New Heuristics for Flowshop Scheduling

Abstract

In the flowshop scheduling problem, n jobs are to be processed on m machines. The order of the machines is fixed. We assume that a machine processes one job at a time and a job is processed on one machine at a time without preemption. Let t _p (i, j) denote the processing time of job j on machine i and t _c (i, j) denote the completion time of job j on machine i. Let Jj denote the j-th job and M _i the i-th machine. The completion times of the jobs are obtained as follows. For i = 1, 2,..., m and j = 1, 2,..., n, t _c (M ₁, J ₁) = t _p (M ₁, J ₁);t _c (M _i , J ₁) = t _c (M _i−1, J ₁) + t _p (M _i , J ₁);t _c (M ₁, J _j ) = t _c (M ₁, J _j−1) + t _p (M ₁, J _j ); t _c (M _i , J _j ) = max{t _c (M _i−1, J _j ),t _c (M _i ,J _j−1)} + t _p (M _i ,J _j ). Makespan is defined as the completion time of the last job, t _c (M _m , J _n ). The goal is to obtain the n-job sequence that minimizes the makespan. The search space consists of n! possible job sequences. The problem is NP-hard.