Cutting Stock Problems and Solution Procedures

European Journal of Operational Research 54 (1991) 141-150 141 North-Holland

Invited Review

Cutting stock problems and solution procedures

Robert W. Haessler and Paul E. Sweeney School of Business Administration, The Uniuersity of Michigan, Ann Arbor, MI, USA

Received May 1991

Abstract: This paper discusses some of the basic formulation issues and solution procedures for solving one- and two- dimensional cutting stock problems. Linear programming, sequential heuristic and hybrid solution procedures are described. For two-dimensional cutting stock problems with rectangular shapes, we also propose an approach for solving large problems with limits on the number of times an ordered size may appear in a pattern.

Keywords: Cutting stock, trim loss, linear programming, heuristic problem solving, pattern generation, two-dimensional knapsack

Introduction literature has led Dyckhoff (1990) to develop a classification scheme for cutting stock and pack- The first known formulation of a cutting stock ing problems (packing problems are closely re- problem was given in 1939 by the Russian lated to cutting stock problems but are not consid- economist Kantorovich (1960). The first and most ered here). He classifies problems using four char- significant advance in solving cutting problems acteristics as follows: was the seminal work of Gilmore and Gomory 1. Dimensionality (1961, 1963) in which they described their delayed (N) Number of dimensions pattern generation technique for solving the one- 2, Kind of assignment dimensional trim loss minimization problem using (B) All large objects and a selection of small linear programming. Since that time there has items. been an explosion of interest in this application (V) A selection of large objects and all small area. Sweeney and Paternoster (1991) have identi- items. fied more than 500 papers which deal with cutting 3. Assortment of large objects stock and related problems and applications. The (O) One large object. primary reasons for this activity are that cutting (I) Many identical large objects. stock problems occur in a wide variety of in- (V) Different large objects. dustries, there is a large economic incentive to find 4. Assortment of small items more effective solution procedures, and it is easy (F) Few items of different dimensions. to compare alternative solution procedures and to (M) Many items of many different dimensions. identify the potential benefits of using a proposed (R) Many items of relatively few dimensions. procedure. (C) Many identical items. The large variety of applications reported in the Cutting stock problems are introduced with a

