Controlling the Risk for an Agricultural Harvest

MANUFACTURING & SERVICE

Stuart J. Allen Penn State University, [email protected] Edmund W. Schuster MIT Auto-ID Labs, 23 Valencia Drive, Nashua, New Hampshire 03062, [email protected]

athering the harvest represents a complex managerial problem for agricultural cooperatives involved in har- Gvesting and processing operations: balancing the risk of overinvestment with the risk of underproduction. The rate to harvest crops and the corresponding capital investment are critical strategic decisions in situations where poor weather conditions present a risk of crop loss. In this article, we discuss a case study of the Concord grape harvest and develop a mathematical model to control harvest risk. The model involves differentiation of a joint probability distribution that represents risks associated with the length of the harvest season and the size of the crop. This approach is becoming popular as a means of dealing with complex problems involving operational and supply chain risk. Signiﬁcant cost avoidance, in the millions of dollars, results from practical implementation of the Harvest Model. Using real data, we found that the Harvest Model provides lower-cost solutions in situations involving moderate variability in both the length of season and the crop size as compared to solutions based on imposed risk policies determined by management. Key words: harvest risk; agriculture History: Received: July 30, 2002; accepted: December 12, 2003. This paper was with the authors 7 1 months for 2 6 revisions.

1. Introduction Determining the optimum rate to harvest and process Gathering the harvest is a complex managerial prob- the crop is a challenging problem in balancing the lem in controlling risk. Since the dawn of agriculture risk of overinvestment in capital with the risk of not about 12,000 years ago, farmers have dealt with sev- harvesting the entire crop. eral categories of risk in producing the food needed to sustain their lives and to build urban centers of 2. Structure of the Concord industry and learning (Garraty and Gay 1972, p. 50). Grape Industry For an agricultural cooperative in modern times, The Concord grape, a flavorful, aromatic purple vari- the harvest is the primary source of assets needed to ety of grape, is grown in the cooler regions of the run the business. Hence, the harvest represents the United States. Major growing areas include western base raw material and most important link in the sup- New York, northern Ohio, and northern Pennsylva- ply chain for these firms. Before each harvest, man- nia (all three near Lake Erie); western Michigan; and agers must make critical assessments concerning the south-central Washington state. In 1869 Thomas B. capability of harvesting the entire crop at optimum Welch, a dentist from New Jersey, used pasteuriza- maturity. tion as a means to preserve Concord grape juice. By In this article, we analyze the risk associated with heating the grape juice before bottling he was able to the harvesting of Concord grapes through the devel- stop fermentation, creating the first nonalcoholic fruit opment of a mathematical model and the presentation juice product. This event marked the beginning of the of results from practice. Specifically, the model calcu- bottled and canned fruit juice industry in the United lates the optimal rate for a processing plant to receive States. grapes. This is a common situation in agriculture During the 130-plus years since this genesis, the where growers deliver crops to a central receiving market for Concord grapes has grown into a billion- area for further processing and storage prior to sale. dollar industry with an annual harvest of about

