Study on the Analysis and Optimization of Brake Disc: A Review

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Volume-6, Issue-6, November-December 2016

International Journal of Engineering and Management Research

Page Number: 19-23

A New Simple Method of Finding an Optimal Solution for the

Transportation Problem

V. Veena Shalani1, Dr.N. Srinivasan2

1_{Assistant Professor, Department of Mathematics, ,St.Peter’s College of Engineering and Technology, INDIA} 2_{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 1941.

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:

Minimize Z =



 

m

1 i

ij n

1 j

ija

C

Subject to

j and i all for 0 a ), Demand ( n ,... 3 , 2 , 1 j , y a

) Supply ( m ,... 3 , 2 , 1 i , x a

ij m

1 i

j ij n

1 j

i ij

 

  



 

where aij – the amount of goods moved from origin i to destination j.

Cij – the cost of moving a unit amount of goods from origin i to destination j.

xi – the supply available at each origin i yj – 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 2:

Ensure whether the Transportaion problem is balanced if not make it to be balanced

Step 3:

Select minimum odd cost from transportation table and subtract the same from each of the odd cost valued cells of the transportation table.

Step 4:

Ensure now all the cost values in the transportation table with only even numbers and zeros.

Step 5:

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Step 6:

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 6 & 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 plants P1, P2, P3, P4 from which it supplies to three markets M1, M2, M3. Determine the optimal transportation plan from the following data giving the plant to market shifting costs, quantities available at each plant and quantities required at each market

Market Plant Supply

P1 P2 P3 P4

M1 21 16 25 13 11

M2 17 18 14 23 13

M3 32 17 18 41 19

Demand 6 10 12 15 43

Solution:

Consider the Transportation table:

P1 P2 P3 P4

M1 21 16 25 13 11

M2 17 18 14 23 13

M3 32 17 18 41 19

Demand 6 10 12 15 43

Choose the minimum odd cost M1 P4 = 13 from all the cost and subtract it from all the odd cost and allocate 11 (supply 11 and demand 15) in the place of zero and delete the row M1 after the allocation at M1 P4 = 11

P1 P2 P3 P4

M1 8 16 12 0

11

11 0

M2 4 18 14 10 13

M3 32 4 18 28 19

Demand 6 10 12 15 4

Remaining values in the table:

Market Plant Supply

P1 P2 P3 P4

M2 4 18 14 10 13

M3 32 4 18 28 19

Demand 6 10 12 4

Choose minimum even number 4 and subtract it from all the cost:

P1 P2 P3 P4

M2 0 14 10 6 13

M3 28 0 14 24 19

Demand 6 10 12 4

Allocate to the position M2P1 (demand 6) and after allocation M2 P1 = 6 and delete the column

Market

Plant

Supply P1 P2 P3 P4

M2 0 6

14 10 6 13

7

M3 28 0 14 24 19

Demand 6

10 12 4

Market

Plant

Supply P2 P3 P4

M2 14 10 6 7

M3 0 14 24 19

Demand 10 12 4

Again allocate M3 P2 demand 10 and delete the column P2

Market

Plant

Supply P2 _P

3 P4

M2 14 10 6 7

M3

0

10 14 24

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Demand 10 ₀ 12 4

Remaining values in the table

.

Market

plant

Supply P3 P4

M2 10 6 7

M3 14 24 9

Demand 12 4

Again choose a minimum even number 6 and subtract it throughout the table cost and then allocate at M2 P4 demand 4 and delete column P4

Market

plant

Supply P3 P4

M2 4

0 4

7 3

M3 8 18 9

Demand 4 ₀

Market

plant

Supply P3

M2 6 3

M3 8 9

Demand 12

Allocation table:

P1 P2 P3 P4

M1

21 16 25

13 11

11

M2

17 6

18

14 3

23 4

13

M3

32

17 10

18 9

41 19

Demand

6 10 12 15 43

No. Of Allocation :

m+n-1= 3+4-1=6

(13*11) +(17*6)+(14*3)+(23*4)+(17*10)+(18*9)=711

143 + 102 + 42 +92 + 170 + 162 = 711 Comparison:

New Method Modi Method

711 711

Example 2:

Given below the unit costs array with supplies ai; i= 1,2,3 and demands bj; j=1,2,3,4,5

bj ai

1 2 3 4 5 supply

1 4 1 2 4 4 60

2 2 3 2 2 3 35

3 3 5 2 4 4 40

Demand 22 45 20 18 30 135

Find the optimal solution to the above Hitcock Problem?

