Thursday, July 11, 2013

Vogel's Approximation Method (VAM)

The Vogel's Approximation Method

In addition to the North West Corner and Intuitive Lowest Cost Methods for setting an initial solution to transportation problems, we can use another important technique - Vogel's Approximation Method (VAM).

Though VAM is not quite as simple as Northwest Corner approach, but it facilitates a very good initial solution, one that is often the optimal solution.

Vogel's Approximation Method tackles the problem of finding a good initial solution by taking into account the costs associated with each alternative route, which is something that Northwest Corner Rule did not do.

To apply VAM, we must first compute for each row and column the penalty faced if the second best route is selected instead of the least cost route.

To illustrate the same, we will look at the Bengal Plumbing transportation problem.

Transportation Matrix for Bengal Plumbing
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A

Rs.50

Rs.40

Rs.30
100
Plant B

Rs.80

Rs.40

Rs.30
300
Plant C

Rs.90

Rs.70

Rs.50
300
Warehouse Requirement
300
200
200
700








We will take the following steps to find an initial solution via VAM method.

VAM Step 1 : For each row and column of the transportation table, find the difference between the two lowest unit shipping costs as follows. 
These figures represent the difference between the distribution cost of the best route and the second best route, or the per unit penalty for choosing the second best route instead of the lowest shipping route and hence are called the Opportunity Cost.  

30
0
0
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A

Rs.50

Rs.40

Rs.30
100
10
Plant B

Rs.80

Rs.40

Rs.30
300
10
Plant C

Rs.90

Rs.70

Rs.50
300
20
Warehouse Requirement
300
200
200
700


In the above table, the numbers at the heads of the columns and to the right of the rows represent the differences between two lowest shipping costs for the respective row or column.
For example in the Column E the three transportation costs are Rs.50, Rs.80 & Rs.90. The lowest costs are Rs.50 & Rs.80. The difference between these two is Rs.30.
Hence Rs.30 is the Opportunity Cost for Column E.

VAM Step 2  Identify the row or column with the greatest Opportunity Cost, or difference. In this case the row or column selected is Column E with an Opportunity Cost of 30.

VAM Step 3 : Assign as many units as possible to the lowest cost route (A - E) with a per unit shipping cost   of Rs.50. In this case 100 units are assigned to that square as anything more would exceed Plant (A)'s availability.

VAM Step 4 : Eliminate any row or column that has been completely satisfied by the assignment just made. This can be done by placing Xs in each appropriate square as shown below.

30
0
0
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A
100
Rs.50
 X
Rs.40
Rs.30
100
10
Plant B

Rs.80

Rs.40

Rs.30
300
10
Plant C

Rs.90

Rs.70

Rs.50
300
20
Warehouse Requirement
300
200
200
700


VAM Step 5 : Recompute the cost differences for the transportation table, excluding the rows or columns left satisfied or crossed out in the preceding step as shown below.

10
30
20
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A
100
Rs.50
 X
Rs.40
Rs.30
100
Plant B

Rs.80

Rs.40

Rs.30
300
10
Plant C

Rs.90

Rs.70

Rs.50
300
20
Warehouse Requirement
300
200
200
700

As a result of the elimination of the first row the Opportunity Costs for the rest of the rows and columns have changed.
For example the Opportunity Cost for Column E has changed from 30 to 10.

VAM Step 6 : Return to Step 2 and repeat the steps until an initial feasible solution which satisfies the demands and supplies of all the rows and columns are obtained.

At the current scenario the largest Opportunity Cost for the transportation table is 30 for Column F. We will assign as many units as possible to the lowest cost route (B - E) with a per unit shipping cost of Rs.40. We will assign 200 units satisfying Warehouse (F)'s demand.
And will cross out the leftover square in Column F as it will surpass Warehouse (F)'s demand. 

10
30
20
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A
100
Rs.50
 X
Rs.40
Rs.30
100
Plant B

Rs.80
 200
Rs.40

Rs.30
300
10
Plant C

Rs.90
 X
Rs.70

Rs.50
300
20
Warehouse Requirement
300
200
200
700

We will now recompute the Opportunity Costs and assign as many units as possible to the row or column with the lowest unit shipping cost having the largest Opportunity Cost, and will cross out the unnecessary squares.

The largest Opportunity Cost in the current transportation table is 50 for Row B. The lowest per unit shipping cost is Rs.30. We can only assign 100 units to the route (B - G) to satisfy the capacity constraint of Plant (B). This is illustrated in the following table.

10
20
From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A
100
Rs.50
 X
Rs.40
Rs.30
100
Plant B
X
Rs.80
 200
Rs.40
100
Rs.30
300
50
Plant C

Rs.90
 X
Rs.70

Rs.50
300
40
Warehouse Requirement
300
200
200
700

VAM Step 7 : The final two allocation (C - E) and (C - G) can be made by inspecting the supply restrictions (in rows) and demand requirements (in columns). We see that an assignment of 200 units has been agreed to route (C - E) and another assignment of 100 units has been agreed to (C - G) route, satisfying all the demands and supply constraints.

From \ To
Warehouse E
Warehouse F
Warehouse G
Factory Capacity
Plant A
100
Rs.50
 X
Rs.40
Rs.30
100
Plant B
X
Rs.80
 200
Rs.40
100
Rs.30
300
Plant C
200
Rs.90
 X
Rs.70
 100
Rs.50
300
Warehouse Requirement
300
200
200
700


The total computed shipping cost according to the Vogel's Approximation Method.


Initial Feasible Transportation Route Selection based on VAM Method
From
To
Units Shipped
Cost per unit
Total Cost
Plant A
Warehouse E
100
Rs.50
Rs.5000
Plant B
Warehouse F
200
Rs.40
Rs.8000
Plant B
Warehouse G
100
Rs.30
Rs.3000
Plant C
Warehouse E
200
Rs.90
Rs.18000
Plant C
Warehouse G
100
Rs.50
Rs.5000
Total Shipping Cost
Rs.39000

The total cost of this initial solution is Rs.39000.00 which is less than both the Northwest Corner Rule and the Intuitive Lowest Cost Method.
Even though VAM takes many more calculations to find an initial solution as compared to the other two methods, it almost produces a much better initial solution. 
Hence VAM tends to minimize the total number of computations needed to reach an optimal solution.