0377-2217/91/$03,50 © 1991 - Elsevier Science Publishers B.V. All rights reserved 142 R. 14(. Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures discussion of the one-dimensional problem in tern j. In order for the elements Aij, i = 1..... m, which many items of relatively few sizes are to be to constitute a feasible cutting pattern, the follow- cut from multiple pieces of a single stock size ing restrictions must be satisfied: (1/V/I/R using Dyckhoff's typology). Two-dimensional cutting stock problems are more dif- ~_~AijWi < UW, (4) ficult to solve than one-dimensional problems be- i cause of the greater complexity of defining feasi- A~/> 0 and integer, (5) ble cutting patterns. Hence the focus in two-dimensional problems is on the pattern generation Xj is the number of stock rolls to be slit using process rather than on the cutting stock problem pattern j, and itself. The paper will conclude with a discussion of T/ is the trim loss incurred by pattern j, a possible new approach for generating patterns = UW - EA,jW,. (6) needed to solve two-dimensional cutting stock i problems of type 2/V/I/R. Note that the objective in this example is simply to minimize trim loss. In most industrial appli- One-dimensional problems cations, it is necessary to consider other factors in addition to trim loss. For example, there may be a An example of a one-dimensional cutting stock cost associated with pattern changes and, there- problem is the trim loss minimization problem fore, controlling the number of patterns used to which occurs in the paper industry. In this prob- satisfy the order requirements would be an im- lem, known quantities of rolls of various widths portant consideration. and the same diameter are to be slit from stock Because optimal solutions to integer cutting rolls of some standard width and diameter. The stock problems can be found only for values of m objective is to identify slitting patterns and their smaller than typically found in practices, heuristic associated usage levels which satisfy the require- procedures represent the only feasible approach to ments for ordered rolls at the least possible total solving this type of problem. Two types of heuris- cost for scrap and other controllable factors. The tic procedures have been widely used to solve basic cutting pattern feasibility restriction in this one-dimensional cutting stock problems. One ap- problem is that the sum of the roll widths slit from proach uses the solution to a linear programming each stock roll must not exceed the usable width (LP) relaxation of the integer problem above as its of the stock roll. starting point. The LP solution is then modified in Let R~ be the nominal order requirements for some way to provide an integer solution to the rolls of width W,, i=l ..... m, to be cut from problem. The second approach is to generate cut- stock rolls of usable width UW. RL, and RUi are ting patterns sequentially to satisfy some portion the lower and upper bounds on the order require- of the remaining requirements. This sequential ment, for customer order i reflecting the general heuristic procedure (SHP) terminates when all industry practice of allowing overruns or under- order requirements are satisfied. runs within specified limits. Depending on the situation, R, may be equal to RLi and/or RU,. All orders are for rolls of the same diameter. This Linear programming solutions problem can be formulated as follows: Min Y'. T;.X, (1) Almost all LP based procedures for solving J cutting stock problems can be traced back to s.t. RL, _< Y'.AoX/_< RUt for all i, (2) Gilmore and Gomory (1961, 1963). They de- J scribed how the next pattern to enter the LP basis could be found by solving an associated knapsack Xj >__ 0 and integer, (3) problem. This made it possible to solve the trim where loss minimization problem by linear programming A,j is the number of rolls of width W, to be slit without first enumerating every feasible slitting from each stock roll that is processed using pat- pattern. This is extremely important because a R. V~ Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures" 143 large number of feasible patterns may exist when Xj which satisfy the order requirements. One com- narrow widths are to be slit from a wide stock roll. mon approach is to round the LP solution down Pierce (1964) showed that in such situations the to integer values, then increase the values of Xj by number of slitting patterns can easily run into the unit amounts for any patterns whose usage can be millions. Because only a small fraction of all pos- increased without exceeding RU,. Finally, new sible slitting patterns need to be considered in patterns can be generated for any rolls still needed finding the minimum trim loss solution, the de- using the sequential heuristic described in the next layed pattern generation technique developed by section. Gilmore and Gomory made it possible to solve In order to make this rounding problem as trim loss minimization problems in much less time simple as possible, it is generally useful to place than would be required if all the slitting patterns limitations on the number of times a given size were input to a general purpose linear program- can appear in a pattern. A very obvious restriction ming algorithm. is that A i < RU,. If this is not satisfied, any pat- A common LP relaxation of the integer pro- tern for which A, > RU, will have to be rounded gramming problem given in (1)-(3) can be stated down to zero and a new pattern found by some as follows: other method. The following simple example dem- onstrates the advantage of placing even greater Min •X, (7) restrictions on the values of A,. Let .i R 1=3, W 1=100, s.t. Y'Ai/X: > R, for all i, (8) / R2=3, W~=90. X,>0. (9) UW = 200. Solving this problem using the Gilmore-Gomory Let U, be the dual variable associated with algorithm yields the following solution: constraint i. Pattern 1. 2-100 and 0-90, X 1 = 1.5. The dual of this problem can be stated as Pattern 2. 0-100 and 2-90, X 2 = 1.5. Max Y'~R,Ui (10) In virtually all situations, the preferred solution i would be: Pattern 3. 1-100 and 1-90, X 3 = 3. s.t. Y" A,jU, _< 1, (11) Haessler (1980) demonstrated how placing restric- t tions on the values of A, in the knapsack problem _> 0. (12) led to LP solutions with fewer patterns which are The dual constraints in (11) provide the means easier to round to integer values. for determining if the optimal LP solution has The primary disadvantage of using LP to solve been obtained or if there exists a pattern which cutting stock problems is that the number of ac- will improve the LP solution because the dual tive cutting patterns in the solution will be very problem is still infeasible. close to the number of sizes ordered. This may be The next pattern A = (A a..... Am) to enter the acceptable only if controlling trim loss is very basis, if one exists, can be found by solving the difficult and LP is the only way to find a low trim following knapsack problem: loss solution.

