 0.0 6.75 6.90 0.0
0.40 0.252 5.217 0.08 0.63 0.404 0.0 21.10 0.08 0.0 1.36
0.25 0.80 5.71 0.08 0.40 0.40 0.0 2.40 0.11 0.0
0.876 0.337 1.768 0.410 0.217 0.171 0.0 0.25 0.56 100.0 2.66
5.09 1.25 5.89 0.41 0.98 ' 0.25 0.0 0.21 0.96 10.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 .0 1.0 1.0 1.0 1.0 0.0 ' 4.95
5.88 2.90 9.55 0.10 0.90 1.80 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 > 3.00
0.0 0.0 1 .0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ; 0.15
0.0 1 .0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 . 0.15
0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.05
0.0 0.0 0.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 - 0.015
Pr j, *• 0,1-1, .... 3, J - I, U, Listed on tlie tirst ten rows are s ^ ^ x 10 and b ^.
117
TABLE 20
Bonferroni inequality with unequal weight on row's probabi lity (BUW). Model for the total mixed ration (TMR).
INGREDIENTS^
1 2 3 4 5 Rows HAY HLG CSL GEC SB]
1.251 1. 139 1.480 1.850 1.811 Fd)" 9.4435 3.6740 2.2650 0.6236 0.51: = ir 15 . 894 14.590 7.875 9 .445 54.9! F (2) 8.7593 10.304 1.7989 2.5418 1.92 ° 2 r S umprob 0.0 0.0 0.0 0.0 0.0
ADF 36.369 40 .244 28.239 9.576 4.5!
CA 0.9 14 0.894 0.274 0.052 0.28!
P 0.298 0.287 0.2 30 0 .314 0 . 70(
Quanti ty 1.0 1.0 1. 0 1.0 1.0
Abbreviations are as reported in the text.
(j 'ij \
1 (- J u^) F(i) e du - Pr J
k = 1 + desired overall probability level.
117
's p r ob ab i- >1R) .
INGREDIENTS^
3 4 5 6 CSL GEC SBM DFT MIN RHS ; :^2
9 1. 480 1.850 1.810 2 . 700 29 . 1
40 2 .2650 0.6236 0.5129 0 .0183
90 7 . 875 9.445 54.999 0 .0 275 .5
04 1.7989 2.5418 1.9219 0.0
0.0 0 .0 0.0 0.0 1 .0 1 .0 0 . 0 =
144 28.239 9.576 4.599 0.0 0.0 0.0 0 .0 > 399 .0
14 0.274 0.052 0.289 0.0 0.0 0.0 18.0 > 11.4
37 0.230 0.314 0 . 700 0.0 0.0 0 . 0 10 .0 > 7.6
0 1. 0 1.0 1.0 1.0 0.0 0.0 1.0 = 19.0
reported in the text.
(- j "2) e du - Pr. = 0 ; i = 1, 2 ; j = 1, 6 1
rail orobability level.
1 18
5: Overall probability by multivariate integration
(MI) .
Nutrients are assumed to follow a multivariate normal dis tribution in the final mix. For reasons discussed earlier, only 2 nutrients can be approached in this fashion. Over all probability of success is achieved through the computa tion of a bivariate normal probability integral. A tetra- chloric series expansion was used as a numerical method for the evaluation of the bivariate probability integral (Gup ta, 1963). The model was applied to the TMR feed since this is a case where it seems reasonable to consider uncer tainty on only two nutrients (Table 21).
Level of complexity increases in ascending order from model 1 to 5. Therefore, accurate assesments of human and computer resources needed for the solution of the problem were recorded and used in comparison of models. Post-opti mization procedures were performed in order to assess the true overall probability of success of the final mix. For each solution obtained, a variance-covariance matrix was calculated from the amount of each feedstuff used and their respective variance-covariance matrices. The solution var iance-covariance matrix (Z) plus the vector of expected nutrient levels (p) were used to generate 10,000 vectors from a multivariate normal distribution (p, Z), using the algorithm described by Fishman (1978). In each instance. 119
TABLE 2 1
Overall probability by multivariate integration algorithm (MI). Model for the total mixed ration (TMR).
INGREDIENTS^
1 2 3 4 5 6 Rows HAY HLG CSLGEC SBM DFTMIN RHS
1.251 1 . 139 1.480 1. 850 1.810 2 . 700 29 . 1
15.B94 14.590 7.875 9.445 54.999 0.0 275 .5
9.4435 3.6740 2.2650 0.6236 0.5129 0.0183
s 2 . 8.7593 10.304 1.7989 2.5418 1.9219 0.0 Z2j • 0.1378 0.2432 0.0062 0.0097 0.0191 0.0 ® 12j • ADF 36.369 40.244 28.239 9.576 4.599 0.0 0.0 = 399 .0
CA 0.914 0.894 0.274 0.052 0 . 289 0.0 18.0 11.4
P 0.298 0.287 0.230 0.314 0. 700 0.0 10.0 7.6
Quantity 1.0 1.0 1 .0 1 . 0 1.0 1 .0 1.0 = 19.0
Abbreviations are as reported in the text.
“ ■J - h (j'%1 (j 'Lj
Ftl): dvdu - Pr ■ 0
J 12J "j 1. 2; J - I. 2.
Reported in the first five rows are a "llj " *“ • »22j :|2j- 120 the achieved level of each nutrient was compared to the de sired level and the result was coded as success if it was larger, and failure otherwise. The Monte Carlo estimate of the overall probability of success is given by the number of vectors passing the test (success) for all nutrients, divided by the total number of vectors generated. The re sulting statistic follows a binomial distribution and was tested using the large sample approximation given by Hol lander and Wolfe (1973, page 16). The desired overall suc cess rate was set arbitrarily at 95% for the supplement and the grain mix. In the TMR case, probability levels of 50,
60, 70, 80, and 90% were investigated. This allows for a crude description of the relationship between certainty level and cost of ingredients in the ration. 121
3.4 RESULTS AND DISCUSSION
3.4.1 Composition of feedstuffs
Univariate statistics on feed composition are reported In table 22 . In all hays and silages, there Is an Increase
In quality from the first cutting to the third cutting, as shown by an increase In the means of CP and NE(1). Mean Ca level also Increases with cutting number, while levels of other nutrients remain fairly constant. Mean compositions are generally close to the NRC tabulated values (NRC,
19 78 ) .
Variations In composition within each forage are sub stantial. Across all forages, coefficients of variation
(C.V.) are relatively low for NE(1) (less than 20%), mod erate for CP, P, K, Ca, Mg and Zn (20 to 30%), and high for
Mn, Fe, Cu (greater than 30%). The effect of soil composi tion and fertility on crops' composition is well documented and explains the variation observed in levels of minerals. 122
TABLE 22
Univariate statistics on feed composition.
ITEM^ MEAN S.D.b c.v.c SKEWNESS KURTOSIS P(Ho)d
Legume hay 1st cutting
DM 86.697 4.198 4.842 -1.150 2.487 <0.01 CP 15.046 3.400 22.594 -0.029 -0.196 >0.15 NE(1) 1.282 0.169 13.175 -0.458 -0.192 >0.15 P 0.258 0.060 23.226 0.159 0.339 <0.01 K 2 .350 0.607 25.844 0.146 0.137 >0.15 Ca 0.995 0.242 24.377 0.164 0.372 0.08 Mg 0.239 0.065 27.280 0.820 1.458 <0.01 Mn 37.713 23.004 60.997 2.626 11.211 <0.01 Fe 97.153 67.568 69.534 3.024 12.206 <0.01 Cu 7.949 2.025 25.473 0.031 0.178 <0.01 Zn 25.806 5.816 22.539 1.187 3.739 <0.01
Legume hay 2nd cutting
DM 86.826 3.464 3.990 -0.969 2.773 <0.01 CP 16.821 3.097 18.409 0.171 2,012 0.02 NE(1) 1.330 0.167 12.576 -0.191 0.329 >0.15 P 0.281 0.055 19.656 -0.148 0.398 <0.01 K 2.437 0.617 25.301 0.347 0.483 0.11 Ca 1 .022 0.250 24.486 0.567 0.804 <0.01 Mg 0.250 0.066 26.617 1.165 2.725 <0.01 Mn 39.582 20.662 52.199 2.053 9.071 <0.01 Fe 92.831 54.325 58.521 2.777 9.903 <0.01 Cu 8.763 1.715 19.568 0.019 0.507 <0.01 Zn 26.258 5.836 22.225 2.334 17.349 <0.01
Legume hay 3rd cutting
DM 86.044 3.495 4.062 -0.451 0.673 <0.01 CP 18.413 3.788 20.575 0.667 7.442 <0.01 NE(1) 1.406 0.180 12.796 -0.626 1.247 0.10 P 0.273 0.056 20.517 -0.007 0.721 <0.01 K 2.452 0.634 25.841 0.178 0.256 >0.15 Ca 1.157 0.289 24.967 0.510 0.405 0.03 Mg 0.258 0.064 24.714 0.384 -0.107 <0.01 Mn 40.430 20.083 49.675 1.670 6.684 <0.01 Fe 110.502 76.515 69.243 3.625 18.355 <0.01 Cu 9.133 1.930 21.129 1.263 7.110 <0.01 Zn 26.996 7.503 27.795 2.706 11.467 <0.01 123
TABLE 22 (continued)
ITEM MEAN S .D. C.V . SKEWNESS KURTOSIS P(Ho)
Legume- grass hay 1st cutting
DM 86.663 3.495 4.033 -1.307 3.403 <0.01 CP 12.634 3.290 26.041 0.381 0.049 <0.01 NE(1) 1.150 0. 0. -0.564 0.400 <0.01 P 0.251 0.555 22.043 0.247 0.341 <0.01 K 2.253 0.567 25.183 0.177 0.217 >0.15 Ca 0.766 0.264 34.511 0.532 0.22 5 <0.01 Mg 0.216 0.065 29.887 0.793 1.099 <0.01 Mn 54.006 29.944 55.445 1.406 2.277 <0.01 Fe 87.565 63.276 72.262 3.188 13.090 <0.01 Cu 7.168 2.166 30.217 0.870 3.964 <0.01 Zn 26.085 6.144 23.554 2.061 10.743 <0.01
Legume-,grass hay - 2nd cutting
DM 86.768 2.987 3.443 -1.249 4.368 <0.01 CP 15.894 2.960 18.621 -0.022 1.120 <0.01 NE(1) 1.251 0.307 24.561 -15.265 20.593 <0.01 P 0.298 0.058 19.338 0.065 0.106 <0.01 K 2.479 0.577 23.289 0.178 0.078 >0.15 Ca 0.914 0.231 25.257 0.434 0.989 <0.01 Mg 0.255 0.061 23.800 0.534 0.553 <0.01 Mn 51.932 26.269 50.584 1.475 3.245 <0.01 Fe 95.601 56.351 58.944 3.431 18.717 <0.01 Cu 8.670 1.872 21.591 1.054 8.767 <0.01 Zn 27.424 5.848 21.323 2.089 16.220 <0.01
Legume-]grass hay - 3rd cutting
DM 86.024 3.382 3.932 -0.824 2.581 <0.01 CP 17.504 3.919 22.389 -0.290 1 .092 <0.01 NE(1) 1.306 0.166 12.735 -0.197 -0.657 >0.15 P 0.299 0.058 19.425 -0.123 0.652 0.03 K 2.536 0.593 23.363 -0.205 0.015 0.01 Ca 0.990 0.287 29.023 0.794 1.928 <0.01 Mg 0.258 0.069 26.702 0.935 1.915 <0.01 Mn 50.658 22.530 44.475 1 .465 4.245 <0.01 Fe 103.022 59.424 57.680 3.543 20.665 <0.01 Cu 8.774 1 .848 21.063 0.341 4.716 <0.01 Zn 27.529 5.589 20.302 0.702 1.479 <0.01 124
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Grass hay - 1st cutting
DM 86.652 3.965 4.575 -1.067 1 .388 <0.01 CP 9.699 2.969 30.613 0.892 1.612 <0.01 NE(1) 1.037 0.176 16.983 -0.429 1.024 0.02 P 0.232 0.064 27.517 0.469 0.578 <0.01 K 2.085 0.582 27.894 0.551 0.360 <0.01 Ca 0.496 0.197 39.707 0.588 -0.313 <0.01 Mg 0.178 0.067 37.738 1.621 4.903 <0.01 Mn 65.950 37.360 56.649 1.203 1.349 <0.01 Fe 81.857 74.717 91.278 3.915 20.412 <0.01 Cu 5.391 1.825 33.849 0.292 0.054 <0.01 Zn 24.687 6.85 1 27.751 1.952 10.956 <0.01
Grass hay - 2nd cutting
DM 86.572 4.236 4.893 -1.437 2.310 <0.01 CP 11.373 3.825 33.630 0.649 0.761 0.02 NE(1) 1.065 0.212 19.921 -0.700 0.249 0.18 P 0.257 0.082 31.991 0.808 1.063 <0.01 K 2.124 0.637 30.005 0.300 0.211 >0.15 Ca 0.575 0.192 33.307 0.026 -0.698 0.03 Mg 0 .228 0.087 38.128 0.895 1 .532 0.02 Mn 70.928 35.735 50.383 0.872 0.630 0.05 Fe 99.609 96.476 96.855 3.132 10.870 <0.01 Cu 6 .122 2.291 37.422 0.316 -0.384 <0.01 Zn 26.009 6.824 26.238 0.913 2.219 <0.01
Grass hay 3rd cutting
DM 87.345 4.987 5.709 -0.334 -0.711 0.04 CP 10.614 4.630 43.623 -0.085 -1.230 0.06 NE(1) 1.106 0.224 20.286 -0.004 -1.558 0.35 P 0.256 0.105 41.065 0.329 -0.844 0.34 K 2.148 0.907 42.248 0.165 -1.121 0.33 Ca 0.517 0.231 44.745 0.048 -1.337 0.07 Mg 0.201 0.090 44.883 0.455 -1 .068 0.02 Mn 69.786 45.576 65.309 1.432 2.084 <0.01 Fe 111.310 116.069 104.275 2.107 4.231 <0.01 Cu 6.345 2.768 43.628 -0.288 -1.011 0.24 Zn 26.586 8.373 31.494 0.713 -0.385 0.06 125
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Legume silage - 1st cutting
DM 45.395 12.268 27.026 0.233 -0.659 0.03 CP 16.053 3.234 20.149 0.427 1.608 >0.15 NE(1) 1.225 0.159 12.984 -0.032 0.356 >0.15 P 0.281 0.058 20.775 0.564 0.379 <0.01 K 2.599 0.610 23.487 0.253 0.385 >0.15 Ca 1.066 0.266 24.976 -0.202 -0.311 0.01 Mg 0.253 0.065 25.708 0.935 2.148 <0.01 Mn 51.581 27.408 53.136 1.724 4.122 <0.01 Fe 216.540 131.247 60.611 1.383 1.625 <0.01 Cu 7.717 2.001 25.928 0.794 1.742 <0.01 Zn 30.254 7.545 24.939 2.300 11.969 <0.01
Legume silage 2nd cutting
DM 49.964 11.543 23.102 -0.181 -0.338 0.12 CP 17.268 3.180 18.415 -0.279 -0.037 >0.15 NE(1) 1.259 0.157 12.504 0.055 -0.304 >0.15 P 0.293 0.053 18.133 0.083 0.210 0.05 K 2.636 0.588 22.290 0.069 -0.269 >0.15 Ca 1.067 0 .262 24.607 0.519 1.157 >0.15 Mg 0.261 0.068 25.991 2.080 10.523 <0.01 Mn 48.987 22.126 45.166 1.495 3.980 <0.01 Fe 198.593 120.563 60.709 1.305 1.055 <0.01 Cu 8.315 1 .942 23.358 0.533 0.575 <0.01 Zn 29.679 8.322 28.038 2.649 12.174 <0.01
Legume silage 3rd cutting dm 49.279 12.617 25.603 -0.126 -0.453 >0.15 CP 19.212 3.388 17.634 0.180 0.906 >0.15 NE(1) 1.391 0.159 11.418 -0.096 -0.255 >0.15 P 0.295 0.054 18.204 0.460 0.428 <0.01 K 2.608 0.518 19.863 -0.211 0.284 >0.15 Ca 1 .220 0.330 27.048 1.077 2.856 0.01 Mg 0.274 0.070 25.556 1.434 4.600 <0.01 Mn 51.161 25.739 50.309 0.837 -0.010 <0.01 Fe 206.000 132.685 64.410 1.469 1.703 <0.01 Cu 8.823 1.767 20 .030 0.166 1.010 <0.01 Zn 29.435 7.638 25.950 2.151 6.928 <0.01 126
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS F(Ho)
Legume-grass1 silage 1st cutting
DM 44.959 11.380 25.311 0.223 -0.395 <0.01 CP 14.590 3.210 22.001 0.299 0.488 0.10 NE(1) 1.139 0.192 16.826 0.493 6.903 <0.01 P 0.287 0.056 19.544 0.250 0.225 <0.01 K 2.620 0.587 22.394 0.078 -0.188 0.03 Ca 0.894 0.280 31.338 0.370 0.064 <0.01 Mg 0.240 0.067 27.804 1.589 6.976 <0.01 Mn 61.535 27.178 44.166 1.237 2.334 <0.01 Fe 211.776 127.705 60.302 1.393 1.855 <0.01 Cu 7.336 2.028 27.649 0.998 5.430 <0.01 Zn 29.985 7.130 23.777 1.644 5.854 <0.01
Legume-grass silage - 2nd cutt ing
DM 45.832 11.302 24.660 0.023 -0.438 >0.15 CP 15.790 3.417 21.643 -0.095 -0.197 >0.15 NE ( 1) 1.179 0.201 17.042 1.150 6.576 0.02 P 0.305 0.062 20.265 0.127 -0.198 <0.01 K 2.694 0.631 23.417 0.162 -0.038 >0.15 Ca 0.961 0.275 28.627 0.376 0.329 <0.01 Mg 0.258 0.065 25.267 0.602 0.374 <0.01 Mn 62.323 28.089 45.070 1.262 2.390 <0.01 Fe 229.074 127.501 55.659 1.192 1.035 <0.01 Cu 8.278 2.219 26.807 1.981 17.621 <0.01 Zn 30.124 6.950 23.071 1.805 8.725 <0.01
Legume-grass silage - 3rd cutting
DM 48.772 11.895 24.388 -0.429 -0.350 0.03 CP 18.175 3.232 17.782 -0.060 0.345 >0.15 NE(1) 1.282 0.178 13.846 -0.223 -0.496 0.07 P 0.318 0.056 17.708 0.233 -0.057 0.10 K 2.694 0.571 21.189 0.127 -0.290 <0.01 Ca 1.085 0.255 23.465 0.630 1.314 <0.01 Mg 0.268 0.064 23.949 0,37 3 -0.205 <0.01 Mn 59.972 28.688 47.835 1.404 2.666 <0.01 Fe 224.563 131.676 58.637 1.463 1.804 <0.01 Cu 8.979 1.913 21.304 0.700 1.748 <0.01 Zn 30.924 7.212 23.322 1 .049 1.716 <0.01 127
TABLE 22 (continued)
ITEM MEAN S.D. C.V . SKEWNESS KURTOSIS P(Ho)
Grass silage - 1st cutting
DM 40.468 12.306 30.409 0.306 “0.448 <0.01 CP 11.549 3.421 29.621 1.636 5.674 <0.01 NE(1) 1.049 0.212 20.181 0.088 -0.498 >0.15 P 0.292 0.075 25.712 0.963 1.915 <0.01 K 2.524 0.747 29.573 0.667 1.027 >0.15 Ca 0.531 0.215 40.494 0.389 -0.797 <0.01 Mg 0.195 0.063 32.306 0.881 1.190 <0.01 Mn 70.343 32.617 46.368 0.874 0.774 <0.01 Fe 219.699 140.554 63.976 1.256 1.093 <0.01 Cu 6.162 2.220 36.032 0.651 1.416 <0.01 Zn 29.670 9.259 31.205 1.377 5.012 <0.01
Grass silage 2nd cutting
DM 39.543 14.558 36.816 0.559 -0.204 0.47 CP 12.691 3.305 26.045 0.345 -0.642 0.56 NE(1) 1.172 0.145 12.393 0.246 0.249 0.97 P 0.303 0.063 20.738 0.063 -0.859 0.44 K 2.470 0.622 25.188 -0.367 -1.048 0.27 Ca 0.613 0.243 39.586 0.115 -1.123 0.39 Mg 0.238 0.094 39.649 0.509 -0.993 0.05 Mn 64.783 39.046 60.273 1 .133 0.639 <0.01 Fe 182.714 130.991 71.692 2.638 9.170 <0.01 Cu 6.696 2.162 32.295 1.436 2.364 <0.01 Zn 33.500 9.028 26.948 0.778 -0.297 0.05
Grass silage - 3rd cutting dm 44.812 17.330 38.672 -0.299 -0.824 0.87 CP 9.945 2.703 27.183 -1.492 2.015 0.06 NE(1) 1.152 0.126 10.926 «,. P 0.284 0.065 22.835 -1.188 2.383 0.13 K 2.244 0.476 21.199 -0.420 0.961 0.65 Ca 0.516 0.157 30.327 0.072 2.157 0.51 Mg 0.264 0.106 40.326 0.387 -0.099 0.70 Mn 55.750 24.052 43.143 -0.297 -1.670 0.40 Fe 206.429 216.946 105.095 2.444 6.153 <0.01 Cu 5.875 1 .553 26.428 -1.255 0.238 0.02 Zn 39.625 8.991 22.690 -0.100 -1.283 0.42 128
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Corn silage no NPN added
DM 35.824 6.454 18.016 0.471 0.732 <0.01 CP 7.875 1.341 17.031 0.613 1.021 <0.01 NE(1) 1.480 0.151 10.172 -1.006 2.502 <0.01 P 0.230 0.042 18.425 0.581 0.977 <0.01 K 1.096 0.262 23.874 0.624 0.465 <0.01 Ca 0.274 0.093 33.823 1.728 4.958 <0.01 Mg 0.209 0.054 25.870 0.971 1.509 <0.01 Mn 41.369 22.418 54.191 1.998 6.316 <0.01 Fe 123.529 101.013 81.772 3.594 17 .887 <0.01 Cu 4.429 1.508 34.044 1.358 10.698 <0.01 Zn 33.485 8.846 26.419 1.376 4.101 <0.01
Corn silage NPN added
DM 37.196 6.496 17.465 0.548 0.415 <0.01 CP 12.046 3.225 26.774 1.348 6.795 <0.01 NE(1) 1 . 503 0.137 9.089 -0.303 2.567 <0.01 P 0.288 0.072 24.874 0.485 -0.209 <0.01 K 1.140 0.266 23.356 0.594 0.389 <0.01 Ca 0.321 0.129 40.265 1 .406 1.846 <0.01 Mg 0.222 0 .062 28.047 1.063 1.526 <0.01 Mn 42.644 21.356 50.081 1.675 4.530 <0.01 Fe 141.058 98.207 69.622 2.842 12.051 <0.01 Cu 5.415 2.141 39.532 1.341 4.614 <0.01 Zn 38.778 12.664 32.658 1.217 2.232 <0.01
Sorghum silage
DM 37.467 15.389 41 .073 1.528 2.732 <0.01 CP 12 .089 3.460 28.623 -0.038 -0.292 >0.15 NE(1) ..•« P 0.270 0.084 31.159 1 . 986 10.788 <0.01 K 2.265 0.898 39.658 0.031 -0.875 >0.15 Ca 0.572 0.235 41.091 0.679 -0.543 <0.01 Mg 0.317 0.128 40.2 56 1.276 3.325 0.02 Mn 56.778 33.252 58.566 1.509 2.385 <0.01 Fe 457.481 582 . 183 127.258 2.090 3.018 <0.01 Cu 7 .222 3.112 43.091 0.483 0.699 >0.15 Zn 39.611 16.643 42.017 2.987 15.943 <0.01 129
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Small grain silage
DM 38.363 9.345 24.361 0.270 -0.64 6 >0.15 CP 11 .348 2.909 25.638 0.565 -0.228 <0.01 NE(1) 1.279 0.149 11.628 -0.056 -0.608 0.09 P 0.315 0.069 22.048 0.551 -0.322 <0.01 K 2.649 0.641 24.185 0.234 -0.229 >0.15 Ca 0.398 0.172 43.088 1.506 2.039 <0.01 Mg 0 .184 0.072 39.161 1.600 2.640 <0.01 Mn 60.591 28.240 46.608 1.237 2.086 <0.01 Fe 312.480 270.999 86.725 1.797 3.215 <0.01 Cu 5.431 2.424 44.629 1.487 5.237 <0.01 Z n 30.802 9.222 29.940 0.616 0.399 <0.01
Dry shelled corn
DM 87.636 4.439 5.294 — 1.646 2.941 <0.01 CP 9.611 0.877 9.127 0.605 2.179 <0.01 NE(1) 2.293 0.066 2.884 -1.333 2.519 0.28 P 0.353 0.062 17.614 -0.130 0.709 0.13 K 0.492 0.068 13.907 0.010 0.739 0.09 Ca 0.036 0.035 96.513 1.930 4.424 <0.01 Mg 0.151 0.031 20.196 -0.015 1.300 <0.01 Mn 11 .009 3.828 34.769 1.194 2.037 <0.01 Fe 40.991 16.510 40.277 2.067 5.854 <0.01 Cu 3.220 2.153 66.866 3.264 16.246 <0.01 Zn 37.3822 9.751 26.085 1.602 4.931 <0.01
High mo isCure s he 1led corn
DM 73.887 4.451 6.024 -0.182 0.004 >0.15 CP 9.758 0.996 10.205 0.004 0.021 >0.15 NE(l) 2.324 0.084 3.605 -0.042 -0.896 0.06 P 0.360 0.065 17.945 0.234 -0.291 <0.01 K 0.490 0.079 16.204 -0.723 4.618 0.03 Ca 0.033 0.029 86.847 2.045 5.439 <0.01 Mg 0.154 0.034 21.729 -0.105 1.375 <0.01 Mn 10.989 4.157 37.831 1.464 3.170 <0.01 Fe 45.768 21.424 46.811 2.274 6.941 <0.01 Cu 2.694 1.875 69.607 3.204 14.605 <0.01 Zn 36 . 288 10.383 28.613 1.563 5.279 <0.01 130
TABLE 22 (continuée")
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Dry ground ear corn
DM 80.650 7.773 9.638 -0.597 -0.313 <0.01 CP 10.506 2.632 25.048 1.271 0.868 <0.01 NE(1) 2.196 0.117 5.321 -0.865 1 .018 0.24 P 0.377 0.109 29.040 0.521 -0.434 0.15 K 0.575 0.132 23.018 1.248 1.563 <0.01 Ca 0 .092 0.104 113.306 1.562 1.289 <0.01 Mg 0.149 0.036 23.796 0.245 -0.041 0.14 Mn 19.855 14.027 70.646 1.664 2.732 <0.01 Fe 71.758 54.890 76.494 1 .441 1.388 <0.01 Cu 4.650 3.995 85.914 2.382 7.364 <0.01 Zn 53.306 35.868 67.287 2.092 4.501 <0.01
High moisture ground ear corn
DM 69.418 6.049 8.714 0.911 1.098 <0.01 CP 9.445 1.594 16.880 1.446 3.875 <0.01 NE(1) 2.145 0.174 8.119 -0.209 0.075 >0.15 P 0.314 0.063 20.115 1.138 2.829 <0.01 K 0.524 0.080 15.354 1.263 3.597 <0.01 Ca 0.052 0.052 99.652 3.687 18.356 <0.01 Mg 0.139 0.028 20.257 0.899 1.071 <0.01 Mn 14.100 6.864 48.679 2.356 8.418 <0.01 Fe 55.361 33.670 60.818 2.391 6.852 <0.01 Cu 3.522 3.337 94.767 4.614 26.449 <0.01 Zn 37.564 12.559 33.434 3.104 16.844 <0.01
Barley grain
CP 12.721 0.881 6.926 0.676 1.586 0.04 NE(1) 1.900 0.065 3.421 -0.668 0.374 0.42 P 0.392 0.056 14.405 -0.189 1.276 0.17 VN Ca 0.128 0.059 46.034 1.505 3.147 <0.01 Fat . Sait 0.233 0.098 42.106 1.223 0.776 <0.01 As h , 131
TABLE 22 (continued)
ITEM MEAN S.D, C.V. SKEWNESS KURTOSIS P(Ho)
Brewers dried grains
CP 27.565 2.178 7.900 -0.069 0.752 >0.15 NE(1) .••.•. P ...... Ca ...... Fat ...... Salt 0.242 0.205 84.953 1.739 3.726 <0.01 Ash 4.677 0.735 15.708 0.860 -0.358 0.10
Corn grain
CP 10.256 0.578 5.636 0.762 1.400 <0.01 NE(1) 1.933 0.048 2.483 -1.169 1.615 <0.01 P 0.316 0.070 22.219 0.455 3.092 <0.01 K . •.. Ca 0.091 0.047 51.156 2.256 10.128 <0.01 Fat , •... Salt 0.178 0.174 97.843 2.124 3.669 <0.01 As h
Corn gluten feed
CP 23.184 1.459 6.293 0.215 -0.048 0.04 NE(1) 1.911 0.093 4.867 -0.120 0.655 0.73 P K •••. . . Ca • • • •, . Fat 3.400 0.713 20 . 978 -0.114 -0.205 0.99 Salt 0.467 0.106 22.743 0.962 0.220 0.15 As h 4.979 0.564 11.326 1.290 3 .509 0.03 132
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Corn gluten meal
CP 69.791 4.488 6.431 -0.619 -0.091 0.08 NE(1) p K Ca Fat Salt Ü.236 0.179 75.801 2.043 4.660 <0.01 Ash 1.814 0.224 12.322 -0.031 -0.688 0.76
Distillers dried grains
CP 29 .928 2.640 8.822 0.618 0.957 <0.01 NE(1) 2.052 0.122 5.933 2.017 4.302 0.03 P ...... K ...... Ca •.«... Fat 6.395 1.927 30.135 -0.614 . 0.77 Salt 0.953 0.526 55.218 0.594 -0.835 0.06 Ash 5.486 0.781 14.239 -0.793 -0.025 0.09
Feathe r meal
CP 91.697 1.911 2.085 0.188 1.295 >0.15 NE(i) 1 .547 0.015 0.957 -1 .620 7.035 0.02 P 0.271 0.167 61.461 5.365 36.535 <0.01 K ...... Ca 0.455 0.144 31.582 -0.350 -0.006 <0.01 Fat 0.057 0.003 5.413 -1.732 « <0.01 Salt 0.363 0.162 44.769 1.009 0.238 0.03 Ash 133
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Meat and bone meal
CP 54.038 3.604 5.670 0.995 4.663 0.05 NE(1) 1.664 0.099 5.934 0.763 0.917 <0.01 P 5.550 0.699 12.597 -0.654 1.035 >0.15 VJv Ca 11.998 1 .’3 8 7 11.561 -0.887 2.521 0.06 Fat 10.160 1 .461 14.380 0.616 0.600 0.08 Salt 1.881 1.141 60.663 2.047 5.240 <0.01
Oat grain
CP 13.572 0.985 7.261 0.756 1.075 0.08 NE(1) 1.750 0.130 7.429 -0.632 -0.891 0.04 P 0.440 0.072 16.285 -0.329 0.259 0.56 K ..... Ca 0.200 0.075 37 .583 1.219 1.069 <0,01 Fat •..... Salt 0.149 0.035 23.286 0.005 -0.484 0.74 As h • • • • • •
Rapeseed meal (Canola )
CP 42.817 1.590 3.713 0.657 5.584 <0.01 NE(1) 1.600 0.071 4.438 -0.510 3.187 >0.15 P K 1 [453 o !231 15)888 0)477 1)559 <0)01 Ca •..... Fat 3.413 1 .295 37.951 0.371 0.267 0.09 Salt 0.240 0.124 51 .701 1.285 1 .244 <0.01 As h 9 ,,. 134
TABLE 22 (continued)
item MEAN S .D. C.V. SKEWNESS KURTOSIS P(Ho)
Soybean meal
CP 55.617 1.386 2.493 0.176 0.092 >0.15 NE(1) 2.034 0.072 3.540 -1.746 3.307 <0.01 P .....• K .....• Ca •....• Fat .., Salt 0.244 0.120 49.111 -0.082 -0.024 0.63 Ash 7.244 0.538 7.442 1.809 4.064 <0.01
Sunflower meal
CP 45.222 2.126 4.701 0.177 0.003 0.15 NE(1) 1.971 0.096 4.863 -0.549 2.399 0.02 P .•...• K 1 .604 0.306 19.104 -0.121 -0.297 >0.15 Ca .«. Fat 1.657 0.858 51 .788 0.317 -0.553 0.06 Salt 0.337 0.112 3 3.228 0.822 0.009 <0.01 Ash • • '• • •
Wheat grain
CP 17.107 1 .103 6.449 0.839 1.026 <0.01 NE(1) 2.050 0.039 1.902 -1.349 2.363 <0.01 P 0.454 0.060 13.192 0.631 2.411 <0.01 K ...«• Ca 0.120 0.046 38.511 1.547 4.990 <0.01 Fat .. Salt 0.192 0.036 18.483 -0.580 0.477 0.43 As h 135
TABLE 22 (continued)
ITEM MEAN S.D. C.V. SKEWNESS KURTOSIS P(Ho)
Wheat shorts
CP 19.893 1.100 5.531 2.633 15.087 <0.01 NE(1) 1 .900 0.070 3.684 -0.779 2.143 <0.01 P ...... K 1.118 0.244 21.842 0.868 2.554 <0.01 Ca ....•. Fat •..,•. Salt 0.281 0.148 52.839 1 . 700 3.057 <0.01 Ash
^Abbreviations are as defined in the text
Standard deviation.
'Coefficient of variation.
Probability for the null hypothesis to be true, i.e., that
the nutrient follows a normal distribution. 136
Variations in composition are much larger for forages
than for grains and by-products. C.V. of crude protein in the latter group ranges between 2 and 8%, which is much lower than the 17 to 4 0% range for forages. Similarly,
C.V. of NE(1) ranges between 1 and 6% for grains and by products, compared to a range of II to 24% for forages.
These differences are not as dramatic when the other nut rients are considered. Actually, mineral composition of by-products shows substantial levels of variation which was expected with this kind of feed.
The use of C.V. for expressing pure variation can be misleading since it expresses variation as a proportion of the mean. This is why feeds with a low level of a given nutrient show large C.V. values. However, the introduction of such a feed into a mix would not change much the varia tion in the mix, since it contributes very little to the final level of the given nutrient. For example, C.V. of crude protein in corn gluten feed is about 3 times as large as the same statistic for feather meal (6.3 vs 2.1%), even though its standard deviation is smaller (1.46 vs 1.91%).
Variations in salt composition are extremely large across all grains and by-products. A dramatic example is f"und in meat and bone meal, with a mean salt level of
1.88% and a standard deviation of 1.14 (C.V.=60.7%). To put things in perspective, the desired salt level in a com 137 mercial dairy grain mix is generally 1% . Substantial lev els of meat an bone meal or distillers dried grains in such a mix might raise the variation in salt to an unacceptable level even though expected composition would still be 1%.
Table 22 reports statistics on skewness and kurtosis.
In general, skewness is significantly p'ositive, indicating that frequencies of high nutrient levels are larger than expected from the normal distribution. This implies that the mean is larger than the median composition. Therefore, the probability of a certain feed being lower than the mean value of a given nutrient is larger than 50%. Level of kurtosis is also generally positive, indicating distribu tions of nutrients with heavy tails. The hypothesis that nutrients follow normal distributions is rejected in most cases . This is troublesome, since CCP procedures used in the next section assume normal distributions of nutrients .
However, the central limit theorem can be invoked to justi fy the assumption of normal distributions in the mix, which is the assumption of importance. Also, it appears that in most cases, deviation from normality is a consequence of a very few data which can be considered as outliers .
Estimated variance-covariance matrices of nutrients within feeds tuffs are reported in Appendix table 42. These will be used for post-optimization procedures in the next section. This information is presented in the form of cor relation coefficients in Appendix table 43. In general. 138
correlations between NE(1) and CP are fairly low. As we
shall see later, this will affect performance of the MI al
gorithm .
3.4.2 SyCyCy2" problem
Least-cost solutions under deterministic nutrients con
straints as found in conventional LP, are reported in table
23. In all cases, constraints on CP, P and salt were found
to be limiting. Fat was a limiting constraint 4 times, while Ca was limiting 5 times out of 10. An upper bound on
Ca should have been included in the model. This is because
in some years, Ca was not limiting and limestone was used
as a filler in the mix, resulting in high expected levels
of calcium. Crude fiber never was a limiting constraint.
Of potentially 17 ingredients, 9 were used at least once
during the 10 year period, with an average of 7 ingredients
per solution. The same solution vector was found in five years. A total of 0.68 second of CPU was used to find so
lutions for an average of 0.068 second per solution. This model does not consider the stochastic aspect of feed com
position. Therefore, the probability that the CP level in
the mix is lower than the desired level (i.e., the failure rate) is 50%.
Results for the approximate Rahman-Bender algorithm with
X fixed (REF) are given in table 24 . Of the potential 17 ingredients, 10 were used in at least one solution, with an 1 39
TABLE 2 3
Solution vectors and related statistics for the SCCP pro blem solved under a conventional deterministic LP model.
YEAR
I np.. 1 2 3 6 5 6 7 6 9 10 Mean
FAT 0 0 0 0 0 0 0 0 0 0
BRY 0 0 0 0 0 0 0 0 0 0
BDC 26 0 0 0 0 0 0 0 0 0
CCP 0 0 0 0 0 0 0 0 0 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 0
DCL U 1 16 1 138 138 161 25 137 138 138 136
DDG 0 0 0 0 0 0 0 0 0 0
FTH 286 287 251 251 287 0 260 251 251 251
LHE 365 336 1 1 7 1 1 7 336 289 360 117 117 1 1 7
MAE 0 0 0 0 0 690 19 0 0 0
MOL 60 60 60 60 60 60 60 60 60 60
OAT 0 36 290 290 36 0 0 290 290 290
SLT 126 126 125 125 126 117 123 125 125 125
SBM 0 0 0 0 0 0 0 0 Û 0
WHT 0 , 0 0 0 0 0 0 0 0 0
MPX 60 60 60 60 60 60 60 60 60 60
CP 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25 .0 25 .0
Fat 1 .0 1.0 2 . 1 2 . I 1 .0 6.7 1.0 2. 1 2 . 1 2. 1 1 . 9
Fiber 0 . 76 0 . 77 3.5 3.5 0. 77 I . 1 0.63 3.5 3.5 3.5 2 . 1
Ca 16 . 6 16.0 6 . 6 6.6 16 .0 15.5 15.0 6 . 6 6.6 6 .6 10 .6
P 3. 1 3. 1 3. 1 3. 1 3. 1 3.1 3. 1 3. 1 3 . 1 3. 1 3.1
Salt 12.7 12. 7 12.7 12. 7 12.7 12.7 12.7 12.7 12.7 12.7 12 . 7
CPU time (seconds) 0.068 140
average of 7 ingredients per solution. Solutions for year
3, 4, 9 and 10, and for years 2, 5, and 8 were identical.
The expected CP level averaged 26.05%, i.e., 1.05% points higher than the 25% desired level. As pointed out earlier,
the Rahman-Bender estimate of o is biased upward. This estimate averaged 0.632 (range of 0.551 to 0.728) while the
true estimate computed from the solutions averaged 0.531
( 0 . 524-0 . 537 ). Therefore, the ratio o ^ /a averaged 1.14
(1.03-1.37). This figure indicates that on the average, the Rahman-Bender estimate is 14% larger than what it should be. As a result, the achieved rate of success aver aged 97.2% (95.5-98.8) instead of the desired 95%. Or ex pressed differently, the failure rate for crude protein was at 2.8% instead of the desired 5%, i.e., that the failure rate is l.B times lower than desired. The economic penalty that must be paid for this increased certainty will be dis cussed later. Amounts of computer resources remained very low, essentially at the same levels as the deterministic model (0.061 second of CPU per solution).
