Answer to Question #308175 in Management for Aarti

Question

Consider the transportation problem presented in the following table:

1. D1 D2 D3 D4 Supply
2. S1 19 30 50 10 7
3. S2 70 30 40 60 9
4. S3 40 08 70 20 18
5. Demand 5 8 7 14

Find initial feasible solution using Vogel's Approximation method (VAM),

also find optimal solution using stepping-stone method.                         

Answer 1

Vogel’s approximation (penalty or regret) is preferred over NWCR and LCM methods. In this method, an allocation is made on the basis of the opportunity (or penalty or extra) cost that would have been incurred if the allocation in certain cells with minimum unit transportation cost were missed. Hence, allocations are made in such a way that the penalty cost is minimized. An initial solution obtained by using this method is nearer to an optimal solution or is the optimal solution itself. The steps of VAM are as follows:

Step 1: Calculate the penalties for each row (column) by taking the difference between the smallest and next smallest unit transportation cost in the same row (column). This difference indicates the penalty or extra cost that has to be paid if decision-maker fails to allocate to the cell with the minimum unit transportation cost.

Step 2: Select the row or column with the largest penalty and allocate as much as possible in the cell that has the least cost in the selected row or column and satisfies the rim conditions. If there is a tie in the values of penalties, it can be broken by selecting the cell where the maximum allocation can be made.

Step 3: Adjust the supply and demand and cross out the satisfied row or column. If a row and a column are satisfied simultaneously, only one of them is crossed out and the remaining row (column) is assigned a zero supply (demand). Any row or column with zero supply or demand should not be used in computing future penalties.

Step 4: Repeat Steps 1 to 3 until the available supply at various sources and demand at various destinations is satisfied. Example 9.4 Use Vagel’s Approximation Method (VAM) to find the initial basic feasible solution to the transportation problem using the data of Example 9.1.

Solution The differences (penalty costs) for each row and column have been calculated as shown in Table 9.5. In the first round, the maximum penalty, 22 occurs in column D2. Thus the cell (S3, D2) having the least transportation cost is chosen for allocation. The maximum possible allocation in this cell is 8 units and it satisfies demand in column D2. Adjust the supply of S3 from 18 to 10 (18 – 8 = 10).

The new row and column penalties are calculated except column D2 because D2’s demand has been satisfied. In the second round, the largest penalty, 21 appears at column D1. Thus the cell (S1, D1) having the least transportation cost is chosen for allocating 5 units as shown in Table 9.5. After adjusting the supply and demand in the table, we move to the third round of penalty calculations. In the third round, the maximum penalty 50 appears at row S3.

The maximum possible allocation of 10 units is made in cell (S3, D4) that has the least transportation cost of 20 as shown in Table 9.5. The process is continued with new allocations till a complete solution is obtained. The initial solution using VAM is shown in Table 9.5. The total transportation cost associated with this method is: Total cost = 5 × 19 + 2 × 10 + 7 × 40 + 2 × 60 + 8 × 8 + 10 × 20 = Rs 779