A new bi-objective model for a closed-loop supply chain problem with inventory and transportation times

This paper considers a closed-loop supply chain design problem including several producers, distributors, customers, collecting centers, recycle centers, revival centers, raw materials customers considering several periods, existing inventory and shortage in distribution centers, and transportation cost and time. this problem is formulated as a bi-objective integer nonlinear programming model. the aim of this model is to determine numbers and locations of supply chain elements, their capacity levels, allocation structure, mode of transportation between them, amount of transported products between them, amount of existing inventory, and shortage in distribution centers in each period to minimize the sum of system costs and transportation time in the network. to validate this model and show the applicability of it for small-sized problems, gams software is used. because this given problem is np-hard, a bee colony optimization (bco) algorithm is proposed to solve medium and large-sized problems. furthermore, to examine the eciency of the proposed bco algorithm, the associated results are compared with the results obtained by the genetic algorithm (ga). finally, the conclusion is provided.