Written by: Paul Rubin

Primary Source: OR in an OB World, 12/26/2019

An interesting variation of the 1-D cutting stock problem was posed on OR Stack Exchange. Rather than having a specified demand for various size cut pieces, the requirement is to cut pieces to cover a specified number of units of fixed lengths. Moreover, you can use multiple pieces to cover one target unit, but if you do, you have to allow a specified overlap between adjacent pieces.

There are two common ways to build an integer programming model for a “typical” 1-D cutting stock problem. One is to use decision variables for how many pieces of each defined length are cut from each piece of stock (along with a variable or variables for how many pieces of stock are used). That approach contains a high degree of symmetry, since it uses an index for each piece of stock that is cut and the solution is invariant to permutations in that index. The other is to define cutting patterns, and use decision variables for how many pieces of stock are cut according to each pattern (along with variables for how many cut pieces result). There’s no symmetry issue, but the number of possible patterns blows up pretty quickly. For problems where the number of patterns is too large to comfortably enumerate, Gilmore and Gomory [1] proposed a column generation approach (which has since been improved on by others). They start with a small number of patterns and relax the integrality conditions to get a linear program. Each time they solve the LP, they use the dual values in a subproblem to see if they can generate a new pattern that would improve the LP solution. If so, they add the pattern, solve the modified LP, and repeat. When no new pattern emerges, they restore the integrality restrictions, solve a MILP once, and get a good (though not provably optimal) solution. More recently, branch and price (or branch-price-and-cut as it is sometimes known) has emerged as a method for finding a provably optimal solution using patterns.

For the problem at hand, we actually have two places where we must choose between piece counts and symmetry or pattern counts: how we will cut the stock, and how we will piece together covers for the output parts. I’ll reproduce here the answer I gave on OR Stack Exchange, which mixes the two methods (patterns for cutting, piece counts for output).

Let \(L\) be the length of a piece of stock and \(D\) the length of a part that needs to be covered. (I’ll keep this simple with a single size for stock and a single size item to be covered, but you can easily expand to multiple lengths of either by tacking on subscripts.) \(N\) will denote the number of items needing to be covered. \(M\) will be the required overlap between adjacent cover pieces.

Suppose that we have a set \(\mathcal{P}\) of patterns for cutting the stock, and a set \(\mathcal{K}\) of possible piece sizes (where \(k\in\mathcal{K}\implies k\le \min\{L, D\}\)). Our first decision variable will be the number \(x_p\in \mathbb{Z}_+\) of units of stock cut according to pattern \(p\in\mathcal{P}\). We want to minimize the use of stock. Assuming there is no scrap value for unused cut pieces or trim waste, the objective is just

\(\displaystyle \min \sum_{p\in\mathcal{P}} x_p.\)

Now let \(\pi_{k,p}\) be the number of pieces of length \(k\in\mathcal{K}\) we get from one unit of stock cut according to pattern \(p\in\mathcal{P}\), and define variable \(y_k \ge 0\) to be the total number of pieces of length \(k\in\mathcal{K}\) produced by all cutting operations. (\(y_k\) will take integer values, but we do not need to restrict it to be integer.) The constraints tying the \(y\) variables to production are

\(\displaystyle y_k = \sum_{p\in\mathcal{P}} \pi_{k,p}x_p \, \forall k\in\mathcal{K}.\)

Finally, let \(n=1,\dots,N\) enumerate the items needing to be covered, and let variable \(z_{k,n}\in \mathbb{Z}_+\) denote the number of pieces of length \(k\in\mathcal{K}\) used to cover item \(n\in\{1,\dots,N\}\). To be sure that we do not use pieces we do not have, we add constraints

\(\displaystyle \sum_{n=1}^N z_{k,n} \le y_k \,\forall k\in\mathcal{K}.\)

Note that this is a “single period” model (we fulfill demand once and that’s that). If the cutting and covering operations were to be repeated over time, we would need to add inventory constraints to account for excess cut pieces being carried over to a subsequent period. Finally, we need to be sure that the solution actually covers all the output units, and meets the overlap conditions. We combine both those restrictions into a single set of constraints:

\(\displaystyle \sum_{k\in\mathcal{K}} k\cdot z_{k,n} \ge D + M\left(\sum_{k\in\mathcal{K}}z_{k,n} – 1\right) \,\forall n=1,\dots,N.\)

By using an inequality there rather than an equation, I snuck in an assumption that excess length can be trimmed at no cost.

This formulation uses patterns on the cutting side but not on the covering side. That means the covering side is subject to symmetry concerns. Namely, given any feasible solution the model, if we permute the subscript \(n\) we get another feasible solution, nominally different but with an identical cost. For instance, suppose that \(D=11\) and \(M=1\), and that we have a solution that covers unit 1 with two pieces of length 6 and unit 2 with two pieces of length 5 and one piece of length 3. Change that so that unit 2 gets two pieces of length 6 and unit 1 gets two pieces of length 5 and one of length 3, and you have a “different” solution that really is not different in any meaningful way. There are various ways to reduce or eliminate the symmetry via added constraints. One simple way to reduce (but not eliminate) the symmetry is to require that the units receiving the most pieces have the lowest indices. The added constraints are

\(\displaystyle \sum_{k\in\mathcal{K}}z_{k,n} \ge \sum_{k\in\mathcal{K}}z_{k,n+1} \,\forall n=1,\dots,N-1.\)

(This still leaves symmetry among covering patterns that use the same number of pieces, such as 5-5-3 versus 5-4-4.)

[1] Gilmore, P. C. & Gomory, R. E. A Linear Programming Approach to the Cutting-Stock Problem. Operations Research, 1961, 9, 849-859

#### Paul Rubin

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