400,000 tons. Though the Concord grape comprises eight hours after picking to avoid fermentation and only about 6% of the total grapes grown in the United deterioration of quality. Exceeding this time limit States, it maintains a high profile because of intensive means the grapes must be diverted, at severe cost marketing; it remains a popular ingredient for juices, penalty, to other uses, such as wine making. The fruit jellies and jams, and concentrates. Not suitable economics of refrigerated warehousing, as well as for the fresh market because of its short shelf life, biological degradation at low temperatures, preclude nearly all of the Concord grapes harvested are imme- storage of raw Concord grapes for later pressing into diately converted into juice. juice. The leading player of the industry is Welch Foods, In each growing area, approximately 100 mechan- Inc., which is owned by the National Grape Cooper- ical harvesters pick grapes on a continuous basis. ative Association. Welch Foods controls about 55% of Since the cost of an individual harvester is very the Concord grapes grown in the United States. The high, approaching $200,000 per unit, growers usually next-largest player controls less than 7%. Collectively, pool resources through joint ownership or contract- five business structures comprise the Concord grape ing. A single harvester picks an average of three sep- industry. arately owned vineyards. Coordination of the field (1) Private packers that purchase Concord grape operations becomes critical to ensure a steady feed of juice as a major ingredient for their products, both freshly picked grapes to the processing plant. branded and house brands. (2) Private processors that contract with growers on 2.2. Harvest Scheduling and Capacity an annual basis for delivery of grapes, then process the Prior to the harvest, planners assign a constant rate of grapes into concentrate for sale as a raw ingredient. picking to each harvester operator based on acreage, (3) Private processors with packaging capabilities, pri- estimated yields, the estimated length of the har- marily involved in production of house brands. vest, and the standard pressing rate for the receiving (4) Cooperatives owned by growers that process plant. This practice establishes fairness: Each vineyard grapes into raw ingredients, sell to other companies, owner faces the same risk of losing crop because of and return profits to the growers. poor weather, such as a frost. A hard frost, where tem- (5) Vertically integrated cooperatives involved in peratures dip below 28 F, weakens stem tissue, caus- growing, processing, packaging, distributing, and ing the grapes to fall from the vines. This is called marketing Concord grape products. shelling. Every grower wants to complete the harvest The industry is a mix of both vertically integrated before a hard frost causes shelling in the vineyards. and modular structures (Fine 1998). However, the The total daily schedule for all harvesters equals long-term trend is toward supply chain integration 80% of the standard plant pressing rate. The reserve similar to the transition taking place in the poultry of 20% allows planners flexibility during harvest industry (Kinsey 2001). Product life cycles are long, operations to accommodate individual growers who with few radical product innovations. Intense compe- might experience crop maturity problems, mechani- tition and lack of pricing power in the market com- cal breakdowns, or an abnormally high risk of crop bine to force a focus on cost control. loss due to frost. The reserve is completely used as part of day-to-day harvest operations so that the rate 2.1. Operational Design, Organization, and of picking always matches the rate of pressing at the Financial Implications receiving plant. This implies that the rate of picking All businesses that convert Concord grapes into and pressing is dictated at the same time that capac- juice—cooperatives and private companies alike— ity investment decisions are made (i.e., once per year, follow similar steps during the harvest process. In the projecting five years into the future). fall, usually beginning in late September, fruit grow- In the midst of harvest operations, it is very dif- ers deliver grapes picked by mechanical harvesters ficult to change the rates of picking and pressing to processing plants located near the vineyards. The grapes because of the large number of harvester oper- grapes are then converted into juice through a com- ators and the complexity of receiving plant opera- plex pressing process. This step takes place within tions. When day-to-day pressing or picking rates vary Allen and Schuster: Controlling the Risk for an Agricultural Harvest Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS 227 by even small amounts a classic system dynamics harvest as part of rolling five-year plans because of response occurs, producing substantial loss of effi- the long lead times for capital equipment. A balance ciency that extends many days past the original dis- must be made between the costs of the resources com- ruption. Extensive preharvest efforts focus on testing mitted and the costs of a partially completed task and maintenance of plant equipment to ensure relia- (grapes remaining in the vineyard). With whatever bility in achieving the standard pressing rate per day. methods employed, the derivation of the optimal har- Upon receipt from the vineyards, the grapes are vest rate is critical for the grower in controlling risk heated and pressed into unsettled juice that is stored and in determining the proper capital investment. in large refrigerated tanks. Unsettled juice is the intermediate step in processing. Welch Foods alone 3.1. The Elements of Uncertainty has over 55 million gallons of tank storage capacity. With uncertainty in both the size of future grape crops Throughout the remainder of the year, the unsettled and the time window available for future harvests, it juice is further processed and used for bottling and as is impossible to calculate the optimal rate of pressing an ingredient in other food products. The manufac- using direct methods. Simply dividing the historical turing structure takes on a “V” shape typical in the average crop size by an estimate of the available days process industries where a small number of raw mate- of harvest will result in a solution that ignores the rials are used to produce a large number of end items inherent risk of agriculture. (Umble 1992). Final processing of the entire harvest The essence of this problem can be defined by three takes close to one year. variables: 2.3. Financial Implications Investment in H = crop size in tons, a random variable; Capital Assets L = length of harvest season in days, a random Investment in capital assets to process Concord variable; grapes into juice is substantial. For cooperatives this R = pressing rate in tons per day, the decision variable. investment in pressing capacity, that remains idle for 10 months per year, is a drain on resources. Despite From these variables two sets of expressions can be tax advantages, cooperatives involved in processing defined, one set each for the amount of crop loss (ACL) raw fruits and vegetables typically have a low return and the excess harvest rate (EHR). on capital compared with the rest of the food industry. In this case, the proper sizing of harvest processing H − RL if H>RL ACL = (1) capacity becomes an important strategic decision with 0ifH ≤ RL significant financial impacts. 0ifH ≥ RL EHR = (2) 3. ProblemDefinition RL − H if H