Least-cost solutions from the iterated Rahman-Bender al gorithm (IRB) are given in table 25 . Again, 10 out of 17 possible ingredients were used at least once. The average expected CP level, 25.85% (25.81-25.89), was a reduction from the previous algorithm (RBF). Under IRB, a is still overestimated by an average of 19% (2-37). However, the 14 1
TABLE 24
Solution vectors and related statistics for the SCCP pro blem (premix) solved with the RBF algorithm.
YEAR
Ing. " I 2 3 4 S 6 7 8 9 10 Mean
FAT 0 0 0 0 0 0 1 0 0 0
BRY 0 0 0 0 0 0 0 0 0 0
BDC 20 0 0 0 0 0 0 0 0 0
CCF 0 0 0 0 0 0 0 0 0 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 0
DCL 141 14 1 1 38 1 38 141 138 142 141 1 38 138
DOC 0 0 0 0 0 0 0 0 0 0
FTH 297 299 267 267 299 294 302 299 267 267
LME 338 329 116 116 329 350 35 1 329 1 16 116
MAH 0 0 0 0 0 15 0 0 0 0
MOL 40 40 40 40 40 40 40 40 40 40
OAT 0 27 274 274 27 0 0 27 274 274
SLT 124 124 125 125 124 123 124 124 125 125
SBM 0 0 0 0 0 0 0 0 0 0
WfiT 0 0 0 0 0 0 0 0 0 0
MPX 40 40 40 40 40 40 40 40 40 40
CP 25.96 25.94 26 . 20 26 .20 25.94 25.96 25.91 25.94 26.20 26.20 26.20
0 0.529 0.532 0.532 0.532 0.532 0.524 0.537 0.532 0.532 0.532 0.531 ^ b 0 .582 0 .563 0.728 0.728 0.568 0.579 0.551 0 . 568 0.728 0.728 0.632
a ^ /o 1 . 10 1 .07 1 . 37 1 . 37 1.07 1 .10 1.03 1.07 1 . 37 1 . 37 I . 14
0.9652 0.9610 0.9880 0.9880 0.9610 0.9658 0.9545 0.9610 0.9880 0.9880 0.9720
CPU time (seconds) 0 .06 1
Abbreviations are as defined in the text.
Rahman-Bender estimate of o, the true standard deviation for protein (CP).
^ Probability of success 142
adjustment on \ produces solutions with probability of suc
cess equal to the desired probability. Clearly, the ad
justment on X was time dependent and ranged from -1.198 to
-1.614. In the previous algorithm, X was fixed at -1.65.
The improvement in probability level was accompanied by
an increase in computer resources needed, for an average of
0.275 seconds of CPU per solution. The number of itera
tions needed in the process averaged 5.8 (2 to 7), which
implies that an average of 11.6 (4 to 14) LP problems had
to bp solved in the iteration process. Due to their rela
tively small sizes, these LP problems were solved rather quickly. In future applications, the range of search for X could possibly be reduced (-0.70 to -1.65) resulting in im proved performance from this algorithm.
Results from the iterated Taylor series expansion algor ithm (ITS) are shown in table 26 . Again, 10 ingredients out of the possible 17 were used at least once. Level of
CP in the final solution was quite stable, averaging 25.87%
(25.84-25.92). The standard deviation of CP in the solu tion averaged 0.529% (0.512-0.559). Of course the success rate for CP was 95% in all cases since this is an explicit constraint in the model. The number of iterations to reach convergence averaged 12.6 (2-31) which is only slightly larger than what was observed with the IRB algorithm. How ever, the size of the LP problem which had to be solved at 143
TABLE 25
Solution vectors and related statistics for the SCCP pro blem (premix) solved with the IRB algorithm.
YEAR
Ing. « I 2 3 4 5 6 7 8 9 10 Mean
^AT 0 0 0 0 0 0 1 0 0 0
BRY 0 0 0 0 0 0 0 0 0 0
BDC 21 0 0 0 0 0 0 0 0 0
CCF 0 0 0 0 0 0 0 0 0 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 0
DCL 141 141 1 38 1 38 141 1 38 142 142 138 1 38
DDC 0 0 0 0 0 0 0 0 0 0
FTH 296 298 262 262 298 293 302 298 262 262
LME 338 329 I 16 1 16 329 351 351 329 116 116
MAB 0 0 0 0 0 15 0 0 0 0
MOL 40 40 40 40 40 40 40 40 40 40
OAT 0 27 279 279 27 0 0 27 279 279
SLT 124 124 125 125 1 24 123 124 124 125 125
SBM 0 0 0 0 0 0 0 0 0 0
WHT 0 0 0 0 0 0 0 0 0 0
MPX 40 40 40 40 40 40 40 40 40 40
CP 25 . 87 25.87 25.87 25.87 25.87 25.87 25 . 87 25.87 25.07 25.87 25.87
C 0.527 0.530 0.527 0 .527 0 .530 0 .522 0.537 0.530 0.527 0.527 0 .528
r / 0.581 0.567 0 . 724 0.724 0.567 0.578 0.550 0.567 0 .724 0 . 724 0.631
1 . 10 1 .07 1 . 37 1 . 37 1.07 1.11 1 .02 1 .07 1 . 37 I . 37 1.19
Pr 0 .9504 0.9503 0.9503 0.9503 0.9 50 3 0.9512 0.9509 0.9503 0.9503 0.9503 0.9504
I t e r ^ 2 7 7 7 7 2 5 7 7 7 5 . 8
if 1.49 7 1.541 1 . 198 1 , 198 1.541 1 .500 1.614 1.541 1.198 1 . 198 1 .403
CPU^ 0 .275
Abbreviations are as defined in the text.
^ Rahman-Bender estimate of c, the true standard deviation for protein (CP).
^ Probability of success.
^ Number of Iterations .
^ Discounting factor as described in section 3.3.3. f CPL’ time in seconds. 144
each iteration was much larger. Therefore, the average,
number of seconds of CPU needed to reach the final solu
tion, 2.02, was over 7 times larger than for the IRB algor
ithm.
Finally, tahle 27 reports results obtained from MINOS.
As with the last two algorithms (IRB, ITS), the expected CP
level in the final mix was quite stable at an average of
25.87% (25.85-23.92). The exact 95% probability of success was implied by the model. On the average, 3.4 major itera
tions and 18.3 total iterations were needed for conver
gence. Both types of iterations are described by Murtagh and Saunders (1980). The coded algorithm appears to be very efficient coraputerwise, as only 0.22 second of CPU were needed on the average to find the solution. This fig ure does not include the time required to compile the user defined CALCFG and CALCON subroutines.
In all cases, ITS and MINOS converged to the same solu
tion point. This is not surprizing because ITS is actually a non-linear algorithm specifically designed for the prob lem. Therefore, both algorithms should theoretically con verge to the same solution.
Cost of ingredients for all algorithms are reported in table 28 . In every year, cost per ton of premix decreased in the order FRB, IRB, ITS, NLP. The NLP algorithm can be used as a standard since it finds the exact minimum-cost 145
TABLE 26
Solution vectors and related statistics for the SCCP pro blem (premix) solved with the ITS algorithm.
YEAR
Ing. ^ 1 2 3 6 5 6 7 8 9 10 Mean
FAT 0 0 0 0 0 0 0 0 0 0
BRY 0 0 0 0 0 0 0 0 0 0
BDC 9 0 0 0 0 0 0 0 0 0
CCF 0 0 0 0 0 0 0 0 0 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 0
DCL 1 39 161 1 38 138 161 109 138 138 1 38 138
DDC 0 0 0 0 0 0 0 0 0 0
FTH 287 298 262 262 298 220 29 3 266 262 262
LME 36 I 329 1 16 116 329 331 351 1 2 7 116 1 16
MAB 10 0 0 0 0 I 39 15 0 0 0
MOL 60 60 60 60 60 60 60 60 60 60
OAT 0 27 279 279 27 0 0 266 279 279
SLT 123 126 125 125 126 122 123 125 125 125
SBM 10 0 0 0 0 0 0 0 0 0
WHT 0 . 0 0 0 0 0 0 0 0 0
MP X 60 60 60 60 60 60 60 60 60 60
CP 25.86 25.88 25.87 25.88 25.92 25.86 25.87 25.87 25.87 25.87 25.87
0 b 0.512 0.530 0.527 0.527 0.530 0.559 0.552 0.526 0.527 0.527 0.529
Pr c 0.950 0 .950 0.950 0.950 0 .950 0.950 0.950 0.950 0.950 0.950 0 .950
I te r ^ 2 6 30 3 25 15 9 31 6 3 12.6
CPU ® 2 .02
Abbreviations arc as defined in the text.
^ Standard deviation of crude protein (CP).
^ Probability of success.
Number of Iterations.
^ CPU time in seconds. 146
TABLE 27
Solution vector and related statistics for the SCCP pro blem (premix) solved with the NLP algorithm (MINOS).
YEAR
Ing. * 3 4 5 6 7 8 9 10 Mean12
FAT 0 0 0 0 0 0 0 0 0 0
BRY 0 0 0 0 0 0 0 0 0 0
BDC 1 1 0 0 0 0 0 0 0 0 0
CCF 0 0 0 0 0 0 0 0 0 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 0
DCL 1 39 1 4 I 1 38 1 38 14 1 109 138 138 1 38 138
DDG 0 0 0 0 0 0 0 0 T) 0
FTH 288 29 1 262 262 298 221 293 263 262 262
LME 34 1 290 1 16 1 16 329 332 351 1 19 1 16 1 16
MAB 1 1 0 0 0 0 136 1 5 0 0 0
MOL 40 40 40 40 40 40 40 40 40 40
OAT 0 75 279 279 27 0 0 276 279 279
SLT 123 124 125 125 124 127 123 125 125 125
SBM 6 0 0 0 0 0 0 0 0 0
WHT 0 0 0 0 0 0 0 0 0 0
MPX 40 40 40 40 40 40 40 40 40 40
CP 25.85 25.86 25.87 25.86 25.87 25.92 25.86 25.87 25.87 25.87 25.87
0 ^ 0.5 14 0.522 0.527 0.527 0.5 30 0.555 0.522 0 .526 0.527 0.527 0.528
Pr ^ 0.950 0.950 0.950 0.950 0.950 0.950 0,950 0.950 0.950 0.950 0.950
MIT ^ 4 4 3 3 3 5 3 3 3 3 3.4
TIT ® 24 20 17 18 16 24 15 1 7 16 16 18.3
CPU ^ 0 . 26 0.22 0.2 1 0 . 20 0 .24 0 . 20 0.21 0.21 0.21 0.21 0 .22
Abbreviations arc as defined in the text.
Standard deviation of crude protein (CP).
Probability of success.
Number of major iterations.
Number of total Iterations.
CPU time in seconds. 147 solution for a 9 5% rate of success on protein level. The added cost due to approximation methods averaged $0.57,
$0.08 and $0.01, respectively, for FRB, IRB and ITS algor ithms. On a percentage basis, these represent 0.43, 0.08 and 0.0% higher costs for the three algoritms. The average cost of NLP's solutions was $99.50. The same figure was
$97.19 under the deterministic model. The difference
($2.31) is the cost associated with the change from 50 to
95% in the rate of success for crude protein.
Clearly, NLP (MINOS) is the best algorithm for solving this type of problem. Computerwise, the code is very effi cient and convergence was easily reached in every case.
However, people from the feed industry are used to custom- made LP algorithms. It seems unlikely that such people would find MINOS 'friendly' enough to be used on a commer cial basis. A file editor program which would translate simple keyword commands into a MINOS model could circumvent the problem. Meanwhile, the iterated Rahman-Bender algor ithm could be implemented easily with almost the same per formance as the NLP. TABLE 2 8
Cost of ingredients for the solution vectors of the SCCP problem (premix).
YEAR
Va r1able Algorithm 1 2 3 4 5 e 1 8 91 10 Mean
DET* 67 ., 13 58 . 34 63 .. 6 1 1 29 ..92 102 .. 86 94 ..80 106 ..31 117 .61 110..99 115.. 29 97 ..19
FRB 68 .63 59 ..47 70 . 75 135..36 105 ..22 98. 97 108 ..51 120 .48 114. 20 119..08 100 ..07
IRB 68..49 59 ., 40 70 . 16 133 .85 105 ,.06 98..77 108.. 46 120. 30 113.. 30 118 .03 99 ..58
ITS 68 ,,48 59 .,40 70 . 16 133.. 85 105..06 98.,33 108 ..35 1 20 .07 113.. 30 118 .03 99 ..50
NLP 68 ,. 48 59 . 38 70 . 16 133 .85 105 ..06 98 .. 33 108 . 35 1 20 .07 113.. 29 118..03 99 .. 50
FRB 0 ., 15 0 ..09 0 ..59 1 . 5 1 0 . 16 0 ..64 0 . 16 0 .4 1 0 ..9 1 1 .05 0 .57
IRB 0 ,,01 0 ..02 0 ..00 0 .00 0 .00 0 ..44 0 . 1 1 0 .23 0 ..0 1 0 .00 0 .08 ^1 ■ *"NLP ITS 0.. 00 0 .02 0 .,00 0 .00 0 .00 0 ..00 0..00 0 .00 0..01 0 .00 0 .01
FRB 100.22 100.15 100.84 101.13 100.15 100.65 100.15 100.34 100.80 100.89 100.43
''l^^NLP * IRB 100.01 100.03 100.00 100.00 100.00 100.45 100.10 100.19 100.01 100.00 100.08
ITS 100.00 100.03 100.00 100.00 100.00 100.00 100.00 100.00 100.01 100.00 100.00
^ Abb re V la i: 1 ons are as defined in the text.
^ C “ Cos': of ingredients for algorithm 1 ($/metrlc ton)
00 149
3.4.3 problem
Results for the Industry standards model are reported in table 29 for the supplement and table 30 for the grain mix.
Upper bounds on ingredients resulted in a greater average number of ingredients per solution for the grain mix than for the supplement. This not only was expected, but was the basic reason for comparing algorithms with two differ ent feeds. Calcium, phosphorus and salt were binding con straints in every case. Due to the discounted ingredients' levels of CP and NE(1), expected levels of these two nut rients were much higher than the desired levels. In all cases, optimal solutions were readily found, using less than 0.35 second of CPU and 20 iterations, on the average.
These solutions will be used for comparison with those of our J.C.C.P. models. As mentioned earlier, these industry standard models were chosen among many alternatives cur rently used by the feed industry. Other alternatives would likely produce slightly different results, but conclusions would not differ.
The RBB model use Rahman-Bender estimates of , i=l,...,5, and insured a 93% global rate of success by a
Bonferroni inequality, i.e., each of the 5 nutrients' rate of success was set at 99%. Results differed very slightly from the industry standard models. For the supplement (ta ble 31), the average expected levels of nutrients were a 1- 150
TABLE 29
Solution vectors and related statistics for the JCCP pro blem (supplement feed) solved with the IS model.
YEAR
log. ® 1 2 3 4 5 6 7 8 9 10 Mean
BDC 0 0 0 0 0 0 0 0 0 0
CCF 307 544 538 481 42 1 421 544 0 467 0
CGM 0 0 0 104 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 424
DDC 0 0 0 0 0 0 0 0 0 0
FTH 0 1 30 1 50 150 0 0 1 30 1 50 1 50 1 50
MAB 0 1 40 105 0 1 49 150 140 1 50 145 1 33
SBM 451 0 0 0 264 263 0 54 0 109
WHT 0 0 0 0 0 0 0 433 0 0
FAT 63 68 73 64 52 52 68 63 89 53
DCL 56 25 34 59 2 1 21 25 46 4 5 36
LHE 28 0 6 26 0 0 0 7 7 0
SLT 31 28 28 30 28 28 28 33 32 2?
NING ^ 7 7 8 6 7 7 7 9 8 8 7.6
NE 1.74 1 1 . 766 1 . 760 1.75 1 1 . 768 1 . 768 1 . 766 1 .728 1 . 780 1.721 1.755
CP 28.90 29 ,67 29.52 29 .62 29. 58 29 . 59 29 .66 29.8 1 30 . 06 26.99 29 . 54
CA 2 . 20 2 . 20 2 . 20 2.20 2 . 20 2 . 20 2 .20 2 . 20 2 .20 2 . 20 2 . 20
P 1 . 70 1 . 70 1 . 70 1 . 70 1 . 70 1 . 70 1 . 70 1 . 70 I . 70 1 . 70 1 . 70
SALT 3.33 3.33 3.33 3.33 3 . 33 3.33 3 . 33 3 . 33 3.33 3.33 ). 33
S " 79 .23 74 . 54 85 . 34 168.56 148.84 127.59 148.31 170.39 166.31 183.55 133.27
CPU 0.35 0.35 0.34 0.34 0.35 0.34 0. 34 0.35 0.35 0.34 0.345
TIT ® 19 19 20 IB 1 8 19 20 23 19 1 7 19 . 2
Abbreviations are as defined in the text.
Number of Ingredients In the solution vector.
Cost of Ingredients ($/mecrlc ton).
CPU time (seconds).
Total number of Iterations. 151
TABLE 30
Solution vectors and related statistics for the JCCP pro blem (grain mix) solved with the IS model.
YEAR
Ing. 1 2 3 6 5 6 7 8 9 10 Mean
BDC 200 200 69 200 200 200 0 0 0 0
CCF 200 200 200 200 200 200 200 69 200 0
CCK 0 0 0 0 0 0 0 0 0 0
CRN 62 6 6 569 227 223 228 603 36 3 603 753
DDC 0 0 0 250 250 250 0 0 0 0
FTH 0 0 50 5 0 0 50 50 50 50
MAB 0 60 68 0 0 a 50 50 50 68
SBM 66 5 0 0 9 0 21 0 2 1 76
WHT 600 600 0 0 0 0 0 600 0 0
FAT 29 25 2 1 38 38 37 16 6 1 6 7
DCL . 9 0 1 1 1 10 9 1 2 1 5
LHE 10 2 0 9 9 7 0 0 0 0
SLT S 7 7 6 6 6 7 7 7 7
NING ^ 10 10 9 10 10 10 9 9 9 8 9.6
CP 16,26 18.39 17.80 18.75 18.75 18.78 17.65 17.79 17.65 17.60 16. 12
CA 0 . 70 0 . 70 0. 70 0 . 70 0 . 70 0. 70 0. 70 0 . 70 0 . 70 0 . 70 0 . 70
P 0,65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65
SALT I .00 1 . 00 1 .00 1 . 00 1 .00 1 .00 1 . 00 1 .00 1 .00 1 .00 1 .00
S ' 58 .25 59 . 22 63.09 111.89 126.67 1 12.68 119.17 110.11 112.21 i 30 .96 100.6 1
CPU ^ 0.35 0. 35 0.35 0. 35 0 . 35 0.36 0. 35 0 . 36 0 . 35 0 .36 0 . 369
TIT ^ 19 19 20 18 18 19 20 23 19 1 7 19 , 2
Abbreviations are as defined in the text.
Number of inp.redlcnts in the solution vector.
Cost of ingredients (S/metric ton).
CPU time (seconds).
Total numhpr of tiens. 152
most identical to those obtained under the IS model. Prob
lems were solved with about the same amount of CPU time, in
a slightly smaller number of iterations. Cost of ingredi
ents averaged $0.45 less when compared to IS. Differences
in the solutions given by the two models were more notice
able in the case of the grain mix (table 32). Expected
levels of CP, NE and salt averaged larger values, while Ca
and P averaged lower expectations. This resulted in solu
tions that were on the average $1.84 more expensive than
those from IS. Achieved rates of success, both individuals and global, will be discussed later.
Much better performance was observed from the model where each was directly computed and overall success rate insured by a Bonferroni inequality, with equal weight on the success rate of each nutrient (BEW). In both the supplement (table 33) and the grain mix (table 34) the av erage number of ingredients used showed a marked increase.
This should contribute to lower standard errors. As a re sult, expected levels of nutrients were markedly lower then those from the IS and RBB. In the supplement, NE(1) and CP were reduced from over 1.75 Mcal/kg to 1.715 Mcal/kg and from over 29.5% to 28.5%, respectively. Decreases of lower magnitudes were also observed for calcium, phosphorus and salt. Even though the model was nonlinear and solved by nonlinear techniques (MINOS), computer resources remained 153
TABLE 31
Solution vectors and related statistics for the JCCP pro blem (supplement feed) solved with the RBB model.
YEAR
I n p . ® 1 2 3 6 5 6 7 8 9 to
BDC 0 0 0 0 0 0 0 0 0 0
CCF 0 665 665 685 318 388 665 0 522 0
CCM 0 0 0 1 1 3 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 0 366
DDC 0 0 0 0 0 0 0 0 0 0
FTH 0 1 50 150 1 50 0 0 150 1 50 1 50 1 50
MAB 0 0 0 0 0 150 0 150 123 0
SBM 695 i 38 138 0 655 287 138 37 0 256
VHT 293 0 0 0 0 0 0 693 0 0
fat 65 80 80 85 62 59 60 69 78 62
DCL 69 52 52 53 68 22 52 29 32 55
LME 27 26 26 23 26 6 26 2 5 26
SLT 27 26 26 26 26 26 26 26 25 27
NING ^ 7 8 8 8 7 8 8 9 8 8 7.9
NE 1 .727 I .763 1.763 1.776 1 .763 1 . 790 1 .763 1 .723 1 .785 1.719 1.757
CP 29. 13 29,63 29 . 68 30. 21 29. 32 30,05 29 .68 29.90 30.08 28. 82 29.60
CA 1 .969 2.025 2.025 2.018 2.002 2.3 58 2.025 2.32 5 2.317 1.973 2.122
P 1 .662 1 . 580 1 . 580 1. 582 1 . 532 1.723 1 . 580 1 . 656 1 .737 1 . 685 1.59 2
SALT 2. 869 2,910 2.910 2. 908 2. 909 3.086 2.910 3.018 3.056 2.916 2.969
S ' 78.61 76. 07 86.78 170.67 169 . 28 130.29 151.05 162.50 159.65 183.29 136.82
CPU ^ 0.33 0. 36 D. 33 0.33 0.33 0.33 0.33 0. 33 0.33 0.33 0.33
TIT ® 1 6 1 7 17 1 6 16 17 1 7 1 8 18 1 7 16.9
Abbreviations are as defined in the text.
^ Number of ingredients in the solution vector.
^ Cost of ingredients ($/oetric ton).
^ CPI' time (seconds).
^ Total number of iterations. 154
TABLE 32
Solution vectors and related statistics for the JCCP pro blem (grain mix) solved with the RBB model.
YEAR
Ins. ^ 1 2 3 6 5 6 7 8 9 10 Hear.
BDC 0 200 0 200 200 200 0 0 0 0
CCF 200 200 200 200 200 200 200 200 200 0
CCM 0 0 0 0 0 0 0 0 0 0
CRN 1 69 32 567 213 203 203 567 317 567 723
DDC 0 0 0 250 250 250 0 0 0 0
FTH 0 0 50 1 3 0 0 50 50 50 50
MAB 0 0 0 0 0 0 0 50 0 0
SBM 1 2 1 59 81 0 26 26 8 1 0 81 131
WHT 600 600 0 0 0 0 0 600 0 0
FAT 1 2 32 22 66 66 66 22 1 0 22 1 6
DCL 7 6 1 0 7 7 7 10 1 1 0 1 1
LME 9 9 8 8 8 8 8 1 8 8
SLT 7 7 8 9 9 9 8 7 8 8
NING ^ 9 10 9 10 1 0 10 9 10 9 5 9 . 6
NE 1.767 1.7 59 1 .750 1 .805 1 . 807 1 . 807 1 .750 1 . 760 1.750 1.731 1.765
CP 18.16 18.70 17.76 19.30 19.31 19.31 17.76 18.20 17.76 17.66 16.37
CA 0 .620 0.623 0.619 0.609 0.608 0. 608 0.619 0.728 0.619 0.59 7 0.625
P 0 . 588 0. 580 0.600 0.575 0.573 0.573 0.600 0.636 0.600 0.559 0.5 56
SALT 0.955 0.967 1.012 1 . 270 1.270 1 . 270 1.012 1 .016 1.012 I .016 1 . 080
S ' 58.91 59 . 99 66 .02 155.55 128.83 113.90 120.88 112.68 115.19 132.53 102.25
CPU ^ 0.33 0.33 0.36 0. 36 0.36 0.36 0.33 0.33 0 . 36 0.33 0.335
TIT ^ 1 7 18 17 18 18 18 1 7 20 1 7 1 6 17.6
Abbreviations are as defined in the text
Number of ingredients in the solution vector.
Cost of ingredients (S/metric ton).
CPU time (seconds).
Total number of iterations. 155
fairly low at an average of 0.458 seconds of CPU per solu
tion. The model was easily implemented on the system and
convergence was readily achieved even with initial solution
points far from the optimum. The number of major and total
iterations needed remained quite constant over the 10
years. Cost of ingredients averaged $127.05. This figure
represents a $8.22 (6.1%) reduction from the IS, and a
$7.77 (5.8%) reduction from the RBB. The same type of re
sults were also observed with the grain mix. Average ex
pected levels of nutrients showed reductions from those of
IS and RBB, especially in the case of CP which dropped from over 18.1% to 17.3%. Again, solutions were readily ob tained with very little resources used: less than 0.5 sec onds of CPU time per solution. Savings from the two previ ous algorithms were substantial. The average cost of ingredients, $95.40, is $5.01 (5.0%) and $6.85 (6.7%) lower than with IS and RBB, respectively.
In the previous model, rate of success for each of the 5 nutrients were preset at 99%. Additional savings can be achieved by allowing these rates to vary between nutrients, as long as their sum meets the Bonferroni inequality. In this fashion, rates of success for 'expensive' nutrients are lowered while those for 'cheap' nutrients increased, resulting in the same global probability of success but at a lower cost. This was the essence of the BUW model whose 156
TABLE 33
Solution vectors and related statistics for the JCCP pro blem (supplement feed) solved with the BEW model.
YEAR
Ing. ^ 1 2 3 6 5 6 7 8 9 10
BDC 5Ï 58 100 91 0 63 15 0 59 53
CCF 185 680 662 360 356 395 681 1 16 230 73
CCM 0 0 0 56 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 193 31
DDC 38 0 66 1 30 107 66 100 30 95 86
FTH 0 59 165 136 0 0 105 1 50 1 50 1 50
MAB 57 1 50 60 0 66 1 50 129 112 92 56
SBM 335 9 1 0 0 299 198 0 0 0 87
WHT 160 0 0 0 0 0 0 627 0 0
FAT 35 56 69 68 60 60 55 38 51 68
DCL 3Ù 1 7 37 50 32 16 23 30 33 6 1
LME 1 5 1 12 21 16 3 6 6 8 1 3
SLT 26 26 26 25 26 23 23 26 26 25
SINC ^ I 1 10 10 10 9 10 10 10 1 1 1 2 10.6
NE 1.69 7 1.761 17.30 1 .720 1 .723 1 .728 1.761 1 .689 I . 699 1 .686 1.715
CP 28 . 26 28 . 86 28 .62 28 .58 28.57 28. 72 28.85 28.63 28 . 29 28.03 26 . 52
CA 1.965 2.211 1.96 5 ] .873 1.970 2 .208 2.152 2.09 5 2.061 1 .937 2.060
P I . 666 1 .612 1.52 5 1.681 1.505 1.597 1.591 1.522 1 . 508 1.651 1 .526
SALT 2.802 2 .902 2.819 2.851 2.857 2.910 2 .908 2.86 1 2 .858 2.826 2.857
S ' 75.07 72 - 52 82.69 258.59 161.96 122.92 166.27 151.55 168.07 172.86 127.05
CPU ^ 0.66 0 . 68 0.52 0. 39 0.63 0.66 0.50 0.61 0.51 0.66 0.658
MIT ^ 6 6 6 5 5 6 6 5 6 5 5.6
TIT ^ 60 63 50 31 38 37 69 36 68 36 60. 6
Abbreviations are as defined In the text.
Number of Ingredients In the solution vector.
Cost of Ingredients (S/metric ton).
CPU time (seconds).
Number of major Iterations.
Total number of iterations. 157
TABLE 34
Solution vectors and related statistics for the JCCP pro- lem (grain mi x) solved with the BEW mode 1
YEAR
Ing. " 1 2 3 6 5 6 7 8 9 10
BDC 165 173 168 200 195 200 102 2 76 69
CCF 200 200 200 200 200 200 200 172 200 89
CCK 0 0 0 0 0 0 0 0 0 0
CRN 85 1 1 6 639 312 313 318 366 297 671 613
DDC 65 1 5 73 195 199 188 101 0 100 99
FTH 0 0 16 0 0 0 6 50 22 50
MAB 5 16 25 0 0 2 22 5 28 23
SBM 0 7 0 0 0 0 16 0 0 0
WHT 600 600 0 0 0 0 117 600 33 0
FAT 9 9 1 5 19 19 19 5 0 6 7
DCL 2 0 0 5 5 6 0 5 0 6
LME 8 6 6 6 6 8 5 7 6 6
SLT 6 6 6 7 7 7 6 6 6 7
NING ^ 11 1 1 10 9 9 10 12 10 1 1 1 1 10.6
NE 1 .700 1 ,700 1 .707 1.718 1.718 1.717 1 . 703 1 .695 1 . 708 1.710 1 . 708
CP 17.39 17.37 17.21 17,61 17.61 1 7 . 58 17. 10 17.13 17 .09 17.03 17.31
CA 0 .552 0.567 0.567 0 .567 0,567 0 . 567 0.578 0 . 569 0 . 596 0 .576 0 . 566
P 0 . 526 0.528 0.538 0.525 0 . 525 0 .525 0.538 0 .525 0.561 0.531 0 . 530
SALT 0.905 0.879 0.966 1.050 1 .050 1 .062 0.956 0.877 0.966 0.975 0.966
S ' 55.09 56.67 60.00 103.59 121 .06 107.95 1 16.06 106.18 106.05 125.56 95.60
CPU ^ 0.68 0 . 66 0 . 59 0 . 39 0.61 0 .66 0.52 0.66 0 .68 0 .66 0.669
MIT ^ 5 5 6 5 5 6 5 5 5 5 5.2
TIT ^ 66 60 62 26 32 37 51 39 66 62 6 1.9
Abbreviations arc as defined in the text.
Number of ingredients In the solution vector.
Cost of ingredients (S/metric ton).
CPU time (seconds).
Number of major iterations.
Total number of iterations. 158
results are reported in table 35 for the supplement. The
average number of ingredients per solution did not change
from the previous algorithm. Actually, solutions were
quite similar in terms of ingredients used, while their
levels were slightly adjusted. Average expected levels de
creased for NE and CP but increased for CA, P and salt.
The average cost of ingredient, $126.24, is a reduction of
$0.81 (0.6%) from BEW, $9.03 (6.7%) from IS and $8.58
(6.4%) from RBB. However, the amount of computer resources needed to solve the problem showed a marked increase from
the previous three algorithms. An average of 0.75 second of CPU time was needed per solution. This increase does not seem large enough to preclude its utilization since the absolute level of resources needed is still fairly low.
Problems in convergence were observed, however. Upper bounds of 0.999 had to be set for each individual rate of success in order for the algorithm to converge. Also, the initial solution vector had to be set relatively close to the optimal point for convergence to occur.
Results were similar with the grain mix (table 36).
Changes in the BEW solution vectors were generally small.
However, the average cost of ingredients, $94.53, is $0.87
(0.9%), $5.88 (5.9%) and $7.72 (7.6%) lower than the same figures from BEW, IS and RBB respectively. Again, a no ticeable increase in CPU time was noted, but without sig- 159
TABLE 35
Solution vectors and related statistics for the JCCP pro blem (supplement feed) solved with the BUW model.
YEAR
Ing. ^ ] 2 3 4 5 6 7 8 9 10
BDC 47 24 85 77 0 7 0 0 51 50
CCF 198 523 465 371 357 429 502 120 256 73
CCM 0 0 0 59 0 0 0 0 0 0
CRN 0 0 0 0 0 0 0 0 172 302
DDC 37 0 37 1 34 101 59 9 1 29 101 92
FTH 0 74 148 1 30 0 0 1 10 150 150 150
MAB 48 150 55 0 58 150 123 109 85 49
SBM 342 6 6 0 0 306 208 0 0 0 89
WHT 150 0 0 0 0 0 0 427 0 0
FAT 35 52 68 66 37 37 53 38 51 46
DCL 36 18 38 51 34 16 25 30 34 42
LHE 18 4 1 4 22 16 5 7 ? 10 15
SLT 25 24 25 25 25 24 24 25 25 25
NING ^ 1 1 10 1 0 10 9 10 9 10 1 1 12 10.3
NE 1 .689 1.7 29 1 .720 1.712 1 . 706 1.717 1.725 1.683 1 . 692 1.677 1 . 708
CP 28.16 28 . 86 28. 46 26.36 28 .52 28.81 28.81 28.28 28.26 27 .97 28.45
CA 1 .966 2 . 306 2 .020 1.912 2.009 2 . 306 2 . 229 2.127 2.070 1 .959 2 .090
P 1 .468 1 .625 1.541 1 . 509 1 . 520 1 . 605 1 . 602 1.526 1.514 1 .455 1 . 537
SALT 2.850 3.003 2 .884 2.930 2.925 3.008 2.996 2.897 2.923 2.878 2.935
S C 74, 71 72.02 82 . 16 157.12 140.77 122.10 143. 16 150.89 147.34 172.14 126.24
CPU d 0.72 0.65 0.77 0.9 1 0.69 0 .64 0.69 0.75 0.73 1 . 10 0.765
MIT ® 6 7 9 9 8 8 9 8 7 9 8
TIT ^ 72 6 1 75 94 68 60 64 74 73 121 76.2
Abbreviations are as defined in the text.
Number of Ingredients In the solution vector.
Cost of ingredients (?/nctrlc ton).
CPU time (seconds).
Number of major iterations.
Total number of Iterations. 160
nifleant practical consequences. However, the same conver
gence problems as those described for the supplement were
encountered.
Algorithms discussed so far insure a 95% rate of success
via a Bonferroni inequality. However, the achieved prob
ability level might be much larger than 0=95= There is an
economic cost that must be paid for an increase in the cer
tainty level. Therefore, we would like the achieved rate
of success to be as close as possible to 95%. Table 37 re ports individual rates of success (per nutrient) as well as the global ones, estimated by Monte Carlo techniques from each solution vector. For both mixes, global rates of suc cess from IS and RBB algorithms were much larger than the desired 95%. In the IS case, this is inherent to the meth od itself which sets RHS at the exact desired levels, while by law, a success is defined to be 80% of the desired (tag) level. In the RBB case, the high probability is directly attributable to the over-estimation of each by Rahman-
Bender estimates. Therefore, individual rates of success are much larger than the prescribed 99%, which causes the global rates of success to exceed 95%. BEW yielded global rates much closer to 95%. In the supplement case, the achieved rate was significantly larger than 0.95, even though the difference was only 0.49% point. In the grain mix, the difference between the achieved and desired rates 16 1
TABLE 36
Solution vectors and related statistics for the JCCP pro blem (grain mix) solved with the BUVJ model.
YEAR
Ing. ^ 1 2 3 6 5 6 7 8 9 10 Mean
BDC 171 187 186 200 1 79 200 109 9 76 66
CCF 200 200 200 200 200 200 200 165 200 89
CGM 0 0 0 0 0 0 0 0 0 0
CRN 83 1 1 2 6 20 328 302 299 376 281 696 612
DDC 63 7 78 179 230 210 1 16 0 1 10 108
FTH 0 0 10 0 0 0 1 67 20 50
MAB 1 18 23 0 0 0 25 0 32 2 1
SBM 0 0 0 0 0 0 13 0 0 0
WHT 600 600 0 0 0 0 93 600 0 0
FAT 8 9 16 17 13 15 3 0 3 3
DCL 6 0 1 6 5 5 0 6 0 5
LME 9 6 5 8 9 9 5 9 6 5
SLT 7 6 7 8 8 B 7 7 7 7
NING ^ 1 1 1 1 1 1 9 9 9 13 10 1 1 1 1 10.5
NE 1 .689 1 .689 1 .698 1 . 706 1.701 1 .700 1 .692 1 . 689 1 .696 1.69 6 1.69 5
CP 17.29 17.29 17. 10 17.33 17.95 17.92 17.09 16.97 17.12 17.05 17.31
CA 0.576 0.606 0.6 18 0.570 0.575 0.576 0.622 0.576 0.637 0. 596 0.595
P 0.535 0.56] 0.567 0 .566 0.560 0.536 0 .568 0.535 0.559 0.563 0.563
SALT 0.952 0.922 1.010 1.123 1.210 1.183 1.035 0.913 1 .038 1 .066 1.06 3
s ' 56.67 56.07 59 .58 103.28 119.76 107.02 113.22 103.53 106.98 126.21 96 .53
CPU ^ 0,63 0.67 0.7 5 0 .60 0 .62 0 .56 0.80 0.68 0 .76 0.9 3 0 .693
MIT ® 7 8 8 8 8 B 8 8 8 1 1 8 . 2
TIT ^ 58 60 79 56 59 50 81 6 6 76 105 68.8
Abbreviations are as defined In the text.
Number of ingredients in the solution vector.
Cost of ingredients (S/metric ton).
CPU time (seconds).
Number of major iterations.
Total number of Iterations. 162
for BEW was only 0.20% point and nonsignificant. The BUW
gave even better results. From table 37, one can see that
individual rates of success were lowered for NE(1) and CP
but raised for Ca, P and salt, when compared to BEW. Since
the former two nutrients are marginally more expensive than
the latter three, this adjustment resulted in a 0.6 to 0.9%
lower cost for BUW. Global rates of success for BUW were
very close and not significantly different from the desired
95%. In all cases, the marginal cost of certainty, at the
same individual rate of success, was larger for NE(1) than
for CP. This is why the final individual rates of success
from the BUW were lower for NE(1) than for CP, while their
marginal cost are equal.
An algorithm based on the integral of a multivariate
probability distribution function would lower the cost of
ingredients a step further,
1. by setting the global rate of success at the exact
desired rate, and
2. by utilizing the information from the covariances,
i.e., with everything else constant, the higher the
correlation between two nutrients within the same
ingredient, the greater the value of the ingredient.
Currently, there is no practical way of computing the lnte= gral of a multivariate normal probability distribution function of more than 2 variâtes. This is why Bonferroni 163
TABLE 37
Average achieved rates of success found by Monte Carlo experiments for the solution vectors of the JCCP problem (supplement feed and grain mix).
NUTRIENT
Model* NE CP CA P SALT GLOBAL Pr(H 0
— - Supplement --
IS 0.9976 0.9992 0.9948 0.9998 1.000 0.9915 <0.0001
RBB 0.9995 0.9998 0.9997 0.9997 1 .000 0.9987 <0.0001
BEW^ 0.9900 0.9900 0 .9900 0.9900 0.9900 0.9549 0.0302
BUW 0.9739 0.9854 0.9977 0.9938 0.9994 0.9516 0.2781
— - Grain mi X ----
IS 0.9998 0.9999 0.9995 1 .000 0.9786 0.9777 <0.0001
RBB 0.9998 1.000 1 .000 0.9999 0.9999 0 .9996 <0.0001
BEW 0.9900 0.9900 0.9900 0.9900 0.9900 0.9520 0.2505
BUW 0.9696 0.9857 0.9995 0.9964 0.9994 0.9515 0.2919
^ Abbreviations are as defined in the text .
^ Probability that the global rate of success equals 0 .95 .
