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Abstract

The presented study investigated the scheduling regarding ��jobs on a single machine. Each ��job will be processed with no interruptions and becomes available for the processing at time 0. The aim is finding a processing order with regard to jobs, minimizing total completion time∑����, total late work ∑����, and maximal tardiness ��������which is an NP-hard problem. In the theoretical part of the present work, the mathematical formula for the examined problem will be presented, and a sub-problem of the original problem of minimizing the multi-objective functions ∑����+∑����+��������isintroduced. Also, then the importance regarding the dominance rule (DR) that could be applied to the problem to improve good solutions will be shown. While in the practical part, two exact methods are important; a Branch and Bound algorithm (BAB) and a complete enumeration (CEM) method are applied to solve the three proposed MSP criteria by finding a set of efficient solutions. The experimental results showed that CEM can solve problems for up to n=11jobs. Two approaches of the BAB method were applied: the first approach was BAB without dominance rule (DR), and the BAB method used dominance rules to reduce the number of sequences that need to be considered. Also, this method can solve problems for up to ��=20, and the second approach BAB with dominance rule (DR), can solve problems for up to ��=60jobs in a reasonable time to find efficient solutions to this problem. In addition, to find good approximate solutions, two heuristic methods for solving the problemare proposed, the first heuristic method can solve up to ��=5000jobs, while the second heuristic method can solve up to ��=4000jobs. Practical experiments prove the good performance regarding the two suggested approaches for the original problem. While for a sub-problem the experimental results showed that CEM can solve problems for up to ��=10jobs, the BAB without dominance rule (DR) can solve problems for up to ��=15, and the second approach BAB with dominance rule (DR), can solve problems for up to ��=30jobs in a reasonable time to find efficient solutions to this problem. Finally, the heuristic method can solve up to ��=4000jobs. Arithmetic results are calculated by coding (programming) algorithms using (MATLAB 2019a)

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