This paper presents a mathematically enhanced genetic algorithm (MEGA) using the mathematical properties of the single-machine scheduling of multiple jobs with a common due date. The objective of the problem is to minimize the sum of earliness and tardiness penalty costs in order to encourage the completion time of each job as close as possible to the common due date. The importance of the problem is derived from its NP-hardness and its ideal modeling of just-in-time concept. This philosophy becomes very significant in modern manufacturing and service systems, where policy makers emphasize that a job should be completed as close as possible to its due date. That is to avoid inventory costs and loss of customer’s goodwill. Five mathematical properties are identified and integrated into a genetic algorithm search process to avoid premature convergence, reduce computational effort, and produce high-quality solutions. The computational results demonstrate the significant impact of the introduced properties on the efficiency and effectiveness of MEGA and its competitiveness to state-of-the-art approaches.
In this work we propose to specify, describe and test a variant of a more powerful and flexible genetic algorithm that could be better suitable to tackle complex optimization problems such as in dynamic, stochastic or robust optimization. Our main goal is to give a new strong tool more efficient in terms of both solution quality and time processing for complex NP-hard optimization problems, which know great importance these past few decades in economy, management, manufacturing and many other fields. This algorithm gives a significant improvement to the basic genetic algorithm of J. Holland in order to imitate and simulate as close as possible the naturel selection phenomenal established in the theory of C. Darwin. Thus, in the evolution process of generations, the population should not keep a fixed size, but it should evolve over the generations. In the other hand, the population should contain several breeds of the species under study. Therefore, much kind of crossovers could be applied randomly such as crossover of pure or hybrid breeds. In addition, many types of mutation would be possible such as substitution, addition or deletion which could also happen randomly in the nature. The main idea is based on the maximal projection of the evolution theory on the optimization field to tackle complex problems. We aim to design flexible genetic algorithm by looking empirically for good compromise of adjusting the genetic parameters on sample cases.
In this paper, we propose a new linear algorithm to tackle a specific class of unrelated machine scheduling problem, considered as an important real-life situation, which we called Batch Scheduling on Unrelated Machine (BSUM), where we have to schedule a batch of identical and non-preemptive jobs on unrelated parallel machines. The objective is to minimize the makespan (Cmax) of the whole schedule. For this, a mathematical formulation is made and a lower bound is computed based on the potential properties of the problem in order to reduce the search space size and thus accelerate the algorithm. Another property is also deducted to design our algorithm that solves this problem. The latter is considered as a particular case of RmCmax family problems known as strongly NP-hard, therefore, a polynomial reduction should realize a significant efficiency to treat them. As we will show, Batch BSUM is omnipresent in several kind of applications as manufacturing, transportation, logistic and routing. It is of major importance in several company activities. The problem complexity and the optimality of the algorithm are reported, proven and discussed.
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