Individual rates of success for BEW were implicitely set at
0.99 in the m o d e l . 164
inequalities had to be used in the supplement and grain mix
cases. However, the BUW algorithm generated global rates
of success that were, for all practical considerations,
equal to the desired rates. Therefore, the only way that
an algorithm based on a multivariate probability integral
could lower substantially the cost is if correlations be
tween nutrients are markedly positive. Judging by the mag
nitude of the correlations reported in Appendix table 43,
this seems unlikely. This conclusion could be different at
a global rate of success lower than 95%- This, as well as
the performance of a multivariate probability integral al
gorithm, were investigated on a total mixed ration (TMR)
with NE(1) and CP considered stochastic.
Table 38 reports results on the TMR for the BEW algor
ithm as well as for a fully deterministic model. Under
such a model, only four ingredients are required in the
least-cost solution. When the global rate of success is
forced to be 50% or higher with a BEW model, all ingredi
ents, including dry fat, enter the solution. NE(1), CP and
cost of ingredients increase as expected with global rate
of success. Level of computer resources remained fairly
low and convergence was achieved without any problem in all
Remember that BEW was putting an equal weight on the in dividual rates of success, i.e., at a 50% global rate of 165
TABLE 38
Solution vectors and related statistics for the JCCP pro blem (TMR) solved with the deterministic and BEVJ models.
DET BEU: Global rate of success
Ing. ^ 50 60 70 80 90
HAY 0 1.41 1.43 1.46 1. 51 1 .58
HLG 0 1.31 1.96 2.44 2.85 3.31
CSL 14.22 8.55 7.61 6 . 97 6.48 6.05
GEC 1.74 4.30 4.31 4.04 3.52 2. 50
SBM 2.67 2.47 2.47 2.53 2.63 2.84
DFT 0 0.64 0.92 1. 28 1.74 2.48
MIN 0.37 0.31 0.30 0.28 0.26 0.24
NE 29.10 30.07 30.23 30.44 30.70 31.16
CP 2.755 2.856 2.880 2.909 2.946 3.007
CPU ^ 0.23 0.31 0.51 0.39 0.39 0.41
MIT ^ - 6 6 8 8 6
TIT ^ 7 25 54 29 31 42
$ ® 1 .90 2. 27 2.37 2.49 2. 63 2.85
^ Abbreviations are as defined in the text.
^ CPU time (seconds).
^ Number cf major iterations.
^ Total number of iterations.
® Cost of ingredients ($/metric ton). 166
success, both NE(1) and CP's rates of success were predet
ermined at 75%. In contrast, the BUW algorithm lets indi
vidual rates fluctuate, for additional savings (table 39).
Again, NE(1), CP and cost of ingredients increase with
probability level. Compared to BEW, there is a marked re
duction in cost at all rates of success, the magnitude of
the difference being larger at lower probability levels.
Like with the BEW, an increase in the rate of success is
accompanied by an increase in the number of ingredients
in the solution vector. Convergence was achieved only when
the starting vector was fairly close to the optimum. To
alleviate this problem, solution vectors from BEW were used
as initial vectors for BUW. Therefore, reported values for
CPU time and number of iterations (Table 39) are substan
tially biased downward.
Table 40 reports results obtained with the bivariate
normal probability integral model (MI). Again, the number
of ingredients in the solution increases as the rate of
success is raised. As expected, the cost of ingredients was the lowest under this model. When compared to the re
sults from the best algorithm so far (BUW), MI solutions were $0.02 (1.0%), $0.01 (0.48%), $0,006 (0.29%), $0,005
(0.20%) and $0,003 (0.11%) cheaper at global rates of suc cess of 50, 60, 70, 80 and 90% respectively. Clearly, the benefits are a function of the the desired rbtes of sue- 167
TABLE 39
Solution vectors and related statistics for the JCCP pro blem (TMR) solved with the BUW model.
Global rate of success
Ing. ^ 50 60 70 80 90
HAY 0 0.65 1.47 1 . 53 1 . 60
HLG 0 0 0.96 2.10 2.91
CSL 13.45 11.79 9 . 20 7.48 6.48
GEC 1. 85 2.81 3.47 3.54 2.79
SBM 3.33 3.40 3.08 2.99 3 .07
DFT 0 0 0.52 1 .08 1.91
MIN 0.37 0.36 0.31 0.28 0.25
NE 29 . 36 29. 60 29.93 30.25 30.76
CP 3.065 3.164 3.120 3.117 3 .140
CPU ^ 0.21 0. 25 0.30 0.33 0.32
MIT ^ 5 7 8 9 9
TIT ^ 11 18 35 42 39
$ " 1 . 996 2.075 2.264 2.453 2 .702
^ Abbreviations are as defined in the text.
^ CPU time (seconds).
^ Number of major iterations.
^ Total number of iterations.
^ Cost of ingredients ($/metric ton). 168
cess. When a relatively high level of confidence is de
sired (90%), there is very little to gain in switching from
a BUW model to a MI model. At low levels of confidence
(50%), possible savings may warrant an MI algorithm. How
ever, the Ml model demands much greater computer resources.
Between 5 and 12 seconds of CPU time were needed per solu
tion. The number of total iterations went over 300. Con
vergence was a real problem and solution vectors from the
BUW had to be used as initial vectors. Parameters in MINOS
had to be changed from their default values. The penalty
parameter, p, had to be increased substantially. Because
of these complications, additional savings from using the
MI model might not be large enough to favor it over the BUW
model in practical applications.
Table 41 reports achieved rates of success as estimated
by Monte Carlo techniques. The solution vector of a fully deterministic model has only a 25% rate of success. This
is to say that in 75% of the cases, achieved levels of ei
ther NE(1) or CP or both will be lower than the desired
levels. This does not say anything about the magnitude of
the difference which might be small enough so that no ad verse effects are observed in the animals. The desired level of nutrient is also a subject of controversy. Animal scientists have addressed the problem of uncertainty of feed composition and nutrient requirements by adding safety 169
TABLE 40
Solution vectors and related s tatis tics for the JCCP pro- blem (TMR) solved with the MI model.
Global rate of success
a Ing. 50 60 70 80 90
HAY 0 0.54 1.49 1.51 1.60
HLG 0 0 1. 03 2.10 2.97
CSL 13.87 11.93 9.01 7 . 49 6.36
GEC 1 . 37 2.84 3.66 3 . 59 2.89
SBM 3. 39 3.33 3.00 2.95 3.04
DFT 0 0 0.50 1 . 07 1.89
MIN 0. 36 0.36 0.31 0 . 28 0.25
NE 29 . 36 29.60 29.93 30 . 25 30.76
CP 3.065 3.164 3.120 3.117 3.140
CPU ^ 5.17 6.66 9.73 14.01 12.02
MIT ^ 9 17 21 18 15
TIT ^ 122 190 261 301 250
$ ' 1.976 2.065 2.258 2 . 448 2.699
^ Abbreviations are as defined in the text
^ CPU time (seconds).
^ Number of major iterations.
^ Total number of iterations.
^ Cost of ingredients ($/metric ton). 170 margins to the true requirements. However, very seldom are the magnitude of these margins published. The NRC does not even state clearly whether safety margins were added to its suggested requirements or not.
Achieved rates of success for the BEW were substantially larger than the desired rates for all certainty levels low er than 90%. The BUW model resulted in achieved rates that were much closer to the desired ones. Only at a certainty level of 50% was the achieved rate markedly higher than de sired. Of course, with the MI model, achieved rates were equal to desired rates since this is an implicit constraint in the model.
In addition to the problem of safety margins added to the requirements, the 'best' rate of success for a TMR has yet to be determined. If returns-over-feed-costs is ac cepted as a criterion of optimality, then a nutrients' re sponse function is needed in order to estimate the optimal rate of success. Figure 13 should help one to understand the concepts involved. In this figure a fixed-input re sponse function has been used for the sake of simplicity but results would be the same for a smooth function. If input N is perfectly controlled (deterministic model), then level Nq is the optimal level of nutrient N and results In level Oq of output. However, because of the uncertainty in the composition of feedstuffs, the actual level of N fed is 17 1
TABLE 41
Achieved global rates of success found by Monte Carlo ex periments for the solution vectors of the TMR problem.
Implied global rate of success (%)
Model ^ 50 60 70 80 90
DET 25. 11 - - --
BEW 57.29 65.15 72.48 81.62 90 .54
BUW 52.85 60.82 70.23 79.95 90.24
MI 49.54 60 . 39 69.56 79.86 90.12
^ Abbreviations are as defined in the text 172 not known but its expectation (p^) and variance (o^) are known. Two things can happen if a ration with expectation
is fed. N o 1. Achieved N > ; output = 0^
2. Achieved N = Ni < N ; output = 0 < 0 ^ o 1 o If the distribution of nutrient N in the ration is symme trical, then case 1 and 2 each happen 50% of the time, re sulting in an expected level of output, 0^, lower than 0^.
If a stochastic model is used so that
Pr(achieved N > N^) = > .5, then a ration with expecta tion N2 will be fed. By chance, the achieved N will some times be less than , resulting in lower output levels, but this case will happen much less frequently than with the deterministic model. Therefore, the expected level of output is raised to IT2 and revenues are increased by
P 0„ - P 0,. But the increase in level of probability re- o 2 o 1 suited in higher feed costs equal to P^N2 " ^n^o* There foreprobability level should be increased until
P o/olN2No0„ - P 0, = P N_ - P„N . This is the optimal r rate of success that maximizes returns over feed costs. This meth odology presents some interesting challenges in practical quantitative problems:
I. Nutrients' response function must be acurately esti
mated. In chapter 2, we have shown that this is not
without problems. 173
Let N ^ be a random variable and 0 = f (N ^ jN • • • » N
a nonlinear function. Then
E(f(N^|N^ N ) = f (E(N^) |N^, . . . ,Np) , where E
stands for expectation. Actually E(f(N^ | N ,N
is complicated to derive even in simple cases. But
it is the function of importance in the outlined methodology.
When there is randomness in the level of nutrient,
the optimum level is higher than what is found under
a deterministic model. However, under uncertainty
in the response function, the optimal level of input
is lower than the value found under a deterministic model (Doll and Orazem, 1978). Therefore, it is not clear exactly what happens to the optimum level of input (nutrient) when both nutrient levels and re sponse functions are stochastic. 174
3 CL
3 O
tu 0 > o tu 0 2 0 1 0 1
N N N
Level of nutrient N
Figure 15. Illustration of the effect of a random input when output is a nonlinear function of input level. 175
3.5 CONCLUSION
This research provided clear evidence of significant varia
tion in levels of nutrient within feedstuffs. However,
riata used did not allow the partitioning of the total vari
ation into components of variance. The goal of feed analy
ses should be to reduce the variation to only the variance
from sampling and the variance from laboratory procedures.
Knowledge of such components would allow the determination
of optimal frequencies of sampling and analysis per feed
type. The economic value of a feed analysis could also be
determined.
The Rahman-Bender estimate of variance is clearly an
over-estimate of the true variability of nutrients in feed
mixes. However, an iterated procedure using this estimate
(RBI) yields satisfactory solutions when applied to a sim
ple feed like a premix. For more complex feeds (supple ments and grain mixes), the uncertainty in more than one
nutrient must be taken into account. A model where indi vidual rates of success are computed and constrained by a
Bonferroni inequality (BUW) appeared to be 'best'. With this algorithm, reduction in costs in the order of 5% can be expected as compared to a model representing the actual industry standards.
An algorithm based on the computation of a bivariate normal probability integral (MI) does not appear to be 176 promising, unless the desired rate of success is relatively low (< 70%). Optimal levels of confidence for the two big gest nutritional inputs in livestock production, energy and protein, remain to be determined. Chapter IV
CONCLUSION
Research reported here has shown that:
1. There is evidence of the applicability of the prin
ciple of energy-protein substitution in the growth
of cattle. The use of tables of nutrient require
ments may result in a lost profit of 0.5 to 2.5
cents/head/day when used on growing beef cattle
weighing 200 kg. Mathematical models should be add
ed to tables for a better definition of minimum nut
rient requirements.
2. Many functional forms can give a good statistical
fit to a given biological system. Biological as
well as economic theory should be used as additional
criteria in the determination of the 'best' func
tion. Past research in agricultural economics may
have yielded wrong conclusions due to the over-use
of functions unacceptable for biological systems.
3. The variation in composition of feedstuffs is sub
stantial and the use of a deterministic LP model re
sults in unacceptable 'least-cost' solutions.
- 177 - 178
A Rahman-Bender estimate of variance can be used in simple feed models like a premix, i.e., when there are less than three sources for the stochastic nut rient in the final solution. An algorithm based on a Rahman-Bender estimate iterated on the penalty pa rameter (IRB) presents the best compromise between simplicity, computational costs and closeness to the true optimal solution.
Rahman-Bender estimates of variance should not be used in joint-constrained programming problems like supplement feeds and grain mixes.
In such cases, non-linear programming models are es sential and should not present enormous difficulties in application provided that an actual efficient al gorithm (MINOS) be supplemented with an editor code.
Among different models investigated, BUW, based on a
Bonferroni inequality without pre-assigned probabil ity levels per nutrient, appears to be 'best'.
The future of a model based on a multivariate inte gral (MI) does not appear to be promising. Because correlation coefficients between nutrients within feedstuffs are generally low (-0 . 5 Additionally, convergence with Ml is uncertain and relatively huge amounts of computer resources are nee de d. 179 Much research is still needed in this field. We suggest: 1. Contemporary functions for the growth of animals with factors other than nutritional inputs added to the model, should be developed. Applicability of a dynamic programming approach should also be investi gated. 2. There is an obvious need for full and reliable vari ance-covariance matrices of nutrient within feeds tuffs. The total variance should be partitioned into useful variance components in order to deter mine the value of a feed analysis. 3. The 'best' level of certainty in composition of livestock rations needs to be determined as well as factors affecting such a level. 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APPENDIX The following abbreviations are used in table 42 and 43 DM = Dry matter (%) CP = Crude protein (%) ADFNN = Nitrogen in acid detergent fiber (%) ADF = Acid detergent fiber (%) NEADF = NE(1) estimated from ADF (Mcal/kg) P = Phosphorus (%) K = Potassium (%) CA = Calcium (%) MG = Magnesium (%) MN = Manganese (ppm) FE = Iron (ppm) CU = Copper (ppm) ZN = Zinc (ppm) FAT = Ether extracts (%) SALT = NaCl from Na (%) ASH = Ash content (%) 190 TABLE 42 Variance-covariance matrices of nutrients within feedstuffs Legume hay - 1st cutting DMCP ADPMN ADPHEADP P K CA MG MM PE CD ZM DM 17.619* -1.53959 -2.632*1 -6.89126 0.18667* -.030053 -.261198 .0053799 .0061377 -6.6623 -*3.0*59 -.37671* -1.0103* CP -1.53959 11.5568 -9.08993 -11.3908 0.308877 0.115663 0.922*05 0.3*5*59 .0*05675 -8.95889 65.0306 2.279*8 2.82527 ADPMB -2.632*1 -9.08993 33.0338 21.7205 -.588836 -.08978* -.856406 -.311039 -.033*58 12.4022 -32.8002 -1.9386* -4.99103 ADP -6.89126 -11.3908 21.7205 38.8299 -1.05287 -.117282 -1.21824 -.6*6223 -.1*3027 -1.33506 -15.429* -3.7*91* -6.72125 MKADP 0.18667* 0.308877 -.588836 -1.05287 .0285*8* .0031801 .0330409 .0175286 .0038787 .035779* 0.418383 0.101677 0.182255 P -.030053 0.115663 -.08978* -.117282 ,0031801 .0036007 .0232862 -1.7E-0* -2.9E-0* .010*568 0.711763 .017*953 .0680577 K -.261198 0.922*05 -.856*06 -1.2182* .0330*09 .0232862 0.368829 -.008981 -.006258 -.2*2062 6.93793 .0307069 0.327638 CA .0053799 0.3*5*59 -.311039 -.6*6223 .0175286 -1.7E-0* -.008981 .05877*3 .0072197 -1.11158 2.93663 0.232213 0.238069 MG .0061377 .0*05675 -.033*58 -.1*3027 .0038787 -2.9E-0* -.006258 .0072197 .00*2*31 .0963576 0.109701 .0*80198 .0725693 MM -6.6623 -8.95889 12.*022 -1.33506 .0357794 .010*568 -.2*2062 -1.11158 .0963576 529.162 2*4.781 -4.599*9 32.501 PS -*3.0*59 65.0306 -32.8002 -15.*29* 0.418383 0.711763 6.93793 2.93663 0.109701 2*4.781 *565.5 14.2596 57.3322 CU -.37671* 2.279*8 -1.9386* -3.7*91* 0.101677 .0174953 .0307069 0.232213 .0*80198 -*.599*9 14.2596 4.0999* 4.62202 ZM -1.0103* 2.82527 -*.99103 -6.72125 0.182255 .0680577 0.327638 0.238069 .0725693 32.501 57.3322 *.62202 33.8309 TABLE 42 (continued) Legume hay - 2nd cutting DMCPADFNN ADF NEADF P K CA HCMN FE CD ZN d m 12.0003 -2.02463 0.462562 1.01814 -.027613 -.019282 -.277896 .0106568 -.004014 -.799633 -23.0645 -.367059 -.343304 CP -2.02463 9.58919 -5.73507 -7.73151 0.209686 .0865716 0.595881 0.326244 .0486097 0.934072 38.0707 1.64278 0.897898 ADPNN 0.462562 -5.73507 34.5577 18.9698 -0.51455 -.058067 -.285324 -.193628 -.002513 8.80937 9.46523 -.697459 .0609532 ADP 1.01814 -7.73151 18.9698 38.0695 -1.03235 -.102923 -.802535 -.526405 -.062589 -8.28549 -28.6586 -1.3351 -.905218 NEADP -.027613 0.209686 -0.51455 -1.03235 .0279947 0.00279 .0217547 .0142778 .0016967 0.224457 0.775334 .0362122 .0243697 P -.019282 .0865716 -.058067 -.102923 0.00279 .0030614 .0194897 -3.4E-04 3.4E-04 0.223003 0.683673 .0145582 .0480905 K -.277896 0.595881 -.285324 -.802535 .0217547 .0194897 0.380268 -.031078 -.011845 1.12966 7.58373 -.003774 0.117801 CA .0106568 0.326244 -.193628 -.526405 .0142778 -3.4E-04 -.031078 .0626492 .0067314 -.783918 1.74273 0.204137 0.223772 HO -.004014 .0486097 -.002513 -.062589 .0016967 3.4E-04 -.011845 .0067314 .0044139 0.233385 .0923906 .0399438 0.103555 HN -.799633 0.934072 8.80937 -8.28549 0.224457 0.223003 1.12966 -.783918 0.233385 426.904 81.1122 -3.13725 24,8118 FE -23.0645 38.0707 9.46523 - 28.6586 0.775334 0.683673 7.58373 1.74273 .0923906 81.1122 2951.26 9.1491 59.5204 CU -.367059 1.64278 -.697459 -1.3351 .0362122 .0145582 -.003774 0.204137 .0399438 -3.13725 9.1491 2.94054 3.58429 ZN -.343304 0.897898 .0609532 -.905218 .0243697 .0480905 0.117801 0.223772 0.103555 24.8118 59.5204 3.58429 34.0559 V£> ho TABLE 4 2 (continued) Legume hay - 3rd cutting DU CPADPNN ADP NEADF PK CA MG MM PE CU ZN DM 12.2166 -3.62777 0.633414 -1.21434 .0328034 -0.04622 -.437309 -.024212 -.011427 -9.74944 -25.0175 -0.99146 -3.20924 CP -3.62777 14.3324 -5.62092 -10.3319 0.265771 .0888054 0.623873 0.3574 .0487671 5.43096 59.327 1.79631 -3.63267 ADPMN 0.633416 -3.62092 27.1362 24.3631 -.665902 .0104133 -.021429 -.364654 -.028662 28.9303 -3.66946 -1.62707 1.34511 ADP -1.21431: -10.5319 24.3631 44.0344 -1.19413 .0516351 0.226947 -1.00408 -.079276 19.3544 -34.6046 -0.22426 7.49316 NEADP .0326036 0.265771 -.663902 -1.19413 .0323627 -.001398 -.006152 .0272427 .0021549 -.524632 0.937236 .0061632 -.203043 P -0.0462% .0688034 .0104133 .0316331 -.001396 .0031464 .0226667 -.002247 4.2E-04 0.272665 0.766412 .0233077 .0127976 K -.4373011 0.623873 -.021429 0.226947 -.006152 .0226667 0.401378 -.051137 -.009376 3.13202 8.18835 0.10479 0.15894» CA -.02421:! 0.3574 -.364634 -1.00406 .0272427 -.002247 -.051137 .0834271 .0039141 -1.17869 3.53536 0.167711 -.114914 HC -.011427 .0487871 -.026662 -.079276 .0021349 4.2E-04 -.009376 .0059141 .0040734 0.177704 0.229746 0.046703 0.103562 MN -9.74944 3.43096 28.9303 19.3344 -.524632 0.272665 3.13202 -1.17869 0.177704 403.343 367.303 -.660916 49.5352 FE -23.0171 39.327 -3.66946 -34.6046 0.937258 0.766412 8.18835 3.53538 0.229748 367.303 3634.47 33.1963 202.036 CU -0.99146 1.79831 -1.62707 -0.22426 .0061832 .0255077 0.10479 0.167711 0.048705 -.660916 33.1963 3.72347 5.09072 ZN -3.20924 -3.6326 7 1.34311 7.49316 -.203045 ,0127976 0.158948 -.114914 0.105562 49.5352 202.036 3.09072 56.3019 VO w TABLE 4 2 (continued) Legume-grass hay - 1st cutting DM CPADFNN ADFNEADF PK CA MO MN FE CU ZN OH 12.2134 -.5*3*06 -3.9162* -7.5**97 0.21791* -.016628 -.121969 -.07298* -.023865 -8.22907 -21.8923 -.686012 -4.16969 CP -.5*3*06 10.82*6 -9.63561 -16.1668 0.466863 0.10*92 0.836557 0.563907 0.101288 -21.3128 37.0376 3.64159 3.20974 ADFNN -3.9162* -9.63561 38.*800 50.0247 -1.4**62 -.102*66 -.83*683 -.323001 -.042833 39.80*2 -3.2913 -1.34224 3.10481 AD? -7.5**97 -16.1668 50.02*7 502.257 -16.81*9 -.112336 -.238212 -.97*828 -.195559 33.283* 21.643 -5.53615 -10.9865 NRAD? 0.21791* 0.*66863 -l.***62 -16.81*9 0.485592 .0032**3 0.006877 .0281531 .0056*81 -.961399 -.623674 0.159892 0.317156 P -.016628 0.10*92 -.102*66 -.112336 .0032**3 .0030619 .021**02 .0031796 8.7E-0* -.015975 0.577867 .0322496 .0715411 K -.121969 Q . 836557 -.83*683 -.238212 0.006877 .021**02 0.321837 .0112969 4.8E-0* -.125999 4.17038 0.100888 0.434981 CA -.07298* 0.563907 -.323001 -.97*828 .0281531 .0031796 .0112969 .0698887 .0102393 -2.4741 2.69204 0.374276 0..293635 MG -.023865 0.101288 -.0*2833 -.195559 .0056*81 8.7E-0* 4.8E-0* .0102393 .00*1825 .0078134 0.538578 .0765858 0.122064 MN -8.22907 -21.3128 39.80*2 33.283* -.961399 -.015975 -.125999 -2.47*1 .007813* 896.645 212.269 -10.6429 31.4179 FÏ -21.8923 37.0376 -3.2913 21.6*3 -.62367* 0.577867 *.27038 2.6920* 0.538578 212.269 4003.89 19.5402 71.4019 CU -.686012 3.6*159 -1.3*22* -5.53615 0. 1 59892 .0322*96 0.200888 0.374276 .0765858 -10.6429 19.5*02 4.69179 5.26558 ZN -*.16969 3.2097* 3.10*01 -10,9865 0.317156 .0725*11 0.43*981 0.293635 0.12206* 31.4179 71.4019 5.26558 37.7476 VO JS TABLE 42 (continued) Legume-grass hay 2nd cutting DM CP ADFNN ADF NEADPP K CAMG MN 88 CU ZN DM «.923*1 -1.12355 -1.99937 -.393752 .0113753 -.009687 -.128007 -0.06119 -.008888 *.*2639 -14.6976 -.667*13 -2.056*1 CP -1.12355 8.75927 -5.48585 -4.76977 0.13781* .05998*8 0.526*92 0.322338 .0*20605 -10.5565 24.39*5 1.7312* -.6658*1 ADFNN -1.99937 -5.*8585 28.0019 13.64*4 -.39*083 -.039128 -.570*16 -.118869 -.001*68 14.9605 10.9953 -.177266 5.17378 ADP -. 39 3752 -*.76977 13.6*** 113.252 -3.27032 -.06*505 -0.39995 -.252627 -.088*93 -7.19*72 10.7085 -0.42765 -2.4302 NEADP .0113753 0.13781* -.39*083 -3.27032 .09**353 .0018637 .011555* .0072932 .0025556 0.207311 -.308808 0.012351 .0701838 P -.009687 .05998*8 -.039128 -.06*505 .0018637 .00332*7 0.019512 -7.78-0* 6.38-0* 0.289105 0.5*5802 .0162881 .05652*4 K -.128007 0.526*92 -.570*16 -0.39995 .011555* 0.019512 0.333*08 -.025166 -.00621* 1.9301 3.61398 .007*827 .039*2*3 CA -0.06119 0.322338 -.118869 -.252627 .0072932 -7.78-0* -.025166 0.0532*2 .0059079 -1.45517 1.7050* 0.199716 0.116217 MG -.008888 .0*20605 -.001*68 -.088*93 .0025556 6.3E-0* -.00621* .0059079 .0036775 0.2*7868 0.312157 .0*29132 .0892839 MN *.*2639 -10.5565 14.9605 -7.19*72 0.207311 0.289105 1.9301 -1.45517 0.2*7868 690.083 195.903 -3.6078 29.1588 PE -1*.6976 2*.39*5 10.9953 10.7085 -.308808 0.5*5802 3.61398 1.7050* 0.312157 195.903 3175.43 11.6665 56.33*4 CU -.667*13 1.7312* -.177266 -0.*2765 0.012351 .0162881 .007*827 0.199716 .0*29132 -3.6078 11.6665 3.50*05 4.5128 ZN -2.056*1 -.6658*1 5.17378 -2.*302 .0701838 .056524* .039*2*3 0.116217 .0892839 29.1588 56.33*4 4.5128 34.1956 VO Ln TABLE 42 (continued) Legume-grass hay - 3rd cutting DM CP ADFNNADFNEADFPK CA MG MN PE CU ZN OH 11.*39* -3.21887 -2.**382 -.022695 6.0E-04 -.019878 -0.20279 -.06830* -.013057 3.9*3*8 -20.338* -.936*27 -2.00735 CP -3.21887 15.3578 -11.265 -11.5*96 0.33368 .0891105 0.989*13 0.51891 1 .05367*7 -5.56*86 30.5667 2.88717 -1.89372 ADFNN -2.**382 -11.265 *2.52* 19.2683 -.55652* -0.08657 -.763135 -.438929 -.07*026 *.12009 -25.6003 -1.3726* 1.26101 ADF -.022695 -11.5*96 19.2683 33.1818 -.958312 -.038*83 -.175589 -.82377 7 -.1*903* 11.1*51 -10.081* -.*81193 1.76009 NEADF 6.0E-0* 0.33368 -.55652* -.958312 .0276768 .0011105 .0050652 .023800* .00*3031 -.322165 0.289936 .0139003 -.050937 P -.019878 .0891105 -0.08657 -.038*83 .0011105 0.003378 .0225327 -0.00201 8.2E-0* 0.20*227 0.35*011 .0309376 .0*291*3 K -0.20279 0.989*13 -.763135 -.175589 ,0050652 .0225327 0.351065 -.035*81 -0.00507 0.75*27 2.71769 0.209662 -.071805 CA -.06830* 0.518911 -.*38929 -.823777 .023800* -0.00201 -.035*81 .0825326 .0062198 -.72*356 1.51077 0.213638 -.013302 MC -.013057 .05367*7 -.07*026 -.1*903* .00*3031 8.2E-0* -0.00507 .0062198 0.00*7*8 0.307029 0.262759 .0387606 0.1*8283 MN 3.9*3*8 -5.56*86 *.12009 11.1*51 -.322165 0.204227 0.75*27 -.72*356 0.307029 507.617 153.839 -2.9*333 35.8*51 FE -20.338* 30.5667 -25.6003 -10.081* 0.289936 0.35*011 2.71769 1.51077 0.262759 153.839 3531.18 25.2223 71.3*6 CU -.936*27 2.88717 -1.3726* -.*81193 .0139003 .0309376 0.209662 0.213638 .0387606 -2.9*333 25.2223 3.*1516 3.48733 ZN -2.00735 -1.89372 1.26101 1.76009 -.050937 .0*291*3 -.071805 -.013302 0.1*8283 35.8*51 71.3*6 3.487 33 31.23* v£) O n TABLE 4 2 (continued) Grass hay - 1st cutting DM CPADPNNADP NEADP PK CAMG MN PE CU ZN OH 15.7U9 -1.18**7 -2.98665 -2.