New mathematical model for the bi-objective inventory routing problem with a step cost function: A multi-objective particle swarm optimization solution approach

Abstract

• Transportation costs are assumed to behave as a step cost function. • A novel bi-objective mathematical model is proposed to consider this assumption. • Despite this assumption, this type of function is not considered in the IRP. • An efficient multi-objective particle swarm optimization algorithm is presented. • The algorithm's efficiency compared with the augmented ε-constraint method. Inventory management and satisfactory distribution are among the most important issues considered by distribution companies. One of the key objectives is the simultaneous optimization of the inventory costs and distribution expenses, which can be addressed according to the inventory routing problem (IRP). In this study, we present a new transport cost calculation pattern for the IRP based on some real cases. In this pattern, the transportation cost is calculated as a function of the load carried and the distance traveled by the vehicle based on a step cost function. Furthermore, previous methods usually aggregate the inventory and transportation costs to formulate them as a single objective function, but in non-cooperative real-life cases, the inventory-holding costs are paid by retailers whereas the transportation-related costs are paid by the distributor. In this study, we separate these two cost elements and introduce a bi-objective IRP formulation where the first objective is to minimize the inventory-holding cost and the second is minimizing the transportation cost. We also propose an efficient particle representation and employ a multi-objective particle swarm optimization algorithm to generate the non-dominated solutions for the inventory allocation and vehicle routing decisions. Finally, in order to evaluate the performance of the proposed algorithm, the results obtained were compared with those produced using the augmented ε-constraint method, thereby demonstrating the practical utility of the proposed multi-objective model and the proposed solution algorithm.