Linear Programming Approach for Solving Balanced and Unbalanced Intuitionistic Fuzzy Transportation Problems

Introduction

In several real-life situations, there is need to transport the homogeneous product from numerous origins (sources) to different destinations and the aim of the decision maker is to find how much quantity of the product from which source to which destination should be supplied so that all the supply points are fully used and demand of all the destinations is fulfilled as well as total transportation cost is minimum. The transportation problems play a vital role in logistics and supply chain management for reducing cost and improving service. In today’s highly competitive market, the pressure on companies to find better ways to create and deliver products and services to customers becomes stronger. How and when to send the products to the customers in the quantities which they want in a cost-effective manner becomes more challenging. Transportation models provide a powerful framework to meet this challenge. They ensure the efficient movement and timely availability of raw materials and finished goods.

Resource allocation is used to assign the available resources in an economic way. When the resources to be allocated are scarce, a well-planned action is necessary for a decision-maker (DM) to attain the optimal utility. If the supplying sources and the receiving agents are limited, the best pattern of the allocation to get the maximum return or the best plan with the least cost, whichever may be applicable to the problem, is to be found out. This class of problems is termed as ‘allocation problems’ and is divided into ‘transportation problems’ and ‘assignment problems’. This type of allocation problems is studied in operations research.

During World War-II, Britain was having very limited military resources; therefore, there was an urgent need to allocate resources to the various military operations and to the activities within each operation in an effective manner. Therefore, the British military executives called upon a team of scientist to apply scientific approach to study the strategic and tactical problems related to air and land defence of the country. As the team was dealing with research a military operations, the work of this team of scientist was named as operations research.

The Transportation Problem (TP) is one of the subclasses of Linear Programming Problem (LPP). The objective of the transportation problem is to transport various quantities of a single homogeneous product that are initially stored at various origins, to different destinations in such a way that the total transportation cost is minimum for a minimization problem or total transportation profit is maximum for a maximization problem. The conventional transportation problem consists in transporting a certain commodity from each of m origins to any of n destinations . The origins are factories with respect capacities and the destinations are warehouses with required levels of demands . For the transport of a unit of the given commodity from the i^th origin to the j^th destination a cost is given for which, without loss of generality, we can assume . Hence, one must determine the amounts to be transported from all the origins to all the destinations in such a way that the total cost is minimized.

The conventional transportation problem can be mathematically stated as follows:Minimize subject to: (Row restriction) (Column restriction)