*8798 .0821132 -.029824 -.161662 -.009808 -.027997 -7.38822 -67.9303 -.*336*7 -1.88*73 CP -1.18*17 8.81531 -6.021*8 -*.93267 0.163175 0.120895 1.02252 0.318851 .096678* -3.06721 33.1603 3.33*8 7.22502 ADPNN -2.986 45 -6.021*8 *2.6672 22.9573 -.759223 -.100239 -.6*0712 .0*635*5 -.05136* -7.53021 32.339* -.688*33 -.*52563 ADP -2.*8718 -*.93267 22.9573 28.3676 -.938292 -0.088*8 -.506973 -.003066 -.051221 -23.2515 14.5776 -.782518 -3.079* HEADP .0821112 0.163175 -.759223 -.938292 .0310351 .0029269 .0167694 1.18-0* .00169*2 0.768927 -.4822*6 0.0259** 0.102189 P -.0298 1* 0.120895 -.100239 -0.088*8 .0029269 0.00*076 .0272281 .00172*7 .001237* .068*216 0.67157* .0*19251 0.1*312* K -.161642 1.02252 -.6*0712 -.506973 .0167694 .0272281 0.338318 .0151707 0.010363 1.7*0*3 6.07512 0.406816 1.0938* CA -.00981)8 0.318851 .0*635*5 -.003066 1.18-04 ,00172*7 .0151707 .0387632 .0075136 -.8*1105 1.12*12 0.216802 0.271016 MC -.0279 17 .096678* -.05136* -.051221 .0016942 .001237* 0.010363 .0075136 .00*5331 0.212263 0.510278 .0603016 0.122301 MN -7.388Ü2 -3.06721 -7.53021 -23.2515 0.768927 .068*216 1.74043 -.8*1105 0.212263 1395.78 154.309 3.19531 *0.0356 PE -67.9303 33.1603 32.339* 1*.5776 -.482246 0.67 157* 6.07512 1.12*12 0.510278 15*.309 5582.7 16.797» 91.2389 CU -.*33647 3.33*8 -.688*33 -.782518 0.025944 .0419251 0.406816 0.216802 .060301* 3.19531 16.7979 1.330*2 5.69*13 ZN -1.88*73 7.22502 -.*52563 -3.0794 0.102189 0.143124 ,1.09384 0.271016 0.122301 *0.0356 91.2389 5.69*13 *6.9361 VO TABLE 42 (continued) Grass hay - 2nd cutting DMCP ADPNN ADP NEADP P K CAMG MN PE CU ZN DM 17.9A 2.00612 -12.5539 -3.71957 0.12305* -.020102 -.400005 .018*522 -.007045 -25.5014 36.5733 0.764661 -.805469 CP 2.00612 U.6275 -12.500* -12.2*6* 0.405159 0.160361 1.18703 0.39*667 0.197026 4.44845 73.121 5.77579 6.2336 ADPNN -12.5539 -12.500* 56.9709 *3.6856 -1.4*467 -.14*418 -.611056 - .0*8*05 -0.06297 52.5247 -5.85641 -3.102 -8.05142 ADP -3.71957 -12.2*6* *3.6856 *1.1552 -1.36108 -.121506 -.832625 .0015976 -.073503 -16.7891 101.074 -3.26742 -4.81515 NEADP 0.12305* 0.*05159 -1 .***67 -1.36108 .0450133 0.004026 .0275706 -4.4E-05 .0024352 0,554987 -3.33259 0.108109 0.159355 P -.020102 0.160361 -.1**418 -.121506 0.004026 .0067586 0.0 12825 .0039413 .0031427 .0260442 1.76917 .0619397 0.214593 E -.*00005 1.18703 -.611056 -.832625 .0275706 0.032825 0.40625* .0145578 .0140087 1.39769 8.87326 0.689021 1.06928 CA .018*522 0.394667 -.0*8*05 .0015976 -4.4E-05 .0039413 .01*5578 .0366774 .0087409 -1.14563 4.67262 0.248488 0.404879 HC -.0070*5 0.197026 -0.06297 - .073503 .0024352 .0031427 .01*0087 .0087409 0.0075*1 0.31527 1.5309 .0903204 0.180461 MN -25.501* *.*48*5 52.52*7 -16.7891 0.554987 .0260442 1.39769 -1.14563 0.31527 1277.01 168.38 -5.41613 -15.7885 PE 36.5733 73.121 -5.856*1 101.07* -3.33259 1.76917 8.87326 4.67262 1.5309 168.38 9307.61 29.2761 169.907 CU 0.76*661 5.77579 -3.102 -3.267*2 0.108109 .0619397 0.689021 0.248488 .090320* -5.41613 29.2761 5.24821 8.4463 ZN -.805*69 6.2336 -8.051*2 -4.81515 0.159355 0.214593 1.06928 0.404879 0.180461 -15.7885 169.907 8.4463 46.5701 VO 0 0 TABLE 42 (continued) Grass hay - 3rd cutting DMCPADPNN ADP NEADP P K CA HG MN PE CU ZN DM 24.8668 3.63261 -17.6293 -12.0439 0.398012 .0697278 -.014924 0.223144 -.097957 12.1167 -257.729 -.694581 -10.9594 CP 3.63261 21.437 -31.7139 -28.4798 0.941403 0.405627 3.17608 0.679005 0.253709 -68.9141 -21.3701 8.69615 7.1102 ADPNN -17.6293 -31.7139 76.7107 53.7655 -1.77737 -.5714 37 -5.14034 -.838853 -.462809 87.432 54.3684 -8.90772 -9.59559 ADP -12.0439 -28.4798 53.7655 46.0839 -1.52355 -.526932 -4.40809 -1.16198 -.367068 58.5045 294.18 -7.49318 -19.6432 NEADP 0.398012 0.941403 -1.77737 -1.52355 .0503695 .0174306 0. 145777 .0384264 .0121397 -1.93379 -9.72936 0.247545 0.649424 P .0697278 0.405627 -. 571437 -.526932 .0174306 .0110394 .0778754 .0147403 .0060345 -.751032 -1.34046 0.162192 0.186084 K -.014924 3.17608 -5.14034 -4.40809 0.145777 .0778754 0.823226 .0998244 0.057107 -10.0937 5.41006 1.47872 1.29111 CA 0.223144 0.679005 -.838853 -1.16198 .0384264 .0147403 .0998244 .0534936 0.008908 -2.04606 -9.08972 0.385394 0.387956 HG -.097957 0.253709 -.462809 -.367068 .0121397 .0060345 0.057107 0.008908 .0081695 -.636138 1.57099 0.160222 0.373091 HN 12.1167 -68.9141 87.431 58.5045 -1.93379 -.751032 -10.0937 -2.04606 -.636138 2077.21 197.127 -22.1058 27.6878 PE -257.729 -21.3701 54.3684 294.18 -9.7 2936 - 1 . 34046 5.41006 -9.08972 1.57099 197.127 13471.9 37.532 210.526 CU -.694581 8.69615 -8.90772 -7.49318 0.247545 0.162192 1.47872 0,385394 0.160222 -22.1058 37.532 7.66256 9.50493 ZN -10.9594 7 . 1102 -9.59559 -19.6432 0.649424 0.186084 1.29111 0.387956 0.373091 27.6878 210.526 9.50493 70.1084 VO VO TABLE 4 2 (continued) Legume silage — 1st cutting DM CP ADPNN ADP NEADP P K CA MG MN PE CUZN DM 150.507 4.91836 6.35525 0.133478 -.003639 -.110026 -.757627 -.005688 -.030617 -67.9338 -208.385 -1.35877 -29.2876 CP 4.91836 10.4619 -1 1 . 2738 -10.0967 0.273742 .0867805 0.919717 0.420231 .0635926 -18.0625 -.713829 1.70651 1.83908 ADPNN 6.15525 -11.2738 59.7762 34.1515 -.926162 -.146642 -1.51397 -.399719 -.010277 41.6778 101.447 .0882599 -2.71576 ADP 0.133478 -10.0967 34.1515 34.3811 -.932358 -.134293 -1 . 2886 -.385863 -.086125 4.58895 18.2978 -1.65003 -7.17939 NEADP -.003639 0.273742 -.926162 -.93235 1 0.025284 .0036396 0.034932 .0104601 .0023344 -.123549 -.504458 .0446511 0.199964 P -.110026 .0867805 -.146642 -.13429 1 .0036396 .0033996 .0247704 1.2E-04 3.0K-04 0. 161392 1.64403 .0196402 0.12545 K -.757627 0.919717 -1.51397 -1.2886 0.034932 .0247704 0.372471 .0029267 -.001292 1.071 10.8158 0.148496 0.856197 CA -.005688 0.420231 -.399719 -.38586 1 .0104601 1.2P.-04 .0029267 .0708821 .0087866 -2.15226 -.367148 0.202116 0.102191 HC -.030617 .0635926 -.010277 -.086125 .0023344 3.0E-04 -.001292 .0087866 .0042401 .0568394 0.424281 .0599965 .098)452 MN -67.9838 -18.0625 41.6778 4.58895 -.123549 0.161392 1.071 -2.15226 .0568394 751.178 795.555 4.62459 54.8058 PE -208.385 -.713829 101.447 18.2971! -.504458 1.64403 10.8158 -.367348 0.424281 795.555 17225.8 43.5292 214.31 CU -3.35877 1.70651 .0882599 -1.65005 .0446511 .0196402 0.148496 0.202216 .0599965 4.62459 43.5292 4.00405 5.03649 ZN -29.2876 1.81908 -2.71576 -7.37939 0.199964 0.12545 0.856197 0.102391 .0983452 54.8058 214.31 5.03649 56.9288 Ni o o TABLE 42 (continued) Legume silage - 2nd cutting DM CPADPNN ADFNEADPP K CA MO HN PECU ZN DM 133.236 2.98902 2.90483 -6.77913 0. 183778 .0063884 0.212604 -.042345 -.053301 -15.431 66.6868 -.423746 -11.7 316 CP 2.98902 10.1116 -10.5992 -11.2211 0.304254 .0821609 0.746498 0.388332 .0240739 -5.41605 -3.43727 2.51596 -.174903 ADPNN 2.90483 -10.5992 52.9669 28.2304 -.765473 -.051313 -1.27 325 -.271188 .0908218 31.1298 271.242 1.22324 17.8839 ADP -6.77913 -11.2211 28.2304 33.6705 -.913226 -.114513 -1.13995 -.339632 .0039505 -8.73665 67.7722 -.595257 8.06747 NEADP 0. 183778 0.30425* -.765473 -.913226 0.024769 .0031055 .0309325 .0092086 -l.lE-04 0.237737 -1.83594 .0161747 -.218754 P .0063884 .0821609 -.051313 -.114513 .0031055 .0028291 .0220381 2.5E-04 2.78-04 0. 183261 1.26318 0.020291 .0882128 K 0.212604 0.7*6498 -1.27325 -1.13995 .0309325 .0220381 0.345357 -.024038 -.007287 1.89413 5.62249 0.029729 0.496536 CA -.0423*3 0.388332 -.271188 -.339632 .0092086 2.5E-04 -.024038 .0688976 .0061268 -1.48801 0.39879% 0.260976 0.286932 HG -.053301 .02*0739 .0908218 .0039505 -l.lt-04 2.7P.-04 -.007287 .0061268 .0045991 0.468331 1.2693 .0588287 0.254*02 HN -15.431 -5.41605 31.1298 -8.73665 0.237737 0. 183261 1.89413 -1.48801 0.468331 489.541 703.489 1.14994 60.2576 FE 6 6■68 68 -3.43727 271.242 67.7722 -1.83594 1.26318 5.62249 0.398792 1.2693 703.489 1*535.5 21.0715 230.5*7 CU -.4237*6 2.51596 1.22324 -.595257 .0161747 0.020291 0.029729 0.260976 .0588287 1.14994 21.0715 3.77221 6.31639 ZN -11.7316 -.174903 17.88 39 8.06747 -.218754 .0882128 0.496566 0.286932 1.254402 60.2576 230.547 6.31639 69.2*88 to o TABLE 42 (continued) Legume silage 3rd cut ting DM CP ADFNNADFNEADFP K CA MG MN FE CU ZN DM 159,185 -0.55925 2.76295 4.50363 -.121861 -.076552 0. 102467 -.108148 -.020625 -78.547 -465.391 -0.89562 -22.0469 C? -0.55925 11.4775 -10.6025 -9.2797 0.251779 .0720941 .0626718 0.420868 .0156995 -7.0608 1.58866 1.60734 -4.90513 ADPNN 2.76295 -10.6025 48.303 30.8878 -.838312 -.117806 -.680827 -.059023 0. 124826 8.6517 146.852 1.63507 9.31018 ADF A. 50365 -9.2797 30.8878 34.2542 -.929282 -.071765 -.088206 -.492329 .0255778 -13.5577 78.3392 -.691265 -.679573 NEADF -.121861 0.251779 -.838312 -.929282 .0252106 .0019477 .0024005 .0133553 -6.9E-04 0.368067 -2.12495 0.01863 .0186977 P -.076552 .0720941 -.117806 -.071765 .0019477 .0028917 .0149853 -.003716 5.1E-04 0.347252 1.11782 -2.5E-04 .0432376 K 0.102467 .0626718 -.680827 -.088206 .0024005 .0149853 0.268397 -.088943 -.011444 -.721968 -8.94101 -.139098 0.229559 r.A -.108148 0.420868 -.059023 -.492329 .0133553 -.003716 -.088943 0.108876 .0100914 -.291613 3.53824 0.235433 -.241184 HC -.020625 .0156995 0.124826 .0255778 -6.9E-04 5.18-04 -.011444 .0100914 .0048987 0.438883 2.37723 .0402859 .0134225 MN -78.547 -7.0608 8.6517 -13.5577 0.368067 0.347252 -.721988 -.291613 0.438883 662.494 1386.35 -3.07684 70.3438 FE -465.191 1.58866 146.852 78.3392 -2.12495 1.11782 -8.94101 3.53824 2.37723 1386.35 17605.3 2.05042 205.521 CU -0.89562 1.60734 1.63507 -.691265 0.01863 -2.SF.-04 -.139098 0.235433 .0402859 -3.07684 2.05042 3.12274 2.26488 ZN -22.0469 -4.90513 9.31018 -.679573 .0186977 .04 32376 0.229559 -.241184 .0134225 70.3438 205.521 2.26488 58.3454 o N) TABLE 42 (continued) Legume-grass silage - 1st cutting DH CPADFNN ADPNEADF PK CAHC MN FECU ZN DM 129.497 2.48201 3.64603 0.676927 -.019514 -.068049 -.655223 0.150029 -.006615 -53.754 -111.809 -.857327 -11.9688 CP 2.48201 10.3041 -7.69928 -8.42275 0.243269 .0855086 0.84852 0.561044 .0906551 -15.6856 52.1331 2.93656 4.6296 ADFNN 3.64603 -7.69928 63.7478 31.9515 -.922757 -.089323 -1.08574 -.149699 -.012886 36.4646 37.9477 0.191442 1.76624 ADF 0.676927 -8.42275 31.9515 44.0512 -1.27225 -.123246 -1.11721 -0.18131 -.067989 1.71885 3.72656 -1.30749 -4.12791 HEADP -.019314 0.243269 -.922757 -1.27225 .0367444 .0035598 .0322784 .0052355 0.001964 -.049536 -.108915 .0377771 0.119228 P -.068049 .0855086 -.089323 -.123246 .0035598 .0031522 .0232184 .0015092 6.48-04 0.253693 1.48217 0.025745 0.107912 K -.635223 0.84852 -1.08574 -1.11721 .0322784 .0232184 0.344196 0.006717 .0017145 1.80083 9.2948 0.14667 0.639798 CA 0.150029 0.561044 -.149699 -0.18131 .0052355 .0015092 0.006717 .0785581 .0112818 -2.28023 2.6709 0.307145 0.294659 HC -.006615 .0906551 -.012886 -.067989 0.001964 6.4E-Q4 .0017145 .0112818 .0044374 -.047703 1.14703 .0659311 0.11748 HN -53.754 -15.6856 36.4646 1.71885 -.049536 0.253693 1.80083 -2.28023 -.047703 738.625 683.592 0.408963 41.3668 FE -111.809 52.1331 37.9477 3.72656 -.100915 1.48217 9.2948 2.6709 1.14703 683.592 16308.5 39.3679 228.673 CU -.857327 2.93656 0.191442 -1.30749 .0377771 0.025745 0.14667 0.307145 .0659322 0.408963 39.3679 4.11471 6.42278 ZN -1 1.9688 4.6296 2.76624 -4.12791 0.11922H 0.107912 0.639798 0.294659 0.12748 41.3668 228.673 6.42278 50.8317 ro o LO TABLE 4 2 (continued) Legume-grass silage - 2nd cutting DM CP ADPNN ADFNEADF P K CA MC MN FE CU ZN DM 127,735 2 .*6758 -28.1622 -1.62859 0.0*6803 -.0*2*4* -.72*309 -.0*3897 -.050625 -18.590* -37.9578 -1.26556 -13.9979 CP 2.*(1758 11.6788 -8.2*503 -11.8961 0.343501 .0937*33 0 .885595 0.531326 0.09031* -15.0976 54.6128 3.75331 2.55632 ADFNN -28.1622 -8.2*503 5911.51 57,8188 -1.66931 -.401*01 -5.40681 0.4 17766 .0628776 -70.1*43 *.*0089 6.6026 -10.3198 ADP -1.62859 -11.8961 57.8188 *8.3566 -1.39668 -.0935*6 -.7 18684 -.4303** -.067*86 31.4103 *0.4931 -1.36863 3.9972* NEADF 0.0*6803 0.3*3501 -1.66931 -1.39668 .0*03*0* .0027026 .0207638 .012*178 .0019*99 -.906775 -1.16952 .0395*31 -.11539* P -.0*2*** .0937*33 -.*01*01 -.0935*6 .0027026 .0038327 .0271025 .0011236 .0010821 0.394996 2.09891 .03558*5 0.1*462* K -.72*309 0.885595 -5.*0681 -.71868* .0207638 .0271025 0.39812 -.0057*6 .0013087 3.39882 16.78* 0.201621 0.736709 CA -.0*3897 0.531326 0.*17766 -.*303** .0124178 .0011236 -.0057*6 .0757281 .0082873 -2.38732 0.865985 0.320092 0.179858 HG -.050625 0.09031* .0628776 -.067*86 .0019*99 .0010821 .0013087 .0082873 .00*2585 , 0,11194 0.53137 .0705967 0.18*107 HN -18.590* -15.0976 -70.1**3 31.*103 -.906775 0.394996 3.39882 -2.38732 0.1119* 789.002 896.353 -1.00713 56.2147 FE -37.9578 5*.6128 *.*0089 *0.*931 -1.16952 2.09891 16.78* 0.865985 0.53137 896.353 16256.6 2*.*935 158.553 CU -1.26556 3.7 5331 6.6026 -1.36863 .0395*31 .03558*5 0.201621 0.320092 .0705967 - 1 . 00713 2*.*935 *.92**8 6.32506 7.N -13.9979 2.55632 -10.3198 3.99724 -.11539* 0. 1**62* U . 736709 U. 179858 0.18*107 56.21*7 158.553 6.32506 48.3027 ro o TABLE 4 2 (continued) Legume-grass silage - 3rd cutting DM CP ADFNN ADFNEADF PK CAMG MN FE CU ZN DM U1.A8I -0.35*71 0.167161 -5.92736 0.170997 -.0*505* -.357857 -0.38*53 -.110333 -35.*898 -20*.98 -*.19728 -15.912* CP -0.35*71 10.**53 -10.6*88 -10.0353 0.289661 .02*0588 0.1*6107 0.3*1632 .050*977 -20.3015 -*5.7658 2.08181 -1.8163 ADFHM 0.167161 -10.6*88 *3.5505 26.9359 -.777598 -.080317 -.907236 .0*35857 -.032*35 6.8728* -*7.*53 -.86839* -7.23363 ADF -5.92736 - 10.0353 26.9359 37.8089 -1.09153 -0.03325 0.309083 -.115736 -.0*026* 5.22917 6.6199* -1.0319* —3.86667 NEADF 0.170997 0.289661 -.777598 -1.09153 .0315123 9.6E-0* -.008896 0.0033* .0011638 -.151207 -.196*86 .0298272 0.11167 P -.0*505* .02*0588 -.080317 -0.03325 9.6E-0* .0031778 .018*026 -.00*31* 7.3E-0* 0.*33912 0.519873 .0126739 0.170012 K -.357857 0.1*6107 -.907236 0.309083 -.008896 .018*026 0.325867 -0.0*739 -.005056 2.78921 3.*5091 -.15977# .0**2692 CA -0.38*53 0.3*1632 .0*35857 -.115736 0.0033* -.00*31* -0.0*739 .06*7792 .0060817 -1.39695 .09973*3 0.187041 -.167836 MC -.110333 .050*977 -.032*35 -.0*026* .0011638 7.3E-0* -.005056 .0060817 .00*1236 0.305851 1.109*5 .0751695 0.232*36 MN - 35.*898 -20.3015 6.8728* 5.22917 -.151207 0.*33912 2.78921 -1.39695 0.305851 822.985 1810.*1 -0.2*363 67.1507 FE -206.98 -*5.7658 -*7.*53 6.6199* -.196*86 0.519873 3.*5091 .0997363 1.109*5 1810.*1 17338.6 20.4957 273.006 CU -*.19728 2.08181 -.86839* -1.0319* .0298272 .0126739 -.159776 0.1870*2 .0751695 -0.2*363 20.*957 3.65929 7.1512 EN -15.912* -1.8163 -7.23363 -3.86667 0.11167 0.170012 .0**2692 -.167836 0.232636 67.1507 273.006 7.1512 52.015 hO o Ln TABLE kl (continued) Grass silage - 1st cutting DHCPADFNNADF NEADFPK CA MC MN FE CU ZN DM 151.4*2 -7.11127 22.714 2.20342 -0.0729 -.244882 -2.35723 0.370937 .0600656 -34.0259 -165.197 -1.65422 -10.267 CP -7.11127 11.7026 -7.13723 -10.135 0.335179 0.132491 1.43496 0.263506 .0722875 -5.83925 15.5838 4.25332 15.3933 ADPNN 22.7 14 -7.13723 106.14 51.6304 -1.707 3 1 -.275998 -2.33744 0.743254 0.136504 47.2273 -68.868 0.531802 0.795197 ADP 2.20342 -10.135 51.6304 4 1.0221 -1.35657 -.272903 -2.20564 0.274736 -.009202 22.6208 1.34843 -1.833 -8.78503 NEADF -0.0729 0.335179 -1.70731 -1.35657 .0448611 .0090248 .0729555 -.009069 3.0E-04 -.747018 -.050174 .0605588 0.289959 P -.244882 0.132491 -.275998 -.272903 .0090248 .0056224 .0386613 -2.7E-04 8.4E-04 -.035948 0.932845 .0703334 0.261951 K -2.35723 1.43496 -2.337*4 -2.20564 .0729555 .0386613 0.55732 .00564 17 .0148779 0.759962 12.4015 0.929169 2.58962 CA 0.370937 0.263506 0.743254 0.274236 -.009069 -2.7F.-04 .0056417 .0461735 .0076231 -.993587 -2.62559 0.216253 0.324291 MC .0800656 .0722875 0.136504 -.009202 3.0E-04 8.4E-04 .0148779 .0076231 .0039521 -.051057 1.03111 .0812618 0.266306 MN -34.0259 -5.83925 47.2273 22.6208 -.747018 -.035948 0.759962 -.993587 -.051057 1063.84 1001.76 -3.90123 39.6213 FE -165.197 15.5838 —68.868 1.34843 -.050174 0.932845 12.4015 - 2.62559 1.03111 1001.76 19755.4 23.1619 316.364 CU -1.65422 4.25332 0.531802 -1.833 .0605588 .0703334 0.929169 0.216253 .0812618 -3.90123 23.1619 4.93008 11.9016 ZN -10.267 15.3933 0.795197 -8.78503 0.289959 0.261951 2.58962 0.324291 0.266306 35.6213 316.364 11.9016 85.7222 O ON TABLE 42 (continued) Grass silage - 2nd cutting DM CP ADPNNADF NEADF PK CA MG MN PE CU ZN DM 211,943 -8.43179 87.9045 33.374 -1.10263 -.013528 -3.52627 0.898745 0. 174261 -46.8628 -261.241 -1.60889 10.9357 CP -8.43179 10.9259 -13.6702 -10.6906 0.354051 .0665464 1.11399 0. 163292 .0553842 -22.2856 65.5513 3.44587 13.6514 ADPNN 87.9043 -13.6702 64.4117 27.4844 -.909556 -.206235 -3.32045 0.725992 .0971591 -56.9705 211.306 .0166667 1.93273 ADP 33.374 -10.6906 27.4844 19.256 -0.63714 -0.0589 -1.3815 0.1807 0.113 81.82 91.7306 0.79 -12.7267 NEADP -1.10263 0.354051 -.909556 -0.63714 .0210817 .0019523 .0457529 -.005976 -.003742 -2.7036 -3.04343 -.026173 0.422156 P -.013328 .0665464 -.206235 - 0.0589 .0019523 .0039383 .0197318 -.006566 -7.3E-05 0.232866 1.607 0.01083 0.290952 K -3.52627 1.11399 -3.32045 -1.3815 .0457529 .0197318 0.387064 -.053595 -.013241- 7.45136 4.73143 0.308636 1.89*32 CA 0.8911745 0.163292 0.725992 0.1807 -.005976 -.006566 -.053595 .0588111 .0109457 -1.01941 -3.86914 0.213373 -0.13811 MG 0.174261 .0553842 .0971591 0.113 -.003742 -7.3E-05 -.013241 .0109457 .0089241 .0896047 5.78907 0.113013 0.342619 MM -46.8628 -22.2856 -56.9705 81.82 -2.7036 0.232866 7.45136 -1.01941 .0896047 1524.63 -1069.49 -2.93281 -38.119 FE -261.241 65.5513 211.306 91.7306 -3.04343 1.607 4.73143 -3.86914 5.78907 -1069.49 17158.7 181.657 784.993 CU -1.60889 3.44587 .0166667 0.79 -.026173 0.01083 0.308636 0.215375 0.113083 -2.93281 181.657 4.67589 9.16667 ZN 10.9357 13.6514 1.93273 -12.7267 0.422156 0. 290952 1.69952 -0.13881 0.342619 -58.119 784.993 9.16667 81.5 ro o 'O TABLE 4 2 (continued) Grass silage - 3rd cutting DM CPAOFNN Aor M F A 0 ¥ P V. CA MC •IN FF. CII 7.K f"' V)n.32 7 -18.0226 4.29667 119.07 -3.9249 -.275196 -3.69648 -.742089 -.06 162 5 3 5 4.261 -78.I 8 1 -6,36964 40.2339 rp - i n .0226 7.30831 2.4295 -1.70 1 0.0 5607 0.14 295 1 . I 2909 0. 29 5993 0.156464 -34.29 126.023 3.77071 -I I . 69 79 Anrr/N 4.2966 7 2.4295 5 0.7633 -12.69 0.4 183 -.2 18167 -1.54517 0.079 -0.77 1 3.48333 288,95 -3.88 3 33 -56.05 Anp 1 19.07 -1.701 -12.69 14 . 58 -0.4806 0, 1 89 -0.108 -0,108 0.567 108 -8.1 0 35.1 NF.AHK -3.9249 0.05607 0.4 183 -0.4806 0,015842 -0.00623 0.00356 0.00356 -0.01869 -3.56 0.267 0 -1.157 p -.275196 0. 14295 -.218167 0.189 -0.00623 ,0041982 .0206839 .006 1161 .0056696 -0.5775 6.57524 .0776786 -. 0 1 55 36 V -3.6964R 1 . 1 290': -1 . 5451 7 -0. 1 (18 0.00356 .0206839 0. 226255 .0279589 .0285554 -7.3075 -11,0062 0.624821 -1.81411 CA -.742089 0.295993 0.979 -0.108 0.00356 .0061161 ,0279589 .0245125 .0039018 -.8 15357 4.23762 0. 1 10893 -.593036 MC -.06 1625 0.156464 -0.771 0.567 -0.01869 .0056696 .0285554 .00390 18 .0113125 -.763214 4.79571 ,0819643 -.036964 MN 344.26 1 -14.29 3.48333 108 -3.56 -0.5275 -7.3075 -.815357 -.763214 578 .5 7.28571 -11.0357 109,036 FK -78.181 12 6,023 288.9 5 -8.1 0,267 6.57 524 -1 1 .0062 4.23767 4 . 7957 1 7.28571 47065.6 91.9286 708.357 rii -6.36964 3.77071 -3.88333 0 0 .0776786 0, 62487 I 0. 1 1 OH 93 .0HI964 J -11.035 7 91.9286 2.4 1071 -2.625 /K 40,2339 -11.6979 -56.05 35.1 - I . i 5 7 -.0 15536 -1.8 14 11 -.593036 -.036964 109.036 708.357 -2.625 80.8 39 3 o 00 TABLE 4 2 (continued) Corn silage - no NPN added DM CP ADFNNADFNEADFP K CA MC MN FF. CU ZN DM A1.657A -.617809 -8.617*3 -7.10474 0.206746 .0 99487 -.346484 -.052767 -.021002 -15.1293 -53.90*3 -1.055* -*.60201 CP -.617809 1.7989 -1.9*367 -0.21128 ,0061606 .0174931 0.114993 .0390779 .0159281 3.8721* 17.4212 0.777238 1.90815 ADFNN -8.617*1 -1.9*367 15161.8 13.6612 -0.39744 .0683473 0.151771 0.291157 -.03*194 22.6328 203.228 *.7*9*7 -6.67999 ADF -7.10*7* -0.21128 13.6612 26.7533 -0.77853 -.031765 0.26077* 0.12463 .0625826 22.097* 14.4759 0.7206*9 2.39572 NEADF 0.2067*6 .0061606 -0.397** -U.77853 .0226556 9.2E-0* -.007585 -.003626 -.001821 -.643017 -.*20835 -.02095* -.069*8* P .0199*87 .017*931. .0683*73 -.031765 9.2E-04 .0017945 .0030638 3,8E-0* 2.68-0* -.09197* 0.325*08 .0085*17 0.110952 K -.3*6*8* 0.11*993 0.151771 0.26077* -.007585 .0030638 .068*768 .0092695 0.002415 1.17218 3.42535 0.155336 0.329255 CA -.052767 .0390779 0.291157 0.12*63 — .003626 3.8E-04 .0092695 .008617 1 .0026016 0.3356*3 1.860*6 .0**1357 0.115008 MC -.021002 .0159281 -.03*19* .0625826 -.001821 2.6E-0* 0.002415 .0026016 .0029128 .057*11* 0.29909 .0138991 .0661761 MN -15.1293 3.8721* 22.6328 22.097* -.643017 -.09197* 1.17218 0.3356*3 .057411* 502.58* 374.1* 6.71192 67.9517 FE -53.90*3 17.*212 203.228 1*.*759 -.420835 0.325408 3.42535 1.860*6 0.29909 374.1* 10203.6 11.2063 152.067 CU -1.055* 0.777238 *.7*9*7 0.720549 -.020954 .0085417 0.155336 .04*1357 .0138991 6.71192 11.2063 2.27389 1.95582 ZN -*.60201 1.90815 -6.67999 2.39572 -.069484 0.110952 0.329255 0.115008 .0661761 67.9517 152.067 1.95582 78.2593 KJ O VO TABLE 42 (continued) Corn silage - NPN added DHCPADPNN ADP NEADF P K CA HO HN PE CU ZN DM 42.2022 -2.73316 3.77565 -1.39714 .0409066 0.019355 -.235962 -.038568 -.024035 -8.01259 -63.1959 -.544481 -1.63093 CP -2.73316 10.4011 -7.30588 0.127823 -.003717 .0925768 0.159534 .0360457 .0266861 14.4462 46.8482 2.87528 14.7084 ADPNN 3.77565 -7.30588 27.6323 14.7876 -.430379 -.060215 .0827792 .0433818 .0195371 1.27519 23.3751 -1.12764 -9.77007 ADP -1.39714 0.127823 14.7876 22.0387 -.641319 .0011703 0.29227 0. 1 1 1465 .0625473 20.7823 107.29 1.52858 5.38507 NEADP .0409066 -.003717 -.430379 -.641319 .0186623 -3.3P.-05 -.008502 -.003243 -.001819 -.604494 -3.12155 -.044481 -.156642 P 0.019355 .0925768 -.060215 .0011703 -3.3E-05 .0051379 .0047938 .0031861 0.001286 0.326961 2.14085 .0578273 0.49287 K -.235962 0.159534 .0827792 0.29227 -.008502 .0047938 .0708833 0.006725 .0013058 1.59567 2.89996 0.249398 0.61559 CA -.038568 .0360457 .0433818 0.111465 -.003243 .0031861 0.006725 .0167366 .0048223 0.657247 4.21382 .0478379 0.276396 HC -.024035 .0266861 .0195371 .0625473 -.001819 0.001286 .0013058 .0048223 .0038765 0.183809 1.73009 .0171774 0.109334 MN -8.01259 14.4462 1.27519 20.7823 -.604494 0.326961 1.59567 0.657247 0. 183809 456.091 534.693 15.1898 113.502 PP. -63.1959 46.8482 23.3751 107.29 -3.12155 2.14085 2.89996 4.21382 1.73009 534.693 9644.61 29.8669 353.293 CU -.544481 2.87528 -1.12764 1.52858 -.044481 .0578273 0.249398 .0478379 .0171774 15.1898 29.8669 4.58298 13.0139 ZN -1.63093 14.7084 -9.77007 5.38507 -.156642 0.49287 0.61559 0.276396 0.109334 113.502 353.293 13.0139 160.38 TABLE hi (continued) Sorghum silage DH CP ADFNNADFNEADF p K CA HC MN FE CU ZN DH 236.814 11.5062 -1 1 .9442 -27.9525 0 0.677648 -.766719 0.165128 -.218313 -7.6434 -2335.08 -4.23019 -3.33302 CP 11.5062 11.9727 -18.5672 -1 1 .0475 0 0.146981 1.18116 0.434748 0.189548 1.43136 165.213 5.45864 10.7982 ADPNN -11.9442 -18.5672 86.1326 41.2515 0 -.347218 -2.29418 -.506921 -.169834 4.52764 -213.766 -11.561 -63.1715 ADF -27.9525 -11.0475 41.2515 85.2214 0 -.402299 2.47007 0.338413 0.409941 34.7813 592.98 0.827746 -32.6759 NEADP C 0 0 0 0 0 0 0 0 0 0 0 0 P 0.677648 0.146981 -.347218 -.402299 0 .0070778 .0144278 .0088667 .0013241 -.056352 3.33361 .0627673 0.336101 K -.766719 1.18116 -2.29418 2.47007 0 .0144278 0.806937 .0750979 .0428808 7.28683 90.0502 1.21901 0.35847 CA 0.165128 0.434748 -.506921 0.338413 0 .0088667 .0750979 .0552803 .0220264 -1.08673 20.7125 0.336918 0,146289 HG -.218313 0.189548 -.169834 0.409941 0 .0013241 .0428808 .0220264 .0163128 -1.12233 -1.5699 0.155912 .0945597 MN -7.6434 1.43136 4.52764 34.7813 0 -.056352 7.28683 -1.08673 -1.12233 1105.72 4219.35 -4.40252 6.10063 FE -2335.08 165.213 -213.766 592.98 0 3.33361 90.0502 20.7125 -1.5699 4219.35 338937 133.929 1048.55 CU -4.23019 5.45864 -11.561 0.827746 0 .0627673 1.21901 0.336918 0.155912 -4.40252 133.929 9.68553 14.0881 EN -3.33302 10.7982 -63.1715 -32.6759 n 0.33610 1 0.35847 0.146289 .0945597 6.10063 1048.55 14.0881 276.997 TABLE 42 (continued) Small grain silage DM CP ADFNN ADF NEADFPK CA MG MN FE CU ZN DM 87.3353 -3.3*763 17.7589 0.82*98 -.022*21 -.089037 -1.22389 -.1*5*41 -.051268 -2.87118 -60.3978 -2.6120* -4.918 CP -3.3A763 8.46*13 -5.51103 -6.65038 0.18055* 0. 108153 1.06252 0.177552 0.07251 -1.35622 198.927 4.13887 12.5* ADFNN 17,7589 -5.51103 51.2308 21.8587 -.592888 -.099108 -1 . 12887 0.115685 .0627238 9.9*7 *.8**36 -2.8*542 -1.42* ADP 0.82A98 -6.6:038 21.8587 30.0666 -.81553* -.092866 -.73**21 -.052269 -.0395*6 -27.0876 -159.952 -3.7283* -4.113 NEADF -.022*21 0.18055* -.592888 -.815534 .0221208 .0025178 .019932* .001*306 .0010772 0.73*859 4.3*8** 0.101198 0.1122 P -.089037 0.108153 -.099108 -.092866 .0025178 .00*8131 .0255076 4.9E-05 3.5E-0* -.11*298 1.071*3 .0580265 0.1913 K -1.22989 1.06252 -1.12887 -.73**21 .019932* .0255076 0.410*09 .01*23*1 0.00502* -1.53851 8.16*47 0.457103 0.91 CA -, 145**1 0.177552 0.115685 -.052269 .001*306 4.9E-05 .01*23*1 .029*277 .00877*1 0.32219 9.39*57 0.210336 0.631* HC -.051268 0.07251 .0627238 -.0395*6 .0010772 3.5E-0* 0.00502* .00877*1 .0051681 -.002065 2.7*353 0.105*39 0.3*55 MN -2.87118 -1.35622 9.9*7 -27.0876 0. 