Earliness and tardiness scheduling with a fuzzy due date and job dependent weights

Abstract

The paper considers the problem of scheduling n jobs on a single machine. Each job j is associated with a weight wj and a processing time pj, and the objective of the problem is to minimize the weighted earliness and tardiness of job completions from a common due date D̃, where D̃ is a fuzzy number, governed by a triangular membership function. A fuzzy distance function is introduced first to measure the deviations of job completions from the fuzzy due date. We then show that an optimal job sequence must be V-shaped in terms of weighted processing time when the problem is agreeably weighted, in the sense that pi < pj implies wi ≥ wj . We further show that, for arbitrary weights, the optimal sequence must be W-shaped if the support of the due date is smaller than p min, the smallest processing time. Algorithms are derived which can find the best V-shaped or W-shaped sequences in pseudo-polynomial time. A polynomial algorithm is also developed for the special case with equal processing times. Numerical experiments were conducted, to examine the applicability of the algorithms proposed to general cases without any conditions.