Sequencing to Minimize the Weighted Completion Time Subject to Constrained Resources and Arbitrary Precedence

Abstract

The primary concern of this thesis is the scheduling of the precedence related jobs non-preemptively on the two resources to minimize the sum of the weighted completion times. The problem is known to be NP-complete.

The problem Pm|prec|ΣwjCj is treated when the m resources are distinct and are of unit availability each. A job may demand the simultaneous usage of any subset of the resources. We develop, in chapter 2, a binary integer program for this problem and use it to solve problems of small size. In chapter 3, we propose an approach based on transforming the precedence graph into a series/parallel (s/p) graph by the introduction of ‘artificial precedence relations (a.p.r), and then reversing these relations to secure the optimum. These reversals of the a.p.r's is of complexity 2m, where m is the number of a.p.r's, and makes the problem NP-hard. To reduce the computational burden, we propose a branch-and-bound approach to search the solution space more efficiently.

The proposed approach is strongly dependent on the work done by Lawler in the field of s/p graphs on one-machine scheduling to minimize the weighted completion time and on the work of Elmaghraby on the construction of a.p.r's and their reversals, which inturn depends on the work of Bein et al on the identification of of cross-arcs in the interdictive graphs. It utilizes the concepts of 'artificial precedence relations' and 'Branch and Bound' to extend the problem for any non-s/p graphs.