Z = Max Y'~ U,A~ (13) i Sequential heuristic procedures s.t. Y'. W,A i _< UW, (14) i With an SHP, a solution is constructed one A, >_ 0 and integer. (15) pattern at a time until all the order requirements are satisfied. The first documented SHP capable If Z _< 1, the current solution is optimal. If Z > 1, of finding better solutions than those found manu- then A can be used to improve the LP solution. ally by schedulers was described by Haessler Once found, the LP solution can be modified in (1971). The key to success with this type of proce- a number of ways to obtain integer values for the dure is to make intelligent choices as to the pat- 144 R.W. Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures

terns which are selected early in the SHP. The accepted and the requirements reduced, the widths patterns selected initially should have low trim remaining at some point in the process may not loss, high usage and leave a set of requirements for have an acceptable trim loss solution. Such would future patterns which will combine well without be the case if only 34-inch rolls are left to be slit excessive side trim. from 100-inch stock rolls. The following procedure is capable of making effective pattern choices in a variety of situations: 1. Compute descriptors of the order require- Hybrid solution procedures ments yet to be scheduled. Typical descriptors would be the number of stock rolls still to be slit In addition to using an SHP to help convert an and the average number of ordered rolls to be cut LP solution to integer values as described earlier, from each stock roll. there are a number of ways in which these ap- 2. Set goals for the next pattern to be entered proaches can be used together to obtain the best into the solution. Goals should be established for possible answer to a given class of one-dimen- trim loss, pattern usage, and number of ordered sional cutting stock problems. Perhaps the most rolls in the pattern. obvious is to use the SHP to generate a solution 3. Search exhaustively for a pattern that meets which is saved and also used as the initial basis in those goals. the LP procedure. Additional LP iterations are 4. If a pattern is found, add this pattern to the then made to reduce the trim loss if that is possi- solution at the maximum possible level without ble. The better of the SHP and rounded LP solu- exceeding R i for all i. Reduce the order require- tions is selected according to the appropriate crite- ments and return to 1. rion for the problem being solved. 5. If no pattern is found, reduce the goal for A second and more powerful hybrid solution the usage level of the next pattern and return to 3. procedure works as follows. The problem is first The pattern usage goal provides an upper bound solved as an LP problem in order to obtain the on the number of times a size can appear in a optimal dual prices. These dual prices are used as pattern. For example, if some ordered width has an additional test before accepting a pattern in an an unmet requirement of 10 rolls and the pattern SHP to ensure that the pattern does not contain a usage goal is 4, that width may not appear more disproportionate share of sizes with relatively low than twice in a pattern. If after exhaustive search dual prices. For patterns in the optimal trim loss no pattern satisfies the goals set, then at least one solution goal, most commonly pattern usage, must be re- laxed. This increases the number of patterns to be z = EA, = 1. i considered. If the pattern usage goal is changed to 3 in the above example, then the width can appear If the value of Z is low (less than 0.97) for a in the pattern three times. Termination can be pattern accepted in an SHP, the total trim loss of guaranteed by selecting the pattern with the lowest the SHP solution may be increased significantly. trim loss at the usage level of one. Although this test makes it possible to avoid mak- The primary advantage of this SHP is its ability ing some mistakes when selecting a pattern using to control factors other than trim loss and to an SHP, it is not foolproof because the SHP may eliminate rounding problems by working only with use the pattern at too high a level. integer values. For example, if there is a cost As the SHP nears completion and the pattern associated with a pattern change, a sequential selection decision becomes more difficult, patterns heuristic procedure which searches for high usage for all