Reducing efficiently the search tree for multiprocessor job-shop scheduling problems

Abstract

The multiprocessor job-shop scheduling problem (JSP) is a generalisation of the classical job-shop problem, in which each machine is replaced by a group of parallel machines. We consider the most general case when the parallel machines are unrelated, where the number of feasible solutions grows drastically even compared with the classical JSP, which itself is one of the most difficult strongly NP-hard problems. We exploit the practical behaviour of a method of the preliminary reduction of the solution space of this general model from an earlier paper by Vakhania and Shchepin (this model works independently and before the application of lower bounds). According to the probabilistic model proposed by Vakhania and Shchepin, this results in an exponential reduction of the whole set of feasible solutions with a probability close to 1. In this paper, this theoretical estimation is certified by intensive computational experiments. Despite an exponential increase in the number of feasible solutions in the instances generated from job-shop instances, we were able to solve many of them optimally just with our reduction algorithm, where the results of our computational experiments have surpassed even our theoretical estimations. The preliminary reduction of the solution space of our multiprocessor job-shop problem is of essential importance due to the fact that the elaboration of efficient lower bounds for this problem is complicated. We address the difficulties connected with this kind of development and propose a few possible lower bounds.