3.2. Important Model Assumptions • Overage cost (CR, dollars per ton). This reflects the and Justifications cost of harvesting too quickly, therefore committing Allen and Schuster (2000, pp. 32–33) report that for an excessive amount of resources to the harvesting one of the Concord grape growing regions a correla- effort. tion coefficient of 0.14 exists between H and L based The formation of the Harvest Model involves on 22 years of data at an observed significance of applying costs to ACL and EHR. Because the Har- 53%. The results of this analysis offer support that H vest Model is probabilistic in nature, expected values and L are statistically independent. This makes sense are needed for both ACL and EHR. Once expressions because H is dependent on many variables such as exist for each, costs can be applied to form a total cost precipitation, soil conditions, temperatures, and avail- function. The minimum of this function will provide able sunlight, which are uncorrelated to the time win- the optimal harvest rate at minimum cost. dow of good weather available to gather the harvest. The remainder of this study will focus on the mat- Then the joint density function can be formed as the ter of harvesting Concord grapes with an uncertain product of two marginal distributions. length of harvest season and an uncertain crop size. Section 4 provides an overview of the literature relat- gH L = fH × f L (3) ing to the Harvest Model. Section 5 gives specifics about the derivation of the expected values and the Another important assumption is the shape of minimization of the total cost function. In §6 detailed the probability distributions used to estimate H and numerical results of the Harvest Model are presented L. Extensive historical information from state agri- based on real data, followed by §7, which provides cultural experiment stations, spanning more than managerial insights. 40 years of history, allows calculation of the mean and standard deviation in each case. Histograms of this data show that both H and L are mound 4. A Review of Cognate Research shaped and reasonably symmetrical. An analysis of The analysis of harvest risk draws upon three gen- data from Allen and Schuster (2000, pp. 32–33), using eral areas of previous research, including the analysis a chi-square goodness-of-fit test, shows insufficient of harvesting operations, agricultural decision making evidence (p = 053 and p = 066) to reject the null involving risk, and the newsvendor model. hypothesis that H and L are normally distributed. In both cases, one degree of freedom exists. The modi- 4.1. Analysis of Harvest Operations fied histograms of H and L contain four observations Thornthwaite (1953) addresses the problem of har- in one and two intervals, respectively. This is out- vesting peas at the peak of maturity. Subsequent side the minimum guideline of five observations for analysis by Kreiner (1994) shows that Thornthwaite’s the chi-square goodness-of-fit test. It is also impor- method was still in use more than 40 years later tant to note that in similar situations researchers use and has reduced harvest and processing costs, and symmetrical distributions (Jones et al. 2001). These lost product. Though Thornthwaite’s work involves observations lead us to assume L and H are normally annual crops (peas), and does not deal with the probability of losing crop as a result of frost, it remains distributed with mean and standard deviation L, L a classic in the application of operations research. To and H , H respectively. our knowledge this is the first reference on the scien- 3.3. The Objective Function tific study of harvest operations. The Harvest Model minimizes a total cost function Porteus (1993a, 1993b) develops a two-part case that contains two costs. study of the National Cranberry Cooperative. In this