residual requirements are generated using patterns may give a solution which has less than LP. If the residual LP solution does not meet one-half the number of patterns required by an LP some target trim value which is based on the solution to the same problem. The major disad- original LP solution to the entire problem, the vantage of an SHP is that it may generate a sequentially generated patterns are dropped one at solution which has greatly increased trim loss be- a time in reverse order of generation and the cause of what might be called ending conditions. expanded residual problem is solved using LP. For example, if care is not taken as each pattern is This process of dropping sequentially generated R. I'lL Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures 145 patterns continues until either a satisfactory solu- s.t. RL, < Z ZAjjkXjk < RUi for all i, (17) tion is obtained or all the patterns are dropped at k j which point the LP solution with the best possible ~Xjk -< Mk for all k, (18) trim loss is generated. J The advantage of this approach is that it in- Xjk > 0 and integer (19) tegrates the ability of the SHP to consider factors such as slitter changes and the LP procedure to where minimize trim loss into a single procedure. This Aijk is the number of rolls for order i to be cut procedure is capable of giving either a pure SHP from stock width k using pattern j, or LP solution depending on which is best. Most Xjk is the number of stock of width k to be importantly, however, is its ability to generate processed according to pattern j, solutions which are part SHP and part LP and ~k is the trim loss incurred by using pattern j therefore likely to be better than either the pure with stock width k, LP or SHP solutions. Haessler (1988) has used this Clk is the dollar value of trim loss per unit for procedure to solve difficult trim loss problems for stock width k, which it is also important to limit the number of C2k ~ is the cost of shipping one roll for order i slitter changes. which is produced from stock width k. It is as- Sweeney and Haessler (1990) used information sumed that the stock width defines the production about optimal dual prices to develop a procedure location. If all the production options are at the for solving a one-dimensional cutting stock prob- same location, this value can be set to 0, and lem with quality variations across the width of the M k is the maximum number of rolls of stock stock rolls. The lower quality material in the stock width k which can be used. roll is not scrapped because there are orders which The LP relaxation of this problem developed can be filled from the lower quality material. by Beged Dov (1970) is Higher quality material can also be used to satisfy any orders for lower grades. A two phase proce- Min E E Cjk Xjk (20) dure is used. In the first phase, each nonperfect ] k stock roll is assigned a value based on the nature s.t. E EAijk Xjk >-- Ri for all i, (21) of its quality variations. Patterns are generated j k and given a value based on the shadow prices X,k >- o. (22) obtained from solving an LP problem. If the value of the selected pattern exceeds the value of the For the most general case with varying costs for stock roll, the pattern is accepted for that one trim loss and freight, Cjk can be represented as stock roll. In the second phase, any order require- follows: ments not met from stock rolls with quality variations are slit from first quality stock rolls based on an LP solution to the residual problem. where C k is the cost excluding material of making one One-dimensional problems with multiple stock sizes stock roll of width k, Cpk is the material cost per inch of that portion A very interesting problem with multiple stock of production roll k actually used, and sizes occurs when the stock sizes are available at C2k~ is the cost of shipping one roll for order i different locations and therefore freight cost also from the location where stock width k is made. influences the choice of which stock size will be In this situation, the shadow prices must be used. adjusted before finding the next cutting pattern This problem can be formulated as follows: that should enter the LP basis, if one exists. The following knapsack problem must be solved for + each stock width. k j i j (16) Z k =max Y'~(Ui - CpkW/ - C2ki)Aij k - Ck, (24) i 146 R. W. Haessler, P.E. Sweeney /Cutting stock problems a-d solution procedures

