Early-tardy scheduling under fuzzy due dates using a genetic algorithm

We consider the problem of scheduling n jobs on a single machine. Each job i has a processing time p_i, a weight w_i, and a due date D_i, which is a fuzzy number with a triangular membership function. The problem is to determine: (i) a job sequence, and (ii) a set of idle times each before one job, so as to minimize the total weighted earliness and tardiness cost under the fuzzy due dates. We first introduce a fuzzy distance function to measure the deviation of the completion time of a job from its fuzzy due date. We show that, given a job sequence, the problem of determining the optimal idle times is a continuous and convex optimization problem with a differentiable objective function, although the objective function after applying the fuzzy distance measure becomes nonlinear. We devise a genetic algorithm (GA) to tackle the problem, using a pigeon-hole coding scheme to represent a sequence and a nonlinear optimizer to determine the idle times. We evaluate the solutions obtained by such a GA as compared to the solutions obtained by treating the due dates as crispy numbers equal to the mean values of its fuzzy partners