Algorithms and Software Packages for Solving Transportation Problems With Intuitionistic Fuzzy Numbers

Introduction Of Traditional Transportation Problem

In many of the real life situations, there is a need to transport the homogeneous product from various sources (e.g., factories) to different destinations (e.g., retail stores or storage centres) and the main aim of the decision-maker (DM) is to find how much quantity of the product from which source to which destination should be supplied so that all the supply points (e.g., a_i, i=1,2,…,m) are fully used and demands (b_j, j=1,2,…,n) of all the destinations are fulfilled as well as total transportation cost (TTC) is minimum or total transportation profit (TTP) is maximum.

TPs play a vital role in industry and network communication for reducing cost and improving services. In today’s very high competitive market, the pressure on organizations to find better ways to create and deliver items and services to customers becomes stronger. How and when to send the items to the customers in the quantities which they want in a cost-effective manner becomes more challenging. The transportation models provide a powerful framework to meet these challenges. They ensure the efficient movement and timely availability of finished goods and/or raw materials.

Resource allocation is a toughest job, 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 decision-makers (DMs) to attain the optimum utility. If the supplying sources and the receiving agents are finite, the best pattern of the allocation to get the minimum cost/the best plan with the maximum return, whichever may be applicable to the problem, is to be found out. Those types of problems are called 'Allocation Problems' and are divided into 2 categories, namely: 1. Transportation Problems (TPs) and 2. Assignment Problems (APs). These categories of problems are studied in Operations Research/Optimization.

During 2^nd World War, Britain was having very less military resources; so, there was an urgent need to allocate resources to the various military operations and to the activities within each operation in an effective manner. So, the British military executives called upon a team of scientist to apply scientific method to study the strategic and tactical problems related to 'land' and 'air' defence of the country. As the team was dealing with research of military operations, the work of this team of scientists was named as Operations Research (OR).

TP is the special case of linear programming problem (LPP) where the objective 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 maximize the total transportation profit or minimize the TTC.

The traditional TP deals with the transportation a certain commodity from each of m origins (O_i or S_i, i=1,2,3,…,m) to any of n destinations (D_j or W_j, j=1,2,3,…,n). Assume that O_i’s or S_i’s are factories with respect capacities a₁,a₂,a₃,…,a_m and D’s or W_j’s are warehouses with required levels of demands b₁,b₂,b₃,…,b_n. For the transport of a unit of the given commodity from O_i to D_j a cost c_ij is assigned with c_ij≥0, ∀i,j. Hence, one must determine the amounts y_ijto be transported from all the sources (S₁,S₂,S₃,…,S_m) to all the destinations (D₁,D₂,D₃,…,D_n)in such a way that the total cost is minimized.

Typically, the traditional TP can be written as follows.