• Underage cost (CH , dollars per ton). This reflects the case, he examines complex trade-offs in capital invest- cost of harvesting too slowly, therefore foregoing sales ment and capacity for the processing of cranber- of grapes left in the field at the end of the harvest ries during a harvest season. Although the harvest season. and processing of cranberries share common elements Allen and Schuster: Controlling the Risk for an Agricultural Harvest Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS 229 with Concord grapes, the author did not address the in raw materials management (Schuster and Allen risk of crop loss from a frost, or from overmaturation. 1998, Schuster et al. 2000) and production planning In aggregate, these four references provide impor- (Schuster and Finch 1990, Allen and Schuster 1994, tant background information on harvesting opera- Allen et al. 1997, D’Itri et al. 1998). Though these mod- tions plus specifics about the uniqueness of this area els yield significant financial savings in practice, none of management. considers the risk associated with harvesting Concord grapes. 4.2. Agricultural Decision Making In a recent article, Allen and Schuster (2000) add- Involving Risk ress harvest risk for Concord grapes through a work- Maatman et al. (2002) note that “various methods ing model with the successful outcome of a closed of risk reduction exist, including those aimed at pre- form solution. Using this approach, senior manage- vention of risk (e.g., irrigation), dispersion of risk (by ment specifies risk levels as policy. An example would diversification of risky activities such as the cultiva- be “Harvest 100% of the crop, 85% of the time.” tion of different varieties of crops), control of risk (e.g., The policy level becomes an important input. Despite by sequential decision making) and ‘insurance’ against offering a large improvement over previous methods risk” (p. 400, italics in original). that involved heuristics, this model does not include Much of research in agricultural decision making focuses on the dispersion of risk through improved an explicit treatment of the cost of lost crop or sea- individual farm planning. Mathematical program- sonal operating costs. ming is often the tool used for modeling farmers’ 4.3. The Newsvendor Model strategies under uncertain conditions (see Anderson The newsvendor approach is effective in a num- et al. 1977, Hazell and Norton 1986). These models are ber of applications, including the defense industry static in nature, assuming that probabilities of occur- (Masters 1987), and is the underpinning of our har- rence are known in advance and that all decisions are vest model. Nahmias (1997, pp. 272–280) provides made at one time. a comprehensive review of the newsvendor model This class of models is similar in spirit to what is including derivation, historical context, and several presented in this article. However, the Harvest Model does not focus on the dispersion of economic risk and important extensions. does not employ traditional mathematical program- The process of harvesting grapes shares character- ming. Rather, we concentrate on control of risk for a istics with the management of style goods such as pooled harvest situation (a cooperative) through opti- fashion items in the clothing industry. As is true of mization based on a generalization of the newsvendor style goods, grapes are perishable. Concord grapes problem. have a limited time for peak maturity combined with Jones et al. (2001) study the control of risk for pro- a short time window for harvesting during the fall. In duction of hybrid seed corn from the perspective of this regard, the harvest of Concord grapes shares sim- both random yields and demands. A follow-up study ilarities to seasonal sales of clothing where there are reports results in practice (Jones et al. 2003). The substantial costs of obsolescence and limited oppor- authors’ work is similar in style to the Harvest Model. tunities to store for future sale. However, the authors do not address uncertain start An important body of literature addresses the style and end times for a harvest, choosing instead to focus goods problem. Fisher and Raman (1996) develop a on a two-stage production model that involves two response-based production strategy to deal with the different regions (North America and South America) uncertain demand characteristic of style goods. Other with staggered growing seasons. In this regard, their authors improve style goods forecasts by using a work differs from the Harvest Model, and will no Bayesian approach (Eppen and Iyer 1997, Murry and doubt lead to further work in controlling agricultural Silver 1966). However, with the harvest of Concord risk. grapes, little opportunity exists to change the harvest Previous research involving Concord grapes and rate during seasonal operations, rendering Bayesian agricultural cooperatives deals with decision making methods less effective. Allen and Schuster: Controlling the Risk for an Agricultural Harvest 230 Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS

As a concluding comment, the distinguishing Finally, we define the two coefficients of variation aspect of the Harvest Model is that it contains two random variables (H and L) as compared to the tradi- kH = H / H kL = L/ L tional newsvendor model that contains only a single Now Q can be expressed entirely in terms of the two random variable (demand). In this way, the harvest coefficients of variation, the standard variate, z , and risk model shares similarities with the work of Song L the dimensionless harvest rate, r: et al. (2000) that considers two random variables. r1 + k z − 1 Q = L L (10) 5. Mathematical Development kH We now turn our attention to expressions for the Then the expected ACL becomes expected ACL and EHR. E(ACL) = H kH GQ zL dzL (11) 5.1. The Expected ACL − Since the ACL is discontinuous at H = RL, the expec- Silver and Smith (1981) provide the tools to carry ted value is given by out the integration of this expression and subsequent L = H = results. First note that Q in Equation (10) can be E(ACL) = H − RL f H dH f L dL expressed in the form Q = az + b, where L =− H = RL L (4)

