Universal Journal of Electrical and Electronic Engineering 6(2): 46-60, 2019 http://www.hrpub.org DOI: 10.13189/ujeee.2019.060202 Particle Swarm Optimization Algorithms for Two-stage Hybrid Flowshop Scheduling Problem with No-wait Mageed A. Ghaleb, Ibrahim M. Alharkan * Industrial Engineering Department, King Saud University, Saudi Arabia Copyright©2019 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Hybrid flowshop scheduling problems have attracted much attention owing to their wide applications in a variety of real-world problems. In some industries, products cannot be allowed to wait between any consecutive productions stages, which illustrates the importance of studying the no-wait constraint. One of the reasons for such a constraint is that, for some products, the waiting time could cause permanent damage. However, the no-wait constraint was neglected in many prior studies, which in some production environments may not be allowed. Minimizing the total tardiness plays a key role in making scheduling decisions to meet customers’ due dates. In this study, we solve the no-wait two-stage hybrid flowshop scheduling problem with total tardiness minimization as an optimizing criterion. We formulated the problem mathematically and proposed two discrete versions of the particle swarm optimization (PSO) to solve it. Moreover, three discrete versions of PSO are adopted from previous studies and used as benchmarks to test the effectiveness of the two proposed algorithms. Compared to the benchmark algorithms, the results showed that the two newly proposed algorithms were effective and performed better than the benchmark algorithms in terms of the average relative error. The current study represents one of the few attempts to investigate the considered problem with total tardiness minimization, as well as introducing new and effective discrete versions of the PSO algorithms to solve the problem under investigation. Keywords Scheduling, Hybrid Flowshop, No-wait, Total Tardiness, Particle Swarm Optimization 1. Introduction Consider a two-stage hybrid flowshop with a set of  jobs (or operations) needed to be scheduled, taking in account that in each stage there is more than one machine. The machines are identical and in parallel. At each stage, each job is performed on only one of the existing machines. Each job has a completion time on stage one that is equal to its start time on stage two. In other words, jobs are not allowed to wait between the two stages and have to be moved from stage one to stage two without any interruptions. This problem is known as the two-stage no- wait hybrid flowshop scheduling problem (NWHFSP). The two-stage NWHFSP has several industrial in industry. Therefore, it has received much attention from many researchers. For instance, [1] presented an application to the synthetic fiber industry. [2] described an application to a computer system that has a single server and two parallel processors. [3] described an application to an annealing roller in a case steel plant. [4] addressed an application to semiconductor packaging. Generally, in any manufacturing firm with a production line that consists of consecutive production stages with parallel identical machines in at least one of those stages, we can consider the allocation of items to be produced in that line with a no- wait restriction between its consecutive stages as a NWHFSP. Hence, the two-stage NWHFSP is realistic and occurs in many real-life applications. The simplest hybrid system, which is a two-stage hybrid flowshop with only one machine in one of its stages and several identical parallel machines in the other stage, has been addressed in many previous studies. For instance, [5] developed a greedy algorithm, named least deviation algorithm (LDA), to minimize the makespan in the simplest two-stage NWHFSP. [6] developed two heuristic algorithms to minimize the makespan in the simplest two- stage NWHFSP with separate setup and removal times. [7] designed a genetic algorithm (GA) to minimize the makespan in the simplest two-stage NWHFSP. [8] developed a branch and bound (B&B) algorithm to minimize the makespan in the simplest two-stage NWHFSP. Problems with setup times, random machines breakdowns, and rework probability have also been investigated in previous studies. For example, [4] formulated the problem as an integer programming model and adopted an ant colony optimization (ACO) algorithm to minimize the total completion time in a two-stage NWHFSP with setup times. [9] Utilized a particle swarm