A Branch and Cut Approach for Workload Smoothing on Assembly Lines

This paper presents a strong cutting plane method implemented by branch and cut to solve the assembly line workload smoothing problem which minimizes the maximum idle time for a specified number of stations in order to balance workloads assigned to all stations. The approach exploits a problem formulation that embeds the assembly line balancing polytope. Thus, inequalities that are known to be valid for the line balancing polytope are also valid for workload smoothing. This paper describes an approach for implementing a strong cutting plane method that employs such valid inequalities, including separation algorithms. Preprocessing methods are described to decompose and reduce a precedence graph as well as to estimate bounds on parameters that are involved in valid inequalities. Finally, computational experience that evaluates the efficacy of the approach is presented. Given a set of tasks, task processing times, task precedence relationships, and desired cycle time, the objective of the assembly line balancing problem (ALBP) is to assign tasks to the minimum number of stations while observing precedence relationships. Solutions do not necessarily assign equal workloads to stations and may, thus, create operating inefficiencies. The workload smoothing problem (WSP) deals with this issue, using the optimal number of stations and assigning tasks with the objective of minimizing the maximum idle time on any station in order to balance workloads assigned to all stations. Since the ALBP is NP-hard [7], heuristics have appeal as fast and convenient solution techniques, and considerable progress has been made in developing and improving heuristics [7]. A reasonable level of research effort has also been directed toward developing optimization approaches. Baybars [2] and Ghosh and Gagnon [7] present recent reviews of research on the ALBP. In contrast, research on the WSP has been rather limited. No optimization methods for the WSP have, apparently, been proposed. Tonge [27] introduced a heuristic to smooth line balance by transferring tasks among stations until the workload distribution is as even as possible. For lines with either single or parallel machines, Sarker and Shanthikumar [25] further improved line balance by the "Trade and Transfer" heuristic to smooth station idleness. Rachamadugu and Talbot [23, 24] formulated the workload smoothing problem as an integer program and developed an iterative heuristic to reduce workload differences among stations. Recently, cutting plane methods have performed successfully in a variety of applications. Even though valid inequalities used as cutting planes are problem specific, the polyhedral characteristics of embedded structures can be applied in a more complex application [4]. This relationship motivated this research, which describes some characteristics of the polyhedral structure of the ALBP and applies that knowledge to resolve the WSP. Specifically, branch and cut has resulted in successful applications including those by Padberg and Rinaldi [17] for symmetric traveling salesman problems (TSP) and Hoffman and Padberg [9] for airline crew scheduling problems. In a related paper [20], we introduced families of valid inequalities for the ALBP and showed conditions under which they define facets for a certain relaxation of the ALBP. This paper, a continuation, presents a separation algorithm for each family of cuts. Preprocessing methods are described to decompose and reduce a precedence graph as well as to estimate bounds on parameters that are involved in our valid inequalities. Recognizing that the ALBP polytope is embedded in the WSP polytope, we implement a branch and cut approach to solve the WSP, using the inequalities we showed to be valid for the ALBP polytope. Finally, computational experience that evaluates the efficacy of the approach is presented. The paper is organized as follows. Section 1 formulates the ALBP and the WSP. Section 2 provides a brief review of valid inequalities. Sections 3 and 4 describe separation algorithms and preprocessing methods, respectively. Section 5 discusses our implementation of branch and cut and its computational evaluation. Finally, Section 6 gives concluding remarks and suggestions for future research. 1. PROBLEM FORMULATION We use a precedence graph to represent tasks and their precedence relationships using notation T, t, H, T, and M(j) as defined in Figure 1. Since nodes represent tasks in H, we use the terms interchangeably. We invoke two assumptions about the precedence graph: (1) H is connected and simple (i.e., no loops and at most one arc joining two nodes), and (2) H has a single source (node 1) and a single sink (node t). To formulate the ALBP, we use parameters pt, SL, SU, Ej, Lj and T(s), and decision variables xsi as defined in Figure 1. The ALBP may be formulated as the following 0/1 integer program.