The inner integral is the familiar normal loss function. a = rkL/kH b = r − 1 /kH (12) We define the standard normal variate for H as In the interest of notational conciseness, we define H − H another recurring parameter as zH = (5) H c = b/ 1 + a2 (13) and RL − Q = H (6) Using Equations (12) and (13), the integrated form of H Equation (11) becomes Then the expected ACL becomes 2 E(ACL) = H kH 1 + a Gc (14) E(ACL) = H GQ f L dL (7) − In Equation (14) and all subsequent results, G· is where defined by Equation (8) and we emphasize that · is GQ = Q − Q1 − Q (8) the standard normal density function (i.e., zero mean and · is the standard normal density function and unit variance) and · is the cumulative of the while · is the cumulative of the standard normal standard normal density function. density function. The form of the expected ACL can 5.2. Expected EHR be simplified by introducing the nondimensional har- Similar to ACL, this function is also discontinuous at vest rate, r. In a risk-free environment, the risk-free H = RL so harvest rate becomes R = / . In this situation, 0 H L R is known because both H and L are known in L = H = RL E(EHR) = RL − H fH fL dHdL advance. Of course in reality uncertain crop size and L =− H =− uncertain harvest start and end times exist, so R is Now add and subtract the complement to the inner different from R . The definition for the nondimen- 0 integral and rearrange to get sional harvest rate becomes r = R/R . Next, we define 0 the standard normal variate for L as L = H = E(EHR) = H GQ + RL − H fH dH L − L =− H =− z = L (9) L · f L dL L Allen and Schuster: Controlling the Risk for an Agricultural Harvest Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS 231 where Q is as previously defined in Equation (10). 5.4. Expected Percentage Crop Recovery ∗ The inner integral simplifies to RL − H and on sub- Beyond r , the Harvest Model also permits calcula- sequent integration over L this becomes R L − H . tion of the expected amount of crop recovery. As men- This should be in nondimensional form as well, so tionedpreviously,the introductionofriskintoplanning using earlier definitions, the result is means that the harvest will be less than 100%.

We deﬁne the expected percentage crop recovery by E(EHR) = H kH GQ zL dzL + H r − 1 (15) − E(PCR) = expected amount of crop harvested The integrated form of Equation (15) becomes /mean harvest size] × 100 2 The expected proportion of ACL is E(ACL)/ . Then E(EHR) = H kH 1 + a Gc + H r − 1 (16) H

5.3. Expected Total Cost and Optimal Harvest Rate E(PCR) = 1 − E(ACL)/ H × 100 (21) The integral form of the total cost (TCP) per ton of harvest can now be expressed as an expected value 5.5. An Alternative Approach to Harvest Risk using Equations (11) and (15): Management Some decision makers prefer a policy approach (as opposed to a cost-based method) for dealing with risk E(TCF)/ H = CH kH GQ zL dzL − management. In an earlier study, Allen and Schuster (2000) examine a policy approach. The objective here + CRkH GQ zL dzL − is to explore the connections between that study and

+ CRr − 1 (17) the cost-based analysis in the present study. In this alternative, instead of specifying cost penalties man- The integrated form of Equation (17) is agement speciﬁes a policy for the probability of harvesting the entire crop. It is important to emphasize E(TCF)/ = C + C k 1 + a2Gc H R H H that this is not at all the same as specifying the pro-

