Integrated production and transportation scheduling considering vehicles with different capacities

becomes an important subject addressed by many researchers. A supply chain represents all stages that have added value to a product.Integration and synchronization of information and material flows of manufacturing sites belonging to a supply chain (SC) has become more practical and have attracted the attention by both industry practitioners and academic researchers. In this study after presenting previous works on scheduling in supply chain, the problem is described and mathematical model of the problem is presented. Then a genetic algorithm is proposed for solving the problem that has chromosomes with variable structures. Finally we provide concluding remarks and some scopes for future researches.This paper studies a 3-stage supply chain scheduling problem in which the first stage is composed of multiple suppliers with different production speeds that produce parts ordered by a manufacturing company at third stage. In the second stage vehicles with variable speeds and variable capacities convey the jobs from the suppliers to the manufacturing company at third stage. The main focus of this study is on the integration of the production and the transportation scheduling.For simplicity, it is assumed that all suppliers are in one geographical zone and the transportation times between them are negligible in comparison with the transportation time from the suppliers to the manufacturing company. However, in some realistic situations the suppliers may be located in multiple geographical zones, in sake of reducing complexity, it can be assumed that original problem can be divided in multiple sub-problems whereas each group of suppliers are located in one geographical zone and sharing the vehicles between the suppliers in different sub-problems are not allowed. Each vehicle after delivering a batch to the manufacturing company back haul empty to the suppliers’ zone for the next dispatching. The objective function of the problem is to minimize delivery time of a set of jobs to the manufacturing company that herein we address it as minimizing maximum completion time of all jobs, i.e, makespan.Since such problems have NP-hard structure, thus Genetic algorithm can be mentioned as an approach that frequently used for solving them. In this study a genetic algorithm named dynamic genetic algorithm (DGA) that has chromosomes with different structure is developed.DGA has six parameters as follows: 1) population size (popsize), 2) crossover rate (percross), 3) mutation rate (permut), 4) percentage of the best chromosomes are selected to the next population (best), 5) number of times with no improvement in fitness function for terminating the algorithm (termination) and 6) a parameter in crossover operation (r).After solving various test problems we empirically have found that values of 100 for popsize, 0.6 for percross, 0.8 for permute,0.7 for best,10 for termination and 0.7 for r may lead to better solutions.According to the author's knowledge, this problem has not been studied previously. Since there is no algorithm to compare with dynamic genetic algorithm in the literature, we compared the results of DGA with those of two algorithms: a random search algorithm and an adapted algorithm based on the nearest problem in the literature to our problem (namely H1).In sake of comparisons, many test problems are produced randomly according to a defined structure and solved by DGA, the random search approach and H1. In order to compare DGA with the random search approach 4 critoria were used as follows: 1) Mean of solutions of DGA, 2) Mean of solutions of the random search, 3) Percentage of runs that DGA get better result than the random search and 4) Percentage of runs that DGA get equal result to the random search. Results show that DGA outperforms the random search approach in all cases. Also with increasing the number of jobs mean of solutions increased. Increasing the number of suppliers causes mean of solutions decreased but when there exist a bottleneck in this stage mean of solutions increased. Also by decreasing the process times of jobs in the first stage, the mean of solutions decreases. Increasing the vehicle's capacities causes the mean of solutions to decrease. DGA is also compared with an adapted algorithm based on the nearest problem in the literature to our problem, namely H1, proposed by Chang and Lee (2004).Their scheduling problem is the same as problem considered in this paper but they assumed that only one vehicle exists in the transportation stage while in our problem transportation fleet is composed of l vehicles with different speeds and capacities. Also they considered at most two suppliers with identical production speed but in our problem m suppliers exist with different production speeds. For the case of two suppliers, they proposed a heuristic algorithm and proved that it could cause at most 100% error under the worst-case with the bound being tight. In order to compare DGA with H1 four criteria for both algorithms are used as follows: 1) Mean of solutions, 2) Percentage of runs that an algorithm gets better result than the another one (PBR), 3) Percentage of runs that an algorithm get equal result to the another one (PER) and 4) Mean of solving time. Experimental results show DGA performance is much better than that of H1.