Solution:

Consider the transportation problem

bj ai

1 2 3 4 5 supply

1 4 1 2 4 4 60

2 2 3 2 2 3 35

3 3 5 2 4 4 40

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1 2 3 4 5 supply

1 4 0

45

2 4 4 60

15

2 2 2 2 2 2 35

3 2 4 2 4 4 40

Demand 22 45 0

20 18 30

1 3 4 5 supply

1 4 2 4 4 15

2 2 2 2 2 35

3 2 2 4 4 40

Demand 22 20 18 30

1 3 4 5 supply

1 2 0 2 2 15

2 0 0 0 0 35

3 0 0 2 2 40

Demand 22 20 18 30

1 3 4 5 supply

1 2 0

15

2 2 60

0

2 0

12 0 5

0 18

0 35 0

3 0

10

0 2 2 40

30 Demand 22 20 18 30

5 Supply

3 2

30

30 0

Demand

30 0

135

Allocation table

bj aj

1 2 3 4 5 supply

1 4 1

45

2 15

4 4 60

2 2

12

3 2 5

2 18

3 35

3 3

10

5 2 4 4

30 40

Demand 22 45 20 18 30 135

m+n-1 = 3+5-1 = 7

(1*45)+(2*15)+(2*12)+(2*5)+(2*18)+(3*10)+(4*30)=295

45 + 30 +24 +10 + 36+ 30+ 120 = 295

Comparison:

295 295

Example 3:

Solve the following transportation problem

A B C D

1 3 1 7 4 300

2 2 6 5 9 400

3 8 3 3 2 500

250 350 400 200

Solution:

A B C D supply

1 3 1 7 4 300

2 2 6 5 9 400

3 8 3 3 2 500

Demand 250 350 400 200

A B C D supply

1 2 0

300

6 4 300

0

2 2 6 4 8 400

3 8 2 2 2 500

Demand 250 350 50

400 200

A B C D supply

2 2 6 4 8 400

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Demand 250 50 400 200

A B C D supply

2 0

250

6 4 8 400

150

3 8 0

50

0 250

0 200

500 0

Demand 250 0

50 0

400 150

200 0

C

supply

2

4

150

Demand

150

C

supply

2

0 150

150 0

Demand

150 0

Allocation table:

A B C D supply

1 3 1

300

7 4 300

2 2

250

6 5

150

9 400

3 8 3

50

3 250

2 200

500

250 350 400 200

m+n-1= 3+4-1 = 6

(1*300) + (2*250) + (5*150) + (3*50) + (3*250) + (2*200) = 2850

300 + 500 + 750 + 150 + 750 + 400 = 2850

Comparison:

2850 2850

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,2016,5,22-30

[2] N. Srinivasan, D. Iranian, An innovative approach for finding the optimal solution for transportation problems. Journal International Journal of Mathematical Archive-6(8), 2015, 2-23

[3] Hamdy,A.T(2007) Operations Research : An Introduction 8th Edition,Pearson Prentice Hall, upper saddle River.

[4] J. K. Sharma, Operations Research- Theory and applications (Macmillan India (LTD), New Delhi, 2005). [5] H. A. Taha, Operations Research- Introduction (Prentice Hall of India (PVT), New Delhi, 2004).

[6] P. K. Gupta, D. S. Hira, Operations Research, S. Chand & Company Limited, 14th Edition, 1999.