Facets and Reformulations for Solving Production Planning With Changeover Costs

We study a scheduling problem with changeover costs and capacity constraints. The problem is NP-complete and combinatorial algorithms for solving it have not performed well. We identify a general class of facets that subsumes as special cases some known facets from the literature. We also develop a cutting plane based procedure and reformulation for the problem, and obtain optimal solutions to problem instances with up to 1200 integer variables without resorting to branch and bound procedures. A key issue in scheduling is the effective allocation of shared resources to multiple products, for instance, for a facility that incurs a changeover cost whenever it switches production from one product to another. For example, in producing printed circuit boards, a plant might include a machine that places a set of components on a board. Typically, the plant will produce different types of boards, each with a different set of components. If it switches from one product to another, the machine needs to change over to a new set of tools and thus incurs a fixed cost. Each time it produces, the machine might also incur an additional set up cost for placing components. The resource allocation problem in this product cycling model must trade off changeover and set up costs against production and inventory holding costs. We study the polyhedral structure of a dynamic, deterministic version of the problem. This problem is NP-hard. As a result, the running time of all solution methods increases exponentially with the number of time periods and products. In the next section, we present an integer programming formulation of the problem. We then describe valid inequalities and facets for the problem, solve the separation problem, and present computational results for problems with up to 4 products. Magnanti and Vachani (1990), who give many further references to the literature on the problem, developed a solution technique based on cutting planes for the constant capacity case. This approach performed well on problems having up to 300 integer variables. Our results generalize those of Magnanti and Vachani by providing a more extensive set of valid inequalities and facets for the problem. We are able to solve larger problems to optimality with up to 1200 integer variables. For single item versions of these problems, the linear programming gaps (i.e., ratio of 100x(IP value LP value)/IP value for a 'natural' formulation of the problem is between 75% and 83% and for multi item problems, the gaps are between 6% and 20%. In each case, we are able to eliminate this gap completely by adding valid inequalities. Several researchers have used a polyhedral cutting plane approach for the lotsizing problem with start up costs. Wolsey (1989) used a cutting plane method that performed well for an uncapacitated version of our model. Van Hoesel, Wagelmans and Wolsey (1994) described the convex hull of this uncapacitated model. Van Hoesel (1991) and Van Hoesel and Kolen (1993) studied a capacitated version of the problem with start-up costs, but without setup costs, which they call the discrete lot sizing problem (DLSP) with start up costs. They introduced a class of strong valid inequalities. Our results differ from those in Van Hoesel and Kolen in two ways (i) we consider set up as well as changeover costs, and (ii) we derive valid inequalities and facets with arbitrary integer coefficients whereas Van Hoesel and Kolen consider valid inequalities and facets with 0-1 coefficients. Van Hoesel and Kolen (1994) also provide a complete linear description of DLSP with start up costs and no set up costs using an enhanced set of variables. Pochet and Wolsey (1994) provide a detailed survey of lot sizing algorithms and reformulations. They provide many citations to the literature which we will not repeat. They classify the problems into five categories (i) uncapacitated lot-sizing (ii) capacitated lot-sizing (iii) lot-sizing with start-ups (iv) discrete lot-sizing and (v) multi-level lot-sizing. In this taxonomy, the model we investigate is a discrete lot sizing problem. 1. Problem Formulation We consider a single machine, multi-product, production planning model. Let T denote the finite time horizon over which the facility is scheduled, P the number of products, dip the demand in period i, and np the total demand for item p. We assume a constant capacity and follow a discrete production policy, i.e, we either do not produce at all or produce to capacity in each time period. This policy is reasonable when it is expensive to run the facility at less than full capacity, or when demand is high and the facility is capacity constrained. It is also easily implemented. As shown in Magnanti and Vachani (1990), without loss of generality we can assume that capacity in each period is 1 unit and that demand is either 0 or 1. We assume that the relevant costs for each product p in period i are the changeover cost Fpi,