Cutting stock with no three parts per pattern: Work-in-process and pattern minimization

Abstract

a b s t r a c tThe Pattern Minimization Problem (PMP) consists in finding, among the optimal solutions of a cutting stock problem, one that minimizes the number of distinct cutting patterns activated. The Work-in-process Minimization Problem (WMP) calls for scheduling the patterns so as to maintain as few open stacks as possible. This paper addresses a particular class of problems, where no more than two parts can be cut from any stock item, hence the feasible cutting patterns form the arc set of an undirected graph G. The paper extends the case G = K n introduced in 1999 by McDiarmid. We show that some properties holding for G = K n are no longer valid for the general case; however, for special cases of practical relevance, properly including G = K n , quasi-exact solutions for the PMP and the WMP can be found: the latter in polynomial time, the former via a set-packing formulation providing very good lower bounds.

Pricing-based primal and dual bounds for selected packing problems

“…Thus, the CSP is trivial, whereas the PMP 2.4 The CSP with due-dates (corresponding to maximizing the number of times two jobs end at the same time) is proved to be NP-hard [93]. The last observation leads to formulate the PMP in terms of SET PACKING, see [4].…”

“…The CSP becomes then a b-matching, and hence can be solved in polynomial time. The PMP loses instead the nice properties it has when G n = K n , but, resorting to a flow model, can still be solved in polynomial time when G n is a split graph [4].…”

“…With this schedule, no more than 2 distinct part types are in-process at any time: so at most 2 stacks are maintained open throughout production. The property still holds if the compatibility graph G n is a threshold graph, but does not hold in general for a split graph [4].…”

New lower bounds based on column generation and constraint programming for the pattern minimization problem

“…The nonlinearities of this model come from the demand constraints (2). The sum of all the activation levels is equal to the number of rolls that are used.…”

“…In [2], the same authors addressed the PMP for special instances where no more than two items fit in a stock roll. They explored two formulations for the problem, and derived various results concerning the existence of specific solutions.…”