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Cutting Stock Problems and Solution Procedures

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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-di- mensional 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-di- mensional 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 sim- ply 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.
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