This paper develops a formulation of the one-dimensional trim problem when there is a fixed charge associated with using a cutting pattern. The purpose of the fixed charge is to limit the number of pattern changes that must be made. Because of the large number of possible cutting patterns, the resulting problem is a combinatorial program that is far beyond the capability of existing algorithms. As a result, we develop a heuristic procedure that can be experimentally tuned to balance the potentially conflicting objectives of minimizing both trim loss and pattern changes. The heuristic procedure is organized around a sequential search that relies on descriptors of the unscheduled orders to set goals on factors such as trim loss and pattern usage for the next pattern to enter the solution. A sample problem is presented along with a discussion of the scope of the application of the heuristic procedure.

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.

A heuristic procedure for scheduling production rolls of paper through a finishing operation to cut them down to finished roll sizes is described. The ratio of service time to interarrival time of production rolls at the initial cutting station is large so that insufficient time is available to set it up unless a minimum number of production rolls are to be processed in the same manner. Otherwise, some portion of each production roll must go through a reprocessing operation to complete the cutting of finished sizes. The objective is to minimize the cost of trim loss and reprocessing. The procedure generates cutting patterns and usage levels sequentially until all the requirements are satisfied. At each step the search is dependent upon the characteristics of the unsatisfied requirements. A maximum of three solutions is generated for each problem. If none satisfies a predetermined aspiration level, the best of the three is chosen. The procedure was evaluated by scheduling a specific paper production facility and observing the results for a set of 15 problems. For each problem, the best solution was recorded. The overall results from this set of problems were then compared to previously recorded results on problems solved manually. There was a 16% improvement in solution quality for the heuristic procedure relative to the manual method.

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This note describes modifications to the Gilmore-Gomory cutting stock algorithm that improve the characteristics (other than trim loss) of the solutions generated. Changes are proposed for the procedures that generate the initial solution and subsequent basis entering patterns. The major point is that controlling the pattern generation by using a more restrictive and therefore less efficient formulation of the knapsack problem may reduce rounding problems and pattern changes, and generally broaden the scope of applications of the algorithm. A set of sample problems is solved and computational times are provided.