A flexible flowshop problem with total flow time minimization

Abstract

In this study, we consider total flow time problem in a flexible flowshop environment. We develop a branch and bound algorithm to find the optimal schedule. The efficiency of the algorithm is enhanced by upper and lower bounds and a dominance criterion. Computational experience reveals that the algorithm solves moderate sized problems in reasonable solution times.

Introduction

This paper addresses the problem of scheduling n jobs on w serial stages, each stage including several parallel identical machines. A job should be processed on any one of the parallel machines at each stage. Such an environment is called a flexible flowshop. Our scheduling objective is to minimize total flow time.

Flexible flowshops are generalizations of flowshops. The literature for the flowshop problem has grown after Johnson’s (1954) well-known algorithm for the two stage maximum completion time, i.e. the makespan problem. Many of the studies on flowshops consider the minimization of makespan and total flow time. The studies by Gupta and Dudek (1971) and Panwalker et al. (1972) have revealed that total flow time problem is more representative of scheduling costs than makespan. The branch and bound algorithms by Ignall and Schrage, 1965, Bansal, 1977 and Ahmadi and Bagchi (1990) and heuristic approaches by Ho and Chang, 1991, Miyazaki et al., 1978 and Rajendran and Chaudhuri (1991) are among various attempts to solve the total flow time problem on flowshops.

Recent studies have recognized the importance of flexible flowshops to reduce the delays caused by bottleneck stages. Flexible flowshop problems arise in a number of different settings including polymer, chemical, and petrochemical industries (Salvador, 1973). It has been encountered in certain manufacturing systems (Zijm and Nelissen, 1990) and in assembly lines with parallel machines at workstations (Brah and Hunsucker, 1991), electronics industry (Guinet and Solomon, 1996) and textile industry (Guinet, 1991).

Many of the studies on flexible flowshops consider makespan minimization. Some special cases of the makespan problem are studied by Arthanari and Ramamurthy, 1971, Mittal and Bagga, 1972, Murty, 1974, Rajendran et al., 1986, Gupta, 1988, Gupta and Tunc, 1994 and Kim et al. (1998). The studies by Sriskandarajah and Sethi (1989) and Brah and Hunsucker (1991) consider the general flexible flowshop makespan problem. Sriskandarajah and Sethi (1989) study a number of heuristics in terms of their worst and average case performances, whereas Brah and Hunsucker (1991) propose a branch and bound algorithm to find an optimal schedule.

To the best of our knowledge, the only reported research on total flow time problem in flexible flowshops is due to Rajendran and Chaudhuri (1992). The study proposes a branch and bound algorithm to obtain an optimal permutation schedule. In this study, we propose a branch and bound algorithm to find an optimal solution for the total flow time problem in flexible flowshops. This optimal schedule need not be a permutation schedule. The lower bounds and the dominance rule developed considerably reduce the size of the branch and bound tree.

The rest of the paper is organized as follows. In the next section, we define our notation and describe the problem. In Section 3, we present the branch and bound algorithm along with the lower and upper bounds and the dominance theorem. In Section 4, we give the results of our computational experience. We conclude in Section 5.