Fast parallel heuristics for the job shop scheduling problem

Abstract

The paper is dealing with parallelized versions of simulated annealing-based heuristics for the classical job shop scheduling problem. The scheduling problem is represented by the disjunctive graph model and the objective is to minimize the length of longest paths. The problem is formulated for l jobs where each job has to process exactly one task on each of the m machines. The calculation of longest paths is the critical computation step of our heuristics and we utilize a parallel algorithm for this particular problem where we take into account the specific properties of job shop scheduling. In our heuristics, we employ a neighborhood relation which was introduced by Van Laarhoven et al. (Operations Research 40(1) (1992) 113–25). To obtain a neighbor, a single arc from a longest path is reversed and these transition steps always guarantee the feasibility of schedules. We designed two cooling schedules for homogeneous Markov chains and additionally we investigated a logarithmic cooling schedule for inhomogeneous Markov chains. Given O( n 3) processors and a known upper bound Λ= Λ( l, m) for the length of longest paths, the expected run-times of parallelized versions are O(n log n log Λ) for the first cooling schedule and O(n 2 ( log 3/2n)m 1/2 log Λ) for the second cooling schedule, where n= lm is the number of tasks. For the logarithmic cooling schedule, a speed-up of O(n/( log n log Λ)) can be achieved. When Markov chains of constant length are assumed, we obtain a polylogarithmic run-time of O( log n log Λ) for the first cooling schedule. The analysis of famous benchmark problems led us to the conjecture that Λ⩽ O( l+ m) could be a uniform upper bound for the completion time of job shop scheduling problems with l jobs on m machines. Although the number of processors is very large, the particular processors are extremely simple and the parallel processing system is suitable for hardware implementations. Scope and purpose In our paper, we are dealing with parallel algorithms for the job shop scheduling problem. The problem is defined as follows: Given a number of l jobs, the jobs have to be processed on m different machines. Each job consists of a sequence of m tasks, i.e., each task of a job is assigned to a particular machine. The tasks have to be processed during an uninterrupted time period of a fixed length on a given machine. A schedule is an allocation of the tasks to time intervals on the machines and the aim is to find a schedule that minimizes the overall completion time which is called the makespan. The scheduling problem is one of the hardest combinatorial optimization problems (cf. M.R. Garey, D.S. Johnson, Complexity Results for Multiprocessor Scheduling under Resource Constraints, SIAM Journal on Computing 1975;4(4):397–411). Many methods have been proposed to find good approximations of optimum solutions to job shop scheduling problems; for an overview, see E.H.L. Aarts, Local Search in Combinatorial Optimization. Wiley & Sons, New York, 1998. Since scheduling problems are hard to solve, one can try to obtain a certain speed-up of computations by using multiprocessor systems. In the present paper, we apply parallel algorithms to the most time-consuming part of several heuristics that have been developed by the authors earlier for the job shop scheduling problem (cf. K. Steinhöfel, A. Albrecht, C.K. Wong, Two Simulated Annealing-Based Heuristics for the Job Shop Scheduling Problem, European Journal of Operational Research 1999;118(3):524–548). We prove that a significant speed-up can be obtained compared to heuristics running on a single processor, however, at the cost of a large number of processors. Furthermore, we discuss a conjecture about an upper bound for the makespan that depends on the sum l+ m only. The conjectured upper bound has an immediate impact on the speed-up of parallel algorithms and is also of importance to other, single processor methods solving job shop scheduling problems. We provide evidence that our conjecture indeed might be true by the analysis of a large number of benchmark problems.