Valid inequalities for a class of assembly system problems

Abstract

This paper establishes a foundation for a cutting plane approach to solve a class of problems that are important to assembly systems, including assembly line balancing (ALB), workload smoothing (WS) and assembly system design (ASD). We describe relationships among the integer programming formulations of these problems and recast their relaxations as node packing problems (NPPs). Exploiting the well-known structure of the NPP and other relationships among the assembly problems, we derive several families of valid inequalities and show conditions under which they also define facets for the associated ALB, WS and ASD polytopes.

Introduction

Cutting plane methods, which employ (strong) valid inequalities, have performed successfully in a variety of applications. The validity of an inequality must be evaluated with respect to a specific polyhedron. Inequalities that are valid relative to a polyhedron that contains another one are also valid for the latter (Crowder et al., 1983). This relationship motivated our research, which addresses a class of problems that are important to assembly systems, including assembly line balancing (ALB), workload smoothing (WS) and assembly system design (ASD).

The purpose of this paper is to establish the theoretical foundation for a cutting plane approach for this class of assembly problems. The ASD problem (ASDP) prescribes the minimum cost system design. It integrates design decisions such as (D1) the number of stations and (D2) type of machine located at each station with operating issues such as assigning tasks to each station (D3) while observing (O1) precedence relationships among tasks and (O2) cycle time restrictions at all stations. Closely related to the ASDP, the classical ALB problem (ALBP) prescribes the minimum number of identical stations, integrating (D1) and (D3) while (O1) and (O2). Solutions to the ALBP do not necessarily assign equal workloads to all stations and may, thus, create operating inefficiencies. The WS problem (WSP) deals with this issue, assigning tasks to a given number of stations with the objective of minimizing the maximum idle time on any station.

Prior research works concerning the ALBP, WSP, ASDP and to cutting plane methods are related to the current study. We give a brief review of each topic in this section.

Ghosh and Gagnon (1989) provided a recent survey of prior research works on the ALBP, which is well known to be NP-hard (Baybars, 1986). An extensive literature has dealt with heuristics and specialized branch-and-bound optimizing methods, but the current study appears to be the first one to develop a strong-cutting plane approach. It should be noted, however, that 0–1 integer programming (IP) formulations have been evolved over time (e.g., Bowman, 1960, Patterson and Albracht, 1975, Talbot and Patterson, 1984).

In contrast, research on the WSP has been rather limited. No optimizing methods for the WSP have, apparently, been proposed. Rather, heuristics such as those developed by Sarker and Shanthikumar (1983) and Rachamadugu and Talbot, 1987, Rachamadugu and Talbot, 1991 have predominated.

Research on the ASDP has typically sought to minimize the total cost of the system design. Ghosh and Gagnon (1989) have suggested that design issues (e.g., line size and machine selection) be integrated with operating issues (e.g., task-related considerations) as does our model. Ghosh and Gagnon (1989), and Soni (1990) have summarized important design issues. Researchers have studied various versions of the ASDP, typically applying branch and bound to prescribe optimal solutions (e.g., Pinto et al., 1983; Lee and Johnson, 1991). In particular, Graves and Lamar (1983) and Graves and Redfield (1988) have developed successful methods to solve actual, single- and multiple-product ASDPs.

Over the last two decades, a substantial effort has been directed towards the study of polyhedral structures and of cutting plane methods. Nemhauser and Wolsey (1988) describe this work and an extensive set of applications. In an early study, Padberg (1979) developed sets of valid inequalities and facets for covering, packing, and knapsack problems. Crowder et al. (1983) reported remarkable computational results in solving large-scale 0–1 integer problems using strong cutting plane methods. Their results motivate our use of valid inequalities.

In this paper, we establish a foundation for a cutting plane method by deriving strong valid inequalities related to the NPP and describing conditions under which they represent facets of polyhedra associated with the ALBP, WSP and ASDP. These inequalities can be extended to form inequalities that are valid for the ASDP by exploiting the relationships between the NPP and the ASDP that we describe in this paper. Pinnoi and Wilhelm (1998) describe this extension and incorporate the resulting inequalities in a branch-and-cut approach to solve the ASDP; they also report successful computational experience. Pinnoi and Wilhelm (1997a) describe a branch-and-cut approach to solve the WSP that incorporates the inequalities derived in this paper; they focus on pre-processing methods and separation algorithms and also report successful computational experience. In this paper, we establish the theoretical foundation for those implementations.

We have organized the body of this paper in five sections. Section 2 introduces mathematical models and discusses relationships among them. In Section 3, we show that the node packing problem (NPP) is a relaxation of the ALBP, WSP, and ASDP and identify some relationships among the associated polytopes. Based on node packing and knapsack structures, we derive families of clique inequalities and packing cover inequalities in 4 Families of valid inequalities based on cliques, 5 Families of packing cover inequalities, respectively (we establish conditions under which these inequalities define nontrivial facets in Appendix A). Finally, Section 6 gives concluding remarks and suggestions for future research.