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A multi-criteria group decision-making model based on interval TOPSIS and grey theory for solving the problem of distribution
rules
Leila Mokhtari, Reza Tavakkoli-Moghaddam, and S.Meysam Mousavi
1Leila Mokhtari, Postgraduate, School of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran
2Reza Tavakkoli-Moghaddam, Professor, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran
3S.Meysam Mousavi, Assistant Professor, Industrial Engineering Department, Faculty of Engineering, Shahed University, Tehran
ABSTRACT
Selecting the dispatcher is a multi-criteria decision problem which it will strongly affect the performance of the production system. But the options and criteria are not clear. They cannot be represented by numerical values. To deal with the complexity of these decision problems using a combination of approaches is necessary. So in this paper, a new group decision-making method is presented based on a grey theory, grey relational analysis (GRA) and TOPSIS concepts to select the distribution rules. Finally, an application example from the literature in the flexible environment is presented and considered in order to show the performance of the presented model.
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: 82084183 E-mail: [email protected]
www.iiec2015.org
Keyword:
Grey theory, GRA, TOPSIS, distribution rules
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Wang,Y.-C., Usher, J.M., "Application of reinforcement learning for agent-based production scheduling", Engineering Applications of Artificial Intelligence ,18(2005): 73–82.
[1]
El-Bouri, A., Shah, P., " A neural network for dispatching rule selection in a job shop", Int J AdvManufTechnol 31(2006):342–349
[2]
Mouelhi-Chibani, H., Pierreva" Training a neural network to select dispatching rules in real time", Computers & Industrial Engineering ,58 (2010) 249–
256.
[3]
Baykasoglu,A., Göçken,M., "Genetic Programming Based Data Mining Approach to Dispatching Rule Selection in a Simulated Job Shop", SIMULATION 2010,86,715-728
[4]
Yamaguchi, Li, G.; Nagai, D.; “A grey based decision making approach to the supplier selection problem”, Mathematical and Computer Modeling, 36(2007), 573- 581
[5]
Yin, M.-S.."Fifteen years of grey system theory research: A historical review and bibliometric analysis, Expert Systems with Applications, 40 2767–2775,
[6]
Deng, J.L.."The introduction of grey system". The Journal of Grey System, 1(1) (1989) 1–24
[7]
Dong, G., Yamaguchi, D., M. Nagai. "A grey-based decision – marking approach to the supplier selection problem", Mathematical and Computer Modeling, 46(2006) 573-581.
[8]
Liu, S., Guo, B., Dang Y.."Grey system Theory and Applications", (Edition 2) (1999): 45–49
[9]
Chamodrakas, I. Leftheriotis, I. and Martakos. D. "In- depth analysis and simulation study of an innovative fuzzy approach for ranking alternatives in multiple attribute decision making problems based on TOPSIS"
Applied Soft Computing, 11(1) 2011:900–907.
[10]
Kadipasaoglu, S.N., Xiang, W., Khumawala, B.M.."A comparison of sequencing rules in static and dynamic hybrid flow system". International Journal of Production Research, 35(1997): 1359–1385.
[11]
.