A New Simple Method of Finding an Optimal Solution for the Transportation Problem

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1 ISSN (ONLINE): 5-758, ISSN (PRINT): 9-9 Volume-, Issue-, November-December 1 International Journal of Engineering and Management Research Page Number: 19- A New Simple Method of Finding an Optimal Solution for the Transportation Problem V. Veena Shalani 1, Dr.N. Srinivasan 1 Assistant Professor, Department of Mathematics,,St.Peter s College of Engineering and Technology, INDIA Professor, Department of Mathematics, St. Peter s University, INDIA ABSTRACT In this paper a simple method for finding an optimal solution for Transportation problem.this method finds an optimal solution without requiring an initial basic feasible solution.in this method the number of allocation m+n-1 is satisfied for all problems. Numerical examples for the new simple method is explained and compared with the results of Modi method. This method does not require arithmetical and logical calculation. It is easy to understand and this method is very efficient for those who are dealing with Transportation problem.it can easily adopt an existing method. Keywords Balanced Transportation, Minimum odd cost, Minimum even cost,transportation I. INTRODUCTION Transportation model is the special case of linear programming problem(lpp).it plays an important role in logistic.the common objectives of transportation problem are (i) minimising the cost of shipping of m units and n destination.(ii)maximum the profit of shipping m units to n destination.this new proposed method for finding an optimal solution directly without using initial basic feasible solution with minimum number of steps and easy computations. Three numerical examples are provided to prove the claim with step wise procedure of the new proposed method. II. MATHEMATICAL REPRESENTATION The Transportation Pronblem (TP) was first developed and proposed by F. L Hitchcock since 191. It usually aims to minimize the total transportation cost and to the maximise the profit. The Hitchcock-Koopman s transportation problem is expressed as a linear transportation model as follows: Subject to m Minimize Z = C n m i 1 n j 1 a x, i 1,,,... m j 1 a y, j 1,,,... n i 1 i j a () (), a for all i and j where a the amount of goods moved from origin i to destination j. C the cost of moving a unit amount of goods from origin i to destination j. x i the supply available at each origin i y j the demand available at each destination j m total number of origins (Sources) n total number of destinations (Sinks) III. PROPOSED ALGORITHM Step 1: Construct Transportaion table for the given transportation problem. Step : Ensure whether the Transportaion problem is balanced if not make it to be balanced Step : Select minimum odd cost from transportation table and subtract the same from each of the odd cost valued cells of the transportation table. Step : Ensure now all the cost values in the transportation table with only even numbers and zeros. Step 5: Allocation is done for zeros, Consider (i, j)th zero position where there is minimum demand / minimum supply. After allocation delete the row / column 19 Copyright 1. Vandana Publications. All Rights Reserved.

2 ISSN (ONLINE): 5-758, ISSN (PRINT): 9-9 Step : Identify the minimum even cost from the table and subtract the same cost from all the cost cells. Step 7: Identify the zero s and apply step 5. If there are more than one zero positions identify the cell ( for allocation) where minimum demand / minimum supply of the transportation table. Step 8: Repeat step & step 7 until the demand and supply are exhausted. Now it can be verified m+n-1 allocations are allotted. Step 9: Calculate the total minimum cost from the transporation table. IV. NUMERICAL EXAMPLES Example 1: A company has four s, P, P, P from which it supplies to three markets M 1, M, M. Determine the optimal transportation plan from the following data giving the to market shifting costs, quantities available at each and quantities required at each market M M M Consider the Transportation table: M M M Choose the minimum odd cost M 1 P = 1 from all the cost and subtract it from all the odd cost and allocate 11 (supply 11 and demand ) in the place of zero and delete the row M 1 after the allocation at M 1 P = M M Choose minimum even number and subtract it from all the cost: M M Allocate to the position M (demand ) and after allocation M = and delete the column M M M M column P Again allocate M P demand 1 and delete the P P P M M M M M Copyright 1. Vandana Publications. All Rights Reserved.

3 ISSN (ONLINE): 5-758, ISSN (PRINT): Remaining values in the table. P P 1 M 1 7 M Again choose a minimum even number and subtract it throughout the table cost and then allocate at M P demand and delete column P P M P M P 7 M M Allocation table: M M 17 M m+n-1= +-1= (1*11) +(17*)+(1*)+(*)+(17*1)+(18*9)= = Example : Given below the unit costs array with supplies a i ; i= 1,, and demands ; j=1,,,,5 a i 1 5 supply Find the optimal solution to the above Hitcock Problem? Consider the transportation problem a i 1 5 supply Copyright 1. Vandana Publications. All Rights Reserved.

4 ISSN (ONLINE): 5-758, ISSN (PRINT): supply supply supply supply Allocation table a j supply m+n-1 = +5-1 = 7 (1*5)+(*)+(*1)+(*5)+(*18)+(*1)+(*)= = Example : Solve the following transportation problem A B C D Copyright 1. Vandana Publications. All Rights Reserved.

5 ISSN (ONLINE): 5-758, ISSN (PRINT): Allocation table: C 5 C supply supply V. CONCLUSION The New method proposed here solves transportation problems.this method can be applied to all transportation problems (Balanced and Unbalanced). A systamatic procedure and easy way to find optimal solution for transportation problem without degeneracy and basic feasiable solution. While comparing to other methods, it is easy to calculate and we get the required solution in few steps. REFERENCES [1] Mollah Mesbahuddin Ahmed, Aminur Rahman Khan,Md.shavif uddin,faruque Ahmed,A new Approach to solve transportation problems, open journal of optimization,1,5,- [] N. Srinivasan, D. Iranian, An innovative approach for finding the optimal solution for transportation problems. Journal International Journal of Mathematical Archive- (8),, - [] Hamdy,A.T(7) Operations Research : An Introduction 8 th Edition,Pearson Prentice Hall, upper saddle River. [] J. K. Sharma, Operations Research- Theory and applications (Macmillan India (LTD), New Delhi, 5). [5] H. A. Taha, Operations Research- Introduction (Prentice Hall of India (PVT), New Delhi, ). [] P. K. Gupta, D. S. Hira, Operations Research, S. Chand & Company Limited, 1th Edition, m+n-1= +-1 = (1*) + (*5) + (5*) + (*5) + (*5) + (*) = = 85 Copyright 1. Vandana Publications. All Rights Reserved.