Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under an Uncertainty Condition

Document Type : Research Paper


1 Professor, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

2 M.Sc. Student, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

3 M.Sc. Grad., Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran


This paper presents a novel bi-objective multi-product capacitated vehicle routing problem with uncertainty in demand of retailers and volume of products (UCVRP) and heterogeneous vehicle fleets. The first of two conflict fuzzy objective functions is to minimize the cost of the used vehicles, fuel consumption for full loaded vehicles and shortage of products. The second objective is to minimize the shortage of products for all retailers. In order to get closer to a real-world situation, the uncertainty in the demand of retailers is applied using fuzzy numbers. Additionally, the volume of products is applied using robust parameters, because the possible value of this parameter is not distinct and belongs to a bounded uncertainty set. The fuzzy-robust counterpart model may be larger than the deterministic form or the uncertain model with one approach and it has with further complexity; however, it provides a better efficient solution for this problem. The proposed fuzzy approach is used to solve the bi-objective mixed-integer linear problem to find the most preferred solution. Moreover, it is impossible to improve one of the objective functions without considering deterioration in the other objective functions. In order to show the conflict between two objective functions in an excellent fashion, a Pareto-optimal solution with the ε-constraint method is obtain Some numerical test problems are used to demonstrate the efficiency and validity of the presented model.


-Agra, A., Christiansen, M., Figueiredo, R., Hvattum, L. M., Poss, M. and Requejo C. (2013) “The robust vehicle routing problem with time windows”, Computers & Operations Research, Vol. 40, pp. 856-866.
-Bertsimas, D. and Sim, M. (2004) “The price of robustness”, Operations Research, Vol. 52, pp. 35-53
-Cao, E., and Lai, M. (2010) “The open vehicle routing problem with fuzzy demands”, Expert Systems with Applications, Vol. 37, pp. 2405- 2411.
-Chen, H.K. and Chou, H.W. (1996) “Solving multi objective linear programming problems - A generic approach”, Fuzzy Sets and Systems, Vol. 82, pp. 3538.
-Cheng, R., Gen, M., and Tozawa, T. (1995) “Vehicle routing problem with fuzzy due-time using genetic algorithms”, Japan Society for Fuzzy Theory and Systems, Vol. 7, pp. 1050- 1061
- Coelho, L. C., Cordeau, J., and Laporte, G. (2012) “The inventory-routing problem with transshipment,” Computers & Operations Research, Vol. 39, pp. 2537-2548
- Golden, B., Assad, A., and Dahl, R. (1984) “Analysis of a large scale vehicle routing problem with an inventory component”, Large scale systems, Vol. 7, pp. 181-190.
-Ghannadpour, S. F., Noori, S. and TavakkoliMoghaddam, R. (2013) “Multiobjective dynamic vehicle routing problem with fuzzy travel times and customers’ satisfaction in supply chain management”, IEEE Transactions on Engineering Management, Vol. 60, pp. 777-790.
-Ghannadpour, S. F., Noori, S., TavakkoliMoghaddam, R., and Ghoseiri, K. (2014) “A multi-objective dynamic vehicle routing problem with fuzzy time windows: Model, solution and application”, Applied Soft Computing, Vol. 14, pp. 504-527
- Gounaris, C. E., Wiesemann, W., and Floudas C. A. (2013) “The robust capacitated vehicle routing problem under demand uncertainty”, Operations Research, Vol. 61, pp. 677-693.
-Gupta, R., Singh, B., and Pandey, D. (2010) “Fuzzy vehicle routing problem with uncertainty in service time”, International Journal of Contemporary Mathematical Sciences, Vol. 5, pp. 497-507.
-He, Y. and Xu, J. (2005) “A class of random fuzzy programming model and its application to vehicle routing problem”, World Journal of Modelling and simulation, Vol. 1, pp. 3-11.
-Kara, I., Kara, B. Y., and Yetis, M. K. (2007) “Energy minimizing vehicle routing problem”, In Combinatorial optimization and applications, pp. 62-71. Springer Berlin Heidelberg.
-Kuo, R. J., Chiu, C. Y. and Lin, Y. J. (2004) “Integration of fuzzy theory and ant algorithm for vehicle routing problem with time window”, Processing of the IEEE Annual Meeting of the Fuzzy Information (NAFIPS'04), Vol. 2, pp. 925- 930.
-Li, H., Xu Z. and Zhou F. (2012) “A study on vehicle routing problem with fuzzy demands based on improved tabu search”, Fourth International Conference on Computational and Information Sciences (ICCIS), pp. 73-76.
-Li, K., Chen B., Sivakumar, A. I. and Wu, Y. (2014) “An inventory–routing problem with the objective of travel time minimization”, European Journal of Operational Research, Vol. 236, pp. 936-945.
- Moin, N.H. and Salhi, S. (2007) “Inventory routing problems: a logistical overview”, Journal of Operation Research Society, Vol. 58, pp. 11851194.
-Montemanni, R., Barta, J., Mastrolilli, M. and Gambardella L. M. (2007) “The robust traveling salesman problem with interval data”, Transportation Science, Vol. 41, pp. 366-381.
-Raa, B. and Aghezzaf, E. H. (2009) “A practical solution approach for the cyclic inventory routing problem”, European Journal of Operational Research, Vol. 192, pp. 429-441.
-Sahin, B., Yilmaz, H., Ust, Y., Guneri, A. F. and Gulsun B. (2009) “An approach for analysing transportation costs and a case study”, European Journal of Oprational Research, Vol. 93. pp.1-11.
-Solano-Charris, E. L., Prins, C. and Santos, A. C. (2014) “A robust optimization approach for the vehicle routing problem with uncertain travel cost”, International Conference on Control, Decision and Information Technologies, pp. 098- 103.
-Sun, L. and Wang, B. (2015) “Robust optimization approach for vehicle routing problems with uncertainty”, Mathematical Problems in Engineering, Article in Press.
-Sungur, I., Ordónez F. and Dessouky, M. (2008) “A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty”, IIE Transactions, Vol. 40, pp. 509- 523.
-Sungur, I., Ren, Y., Ordóñez, F., Dessouky, M. and Zhong, H. (2009) “A model and algorithm for the courier delivery problem with uncertainty”, Transportation Science, Vol. 44, pp. 193-205.
-Tang, J., Pan, Z., Fung, R. Y. K., and Lau, H. (2009) “Vehicle routing problem with fuzzy time windows”, Fuzzy Sets and Systems, Vol. 160, pp. 683-695.
-Wang, H. F., and Wen, Y. P. (2002) “Timeconstrained Chinese postman problems”, Computers & Mathematics with Applications, Vol. 44, pp. 375-387. 
-Xu, J., Yan, F. and Li, S. (2011) “Vehicle routing optimization with soft time windows in a fuzzy random environment”, Transportation Research Part E: Logistics and Transportation Review, Vol. 47, pp. 1075-1091.
-Yu, Y., Chen, H. and Chu, F. (2008) “A new model and hybrid approach for large scale inventory routing problems”, European Journal of Operational Research, Vol. 189, pp. 1022-1040.
-Zheng, Y. and Liu, B. (2006) “Fuzzy vehicle routing model with credibility measure and its hybrid intelligent algorithm”, Applied Mathematics and Computation, Vol. 176, pp. 673-683.
-Zimmermann, H. J. (1978) “Fuzzy programming and linear programming with several objective functions”, Fuzzy Sets and Systems, Vol. 1, pp. 4555.