Comparing Multi-Objective Meta-Heuristics for Multi- Commodity Supply Chain Design Problem with Partial Coverage

Document Type : Research Paper


1 Department of Industrial Engineering, Faculty of Engineering, Alzahra University, Tehran, Iran.

2 Department of Industrial and Mechanic Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran.

3 Department of Social Science, Imam-Khomeini International University, Qazvin, Iran.


A three-echelon multi-commodity supply chain including manufacturers, distribution centers (DCs) and customers is considered. Customers may be partially or fully covered by the DCs which should be opened in some candidate locations. A two-objective model is developed to find the locations of DCs and the flows of commodities in the whole supply chain considering a pre-determined number of DCs. The first objective function minimizes the total operation costs including transportation, inventory holding, production and site opening costs while the second objective maximizes the customers’ partial coverages. Since the presented problem is NP-hard in nature, three metaheuristic algorithms of NSGA-II, NRGA and MOPSO are developed to find the Pareto-optimal solutions and are compared using some standard criteria for multi-objective algorithms. Numerical examples are designed to assess the performance of the model and the developed metaheuristic algorithms. Considering different criteria for comparing the algorithms, the superiority of some algorithms against others are reported.


                [1]        Current, J.R., Daskin, M.S., Schilling, D. (2001). Discrete network location models, In book: Facility Location; Applications and theory. Editors: Z. Drezner and H. Hamacher, Springer, Heidelberg, 80-118.
                [2]        Warszawski, A., Peer, S. (1973). Optimizing the location of facilities on a building site.  Journal of the Operational Research Society, 24, 35–44.
                [3]        Syam S.S. (2002). A model and methodologies for the location problem with logistical components. Computers and Operations Research, 29, 1173–1193.
                [4]        Church, R., ReVelle, C. (1974). The maximal covering location problem. Papers of the Regional Science Association, 32, 101–18.
                [5]        Schilling, D., Jayaraman, V., Barkhi, R. (1993). A review of covering problems in facility location. Location Science, 1, 25-55.
                [6]        Galvao, R.D., ReVelle. C. (1996). A Lagrangean heuristic for the maximal covering location problem. European Journal of Operational Research, 88, 114–23.
                [7]        Gendreau, M., Laporte, G., Semet, F. (2006). The maximal expected coverage relocation problem for emergency vehicles. Journal of Operational Research Society, 57, 22–28.
                [8]        ReVelle, C. (2008). Solving the maximal covering location problem with heuristic Concentration. Computers & Operations Research, 35, 427 – 435.
                [9]         Karasakal, O. (2004). A maximal covering location model in the presence of partial coverage. Computers & Operations Research, 31, 1515–1526.
              [10]       Mestre, J. (2008). Lagrangian relaxation and partial cover. In: Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 539–550.
              [11]      Melo, M.T., Nickel, S., Saldanha-da-Gama, F. (2009). Facility location and supply chain management a review. European Journal of Operational Research, 196, 401-412.
              [12]      Pereira, M.A., Coelho, L.C., Lorena, L.A.N., De Souza, L.C. (2015). A hybrid method for the probabilistic maximal covering location-allocation problem. Computers & Operations Research, 57, 51-59.
              [13]      Seifbarghy, M., Soleimani, M., Pishva, D. (2016). Multiple commodity supply chain with maximal covering approach in a three layer structure. International Journal of Mathematical Modeling and Numerical Optimization, 7 (2), 138-161.
              [14]      Li, X., Ramshani, M., Huang, Y. (2018). Cooperative maximal covering models for humanitarian relief chain management. Computers & Industrial Engineering, 119, 301-308.
              [15]      Eidy, A., Torabi, H. (2019). A maximal covering problem in supply chain considering variable radius and gradual coverage with the choice of transportation mode. Arabian Journal for science and Engineering, 44, 7219-7233.
              [16]      Vatsa, A.K., Jayaswal, S. (2021). Capacitated multi-period maximal covering location problem with server uncertainty. European Journal of Operational Research, 289 (3), 1107-1126.
              [17]      Karasakal, E., Silav, A. (2016). A multi-objective genetic algorithm for a bi-objective facility location problem with partial coverage. TOP, 24, 206-232.
              [18]      Cordeau, J-F., Furini, F., Ljubic, I. (2019). Benders decomposition for very large scale partial set covering and maximal covering location problems. European Journal of Operational Research, 275(3), 882-896.
              [19]      El-Hosseini, M., ZainEldin, H., Arafat, H., Badawy, M. (2021). A fire detection model based on power-aware scheduling for IoT-sensors in smart cities with partial coverage. Journal of Ambient Intelligence and Humanized Computing, 12, 2629–2648.
              [20]      Deb, K., Agrawal, S., Pratap, A., Meyarivan, T. (2000). A fast elitist non-dominated sorting genetic algorithm for multi- Objective optimization: NSGA-II. In: Proceedings of the parallel problem solving from nature VI (PPSN-VI) conference, 849-858.
              [21]      Al jadaan, O., Rao, C.R., Rajamani, L. (2008). Non-dominated ranked genetic algorithm for solving multi-objective optimization problems: NRGA. Journal of Theoretical and Applied Information Technology, 60-67.
              [22]      Coello Coello, C., Lamont, G.B., Van Veldhuizen, D.A. (2007). Evolutionary algorithms for solving multi-objective problem. 2nd end, Springer, Berlin.
              [23]      Sharaf, A.M., Elgammal, A. (2009). A multi objective multistage particle swarm optimization MOPSO search scheme for power quality and loss reduction on radial distribution system. In Proceedings of the International Conference on Renewable Energies and Power Quality (ICREPQ’09).
              [24]      Zitzler, E., Thiele, L. (1998).  Multiobjective optimization using evolutionary algorithms- a comparative case study. Fifth International Conference on Parallel Problem Solving from Nature (PPSN-V), Berlin, Germany, 292-301.