A branch and bound algorithm to minimize completion time variance on a single processor

Abstract

In this paper we discuss a single machine scheduling problem with the objective of minimizing the variance of job completion times. The CTV problem has been proved to be NP hard (Oper. Res. Lett. 14 (1993) 49) and no polynomial time algorithm exists to find an optimal solution for CTV minimization on single machine. Hence enumerative techniques and heuristics are used to get optimal and near optimal solutions, respectively. We present a branch and bound algorithm and extend the same algorithm to generate epsilon optimal solutions for large sized problems (i.e., number of jobs >30). The algorithm has been computationally tested, with randomly generated problems involving up to 100 jobs, using a personal computer (PC) with a 64 MB RAM capacity. The computational time required for generating optimal solutions are in few seconds for problems with jobs between 25 and 30. The performance of the branch and bound algorithm is compared with the pseudo-polynomial algorithm (Oper. Res. 40 (1992) 1148) for small sized problems. For problems with greater number of jobs, the epsilon optimal solutions obtained using branch and bound algorithm are compared with results of simulated annealing (Single machine scheduling with some non-regular objectives, M.S. Thesis, IIT Madras, 1997), tabu search (Proceedings of Operations Management Conference, IIT Madras, 2000) and heuristic proposed by Manna and Prasad (Eur. J. Oper. Res. 114 (1999) 411). Scope and purpose Scheduling jobs in manufacturing systems considering non-regular measures of performance have gained importance in recent years. The objectives of minimizing completion time variance ( CTV problem) and minimizing squares of deviations of job completion times from a given common due-date ( MSD problem) are the two problems that use non-regular measures, and have attracted attention in single machine scheduling and flowshop scheduling when jobs require uniform treatment i.e., each entity spends approximately same time in the system or waits for service as every other entity. This problem is very much relevant when a manufacturing/service system places emphasis on Just-in-Time philosophy. The purpose of this paper is to present an algorithm for minimizing completion time variance for a set of jobs to be processed on a single machine. The proposed algorithm is computationally evaluated with randomly generated problems involving up to 100 jobs and results are reported.