73*869 -.11*298 -1.53851 0.32219 -.002065 797.51 2378.38 6.51286 *1.16 FE -60.3978 198.927 * .8**36 -159.952 *.3*8*4 1.071*3 8.16**7 9.39*57 2.7*353 2378.38 734*0.6 136.282 698.7 CU -2.6120* 4.13887 -2.8*542 -3.7 283* 0.101198 .0580265 0.457103 0.210336 0.105*39 6.51286 136.282 5.8752* 14.40 ZN -*.91658 12.5*36 -1.42*6* -4.1139* 0.112209 0.19132* 0.9168 0.631*62 0.3*5575 *1.16*4 698.765 I4.4058 85.0 N) Nî TABLE kl (continued) Dry shelled corn DM CP A"FN N AOK N KAOF 1» Y CA "C MN FF. CU 7.N I'M 21.5200 -.053837 -20.1683 -6.58889 0.12 5882 .0018308 -0.03769 -.021116 -.010835 1.7061 7.97009 -.135168 -2.07072 rp -.053817 0.7606 52 0.160095 0. 18 1889 -.003518 .0276002 .0213567 3.8F-06 .0150063 0.567069 6.06686 0.350836 2.51258 AnFtJN - 2 0 . I4R1 0 . I 60095 88.5087 12.3107 -.2 36642 -.102026 0.0606 -.020965 -.06 2065 6,66607 36.9669 15.2030 46.8 33 3 ADF - A . 58889 0.18 1889 12.3107 2.53833 -.068346 -.012278 -0.010 .0061607 -.0 16667 1 .027 78 0.638889 5.I 1667 12.3667 MBADF 0, 1 25882 -.003518 -. 2 36662 -.068366 0.28-06 2.6E-06 3.6E-06 -7. 5F-05 3.2K-06 -.020022 -.010778 -.097889 -.236689 P .0018308 .0276002 -. 102026 -.012278 2.6F-06 .0038728 .0031165 - 6 , 6K-06 .00 16786 .0701668 0.335555 .05136 56 0.36666 S. -0.01760 .0211567 0.0606 -0.010 1.6F.-06 .0011165 .0066815 -1.6F.-05 .0012531 0.050283 0.201762 .0372291 0.306011 CA -.021 lift 3.88-06 -.020965 .0061667 -7.5F.-05 -6.6F. - 06 -3.6F.-05 .001 2036 2.0F-05 .0600667 0. 191016 .0107377 0.079888 MH -.019835 ,0150063 -.062065 -.016667 3.2F.-06 .00 16786 ,0012531 2.0F-05 0.3E-06 .0173206 0.150936 .0096426 0.165176 MK 1 ,7061 0.567060 6.66607 1.02778 -.020022 .0701668 0.050283 .0600667 .0 173296 16.6513 31,6102 3.61111 19.602 FF. 7.97090 6.06686 36.0660 0.618889 -.010778 0.135555 0.201762 0. 101916 0.150036 31.6102 272.578 12.3531 75.7255 CU -, 135I6R 0.350836 15.2039 5. 11 66 7 -.0 0 788 0 .0513656 .0372291 .0107377 .00966 26 3.6 I I 1 1 12.3531 4.63626 12.8907 7.V -2.07072 2,51258 66.8 333 12.3667 -.2 36689 0.36666 0, 30601 I 0.070888 0.165176 10.602 75.7255 12.8907 05.0822 to w TABLE 4 2 (continued) High moisture shelled corn DM CP ADFNN ADF NEADF P K CA HG HN FE CU ZN d m 19.808 0.219471 0.472877 -.872261 .0167357 0.042515 -.003408 .0062349 .0175412 2.65925 -12.2289 0.552935 2.00596 CP 0.219AM 0.991687 -.171751 0.428038 -.008105 0.025951 .0213384 .0049024 .0144784 1.24637 3.49844 0.502955 2.56167 ADFNN O.A72877 -.171751 42.8005 6.11411 -.116187 -.005803 .0115369 .0232019 0.031556 2.46967 15.3585 -.078302 -0.68257 ADF -.872241 0.428038 6.11411 3.96972 -.075598 0.065666 .0646504 .0033574 .0326181 2.63377 7.64521 1.29719 7.1951 NEADF .0167337 -.008105 -.116187 -.075598 .0014398 -.001254 -.001234 -6.5E-05 -6 . 2E-04 -.050529 -.145996 -.024843 -.137675 P 0.0423:5 0.025951 -.005803 0.065666 -.001254 .0041675 .0035986 2.8E-04 .0016832 0.130188 0.321927 .0435603 0.395343 K -.003408 .0213384 .0115369 .0646504 -.001234 .0035986 .0063059 2.4E-04 .0016807 0.123678 0.400174 .0543467 0.388948 CA .0062349 .0049024 .0232019 .0033574 -6.5E-05 2.8E-04 2.4E-04 8.2E-04 3.6E-05 .0418251 0.104052 .0190746 0.108471 HC .0175412 .0144784 0.031556 .0326181 -6.2E-04 .0016832 .0016807 3.6E-05 .0011255 .0688251 0.154043 0.016588 0.18382 MN 2.65925 1.24637 2.46967 2.63377 -.050529 0.130188 0.123678 .0418251 .0688251 17.2818 26.2067 3.76218 28.706 FE -12.2209 3.49844 15.3585 7.64521 -.145996 0.32 1927 0.400174 0.104052 0.154043 26.2067 458.994 10.9684 87.8672 CU 0.552935 0.502955 -.078302 1.29719 -.024843 .0435603 .0543467 .0190746 0.016588 3.76218 10.9684 3.51652 11.6609 ZN 2.00596 2.56167 -0.68257 7.1951 -.137675 0.395343 0.388948 0.108471 0.18382 28.706 87.8672 11.6609 107.806 ro is TABLE 42 (continued) Dry ground ear corn DMCP ADFNN ADFNEADF PK CA HC MN FECU ZN DM 60.AI83 7.11515 -7.3*989 -12.372* 0.235542 0.274156 0.24:172 0.185304 .0608705 10.5*67 U . 3 5 1 6 1.72076 27.*07* CP 7.11515 6.92*9* -2.5***9 -3.63529 .0690077 0.163213 0.280655 0.168292 .0*23737 21.6317 59.2096 *.38603 3*.7639 ADPNN -7.5*989 -2.5***9 61.0*13 11.5*97 -.220837 .020747* -.141947 -.156413 0.101*13 -23.9163 -56.5261 -.526901 -8.88921 ADF -12.372* -3.63529 11.5*97 7.8*09* -.149463 -.120901 -.077102 -.103509 -.009155 -8.68333 2.1216* -.656863 18.575* NEADF 0.2355*2 .0690077 -.220837 -.1*9*63 .0028491 .0022982 .0014612 .0019638 1.7E-0* 0.16*778 -.0*228* 0.0125*9 -.35*912 P 0.27*156 0.163213 .0207*7* -.120901 .0022982 .0119823 .0088166 0.004948 .0027283 0.839712 3.65318 0.202585 1.68276 K 0.2*1172 0.280655 -.1*19*7 -.077102 .0014612 .0088166 .0175368 .0056578 .0023008 1.012*2 3.11918 0. 168966 i.*79 CA 0.18530* 0.168292 -.156*13 -.103509 .0019638 0.0049*8 .0056578 .0107737 .001*593 0.83573* 2.3878 0.2036*6 1.36923 HC .0608705 .0*23737 0.101*13 -.009155 1.7E-04 .0027283 .0023008 .0014593 .0012629 0.1*55** 0.396923 0.03*097 0.361855 HN 10.5*67 21.6317 -23.9163 -8.68333 0.164778 0.839712 1.01242 0.835734 0.1455** 196.7*9 557.55* 38.3805 327.389 PE M . 3 5 1 6 59.2096 -56.5261 2.12164 -.042284 3.65318 3.11918 2.3878 0.396923 557.55* 3012.9* 108.353 786.092 CU 1.72076 *.38603 -.526901 -.656863 0.012549 0.202585 0.168966 0.203646 0.034097 38.3805 108.353 15.9602 104.1*9 ZN 27.*07* 3*.7639 -8.88921 18.5754 -.354912 1.68276 1.479 1.36923 0.361855 327.389 786.092 104.1*9 1286.54 N) I—• Ln TABLE 42 (continued) High moisture ground ear corn DM CP ADPNN ADP NEADF PK CA MC MN PE CU ZN DM 15.587* 1.7017* 10.29*5 -.598201 .011010* 0.111781 -.00*062 .0021008 .0502591 *.*0801 -29.1999 2.19502 18.1199 CP 1.7017* 2.5*181 -0.89651 -.512109 .0097*68 .0571*16 .0111891 .0289*76 .0192*76 *.*7915 6.08579 2.17222 9.99175 ADPNN 10.29*5 -0.89551 17.871* 16.2591 -.109678 .0278221 0.107122 .01519*5 .028*111 7.11821 -27.5111 -.72*558 8.05767 ADP -.598201 -.512109 15.2591 17.1908 -.127*1* -*.8E-0* 0.115826 .0*70*7* .0111191 11.5811 -17.5211 -.180279 -1.758*7 NEADP .011010* .0097*68 -.109678 -.127*1* 0.006216 9.8E-06 -.002587 -9.0E-0* -2.1E-0* -.220221 0.115121 .0015908 .0116985 P 0.111781 .0571*16 .0278221 -*.8E-04 9.8E-06 .0019856 .002*56* .0011886 .0012295 0.1*11*9 0.*59521 .0858808 0.511526 K -.00*062 .0111891 0.107122 0.115826 -.002587 .002*56* .006*771 .001*8** 8.0E-0* 0.2*9271 0.51620* .08*1215 0.*116*5 CA .0021008 .0289*76 .01519*5 .0*70*7* -9.0E-0* .0011886 .001*8** .0027271 1.4E-0* 0.181521 0.510278 0.081788 0.111727 HC .0502591 .0192*75 .028*111 .0111191 -2.lE-0* .0012295 8.0E-0* l.*E-0* 7.9E-0* .0500705 .0*71229 .0201578 0.155007 MN *.*0801 *.*7915 7.11821 11.5811 -.220221 0. 1*11*9 0.2*9271 0.181621 .0500705 *7.1122 51.51*8 9.05587 *1.0955 PE -29.1999 6.08679 -27.5111 -17.5211 0.115121 0.*69521 0.51620* 0.510278 .0*71229 51.61*8 11 ,1.65 21.27** 11*.*7 CU 2.19502 2.17222 -.72*558 -.180279 .0015908 .0858808 .08*1215 0.081788 .0201578 9.05687 21.27** 11.1185 25.9758 ZN I N . 1199 9.99175 8.06767 -1.758*7 .0116985 0.511526 0.*11645 0.111727 0.155007 *1.0955 11*.*7 25.9758 157.71 ro ON TABLE 4 2 (continued) Barley grain CP ADFNRADF P K CAFAT SALTASH CP n.776 ion 0.448 -.018812 — 5.68 — 04 0 .0052961 0 0.020494 0 ADF 0.448 2.40877 -.101149 .0323481 0 -.005237 0 0 0 NKADF -.018812 -.101149 .0042475 -.001358 0 2.2F-04 0 0 0 P -5.6E-04 .0123481 -.001358 .0031885 0 — 2.6F —04 0 0 0 K 0 0 0 0 0 0 0 0 0 CA .0052961 -.005237 2.2F-04 -2. 6F.-04 0 .0034783 . 0 0 0 FAT 0 0 0 0 0 0 0 I) u SAtT U.020494 0 0 (1 (I 0 0 .0096368 0 ASM 0 0 0 0 0 0 0 0 0 (V) >—• TABLE 42 (continued) Brewers dried grains CP ADF NF.ADF p K CA FAT SALT ASH CP 6.74178 0 0 0 0 0 0 -.221831 -.619671 ADF 0 0 n n 0 0 0 0 0 NEAOF 0 0 0 0 0 0 0 0 P n n n 0 0 0 0 0 K 0 n n n 0 0 0 0 0 CA 0 0 0 0 0 n 0 0 FAT 0 n 0 n 0 n 0 n 0 AA1.T -.221831 n n n 0 0 f) .0621569 0 ASH -.619671 0 n n 0 n 0 0 0. 53982 3 N) 00 TABLE 42 (continued) Corn grain CP ADFNRADF P K CA FAT SALT ASH CP 0.334096 -.113556 .0047684 .0107122 0 .0039439 0 -.020133 0 ADF -.113556 1.324 11 -.055602 -.004712 0 .0018925 0 0 0 NEADF .0047684 -.055602 .0023348 2.0F.-04 0 -7.9F.-05 0 0 0 P .0107122 -.004712 2.0R-04 .0049333 0 6.9F.-04 0 .0042924 0 K 0 0 0 0 II 0 0 0 0 CA .0039439 .0018925 -7.9R-05 6.9R-04 0 .0021646 0 -.020214 0 FAT 0 0 0 0 0 0 0 0 0 SALT -.020133 0 0 .0042924 0 -.020214 0 .0301715 0 ASM 0 0 0 0 0 0 0 0 0 N) VO TABLE 42 (continued) Corn gluten feed CPADF NFADFP K CA FATSALT ASH CP 2.12«2 0.318509 -.013375 0 U 0 -.646384 -.022963 0.299764 ADF 0.318509 4.90326 -.205898 0 0 0 0. 52641 1 -.017284 0 NF.ADF -.013375 -.205898 .0086461 0 0 0 -.022105 7.3F.-04 0 P 0 0 0 0 0 0 0 0 K 0 0 0 0 0 0 0 0 0 CA 0 0 0 0 0 0 0 0 FAT -.84638 6 0. 5264 1 I -.022105 0 0 0 0.508748 -.038148 0 SALT -.022963 -.017284 7 . 3F-04 0 0 0 -.038148 .0112829 0 ASM 0.299766 0 0 0 0 0 0 0 0.318018 N 3 hO O TABLE kl (continued) Corn gluten meal CP AOF NF.AOF p K CA FAT SALTASH CP 20.1457 0 n 0 0 0 -.444954 -.104047 ADF 0 0 0 0 0 n 0 0 0 NEADF 0 0 n 0 0 0 0 P 0 0 0 0 0 n 0 0 K 0 0 0 0 n n 0 0 0 CA 0 0 n 0 n n n 0 FAT 0 0 0 0 0 0 0 0 0 SALT -.4 44 954 0 0 0 0 0 0 .0320734 0 ASH 104067 0 0 0 0 u 0 0 .0499792 ro to TABLE 42 (continued) Distillers dried grains CP AOF NF.AOF p K CA FATSALTASH CP 6.9702 -1.15386 .0484528 0 0 0 2. 1 7608 -.254037 .0817563 ADF -1.15386 8.40865 -.353096 0 0 0 6.05038 0 0 NKADF .0484 5 28 -.353096 .0148272 0 0 0 -;254068 0 0 P 0 0 0 0 0 0 0 0 K 0 0 0 0 0 0 0 0 0 CA 0 0 0 0 0 0 0 0 PAT 2.17608 6.05038 -.254068 0 0 0 3.7 1365 0 0 SALT -.254037 0 0 0 0 0 0 0.276662 0 ASH ,0817563 n n 0 0 0 0 0 0.610199 w NJ to TABLE 42 (continued) Feather meal CPADFNKADFPK CA FAT SALT ASH CP 3.65257 -.027604 .0061458 -.028801 0 — .078066 -.002601 -.180033 0 ADF -.027606 .0019469 -7 . 2F.-04 .0038347 0 .0085783 0 4. 3F.-04 0 NKADF .00614 58 - 7.2K-04 2.2F-04 -.002108 n -.001623 -I .8K-05 —6.6R— 04 0 P -.028801 .0038347 -.002108 .0277422 0 .0075705 1.8K-04 .0038763 0 K 0 0 0 0 0 0 0 0 0 CA -.078066 .0085783 -.001623 .0075705 0 .0206563 1.1F.-04 .0075559 0 FAT -.002601 0 -l.RK-05 1.8K-04 0 I . lK-04 9.6K-06 0 0 SALT -.180033 4.3F.-04 - 6 . 6 K - 0 4 .0038763 0 .0075559 0 .0263964 0 ASH 0 0 0 0 0 0 0 0 0 N) N) U) TABLE 42 (continued) Meat and bone meal CP AHF NF.Anp CA f a t SALT ARH CP 9.38B42 0 0.115547 -1.00 27 0 -2.18173 -.553301 -.129484 ADF 0 0 0 0 0 0 0 0 NFADF n. 1 1 5547 0 .0087547 -.082225 0 -.126233 .0885808 .0058084 P -1 .0027 0 -.082225 0.488853 0 0.834491 -.371448 -.023858 K 0 0 0 0 0 0 0 0 CA -2.18173 0 -.128233 0.834491 0 1.92422 -.822851 .0260065 FAT -.553301 0 .0895808 -.371448 0 -.822851 2.13454 0.29095 SALT -.128484 0 .0058084 -.023858 0 .0280065 0.28095 1.30187 ASH 0 0 0 0 0 0 0 0 to to is TABLE 4 2 (continued) Oat grain CP ADF MOAOF P K CA FAT SALT ASH CP 0. 97 1 184 -1 .44458 .0606607 0.028902 0 .0053684 0 0.011242 0 ADF -1.44458 9.6275 -0.40427 -.008056 0 0.056149 0 0 0 NEADF .0606607 -0.40427 .0169761 3.4F.-04 0 -.002358 0 0 0 P 0.028902 -.008056 3.4R-04 .0051377 0 .0021698 0 0 0 K 0 0 0 0 0 0 0 0 0 CA .0053684 0.056149 -.002358 .0021698 0 .0056697 0 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT 0.011242 0 0 0 0 0 0 .0012084 0 ASH 0 0 0 0 0 0 0 0 0 N> to Ln TABLE 4 2 (continued) Rapeseed meal (canola) CP Anp NEADF CA FAT SALT ASH CP 2.52796 -0.35039 .0 147 136 0 -.008659 0 -.370993 .0124988 ADF -0.35039 2.82968 -. 1 1 8824 0 .0253388 0 -. 171556 .0091032 NEADF .0147136 -.118824 .0049897 0 -.001064 0 0.007204 -3.8E-04 P 0 0 0 0 0 0 0 K -.008659 .0253388 -.001064 0 .0532972 0 -.060935 .0010942 CA 0 0 0 0 0 0 0 FAT -.370993 -.171556 0.007206 0 -.060935 0 1.67739 -0.0197 SALT ,0124988 .0081032 -3.8F-04 0 .0010942 0 -0.0197 .0153774 ASH 0 0 0 0 0 ro N) O' TABLE 4 2 (continued) Soybean meal CP ADF NKADFP V CA FAT SALT ASH CP 1 .921RR -.453706 0.019052 0 0 0 0 .0461 161 -0.00796 ADF -.453706 2.908R5 -.122 143 0 0 0 0 -.379161 n NKADF 0.019052 -.122143 .0051293 0 0 0 0 .0159217 0 P 0 0 0 0 0 0 0 0 K 0 0 0 0 0 0 0 0 0 CA 0 0 0 0 0 0- 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT .0461161 -.379161 .01592 17 0 0 0 0 .0143323 0 ASM -0.00796 0 0 0 0 0 0 0 0.233987 ro TABLE 42 (continued) Sunflower meal CPADF NFADF CA FAT SALT ASH CP 4.51993 -3.09573 0.129996 0 -.004177 0 -.3 52366 .0495005 ADF -3.09573 5.21073 -.218309 0 -.029598 0 0.217197 -.042488 NEADF 0.129996 -.218809 .0091882 0 .00124 29 0 -.009121 .0017842 P 0 0 0 0 0 0 0 K -.004177 -.029598 .0012429 0 .0938858 0 -.093331 -.012016 CA 0 0 0 0 0 0 0 FAT -.352366 0.217197 -.009121 0 -.093331 0 0.736163 .0262207 SALT .0495005 -.042488 .0017842 0 -.012016 0 .0262207 .0125373 ASH 0 0 0 0 0 0 0 0 N5 N 5 00 TABLE 4 2 (continued) Wheat grain CP ADF NEADF p K CA FAT SALT ARH CP 1 .21726 0.208109 -.008763 0.004156 0 .0200796 0 -.013629 0 ADF 0.208199 0.879826 -.036966 .0160752 0 .0060836 0 0 0 NEADF -.00876] -.036966 .0015516 - 6 . 8 E - 0 6 0 -2.6E-06 0 0 0 P 0.006156 .0160752 —6.88-06 .0035921 0 6.5F-06 0 0 0 K 0 0 0 0 0 0 0 0 0 CA .0200796 .0060836 -2.6E-06 6.5E-06 0 .0021521 0 0 0 FAT 0 0 ' 0 0 0 0 0 0 0 SALT -.013629 0 0 0 0 0 0 .001261 1 0 ASH 0 0 0 n 0 0 0 U (1 N> ro kO TABLE 4 2 (continued) Wheat shorts CP ADF NEADF P K CA FAT SALT ASH UP I.21078 -.322218 .0135306 0 .0113526 0 0 .0361835 0 ADF -.322218 2.77649 -.116506 0 .0168909 0 0 .0152017 0 NEADF .0135306 -.116506 .0048923 0 -7.lE-04 0 0 -6,4E-04 0 P 0 U 0 0 0 0 0 0 K .0113526 .0168909 -7.1E-04 0 .0596154 0 0 -.004536 0 CA 0 0 0 0 0 0 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT .0361835 .0152017 -6.4F.-04 0 -.004536 0 0 .0220127 0 ASH 0 0 0 0 0 0 0 0 0 N > LO o TABLE 43 Correlation coefficients between nutrient within feedstuffs. Legume hay - 1st cutting rr CP <1888 *08 M ADF r K CA MG MN FF (U 7N CM 1 «ccooo -C.1C7F9 -(.104 80 -0.27258 C .27231 "C .11894 -0.10714 C.00527 C.02238 O.CI 00 -0.06942 -0.15121 -0.04416 -0.04157 0.0024 0 .0024 o.o;% ; 0.1964 0.0045 0.4088 3* ? '"'SIS * 122 122 '"'SIS o-'SSS 348 352 357 CP -0* )C7PS 1.CCOOO -0.43993 -0.50953 0.50956 C.568C0 0.44757 (.41991 (.18357 -0.11690 O.G4I( (.0 0 0 0 0 .0 0 0 1 0 .0 001 0.28336 0.33144 0.14306 C.OOOl C.COCl O.COOl 0 .0 0 0 1 0.0005 0 .0 0 0 1 o.oogi 0.00 73 3?? 357 169 122 122 35? 353 353 353 352 351 ADFNK -O.ICAfO -C .43993 1.0 000 0 0.58713 -C.5P700 -0.75614 -0.75163 -(.710 67 -0.09438 -0.07823 -0.15879 -0.15871 (.0 0 0 1 0 .0 0 0 0 o.o ^g ! 0 .0 0 0 1 o.ooce 0 .0 0 1 0 0.0061 0.7736 U 9 169 0.0398 0.0405 ICl 168 168 168 168 168 168 167 *Df - o . ; 7 iî e -C .50953 0.5*713 1 .coooo -1 .COOOO -(.3 0 8 (7 -0.37364 -(.41764 -(.407 76 *0.00876 (j.cc; * 0 .0 001 0 .0 001 -0.03333 -0.76377 -0.19976 0 .0 0 0 0 0 .0001 0 . 0 0 (6 0.0003 0 .0 0 0 1 o .c g g j 0 . 9 ^ J 0 .7 J |6 «.0 0 |J 0.07 80 122 122 101 1 2 2 122 Î77 177 177 171 0.27231 (.50956 -0.58700 - 1 . COOOO 1 .COOOO 0.308(7 0.377,77 (.41779 (.40781 0.00866 0.03333 0.76337 0.19977 0.0124 0.0 001 0 .0 0 0 1 O.COOl C.OOOO C.C0 C6 0.C003 0 .0 0 0 1 0 .0 0 0 1 0.9746 101 0.7155 0.0034 0.0780 122 122 1 2 2 122 177 177 177 177 177 177 171 P -0.11P94 0 .56800 -0.25614 -0.30802 0.30802 1.00 000 0.63899 -0.01173 -0.07478 0.00775 0,14379 0,19575 O.O.M 0.0 001 0.0008 C.C006 C .0006 C.OOOO O.O^O, 0.8767 0.163J 0.8855 '6l3||8 6 . o o || Ô.OOOJ 353 168 1 2 2 122 353 353 348 K 0.44757 -0.25163 -0.32364 0.32372 0.63899 1 . 0 (0 0 0 -(.061 00 - C .1*820 -0.01767 0.16905 0.07497 0.09304 -"ô!gi43 0.0 001 0 .0 0 1 0 0.0003 I .0003 O.COOl 0 .0 0 0 0 :b3 0.7530 0.0029 0 .0 0 |5 0.6406 0.0817 353 168 122 122 353 353 353 357 351 CA O.CCf?7 ( .41991 -0.21067 -0.41764 0.41779 -0.06100 1.00000 •0.20284 0.18005 0.47510 0.16947 O.V^IA 0.0 001 0.0061 0 .0 0 0 1 C.CCOI 0.7530 0 .0 0 0 0 0 .0 0 0 1 0.0007 0 .0 0 0 1 0.0014 353 168 122 122 353 353 353 353 348 352 357 351 MG o .c z ia e 0.18352 -0.09438 -0.40276 0.40281 -0.07478 -0.15870 (.45718 0 .( 7t2 1.00000 0.06423 0.36401 0.19130 0.0005 c .c o o j O.COOl 0.1 6 ;7 0.0079 0 .0 0 0 1 0 .0 0 0 0 0.0 001 0.0003 :'t3 353 122 122 353 353 353 “' " 1 1 1 357 351 MN •O.tf'yA? -0.11690 0.10425 -C.C0876 0.00866 (.00775 -0.01767 -(.70789 C.1964 0.06473 1 .00000 0.15683 -0.09879 0.76335 0.0292 0.1787 C . « 3 J 0.9246 0.6855 0 . M 3 3 0 .0 0 0 1 0.7370 0 .0 0 0 0 0.0034 0.0657 o .o g o i 348 168 122 348 348 348 348 348 348 Ft (.28336 -0.07823 -0.C3333 0.03333 0.17530 0.16906 (.18005 (.07497 0.15683 1 .COOOO 0.10473 0.14570 0.004b 0.0 001 0.3135 0.7156 C.7155 C.OOlO 0.C015 0.0007 0.6417 0.0034 0 .0 0 0 0 0.0507 0.0067 352 168 122 122 357 352 357 357 348 357 357 Adt CU -0.04416 ( .33144 - C . 26327 0.26332 (.14379 0.C2497 0.36401 -0.09879 1 .0 0 0 0 0 0.39201 c .c c o i C.0034 0.0024 0.0069 0.6406 O.COOl 0.0657 °5:s;g; 0 .0 0 0 0 0 . 0 0 0 : 352 168 122 122 357 352 352 357 348 357 357 35: Zh (.14308 -0.15871 - C . 19976 0.19977 (.19575 0.09304 C.16942 0.19130 0.26335 0.14570 0.39701 1 .0 0 0 0 0 (.0073 0.0405 0.0280 0 .0280 0.0007 0.0014 O.COg3 0 .0 0 0 1 o .o o |i 0.0 001 0 .0 0 0 0 .1! 1 35 1 167 121 121 351 351 347 351 35: TABLE 43 (continued) Legume hay - 2nd cutting Ih CP ACFNN AOF ht ADF PK C« hG MN Ft CU ih [M I.CCCCO -0.1*874 0.C4746 -0.C4747 - C . 10076 -0.12998 (.01231 -C.01746 - 0 .0 1 1 2 0 -0.12213 -0.06153 -0.01692 c.ucoo O.GCOl 0.4477 C.0126 0.C013 0.7610 0.6665 0.7840 0.0026 0.1309 0.6778 # Iff 618 21: 258 6)3 609 6 l3 6 )2 601 605 604 605 CP - 0 . I f f 74 1.CCOOO -C .3 J ÎÎ0 -0.41252 0.41257 0 .50434 0.3)242 (.42014 €.23704 0.01485 0.22589 0.30861 0.04959 C.C(01 0 .0 0 0 0 O.OCOl 0 . 0 0 0 ) 0.0 001 O.OOCl O.COOl 0 .0 0 0 1 0 .0 0 0 1 0.7164 0 .0 0 0 1 0 .0 0 0 1 1 )P 618 218 258 258 613 609 613 612 601 605 604 ADfNh o . c m i -0.22550 1.CCOOO 0.48019 -0.48034 -0.C8030 -C .12871 0.07026 0.02*84 -0.07137 (.7U77 C.CCOI 0 .0 0 0 0 0 .0 001 C.ODCl *'cl2S o ’ ? 0.0217 0.2144 0.6107 0.2080 31* 3)8 318 22C 220 318 318 318 314 314 ADF O.C474b -0 .4 1252 0.48CI9 1 .1 000 0 - 1.00000 -0.32140 -0.22691 - t . 34199 -C .18400 -0.06818 -0.08029 -0.02321 C.44 76 0.0 001 O.COOl 0 .0000 C .0001 O.OOCl 0 .0 0 0 2 0 .0 0 0 1 O.C030 0.2004 • “ o .'S IIS 25C <20 256 256 2*8 258 258 258 256 255 hh ADF -D.C4747 C .41257 -0 .4 *0 3 * - 1 . COOOO 1 .0 000 0 0.32128 0.22*82 (.34207 (.10394 0.08010 0.13729 0.02304 0.44 77 0.0 001 O.COOl 0.0 001 C .0000 O.COOl 0 .0 0 0 2 0 .0 001 0.0030 0.2014 0.0284 i*-.e 256 220 256 258 258 258 258 258 256 256 255 K " C .10076 C .50434 -0 .1(4 72 -0.32140 0.32126 l.COOCO 0.5*968 -(.02462 0.19883 0.22633 0.15256 0.14821 C.CI76 0.0 001 0.0005 0 .0 0 0 1 0 .0 001 0 .0 0 0 0 O.COOl 0.5428 0 .0 001 0 .0 0 0 1 0 .0 0 0 2 0.0003 f 13 613 318 258 258 613 609 613 612 601 605 604 603 K - 0 . 1ZS96 C.21242 -0 .0(0 30 -0.22691 0.22682 0.569*8 1.CCOOO -(.20297 -C .28073 0.08939 0.22651 -0.00357 0 .00)3 O.COOl C.0002 C .0002 0 .00^1 0 .( 0 0 0 0 .0 0 0 1 O.COgl tC9 609 '-'SI: 258 256 *09 609 O 'O gSl « " 'a s ; CA 0.C K 31 C.42C14 -0.34199 0.742C7 -0.20297 1.00000 0.40652 0.12994 0.48192 0.15532 0.7( 1C C.OOOl 0 .0 001 C.OOOl O.COOl O.OOOQ 0 . 0 0 0 ) O.OOgl o .o o o i < 13 612 318 256 256 613 609 613 612 601 605 » ro w to TABLE 43 (continued) Legume hay - 3rd cutting I f CF »rt NN 407 K( >0’ 1 K CA KG HN 7t CU 2N CM 1 .tC(CO -C .2 ,297 C . C ’iSJ -0.(9419 C .09398 -C .23529 -0.19739 -(.0 2 494 -0.05112 -0.14205 -0.09387 -0.14672 -0.12213 O.C( 10 C.OOOl 0.6199 0.9704 0 .9 7|9 O.OOCl 0.0019 0.7031 0.0290 0.0188 0.0910 ; ' 7 257 134 112 256 296 256 256 2 96 CP -0..7297 ) •( cocc -0.3*640 -0.93632 0,5266? C .41712 0.29949 C.32601 0.24994 -0.13498 ( .(1 0 1 C.OOOO 0.000) 0.0001 0,0001 C.OOCl 0.0001 C.OOOl w s «ô!8S?3 0 .0 001 0.0314 ; ‘ 7 257 134 112 112 256 296 256 296 249 299 296 2 96 Aiir Nh 0.C3-S7 -C .25(40 1.00000 0.63947 -0 .(35 35 C.04056 -0.00699 -(.22694 -0.09198 0.25928 -0.01126 C.COCl 0.0000 0.0001 C .0001 0.74J7 0.9402 0.0084 0.2905 0.0026 0.8977 -°ôî§lî5 1 24 134 93 53 134 134 134 133 133 134 134 /OF -0.(5419 -C .53632 0.63547 1 .00000 -1 .00(00 C .14964 0.09337 (.40146 -C .20492 0.13441 •0.10287 -0.01707 0.20968 (.5 7 (4 C.COCl O.COOl 0.0000 O.OOOI 0.1148 0.9 76| o.o^^l 0 .0 3 0 | 0.1J96 0.28J6 0.8982 0.0J96 n ; 112 93 112 112 NEfUF o . t ‘ r9e C .52663 - 1 . COOOO 1 .COOOO -0.09339 (.48110 C.2C500 -0.20552 0.5 719 C.OOOl 0.0001 C .0000 0.9764 C.OOOl O.C301 °ôîl99B v i i n 0.0297 ) Ic 112 93 112 112 112 112 112 112 111 117 112 r -O.Z3!29 C.41712 0.04C56 0.14984 -0.14959 1 .COOCO 0.63782 -(.138 69 C .11625 0 .2 6 2 1 6 0.18003 0.23566 0.03041 O.OCOl C.OOOl 0.64,7 0.1148 C.1154 CUOOCO O.COOl 0.0265 0.0633 0 .0 0 0 1 0.0039 0 .0 0 0 1 0.6262 ; ‘t 256 112 256 296 256 256 249 299 256 256 K -0.1^719 (.25945 0.09337 -0.C5235 0 .0 7 0 2 1 .CCOOO -(.279 45 -C. 23186 0.29747 0.16906 0.06572 C.0015 C.OOOl 0.9763 0.5764 C.OOOl C.OOOO 0 .0 0 0 1 0.C002 0*0 jo| O.OJ6§ ”ô?llîî i •( 256 112 112 256 296 256 256 *''342 256 CA -0.12394 C .22603 -0.48146 0.401/0 -0.13669 -0.27945 1.00000 (.32062 -0.20291 0.19989 0.30091 -0.09302 0.7(31 c.cooS 0.0 001 C.OOOl 0.0265 0 .0 0 0 1 C.OOOO O.COOl 0 .0 0 Ï3 0.0106 0 .0 0 0 1 0.3982 i5( 256 134 112 112 2 56 296 256 256 299 256 296 KG -0.05112 C.20MC - C .09198 -0.20492 C .20500 C .11625 -0.23188 (.32062 l.OCOOO 0.04699 0.39546 0.22043 (.4154 C.C012 0.2909 0.0309 C.OJOl 0.0633 0 .0 0 0 2 O.OOOI o.co^o V è ! î l 0.4990 0 .0 0 0 1 0 . 0004 . 50 256 134 112 112 256 256 256 249 256 296 KN -0.142C5 ( .0 7 5 6 0.2*928 0.13441 -0 .1 3 '3 5 C.26216 0.29747 -C .20291 1.0 000 0 0.39042 0.02 50 0.0026 C.,996 0.I59P C.OOCl 0 .0 0 0 1 0.0013 ^olc'^gl 0 .0 0 0 0 °ÔÎ888Î 0 .0 0 0 1 2*9 c.'SiS 111 249 249 249 249 249 248 249 249 ft -0 .(5 3 *7 0.102(4 C.180C3 0.1(906 (.15985 (.04699 0.26002 1.0 0 0 0 0 0.23809 0.39790 (c!%3?6 -'cfsasz 0.2633 0.0039 0.0068 0.0106 0.4550 O.OOOI 0 .0 0 0 0 O.OOOI O.OOOI iii 133 111 255 299 255 255 248 299 299 CU -0.14( 72 C -C . 16344 -0.01707 ( .23566 0.08972 (.30091 C .39548 -0.01729 0.23809 1.00000 0.39160 1 .CJ(f C.CCOI 0.0992 0.8982 °C?A{|| O.OOCl 0.0 0 0 1 O.COOl 0.7869 o .o o o i 0 .0 0 0 0 0 . 0^01 . ?( 2 i t 112 256 ° '’Ul 256 256 249 299 296 7N - C .12213 -C .J3.ÎC 0.09439 0.20968 -C.2C552 C .03041 0.03344 -(.05302 C .22043 0.39042 0.39790 0.39160 1.0 0 0 0 0 0.C5 1Ü 0.9329 0.0296 C .0297 0 .7 2 » (.3982 0.C004 0 .0 0 0 1 0 .0 0 0 1 0.0 0 0 1 0 .0 0 0 0 « • t 112 112 256 256 249 299 296 296 ro w w TABLE 43 (continued) L e g u m e g r a s s h a y 1st cuttini 17 Cl 4C16M * 0 F MAt't 1’ ft, Mh 88 CU 7N Ch 1 .ICCOÜ -C .C4726 - c . 18446 -C.C9359 0.C936C -C.CP596 -0.0613? - c . 07897 -C.1C553 -0.07909 -0.09890 -0.09052 -0.19407 t.ccco C.C91P O.COOl 0.0350 0.0349 0 .( 0 2 2 0.C295 0.(1049 O.C002 1< 74 1274 683 508 0.Ç05J 0.0005 0 .0 0 0 1 1265 1?98 1265 1263 1247 *’*1248 tp -0.14726 l.CCOOC -0.45531 -0.19873 0. 196 n C .57527 0.446S6 ( .64728 (.47497 -0.21643 0.17751 0.50970 0.15844 ( .0917 C.0000 C.OOOl 0 .0 0 0 1 0.0001 O.OOCl O.COOl 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 i; 74 1274 683 50f see 0 . 0 0 0 ^ 0 .0 001 0 .0 0 0 1 12(5 libfi 1265 1263 1241 1248 1249 *UfNt -0.1* 446 - t . 45531 l.CCCOO 0.21522 -C .3IS 2! -C .30865 -0.24355 -(.19095 -(.104 29 -0.10349 (.0 0 0 1 C.COOl O.CCOO C.OOOl C.OOOl O.OOCl O.COOl o.o ooi <73 683 663 452 4S2 0.0066 0.0073 °6°SiiS 679 670 679 678 671 6 72 673 t C f -0.C93S9 - C . 15873 0.31522 1.COCOC -1 .coooo -0.0(471 -0.01766 -C .15229 - C .12453 C .01370 -0.11061 -0.06955 O.C3'C C.COOl O.COOl 0 .0 0 0 0 c .0 0 0 ) 0 .0 5(6 0.6919 C.0006 0.C050 « et 508 452 o . î j o i 0 .1 2 0 0 508 sue 5(7 506 507 506 501 M IDF 0.093(0 0.19872 -0.31521 -1 .COOOO .coooo 1 €.0(472 0.01765 (.15229 C .12454 -0.04540 -0.01367 0.11062 0.06952 0.C349 C.OOOl O.COOl 0 .0 0 0 1 0 .0000 0.0 5(6 0.6920 C.0006 0.0050 rcf 508 452 508 0.7607 0.0133 0 .1 2 0 2 SOf* 5(7 506 507 506 0-3UI 499 500 501 P -O.CC*96 0.57537 -C.2C885 -0.C8471 0.Cfî472 1 .COOCO 0.68351 (.21736 C.24226 -0.00969 0.16524 0.26924 0.21381 C..0C'22 C.COOl O.COOl 0.0566 C.0S66 o.ooco O.COOl 0 .0 0 0 1 0 .0 0 0 1 12(6 1265 6 79 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 507 so? 12(5 1258 1265 . 126 3 1247 1248 1249 K 0.44656 - C .24255 -0.C1766 0 .r i7 (S €.(8351 1 .CCOOO (.07532 C.01297 0.12466 C.OOOl C.CCOI 0.6919 0 .6920 C.OOCl o.cooo 0.0075 l i i t 1258 6 78 0.6457 -°g?93S; °g!&2&? 0 .0 0 0 1 506 SC( 1256 1258 1258 1258 1241 1247 12*6 1249 CA -0 .6 7>97 C.64728 - C . 19095 -0.15229 0. 15229 €.21736 0 .07532 1.00000 0.59869 -0.31200 0.16056 0.65238 0.18048 C.C049 C.OOCl O.COOl 0.0006 C .0006 O.OOCl 0.C075 K 61 C.OOOO 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 1265 6 79 50 7 sc? 12(S 1258 1265 1263 124 1 1247 1248 PC -0.1C553 0.4 7497 -0.10429 -0.12452 0. 124S4 €.24226 0.01297 (.59869 l.OCOOO 0.00401 0.13099 0.54449 0.30603 (.0 0 0 2 0 .0 0 0 1 O.OC66 0.0050 C .0050 O.OOCl 0 .0 001 1263 Ï262 o.cooo 0.8878 O.OOOI 0 .0 0 0 1 O.OOOI 6 78 506 S06 12(3 '’ *1258 1263 1263 1241 1247 1248 1249 KN -0.(7509 -C .21643 0 .2 1 2 1 0 0.C4539 -0.C4S40 -€ .009 (9 -0.0C754 (.31200 C.00401 1.0 000 0 0.11218 -0.16399 C.C053 C.COOl O.COOl 0.3115 0.17028 1: *1 0 .0 001 0.8878 0 .0 0 0 0 O.OOOI 1241 « 70 499 1241 1241 1241 l238 °*?SS4 ft -0.(91 90 0 .17751 0.C1370 -0 .013(7 C .16524 c . 16056 C .13099 0.11218 1 .COOOO 0.14334 0.18361 C.OOOl 0 .0001 0.7602 C.7607 o.çoci C.OOOl 1.47 °g!A3&; 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 0 O.ÇOOl 0 .0 0 0 1 1247 499 499 I2%7 1247 1247 1238 1247 1246 CU -0.