Scheduling Precedence Related Jobs on Identical Parallel Processors

Abstract

The problem of concern to us in this thesis is the scheduling of precedence-related jobs non-preemptively on two identical parallel processors to minimize the sum of the weighted completion times. The problemis known to be NP-hard.We develop, in chapter 2, a binary integer program which is capable of solving only small size problems (no larger than 12 jobs) to optimality at the present time. We also present a linearprogramming (LP) model adopted from the literature todetermine the lower bound on the optimum. This LP stands us in good stead when we perform the optimization via the Genetic Algorithm approach (which is the subject matter of chapter 3). Wealso present a dynamic programming formulation based on the approach used for solving the 'weighted earliness-tardiness' problem. Although DP expands somewhat the size of the problemsthat can be solved to optimality, its computing burden becomesonerous for more than 25 jobs.In an attempt to solve larger, and more realistic problems, a Genetic Algorithm (GA) is presented in chapter 3. The salient feature of the GAmodel is that the 'initial population' of trial solutions are not allrandomly generated but are constituted from a set of priority rules whichare known to be 'good' relaxation (in the sense of being 'close' to theoptimum) of the original problem. Also, generation of infeasible solutionsis avoided by the use of post-processing procedures after crossover and mutation operations. Computational results show that the GA approach arrivesto within 20% of the lower bound (and hence of the optimum) in very few iterations.