Heuristic methods for gang-rip saw arbor design

Abstract

We consider the problem of design and scheduling of arbors for a computer assisted gang-rip saw system. Such systems are typically used within the furniture manufacturing industry for processing lumber. We discuss the computational complexity of the problem, and develop several heuristic procedures for solving it. These procedures are based on the principles of local improvement, simulated annealing, and genetic algorithms. The results of a computational experiment are also presented.

Introduction

A typical lumber processing method within the furniture manufacturing industry is the cutting of lumber along the length (parallel to the grain) into strips of different widths. These strips are then cut to the required lengths and used in manufacturing.

A commercial saw with multiple cutting channels has been developed to carry out this task. This saw has fixed blades at specific positions on a rotating shaft which rip incoming lumber boards lengthwise into desired finished widths. This method is commonly referred to as the “gang-rip method” of lumber cutting, and the setting of the blades (i.e., the pattern of cutting channels) along the saw shaft is called an “arbor” (see Fig. 1).

During the cutting operation, uncut boards are pulled randomly from an incoming lot and fed into the saw; no presorting by width is performed. The probability distribution of the width of an incoming board is estimated based on the empirical frequency distribution derived from a sample tabulation.

The amount of “waste” produced over the course of a cutting job is the key issue in the problem. Waste is produced as a result of (1) sawdust loss, and (2) strips cut at the edges of an incoming board which do not satisfy any finished width in the demand schedule (edge strips). The saw used in the cutting job has the additional capability that once the usable width of an incoming lumber board is registered, it moves the board (parallel to the shaft) to the block of contiguous cutting channels on the arbor that generate the minimum waste for the board.

Aside from minimizing the waste, there are two other issues of concern in the operation of a gang-rip saw. The first issue is the “overage”, i.e. overproduction in any one of the finished widths in the demand schedule. It is desirable (but not required) that the quantity of each finished width produced does not significantly exceed its corresponding demand. The second issue is the number of arbors employed to do a cutting job. Naturally, a larger number of arbors allows for more flexibility, and possibly leads to a higher yield. However, in order to exchange an arbor we need to stop the saw operation for approximately 15 minute, resulting in a corresponding loss of production. Thus, it is generally preferable to use the least number of arbors (possibly one) for a particular cutting job to avoid such production loss, while maintaining a high yield.

There are two units of measure which are commonly used to express the amount of lumber in this context. A linear foot is a unit of measure which is concerned with length only; a board which is ten feet long measures ten linear feet regardless of its width or thickness. A board foot, on the other hand, is a measure of volume of lumber and as such it is concerned with all three dimensions simultaneously. For ease of presentation throughout this paper we use the term board foot to refer to one square foot of lumber regardless of its thickness. Since all cutting stock in a cutting job is assumed to be of the same thickness, this change of terminology does not affect the validity of our results.

Considering the situation described above we define the “Gang-Rip Saw Arbor Design and Scheduling Problem” (GRSADSP) as follows:

• GRSADSP: Given (1) a set of desired finished widths (cut-widths) and their corresponding quantities (demand schedule), (2) the frequency distribution of incoming boards (raw material), (3) the shaft length, and (4) the blade width, find a set of (one or more) arbors (i.e. blade settings) and the corresponding quantity of incoming lumber to run through each arbor, such that the demand is met while the total board feet of waste is minimized.

Note that in this definition, the number of arbors is not specified, and the problem solver is free to choose as many arbors as he/she finds appropriate.

In the context of the literature on cutting and packing, this problem can be categorized as a one-dimensional stock cutting problem (or SC1 according to the notation proposed in [3]). Due to its wide range of applications, SC1 is one of the most frequently referenced problems in the open literature on cutting and packing. See [2], [4], [5], [8], [15], [16] for several variation of SC1.

The problem we study here, however, has the distinguishing feature of ‘arbor design’ associated with it. Each incoming board width is cut according to the best cutting pattern for that width that is available on the arbor as a contiguous block of cutting channels. Thus, the main challenge in this problem is to design a set of one or more arbors, with their corresponding run time, so that each incoming board is cut in an efficient manner within the context of meeting the demand and minimizing the total waste. It is the combinatorial nature of the problem of arbor design (i.e., arrangement of the cutting channels across the length of the saw shaft and the corresponding cut-width weights) that distinguishes this problem from other SC1 problems.

In an earlier article [7] we proposed a linear programming (LP) model for solving GRSADSP and used the concept of column generation to solve this model. Using this approach we obtained good results (low waste and low overage) within a reasonable amount of CPU time. However, in most instances the total number of arbors used in the optimal solution is relatively large (usually as many arbors as the number of finished widths required). This, in turn, results in important loss of production time as mentioned above. Obviously we always prefer to have a solution that achieves low waste and uses a relatively small number of arbors (ideally only one arbor). To limit the number of arbors within the context of the LP model [7], however, is somewhat problematic. This number is associated with the size of the basis in the LP model, and to limit this number requires the introduction of a relatively large number of binary variables. The resulting mixed integer linear programming model is likely to be significantly more difficult to solve. In this article we propose a different approach for solving the problem when the number of arbors is limited.

More specifically, we define the “Gang-Rip Saw Arbor Design and Scheduling Problem With 1 Arbor” (GRSADSP/1) as: Solve GRSADSP using only 1 arbor. We devise several procedures for solving GRSADSP/1 and carry out a computational experiment to investigate their effectiveness. In Section 2 we define various notation and terminology required to define an instance of the problem. Section 3 is on the complexity of GRSADSP/1. The next three sections are devoted to the descriptions of the three heuristic procedures that we have developed for this problem. Finally, in Section 7 we present a summary of the results of a computational experiment with these procedures and make some concluding remarks.