s.t. Y~ W,A,j k < UWk, (25) a subset of orders, can be completely determined i except for the cost of the roll stock changeover A,j k _> 0 and integer. (26) which depends on which other setups are selected. In the second stage the least cost means of The pattern for which Z k has the largest value producing each order can be determined by solv- greater than zero is the one which enters the ing the following set partitioning problem: solution. Min ~ CjXj + ~ SkYk (27) j k

1.5-dimensional problems s.t. ~Aijx j-- 1 for all i, (28) J There are a number of cutting stock problems ~ FjkXj <__M~Y k for all k, (29) which are more complex than the one-dimensional J problems discussed previously, but are not true Xj, Yk=0orl for allj, k (30) two-dimensional problems. These are generally re- ferred to as 1.5-dimensional. Haessler and Talbot where (1983) discussed an example of this type of prob- Xj is 1 if element j is used and 0 otherwise, lem which occurs in the production of corrugated Yk is 1 if stock size k is used and 0 otherwise, shipping containers. In this situation a customer A,j is 1 if order i appears in element j and 0 order will be for R i blanks of width W, and length otherwise, L,. The corrugated material is produced continu- Fjk is the lineal quantity of stock size k re- ously out of a variety of available rollstock widths. quired by setup j, Slitters and cutoff knives can be set to produce M k is the lineal quantity of stock size k availa- appropriate size blanks. The number of cutoff ble in inventory, knives, which is commonly two, determines the Cj is the total cost of using element j exclusive number of different orders which can be com- of the cost of changing to the required roll stock bined across the width of the corrugator. size, and Although rectangular blanks are being cut, it is S k is the cost of changing to stock size k. not a two-dimensional problem because trim loss The value ~ includes the cost of corrugator along the length of the corrugator is not a dimen- time and paper used plus the cost of pattern sional issue. The problem is more complex than changes. If any order is not produced at the maxi- the one-dimensional problem because of the desire mum quantity, this value is scaled to reflect the to match up orders both across the width and cost of producing the order at its maximum quan- along the length dimensions. The total set of fac- tity, which typically is 110% of the quantity tors to be considered is: ordered. This scaling is needed to avoid selecting -roll stock changes elements simply because the order quantities pro- -slitter changes duced are at the lower end of the acceptable -minimum runs of a slitter setup range. It is assumed that if two or more elements -order contiguity use the same stock size, they will be run sequen- -corrugator width utilization tially so there will be only one setup for each stock -roll stock availability size. -side trim If the problem is formulated with pattern usage variables as shown earlier, the resulting formula- Rectangular two-dimensional problems tion is extremely complex. A more effective way to deal with this problem is to solve it using a two- stage process. In the first stage, all possible corru- The formulation of a higher dimensional cut- gator setups which will completely produce one, ting stock problem is exactly the same as that of two or three orders from a single roll stock size are the one-dimensional problem given in (1)-(3). The generated. All the restrictions on setup feasibility only added complexity comes in trying to define must be met. The cost of any setup for producing and generate feasible cutting patterns. The sire- R. W. Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures 147

L showed how cutting patterns can be generated by //+ ,,///J/ solving two one-dimensional knapsack problems. .H To simplify the discussion, assume that the orien- tation of each ordered piece is fixed relative to stock piece and the first guillotine cut on the stock (a) 2-STAGE GUILLO'RNE (b) 2-STAGE GUILLOTINE NO TRIMMING WITH TRIMMING pieces must be along the length (larger dimension) of the stock piece. For each ordered width W k find the contents of a strip of width W~ and length L which gives the maximum contribution to dual infeasibility.

Z k =Max ~ U, Ai~ (31) i~l~

s.t. ~ LiAik < L, (32) (d) GENERAL GUILLOTINE (C) 3-STAGE iEl~ GUILLOTINE Aik > 0 and integer, (33) Ik = {ilWi_<_< Wk }. (34) Next find the combination of strips which solve the following problem:

