A new mathematical model for the closed-loop supply chains considering pricing for product, a fleet of heterogeneous vehicles, and inventory costs

Document Type : Original Manuscript


1 Professor of Industrial Engineering at University of Kurdistan, Sanandaj, Iran

2 Department of Industrial Engineering, Payame Noor University , Tehran ,Iran.

3 Assistant Professor at Department of Industrial Engineering, Payame Noor University, Tehran, Iran

4 Associate Professor at Department of Industrial Engineering, kordstan University, Sanandaj, Iran


Mathematical models have been used in many areas of supply chain management. In this paper, we present a mixed-integer non-linear programing (MINLP) model to solve a multi-period, closed-loop supply chains (CLSCs) with two echelons consist of producers and customers. In order to satisfy the demands, the producers are be able to order for materials in the beginning of each period for one or more periods. A fleet of heterogeneous vehicles are routed to deliver the products from producers to customers and to pick up defective products from the customers and move them to the collection-repair center. Also, it is assumed that the rate of defective products is related to the price. In the other words, the more expensive product, the less rate of defect. The objective function maximize the profit which comes from total cost which is subtracted from income. The income is divided two part, selling products and wastes, and total cost consists of several part such as, costs of defective products, ordering cost, cost of holding in producers and collection-repair center, transportation costs, and the cost of assigning place for collection-repair center. Finally, computational results are discussed and analyzed for a numerical example in order to demonstrate the effectiveness of the proposed model.


  • Presenting a mixed-integer non-linear programming (MINLP) model to solve a multi-period, closed-loop supply chain (CLSC).
  • The objective function maximizes the profit, which is equal to total cost minus income.
  • Presenting and discussing two numerical examples with their computational results.
  • Presenting a solution approach which is analyzed by applying the examples to show the efficiency of the proposed method. 


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