Lecture_17_Vogal s Approximation method

- Vogel’s Approximation Method (Unit Cost Penalty Method)

Step1
For each row of the table, identify the smallest and the next to smallest cost. Determine the difference between them for each row. These are called penalties. Put them aside by enclosing them in the parenthesis against the respective rows. Similarly compute penalties for each column.
Step 2

Identify the row or column with the largest penalty. If a tie occurs then use an arbitrary choice. Let the largest penalty corresponding to the ith row have the cost cij. Allocate the largest possible amount xij = min (ai, bj) in the cell (i, j) and cross out either ith row or jth column in the usual manner.

Step 3

Again compute the row and column penalties for the reduced table and then go to step 2. Repeat the procedure until all the requirements are satisfied.
Find the initial basic feasible solution using vogel’s approximation method

1.

W1 W2 W3 W4 Availability
F1 7
F2 9
F3 18
Requirement 5 8 7 14

Solution
F1 F2 F3



W1 W2 W3 W4 Availability Penalty
7 19-10=9
9 40-30=10
18 20-8=12

Requirement Penalty 5 8
40-19=21 30
-8=22 7
50-40=10 14
20-10=10
W1 W2 W3 W4 Availability Penalty
F1 7 9
F2 9 10
F3 18/10 12
Requirement 5 8/0 7 14
Penalty 21 22 10 10



F1 W1 W2 W3 W4 Availability 7/2 Penalty 9
F2 9 20
F3 18/10 20
Requirement 5/0 X 7 14
Penalty 21 X 10 10


W1 W2 W3 W4 Availability Penalty
F1 7/2 40
F2 9 20
F3 18/10/0 50
Requirement X X 7 14/4
Penalty X X 10 10



F1 W1 W2 W3 W4 Availability 7/2/0 Penalty 40
F2 F3 9
X 20
X
Requirement X X 7 14/4/2
Penalty X X 10 50


F1 W1 W2 W3 W4 Availability
X Penalty
X
F2 X X
F3 X X
Requirement Penalty X X X X X X X X

Initial Basic Feasible Solution
x11 = 5, x14 = 2, x23 = 7, x24 = 2, x32 = 8, x34 = 10
The transportation cost is 5 (19) + 2 (10) + 7 (40) + 2 (60) + 8 (8) + 10 (20) = Rs. 779





2.





A
Warehouse B C

Stores Availability
I II III IV
11
13
19

Requirement 6 10 12 15



Solution





A
Warehouse B C

Stores Availability Penalty I II III IV
11 2
13 3
19 9

Requirement 6 10 12 15
Penalty 4 2 1 10






A
Warehouse B C



Stores Availability Penalty I II III IV
11/0 2
13 3
19 9

Requirement 6 10 12 15/4
Penalty 4 2 1 10






A
Warehouse B C



Stores Availability Penalty I II III IV
X X
13/9 3
19 9

Requirement 6 10 12 15/4/0
Penalty 15 9 4 18















A
Warehouse B C













Stores Availability Penalty I II III IV
X X
13/9/3 3
19 9

Requirement 6/0 10 12 X
Penalty 15 9 4 X







A
Warehouse B C

Stores Availability Penalty I II III IV
X X
13/9/3/0 4
19 9

Requirement X 10/7 12 X
Penalty X 9 4 X






A
Warehouse B C

Stores Availability Penalty
I II III IV
X X
X X
X X

Requirement X X X X
Penalty X X X X


Initial Basic Feasible Solution
x14 = 11, x21 = 6, x22 = 3, x24 = 4, x32 = 7, x33 = 12
The trans