(5052 C .50970 -0.11061 G. 110(2 O . U 3 I 7 (.652 38 (.54449 -0.16399 0.14334 1 .0 000 0 0.396 28 t .00 1 4 C.0001 0.0)33 C.0133 (g?S3li O.COOl C.OOOl 0 .0 0 0 1 12 48 1248 o .o o o i 0 .0 0 0 1 0 .0 0 0 0 0 .0 0 0 1 SOC soo 1248 1246 1248 1239 1245 1248 1247 2 H -0.15407 C.15844 -C.C6955 O.C(9S2 c.21381 0.12466 (.18048 C.COOl C.30603 0.17028 0.18361 0.39628 1 .0 0 0 0 0 C.OOOl Ce 1200 C.12C2 O.COOl 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 0 1.’49 1249 " " l i s SOI SCI 1249 1249 1249 1240 1246 1247 1249 LO TABLE 43 (continued) Legume-grass hay - 2nd cutting Cl ACFNN Aor hi ACf 1 K LA MG M6 f t CU If* rn 1 .tocco -0 .12701 -0.12664 - 0 .0 122 2 0.01/23 -C .05625 -0.07410 -C.088 79 -C .04906 C.05638 < .Cl 00 C.OOOl C.CC03 0.7632 -0.08712 -0.11917 -0.11754 C.7631 C.U278 0.(038 0.0005 0.0283 0.0007 O.OOOI 0 .0 0 0 1 15Jt 795 61C ( 10 1520 1522 1530 1513 1516 1518 1518 cr -C.)Z7Cf 1 .CCOOO -C.35C60 -0.15139 0.15148 C .35181 0.3C826 (.4 7242 (.23448 -0.13696 0.14646 0.31288 O.GCOl O.COOO O.COOl C.0002 ( .0002 O.OOCl 0 .(0 0 1 0.0 0 0 1 -0.03853 !• 3( 0 .0 0 0 1 0 .0 0 0 1 0.0 001 153(. 795 610 6 )0 1530 1530 1513 1516 1518 AOFHK -0.121(4 -C .35060 1,00000 0.25348 -0.25353 - C .13332 -0.1547» -(,10271 0.10491 •0.01880 0.17025 o.cco? C.OOOl 0 .0 0 0 0 0.0 0 0 1 C.OOOl 0.00C2 O.COOl 0 .0 0 38 ’ ^0^6929 0.0032 7 " 795 795 543 54 3 0.6016 0 .0 0 0 1 753 791 793 792 788 790 790 AUf - 0 . 1 1 /2 2 - C .15139 C .25248 1 .(0 0 0 0 - 1 .00 000 -C .10749 -0.06644 -C .10810 -0.02199 -0.03825 0.7C22 C.0002 O.OOOI 0 .0 0 0 0 0 .0001 0 .0 0 8 0 0 .1 0 2 0 0.0077 t 10 -=a!3S2; ®ÔÎ670Î 0.5888 0.3469 610 543 610 610 6C7 607 607 607 606 607 607 607 r,t)2 2 3 C.1514E - C .25353 - 1 .(0 0 0 0 1 .COOOO 0.10755 0.0(647 C.10808 C.14253 0.02556 -0.01730 0.03825 0.7< 31 C.C002 O.OCOl 0 .0 0 0 1 C.OOOO 0.0080 0.1018 0.0077 0.0004 ( IC (1C 0.5299 0.6705 VAÏÎ] 0.3468 543 61C t 10 6 C 7 607 607 60 7 606 607 6C7 607 F -O .C 5(25 C .35181 - C . 12332 -0.10749 0.10755 l.COOCO 0.55061 (.05767 (.18154 0.19362 C .16906 0.15191 0.16889 0.02 78 C.OOOl 0.0C02 0.0080 0.0080 O.COCO O.COOl 0.0241 0 .0 0 0 1 0 .0 0 0 1 0 .Ç001 0 .0 0 0 1 O.ÇOJ1 153C I53C 793 607 tC7 1520 1522 1530 1529 1513 15l8 K -0.C74 10 C .20826 - C . 19477 -0.06644 0.06647 C.59081 1.CCOOO -C .18891 -C .17757 0.12786 0.11096 0.00692 C .00 38 C.OOOl O.OCOl C.1920 C.IOIP O.OOCl 0 .(0 0 0 0 .0 0 0 1 0 .0 0 0 1 0 . 0001 0 .0 0 0 1 0.7877 1522 1522 791 607 (07 1522 1522 1522 1522 1513 1516 1518 1518 CA -0.088 79 (.47242 -C .IC 2 7I -0.10810 0.10808 -C .05787 -0.18891 1.00000 (.42209 -0 .2)9 73 0.13099 0.46185 0.08628 0 .000» C.OOOl 0.C03B C .0077 O.ÇOOJ 0 .0 0 0 0 0 .,JÇÇ, 1! 3C 1530 793 607 1530 " • m l "•?%?! "'Mis KG -O.CA'CC C .23448 -C.CC479 -0.14252 0.14253 C .ll- l! * -0.17757 (.42209 1.00000 0.15552 0.09125 0.37782 0.25225 0.0». 51 C.OOOl 0.8929 0.0004 0 .0004 O.OOCl 0 .0 0 0 1 C.OOOl 0 .0 0 0 0 0 .0 0 0 1 0.0004 0.0 001 1* 2*; 1529 792 607 CO? liJ 9 1522 1529 1529 1513 1516 1^18 ° * m i KN G.C5I 38 -C .13(96 C.1C491 -0.(2562 0.C2556 C .193(2 0. 12786 -(.2 3 9 73 C.1Î552 1 .00 000 0.13233 -0.07388 C .0283 ( .0001 C.0C32 C.5290 C.5299 0 ooci O.COOl 0 .0 0 0 1 O.^OJl 0 .0 0 0 0 0 .0 0 0 1 0.0040 °ô!286l J»13 1513 788 606 6C6 lb l3 1513 1513 1513 1510 IM 2 1512 FE -0,08712 C .14646 0.03513 0.C1733 -0.01730 C .109(6 0.11096 (.13099 (.09125 0.13233 1.0 000 0 0.1714? 0.0007 O.COOl C.6701 0.C705 O.OOCl O.OOOI 0 .00 00 "o!&88? 1» l( 1516 °*"789 607 6C 7 1616 1516 1516 1515 *1515 CU -0,1 IM 7 (.31288 -C.CI860 -0.02199 0.02199 C .lM S l 0.CC692 (.46185 -0.07388 0.11060 1.0 000 0 0.41472 O.OCOl C.CCOI 0.6016 0.5866 0.5887 O.OOCl 0.1877 0 ,0 0 0 1 V . U î \ 0.0040 0 .0 0 0 1 0 .0 0 0 0 O.OOOJ It 18 1518 790 607 C07 1516 1518 1516 1518 1512 1515 1518 7h -0,11754 -C .03852 C.17C25 0.C3825 C .16889 0.0)167 C .08628 €.25225 0.19011 0.17149 0.41472 1.0 000 0 0 .0 0 0 1 C.I335 O.CCO] -"ôÇSlZS C.346fl C.OOCl 0.0006 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 O.OOOI 0 .0 0 0 0 1*18 1518 790 607 6C 7 1518 1516 1518 1512 1515 1517 1518 N3 W Ui TABLE 43 (continued) Legume-grass hay - 3rd cutting ( t cr «CFNN AOF HE ACF p K CA PC HP FE (0 2N TH 1 •<0 f 00 - c .f A z r t - C . 12105 -0 .(0128 C.C0118 -C .101(3 -O .lC llO -C.0 7D23 -C .04597 0.05255 - 0 .1 0 1 1 0 -0.1*968 -0.10610 0*C( CO C.COOl 0.1261 0.9893 C.9902 0.0728 0.(736 c . n * s 0.3228 0.3565 0.0736 0.0079 0.060* 314 161 111 M l 314 314 310 31* 31* 31* CP -C.;42P5 1.CCOOO -€ .*6 3 *5 0.56*69 0.46489 (.3 9 0 (3 0.*25*5 (.46020 C .19646 -0.06484 0.13105 0.3980* -0.08633 (.coo; C.OOOO C.OOOl 0 .0 0 0 1 O.COOl O.OOCl 0 .(0 0 1 O.OOOI O.C004 0.2550 0 .0 2 0 2 0.0 001 0.1269 / I t 314 1(1 111 111 314 31* 314 314 310 31* 31* 31* fUfNh - 0 . IJICS -C .46344 1 .COOOO 0.65*37 -0.(4426 -C .25127 -0.20818 -C .21977 - C . 17105 -0.07100 0.03705 0 • I <' 61 C.COOl c.ocoo o.oooi C.OOOl C.0014 0.C0R2 0.0052 0.0306 **0?7333 0.6*18 If 1 161 161 105 104 1(0 16( 160 160 159 160 160 ACf -o.coi?e - c . 5(469 (.65*31 1 .(0 0 0 0 -1 .(CCOO -C .13324 -0 .0 5 7 ** -{.50421 -C .42513 0.08767 -0.02783 -0.0*780 0.05*99 0.9(93 C.OOOl O.OCOl 0 .0 0 0 0 0 .0 0 0 1 0.1649 0 .0 0 0 1 0 .0 0 0 1 0.3647 0.6200 0.5683 M l M l 105 111 M l l i e 110 110 109 "''1(2 110 110 HE ADI 0 . CCI I t (.5 (4 8 9 -C .65*36 - 1 .(0 0 0 0 1 .00000 C .13324 0.05737 (.50451 C.42503 -0.08775 0.02771 0.0*781 -0.05510 C.99C2 C.OOOl O.COOl O.O^Oj C .0000 0.5516 0 .0 0 0 1 O.COOl 0.3642 M l 111 105 111 '"'tig lie 1 10 lie 109 *'*112 P -0.10103 C .39062 -C .25127 -0.13335 0.13324 I«0 0 0 (0 0.65*32 -C .12040 C.20448 0.16329 0.10250 0.2880* 0.13212 O.C73t C.COOl O.CCl* 0.16*9 Q.OOCO O.COOl 0.0329 0.0003 0.0039 0.0697 0 .0 0 0 1 0.0192 /M 314 160 110 314 31* 314 314 310 31* 31* ¥ - 0 .1 0 1 1 0 C .42445 -0 .2(6 18 -C.C57** C .04737 C.64422 1 . 0 (0 0 0 -(.20845 -C .12419 0.05798 0.191*6 -0.02168 0.0736 C.OOOl 0.0082 0.4416 0 .0 0 ^, o.cggg 0 .0 0 0 2 O.C276 0.3089 0.0006 0.7019 : 14 314 160 MO 314 314 310 31* 31* 31* CA -0.C7023 -C .21577 t . 40441 - 0 .2(8*5 1.00000 C .31420 0.08850 0 .* 0 2 *0 -0.008 29 0.Z146 0.0052 -*ô!833l 0 .0 0 0 1 0.CC02 O.OOOQ 0 .0 0 0 1 0.1J76 o.ogoi 0.8837 314 314 160 110 110 314 31* 314 314 31* PC -0.Ct*>S7 (.19846 -0.17105 -0.42513 0.424C3 C.20448 -0.12*19 (.31420 l.OCOOO 0.19901 0.06*17 0.30*39 0.36505 0.3; C.C004 0.C306 O.OOOI C.OOOl O.OOC3 0.C278 O.OOOI o.cooo 0 .0 0 0 * 0.2569 0 .0 0 0 1 O.OOOI 314 314 160 1 10 MC 314 31* 314 314 310 31* 31* 31* PH o.ct;55 -C.0(484 C.0272* C.C8767 -0.CB774 C.16329 0.05798 -(.11136 (.19901 1 .00000 0.11*61 -0.07189 0.29060 C-».33 0.3647 0.3642 0 .0 0 3 9 0.3089 0.0501 0.C004 0 .0 0 0 0 0.2069 0 .0 0 0 1 1 09 109 3 I0 310 310 310 310 310 ft - C .l( 110 C .13105 0.02771 C .10240 (.08850 C.06417 1.coooo 0.22968 0.21*83 ( .C736 (. 0 /0 2 -:(Ç3)2S 0.7738 0.0697 0.1176 0.2569 o.ogoo 0 .0 0 0 1 0 .0 0 0 1 314 3M 160 11C n o 314 214 314 314 310 31* 31* CU -0.14968 C.3980* -0 .121*3 -0.04780 0.04781 C.2P6C4 0.19148 (.40240 (.30439 -0.07189 0.22968 1.0 000 0 0.33766 ( .C079 C.OOOl 0.1261 C.620C C .6199 O.OOCl 0.C006 0 .0 0 0 1 O.COOl 0.2069 0 .0 0 0 1 0 .0 0 0 0 0 .0 0 0 1 ; 14 31* 160 110 lie 314 314 314 314 310 31* 31* 31* /H -C. 1 0 (10 -t.C tt3 3 C.037(5 C.C5499 -C.C451C C .13212 -0.02168 -C .00829 C .38505 0.29060 0.21*83 0.33766 1 .0 000 0 (.((04 C. 1209 0.6*18 C .5683 0X 674 0.0142 0.7019 0.8637 O.COOl O.OOOI 0 .0 0 0 1 0.0 001 0 .0 0 0 0 314 31* 160 lie n o 314 314 314 314 310 31* 31* 31* N3 u> ON TABLE A3 (continued) Grass hay - 1st cutting rr c r ATPhN *Df M/t.F FK t* MO HK Ft CU 2 h Lr- ] .toooc -C.I2C49 -0.17113 C .12186 -C.11792 -0.07014 -0.01758 -0.10597 -0.04993 -0.22919 -0 .05990 -0.06935 r.c u c i 0.0561 1.0871 C .0628 0.01C3 0.1286 0.7857 0.0726 0.2868 0 .0 0 0 1 0.1948 •.7 7 477 252 701 2 d 472 471 5 77 467 4 70 4 70 *"*4 70 CP -C. ICt< 3 1 .CCOOO -C .29437 -0,156 35 0.29643 C.(3528 0.99129 (.5 (1 1 3 -0.02764 0.14914 0.61408 0.35440 o .l;» 0 C.OOOO O.OCOl C.OOOl O.OOOI O.OOCl O.COOl 'c f 3 3 g ; O.COOl 0.6670 0 .0 0 1 2 0.0 0 0 1 O.OOOI «, n 477 252 701 2 d 4 12 471 517 5 77 467 470 470 4 70 *CfNA -0.1:049 -C .25437 1 .COCOO 0.(5785 -0.(9774 -C .24699 -C .17877 0.03505 -(.117 07 -0.03092 -0.05609 -0.01050 C.CÎJ3 C.COOl c.cooo o.oooi C.OOOl O.OOCl 0.5806 0.0763 0.6316 VAÎII 0.3762 0.0605 252 252 163 163 2 t l 251 243 251 261 251 4Df -C .25639 C.69784 1.00000 - I .CCOOO -C .26923 -0.11856 -0.00795 - C . 13679 -0.12546 0.03299 -0.00404 -0.10504 0 .0 1 :1 C.OOOl C.CCOI 0 .0 0 0 0 C.OOCl C.OOCl 0.0105 0.9667 0.Q 50| 0 .6 |» 7 « f V 207 183 701 2 d 2C‘ 705 705 MADP o .ijc e c C.25643 -C .69774 - 1.00000 1 .COOOO C.2692* 0.11856 0 .00306 0.13679 -0.03799 0.08505 0.10618 I'.cr :e C.OOOl C .0001 O.COOl C .0000 O.OOCl 0.0105 0.9653 0.0505 0.6386 /C I 2C7 183 701 2C7 2C9 705 205 705 702 705 f - C .11792 C.63528 -0.24699 -0.76573 0.26929 1 .rooco 0.13757 C.13771 C.78787 0.07900 0.15163 0.36 700 0.37919 C.0103 C.OOOl C.OOOl 0.0 0 0 1 0.0 0 0 1 C.OOOO O.COOl 0.0078 o . o o o i 0.5365 O . O O ^ J 0 .0 0 0 1 0.0 0 0 1 4 7: 472 251 705 209 4 72 571 5 17 577 510 570 K -0.07014 -O .I785( 0.17856 C .737*2 l.CCCOO C.13255 C.26535 0.38785 0.77570 O.lîOt^ 0.0105 C.0104 O.OOCl O.CCOO 0.0050 0 ,0 0 0 1 V . V o t l 0 .0 0 0 1 0 .0 0 0 1 4 71 471 251 705 205 471 571 571 5 11 457 470 570 5 70 CA -0 .01:5 8 C.54332 C.C350. -0.00755 0.00306 C .137:1 0.13755 1 .00000 0.5(681 0.60373 0.20087 C.COOl 0.5t06 0.5661 0.0028 0.0050 0 .0 0 0 0 O.COOl *‘’6Î S Î | Î 0 .0 0 0 1 O.OOOI 472 251 705 4 72 571 5 77 572 467 4 70 510 5 70 r t -0.IC4S7 C .46173 - C .IU O l -0.13675 0.13615 0.28787 0.7(535 C.56681 1.0 0 0 0 0 0.00616 0.58978 0.26561 c .o: : 6 C.OOOl C.0763 0.0505 0.0505 O.OOCl O.COOl 0 .0 0 0 1 0 .0 0 0 0 0.0597 O.OOCl o . o o o i 4 7: 472 <51 705 705 4 72 512 577 457 470 570 5 70 -0.14993 -0.03057 0.17555 C.029(0 -(.11367 (.0 (8 1 6 1 .0 0 0 0 0 0.17375 0 . : f 68 ^C°557C 0 .0 1 5 0.0753 ° ô î o * i n 0.0151 0.0597 0 .0 0 0 0 * ô ? 5 î | l 0 .0 0 0 2 457 457 2H3 707 707 457 557 557 457 457 557 M - o . : : 5 i9 1 . 14914 C.06077 0.03755 -0.03755 C.O 163? 1.00000 0.17875 O.CCO) 0 .0 0 1 2 0.3380 0.6387 0.6386 ° ô ! 2 3 S ; 0.0981 ' ô l ? P o l 0 .0 0 0 0 * o ! g 3 i ? 0 . 00^1 4 7C 470 751 705 705 4 70 5 70 5 70 470 457 470 4 70 CU -0.1599C C.61408 -0 .05(05 -0.08585 0.08505 C.362C0 0.3(785 C.60373 €.46976 0.04683 0.12319 1.0 0 0 0 0 0.55553 (,1 9 4 t C .0001 0.3767 O.OOC 1 C.CCOI o.o o o i 0 .0 0 0 1 0.0075 O.OOCO 0 .0 0 0 1 4 70 47C ” ■^705 4 70 570 5 10 470 O':;;? 4 70 470 5 70 7N -0.0(535 C .35440 -0.01050 - C . 10684 C.32919 0.71570 C.70087 C.26461 0.17374 C .17024 0.45543 1.0 0 0 0 0 0.1223 ( .COOl 0.8(85 C.I31C C.COCl C.CCOI 0 .0 0 0 1 O.COOl 0 .0 0 0 2 O.OOOI O.OOOI o.oogo • 71 47C i5 l 209 2C5 4 71 570 5 70 470 467 470 470 570 ro w •vj TABLE 4 3 (continued) Grass hay 2nd cutting 1 f cr «CfKN ADF KF ADF PK CA MG MK fE CU Itt I f 1 .ICCOG -C.39C45 -0 .14231 ^ . 1 4236 -C.C574I -0 .1 4 7 3 5 C.02262 - C .01905 -0 .1 6 9 9 6 0 .06668 0.07606 •0.02761 O .C loc c i f l î ? C .C O ll C .3454 ( .3453 0.54C5 0 . 1145 C.80V5 0.8392 0.0745 0 .3 9 60 0.9069 0.7696 1 \f IK! 61 46 46 l i e 116 116 l i t 111 IK 119 119 CP Ü.1/3P4 .CCOOO - C . 43171 -0 .4 4 7 1 5 0.44731 C.507C4 0 .4 (4 1 0 (.5 3 5 66 (.5 6 9 76 0.19726 0.69619 0.23779 C'. H 11 C.COOO C.0003 o .o o ie c .o u ie C.OOCl O.COOl O.OOOI O.COOl 0 .0 3 96 0.0001 0.0109 J 11 IIP 61 46 46 l i t 116 116 116 IK 119 119 - .43171 C .f686e •C .86863 - C .23368 -(.0 3 4 3 3 -C.1C254 •0.00661 •0 .2 1 11 0 •0 .1 (8 9 9 O .Cl 11 C.C003 'c'SSSS 0.0001 C.COOl O.C57G - * a ! 3 ; 3 3 0 .7 8 2 ^ 0 .4 0 9 ^ 0 .9 (8 0 0 .0889 e 7 67 44 44 67 67 66 ADF -0.14,31 - .44715 C.86B68 1 .(0 0 0 0 - 1 .COOOO -0 .2 3 65 4 -0 .2 3 37 1 C.0U122 -0 .1 2 4 9 8 •0 .0 7 0 2 9 0.11933 '0 .2 9 6 9 9 •0.1 0 86 6 0.2454 c .o o ie O.COOl 0.0000 C.COOl 0.1135 C.1160 0.9936 0.4079 0 .6 (9 9 0 .9 (0 6 0 .Î0 2 7 0.9766 4( 46 44 46 46 46 46 46 46 93 99 9( 99 NE ADF 0.14236 .44731 •0 .6 (8 6 3 rl .(OOOC 1 .COCOO 0.23698 0.23400 -C.O O lO l C .12521 0.07021 •0 .1 1 9 9 7 0 .2 9 6 (9 0.10893 0.24&3 C.0016 C.CCOI O.OOOQ r .0000 0.112B O .II7 5 C.9947 0 .4 0 70 0 .6 (9 6 0 .9 9 2 0 0.1026 0 .9 7 6 | 46 46 44 46 46 46 46 93 F -0.15741 .50704 - C . 23368 0.23698 1.0 0 0(0 0.62666 (.2 5 0 33 C .44021 0.00886 0 .3 2 8 (8 0.36216 0.5405 C.CCOI 0 .0 (7 0 0.1128 0.0000 O.COOl 0.0067 O.COOl 0 .9 2 6 ( 0.0003 O.OCOl 111 116 67 46 46 1 16 116 1 16 116 111 119 119 K -0 .1 4 73 5 .43410 •0 .1 4 2 7 9 - C . 23371 0.23400 0.62644 l.OCOOO (.1 1 9 2 6 C .25309 0.06327 0.97008 0.29990 0.1145 C.OOOl 0.2490 C .I18C C .1 1Î5 O.OOCl c .c o o o 0.2023 0.0061 0 .(0 9 9 VMU o .o o g ^ 0 .0 0 8 | l i e l i e 67 46 46 1 16 116 116 116 111 119 CA 0.12262 .53569 •0 .0 3 4 3 3 0.G0122 -O .CO lO l C .25033 0.11926 1.00000 C.52558 •0 .1 6 61 1 0 .29293 0.96933 O.P(;95 C.COOl 0.7627 0.9936 0.9947 0 .0067 C.2023 0.0000 0.0001 0 .0819 0 .0 0 69 0.0001 116 l i e 46 46 l i t 116 1 16 116 111 119 119 119 HG - C .11505 .50576 0.12521 C.44021 0 .2 (3 0 9 (.5 2 5 5 8 l.OCOOO 0.30391 0.6392 C.OOOl '= c !S !3 3 - ° ô ! î S ? l 0 .4 0 /0 O.OOCl 0.C061 0.0001 O.COOO •ôiSfll 0 .0 0 |0 J16 l ie 67 46 46 116 116 116 l i e 111 115 KN -0.16596 •03259 0.16277 -C.C7024 0.07021 (.0 0 6 8 6 0 .06327 - C . 16611 (.1 0 7 94 1.00000 •0 .0 6 93 7 O.C745 0.1463 0.6544 C .6546 0 .9 2 ( ( 0 .(0 9 6 0.0814 0 , 2 ( 9 ( 0.0000 0 .9 9 (9 M l : 'ili 64 43 43 111 111 111 - M i l Ft 0 .1 (6 6 6 .19726 •0 .0 0 (6 1 C .11533 -0.11497 C .22286 0 .1 6 3 7 ( (.2 5 2 43 1.00000 0 .2 (6 0 7 (.3 4 6 0 0 .9 (6 0 0.4506 C.452C 0.0167 0.0065 0 .0000 “ôiîis: 0 .0 0 (9 115 66 45 45 115 115 115 111 IK IK 119 CU C .(7 I0 6 -0 .2 4 64 4 0.24654 C .328Î8 0.67C06 (.5 8 5 33 C .45250 1 .00000 0 .(9 0 2 7 0.4C69 otooci 0.1027 C.1026 0.0003 O.COO^ 0.0001 0.0001 «ôlîlil 0.0000 o.o o g^ M5 I1 ‘ 66 45 44 115 115 115 111 IK ZN -0 .(2 7 6 1 .23775 -0 .1 0 66 6 0.10893 0.38216 0 .24490 C .30922 C .30351 -0 .0 6 5 3 7 0 .2 (8 0 7 0.99027 l.OOCOO C .7.S6 C .0 |0 ‘, -'0:3:3% C.4766 C .4 763 0.0001 0.CO63 0.0006 O.COlO 0 .9 9 ( ( 0 .0 0 (9 0 .0 ^ 0 ^ o.oogo 66 45 45 1 15 115 ; Î5 115 IK K3 W 0 0 TABLE 43 (continued) Grass hay - 3rd cutting ( ^ cr in» NN *07 Kl AC! P K CA MC MM (E CO 7N r M 1.cccco - 0 . 4 ; 7te -0.42184 C.42167 C.I33C6 -0.CC330 C.19347 0.05326 -0.44528 -0.05032 -0.26248 r . c ( cc 0.0868 0.1720 C.1722 0.4913 0.9865 C.3146 0.7676 0.0155 0.7955 0.1690 2*i 2S n 12 12 29 29 29 29 26 29 29 29 CP 0.1*73^ I .occoc -0.83171 -0.82335 0.62322 €.63362 0.75605 (.63407 C.60626 -0.32308 -0.03977 0.67851 0.18341 C.41M c.co o o C.OOOl C.OOJC C.OOlO O.OOC] o . c o | i o .o o u ^ O.C005 0.09J5 0 .8 3 |7 o.oogi 0.34 09 7S 29 12 29 29 29 forwN -C.4P766 -C.E3171 1.00000 0.90106 -0.90088 -C .64209 -0 .(65 96 -(.44796 - C .68466 0.23673 0.07170 -0.40899 -0.15747 C-Citft C.OOOl o.oogo o .o o g i C ,0002 0.0055 0.C035 0.0714 0 . 002^ 0.360^ 0.784g o.io|^ 0.546^ 1 7 17 11 17 AÜ» "0.421P4 -(.B2335 0.90106 1 .COOOO -I.COOOO -C .66012 - C .78425 0.54631 -0.51977 0,1720 C.COlO 0 .0 0 0 2 C.OOOO C.COOl 0,0155 -°ôî258i 0.0025 “ôîS îiî 0.0661 0.0833 12 12 11 12 12 12 12 12 12 12 12 12 HEAD* 0.421C7 C.E2322 -0.90088 -I.COOOO 1 .COOOO 0.66050 0.65503 (.64725 C.7845'2 -0.20913 -0.54652 0.51978 i . , 1 2 2 C.COIC C.0002 0 .0 0 0 1 C .0000 0.0154 0.C208 0.0005 0.0025 0.514J 0.066g ‘’ôîî??5 0.0833 12 11 12 12 12 12 12 12 12 P C.fl33E2 -0 .(42 09 -0.(6012 0.66C5C 1.COOCO 0.81690 C.60657 C.63543 -0.15827 -0.10992 0.59766 °ô!;3f§ C.OOOl 0.0095 0.0195 0.0194 O.OOCO 0 .0 0 0 1 0.0005 o.cogi 0.42J2 0.5703 0.0017 29 29 17 12 12 29 29 29 29 29 29 K -C.CC330 C .7*605 -0.66596 -0 .(54 83 0.65503 C.81650 1 .QOCOO (.4 7569 C.69636 0.58676 0.16995 0 , 9 ff 1 C.COOl 0.0035 0.0208 C .0208 O.OOC1 O.COOO 0.009^ 0 . 000| -vm ®5?79|| 0.0008 0 .37|1 29 29 1 7 12 12 29 29 29 C* 0.19347 C*63407 -0.44796 -0.84701 0.P4725 C .60657 1.00000 -0.33860 0.60196 0,3146 t .0002 0.0714 C .0005 0.0005 0.0005 °ôtc091 C.OOOO ■ °ôîlîli 0.0724 0.0006 2' 29 12 12 29 29 29 29 28 29 29 29 PC - C . . 1733 (.60626 -0 .(84 66 -0.78425 0. 76452 (.63543 0.65636 (.4 2612 l.OCOOO -0.15697 0.14975 0.64038 0.49298 0,,b74 0,0005 0.0024 0.0025 0.0025 0 .0 0 0 2 O.COO^ 0 .0 2 1 2 O.COOO 0.425J 0 .4 3 8 | 0 . 000| 0.0066 2'+ 29 1 7 12 12 29 29 29 29 PN 0 .t* 3 2 t -C .32306 0.23(73 -0.20913 1.00000 0.03672 0.07863 C.7I ?t C.0935 0.3603 C.5142 0 .0 0 0 0 0.8528 •“o llijl 0.6909 2 # 2P 17 12 12 i f 28 28 26 28 28 28 f t -0.44S2E -C.C3977 0.07170 0*54631 -0.54652 -C .10992 0.05137 -C .33860 C.14975 0.03672 1 . COOOO 0.21662 o.ci;% C.6377 0.7845 0 *Q6 6 | C .0660 C.57C3 0.79J3 C.0724 0.43|1 0.85J8 0 .0 0 0 0 0.2590 29 12 29 29 29 29 CU C.67851 -0.40899 -0.41925 C .41894 C.55766 0.5(876 C.60196 C.64038 0.11682 1.0 000 0 0.41009 C.COO! 0.1031 C .I749 t .175? C.Ü017 0.(008 C.0006 0 .0 0 0 2 0 .5 4 6 | 0 .0 0 0 0 0.0271 29 29 12 12 29 29 29 25 29 29 ZN -0./C24P -0.15747 -0.51977 0.5 15 78 C.45298 0.07863 0.21662 0.41009 1 .0 0 0 0 0 0, J19C :c!S(3! 0.5461 0,0*33 C .0633 0.C066 0.6909 0.2590 0.027^ 0 .0 0 0 0 <9 2*) 17 12 1 2 29 29 29 26 29 29 ro U) VO TABLE 4 3 (continued) Legume silage - 1st cutting ( f cr M AM AGFNN ADF f K LA 7!C HN FF (U 7M LH 1 •ctcoc t . 12295 C .06767 0.(0189 -0 .00190 -C.1543C -0.1(151 - c . 00175 -C .03845 -0.20 301 -0.12931 -0.13731 •0.31768 c .rioc C.0076 0.975? C .9756 O.OOC9 0.0302 C.9703 0.4128 O.OOOI 459 262 26? 0.0071 U.00 34 0 .0 0 0 1 456 456 4 56 456 453 933 453 453 f P o . iz r s t 1 .ccooc - 0 .^ ;i3 7 -0.54768 0.547*5 C.46054 0.46670 (.48883 C.3C245 -0.20346 -0.00168 0.26328 0.07525 C.Ct7F c.cooo O.OCOl 0 .0 0 0 1 C.000 1 O.OOCl O.COOl 0 .0 0 0 1 O.COOl O.OOOI 459 3C4 262 262 0 .9 722 O.OOOI 0.1097 456 456 456 456 453 453 453 ACfNk 0.t< 767 -C .45137 1 .occoo 0.69524 -0.69525 -C ,32361 -0.32670 (.19324 -C .02085 0.20274 0.10600 0.00587 -0.04388 C.OOCl c.ocoo C .0001 C.OOOl O.OOCl O.COOl 0.0007 0.7178 0.0004 204 303 239 239 0.0730 0.9191 0.44 74 3(3 303 303 303 302 287 302 302 AOf -C .54766 C.69523 1 .(0 0 0 0 - 1 .COOOO -C .38344 -0.35982 -(.24432 -C .23749 0.03091 -0.14658 -0.17187 "otSiS? C.OOOl 0.0 0 0 1 0 .0 0 0 0 C.OOCl O.OOCl OuCOOl C.OOOl O.COOl 0.6205 f ff 262 239 262 26? VAtïl 0.0183 0.0056 260 260 260 260 259 249 259 259 NEADf "0.C0)90 C.54755 -C .69925 - 1 . CCOOC 1 .COOOO C .38322 0.35970 («24424 C.23738 -0.03069 -0.02449 0.14627 0.17174 C.OOCl C .CCOl O.COOl C .0000 O.OOC] C.CCOI O.OOOI O.COOl 0.6230 26? 2 39 262 0.7005 0.0185 0.0056 262 26C 260 260 260 259 249 259 2 59 P -0.15430 C .46CS4 -0.32361 -C .38344 0.38322 1 .COOCO 0.69611 (.00790 C.08024 0.0009 C.OOOl 0.10074 0.21702 0.16791 0.28444 C.OJOI o.ooco O.CJgl 0.0870 0 . 0 J| 1 o.ogg^ 0.0003 O.O^^I 45t 456 456 456 K -0.10151 C .46670 -0-,32GJQ -0.25982 c . 35970 (.696 11 l.OCOOO (.01801 -C.03250 0.06919 0.13626 0.12182 0.18627 0.0302 C.OOOl O.C^OÏ 0 .0 0 0 1 C.OOOl O.OOCl o.oogo 0.4888 456 456 0 . , 7 | , 0.0045 0.0095 o.oooi 26C 26C 456 456 456 433 453 453 CA -0.00175 C .48683 -0.24432 0.24424 (.00790 l.COOCO C.50683 -0.29959 -0.01065 0.38039 0.05108 C .5703 c .o c o i 0 .0 0 0 1 C.OOOl 0.8664 0 .0 0 0 0 456 O.OOOI o.ogoi 0.8250 O.OOOI 0.2780 456 2C3 ?6 C 260 456 456 456 433 453 453 PC -0.C3F45 C .20245 -C.02C85 -0.23749 0.23738 C .08024 -0.C3250 (.50683 l.OCOOO 0.03178 0.04921 0.45948 0.19974 0 .4 1?P C .COOl 0.7I7B C.OOOl C .COOl 0.C870 0.4888 O.OOOI O.COOO 451) 456 0.9999 0.3070 O.OOOI o.oooi :C3 260 260 456 456 466 456 993 433 453 453 PN -0.iC3C| - c . 20246 C .202 73 0.C3091 •0.03069 C .10074 0.06414 -(.ÎS 5 5 9 C.03178 1.0 000 0 0.23208 0.08432 0.26503 O.OOCl C.COOl C.C003 0.6205 0.6230 453 0.0321 0.1729 O.OOOI 0.4999 0 .0 0 0 0 O.OOOI 0.0730 O.OOOI 452 3C2 259 259 453 453 453 453 993 433 453 453 Ft -0.12521 -C .00168 C.IC600 0.(2419 -0 .02449 C.217C2 0.13626 -(.010 65 0.C071 0.23208 l.OCOOO 0.16526 0.21330 C.972? C.C730 C.7005 C.OOCl 0.CO45 0.8250 0 .0 0 0 1 0 .0000 0.0006 O.OOOI 4 33 433 297 249 433 433 433 4 33 933 433 433 4 33 CU -0.I273C C .2632r 0.00587 -0.14658 0.14627 C .16791 C .12182 (.38039 C.45948 0.06932 0. 16526 1 .00000 0.33359 O.OCOl 0.0183 C .0185 C.C0C3 0.C095 C.OOOl O.COOl 0.0730 0.0006 O.OOCO O.OOOI 452 259 259 453 453 453 453 993 433 453 453 -0.31766 I.C7525 -C.C33B8 -0.17107 0.17174 C.28444 C .16627 (.05108 C .19974 0.26903 0.71330 0.33359 1.0 000 0 O.COOl C.1097 C.0056 C .0056 O.OOCl o.oogi 0.2780 o .o o o i 0 .0 0 0 1 o.ogg^ o.oooi o.ooco 452 452 259 259 453 453 453 453 993 453 453 TABLE 43 (continued) Legume silage - 2nd cutting f 7 c r ACrhh A D f K AC ( r K I * MG M6 (( CO IN (h 1 .CCOOO C .C37C9 c . Kfcoe 0.01041 0.03134 -C .01396 -0 .0 (8 0 9 -0 .0 6 07 4 0 .0 4 (0 5 -0 .0 1 84 6 -0 .1 2 2 2 9 O.CCOO C.6290 L .2283 0 .^ 7 3 , 0 .(3 1 2 C.B304 0 .2965 0.3549 0.4674 0 .7 7 2 ^ 0 .0 6 18 . 37 '«•siK 172 126 126 237 237 231 2 34 CP 0.C6143 1 .iCCOC - C . 46285 -0 .(1 5 8 9 0 .6 1570 0.34947 C.46425 0.11163 -0 .0 7 6 9 6 -0 .0 0 8 9 5 0.40738 -0 .0 0 6 6 0 C .2 I K C .0000 C.OCOI C.COOl C.COCl ^ o to o c i o .o o o i C.OOOl 0.0864 0 .2409 0.8924 0 .0 0 0 } 0 . 42 00 ,3 7 237 172 126 126 237 237 237 237 234 231 235 2 34 0 .(3 7 0 5 -C.462P5 1 .OCCOC C .(?907 ■0.623S3 •C .I32C 7 -0 .2 4 62 4 - C .14213 0.20516 0.30937 0.32751 ( .62 90 C .CCC 1 C .OCOQ C.OOOl C.OOOl 0.0842 O.COOl 0 .0624 0 .0064 0.0001 0 .0001 1 72 172 172 119 119 112 172 172 172 172 170 172 172 fiVf - 0 . ICI 13 - C . t 1*89 0.62401 1 .(0000 I.COOOO - 0 .3 4 7 (6 -0.3C 644 -C .21768 0.01087 -0 .0 6 9 7 0 0.09519 -0 .0 5 3 0 0 0 .17943 (.< 2 1 1 C.OCOI C.OOOl C.OOOO O.COOl O.OOOI 0.0004 0 .0143 0 .9038 0 .4 J8 0 0 .2 ^ |0 0.5556 0 .0444 126 126 119 126 126 126 126 126 126 126 126 KE /o r O .IC I08 0 ,6 1570 - -0 .6 2 2 9 3 1.(0000 1 .00000 0 .3 4 7 (2 0 .30704 - C . 01044 0 .05309 -0 .1 7 9 3 8 (.2 2 8 3 C.COOl C.OOOl O.OOOI C .0000 O.OOCl 0.0004 0 .9 0 32 0 .0 4 4 5 J2( 126 119 126 126 126 126 126 126 126 124 "''332 126 r C .01041 C .4(577 - C . 13207 -0 .35786 0.35782 I.o o o c o 0.70404 C.01774 0.07385 0.15511 0.19596 0.6734 C.OOOl O . C t .| C.OOOl C.COOl C.OOOO O.COOl 0.78 43 0 .2475 0.0176 "6:223; 0.0025 *6:23% 237 237 126 126 237 237 237 234 231 235 2 34 K 0.C3134 C .39947 -0 .2 9 6 2 4 - C . 20695 0.30709 C.705C4 l.OCOOO (.1 4 5 6 4 -0 .1 (2 8 5 0.14665 0.08042 0.02618 0 .10193 0.6312 C.OOCl C.CCOI 0.0005 0 .0005 0 .0001 0.0000 O.OJ63 0 .0047 0 .0249 0 .2 2 |4 0 .6897 0 .1200 237 237 172 126 126 237 237 23 7 234 235 2 34 CA -0.C 1398 0.46525 -0 .1 4 2 1 3 -0 .2 1 76 8 0.21761 1.00000 0.34419 -0 .2 5 9 3 3 0.01266 0.51718 0.13270 0 .8 3 0 ! C.OCOI o . c ^ i 0.0144 ” fi!7843 “ ’ôîélfS 0.0000 o .o g o ^ 0.0001 0 .8482 0.0001 0 .0 4 26 2 37 231 126 237 237 234 231 235 234 HG -0 .1 6 (0 9 (.1 1 1 6 3 C .2 C ÏI6 O.C1087 '0.01094 0 .0 7 3 (4 -0 .1 (2 8 5 (.3 4 4 19 1.00000 0.31108 0.18375 0.44537 0.44929 0.2965 C.C664 o .c c ts C .9038 C.9022 0 .2 4 p ^ 0.0041 0.0001 0 .0 0 00 O.OOOI 0.0051 0.0001 0.0001 237 237 172 126 126 231 237 237 234 231 235 234 PN -0 .0 (0 7 4 - ( .07696 C.2CZ98 -0.C 69IC 0.06993 0.14665 25933 C .31108 1.00000 0.28457 0.02671 0.336 4 2 0 .3 ! 49 C.C^76 0.9380 0.4365 0 . 0 | 4 , o . o j o | 0 .0 0 ^ 1 o .o o g i o.oooi 0.6844 0 .0001 4 34 126 126 234 230 234 2 33 K 0 .0 4 (0 5 0.C9519 0.09508 0.19840 C.O 1266 0 .1 (3 7 5 0.28457 1. coooo 0.09077 0 .2 3934 0 .4 ( 74 0.2930 0.2935 0.0024 VAIW 0.8482 0.0051 0.0001 0 .0000 0.1691 0.0 0 02 231 17C 124 124 231 231 231 230 231 231 230 CU -O.C 11 9( C .40736 0.06733 -0.C530C 0.C5309 C .14446 0.02618 C.51718 C .44537 0.02671 0.09077 1.00000 0.39000 0.7725 ( .COOl 0.2546 0.5549 0.0024 0.6897 C.OOOl 0.0001 0.6844 0 .1 6 9 } 0.0000 0.0001 2 3! 235 172 126 234 235 234 234 234 235 234 Zh - C . 12229 - ( .CC660 0.32751 0.17943 •0.1793P 0 .1 4 8(1 0.10193 C .13270 0.44929 0.33642 0.23934 0.39000 1.00000 ( . r i J( 0.9200 C.OCOI 0.0444 C .0445 0 .0023 0.1200 0.0426 0.0001 0 .0 0 0 ] 0.0002 0.0001 0 .0 0 00 . 34 234 172 126 126 234 234 234 234 233 230 234 234 N) TABLE 4 3 (continued) Legume silage - 3rd cutting r» cr 9r86N *08 M A l;f P K I* MG MN FE CU 2N CM 1 .iCCOÜ -C .C13CF 0.031(8 C.C6771 -0 .062 (5 -C .11263 0.01968 -C.07998 -C .07336 -0.29187 -0.27886 -0.04017 -0.77877 Ü.((CO C.F853 0.1(99 0.9909 0.5919 0.2121 0.8678 0.7796 0.7968 0.0068 0.0020 0.6576 0.0106 i;4 124 97 77 77 124 129 129 129 129 120 124 179 CP -O.C)2CF 1 .(CCOO -0.99789 -0.9793( 0.47442 C.39573 0.C3971 (.37699 (.06621 -0.08097 0.00360 0.26846 -0.18955 c . ; t ' 3 C.0000 0.0001 c.o o o j O.COOl O.OOCl 0.(938 O.OOOI 0.9650 0 .37,3 0.9689 0.0026 0.0 |5 0 124 97 77 124 179 179 129 120 124 fOfNN 0.C3WF (.4 5769 l.OOCOO 0.73901 -0.73427 -C .31313 -0.19987 -C .07787 C.29509 0.04892 0. 16972 0.13787 0.17929 0 •?(44 C.CCOJ c.coog C.COCl 0.000 1 0.0024 C .C 6|| 0.79|9 o.oieg 0 .6 4 || 0.1098 0.1900 0.0966 c? 70 7C 52 90 92 92 ACf 0.t6?2J -C .47436 0.73901 1.(0000 -l.OCOOO -0.225(3 -0.03009 -( .7 7993 (.05829 -0.08914 0. 10475 -0.07175 -0.01553 U*tV09 C.OOOl C.COOl 0.0000 0.000 1 (.0465 0.7993 0.0139 0.6198 0.4407 0.3678 0.5352 0.8933 77 77 7: 77 77 77 77 77 77 77 76 77 NEADf - o . t f rr.5 C.47442 -0.73977 - 1 . COOOO 1.COOCO (.77990 -C .09813 -0.10474 0.01575 0 .