Mathematical Model for Multi Depot Simultaneously Pick Up and Delivery Vehicle Routing Problem with Stochastic Pick Up Demand

In a classical vehicle routing problem (VRP), customer demands are known with certainty. On the other hand, in real-life problems, customer demands may change over time. Therefore, in the classical VRP, the assumption that customer demands are stochastic should be taken into account. To expedite consumer demands and minimize fuel use and carbon emissions, organizations must concurrently address client distribution and collection requirements. Customers' distribution requirements can be predicted, but it is impossible to predict in advance the product requirements they will send for recycling. Hence, in this study, a mathematical programming model is developed for the multi-depot simultaneous pick-up and delivery vehicle routing problem under the assumption that customers' picking demands are stochastic. However, there are non-linear constraints in the developed model. Thereby, firstly, the stochastic model is linearized, and then the effectiveness of the model is analyzed. The efficacy of the linearized model is ascertained by generating test problems. The study investigated the impact of varying reliability levels and the number of depots on the model. As a result of the sensitivity analysis, it was determined that by decreasing the reliability level, the solution time of the problems decreased and the number of problems reaching the best solution increased. In the study, 135 test problems were solved by changing the reliability level, and the best result was achieved in 105 of these problems within 7200 s. The increase in the number of depots both reduced the solution time of the problems and was effective in reaching the best solution for all solved test problems.