Fuzzy Multi-objective Mathematical Modelling for Distribution Planning

This paper discusses a distribution planning model with n distribution centers (DCs) and m customers. Ideally, each DC serves customers in its zone. Each customer order may include multiple items. However, as the need arises, customers can also be served from other DCs to avoid delays to the customers. Besides, items in a customer’s order may also come from multiple DCs not to violate safety stock values established for each item in each warehouse. Furthermore, there are alternative transportation modes available with different costs and CO2 emission. Environmental issues have become increasingly popular and various studies have been proposed in the literature in recent years. Many corporates are increasing their sustainability efforts and becoming more responsible members of the society. We assume stochastic transportation leadtimes. They are assumed to be normally distributed with known mean and standard deviation values. There are multiple objectives considered in this study and they are: (1) minimize total shipping cost, (2) minimize probability of tardiness, (3) minimize the environmental impact of transportation, (4) minimize safety stock violations. A multi-objective fuzzy mixed-integer-non-linear programming model is developed to handle all these objectives such that satisfaction level is maximized. The task is to determine which DC(s) will serve each customer. Two experiments are conducted, (i) the same network is experimented with multiple periods where inventory levels of items are revised at the end of each period based on the shipment made, (ii) multiple experiments are conducted with various parameters including number of customers, number of items in orders, transportation leadtime distributions, starting inventory levels, safety stock values and fuzzy operators. One of the important features of this study is that it combines both stochastic and fuzzy parameters in the same mathematical model, which has been rarely found in the literature.