* 9 I9 C.COOl O.OCOl C.OOOl C.COOO 0.0139 0.6156 0.3679 0.89^8 Î7 77 70 77 1 1 77 77 77 76 F -0 .1 Ü F 3 C .39573 -0.31333 - C .27963 0.22572 l.COOCO 0.53790 -(.70997 C .13567 0.25089 •0.00266 0.10526 C.OOOl 0.C029 0.0989 0.0464 C.OOCO O.COOl 0.0196 0.1330 0.0^99 v - m i 0.9766 0.29J6 124 92 77 179 124 179 129 124 ¥ O . U U f t C.03571 -0.19987 -0.C3009 C .03014 C.93790 1 .CCOOO -C .97031 •C .31567 -0.05414 -0.15194 c.tot ?e C .6936 O.C67.7 0.7993 0.7941 O.COOl O.COOO C.OOOl 0.0009 0*5503 0.0921 K 4 124 92 77 129 124 129 129 124 124 CA -0.0259F C .37649 -0.07787 -0.21943 C.2794C -C .70997 -0.52031 1.OOOCO C.93696 -0.03434 0.08203 0.40377 -0.09569 C.COOl C..7979 0.0139 C.C139 0.0196 O.COOl 0.0000 O.COOl 0 .fç 5 0 o.oooi 0.7909 124 97 77 77 179 124 129 129 124 129 PC -0.L2336 C.C6621 C .79909 0.C5624 -0.05613 0.139(7 -0.31562 (.9 3696 l.OCOOO 0.29362 0.25592 0.32572 0.7966 (.4650 C.C1^6 0.6146 C.61^g 0.CÇ09 0.0001 O.COOO 0.0069 o.ojje 0.0Ç0J 124 124 179 129 - H i PN -0.24167 -C .C6C97 C.C9897 -C.C8914 0.C6921 C.79089 -O.C9919 ” (.039 39 C.79362 1.00000 0.91599 -0.06765 0.35779 O.OOtb C.VM3 C .6 9 y 0 .4 4 0 j 0.4404 C.C099 0.9903 C.7090 O.0O|9 0.0000 O.O^OJ 0.9559 O.OOOI 124 1 29 129 179 179 179 fF - o . / 7 f e t C .003(0 C .16977 C .10475 -C. 10474 (.08703 0.91599 l.OCOOO 0.00877 0.71195 O.C020 0.9609 0.1C98 0.3676 C .3679 0.0001 0.0000 0.OÎ09 12C 170 90 76 76 1 20 120 120 120 120 CU -0.14017 C .76898 0.13787 -0.C Î175 0.07128 -C .007(6 (.90377 C .32572 -0.06765 0.00877 1.00000 0.16779 C.f ! 76 C.0076 C.1900 €.5352 0.5379 C.97(6 ■°ôîél?î 0.0001 0.0002 6.9559 C.OOOO 0.0675 124 179 97 77 1 1 179 124 179 124 179 179 /N -0..21.77 - C . 18939 C .17979 -0 .(15 53 0.01575 C .10976 -(.09569 0.35779 1.00000 o.oiot C.039C 0.0966 C.6933 0.6910 0.7996 C.7909 ' ô ° f a i 4 O.OOOI 5.0000 124 179 97 77 77 '29 179 129 129 124 120 179 179 ro TABLE 43 (continued) Legume-grass silage - 1st cutting I } cr »rrN6 AUl M *Cf F K I* 80 HN K (U 7N rn 1 •< COCO C .0(795 C.C3934 0.(089? -c.roa96 -C .10624 -0.C9787 C.04690 -0.00870 -0.17360 -0.07656 -0.03704 -0.14699 ( .CC< 0 0.7773 0.8099 C .8 102 C.OOCl O.COOl 0.0697 0.7363 0.0001 0.0035 0.0001 1! I 7 ("SSW 721 l i t I5C2 1507 1507 1507 1497 1457 “ • i U S 1496 CP C.Ct79^ 1 .coooo -C .30(57 -0.41014 0.41016 0.47245 0.44654 C.67070 (.47187 -0.17924 0. 12656 0.44895 0.20181 r.O UF1 c.ocoo O.OCOl O.COOl C.OOCl O.OOCl O.COOl 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 )\ 17 7 544 721 721 15C2 1507 1507 1502 1497 1452 1500 1496 >urNN 0.13^34 -C .30(57 l.CCCOO 0.58522 -0.585 19 -C .19819 -0.23547 -C .06764 - t . 07398 0.16450 0.03589 0.01166 0.04719 C .;f7 2 C.CCOI C.COOO C.OOOl C.OOOl C.OOCl O.COOl 0.0384 0.4635 0.0001 0.2806 0.1500 «,44 S44 944 673 673 937 937 937 937 937 906 937 *0 f O.lCfS7 -C .41014 C.58522 I .COOOO *1 .00000 -C.333(0 -0.79036 -C .10181 -C .16446 0.00960 0.00427 -0.0944 7 -0.08421 U.ECS9 C.OOOl C.OOCl O.OOOQ O.OOCl O.COOl 0.0065 O.OOOQ 0 .9 I0 | 0 .0 |J 6 0.0247 7?l 721 721 714 711 AEADF -0.CCF96 C .4 1016 *1 .COOOO 1 .ooooc (.10179 -0.00958 -0.00432 0.09451 0.084 77 C .l 10? C.OOOl *°c !o o o i 0.0001 0.0000 0.0065 ‘ÔÎS23? 0.7988 0.9098 0.0^15 0.0247 v; 1 771 673 721 721 714 714 714 714 711 691 711 P -0 . 1C< 24 C .47745 - C . 19619 -0.33360 0 .733(3 1 .COOCO 0.7C489 (.09590 C .17014 0.16610 0.20893 0.26985 C.COOl C.OOOl C.OOOl C.COOl (.0001 O.OOCO O.COOl 0 .0 0 0 | O.OOOI o .o o o i ÎVC2 1507 937 714 714 1517 1507 1462 1510 “•MSI K -O.C9 7f 7 C .44054 -C .73547 -0.29036 0.79047 C .70469 1 .CCOOO (.04085 C.04381 0.11283 0.12527 0.12319 0.15304 0.0001 C.COOl O.CCOJ O.OOOI C.COOl O.OOCl O.COOO 0.0881 0.0001 o .o o o i 0.0001 0.0001 It0 2 15C7 714 714 1517 1517 1517 1507 1462 1510 1506 CA 0.C4C9O C.C 7070 *0.06764 -0.10181 O .K I 19 C.C9590 0.04085 1.00000 C.60426 -0.29915 0.07476 0.54014 0.14849 0.0092 C.OOOl C.C384 0.0065 0 .0065 0.0002 C.OOOO O.COOl 0.0001 0.0001 1‘ C? 15C2 937 714 714 1517 1517 1512 1510 1506 MC -O.tOblC C .42167 -0.16446 0.16449 C .17014 0.04387 (.60426 l.OCOOO -0.02633 0.48780 0.26977 0.7363 C.CCOI C.COOl C .000 1 O.OOCl 0.C881 C.OOOl O.COOO 0.3071 0.0001 r 0? 1507 714 714 1517 1517 1512 1517 1507 1462 1506 MM -0.17360 - C . 17974 0.CC96C -Ü.C095H C .16610 0.11283 -(.299 15 1.00000 0.21361 O.CCOl C.COOl °c:*23% C.7964 0.7988 C.COCl O.COOl 0.0001 0.0000 °ô:38S! 0.0001 1497 1497 937 711 711 15C7 1501 1507 1507 14tO 1506 1502 FE -0.17656 (.17656 0.03589 C.C0427 -C.CC432 C .70893 0.17527 (.0 74 76 C .19855 1 .COOOO 0.15223 0.25172 O.C03» C.COOl 0 .7606 0.9109 C .9098 O.OOC] O.ÇOOJ 1452 1457 9C6 691 651 146? O 'tS S i C'SSH “• m s “• m i CU -C .13704 C.44895 0.C1I66 -0.C9447 0.C9451 C .77593 (.54014 (.46780 I . 00000 0 .1 1 1 ( C.COOl C.0116 c . c i i g O.OOCl 0.0001 O.COOl 0.0000 15*0 1500 713 1510 1510 1510 1510 1506 1461 1510 1505 2N -C.14C 99 C.70161 C .04719 -0.C8421 0.06477 C .76965 0.15304 (.14849 C.26977 0.71361 0.25172 0.44590 1.00000 C.COCl C.COOl 0.1^00 0.0247 C.C74J C.OOOl O.COOl C.OOOl o .o ^ g i O.OOOI 0.000] 0.0000 I4«,t 1496 711 15C6 1506 I5C6 *1506 1458 1505 1506 ro LO TABLE 43 (continued) Legume-grass silage - 2nd cutting 1 t CP ALFNN AOF Nf AUF 1 K lA PC M6 H CU IH CH 1.tc c c o C .C t’ fiS -0.C3250 •0.0201P 0.C2C06 -C .06069 -0 .IC I4 1 -C .01*09 -(.06855 -0.05850 -0.02605 -0 .0503* -0.17802 ().C( cc C. 1497 0.5575 (.7526 C .7540 0.1/24 0.C224 C. 1516 0.1232 0.1889 0.5679 0.258* 0.0001 *. IC •j IC 228 246 246 5(7 507 507 507 506 *83 5C6 506 cr C.C63P9 1 .CCOOO -C.03C91 -9,50135 0.5C121 C.444Î9 0.41122 C.56576 (.*0561 -0.15795 0.12*11 0.*9632 0.10808 C .0000 C .0001 O.OOC] C.OOOl O.COOl C.OOOl o.oooi 0.000* 0.0063 0.0001 0.0150 *>K 246 246 5(7 507 507 506 *83 506 506 APfNh l.CCCOO 0.10389 -0 . 10’ P5 C.01881 (.01179 -0.03339 0.00391 0.03610 -0.02170 -°ôi2:SS 0.0000 0 .1 Î95 C.7351 0.8320 0.5*87 0.9*52 0.5166 0.6968 32t 326 328 226 3<6 326 326 326 325 325 325 ACF -o .c z c ie -C.t013S 0.I03E9 1.(0000 -) .CCOOO -0.725*6 -(.21956 - ( .1 * * 7 3 0.16901 0.05004 -0.0823* 0,09033 o.75?e C .0001 0.118* 0.0000 C.OOOl O.OOC* -°a!ZSS; 0.0306 0.C 23| 0.0080 0.4481 0.1990 0.1587 ; 4 t 246 22b 246 246 2*5 245 2*5 2*5 232 2*5 2*5 NC Aor 0.020CP C.50121 -0.10385 -1 .(00 00 1 .COOOO 0.225*2 0.17042 (.210 35 (.1**78 -0.16892 -0.05004 0.08236 -0.090 28 0.7^40 C.COOl C.OOOl C.OOOO 0.000* C.C075 0.0006 0.0081 0 .*J8 J 0.1989 24t 246 246 246 2*5 245 2*5 2*5 2*5 p -0.C6C69 (.44459 -0.0(328 -0.22546 0.22552 1 .00000 0.69282 (.26785 0.22699 0.26500 0.25885 0.33390 C.OOOl 0.0004 0.0004 0.0000 0.0001 0.0001 0.0001 0.0001 0.0^01 0.0001 507 245 245 508 508 508 508 50 7 484 307 n -0.101*1 C .4 1122 -0.11219 -0.17037 0.17042 (.(9 3 (2 l.CCCOO -(.03309 C.02178 0.19187 0.20899 0.14407 0.16008 C .CCOl 0.0*18 0.0075 0.0075 0.0001 o.cooo 0.0001 O.OOOJ o .o g i^ o.oooi 507 326 245 245 508 508 **'SSS 507 307 CA -O.OI4C9 C.56576 0.01(81 -C .21856 0.21635 0.06595 -0.02309 1.00000 C .46148 -0.3085* C.OOOl C.0006 0 .0006 0.4568 C.OOOO 0.0001 o.oooi 5C7 245 245 ° ' ^ U l 508 508 508 507 *8* 507 PC -O.CfPtb 0 .40561 -0.144 73 0.14478 0.02178 (.46148 l.OCOOO 0.06101 0.06310 0.48703 0.4055* C.OOOl 0.0235 C.C,3* 0.4748 c.ooog O.COCO 0.1657 0.0001 o.oooi 507 226 245 5ce 506 508 o - ' B i *8* 507 507 PN -O.CtfbO •(.15795 -0.02339 0.16901 -0 . 16892 (.22699 0.19187 -(.3 0 8 5 * 1.00000 0.25171 -0 .01616 0.28796 O.lt P9 C .0004 C .M 81 0.0080 C.OOM O.OOCl 0.0001 0.0001 0.0000 0.0001 0.7167 0.0001 ! Ct 5C6 245 245 5(7 50 7 507 50 7 50 7 *8* 507 507 Ft -O.C2#Ct C.124 11 0.00391 (.(500 4 -0.05004 C.265C0 0.2C699 (.0 2*56 (.06310 0.25171 1.00000 0.0855* 0.188 72 0,5( 79 C.C063 0.4461 C.44P1 O.OOCl O.COOl 0.5899 0.1657 0.0001 0.0000 0.0601 0.0001 413 4P3 232 232 4 8 4 484 *8* *8* *8* *8* *8* *8* CU -0.C5C34 C .49632 0.02610 -0.06234 0.08236 0,25685 (.52365 (.*87 03 -C.01616 0.0855* 1.00000 0.41011 0 .,!P 4 C.COOl 0.5166 0.1990 0.1989 C.OOCl 'ôîîoîj 0.0001 0.0001 0.0001 0.0000 O.OOOQ ! CC 506 225 245 245 5C7 50 7 *8* 507 7N -0.17102 ( , loeoe -C .021Ï0 0.(9033 -0.09028 0.33590 0.1(808 (.*C 55* 0.28796 0.16872 0.41011 1. coooo C.CCOI C .C15C 0*6966 0.1587 (.151*9 O.OOCl O.COOl W l O.COOl 0.0001 0.0001 o.oooi 0.0000 • Ct 506 225 245 2*4 *. 5C? 507 507 507 507 *8* 507 507 N3 TABLE 43 (continued) Legume-grass silage - 3rd cutting Î F CF ACFKh ADF K£ ADI r K CA *c HK FE Ct IH r H 1.COCOO -C.CC923 C .00217 - 0 .(7 8 0 0 0.07794 - C .06790 -0 .0 5 2 5 2 - C . 12658 -0 .1 4 40 4 -0 .10371 -0 .1 2 9 6 0 -0 .18401 -0 .1 8 4 8 6 (.C 0 (0 C .9I2C 0 .9635 C.5119 0.5122 o . M j e 0.5318 € .1306 0.CB5O 0.2161 0 .1 J5 6 0.0273 0 .0 2 65 144 144 94 73 73 144 144 144 144 144 144 CP -0.CCS23 1 .(COOO -0.55C 06 -0.60902 0.60891 0.07892 (.4 1 3 8 6 C .24263 -0 .2 1 83 4 -0 .1 0 7 8 0 0.33589 -0 .0 7 7 6 6 C.OOOO C.OOOl C.OOOl 0.0001 0 .3 4 7 , 0.0001 0.C034 0.0086 0.2033 O.OOCl 0 .3 5 4 9 144 144 94 73 72 144 144 144 144 141 144 144 ADFNt C ,( C ;i7 “ C .55006 l.CCCOO 0.78197 -0 .7 6 1 9 9 - C . 20686 C.02790 0.03457 -0 .0 7 8 5 8 -0 .1 6 7 9 8 O.V( 3 t C.COCl C.OOOO 0.0001 O.OOOI 0 .0455 0 . 7896 0.74§8 0 .4 5 |5 0 .1 0 56 44 94 66 66 44 94 94 94 ADF -0.C7KC - C . 60902 C .78197 1 .COOOO -1 .0 0 00 0 "0 .09 7 2 0 0.08200 -0 .1 0 7 4 5 0.03059 -0.09921 •0 .0 9 0 2 6 O.b 119 C.OOOl C.COOl 0 .0000 C.OOOl C .4 1 28 0.4904 " o ?2 I S 2 0 .3 6 56 0 .7973 V M l t 0 .4037 0 .4 4 76 73 73 66 73 73 73 73 73 73 73 71 73 73 KFAOf O.C7794 c .c c e s i -0 .7 6 1 9 9 -1 .COOOO 1.COOCO C .04746 -0 .0 6 17 4 (.0 1 3 6 6 C .10758 -0 .0 3 06 4 -0 .0 0 74 4 0.09932 0 .09029 0 .5122 C.COOl C.CCOI o.rool C.COOO 0.4121 0.4911 0 .5357 0 .3 6 50 0 .7969 0 .9509 0.4031 0 .4 4 74 73 73 66 73 73 73 73 73 73 73 71 73 73 F -0.C 6790 C .13344 -C.2C666 -0 .(9 7 3 0 0 .09746 1 .COOCO 0 .57167 - (.3 0 0 6 6 C .20268 0.26831 C.06962 0.11753 0 .41817 0 . 4 1 f t c . i l o e 0.C455 0.4128 0.4121 0 .0000 O.OOOQ 0 .0 0 0 2 O .O J4| 0 . 4 } 0 | 0 .1 5 9^ o.oooi 144 144 94 73 73 1 45 145 145 K -0.C 5252 I .17092 -C .24(81 O.C8200 -O .O e 175 0 .57167 1 .OCOOO - C .32617 0.17032 0.01075 0 - . 3 ! f 0.0194 0.4904 0 .4917 O.OOOI o.cooo O.OOOI 0.0405 0 .8 9 79 94 73 145 145 145 145 145 142 145 145 CA - 0 .1 2 1 5C C.41268 C.C279C -0.C 7369 C.C7366 -C .30066 -0 .3 2 61 7 1.00000 C .37211 -0 .1 9 1 3 2 0.00296 0.38417 -0 .0 9 1 4 3 0 . J30C C.OOOl 0.7696 0.5355 0.5357 0 .0002 O.COOl 0 .0000 O.OOOI 0 .0212 0.9721 0.0001 0.2741 144 144 94 73 73 145 145 145 145 145 145 FO -0 .1 4 40 4 C .24262 -C.C6299 0.10758 C .20268 -0 .1 3 79 1 C .37211 l.OCOOO 0 .16603 0.61194 0.50231 o.c.;o C.C^3J 0.4265 0 .3650 0 . 0 J « O .C |5 , c.o6o. 0 .0000 0 .0460 0 .0 0 0 ^ 94 73 145 145 hN -0 .2 1 6 3 4 0.03457 -0 .0 3 06 4 C .26831 0.17032 C .16603 1.00000 0 .4 8 87 7 -0 .00444 ■'’c'Jf/l C.C066 0 . 7408 0.7969 0.0011 0.C405 0 .0 4 60 0.0000 o.oooi *5?ooot 144 144 94 73 73 145 145 145 145 145 142 145 FE -C.129 ro Ln TABLE 43 (continued) Grass silage - 1st cutting u Cf 4Cf NN ADF K( AOF 1 K CA MG Mh fl CU 7N r M 1«lOtOO - c , 1(892 0.10628 0 .( 2803 -0.02804 -C .26826 -0.25780 C.14037 C.07826 -0.GP426 -0.09479 -0.06187 -0.09080 U .0 (00 C.C219 O.C*. 36 C.7978 C.7977 0.U0C2 0.C004 0.U561 0.2621 0.2J34 0.4054 0.2216 If A 18 4 ICH 86 86 18 3 183 163 " ' 1 1 1 179 183 183 ( p -C.1CI9? 1 .CCOOC - c . 21613 -0.4449C 0.44492 C.53759 0.58128 (.36935 (.34886 -0.05348 0.03308 0.58921 0.50422 O.Ci I® C .COOO C.OOOl C .0001 O.OOCl O.COOl O.OOOI O.COOl 0.6648 0.0001 0.0001 If A 184 C'Cia; 86 86 18 3 183 183 183 174 183 183 ADFNN 0.16(26 -(.216 13 1. occoo 0.75360 -0.75357 -C.363C6 -0.31461 C.35341 C.2C349 -0.05022 0.02531 0.00955 0.0L3C c.cooo 0.0001 C.OOOl O.OOCl O.C009 0.0002 0.0347 0.6127 0.7946 0.9219 1C& ICO 83 83 1(8 108 1C8 108 “" • i s ! 104 108 108 AUF 0.C 2 I03 -C.4449C 0.75360 1.(0000 - I .CCOOO -C .571(0 -0.4*435 (.21022 -C .02375 0.00166 0.79 7C C.COOl 0.000) 0.0000 C.OOCl O.OOCl O.COOl 0.0520 0.8282 0.988J -"ÔÎI8II -"ôîîSII 6( 86 63 86 86 86 86 86 86 86 86 NEADF -0.C2F04 C .44492 -0.75357 - 1 . COOOO 1 .CCOOO 0.571C0 0.4*445 -(.21023 (.02364 -0.10504 -0.00187 0.13686 0.14439 0.7977 c.oooj C.OOOJ C.OOOl C .0000 C.OOCl O.CCOJ 0.05^0 0.0290 0.9866 0.2089 0.1847 86 86 Kt) 86 83 86 86 P -0.2(826 C .Î37ÎS -C .36206 -0.57100 0.57100 1 .OOOCO 0.69066 -(.01693 (.17815 -0.01464 0.08992 0.42245 0.37732 O.C002 C.OOOl O.OOOI C.OOOl C.OOOl 0.0000 O.COOl 0.0001 If 3 102 108 86 86 185 185 *'"131 " ' 1 1 1 185 K -0.25780 C .50120 -0.31461 -0.45435 0.45445 C.69066 1.CCOOO (.03517 0.03096 0.56055 0.37466 0.0C04 C.COOl 0.C009 C.OOOl C.OOOl O.OOCl O.COOO 0.6346 'ôîaisi 0.6791 "ôiisiî 0.0001 0.0001 if 3 102 IC8 86 66 165 165 165 185 181 176 185 185 CA 0.14037 C .30935 0.35241 0.21022 -0.21023 -(.01693 0.03517 1.00000 (.56431 -0.08630 0.45325 0.16300 O.C* 81 C.OOOl C .0520 0.0520 0.8191 0.(346 C.OOCO 0.0001 ‘“ÔÎSSII 0.0001 0.0266 183 103 86 66 185 185 165 185 181 " ' 1 1 1 185 185 PO O.C 7* 26 C.24006 C.2C349 -C.C2375 0.C2364 (.1 7 8 )5 0.31701 (.56431 1.0(000 -0.02497 0.11684 0.58216 0.45753 C.COOl 0.8282 C .6290 0.0153 O.COOl 0.0001 o.cooo 0.7386 O.OOOI O.OOOI 10 3 86 86 185 185 165 185 181 "'•HI 185 185 PN -C.C5340 C .14293 C .10519 -C. I05C4 -C .014(4 -C .02497 1.00000 -0.05377 0.13103 C.4771 0.3351 C.3358 0.8449 0.7386 0.0000 0.0787 179 86 86 18) 181 181 181 174 "'MIÎ 181 FE -0.69479 t .03300 0.C0166 -C .C C lf7 -(.086 30 0.22317 1.00000 0.239 35 0.6641 C.9881 C .9866 'ôllSSÎ o .o o il 0.0000 0.0014 * 1C4 83 83 17*. 176 176 176 176 176 CU - 0 . ( 6 I f 7 C.02531 -0.13699 0.13686 C.42245 0.5(055 (-45325 (.58216 -(.05377 0.07273 1 "0000 0 . 578 94 0.4654 'c'Sgg! C. 1948 0.2085 C.2089 O.OOCl O.CCOl C.OOOl 0.0001 1 0000 0.0001 113 103 1C8 86 86 175 185 185 185 185 185 ; n -O.C9< f 0 C.5C422 C .14439 0.37466 (.16300 C.45753 1.00000 U.2i 1( C.OOOl "«?§2i9 C.1847 ^ c : U U O.COOl 0.0266 O.COOl *a!g)g; v j é n "ôllSIÎ 0.0000 If 3 10 3 lOH 86 86 185 185 185 185 181 176 185 185 ro ON TABLE 43 (continued) Grass silage - 2nd cutting t f cr ALINN *DF NE ACF P K 0* 80 M6 F( 00 7N CH l.CCOCC - C . 17522 0 . 7C546 0.45401 -0.45340 -C.014(1 -1 .38933 0.25456 C. 12671 -0.08244 -0.14001 -0.05111 0 .08140 o.cooo C.4239 C.0104 0.1606 0.1613 0.9465 0.C663 (.2411 0.5645 0.7084 0.5450 0.8169 0.1188 22 12 11 1 1 23 23 23 23 23 21 23 22 CP -0.17^22 1 .ccocc -C . 59 322 0.19577 C .320(0 0.5417C (.20311 0.11137 -0.17267 0. 14497 0.48210 0.44879 (-.a ; 39 c.coco C.0403 -(ctSISI 0.0175 C.1356 0.0016 0.3512 0.4181 0.4308 0.5307 0.0198 0.0362 ;3 23 12 11 11 <3 23 23 23 23 21 23 22 ADfNN 0.7CÎAI -C .59722 l.COCOO 0.14050 -0.74093 -0.50669 -0.64914 0.32609 0.14524 -0.15593 0.39946 0.00131 0 .05796 C.CICA C.C403 c.ccoo 0.0143 C .0142 0.09|l 0.0224 0.3009 0.6524 0.6284 0.25|8 0.9968 0.8656 12 12 10 10 12 12 12 12 12 11 ADf O.AtAC? -C.t94fc4 0.74C50 1.00000 -1.coooo -C.26541 -0.46533 0.11028 0.38272 0.39183 0.28677 0.12847 -0.34879 0.11 Ob C.0I77 0.0143 0.0000 0.0001 0.4301 0.1492 0.6161 0.2454 0.2333 0.4544 0.7066 0 .3233 11 11 10 11 It 11 11 11 11 9 11 10 NEAPf -0.4*340 C.69527 • C . 74093 -1.(0000 1 .00000 C . 26594 0.4(515 -0.11020 -0.28754 -0.12864 0.34968 O . U 13 C.0175 0.0001 c.cooo 0.4253 0.1488 0.6168 0.4531 0.7062 0 .3219 n 11 11 11 11 11 11 11 9 11 10 P - 0 . C 1 4 M C.32060 -0.26541 0.26594 1.00000 0.50538 -0.43145 0.09503 0.50267 0.9465 0.1356 0.4301 C .4293 0.0000 0.0139 0.0398 0.666| v . n u 0.0171 23 23 11 11 13 23 23 23 21 23 K -0.3f933 C.Î417C -0.64914 -0.46533 0.4C575 0.50538 1.00000 -0.35523 -0.22529 0.30673 0.05560 0.296 8 7 0.CI63 C.C076 C.0Z^4 C.,492 0.14P8 0.0139 0.0000 0.09|| 0.30|3 0.1546 0 .8108 0.1197 Î3 23 11 23 23 23 C« 0.25456 C . 32609 0.11028 -0.17020 -C.43145 -0.35523 1.00000 -0.10766 -0.11766 (.24 11 0.3009 0.6161 C .(168 0.C398 0.0962 0.0000 (ôîo'JÎ? 0.6249 0.6114 23 23 12 11 1 1 23 23 23 23 23 21 23 22 PC 0.12671 C . 17737 0.14524 0.38212 -0 .3630* -0.01223 -0.22529 0.41718 KOCOOO 0.02429 0 .45568 0.55358 0.40091 0.5145 C • 4 1 6 1 0.6524 0.2454 L .2449 0.9556 0.3013 0.021^ 0*0000 0.9124 0 .0379 0.006| 0.0644 23 23 11 11 23 23 23 23 22 hK -o.ce;44 -C.l 3267 -0.15593 0.39183 -0.39:30 C.055C3 0.30613 -0.10766 (.02429 1 .00000 -0.20354 -0.034 74 -0.16118 C.7CP4 C.430C 0.6284 0.2333 0.2340 0.6662 0.1546 0.6249 0.9,24 0.0000 0.316^ 0.8750 0.4736 23 23 11 1 1 23 23 23 23 23 22 Ft -0. 14001 C . 1 4497 0.39946 C . 28611 -0.2B754 C . 19912 0.05560 -0.11768 -0.20354 1.00000 0.656 36 0.5450 C..30, 0.2528 C.4544 C.4531 0.38}4 0.8108 0.6114 0.3762 0 .000^ cgfg&ll 0.00 21 10 9 9 < 1 21 21 21 21 CU -0.C511I C .4 6210 0.00131 0.12841 -0.12664 0.019(1 0.22942 (.41071 0.55358 -0.03474 0.63110 1.00000 0.46563 C.f 169 C.019P 0.9968 0 .106j C.1C6^ C.1114 0.29|3 0.0516 0.0062 0.8750 0 .0022 0.0000 0.0290 23 23 23 23 23 23 7N 0.CP140 (.05196 0.34968 (.502(1 0.29681 -0.06210 C . 40091 - C . 16118 0.65636 0.46563 1.00000 0.7166 0.8656 C.3219 0.0111 0.1191 1.1816 0.0644 0.4136 0.00j2 0.0290 0.0000 22 22 11 1C 1C 22 22 22 22 22 22 22 Table 43 (continued) Grass silage - 3rd cutting 1 ^ cr «CFNN ADf Nf AO# r R CA HO M8 6k CU 76 CP 1 .cccoo -C .3b4fcS 1 .COOOO - 1 .CCCOO -C .27350 -0.23673 C.C( CO C.347 7 'gÇlfS? -°g:32s; 0.5122 °g!32?* -*g5ias| 0.4724 I b 3 ? ? p e 8 B e t 8 CP - 0 .]( 1 .COOOO 0.84641 -1 .COOOO 1 .00000 0.81610 0.R1605 C.69932 C.44416 -0 .42736 0.20374 0.89834 -0.48127 ( .3 4 4 1 C.OCOC C.3Î75 0.0135 0.0041 0.05 36 0.1632 0.1792 0.6612 0.0024 0.2273 1 1 2 ? b e P 6 8 8 8 AOfNK C .f 4641 1 .cccco -1 .COOOO 1 .coooo -C .75766 -0.91292 C.98148 0.02289 0.94410 -0.94404 -0.88409 f .S» 27 (.3575 0.0000 0.452 7 0.2676 0.1227 •'g«!?5| 0.9844 0.1914 0.2140 0.3082 3 2 2 3 3 3 3 «DI I.ICUOO -I.CCOOC - 1 .occoo 1 .COOOO -1.00000 1 .00000 -1 .00000 - I . 00000 1.00000 1.00000 -1.00000 0.00000 1.00000 2 2 2 2 2 2 2 2 2 2 2 2 KC Aor - 1 .cocoo 1 .00000 1 .00000 -1 .COOOO 1 .coooo - 1 .00000 1.00000 1 .00000 -1.00000 -1.00000 1.00000 0.00000 -1.00000 ; 2 2 2 2 2 2 2 2 2 2 2 2 p -0 ,;45 06 C.PI610 -0.74766 l.COCOO -1.00000 I.OOOCO 0.67112 C.60290 -0.33848 0.77214 -0.02667 O.ttPS C.0I35 0.4427 O.OOCO 0.0684 0.1136 "ofgiZ? 0.4122 “ôîiiîS 0.0248 0.9400 b e 2 2 b 8 8 e 8 V -0.44P43 C.f7GC« -0.S1242 - 1 . COOOO 1 .00000 0.67112 1 .COOOO C.37543 0.46443 -0.63873 0.84603 C.£<5^ C.C04| 0.2676 C.C664 c.ccoo 0.3594 0.144g 0.0 8 8 | ■°6l81S5 0.0 0 8 | 2 2 P e 8 CA -0.5 7350 C.< Y932 0.9(146 - 1 .COOOO 1.CCCOO 0.60250 0.31543 1.00000 C.23431 0.11494 0.44618 -0.42128 C.51Z? C.0536 0.1227 0.1136 0.3594 0.0000 0.4765 -°g!lS2l 0.8044 0.2449 0.2986 P 2 2 P P 8 e 8 8 f-Ù -C.C3343 1 .COOOO -1 .COOOO 0.56443 1. occoo 0.19427 0.49633 -0.03864 C.S374 cgtZii? 0.1450 'g!3i2i 0.0000 0.6764 0.2109 0.9276 1 t 3 2 2 P 8 8 e 8 8 8 PN C .02289 l.COCOO -1 -00000 -0.33848 -C.62P73 -C .21652 1.00000 0.00143 0.40420 C.CC75 0.9854 0.4122 0.0882 C.6065 ^0^4729 0.0000 0.9976 0.2026 f e 2 2 r e e e 8 8 8 n -c .c ; if.p C .;0 37i f .95510 -l.COCOO 1 .COOOO 0-43340 -0.10058 C .11595 0.00143 1 .COOOO 0.28947 0.338 14 C.5t3< C.6612 0.1915 C.3314 0.8301 C.8045 0.9976 0.0000 0.4289 0.4482 7 1 2 2 7 CU ( .1 9f34 - C .94404 O.COÛOC o.coccc 0.772 14 0.84603 C.45618 I.OOOCO (.C0?4 C.2140 0.024P 0.C081 0.2559 r j V c i * g % ? a ; o.oocg *“gÎ6*?§î ( t 3 2 ? P 8 8 e 8 6 2H 0.55(2? -C .47127 - C .88509 1.COOOO -1 .cocoo "0.02667 -C .42126 -C .03865 0.50420 -0.188C4 1.00000 0.1365 (.2272 C.20B2 0.95C0 •“ôîfîî? 0.2986 0.9276 0.2026 0.6447 0.0000 1 y 2 2 8 H e 8 8 e N) 0 0 TABLE 43 (continued) Corn silage - no NPN added { K cr AffNN ADf h i AOF r K 0* HC HK ft CO 7N 1 M 1 .10(00 -C.C7137 -c.cjcei -0.2C834 0.20(36 C.C7324 -0.2(633 -0.08862 -0.06059 -0.1054 3 -0.08333 -0.1092b -0.08134 00 C.COOl 0.6659 0.0001 C.COOl 0.00C7 0.0001 C.OOOl 0.0009 0.0001 0.0001 o.oooi O.OQOl :< ; A 3024 1597 1178 1 1 ?H 3009 2977 2968 3005 2985 2977 2988 29 78 CP •C.C71?7 1 .COOOO -0.01227 -0.03106 0.03114 C . 308(8 0.32859 (.31724 0.22015 0.12848 0.38348 0.16142 u.ctui c.ccoo 0.2866 C .2856 C.COCl 0.0001 C.OOOl 0.0001 0.0001 0.0001 0.0001 3024 1178 7178 30C9 2977 2968 3005 2985 2977 2986 29 78 fDfNH -O.CItKj -1.(1227 1 .cocoo 0.(1726 -0.CI726 C.0725 7 0.0(210 (.024 34 -0.00490 0.00848 0.01683 0.02559 •0.006 28 0.0000 0.6026 1597 O ' t W Z O ' t S W O'ISIS 1585 f ur -o.;ct 3( -C.C3ICI 0.0112» 1.(0000 - 1.coooo -0.13824 0.19999 (.25158 0.09081 r.oooi C.286t 0.5758 0.0000 C.OOOl 0.00(1 0.0001 0.0001 %*!o’o’§I VMU 0.0018 117» m e 1C53 11 78 1178 1176 1163 1167 1176 1172 1167 1174 1169 NEfDf O . / O f 36 C.03114 -0.01726 -1.(0000 1 .CCOCC 0.13833 -0.19990 (.25153 - C . 22299 -0.19566 -0.02790 -0.09073 -0.05120 C .CUCl 0.0001 C .0000 0.0001 0.0001 0.0001 O.ÇOOJ 0.0001 0.3409 0.00J, 0.0802 il7e ( " M i l 11 18 1178 1176 1167 1172 1167 1169 P 0 . U 3 7 4 ( ,3oeoe 0.01297 -0.13824 0.13P33 1.00000 0.27680 (.09560 0.11412 -0.09666 0.07595 0.13347 0.29617 0.6C01 C.COOl 0.6047 O.COCl 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001 0.000, 0.0001 O.OOOJ 3C 09 2009 1595 1176 11 76 3009 2976 2987 3004 2984 2 987 K -0.2C( 33 C.32859 0.08210 C . 19999 -0 .19990 0.2 76(0 1.COOOO 0.17216 0.39903 O.CGOl C.COOl C.OOOl C.OOOl 0.00(1 0.0000 ((fSSSI C.COOl "6:3%3? " SÎÎ38Î 0.0001 ;S77 2977 1163 1163 29 76 2977 2957 2975 2960 2951 2962 2952 ( A •0 -Clb62 C . 31724 0.02434 0.25158 -0.25153 0.095(0 0.38666 1.00000 0.52411 0.16252 0.19798 0.31896 0.13968 U.OOOl O.OÇOJ C .0001 0.000^ 0.0000 o.cogi 0.000» 0.0001 0.0001 i'.'Ke C'ssia O'ilSS 1 16 7 "'SSSl 2988 2Sil « 'SSSl 2957 KG "0.6CC5S C . 22315 -0.0C490 0.22306 -0.22299 0.11412 0.17216 (.52411 1.00000 0.04 74 7 0.05485 0.17095 0.138 80 0.CÜC9 0.0001 0.0001 0.0001 3(0t C'SSgi 1176 2984 ° 'SSS& 29 75 KN -0.10543 C .12P48 C.0Cf48 0.7956? •r. 79566 (.096(6 0.19950 (.16252 C.04747 1.00000 0.16736 0.34356 ('•C( 01 C.OOOl 0.0007 C.COOl O.COCl 0.0001 0.0001 0.0095 0.0000 O.OOQI “ôio’SSÎ O.OOQI 2985 °'!33a 1772 1172 2984 7960 2964 2983 2985 2911 2982 29 72 FI •O.OC333 C . 12821 0.01683 C.C2793 -0.02790 0.07595 0.(7906 (.19798 (.05485 0.16736 1.00000 0.07365 0.17110 0.0(01 c o c o . 0.3405 C.34C9 0.00(1 0.0001 0.C028 O.OOOÏ O.OOOÔ 0.0001 0.0001 ?'-77 ^*1582 7 7 6 7 116? 2976 ° ’295l 2957 2974 2971 2977 2974 2964 CU "C.1C52C ( .3(348 C.02559 C,C90ei -0.C9073 C . 13347 0.35903 C.31896 C . 17095 0.19864 0.07365 1 .00000 0.14706 I.0001 o.ooie C .0019 0.0001 0.0001 0.000Î 0.000^ O.JOgJ O.OOQI ("SSPl G'SEil 17 74 1174 2967 llbZ 2967 2974 °'S8gg ;n -O.CI1?4 ( .16142 - C . 00628 0.(5137 -0.C512C (.296 17 0.14161 (.13968 0.13880 0.17110 0.14706 1.00000 0.0(01 C .CCOl C.6C26 C.0791 0.0802 O . O O C 1 O.OCOI C.OOOl 0.0001 O.OCOI 0.0001 0.0000 ?*’7fe 2S7e 1565 1769 1 169 29 7 7 2952 2957 2975 2972 2964 2975 29 78 NJ VO TABLE 43 (continued) Corn silage - NPN added I »• Cl AtINN ADf h i ACf P X C.f MC MX KCU 7h rn 1 .ictoc -0.1304! 0.11015 -C.C4456 C.C44P3 -0.13626 -(.04588 -C.05960 -0.05766 -0.09890 - 0 «03916 -0.01983 D,(i cr C.COOl 0.000 3 0.2637 C.2608 'aSIAIS O.COOl 0.0773 0.0207 0.0251 0.0001 0*1266 1!>?4 K 8 6 631 631 J95? 1986 1484 1507 1508 1508 1507 fp -0. 1 .CCOCC -0.93289 0.00843 -0 .00842 0.92593 0.1(568 (.08642 C . 12280 0.20926 0.19758 0.42734 0.37982 O.CCC! C.CCOO C.OOOl C.8326 0 .8328 C.UOCl O.CCOl C.0009 O.COOl 0.0001 0.0001 o.oooi 0.0001 1‘ / A 1*>24 1086 631 631 1952 1986 1464 1507 1508 1508 1507 19 96 ALf U h Ü.11U1S • C . 43384 1.00000 0.54986 -C.54996 -0.15976 0.05791 (.06015 -0.09676 -0.19580 0.CU03 C.COOl C.COOl 0.0001 O.OOC] 0.0001 n e t 1086 C'SE8% 605 605 I0( 1 ''•sill ° i S 1S ®*1077 1067 ADf -O.CAASt 0.00642 0.59586 l.COCOO -1 .COOOO 0.00336 0.22949 (.1(990 (.21389 0.20930 0.23223 0.14317 0.09203 C.Z# 37 C.6326 0.0001 0.0000 0.0001 0.9335 O.COOl 0.0001 o.oooi 0.0001 0.0001 0.0003 0.0 2 1 9 <31 631 (05 631 631 620 619 616 627 630 630 629 620 h i ADF O.CAAr.3 -C .00842 -C.59596 •l.COCOO 1.COOOO -C.00322 -0.22941 -C . 1(988 - C . 21379 -0.20921 -0.23219 -0.14317 -0.09199 U.2( Ob O.OCOI 0.0001 C ,0000 0.93(3 O.COOl 0.0001 O.OOQI 0.0001 0.0001 0.0003 0.0220 i 31 (05 631 631 620 619 616 627 630 630 629 620 F o.CAipe - C . 15576 -C.C0322 I.COOOO C.35539 0.22099 0.30893 0.39656 0,56779 o.irtp c.çoo. ' c n i l ‘5 C .9363 0.0000 0.0001 0.0001 O.OOOI 0.0004 6.J001 620 620 1952 ” ' i f S 1966 1985 1985 1985 K -0.12426 t .ie%68 0.05791 0.22949 -0.22941 C.25250 1.CCCOO C . 19591 C.07839 0.28073 0.III20 0.99 363 0.183 79 O.OCOI 0.0001 0.0(09 C.OOOl C.OOOl 0.0001 C.CCOO 0.0001 0.0026 0.0001 o.oooi 1A6C M B 6 1052 619 619 19(9 1486 1955 1979 “ i S 8 i 1980 1979 CA -o.CAtme C.CC6.2 0.06352 0.1849C -0.18488 0.35539 0.19541 I.00000 C.61830 0.29000 0.3333? 0.17923 0.0 773 c.oocs 0.0388 C.OOOl C.OOOl O.OOCl O.COOl 0.0000 0.0001 0.0001 0.0001 O.JOOI 1«FA ).p. 1058 616 616 19(6 1455 1989 1977 1977 1977 1466 hC -0.05960 C .13200 C.C6C15 0.21384 -0.21374 0.299 99 0.07839 (.61830 1.00000 0.13833 0.28287 0.12960 0.14109 0.02 07 C.COOl C.OOOl O.OOCl 0.00^1 o.coog 0.0001 o.oooi o-ooci 0.0001 1ÎC7 ' i ? 8i ' i n s 627 627 19(5 1507 1500 1500 1490 MN -0.C5766 C.20920 0.01155 0.204 30 -0.20421 0.22059 0.2(073 (.24000 C . 13833 1.00000 0.25986 0.33287 0.42315 C .0001 0.0001 C.OOOl O.COOl 0.0001 0.0001 lice '•iTil 63C 630 198C 14 7? ° i § 8 5 ° i § 8 8 1507 ° i § 8 i f r -0.09# 90 0.1975* 0.23223 -0.23219 C. 30853 0.I112C (.33339 (.2*287 (.25986 1.00000 0.28689 C.CCOI C.OOOl C.COOl C.OOOl O.OOCl O.COOl O.OOOl O.COOl o.oooi 0.0000 °ô!3g2l O.OOOI I M > lice 1077 63.? 630 19(5 19BC 1477 1500 1507 1508 1506 1495 CU -0.03916 0.92739 -C . 09676 0.14317 -0.14317 C . 39636 0.99363 (.1 7423 C. 12960 0.33287 1 .00000 0.48911 0.1266 C .0001 C .0003 C.0003 O.OOCl 0-Ç001 0.0001 O.COOl 0.0001 “ôîoooi O.OOOÔ O.OOOI 1 U 7 1507 ^*1077 629 629 19(5 1476 1999 1506 1506 1507 1495 2fi -0.