An Approach for a Multi-Objective Capacitated Transportation Problem

Introduction

The classical transportation problem (TP) is a special type of linear programming (LP) problem. In this problem, a homogeneous product is to be transported from several sources to several destinations in such a way that the total transportation cost is minimum, or profit is maximum. In this problem, while sources can be defined as suppliers, production centers, factories, etc., demands may be customers, warehouses, sinks, etc. Moreover, this problem may have equality or inequality type of constraints. In real-life problems, the constraints of TPs are not generally in equal form. In some cases, the decision-maker may have specified a supply amount that must be provided from a particular source. Then, the corresponding supply constraint will be “greater than or equal to” form. Similarly, when the amount of resources owned by any supplier is limited (i.e. has an upper limit), then the corresponding supply constraint will be “less than or equal to” form. These different types of inequalities may also appear in any demand constraint. Therefore, TP with mixed constraints arises.

Moreover, when the total demand equals the total supply, the TP is referred to be a balanced transportation problem. If a TP has a total supply that is strictly less than the total demand, then the problem has no feasible solution. In this situation, it is sometimes desirable to allow the possibility of leaving some demand unmet. In such a case, we can balance a transportation problem by creating a dummy supply point that has a supply amount equal to the unmet demand and associating it with a penalty.

Furthermore, a STP with two or more fractional objective functions is called as a multi-objective fractional STP. In this problem, maximum profitability - profit/cost or profit/time – as a criterion function is maximized subject to supply, demand, and conveyance constraints.

The transportation problem, in its simplest form, deals with the physical transfer of some goods from sources to destinations but also has different applications in many different fields. Many subproblems that can be defined especially in the field of logistics are based on the logic of transportation problems. From a more general point of view, there are many transportation problems in the field of the supply chain, which includes the movement of the product or service from the supplier to the customer and is defined as the whole systems of organizations, people, technology, and activities. Some of these are the transportation of raw materials to be sent from suppliers to the factories, the transportation of goods to warehouses or distribution centers, the delivery of the products to customers, and transferring used products to recycling centers.

In today's world of globalization, we are faced with the transportation of more products and the diversity in the ways of transportation of the product. To meet this need, we had to define the solid transportation problem (STP) which is one of the important research topics from both theoretical and practical aspects. STP is a special type of the traditional transportation problem in which three-dimensional properties (supply, demand, convenience) are taken into account in the objective and constraint set instead of source and destination. The necessity of considering this special type of transportation problem arises when heterogeneous conveyances are available for the shipment of products. The STP is also applied in public distribution systems. In many industrial problems, a homogeneous product is delivered from a source to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships, etc. These conveyances are taken as the third dimension. An STP may be transformed into a traditional TP by considering only a single type of conveyance.