Optimizing bi-objective solid transportation problem using hierarchical order goal programming technique: a case study problem

The solid transportation problem (STP) is a three-dimensional optimization problem concerned with minimizing total costs while meeting supply and demand limitations by utilizing a variety of transportation modes, referred to as conveyances. This study examines the application of the STP to a logistic problem related to the milk distribution problem. Here, the goal is to efficiently deliver milk from a production facility to multiple retail outlets, considering both the cost of transportation and time of delivery as the two primary objectives of the solid transportation problem. Hence, the bi-objective solid transportation problem (BOSTP) is solved, supposing its constraints are fuzzy due to various unpredictable factors during the transportation period. These uncertain values are initially de-fuzzified into the equivalent crisp values using the α-cut approach. Later, the STP is solved with single objectives at once using Vogel’s approximation method. This mathematical model is then formulated to optimize the total transportation cost and time and solved using the hierarchical order goal programming technique to determine the most cost-effective and time-efficient route. The results show that the hierarchical order goal programming technique minimizes the total transportation cost and time while improving the effectiveness of milk distribution.