Two-echelon multi-product multi-constraint product returns inventory model with permissible delay in payments and variable lead time

• We investigate multi-echelon multi-product multi-constraint inventory problem with shortage backordering, permissible delay in payments and variable lead time.
• The distributor and the warehouse belong to different corporate entities and are interested in minimizing their joint total cost.
• In this study, the defective products are returned back to the warehouse by the distributor and the warehouse recovered those defective products into perfect products having the same value as the procured products.
• We show that the model of this problem is of an integer nonlinear programming type and in order to solve it, a simple Lagrangian multiplier algorithmic approach is employed.
• We determine the order quantity, lead time and the total number of deliveries such that the integrated expected total cost of the supply chain is minimum.

Supply chain is not limited to delivering products to the end-costumers since the defective products that are returned back to the producers by the consumers. The producers should be superior knowledge to utilize the return products effectively so as to maintain our natural resources and to provide better service to customers. In this paper, a distributor and a warehouse consisting of a serviceable part and a recoverable part supply chain problem is considered in which there are several products, the distributor has limited space capacity and budget to purchase all products. In this supply chain, the defective products are returned back to the warehouse by the distributor and the warehouse recovered those defective products into perfect products having the same value as the procured products. The lead-time of receiving products from a warehouse to a distributor is a variable which is controllable by adding extra crashing cost. For each product, a fraction of the shortage is backordered and the rest are lost. A mathematical model is employed in this study for optimizing the order quantity, lead time and total number of deliveries with the objective of minimizing system total cost. We show that the model of this problem is a constrained non-linear programme and present a simple Lagrangian multiplier technique to solve it. Numerical and sensitivity analysis are given to show the applicability of the proposed model in real-world product returns inventory problems.