+ CRr − 1 (18) portion of crop harvested as was addressed in the previous section. Next, an expression is obtained for the optimal The probability of harvesting the entire crop is dimensionless harvest rate. Taking the first derivative found from of Equation (17) with respect to r, setting the result to zero, and rearranging terms yields an implicit expres- Prob(complete crop harvest) sion for the optimal dimensionless harvest rate r∗ : ≡ P(CCH) = PH ≤ RL = 1 − PH > RL L = H = 1 − Q 1 + k z z dz = C /C + C (19) P(CCH) = 1 − fH dH f L dL L L L L R R H L =− H = RL − Again using Silver’s results, the integrated form of The inner integral is simply 1−FRL , so again using Equation (19) yields the definition of Q in Equation (10) employed in our earlier work, we obtain 2 c + kL a/ 1 + a c = 1 − CR/CR + CH (20) P(CCH) = 1 − 1 − Q zL dzL where a and c are functions of the decision variable r − and the parameters kH and kL through Equation (12). We will illustrate the use of this policy-based risk It deserves emphasis that r ∗ depends only on the management method for the case of a specified nondimensional ratios kH kL; and the cost ratio CR/ P(CCH) = 0.85 (85%) in §6.2. Integration of the above CR + CH . Furthermore, the second derivative of yields the following implicit equation for the dimen- Equation (17) with respect to r is positive for all pos- sionless harvest rate, r: sible z values so that Equation (20) will yield the minimum r. c = P(CCH) = 085 (22) Allen and Schuster: Controlling the Risk for an Agricultural Harvest 232 Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS

6. Numerical Results 10% or less and the corresponding capital required to This section draws from real data on the harvest of support the higher pressing rate. This was the proce- Concord grapes documented by Allen and Schuster dure used at Welch Foods. (2000) as a basis for examples and analysis. Their For example, at one grape processing plant located work deals specifically with the decision-making pro- in Pennsylvania, increasing R by 180 tons per day cess involving the harvest of Concord grapes for requires a capital investment of $1.5 million. During Welch Foods. The authors were directly involved the course of an average harvest, this increased rate in planning and managing harvest operations for a means that 5,400 extra tons of grapes can be pressed period of 10 years. before a hard frost. Assuming a 10 year life span, the estimated amortized cost of this extra capacity is 6.1. Data $150,000 per year. If part or all of this extra capacity is The model requires two sets of historical data: H not needed, then CR becomes the cost of the incremen- (tons) and L (days). Both H and L vary from year tal capacity divided by the additional tons harvested to year. In each growing area, the state agricultural at the higher rate ($150,000 per 5,400 tons), or about experiment station keeps detailed records by year on $30 per ton of grapes. This represents the estimated cost yields and crop size. Historical records also show the of incremental capacity occurring around the point in harvest start date (when the crop is at the correct level time when a hard frost might occur. of maturity) and the first 28F day (hard frost) that signifies the end of the harvest. The value of L for 6.2. Results each harvest can be calculated for a growing region We now provide numerical results from the Harvest by counting the days between the start and end dates. Model for the following practical problems: Similar to the crop size, L varies from year to year. • calculation of r ∗ under different scenarios, Besides H and L, the Harvest Model requires two • determination of the percent recovery for a spe- ∗ costs, CH and CR. The calculation of CH is straightfor- cific r , and ward; if a grower does not harvest a ton of grapes, • finding r under conditions of an imposed harvest then the relative value of the loss equals the open policy. market price. This is a conservative approach to val- Using the historical data for H and L along with uation. The examples presented assume C = $250 H CH and CR as inputs, the Harvest Model calculates per ton of grapes. This is a long-term estimate of the r ∗ providing the opportunity to analyze a number of open market price. The market price of grapes will harvest risk scenarios. change a considerable amount on a year-to-year basis Table 1 gives results for r ∗ of various combinations depending on industry crop size, carryover from the of kH and kL for a cost ratio, CR/CR + CH , of 0.10 previous year, consumer demand, and the pricing of using Equation (20). These were obtained by using the alternative raw ingredients. Goal Seek utility in Excel with a tolerance of 0.0001 for ∗ The calculation of CR is a more complex matter. all calculations of r. The data show that r increases Harvesting at an excessive rate means that the entire with increases in variability for H and L. This is a con- harvest will take place well in advance of a hard frost. sistent result. The greater the risk, the greater the r ∗. If this happens, then R is too high and the long-term investment in capital is excessive. In this situation C R Table 1 Optimal r for C /C + C = 01 relates to the amortized cost of the unneeded capacity. R R H

