A new multi-objective mathematical model for a Citrus supply chain network design: Metaheuristic algorithms

Document Type: Original Manuscript


1 Department of Industrial Engineering,Yazd University

2 Yazd University



Nowadays, the citrus supply chain has been motivated by both industrial practitioners and researchers due to several real-world applications. This study considers a four-echelon citrus supply chain, consisting of gardeners, distribution centers, citrus storage, and fruit market. A Mixed Integer Non-Linear Programming (MINLP) model is formulated, which seeks to minimize the total cost and maximize the profit of the Citrus supply chain network. Due to the complexity of the model when considering large-scale samples, two well-known meta-heuristic algorithms such as Ant Colony Optimization (ACO) and Simulated Annealing (SA) algorithms have been utilized. Additionally, a new multi-objective ACO algorithm based on a set of non-dominated solutions form the Pareto frontier developed to solve the mathematical model. An extensive comparison based on different measurements analyzed to find a performance solution for the developed problem in the three sizes (small, medium, and large-scale). Finally, the various outcomes of numerical experiments indicate that the MOACO algorithm is more reliable than other algorithms.

Graphical Abstract

A new multi-objective mathematical model for a Citrus supply chain network design: Metaheuristic algorithms


  • This study considers a four-echelon citrus supply chain, consisting gardeners, distribution centers, citrus storage and fruit market.
  • This study develops a Mixed Integer Non-Linear Programming (MINLP) model which consists of minimizing the total cost and maximizing the profit of citrus supply chain.
  • Due to complexity of proposed model when considering large-scale samples, two well-known meta-heuristic algorithms including Ant Colony Optimization (ACO) and Simulated Annealing (SA) algorithms have been utilized.
  • An extensive comparison based on different measurements has been analyzed to find a performance solution for the developed problem.
  • Finally, a set of discussion and managerial implications has been outlined.


Abarqhouei, N.S., Hosseini Nasab, H., Fakhrzad, MB., (2012), Design of the evaluation model for total ergonomics interventions with fuzzy approach, Sci. J. Pure Appl. Sci., 1, 119-129.

Bortolini, M., Galizia, F. G., Mora, C., Botti, L., & Rosano, M. (2018). Bi-objective design of fresh food supply chain networks with reusable and disposable packaging containers. Journal of Cleaner Production.

Banaeian, N., Mobli, H., Fahimnia, B., Nielsen, I. E., & Omid, M. (2018). Green supplier selection using fuzzy group decision making methods: A case study from the agri-food industry. Computers & Operations Research, 89, 337-347.

Billal, M. M., & Hossain, M. (2020). Multi-Objective Optimization for Multi-Product Multi-Period Four Echelon Supply Chain Problems Under Uncertainty. Journal of Optimization in Industrial Engineering, 13(1), 1-17.

Dorigo, M. (1992). Ant Colony Optimization for vehicle routing problem (Doctoral dissertation, PhD thesis, Politecnico di Milano, Milan, Italy).

Etemadnia, H., Goetz, S. J., Canning, P., & Tavallali, M. S. (2015). Optimal wholesale facilities location within the fruit and vegetables supply chain with bimodal transportation options: An LP-MIP heuristic approach. European Journal of Operational Research244(2), 648-661.

Eslamipoor R., Fakhrzad M.B., Zare Mehrjerdi Y. (2015). A new robust optimization model under uncertainty for new and remanufactured products. International Journal of Management Science and Engineering Management 10 (2)137-143.

Fakhrzad, M. B., Talebzadeh, P., & Goodarzian, F. (2018). Mathematical Formulation and Solving of Green Closed-loop Supply Chain Planning Problem with Production, Distribution and Transportation Reliability. International Journal of Engineering, 31(12), 2059-2067.

Fard, A. M. F., & Hajaghaei-Keshteli, M. (2018). A tri-level location-allocation model for forward/reverse supply chain. Applied Soft Computing, 62, 328-346.

Fakhrzad, M. B., & Goodarzian, F. (2019). A Fuzzy Multi-Objective Programming Approach to Develop a Green Closed-Loop Supply Chain Network Design Problem under Uncertainty: Modifications of Imperialist Competitive Algorithm. RAIRO-Operations Research, 53(3), 963-990.

Fakhrzad, M. B., Goodarzian, F., & Golmohammadi, A. M. (2019). Addressing a fixed charge transportation problem with multi-route and different capacities by novel hybrid meta-heuristics. Journal of Industrial and Systems Engineering, 12(1), 167-184.

Fathollahi-Fard, A. M., Ahmadi, A., Goodarzian, F., & Cheikhrouhou, N. (2020). A bi-objective home healthcare routing and scheduling problem considering patients’ satisfaction in a fuzzy environment. Applied Soft Computing, Vol. 93, 106385.

Gupta, A., & Saini, S. (2018). On Solutions to Capacitated Vehicle Routing Problem Using an Enhanced Ant Colony Optimization Technique. In Networking Communication and Data Knowledge Engineering (pp. 257-266). Springer, Singapore.

Goodarzian, F., & Hosseini-Nasab, H. (2019). Applying a fuzzy multi-objective model for a production–distribution network design problem by using a novel self-adoptive evolutionary algorithm. International Journal of Systems Science: Operations & Logistics, 1-22.

Goodarzian, F., Hosseini-Nasab, H., Muñuzuri, J., & Fakhrzad, M. B. (2020). A multi-objective pharmaceutical supply chain network based on a robust fuzzy model: A comparison of meta-heuristics. Applied Soft Computing, Vol. 92, 106331.

