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Journal for Studies in Management and Planning

Available at http://edupediapublications.org/journals/index.php/JSMaP/

I SSN: 2395-0463

Vol ume 03 I s s ue 10

September 2017

Available online: http://edupediapublications.org/journals/index.php/JSMaP/ P a g e | 339

An Alternative Approach for finding initial basic feasible solution of Balanced

transportation problem.

Meenakshi

Assistant professor , Department of Mathematics,

Govind National College,Narangwal,Ludhiana(PB)

(email id : gargmeenakshi14@gmail.com)

ABSTRACT:

This paper discusses about a new method for finding the intial basic feasible solution of cost

minimization transportation problem with minimum transportation cost.Numerical examples are

also provided to prove that the proposed algorithm gives a better result than existing algorithms.

Keywords:Cost minimization tramsportation problem, basic feasible solution.

I ntroduction:

Transportation problem is a special type of linear programming problem in which goods are

transported from fixed number of sources to fixed number of destinations in such a way that total

cost of transportation is minimized.The basic transportation problem was originally developed by

Hitchcock in 1947[1] and then the systematic solution procedures from the simplex algorithm

were further developed primarily by Dantzig[2] and then by Charnes,Cooper and Henderson in

1953[3]

.

The well recognized methods for finding initial basic feasible solution are North West Corner

Rule(NWCR),Row minima,Column Minima,Matrix minima(Least Cost method) and Vogel’

s

Approximation Method.

Mathematical formulation of transportation problem:

.Let there are m sources S1, S2,_ _ _ _ _ _,Sm. and n destinations D1,D2,D3_ _ _ _ _,Dn.

Transportation problem can be represented mathematically as LPP as follows

Minimize : Z = ∑ ∑ cij xij

n

j=1

m

i=1

Subject to

∑ xij n

j=1 ≤ ai, i=1,2,3....m

∑ xij m

i=1 ≥bj, j=1,2,3......n

x ij≥0 for all i,j

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Journal for Studies in Management and Planning

Available at http://edupediapublications.org/journals/index.php/JSMaP/

I SSN: 2395-0463

Vol ume 03 I s s ue 10

September 2017

Available online: http://edupediapublications.org/journals/index.php/JSMaP/ P a g e | 340

ai = quantity of commodity available at origin i

bj = requirement of commodity at destination j

cij =cost of transportation of one unit of commodity from ith source to jth destination .

xij = number of units of commodity to be transported from ith source to jth destinatination

Algorithm for the proposed method:

Step1:Construct the transportation table from given transportation problem.

Step2:Select the least even cost from all the cost in the table.

Step3:Subtract the least even cost from even costs in the table.

Step4:Compare the minimum of supply or requirement whichever is minimum,then allocate the

minimum supply or requirement at the place of minimum value of related row or column.

If tie at the place of minimum value in supply or requirement, then allocate at minimum cost

corresponding to that row or column.

Step5:After completing step4 delete the row where supply from a given source is exhausted or

delete the column where requirement for a given destination is satisfied.

Step6:Repeat step4 and step5 until all the suppliers are exhausted and all the requirements are

satisfied.

Step7:Finally compute the total transportation cost as the sum of the product of cell allocations

and unit cost.

Numerical Examples:

Example1:Consider the following cost minimization transportation problem

Destinations

Sources D1 D2 D3 D4 Supply

S1 12 17 29 7 8

S2 54 19 24 39 10

S3 29 5 49 9 11

Requirement 4 7 6 12 29

Solution of the problem by proposed method is represented in the following table

Sources D1 D2 D3 D4 Supply

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Journal for Studies in Management and Planning

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September 2017

Available online: http://edupediapublications.org/journals/index.php/JSMaP/ P a g e | 341

S1 12(4) 17 29 7(4) 8

S2 54 19(4) 24(6) 39 10

S3 29 5(3) 49 9(8) 11

Requirement 4 7 6 12 29

Total transportation cost=12×4+7×4+19×4+24×6+5×3+9×8

=Rs.383

Example2:Consider the following cost minimization transportation problem

Sources D1 D2 D3 D4 D5 Supply

S1 4 5 7 9 10 15

S2 3 11 2 6 9 30

S3 8 12 21 41 4 12

S4 3 2 10 15 17 14

Requirement 35 6 8 15 7 71

Solution of the problem by proposed method is represented in the following table

Sources D1 D2 D3 D4 D5 Supply

S1 4(15) 5 7 9 10 15

S2 3(7) 11 2(8) 6(15) 9 30

S3 8(5) 12 21 41 4(7) 12

S4 3(8) 2(6) 10 15 17 14

Requirement 35 6 8 15 7 71

Total transportation cost=4×15+3×7+2×8+6×15+8×5+4×7+3×8+2×6

=Rs.291

Result Analysis:

Comparison among the solutions obtained by Proposed method and the other existing methods

and also with the optimal solution by means of above examples is shown in the following table

Method Total transportation cost(in Rupees)

Ex.1 Ex2

North West Corner Rule 455 830

Row Minima 560 433

Column Minima 447 471

Matrix Minima 487 461

Vogel’

s Approximation Method 383 291

Proposed method 383 291

Optimal Solution 383 291