Metaheuristic Algorithms for Hybrid Flow-Shop Scheduling Problem with Multiprocessor Tasks

This paper is concerned with hybrid flow-shop scheduling problem involving multiprocessor tasks. The hybrid flow-shop problem is a generalization of the classical flow-shop problem by permitting multiple parallel processors at each stage of task processing. The multiprocessor tasks, on the other hand, overcome the restriction of the classical scheduling problems by allowing tasks to be processed on more than one processor simultaneously. The hybrid flow-shop is defined by the set M = {1, ...,m} of m processing stages, in which each stage contains a set Mi = {1, ...,mi} of mi identical processors. There is a given set J = {1, ..., n} of n jobs to be processed in this hybrid flow-shop system. Each job can be viewed as a sequence of m multiprocessor tasks. The i-th task in the sequence has to be processed for pij time units without preemption by sizeij processors of stage i, and can be started only after the completion of the previous task from this sequence. In other words, the i-th task of job j ∈ J is defined by its processing time, pij > 0, and the number of processors required, sizeij. For convenience, we will regard the i-th task of job j to be comprised of sizeij operations, which must be processed simultaneously at stage i. The remaining parameters of the jobs are as follows: rj release date of job j, qj delivery time of job j, tij transport time between stage i and i + 1 of job j. To be in line with transport times, we will refer to job release dates and delivery times as t0j and tmj , respectively. Additionally, there are stageand sequence-dependent set-up times. sijk, i ∈ M , j ∈ J ∪ {0}, k ∈ J , denotes the set-up time between job j and job k at stage i, and si0k is a set-up time, which occurs before i-th task of job k if it is processed first on the processor. We also assume that the set-ups are anticipatory, i.e. the set-up processing can start before the arrival of a task.