Z = Max~-'ZkA k (35) (e) NONGUILLOT1NE (f) NONOR'n.IOGONAL k Figure 1. Sample cutting patterns s.t. Y" WkA k < W, (36) k plest two-dimensional case is one in which both A k > 0 and integer. (37) the stock and ordered sizes are rectangular. Most of the important issues regarding cutting patterns Any pattern for which Z is greater than one will for rectangular two-dimensional problems can be yield an improvement in the LP solution. seen in the examples shown in Figure 1. The major difficulty with this approach is the One important issue not covered in Figure 1 is inability to limit the number of times and ordered a limit on the number of times an ordered size can size appears in a pattern. It is easy to restrict the appear in a pattern. This generally is a function of number of times a size appears in a strip and to the maximum quantity of pieces, RU i, required for restrict the number of strips in a pattern. The order i. If R i is small, it is just as important for problem is that small ordered sizes with small the two-dimensional case as the one dimensional quantities may end up as filler in a large number case that the number of times size i appears in a of different strips. This makes the two-stage ap- pattern should be limited. This becomes less im- proach to developing patterns ineffective when the portant as R, becomes larger and as the difference number of times a size appears in a pattern must between RU~ and RLi becomes larger. be limited. The cutting pattern shown in Figure 1 (a) is an Gilmore and Gomory (1966) also developed example of two-stage guillotine cuts. The first cut dynamic programming recursions for generating can be in either the horizontal or vertical direc- patterns both for staged and general guillotine tion. A section cut perpendicular to the first, yields cuts. Their recursion for generating patterns with a finished piece. Figure 1 (b) is similar except a general guillotine cuts such as shown in Figure 1 third cut can be made to trim the pieces down to (d) is: the correct dimension. Figure 1 (c) shows the G( X, r)= Max {H(X, V), G( Xo, Y) situation in which the third cut can create 2 ordered pieces. +G( X- X o, Y),G(X, to) For simple staged cutting such as shown in Figure 1 (a-c), Gilmore and Gomory (1965) +G(X, Y- Yo)} (38) 148 R. W. Haessler, P.E. Sweeney /Cutting stock problems a-d solution procedures

where x o < ½X and Y0 < ½Y, G(X, Y) is the maximum value that can be 01 obtained from an X by Y rectangle using W, by 0 2 L i rectangles at price shadow U, and any succes- sion of guillotine cuts, and H( X, Y) = Max, { U, I Wi < X and L, < Y ). (a) Horizontal build of 01 and 0 2 Beasley (1985a) presented some computational improvements to the Gilmore-Gomory recursion given above and demonstrated that this procedure was capable of solving problems with as many as 02 50 sizes in under 2 seconds on a CDC 7600. The major problem with this approach is also its inability to limit the number of times a size appears in a pattern. 01 Christofides and Whitlock (1977) used a depth-first branch and bound algorithm to find optimal patterns for the general guillotine cut case as shown in Figure 1 (d) with limits on the num- (b) Vertical build of 01 and 0 2 ber of times a size can appear in the pattern. Even Figure 2. though they took great pains to make the procedure as efficient as possible by eliminating dupli- cate cuts, the average time to generate a pattern (iii) those rectangles 0 i, appearing in T do not with 20 ordered sizes was over 2 minutes on a violate the constraints on the number of times a CDC 7600 computer. This suggests that this pro- size can appear in a pattern. cedure could be used to solve only small to medium b. Set L (K) = L (K-a) U F (K). Remove any equiv- sized cutting stock problems because the number alent (same component) rectangle patterns from of iterations required to find an optimal LP solu- L(K). tion could easily exceed 2 or 3 times the number Step 3. If F (K) is non-empty, set K=K+I of ordered sizes. and go to Step 2. Otherwise go to Step 4. Wang (1983) developed an alternative approach Step 4. a. Set M=K-1. to generating general guillotine cutting patterns b. Choose the rectangle in L (M) which has the with limits on the number of times a size appears smallest total trim waste when placed in the stock in a pattern. She combined rectangles in a hori- rectangle. zontal and vertical build process as shown in Figure 2 where 0 i is an ordered rectangle of width Viswanathan and Bagchi (1988) and Vasko W, and length L~. (1989) have both improved this algorithm by using She used an acceptable value for trim loss, B, better bounds to select the rectangles to be com- rather than the shadow price of the ordered sizes bined. Viswanathan and Bagchi used the uncon- to drive her procedure which is as follows: strained dynamic programming solution to the general guillotine problem in a best-first search Step 1. a. Choose a value for B the maximum algorithm which generates optimal solutions with acceptable trim waste. significantly fewer nodes than required by Wang b. Define L (°) = F (°) = {01, 02 ..... 0" ), and set K (1983) or Christofides and Whitlock (1977). =1. Vasko used the solution to the two-stage cut- Step 2. a. Compute F ~K) which is the set of ting pattern problem to find initial upper bounds all rectangles T satisfying on the general guillotine problem with restrictions (i) T is formed by a horizontal or vertical on the number of times a size can appear in a build of two rectangles from L tK-1), pattern. This procedure is reported to be 25 times (ii) the amount of trim waste in T does not faster than Wang's for generating optimal solu- exceed B, and tions to the pattern generation problem. R.W. Haessler, P.E. Sweeney / Cutting stock problems a-d solution procedures 149

