In this exercise we consider a two-dimensional version of the cutting stock problem

 In this exercise we consider a two-dimensional version of the cutting stock problem.

a) Suppose that we have a W-by-L piece of cloth. The material can be cut into a number of smaller pieces and sold. Let πi j denote the revenue for a smaller piece with dimensions wi by ` j (i = 1, 2, . . . , m; j = 1, 2, . . . , n).

Operating policies dictate that we first cut the piece along its width into strips of size wi . The strips are then cut into lengths of size ` j . Any waste is scrapped, with no additional revenue.

For example, a possible cutting pattern for a 9-by-10 piece might be that shown in Fig.. The shaded regions correspond to trim losses. Formulate a (nonlinear) integer program for finding the maximum-revenue

cutting pattern. Can we solve this integer program by solving several knapsack problems? [Hint. Can we use the same-length cuts in any strips with the same width? What is the optimal revenue vi obtained from a strip of width wi ? What is the best way to choose the widths wi to maximize the total value of the vi’s?]

b) A firm has unlimited availabilities of W-by-L pieces to cut in the manner described in part (a). It must cut these pieces into smaller pieces in order to meet its demand of di j units for a piece with width wi and length ` j(i = 1, 2, . . . , m; j = 1, 2, . . . , n). The firm wishes to use as few W-by-L pieces as possible to meet its sales commitments. Formulate the firm’s decision-making problem in terms of cutting patterns. How can column generation be used to solve the linear-programming approximation to the cutting-pattern formulation?


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