It is unique to each type of business organization. kH

Estimating CR is difﬁcult because it involves capital 0.05 0.1 0.15 0.2 0.25 equipment that is added in steps. There is no question kL 025 13350 13564 13828 14195 14620 that the cost is nonlinear over large ranges of R. How- 030 13880 14019 14291 14618 15004 ever, for small changes of R the cost becomes linear. 035 14306 14449 14662 14954 15305 An estimate can be calculated on a cost-per-ton basis 040 14586 14732 14879 15167 15489 by analyzing small increases of R, e.g., an increase of 045 14713 14860 15008 15239 15532 Allen and Schuster: Controlling the Risk for an Agricultural Harvest Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS 233

∗ Figure 1 Cost Ratio vs. r for kH = 025 Table 3 Total Cost per Harvest Ton at Optimal r

1.0 kH 0.05 0.1 0.15 0.2 0.25 0.9 kL=0.05 kL= 0.15 k 025 1731 1804 1919 2066 2237 0.8 L kL= 0.25 030 2204 2268 2369 2501 2659 k = 0.35 035 2728 2785 2876 2997 3144 ) 0.7 L H 040 3300 3352 3436 3548 3686 C 0 kL= .45

+ 045 3913 3962 4041 4147 4278

R 0.6 C /( R C 0.5 between the cost of unharvested crop and the cost 0.4 of an EHR. Table 2 is an important tool to explain why 100% recovery of a crop is not optimal given Cost Ratio 0.3 the trade-off between crop loss and a consistent early 0.2 completion of harvest operations. The total cost per ton for each value of r ∗ can be 0.1 calculated using Equation (18). These results appear ∗ 0.0 in Table 3. As is the case with r , the data show that 0.0 0.5 1.0 1.5 2.0 2.5 3.0 cost per ton increases with increases in variability for H and L. Optimal Dimensionless Harvest Rate, r* Finally, Goal Seek can be used to find r for specified values of k and k under an imposed policy for crop Additional descriptive results can be observed by H L recovery, Equation (22). Decision makers often use evaluating the left-hand side of Equation (20) over this approach to demonstrate how r changes for pol- ranges of r ∗ for representative values of k and k . H L icy alternatives (suboptimal values of r) approaching The resulting values correspond to the cost ratios at a complete harvest. Since a policy alternative might optimality. The set of graphs in Figure 1 are for a con- not be optimal, it is important to understand the cost stant k of 0.25 and a range of k values. An estimate H L difference between the optimal solution and the cho- of r ∗ can be made directly from the graph without sen policy. In some cases, choosing an r that is not invoking a search engine by specifying the cost ratio optimal is justified based on the timing of capacity and the coefficient of variation for the length of har- additions in relation to legacy equipment, an estab- vest season. This provides planners with a visualiza- lished consensus by growers who are risk adverse, tion of the relationship between the cost ratio and r ∗. and the uncertainty surrounding cost data or a shift in Computations of expected percentage of crop historical trends. By substituting the values of r into recovery using Equations (14) and (21) for the r ∗ of Equation (18) total costs can be computed for each Table 1 (for C /C + C = 010), appear in Table 2 R R H imposed policy alternative. These results appear in below. Tables 4 and 5. The imposed policy for this example Increasing variability in crop size and length of season is “Harvest 100% of the crop 85% of the time.” means lower crop recovery. This reflects the trade-off

Table 2 E(PCR) in % Table 4Policy r for 85%

kH kH 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 kL 025 9716 9708 9695 9679 9661 kL 025 13547 13682 13927 14225 14573 030 9596 9591 9580 9565 9548 030 14570 14684 14883 15144 15458 035 9452 9446 9435 9421 9404 035 15729 15842 16001 16249 16533 040 9275 9271 9256 9244 9227 040 17094 17265 17371 17545 17837 045 9067 9063 9049 9035 9018 045 18716 18903 19002 19192 19384 Allen and Schuster: Controlling the Risk for an Agricultural Harvest 234 Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS

