Answer to Question #286793 in Operations Research for yas

Question

Given a transportation problem with the following costs, supply, and demand, find the initial solution

using the minimum cell cost method and Vogel’s approximation model. Is the VAM solution

optimal?

To

From 1 2 3 Supply

A 6 7 4 100

B 5 3 6 180

C 8 5 7 200

Demand 135 175 170

Answer 1

Solution:

Lowest cost $= 5\times5+8\times130+3\times175+4\times100+7\times70 = \$ 2480$

With the minimum cell cost method, the cost calculated is $2480.

Vogel's method:

TOTAL number of supply constraints : 3

TOTAL number of demand constraints : 3

Problem Table is

Table-1

Table-2

The maximum penalty, 7, occurs in row S₃.

The minimum c_ij in this row is c₃₃=7.

The maximum allocation in this cell is min(70,70) = 70.

It satisfy supply of S₃ and demand of D₃.

Initial feasible solution is

The minimum total transportation cost =4×100+5×5+3×175+8×130+7×70=2480

Here, the number of allocated cells = 5 is equal to m + n - 1 = 3 + 3 - 1 = 5

∴ This solution is non-degenerate.

Thus, the cost calculated by Vogel's Approximation Method is $ 2480. Both the values are the same. So, the solution attained by VAM is optimal even though it achieves the same value as the minimum cell cost method.