Parallel machine scheduling with step deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm

This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary ones. A mixed integer programming model for the optimal solution is derived. Due to its NP-hard nature, a hybrid discrete cuckoo search algorithm is proposed to solve this problem. In order to generate a good initial swarm, a modified heuristic named the MBHG is incorporated into the initialization of population. Several discrete operators are proposed in the random walk of L\'{e}vy Flights and the crossover search. Moreover, a local search procedure based on variable neighborhood descent is integrated into the algorithm as a hybrid strategy in order to improve the quality of elite solutions. Computational experiments are executed on two sets of randomly generated test instances. The results show that the proposed hybrid algorithm can yield better solutions in comparison with the commercial solver CPLEX with one hour time limit, discrete cuckoo search algorithm and the existing variable neighborhood search algorithm.