Cutting Stock Problem - Formulation and Solution Approaches

Formulation and Solution Approaches

The standard formulation for the cutting-stock problem (but not the only one) starts with a list of m orders, each requiring q_j, j = 1,...,m pieces. We then construct a list of all possible combinations of cuts (often called "patterns"), associating with each pattern a positive integer variable x_i representing how many times each pattern is to be used. The linear integer program is then:

minimize

subject to and

, integer

where a_ij is the number of times order j appears in pattern i and c_i is the cost (often the waste) of pattern i. The precise nature of the quantity constraints can lead to subtly different mathematical characteristics. The above formulation's quantity constraints are minimum constraints (at least the given amount of each order must be produced, but possibly more). When c_i=1 the objective minimises the number of utilised master items and, if the constraint for the quantity to be produced is replaced by equality, it is called the bin packing problem. The most general formulation has two-sided constraints (and in this case a minimum-waste solution may consume more than the minimum number of master items):

This formulation applies not just to one-dimensional problems. Many variations are possible, including one where the objective is not to minimise the waste, but to maximise the total value of the produced items, allowing each order to have a different value.

In general, the number of possible patterns grows exponentially as a function of m, the number of orders. As the number of orders increases, it may therefore become impractical to enumerate the possible cutting patterns.

An alternative approach uses delayed column-generation. This method solves the cutting-stock problem by starting with just a few patterns. It generates additional patterns when they are needed. For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program. The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be much more efficient than the original approach, particularly as the size of the problem grows. The column generation approach as applied to the cutting stock problem was pioneered by Gilmore and Gomory in a series of papers published in the 1960s. Gilmore and Gomory showed that this approach is guaranteed to converge to the (fractional) optimal solution, without needing to enumerate all the possible patterns in advance.

A limitation of the original Gilmore and Gomory method is that it does not handle integrality, so the solution may contain fractions, e.g. a particular pattern should be produced 3.67 times. Rounding to the nearest integer often does not work, in the sense that it may lead to a sub-optimal solution and/or under- or over-production of some of the orders (and possible infeasibility in the presence of two-sided demand constraints). This limitation is overcome in modern algorithms, which can solve to optimality (in the sense of finding solutions with minimum waste) very large instances of the problem (generally larger than encountered in practice).

The cutting-stock problem is often highly degenerate, in that multiple solutions with the same waste are possible. This degeneracy arises because it is possible to move items around, creating new patterns, without affecting the waste. This give arise to a whole collection of related problems which are concerned with some other criterion, such as the following:

• The minimum pattern count problem: to find a minimum-pattern-count solution amongst the minimum-waste solutions. This is a very hard problem, even when the waste is known. There is a conjecture that any equality-constrained one-dimensional instance with n orders has at least one minimum waste solution with no more than n + 1 patterns. No upper bound to the number of patterns is known either, examples with n + 5 are known.

• The minimum stack problem: this is concerned with the sequencing of the patterns so as not to have too many partially completed orders at any time. This was an open problem until 2007, when an efficient algorithm based on dynamic programming was published.

• The minimum number of knife changes problem (for the one-dimensional problem): this is concerned with sequencing and permuting the patterns so as to minimise the number of times the slitting knives have to be moved. This is a special case of the generalised travelling salesman problem.