A Heuristic Programming Solution to a Nonlinear Cutting Stock Problem

Abstract

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.

Column generation and sequential heuristic procedure for solving an irregular shape cutting stock problem

“…Haessler (1971) first described a SHP algorithm. He pointed out that the key to success with this type of procedure is to select intelligently the patterns that are used early in the SHP.…”

“…The multi-stage cutting stock problem also appears in the paper industry. Zak Papers that address multi-objective, multi-constrained one-dimensional cutting stock problem (1DCSP) using problem specific mathematical models and solve them using a heuristic include (Antoni et al (1999); Burkard and Zelle (2003); Chu and Antonio (1999); Coverdale and Wharton (1976);Haessler (1971Haessler ( , 1975; Haessler and Sweeney (1991); Kasimbeyli et al (2011);McDiarmid (1999); Thorstenson (2000, 2008); Sinuany-Stern and Weiner (1994);Song et al (2006)). The Sequential Heuristic Procedure (SHP) is the most commonly applied.…”

“…When no patterns satisfy these goals, some of the goals are relaxed. Coverdale and Wharton (1976) also extended the work of Haessler (1971) to consider factors such as set up time for the machine operators. Pure SHP has been shown to increase the trim loss because of so-called ending conditions (Haessler and Sweeney (1991)).…”

In situ column generation for a cutting-stock problem

“…For example, we can take β = ⌈log 2 (1 + j∈N x j )⌉, where (x, z, s, t) is any feasible solution to (5)(6)(7)(8)(9)(10)(11)(12)(13). It is desirable to have the β i and β as small as possible.…”

“…This kind of limitation can be quite practical for actual applications, but it is difficult to handle computationally (see [22] and the references therein). Heuristic approaches to this type of limitation include: variations on the simplex-method based GG approach (see [30], [7] and [14]), incremental heuristics for generating patterns (see [12] and [13]), local-search methods for reducing the number of patterns as a post-processing stage ( see [15], [11] and [8]), linear-programming based local search (see [26]), heuristically-based local search (see [27]) and [28]). While a computationally intensive Branch, Cut and Price methodology is explored in [29] and later a Branch and Price approach in [2].…”