An integrated queueing-inventory-routing problem in a green dual-channel supply chain considering pricing and delivery period: a case study of construction material supplier

The current paper presented a stochastic integrated queueing-inventory-routing problem into a green dual-channel supply chain considering an online retailer with a vehicle-routing problem (VRP) and a traditional retailing channel with an M/M/C queueing system. A mixed-integer non-linear programming model (MINLP) is presented to address the integrated VRP and M/M/C queueing system. The suggested model makes decisions about optimal routing, delivery time interval to customers, number of servers in traditional retailers, inventory replenishment policies, and retailers' price. For the first time, this model considers two retailing channels simultaneously under different uncertainty, including demand, delivery lead time, service time, and delivery time interval to customers. The inventory model also follows a continuous-time Markov chain. The small-scale test problems are solved using GAMS software. Since the problem is NP-hard, this study conducts a comprehensive comparative analysis of the performance of 13 different metaheuristics. The ant lion optimiser, dragonfly algorithm, grasshopper optimisation algorithm, Harris-hawks optimisation, moth-flame optimisation algorithm, multi-verse optimizer, sine cosine algorithm, salp-swarm algorithm, the whale optimisation algorithm, grey-wolf optimiser, genetic algorithm, differential evolution, and particle swarm optimization are algorithms that were chosen for this study. Comprehensive statistical tests were conducted to evaluate the performance of these methods. Furthermore, the model is executed for construction material producers as a case study. Finally, sensitivity analyses were conducted on crucial model parameters; and managerial insights were recommended.