Hyland, M. F., Mahmassani, H. S., & Mjahed, L. B. (2016). Analytical models of rail transportation service in the grain supply chain: Deconstructing the operational and economic advantages of shuttle train service. Transportation Research Part E: Logistics and Transportation Review93, 294-315.

Khalifehzadeh, S., Fakhrzad, MB., (2019). A Modified Firefly Algorithm for optimizing a multi stage supply chain network with stochastic demand and fuzzy production capacity, Computers & Industrial Engineering, 133, 42-56

Kirkpatrick, S. (1984). Optimization by simulated annealing: Quantitative studies. Journal of statistical physics34(5-6), 975-986.

Kuo, R. J., & Nugroho, D. Y. (2017, April). A fuzzy multi-objective vehicle routing problem for perishable products using gradient evolution algorithm. In Industrial Engineering and Applications (ICIEA), 2017 4th International Conference on (pp. 219-223). IEEE

Lamsal, K., Jones, P. C., & Thomas, B. W. (2016). Harvest logistics in agricultural systems with multiple, independent producers and no on-farm storage. Computers & Industrial Engineering91, 129-138.

Liu, R., Li, J., Song, X., Yu, X., & Jiao, L. (2018). Simulated annealing-based immune dominance algorithm for multi-objective optimization problems. Knowledge and Information Systems55(1), 215-251.

Mousavi, S. M., Alikar, N., Niaki, S. T. A., & Bahreininejad, A. (2015). Optimizing a location allocation-inventory problem in a two-echelon supply chain network: A modified fruit fly optimization algorithm. Computers & Industrial Engineering87, 543-560.

Maiyar, L. M., Thakkar, J. J., Awasthi, A., & Tiwari, M. K. (2015). Development of an effective cost minimization model for food grain shipments. IFAC-PapersOnLine, 48(3), 881-886.

Masson, R., Lahrichi, N., & Rousseau, L. M. (2016). A two-stage solution method for the annual dairy transportation problem. European Journal of Operational Research, 251(1), 36-43.

Mogale, D. G., Dolgui, A., Kandhway, R., Kumar, S. K., & Tiwari, M. K. (2017). A multi-period inventory transportation model for tactical planning of food grain supply chain. Computers & Industrial Engineering, 110, 379-394.

Musavi, M., & Bozorgi-Amiri, A. (2017). A multi-objective sustainable hub location-scheduling problem for perishable food supply chain. Computers & Industrial Engineering113, 766-778.

Núñez, A., Cortés, C. E., Sáez, D., De Schutter, B., & Gendreau, M. (2014). Multi-objective model predictive control for dynamic pickup and delivery problems. Control Engineering Practice, 32, 73-86.

Nadal-Roig, E., & Plà-Aragonés, L. M. (2015). Optimal Transport Planning for the Supply to a Fruit Logistic Centre. In Handbook of Operations Research in Agriculture and the Agri-Food Industry (pp. 163-177). Springer, New York, NY.

Rong, A., Akkerman, R., & Grunow, M. (2011). An optimization approach for managing fresh food quality throughout the supply chain. International Journal of Production Economics131(1), 421-429.

Reardon, T., & Zilberman, D. (2018). Climate smart food supply chains in developing countries in an era of rapid dual change in agrifood systems and the climate. In Climate Smart Agriculture (pp. 335-351). Springer, Cham.

Soto-Silva, W. E., Nadal-Roig, E., González-Araya, M. C., & Pla Aragones, L. M. (2016). Operational research models applied to the fresh fruit supply chain. European Journal of Operational Research251(2), 345-355.

Sembiring, P., Mawengkang, H., Sadyadharma, H., & Bu’ulolo, F. (2018, January). Mixed Integer Linear Programming model for Crude Palm Oil Supply Chain Planning. In IOP Conference Series: Materials Science and Engineering (Vol. 300, No. 1, p. 012015). IOP Publishing.

Sellitto, M. A., Vial, L. A. M., & Viegas, C. V. (2018). Critical success factors in Short Food Supply Chains: Case studies with milk and dairy producers from Italy and Brazil. Journal of Cleaner Production, 170, 1361-1368.

Song, M., & Chen, D. (2018). A comparison of three heuristic optimization algorithms for solving the multi-objective land allocation (MOLA) problem. Annals of GIS, 1-13.

Sahebjamnia, N., Goodarzian, F., & Hajiaghaei-Keshteli, M. (2020). Optimization of Multi-Period Three-echelon Citrus Supply Chain Problem. Journal of Optimization in Industrial Engineering, 13(1), 39-53.

Waitz, M., Mild, A., & Fikar, C. (2018, January). A Decision Support System for Efficient Last-Mile Distribution of Fresh Fruits and Vegetables as Part of E-Grocery Operations. In Proceedings of the 51st Hawaii International Conference on System Sciences.

Wei, L., Zhang, Z., Zhang, D., & Leung, S. C. (2018). A simulated annealing algorithm for the capacitated vehicle routing problem with two-dimensional loading constraints. European Journal of Operational Research265(3), 843-859.

Xu, Q. L., Cao, Y. W., & Yang, K. (2018, March). Analysis of parameter estimation and optimization application of ant colony algorithm in vehicle routing problem. In MIPPR 2017: Parallel Processing of Images and Optimization Techniques; and Medical Imaging (Vol. 10610, p. 1061009). International Society for Optics and Photonics.

Zhang, H., Xiong, Y., He, M., & Qu, C. (2017). Location Model for Distribution Centers for Fulfilling Electronic Orders of Fresh Foods under Uncertain Demand. Scientific Programming2017.