Credit linked two-stage multi-objective transportation problem in rough and bi-rough environments

With a rapid growth of research in multi-objective transportation problem, rough and bi-rough sets are two new mathematical ideas for formulating real-world-based problems involving uncertain data. In this study, we have investigated a two-stage multi-objective transportation problem by considering credit period policy under rough and bi-rough environments. In this regard, three conflicting objective functions have been optimized simultaneously under the same restrictions. In first objective function, we have presented the minimization of transportation cost of a production house. In second objective function, total transportation cost of retailers has been minimized. But, in last one, we have maximized total profit of distributors. Besides, due to existence of different types of uncertainties in our real-life problems, in the proposed model, independent parameters (including, actual transportation cost, requirement of the retailers, and cost per unit distance) have been considered as rough in nature and dependent parameters such as demanded transportation cost and demand of the distributors have been considered as bi-rough in nature. Moreover, to convert the uncertain model into an equivalent deterministic form, a rough and bi-rough programming approach has been derived along with the expected value approach. Finally, by using these ideas, the mathematical model of our considered transportation problem has been illustrated. After that, the proposed model has been solved by applying NSGA-II algorithm (elitist non-dominated sorting genetic algorithm) with some simulated numerical data. Some sensitivity analysis associated with our proposed model has also been discussed to show the effectiveness of the model.