## Abstract

This paper integrates the concept of pricing into the location–routing problem. The problem consists of a firm trying to optimize its multi-zone network in order to maximize its profit. It divides its big market into some zones and tries to determine the optimal zonal price of the products. It then dispatches its products from its position to each zone. After receiving the products to depots, a vehicle should distribute the products between customers. This matter is compatible with real-world distribution systems such as fruits, textile products, and leather. In other words, the main objectives of the firm are to have the optimal price, vehicle routes, and location of the depot in each zone. Therefore, a flow-based model as a mixed-integer nonlinear programming is proposed to solve the problem. In the light of this nonlinearity, we employ a piecewise linearization approximation method. In addition, to adapt with the large-scale problems, two heuristic algorithms with three combinations of operators in the local search are suggested. In order to evaluate their performance, some test instances and a case study are solved. Based on the results, the performance of the algorithms is compared with each other and the flow-based formulation. The results show the outperformance of the algorithms rather than the flow-based model. Moreover, the results show the linearization approach can efficiently approximate the nonlinear model. Furthermore, the impact of the shipping cost ratio on the depot selection, vehicles’ route, and firms’ profit is revealed.

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The authors are grateful for the valuable comments and suggestions from the respected reviewers. Their valuable comments and suggestions have enhanced the strength and significance of our paper.

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Sadeghi Dastaki, M., Setak, M. & Karimi, H. The multi-zone location–routing problem with pricing: a flow-based formulation and two heuristic approaches.
*Soft Comput* **25**, 741–769 (2021). https://doi.org/10.1007/s00500-020-05186-y

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DOI: https://doi.org/10.1007/s00500-020-05186-y