A combined benders decomposition and Lagrangian relaxation algorithm for optimizing a multi-product, multi-level omni-channel distribution system

Document Type : Article


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

2 Department of Business and Management Science, Norwegian School of Economics, Bergen, Norway

3 Department of Industrial and Systems Engineering, Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran


The development of supply chain distribution systems from single- to multi-channel networks for delivering items to end customers has effected many changes in the retail sector. Following the adoption of multi-channel distribution strategies and rapid development of relevant technologies, the Omni-channel approach can yield significant benefits and facilitate trade with customers. This paper aims to optimize a multi-product, multi-level Omni-channel distribution network and shipping flows of products within the network under uncertain conditions. A multi-objective mathematical model is developed that minimizes the costs of supply chain while maximizing customer satisfaction over different scenarios. In order to solve the proposed model, a combined algorithm is developed based on Benders Decomposition (BD) and Lagrangian Relaxation (LR). The presented model and solution approach is implemented in a case study of a distribution system, a large e-commerce startup and online store. Five different scenarios with various service levels are investigated and the numerical results are discussed compared to previous findings. The efficiency of the proposed combined BD-LR solution algorithm is also demonstrated. The results obtained from the case study show that higher service levels are correlated with higher levels of customer satisfaction and lower cost of the system.


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