Beasley (1985b) used Lagrangean relaxation of dimensional problems in 1961 when Gilmore and a zero-one integer formulation of the non-guillo- Gomory published their first paper on cutting tine cut case shown in Figure 1 (e) as a bound in a stock problems. tree search procedure to find optimal cutting pat- Taken together this suggests that there is much terns for the non-guillotine cut problem. Subgradi- more research needed on procedures for solving ent optimization and reduction tests make it possi- two-dimensional cutting stock problems. An alter- ble to solve small problems with 10 sizes on a native worth considering, especially in those cases CDC 7600 in anywhere from 10 to 229 seconds. where there are many different ordered sizes with Hadjiconstantinou and Christofides (1991) have small order quantities, might be to first select a developed a exact-tree search procedure for solv- subset of orders to consider by solving a one-di- ing the non-guillotine cut problem with a con- mensional knapsack problem as in (13)-(15) based straint on the number of times a size can appear on area and then see if the resulting solution can in the pattern. This procedure also uses subgradi- be put together into a feasible two-dimensional ent optimization and reduction tests to make the pattern. Wang's algorithm seems to be ideal for algorithm more efficient. They report computa- this purpose inasmuch as the trim loss in the tional times for 2 problems with 15 ordered sizes pattern would be known. of 3.2 and 65.2 seconds on a Cyber-855 computer. A candidate set of items to be included in the Optimal solutions to two other problems with 15 next pattern could be found by solving the follow- ordered sizes were not found in 800 seconds. ing problem: All the two-dimensional patterns considered to this point involve only orthogonal cuts. Rinnooy Z = Max E U, Ai (39) Kan, de Wit and Wijmenga (1987) demonstrated i that under certain circumstances yield can be sig- s.t. ~AP~A, < UAR for all i, (40) nificantly improved if non-orthogonal cuts can be i used. Figure 1 (f) shows an example where non-or- thongonal cuts must be used because one-dimen- A i ~ b i, (41) sion of the ordered size is greater than both the A~ >_ 0 and integer (42) width and length of the stock size. They derive relationships which indicate under which cir- where cumstances this will be beneficial based on the AR i is the area of ordered rectangle i, and effective height and width of the tilted pieces. bi is the upper limit on the number of times order i can be included in the pattern. This candidate pattern (A 1..... A,,) could then Future directions be tested for feasibility using Wang's procedure. If the AR i are small, the chances are that there will It is clear that moving from one to two dimen- be little trim loss in the candidate patterns gener- sions causes significant difficulty in the pattern ated. This may require that UAR be reduced to generating process. This is all the more alarming force some trim loss to make it more likely that in light of the fact that only rectangular shapes feasible patterns are found. were considered. It must also be noted that most of the two-dimensional papers referenced did not solve cutting stock problems as defined in (1)-(3). References Except for Wang (1983), the primary focus was simply on pattern generation. Wang used her pat- Beasley, J.E. (1985a), "Algorithms for unconstrained two-di- tern generating procedure to solve a cutting stock mensional guillotine cutting", Journal of the Operational problem. However, she did not use the Gilmore- Research Society 36, 297-306. Gomory delayed pattern generating technique but Beasley, J.E. (1985b), "An exact two-dimensional non-guillo- rather generated a large number of low trim pat- tine cutting tree search procedure", Operations Research 33, 49-64. terns all at once and used a standard LP proce- Beged Dov, A.G. (1970), "Some computational aspects of the dure to solve the cutting stock problem. This is the M paper mills and P printers paper trim problem", Jour- same approach which was made obsolete for one- nal of Business Administration 1, 15-34. 150 R.W. Haessler, P.E. Sweeney /Cutting stock problems a-d solution procedures

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