Identical Parallel Machine Scheduling with Dynamical Networks using Time-Varying Penalty Parameters

The classical identical parallel machine scheduling problem can be stated as follows: Given n jobs and m machines, the problem is to assign each job on one of the identical machines during a fixed processing time so that the schedule that optimizes a certain performance measure is obtained. Having numerous potential applications in real life, in recent years, various research works have been carried out to deal with the parallel scheduling problems. The literature of parallel machine scheduling problems has been extensively reviewed by (Cheng & Sin, 1990; Mokotoff, 2001). Among many criteria, minimizing makespan (maximum completion time) has been one of the most widely studied objectives in the literature. Using the three-field classification introduced in (Graham et al., 1976), the problem is denoted in the scheduling literature as P||Cmax where P designates the identical parallel machines, Cmax denotes the makespan. We assume, as is usual, that the processing times are positive and that 1<m<n. The problem is known to be NP-hard in the strong sense (Garey & Johnson, 1979; Sethi, 1977). Although traditional techniques such as complete enumeration, dynamic programming, integer programming, and branch and bound were used to find the optimal solutions for small and medium sized problems, they do not provide efficient solutions for the problems with large size. Having found no efficient polynomial algorithm to find the optimal solution led many researchers to develop heuristics to obtain near optimal solutions. Though, efficient heuristics can not guarantee optimal solutions, they provide approximate solutions as good as the optimal solutions. These can be broadly classified into constructive heuristics and improvement heuristics. Most of the algorithms belong to the first category and have known worst case performance ratio (Coffman et al., 1978; Friesen & Langston, 1986; Friesen, 1987; Graham, 1969; Hochbaum & Shmoys, 1987; Leung, 1989; Sahni, 1976). The LPT rule of Graham, one of the most popular constructive heuristics, has been shown to perform well for the makespan criterion. This rule arranges jobs in descending order of processing times, such that p1 p2 ... pn, and then successively assigns jobs to the least loaded machine. The MULTIFIT algorithm, a classical constructive heuristic developed by (Coffman et al., 1978), determines the smallest machine capacity to find a feasible solution using the LPT scheme. This is achieved by solving heuristically a series of bin packing