# Transportation Model

Pages: 8 (2843 words) Published: May 6, 2015
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.1, 2014

www.iiste.org

A comparative study of initial basic feasible solution methods for transportation problems
Abdul Sattar Soomro1 Gurudeo Anand Tularam2 Ghulam Murtaa Bhayo3 dr_sattarsoomro@yahoo.co.in, a.tularam@griffith.edu.au, gmsindhi@yahoo.com 1
Professor of Mathematics, Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Sindh, Pakistan
2
Senior Lecturer, Mathematics and Statistics, Science Environment Engineering and Technology [ENV], Griffith University, Brisbane Australia
3
Lecturer, Govt. Degree College, Pano Akil, Sukkur, Sindh, Pakistan Abstract
In this research three methods have been used to find an initial basic feasible solution for the balanced transportation model. We have used a new method of Minimum Transportation Cost Method (MTCM) to find the initial basic feasible solution for the solved problem by Hakim [2]. Hakim used Proposed Approximation Method (PAM) to find initial basic feasible solution for balanced transportation model and then compared the results with Vogel’s Approximation Method (VAM) [2]. The results of both methods were noted to be the same but here we have taken the same transportation model and used MTCM to find its initial basic feasible solution and compared the result with PAM and VAM. It is noted that the MTCM process provides not only the minimum transportation cost but also an optimal solution.

Keywords: Transportation problem, Vogel’s Approximation Method (VAM), Maximum Penalty of largest numbers of each Row
1.

Introduction

The transportation problem is a special linear programming problem which arises in many practical applications in other areas of operation, including, among others, inventory control, employment scheduling, and personnel assignment [1].

In this problem we determine optimal shipping patterns between origins or sources and

destinations. The transportation problem deals with the distribution of goods from the various points of supply, such as factories, often known as sources, to a number of points of demand, such as warehouses, often known as destinations. Each source is able to supply a fixed number of units of the product, usually called the capacity or availability and each destination has a fixed demand, usually called the requirements. The objective is to schedule shipments from sources to destinations so that the total transportation cost is a minimum.

There are various types of transportation models and the simplest of them was first presented by Hitchcock (1941). It was further developed by developed by Koopmans (1949) and Dantzig (1951). Several extensions of transportation model and methods have been subsequently developed. In general, the Vogel’s approximation method yields the best starting solution and the north-west corner method yields the worst. However, the latter is easier, quick and involves the least computations to get the initial solution [5]. Goyal (1984) improved VAM for the unbalanced transportation problem, while Ramakrishnan (1988) discussed some improvement to Goyal’s Modified Vogel’s Approximation method for unbalanced transportation problem [6].

Adlakha and Kowalski (2009) suggested a systematic analysis for allocating loads to obtain an alternate optimal solution [7]. However, the study on alternate optimal solutions is clearly limited in the literature of transportation

11

Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.1, 2014

www.iiste.org

with the exception of Sudhakar VJ, Arunnsankar N, Karpagam T (2012) who suggested a new approach for finding an optimal solution for transportation problems [8].

2. Transportation problem and General Computational Procedures The transportation model of LP can be modeled as follows:
Minimize Z 

m

n

 C

ij

i 1 j 1
n

Subject to

x
j 1

ij

m

x
i...

References: Problem, Annals of Pure and Applied Mathematics, Vol. 1, No. 2, 2012, 203-209.
[3] S. K Goyal, Improving VAM for unbalanced transportation problems, Journal of Operational
Research Society, 35(12) (1984) 1113-1114.
[4] P. K. Gupta and Man Mohan. (1993). Linear Programming and Theory of Games, 7th edition,
Sultan Chand & Sons, New Delhi (1988) 285-318.
[6]
Goyal (1984) improving VAM for the Unbalanced Transportation Problem, Ramakrishnan
(1988) discussed some improvement to Goyal’s Modified Vogel’s Approximation method
for Unbalanced Transportation Problem.
[7] Veena Adlakha, Krzysztof Kowalski (2009), Alternate Solutions Analysis For
Transportation problems, Journal of Business & Economics Research – November,Vol 7.
[8] Sudhakar VJ, Arunnsankar N, Karpagam T (2012). A new approach for find an Optimal
Solution for Trasportation Problems, European Journal of Scientific Research 68 254-257.