<1*583 ( .379P2 - C . 14580 0.C9203 -0.C9IS9 C.56779 0.1(374 (.1 7059 C . 19109 0.92315 0.28689 0.98911 1.00000 C.OOOl C.COOl 0.0219 C .0220 O.OOCl O.COOl 0.0001 0.0001 0.0001 0.0004 0.0001 0.0000 I99t IC67 620 62C 197( 1468 14 66 1990 1995 1995 1496 VI o TABLE 4 3 (continued) Sorghum silage cr *C»NN ADF r.( A ll CA ro MF, Ft AN OH 1 .LCOOC C .2 16C9 - C . 06857 -C .2 I9 3 ? C .52342 -0 .0 5 54 6 C.04564 -0 .1 1 1 0 7 -0.01A 82 -0 .2 5 8 6 2 *0.0 87 6 4 -0 .01291 Ü .ccoc 0.5722 0 .2 )2 6 O.OOCl 0.6875 0.7407 0 .4 1 95 0.9153 0 .0590 0.5286 0.9262 43 34 C 55 55 55 55 5A 54 54 54 CP C 3I.C9 1 .ccocc “ C .62999 - C . 29916 C .50492 0.36001 (.5 3 4 3 9 C .42890 0.01269 0.08368 0.51721 0.19132 c . n 31 c.cooo 0.0001 0 .0 )9 4 O.OOCl 0.CO4? c .o o o j 0 .0 0 )1 0.927J 0.5474 0.0001 0.1658 h t 95 43 34 C 55 55 55 54 54 AOrHN - C .C ftt V -C .6 2999 1 .OCCOO 0.51596 - C . 41351 -0 .2 7 3 8 5 -(.2 3 4 2 5 -0 .4 5 46 1 -0 .3 9 01 5 C.COCl c.oooo 0.0021 0.0058 0.C756 C.1305 -'ÔÎ5I8Î 0.0025 0 .0 1 06 4 3 42 43 33 C 43 43 43 A3 42 42 42 42 AÜF - 0 .4 :9 3 7 -C .29916 1 .COOOO -0 .4 3 9 6 6 0.20449 C.15323 C .3 28:9 0.03934 o . z iz e C .0 9 4 c.o o o o 0.0093 O.CPOO t .3869 0.C S8I v . m i 0.8279 34 34 23 34 0 24 34 34 39 33 33 33 KEAOF p 0 . ‘ 2342 C .50492 - c . 41350 •0.43966 1 «oocco 0.19091 C.44B26 C .12323 -0.02005 0.06775 0.23865 0.23896 O.COOl C.OOOl C.0058 0.0093 O.OOCO 0.1627 c.ooog 0.370^ 0.8856 0.6264 0.0B|2 0.08^8 55 55 43 34 C 55 54 54 K -C.C5546 C .’ fCOl -0.27285 0.30449 C .19091 1 .CCOOO (.35557 0.24633 0.17387 0.44030 0.02421 0.6.6 75 C.C042 C.C756 C.0800 0.1627 C.CCOO 0.00^7 0.0726 0.2086 0.0009 0.6620 f 5 55 43 34 C 55 55 55 54 54 54 CA 0 .04564 C. 53439 -C .23425 0.15323 C.44826 0.25557 1.00000 C .73349 -0.13806 0.15029 0.45733 0.03713 C.74C7 C.COOl 0.1305 0.3869 C.00C6 0.CO77 0.0000 0.0001 0.3195 0.2780 0.0005 0.7898 55 55 43 34 0 55 55 55 55 54 54 54 54 HC -0.11107 C .4269C - C . 14604 0.32819 C .12323 0.37375 C.73349 1.00000 -0.26669 -0.02131 0.39585 0.04489 0.4195 C.OOll C.2501 0.058^ C.37JJ 0.C049 O.OOCl O.OOOO 0.0512 0.8785 0 .0 0 |0 0.74^2 55 55 4 3 c 55 55 55 54 54 HN -0.C1482 C .01269 C.01538 C .10724 -C .020(5 0.24633 -(.13806 -C .26669 1,00000 0.21795 -0.04254 0.01102 0.9153 C.9274 C.9230 C.5525 0.6856 0.07|6 0.3195 0.0512 0.0000 0.1134 0.7600 0.9369 54 54 42 33 0 54 54 54 54 54 54 f t -0.256 62 (.(6 3 6 8 -C .04285 0.12696 C.06775 0.17367 (.15029 0.21795 1.COOOO 0.10822 C.C 'fO 0.5474 0.7276 1.44 72 C.6264 0.2086 0.2780 - 'ô ? l* 7 ? î 0.1134 0.0000 0.4360 54 54 42 33 0 54 54 54 54 54 94 94 54 CU -0.CF764 C .51721 -C .45461 C.C3934 C.23865 0.44030 (.45733 C .39585 -0.04254 0.07392 1 .00000 0.27199 C .5266 C.COCl C.CC25 C.8279 0.0822 0.C009 C.0005 0.C 0|0 0.76^0 0.59^3 0.0000 0.0466 54 54 42 33 0 54 54 54 94 54 7N -C.C1291 -C.39CI5 - h . 17937 C .23896 0.02421 (.03713 f . 04489 0.10822 0.27199 1.00000 0.5262 ^ c W d l 0.C1C6 0.3179 C.0H16 0.8620 0. 7696 0 . 7472 VMU 0.4360 0.0466 0.0000 54 54 42 33 c 54 54 54 54 54 54 54 54 ro Ln TABLE 43 (continued) Small grain silage CM CF IC76N AOf M ADf P K (A KG MN 88 CU 2N IM l.CCCOO -C .12312 C .25542 0.C152O -C.C1523 -0.2C5IB -C .09131 -C .07596 -0.01081 -0.02376 -0.11494 -0.05660 c.ccoo 0.0003 0.8742 C.C74 0 ■''o’ ” ’ 4 0.C006 0.1324 0.2100 0.8586 0.6948 0.0565 186 111 '"■'270 278 274 274 275 276 CP - 0 .1 f j| 3 1 .(COOC -0.26266 -0.4C644 0.4C(H2 C.53848 0 .57062 (.36440 0.34704 -0.01639 0.25169 0.58 393 0.46386 c.cocc 0.0C03 C.COOl C.OOOl O.OOCl O.COOl o.oooi C.COOl 0.7871 0.0001 0.0001 0.0001 ?P3 186 111 111 27E 273 274 274 275 276 273 ACfNr. l.COCOO 0.60297 -C.6C2S5 -0.19768 -0.2445C (.05014 (.12824 0.04651 0.00241 -0.17730 -0.020 79 0.C0C3 0.0000 O.OOOI C.OOOl 0.0065 o.cooe 0.0845 0.0163 0.7817 18( le t 186 107 1C? 176 184 182 “•’ISI 183 180 ADF O.CltZO -c . t c t t i C.6C297 1 .COOOO -l.COOOO -0.25673 -C .04960 -0.30881 •0.07552 1.8742 C.COOl O.COOl 0.0000 C.OOOl 0.0085 0.6136 0.00J2 0.4416 i n 111 ICI 111 111 1(3 ■ " • ‘ill 106 106 106 107 106 NE AOF -0.01523 c ..c tr z -0.60295 - 1. coooo 1 .COOOO 0.25661 C.05005 0.17321 0.30903 0.E74Û C.COOl C.OOOl C.OOOl C .0000 0.00 8 9 "êîïîiî 0.6104 0.0758 “ÔÎHH 0.0012 111 111 111 III 1C3 107 106 106 106 107 107 106 P -0.13797 c .S ’ ete - C . 19768 -0.25673 0.25661 1.COOCO 0.51462 (.00428 C.06963 -0.06240 0.05790 0.37187 0.31350 0.0234 O.COOl 0.0099 0.0069 0.0000 o.coog 0.9450 0.2596 o.oooi 0.0001 270 ?7C '•"??? 103 103 270 263 264 266 265 PC -0 .2 0 * IP C.i7C62 -0.20498 0.20510 0.57462 1 ,CCOOO (.12986 (.11021 •0.08847 0.04779 0.29600 0.15534 0.0006 C.COOl - v . m 0.0342 C.0341 O.OOC 1 c.ccoo 0.0326 0.C696 0.1458 0.4316 O.OOOÏ 0.0104 ; 76 27P 107 167 269 276 271 272 272 273 2 74 271 CA -0.C9131 C.36440 -0.C4960 0.05005 0.004 28 0.12986 1.00000 C .77616 0.06898 0.20219 0.57970 0.41573 C.OOOl 6.6104 0.9450 O.C3^6 O.OQOO O.OOOÏ 0.0009 0.0001 0.0001 273 166 263 273 267 o -'S lS 269 269 266 MG -0.C7ÎS6 0 .347C4 0.C9672 C.06963 0.11021 (.77616 1.00000 -0.00104 0.15078 0.53494 0.2100 C.COOl 'Sis:;; 0.3240 0.2596 0.C696 C.OOOl 0.0000 0.9864 0.0130 0.0001 . 74 274 182 106 106 264 272 267 274 270 2 70 MN -O.CIOPI -C .06240 -0.08847 ( .06898 -C .00104 1,00000 0.16452 c .tte e 0.3125 0.2613 0.9864 C.OOOO V J U t 0.006 7 « 74 274 106 IC6 264 272 267 270 274 272 273 270 Ft •0.02376 C .24169 -0.11137 0.11162 C.05750 0.04779 (.20219 C .15078 0.31025 1.00000 0.21930 0.28288 0.6«»4I5 C.OOOl 0.4316 0.0009 O.C130 0.0001 0.0000 O.OOÔ3 0.0001 275 275 ‘ * ” 111 273 269 271 272 275 274 271 CU -0.11494 C .56393 -0.17730 -0.30881 C . 30903 0.25600 (.5 7910 C.61717 0.09519 0.21930 1.00000 0.65946 0.0*65 C.COOl 0.0163 0.0012 0.0012 O.COOl 0.0001 0.0001 0.1166 0.0003 O.OOOO 0.0001 . 76 276 183 107 107 266 274 269 273 273 274 276 273 IH •C.65<60 C .46366 -0.02079 -0.C7552 O.C7595 C .31350 0.15534 (.41573 0.53494 0.16452 0.28288 0.65946 1.00000 C.?t 15 C.COOl 0.7817 C.4416 C.4391 C.OOC] 0.C104 0.0001 0.0001 0.0067 O.OOOÏ o.ooçl 0.0000 - 73 273 18C lot 1(6 265 271 266 2 10 270 211 2 73 NJ Ln N3 TABLE 43 (continued) Dry shelled corn ( f cr «CINN AOF MALI 1 K CA fC M* FE CU Ih 1 .(c ro o -c .CI3Z: -0 .7 9 2 I( C .79953 C .00630 -0 . 11650 -6.12997 - C . l 3891 0.094AG 0.10255 -0.01395 -0.06491 (.0000 r.DBti ^otc213 C.0063 C.0060 0.99 75 0.2277 0. 1859 0 .1 ,91 0 . 3 2 b 2 0.2869 0.8896 0.6005 J IF lie 2 2 IC 10 I 12 109 106 no 110 109 110 CP - 0 . ( I3Î3 1 .ccocc C.02095 0.15829 -0 .1(0 6 7 0.99316 0.39939 6.C0012 C.59811 0.16126 0.28052 0.18318 0.29074 0 .K 69 C.COOC 0.8982 0.6629 C .( 5 75 C.OOOl 0.6002 0.9990 0.0001 0.0924 0.0030 0.0566 0.0021 IJF lie 2 2 IC 10 112 109 106 112 110 110 109 110 ADFNN -C.A!77A C .C7P9Î l.COCOO 0.56827 -0.57359 -0.159 30 O.C8752 -6.05619 -6.13882 0.17999 0.95529 o.c; 13 c .,,; ; C.CCOO 0.1109 C .1069 c .9 , ;o 0.7060 0.8090 0.5378 0.9293 0 .0 3|2 “ôîSl?? ?2 9 21 22 22 22 22 22 ADf -0 .792 )0 C .56827 l.COOOO -0 .99995 -0.129 13 -0.21067 (.05313 -6.30259 0.23661 0.01593 0.70931 0.61830 0.CUC3 C .l I 09 C .0000 C.OOUI C.7326 0.5591 0.8891 0.3955 0.5109 0.9662 0.0216 0.0567 )C 1C 9 IC 10 IC 10 10 10 10 10 10 10 HE ADF 0.794 53 - C . 16067 - C .57359 -0.99995 I.CCCCO 0.12991 0.20999 -C .05028 0.30935 -0.01367 -0.71296 -0.62131 (1.00(0 C.6575 0.1C69 C.OOOl C.0000 0. (310 0.5619 0.8903 0.3926 0.9701 0.0208 0.0552 IC 1C 9 1C 10 10 10 10 10 10 10 10 10 P 0.CC030 C .49316 -0.15930 -0.12913 0.12991 I .60000 0.73972 -6.00299 0.77725 0.29285 0.33973 0.37933 0.60053 C.COCl C.9930 0.7326 0.7310 C.OOOO 0.9761 O.COOJ 0.0019 0.0003 0.0001 o.oooi 112 10 IC 112 106 110 110 109 110 K -O .jjfS O C .34439 C .08752 -0.21067 0.20995 0.739 72 1.CCOOO -(.01993 0.59539 0.19050 0.18961 0.26696 0.95323 0.2277 C.C002 C.7C60 C.559I 0.5619 O.OOOI o.cooo 0.8850 O.OOOI 0.0999 0.0570 0.0057 O.OOOI 109 109 21 IC 10 169 109 103 109 107 107 106 107 CA -0 . 12947 C .0CO12 -C .05619 0.C5313 -0.05028 -0.00299 1.00000 0.01873 0.3362 7 0.38995 0.115? C.999C 0 .8C9C 0.889 I C.8903 0.976 I C.OOOO 0.8988 0.0005 O.COOl 106 K 6 22 10 10 IC( 103 106 106 109 109 103 104 KG -0 . 13691 0.5481 1 -0.13882 -0.30259 0 .T 9 3 5 0.77725 0.59539 (.01873 1.00000 0.30973 0.48370 C.COÇ, 0.5378 0.3955 C.3926 C.COCl O.COOl 0.6988 o.cooo “ôîîJ§l o .o o io “ô îî5 n 0.0001 IC 10 112 109 106 112 110 110 109 110 ► h 0.154(6 C .23661 -0.2926 I C.292(5 0.19050 6.33627 1.00000 0.53532 0.43627 0.62519 0.:*; 52 (c!2i;s 'c!3S2l 0.5109 C.50C5 0.0019 0.0005 0.0000 O.OOOI o .o o o i 0.0001 l i e lie 22 IC 10 110 '’•Toî 109 110 110 109 109 110 FE 0.10:55 C . 2 t O i 2 C . l 7999 0.C1593 -0.CI367 C.33973 (.38995 C.3C973 0.53532 l.COOOO 0.34667 0.47097 0.2164 C.C03C 0.9293 0.9662 C .9701 O.OOC3 C.OOOl 0.0010 0.0001 0.0000 0.0002 O.OOOI 110 EIC 22 10 ir 1 16 107 109 110 109 110 106 109 CU -0.C1345 C .II31E 0.95529 0.70931 -0.71296 0.37933 0.26696 (.19610 0.93627 0.39567 1 .00000 0.61122 0 .0 3 ;; C.0216 0.0208 O.OOCl 0.C057 0.1909 0.0001 0.0002 0.0000 O.OOOI IC 10 IC9 106 103 ' “ ' • î i l i 109 108 109 109 Iti -C.lt49) C.2S074 0.99589 0.6 1830 -C .6 2 I3 I 6 .6005 3 0.95323 (.26709 C.4C370 0.52519 0.97097 0.61122 I.OOOCO C.CO?I 0.0375 C.0567 0.0552 C.COCl O.COOl 6.0061 O.OOOI O.OOOI 0.0001 0.0001 0.0000 lie lie 22 IC 10 I IC 107 109 no 110 109 109 110 Ui w TABLE 43 (continued) High moisture shelled corn IF CF AOF HN AOF M ADF P K CA MC M* FE CU 2N CM 1 .( COCO C.C49Î2 C.01769 -0.(9621 0.09693 0.14797 (.04073 0.11711 0.14247 -0.12776 0.06698 0.09329 O.COOO C.4I34 0.8565 0.4765 0.4732 0.0140 0.0533 0.0199 0.0355 0.2782 0.9789 i 275 107 57 57 275 273 273 267 271 268 271 CP 1 .COOOC -0.02778 0.22577 -0.2244? 0.40387 0.28934 C.17169 (.43203 0.29691 C . 16315 0.26761 0.29650 (.4)34 C .0000 0.7764 0.0913 C .0932 0.0001 O.COOl 0.0096 0.0001 0.0001 0.0071 0.0001 0.0001 , 7* 275 107 57 57 275 273 271 273 26 7 271 2(8 271 ADFNN 0.(1769 -C.02778 1.OCCOO 0.46740 -0.46629 0.02521 (.12295 (.14907 0.08215 -0.00539 -0.01032 c.oooo o.oog^ C .0003 0.7966 0.2093 0.4071 ° 5 !m ; 0 .9567 0.9163 '"'is; 107 55 1(7 107 106 104 106 106 106 AOF -0.C9C2I (.22577 C.4674C 1 . COOOC -0.99997 C.480Î4 0.42493 ( .06287 (.46779 0.26843 0.25805 0.29917 0.35955 0.4765 C.0923 0.0003 0.0000 0.0001 0.0002 O.COOO C.6953 0.C003 0.0435 0.0526 0.06|| 0.0068 57 55 57 57 57 56 56 57 57 57 HEAD* 0.09(93 -0.46629 - C . 99997 1 .CCCOO -(.40102 -0.42575 -C .06520 -(.46835 •0.27042 -0.25876 -0.29555 -0.35623 0.4732 0.0003 0.0001 (.0000 0.0001 0.(010 C.6383 0.0003 0.0419 0.0519 0.0656 0.0065 57 57 55 57 57 57 57 56 56 57 57 57 57 P 0.14797 (.40367 - C . 01414 0.48054 -0.48182 l.COOOO 0.7(307 (.15296 (.78151 0.4i>383 0 .23227 0.58856 0.0140 (.0001 0.8851 0.0002 C .0001 0.0000 O.COOl 0.0117 O.COOl O.OC'r;! o.oogi 0.0001 77! 275 107 57 57 275 273 271 273 167 268 271 K -C.C0962 (.26934 C.02521 0.42493 -0.42575 0.70307 1 .CCOOO (.10253 (.63054 0.40522 0.25986 0.39998 0.538 76 Ü.F742 (.0001 0.7966 0.0010 c.coio 0.0001 O.COOO 0.0927 O.COOl 0.0001 O.OOOI 0.0001 0.0001 ;?3 273 ICI 57 57 273 273 2 70 272 266 269 266 269 CA 0.(4173 C . l 7169 0.12295 0.(6287 - C . 06420 (.15296 0.1C253 1.00000 (.03746 0.39186 0.19310 0 .37999 0.38106 0.4743 (.(048 0.2093 0.6453 (.8383 C.01,7 0.0927 C.OOOO 0 .5400 0.0001 0 .0015 o.oooi 7 » 1 ?71 106 56 56 27C 271 270 264 267 269 O'Oggl HC 0.11711 C.422C3 0.14907 0.46779 -0.48835 (.78151 0.63054 (.03796 l.OCOOO 0.50848 0 .22587 0.27290 0.57187 0.0533 C.OOOl C.C003 C.C003 O.OOCl O.COOl 0.5500 o.oggg O.OQOl 0 .0002 0.0001 0.0001 , 73 273 56 56 273 272 270 266 269 266 269 FN 0.14.47 (.29*91 C.08Î15 0.26843 -0.27042 (.48383 0.40522 (.35186 C.9C848 1.00000 0.30959 0.50631 0.72159 0.0199 C.OOOl 0.04^5 C.C419 C.OOOl O.COOl O.OJOl o.cog^ 0.0000 0.0001 0.0001 O.OOOJ étl 267 57 287 266 26 7 26* 269 f C -0.12776 0.25805 -0.250 76 (.23227 0.25964 0.30459 l.COOOO 0.27550 0,39500 0.0355 0.0526 C .0519 C.OOOl O.COOl (a!2g)S V.llll O.OQOl 0.0000 0.0001 0.0001 7 71 271 106 5 7 57 271 269 267 269 267 271 268 271 CU O . C ( ( 4P (.2878 1 - C . 00534 C.2441 7 -0.24*55 0.35733 0.39498 (.31999 (.27290 0.50631 0.27550 1.00000 0.62893 r .7 7*2 C.COCl 0.9567 (.0672 (.0656 O.COCl 0.(001 C.OOOl 0.0001 0.0001 0.0001 0.0000 O.OOOI 2tF 28 8 106 57 57 266 266 264 266 264 268 268 268 2 h 0.04324 C. 24650 -C.CIC32 C . 35455 -0.25623 C.ÎF856 0.53676 C.381C6 (.57187 0,72159 0.39500 0.62893 1.00000 0.47*4 C.COOl 0.916 3 ( .0068 C .0065 O.OOCl O.COOl 0.0001 O.COOl 0.0001 0.0001 0.0001 0.0000 . 71 271 1(6 5 1 57 271 269 267 269 26 7 271 266 271 ro Ln TABLE 43 (continued) Dry ground ear corn V f CF ATFhN M A D F 1 K CA 80 HN FE CU 7N UH l.tCLOO C .347PÎ " C .11566 -0 .t3 P 2 ) 0.53759 0.32221 1,23430 C.22753 (.21855 0.09673 0.03364 0.09830 n.LLLC C ,(05t C.6273 C.C17A C.0176 O.blCt C.C668 t.0917 0.C906 0.4545 0.7952 0.44 72 62 20 19 19 (2 62 56 61 62 62 60 62 CP C.3A7PÎ 1 .CCCOO -C.tOZ98 0 .4 9 9 9 7 0.5(660 0,eC536 C.67909 (.46040 0.58604 0.40991 0.42166 0.36631 O .C t'f c.cocc C.0063 0.0066 C.COCl C.COOl 0.0001 O.OOOj 0.0009 O.OOCb 0.00 32 f 2 62 19 19 62 62 56 “•'"Si 62 60 62 KOtMh -O.llttb - c . m c 5 1 .CCCOO 0.A6569 • C .41728 0.(3031 -0,17906 -C .18492 -0.24368 -0.17511 -0.03730 0.«.73 C.6221 0.0000 0.0691 0.0680 0.6991 0.4500 0.4351 ‘iîîSl 0.3005 0.4603 -V.HU 0.8799 2C 20 It 16 2( 2C 20 20 20 19 20 *nr -C .60242 1 .tcooo -C.99999 C.0063 '6:s2s? C .0000 C .0001 •'ôtéift '°ô:UU " e f t s # " g ! ; n s 19 19 16 19 19 19 IS 19 19 19 19 18 19 hEAOF 0.53759 C .59997 -C .46728 -0.99999 1 .CCOOO C.451C1 0.38158 0.38962 0.09228 o .c n c C .0066 C.068C C.OOOl c .cooo G.0526 0.1070 0.0992 0 .7 ,57 I® IS 16 19 19 19 19 19 19 19 19 P 0.32221 C .!66(C -0.95226 0.45 101 1.COOCO 0.6(821 C.50072 (.70302 0.54690 0.60800 0.47300 0.42859 O.CJOf C.OOCl 0.0519 0.0526 O.OOCO O.COOJ o.ooo| O.OOOI o.ooo| O.OOOj O.OOOJ 0.0005 f 2 ( 2 20 IS 19 62 61 K 0.23*30 C.PC536 -0.30301 0.3( 15P €.(0821 1 .CCCOO (.44397 (.50054 0.54504 «.42911 O.C((t C .COOl 0.1058 C.10/0 0.0001 O.COOO 0.0006 O.COOl 0.0001 0.0009 62 62 20 19 19 62 62 56 61 62 62 60 62 CA 0..27b3 C.67909 - C . 18492 -0.36089 0.35920 0.50072 0.44397 1.00000 0.38479 0.67236 0.98968 0.93006 0.38455 0.C9 17 C.COOl C . 3 | . C.I29C 0.1310 C.OOC] 0.(006 0.0000 0.0034 0.000! 0.0001 0.000| 0.0034 56 19 19 5( 56 56 56 56 96 56 MG 0 .,]1 5 5 C .46040 -0.(8138 0.CGC63 C.7C3C2 0.50054 (.384 79 l.OCOOO 0.32301 0.20991 0.234 72 0.28206 C.C906 C .0002 r . n i l 0.7405 C .7428 O.COCl O.COOl 0.0034 O.COOO 0.0111 0.1091 0.0736 0.02 76 # 1 6 1 20 19 19 (1 61 56 61 61 61 99 61 KN 0.151 73 C.'66C4 -C .24368 -0.39138 0.38962 C.546S0 0.54504 (.6 7236 (.32301 1 .00000 0.72416 0.68946 0.65072 0.4" 4ÎJ C .0001 0.3C05 0.0975 0.0992 C.COCl O.COOl C.OOOl O .C ll, 0.0000 0.0001 O.OOOI 0.0001 (2 62 20 19 19 62 62 56 62 62 60 62 rc 0.C3364 C.4C991 -0.17511 0.(1963 -0.02053 c.foeco 0.42911 (.58568 0.72416 l.COOOO 0.49993 0.399 2 7 0 .7‘.*2 C.C009 C .4603 0.9364 0.9335 O.OOCl 0.(005 0.0001 0.0001 0.0000 0.0001 0.0013 C. 6 2 20 IS 19 62 62 56 61 62 62 60 62 CL O.C5S?7 C .42166 -C .09209 0 .09220 0.473(0 (.53006 (.23472 0.68948 0.49993 I.OOOCO 0.73410 0 .t7 3 i C .0009 C.7163 C.7157 C.OOCl 0.000^ 0.0736 0.0001 0.0001 0.0000 0.0001 # ( 6C 19 (C 60 59 60 60 60 60 7N 0.(91 3( C .36831 0.428*9 (.38455 (.28206 0.65072 0.39927 0.73410 1,00000 (J .4* 72 C.C032 C.00C5 0.00 34 0.0276 0.0001 0.0013 0.0001 0.0000 6i 62 19 19 62 62 56 61 62 62 60 62 N> Ln Ln TABLE 43 (continued) High moisture ground ear corn ( F CP «0166 *01 M «01 1 K C« MC MM FE tu IM I .cctoo C .17667 0.30201 -0 .(30 98 0.03031 0.29796 -0.00839 C.00737 0.29539 0.10623 -0.19927 0.10885 0.23865 C.CCOO c.cooe O.OCCl C.»” O.OOCl 0.8729 0.8876 O.COOl 0.0919 0.0056 0.0369 0 .0 0 0 1 3 73 373 169 373 373 370 372 369 368 368 369 CP 0.17(6? 1 .CCOOC -0.09079 -0.C8898 0.08888 0.56973 0.29969 (.35816 (.92979 0.90787 0.11283 0.90663 0.99733 C'.OCOe C.OOOO 0.2205 0.2976 0.2961 c.ogc^ O.OOOJ C.OOOl O.OOOJ o.oooi 0.0305 0 .0 0 0 1 0 .0 0 0 1 3 73 373 189 139 139 3 70 369 368 368 369 fOfNH 0.2C201 -C.CS075 1 .CCOOO 0.13192 -0.63132 C .07561 C.09973 -0.13150 -0.02968 0.11937 O.CCOl C.OOOO 0.0 001 C.0301 0.3077 0.5038 0.0776 0.6916 0.1095 if 4 : i?; 120 120 169 189 183 189 ' - « ’ill iei 181 Aur -C.C3CSE C .63192 l.COOOO -0 .09999 - 0 .0 0 2 0 1 0.39098 (.18266 C.IC250 0.40009 -0.01015 -0.03792 0 .0 0 0 1 c.oooo C.COOl C.9BI2 O.COOl 0.0320 0.2298 0 .0 0 0 1 0.9Ç66 0.6611 c 7)31 120 139 1 39 139 139 138 139 136 136 136 136 NE AOF 0.13031 ( .ctcee - C .63132 -0.99999 1 .COCOO 0.00215 -0.39100 -C .18289 -C.1C255 -0.39941 0.119*5 0.01062 0.03816 c . , , , ; O.OCOI 0 .0 0 0 1 0 .0 0 0 0 0.9800 0.0 001 0.0318 0 .0 0 0 1 0.1809 0.9029 0.6592 * 'Î33 120 139 139 139 139 138 136 136 136 r 0.;9796 C.46973 l.OCOOO 0.98396 C.37001 O.OCOI C.COOl 'ô?38i* '(!§g2S O.OOCO 0 .0 0 0 1 C.OOOl 'ÔÎ283Î 3 73 373 189 139 139 373 373 370 372 369 368 368 369 K -0.C O N 34 C.24464 0.22019 0.39098 -0.39100 0.96396 l.OCOOO C.35539 0.35288 0.99951 0.19699 0.92738 C.P724 t.COCJ 0.0027 0 . 0 0 0^ 0 .0 001 O.OOCl o.oogg 0.05^1 o.oj^j O.OOOJ 0 .0 0 0 1 373 189 139 373 368 369 CA 0.C0737 0.09973 0.18266 -0.18289 0.37001 0.35539 I.OOOCO 0.51076 0.97963 0.53996 t . t t 7 t ' u é o ô î 0.5038 0.0320 0.0318 0.0 001 0 .0 0 0 1 c.oooo 'ô!88iî O.OOOI 0 .0 001 0 .0 0 0 1 37C 37C 183 1 38 370 370 370 370 366 364 365 366 ML O.ÎSÎ 34 C.42S74 0.69039 0.35288 (.22817 l.OCOOO 0.30950 0.04946 0.21397 0.936 9 7 O.CCOl C.OOOl 'ôîâisi °0l5l5i C.OOCl C.OOOl 0 .0 0 0 1 0 .0 0 0 0 O.OOOJ 0.0 001 O.OCOI 3 72 372 189 139 139 372 372 370 372 368 367 368 HN 0.10623 C.407P7 C. 17675 0.90009 -0.39991 0.32982 0.99951 (.5 1 0 76 C .30950 1 .00 000 0.23168 0.39559 0.99993 0.0414 C.OOOl 0.0173 0 .0 001 C.COOl O.OOCl O.COOl 0 .0 0 0 1 O.OOOI O.OQOO 0 .0 0 0 1 O.O5 OJ 0 .(2 0 1 369 369 191 131 136 369 369 366 368 369 368 369 f E -0.14427 0 . 1 12F3 -0.13150 -0.11500 0.11555 0.21967 0.19699 (.30725 C.09996 0.23168 1.COOOO 0.20657 0.31763 o.cc‘ t C.030Î C.C776 C.OOCl C.COOl O.OOCl C.OOOl 0 .0 0 0 0 0 .0 001 0 .0 0 0 1 ?(» 360 181 368. 368 365 368 368 367 368 CU C.ICIP" C .60663 -0.02968 -0.01015 0.01062 0.90593 0.31159 C.9 7963 (.21397 0.39559 0.20657 1.0 000 0 0.62357 0.C36») C.OOOl 0.6916 0.9066 0.9029 C.OOOl O.CCOl C.OOOl O.COOl O.OOOI o.oooi O.OQOO O.OOOI J t l 368 181 136 136 368 368 365 367 368 367 368 368 7H 0 .2 3tft^ 0.99733 0.11937 -0.03792 0.03816 0.670C6 0.92738 (.53996 (.93697 0.99993 0.31763 0.62357 1 .0 000 0 C.CCOI C.COOl 0.6611 0.6 512 C.OOOl 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 0 .0 0 0 1 O.OOQI 0 .0 0 0 1 0 .0 0 0 0 : 6' 309 1 36 369 369 366 368 369 368 368 369 N) Ln ON TABLE A3 (continued) Barley grain CP AEF NEAOF p K CA FAT SALT ASH CP loOOCOO -0.29954 -0.01094 0.09888 0.26786 C.OOOO °5 înîi 0.1215 C.9568 0.6237 0.3159 47 28 20 27 0 27 0 16 0 ADF 1.OCCOO -1.00000 0.34446 C.OOOO o.oooi C.C918 28 28 28 25 0 25 0 0 0 NEAOF -0,34446 -^ôî8§8î 's?8g88 C.G918 28 28 28 25 0 25 0 0 0 P -0.01094 0.34446 -C.34446 l.COOOO -0,07955 0.9568 0.0918 0.0918 0.0000 0.6933 27 25 25 27 0 27 0 0 0 0 0 0 0 0 0 0 0 0 CA -0.05775 -0.07955 1,00000 0.7839 C.6933 0.0000 27 25 25 27 0 27 0 0 0 FAT 0 C 0 0 0 0 0 0 0 SALT '5?g888 16 0 0 0 0 0 0 16 0 ASH 0 0 0 0 0 0 0 0 0 N> Ln TABLE 43 (continued) Brewers dried grains CP ADF NEAOF CA FAT SALT ASH CP i.Qogoo O.OOOO 86 17 13 ADF NEAOF CA FAT SAIT 1.00000 0.0000 17 17 ASH -0 ..>60] ^5?88 13 13 N5 Ln 00 TABLE 43 (continued) Corn grain CP ADF NEAOF p K CA FAT SALT ASH CP 1.00000 -0.16971 C. 16971 0.26492 0.14957 0.0000 0.0545 0.0545 0.0011 0.0706 -°gilS|| 184 129 129 149 0 147 0 0 AOF 'g?82S8 - ' 5 8 » -°g5U?8 129 129 129 129 0 127 0 0 0 NEAOF 0.05702 -0.03801 - ‘s?§88? :g?8888 Ü.5210 0.6714 129 129 129 129 0 127 0 0 0 P 0.05702 l.COOOO 0.5210 0.0000 “ô!8ai? 149 129 129 149 0 147 0 3 0 0 0 0 0 0 0 0 0 0 CA 0.14957 0.03801 0.22698 C.C706 0.6714 0.0057 'g?8888 147 127 127 147 0 147 0 3 0 FAT 0 C 0 0 0 0 0 0 0 SALT 1.00000 0.0000 35 C 0 3 0 3 0 35 0 ASF 0 0 0 0 0 0 0 0 0 NJ Ln VO TABLE 4 3 (continued) Corn gluten feed CP AOF NEAOF CA FAT SALT ASH CP -0.67766 -0.15149 0.42300 0.0648 0.4328 0.0313 6& 12 12 0 8 29 26 AOF 0.11993 I.OOOCO -1.ÛOCOO 0.44999 -0,75593 C.7104 C.OCCC C.OCOl 0.3110 0.4544 12 12 12 0 7 3 0 NEAOF -0.11993 -1.00000 1.00000 -0.44999 0.75593 0.7104 O.OCOI ' 0.0000 - - 0.3110 0.4544 12 12 12 0 7 3 0 CA 0 0 0 FAT -0.67766 0.44999 -0.44999 0.0648 0.3110 0.3110 ' o M 8 7 7 0 8 3 SALT -0.15149 -0.75593 C.75593 0.4544 0.4544 29 3 3 0 3 29 ASF l.OOQOO 0.0000 26 26 ho O TABLE 43 (continued) Corn gluten meal CP ADF NEAOF CA FAT SALT ASH C>> ^028888 81 0 16 10 AOF NEAOF CA FAT 0 SALT "o^OooC 16 0 16 ASF ' ° b \ n n '688888 10 0 10 N) Os TABLE 4 3 (continued) Distillers dried grains CP ADF NEAOF CA FAT SALT ASH CP ^258888 °ô!§822 °ô?îi8? 98 5 5 0 3 22 17 ADF °ô«§8?5 0 3 0 0 NEAOF 0.13888 -l.OOQOO 1.00000 -0.82054 0.8237 O.OCCO 0.0000 0.3873 5 5 5 0 3 0 0 CA 0 FAT 'a?8888 3 3 3 3 SALT -0.30889 C.1619 'a?88 22 0 0 22 0 ASh 0j07|08 ' ô M 17 NJ ON hO TABLE 4 3 (continued) Feather meal CP AOF NEAOF P K CA FAT SALT ASH CP l.COOOO C.21891 0.0000 0.0822 67 64 64 0 64 3 20 0 AOF -0.98596 0.96836 0.0140 0.0316 4 4 4 4 0 4 0 4 0 NEAOF -0.85477 '5^8888 0.0001 64 4 64 64 0 64 20 0 P 0.96836 -0.85477 1.00000 0.0316 0.0001 0.0000 64 4 64 64 0 64 3 20 0 0 0 0 0 0 0 0 0 0 CA -0.28648 0.99083 -0.76270 0.31625 l.OOOOQ 0.0217 0.0092 0.0001 0.0109 0.0000 °ô!:î8î 64 4 64 64 0 64 20 0 FAT -0.25545 -C.69338 0.96458 1.00000 0.8356 0.5122 C.1699 O.OOCO 3 0 3 3 0 3 3 0 0 SALT -0.50059 0.05928 -0.34187 0.37936 0.28806 0.0246 0.9407 0.1401 0.0990 0.2181 'ô?888g 20 4 20 20 0 20 0 20 0 ASH 0 0 0 0 0 0 0 0 0 fO w TABLE A3 (continued) M e a t a n d bone meal CP ADF NEADF P K CA FA7 SALT ASH CP -0.46804 '298888 C.OOCl - ° D \ m 88 0 87 88 0 88 87 84 0 ADF 0 0 0 0 0 0 0 0 0 NEAOF 0.38330 l.COCOO -ü.90544 C.0Q02 0.0000 C.OOOl 8? 0 67 87 0 87 87 83 0 P -0.46804 -0.90544 1.00000 0.86041 -0.36538 -0.03070 0.0001 O.OCOI C.OOOO 0.0001 0.0005 0.7816 88 0 87 88 , 0 88 87 84 0 0 0 0 0 0 0 0 0 0 CA -0.51331 l.OOOOQ O.OOOÏ -'29888Î °2!888l O.OCCO -°ô98SI? 88 0 87 88 0 88 87 84 0 FAT 0.62081 1.00000 0.0001 -°ô98SI? 0.0000 87 0 87 87 0 87 87 83 0 SAIT 0.05464 -0.03070 0.6237 C.7E16 v . i n i '298888 84 0 83 84 0 84 83 84 0 ASh 0 0 0 0 0 0 0 0 0 N 3 TABLE A3 (continued) Oat grain CP ADF NEADF P K CA FAT SALT ASH CP 0-40589 0.0546 =o!U9| 38 21 21 23 0 23 0 0 ADF 'a?8288 -'sS888g 2 1 21 21 21 0 21 0 0 0 NEAOF 0,44944 1.00000 0.03495 0.0410 -'ô?8§8? O.OOOO 0.8804 21 21 21 21 0 21 0 0 0 P -0.03495 0.03495 l.COOOO 0.8804 0.8804 c.oooo =5:8191 23 21 21 23 0 23 0 0 0 0 0 0 0 0 0 0 0 0 CA -C.23482 1.00000 0.3C55 0.0000 23 21 21 23 0 23 0 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT 1.00000 0.0000 7 0 0 0 0 0 0 0 ASh 0 0 0 0 0 0 0 0 0 ho Ln TABLE A3 (continued) R a p e s e e d m e a l ( C a n o l a ) CP ADF NEADFP K CA FAT SALTASH CP -0.02526 -0.18016 '628888 0.7460 0.0067 = 6 2 9 H | 225 222 222 0 167 0 225 167 0 ADF '628888 " 6 2 8 8 8 8 °ô25lli -=ô2I§?i = 6 2 1 1 # 222 222 222 0 164 0 222 164 0 NEADF " 5 2 8 8 8 8 '528888 - = 6 2 # 9 i 222 222 222 0 164 0 222 164 0 0 0 0 0 0 0 0 0 0 K -0.02526 0.06295 -0.06295 1.00000 0.7460 0.4232 0.4232 0.0000 = 6 2 2 ! # 1.6 7 164 164 0 167 0 167 167 0 CA 0 0 0 0 0 0 0 0 0 FAT -0.18016 0.07093 1.00000 0.0067 0.2415 0.0000 225 222 222 0 167 0 225 16? 0 SALT -0.04262 ° ô 2 s a % 0.5879 -°ôi8i8l '628888 167 164 164 0 167 0 167 167 0 ASF 0 0 0 0 0 0 0 0 0 NJ ON ON TABLE 4 3 (continued) Soybean meal CP AtF NEADF CA FAT SALT ASH '2?8888 100 31 31 0 15 15 'ô?8888 - * ô M -°ô?2ièè 31 31 31 0 3 0 NEAOF 0.18947 -l.OQOOO 1.00000 0.3073 0.0001 0.0000 ° ô V o ü h 31 31 31 0 3 0 CA FAT 0 0 0 SALT 0.28708 -0.99211 0.99211 0.2995 C.08C0 0.0600 '6?8888 15 3 3 0 15 0 ASF ^o?8888 15 0 0 0 0 15 ro ON '«vj TABLE 43 (continued) Sunflower meal CP ADF NEAOF CA FAT SALT ASH CP - ° 6 2 8 m -*528:11 *5!8%8 101 89 89 69 0 89 ' 59 AOF ^028828 -'528882 - ° 5 2 i m *52^888 -°ô21Hl -hnmo9 89 89 69 0 89 59 NEAOF °c2â38? "52288! '528288 *52113% -*523888 *ô25Hl 89 89 89 69 0 89 59 1,00000 -0.34895 -0.33874 -*528m-*521138 *521H8 5.0000 0.0033■ 0.0087 69 69 69 69 0 69 59 CA FAT -0.34895 *ô 2o 657 *ô2ioo8 *5.3004 0.0033 '52*8*8 *6t04Ï 89 89 89 0 69 0 89 59 SALT -0 *5!82:2 -*5.'tiîi *ô.‘l3îi 087 *5!8S!i '528888 59 59 59 59 59 59 ASh fO O' 03 TABLE 4 3 (continued) Wheat grain CP AOF NEAOF p K CA FAT SALTASH CP '5?8288 “ 582881 88 56 56 60 0 60 0 20 0 AOF '688888 " 5 8 8 8 8 8 56 56 56 53 0 53 0 0 0 NEAOF -'688888 '588888 -“ô'.tgaa 56 56 56 53 0 53 0 0 0 P l.OQOOO O.OOCO 60 53 53 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 CA °5tSg8S ^ôl^ïiî '588888 60 53 53 60 0 60 0 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT l.OCOOO 0.0000 20 0 0 0 0 0 0 20 0 ASh N> On VO TABLE 43 (continued) Wheat shorts CP AOF NEAOF P KCAFAT SALT ASH CP 1 6 2 143 0 155 0 0 155 0 AOF ° b V 6 m 14 J 144 144 0 140 0 0 139 0 NEADF "siSSSl - Y . t m 14i 144 144 0 140 0 0 139 0 0 0 0 0 0 0 0 0 0 K 'ô?8888 155 140 140 0 15b 0 0 153 0 CA 0 0 0 0 0 0 0 0 0 FAT 0 0 0 0 0 0 0 0 0 SALT ‘8?8888 155 139 139 0 153 0 0 155 0 ASH 0 0 0 0 0 0 0 0 0 a Each cell reports the correlation coefficient (r), the probability that r / 0, and the number of observation to compute r. o