OJOpOpen Journal of Optimization2325-7105Scientific Research Publishing10.4236/ojop.2016.51003OJOp-64199ArticlesComputer Science&Communications Engineering Physics&Mathematics A New Approach to Solve Transportation Problems ollahMesbahuddin Ahmed1*AminurRahman Khan1Md.Sharif Uddin1FaruqueAhmed1Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh* E-mail:mesbah_1972@yahoo.com(OMA);030320160501223024 November 2015accepted 1 March 4 March 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Finding an initial basic feasible solution is the prime requirement to obtain an optimal solution for the transportation problems. In this article, a new approach is proposed to find an initial basic feasible solution for the transportation problems. The method is also illustrated with numerical examples.

Transportation Problem Transportation Cost Initial Basic Feasible Solution Optimal Solution
1. Introduction

Transportation problem is famous in operation research for its wide application in real life. This is a special kind of the network optimization problems in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the source and destination, respectively, such that the total cost of transportation is minimized. The basic transportation problem was originally developed by Hitchcock in 1941  . Efficient methods for finding solution were developed, primarily by Dantzig in 1951  and then by Charnes, Cooper and Henderson in 1953  . Basically, the solution procedure for the transportation problem consists of the following phases:

・ Phase 1: Mathematical formulation of the transportation problem.

・ Phase 2: Finding an initial basic feasible solution.

・ Phase 3: Optimize the initial basic feasible solution which is obtained in Phase 2.

In this paper, Phase 2 has been focused in order to obtain a better initial basic feasible solution for the transportation problems. This problem has been studied since long and is well known by Abdur Rashid et al.  , Aminur Rahman Khan et al.  - , Hamdy A. T.  , Kasana & Kumar  , Kirca and Satir  , M. Sharif Uddin et al.  , Mathirajan, M. and Meenakshi  , Md. Amirul Islam et al.   , Md. Ashraful Babu et al.  - , Md. Main Uddin et al.   , Mollah Mesbahuddin Ahmed et al.  - , Pandian & Natarajan  , Reinfeld & Vogel  , Sayedul Anam et al.  , Shenoy et al.  and Utpal Kanti Das et al.   .

Again, some of the well reputed methods for finding an initial basic feasible solution of transportation problems developed and discussed by them are North West Corner Method (NWCM)  , Row Minimum Method (RMM)   , Column Minimum Method (CMM)   , Least Cost Method (LCM)  , Vogel’s Approximation Method (VAM)   , Extremum Difference Method (EDM)  , Highest Cost Difference Method (HCDM)   , Average Cost Method (ACM)  , TOCM-MMM Approach  , TOCM-VAM Approach  , TOCM-EDM Approach  , TOCM-HCDM Approach  , TOCM-SUM Approach  etc.

In this paper, a new algorithm is proposed to find an initial basic feasible solution for the transportation problems. A comparative study is also carried out by solving a good number of transportation problems which shows that the proposed method gives better result in comparison to the other existing heuristics available in the literature.

2. Network Representation and Mathematical Model of Transportation Problem

Generally the transportation model is represented by the network in Figure 1. There are m sources and n destinations, each represented by a node. The arcs represent the routes linking the sources and destinations. Arc (i, j) joining source i to destination j carries two pieces of information: the transportation cost per unit, cij and the amount shipped, xij. The amount of supply at source i is Si, and the amount of demand at destination j is dj. The objective of the model is to determine the unknowns’ xij that will minimize the total transportation cost while satisfying the supply and demand restrictions.

Network representation of transportation problem

Considering the above notations, the transportation problem can be stated mathematically as a linear programming problem as:

Minimize:.

Subject to: ;.

;.

for all and.

The objective function minimizes the total cost of transportation (Z) between various sources and destinations. The constraint i in the first set of constraints ensures that the total units transported from the source i is less than or equal to its supply. The constraint j in the second set of constraints ensures that the total units transported to the destination j is greater than or equal to its demand.

3. Proposed Approach to Find an Initial Basic Feasible Solution

In the proposed approach, an allocation table is formed to find the solution for the transportation problem. That’s why this method is named as Allocation Table Method (ATM) and the method is illustrated below:

・ Step-1: Construct a Transportation Table (TT) from the given transportation problem.

・ Step-2: Ensure whether the TP is balanced or not, if not, make it balanced.

・ Step-3: Select minimum odd cost (MOC) from all the cost cells of TT. If there is no odd cost in the cost cells of the TT, keep on dividing all the cost cells by 2 (two) till obtaining at least an odd value in the cost cells.

・ Step-4: Form a new table which is to be known as allocation table (AT) by keeping the MOC in the respective cost cell/cells as it was/were, and subtract selected MOC only from each of the odd cost valued cells of the TT. Now all the cell values are to be called as allocation cell value (ACV) in AT.

・ Step-5: At first, start the allocation from minimum of supply/demand. Allocate this minimum of supply/ demand in the place of odd valued ACVs at first in the AT formed in Step-4. If demand is satisfied, delete the column. If it is supply, delete the row.

・ Step-6: Now identify the minimum ACV and allocate minimum of supply/demand at the place of selected ACV in the AT. In case of same ACVs, select the ACV where minimum allocation can be made. Again in case of same allocation in the ACVs, choose the minimum cost cell which is corresponding to the cost cells of TT formed in Step-1 (i.e. this minimum cost cell is to be found out from the TT which is constructed in Step-1). Again if the cost cells and the allocations are equal, in such case choose the nearer cell to the minimum of demand/supply which is to be allocated. Now if demand is satisfied delete the column and if it is supply delete the row.

・ Step-7: Repeat Step-6 until the demand and supply are exhausted.

・ Step-8: Now transfer this allocation to the original TT.

・ Step-9: Finally calculate the total transportation cost of the TT. This calculation is the sum of the product of cost and corresponding allocated value of the TT.

4. Numerical Examples with Illustration4.1. Example-1

A company manufactures motor cars and it has three factories F1, F2 and F3 whose weekly production capacities are 300, 400 and 500 pieces of cars respectively. The company supplies motor cars to its four showrooms located at D1, D2, D3 and D4 whose weekly demands are 250, 350, 400 and 200 pieces of cars respectively. The transportation costs per piece of motor cars are given in the transportation Table 1. Find out the schedule of shifting of motor cars from factories to showrooms with minimum cost:

Data of the Example-1
FactoriesShowroomsProduction capacity
D1D2D3D4
F13174300
F22659400
F38332500
Demand250350400200
4.1.1. Solution of Example 1 and Its Explanation

Allocation of various cells in the allocation table for Example-1 is shown in Allocation Table 2.

Allocation of various cells are in the allocation table
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