Table 5 Total Cost in US$ per Harvest Ton for Policy r The Harvest Model gives consistent answers to a number of important questions involving risk. Based k H upon considerable use in practice, the Harvest Model 0.05 0.1 0.15 0.2 0.25 is suitable as a general solution for problems encoun- kL 025 1733 1805 1919 2066 2238 tered in agriculture where a sudden event, such as a 030 2226 2286 2384 2513 2667 fall frost, concludes the harvest season. 035 2798 2852 2936 3051 3191 040 3472 3528 3598 3696 3827 045 4275 4332 4394 4487 4596 7. Managerial Insights The true test of any model is performance in practice. Table 6 shows the cost penalties for the 85% policy The Harvest Model consistently provides reasonable ∗ as compared to r for each case of kL and kH . The cost results in addition to new insights. As an example, penalty for the 85% policy can be computed using the the traditional belief that the Concord harvest sea- following formula: son should last 30 days to ensure complete harvest percent cost penalty = (policy cost–optimal cost) of the crop proves inaccurate when analyzed using · 100/optimal cost. the Harvest Model. Regional climatic variation plays a signiﬁcant role in the probability of a frost. A large

At the extreme (kL = 045), the maximum cost body of water, such as Lake Erie, acts as a heat sink in penalty for the 85% policy is about 9.2%. For lower the fall, thereby extending the probable length of the values of kL, the penalty is less than 3%. It is inter- harvest season for growers in close proximity. Anal- esting to note that in all cases for a given kL the cost ysis using the Harvest Model shows that extending penalty decreases as kH increases. The greater vari- seasonal operations beyond 30 days in parts of the ability in crop size has the impact of narrowing the Lake Erie region can be accomplished with low risk cost difference between r, at 85% policy, and r ∗. Since of crop loss. This translates into a slower harvest rate in all cases the penalties are modest, we conclude that and reduced capital investment for cooperatives in the 85% policy gives an r that is close to optimum in this growing area. terms of cost (for the set of parameters examined). However, similar analysis for growers in Michigan, The Harvest Model also provides results based on who are located a considerable distance from any various policy alternatives that are close to those large body of water, shows that seasonal operations obtained in the study by Allen and Schuster (2000) should be less than 30 days because of signiﬁcant frost with some small differences in the second decimal risk. In this growing region a small probability exists place for the lower values of kH . The more accurate that a frost will occur before the beginning of the har- values are probably those of the prior study that did vest. Given this case, the harvest rate must increase, not rely on Goal Seek. This validates the Harvest which requires more capital investment. Model. The above results are not optimum from the The authors have experience implementing results point of view of minimizing costs but rather are the of the Harvest Model, using normal probability dis- result of an imposed policy. This is an effective way tributions to express harvest size and length of sea- of communicating harvest risk to growers investing son, for growing areas near Lakes Erie and Michigan. in receiving plants through agricultural cooperatives. In total, capital avoidance equaled $2 million through establishing the proper harvest rate by region. This Table 6 Cost Penalty for Policy r in % is one example of the beneﬁts of controlling harvest

kH risk, and of the adoption of precision agriculture in 0.05 0.1 0.15 0.2 0.25 practice. Beyond savings in capital, the Harvest Model pro- kL 025 014 007 003 000 000 030 096 081 063 046 031 vides other organizational beneﬁts. In the case of 035 256 239 207 181 150 Welch Foods, the Harvest Model is an important 040 522 525 472 417 382 tool for corporate planning with results of the model 045 923 934 874 819 743 being presented at meetings of the board of directors. Allen and Schuster: Controlling the Risk for an Agricultural Harvest Manufacturing & Service Operations Management 6(3), pp. 225–236, © 2004 INFORMS 235

Welch Foods is owned by the National Grape Coop- in industries such as project management and style erative and is an example of vertical integration in goods planning. We plan further research in this area. agriculture. The company’s span of control ranges from raw grapes to end-product marketing. A critical Acknowledgments measure of effectiveness is the capability of receiving The authors wish to thank the editor, senior editor, and two the grower’s grapes with little risk of loss. By using anonymous reviewers for the constructive comments pro- vided during the review process. the Harvest Model, management can communicate to growers—the owners of the cooperative—exact levels of risk associated with harvesting the crop. Shifting References dialogue to a fact-based discussion offers great advan- Allen, S. J., E. W. Schuster. 1994. 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