A company has four ware houses W1, W2, W3 and W4. It is required to deliver a product from these warehouses to three customers A, B and C. The warehouses have the following amounts in stock:

Question

Ware house:
W1
W2
W3
W4

No of units:
15
16
12
13

And the customer’s requirements are:
Customer:
A
B
C

No of units:
18
20
18

The table below shows the costs of transporting one unit from warehouse to customer is given below.
Minz=8x11+9x12+6x13+3x14
+6x21+11x22+5x23+10x24
+3x31+8x32+7x33+9x34
Find the initial solution using each of the three methods.
Test optimality using Stepping stone method.

Answer 1

To find the initial solution using each of the three methods (Vogel's Approximation Method, Northwest Corner Method, and Least Cost Method), we will compare the costs of transporting one unit from each warehouse to each customer and allocate the units accordingly.

1. Vogel's Approximation Method:
- Calculate the penalty (difference between the two smallest costs) for each row and column.
- Choose the row or column with the highest penalty and allocate units based on the minimum cost available in that row or column.

2. Northwest Corner Method:
- Start from the top-left corner of the table and allocate as many units as possible while moving horizontally or vertically.

3. Least Cost Method:
- Choose the cell with the lowest cost and allocate units until either the supply from the warehouse or demand from the customer is exhausted.

Given the cost table:
8 9 6 3
6 11 5 10
3 8 7 9

We will now calculate the initial solutions using each of the three methods and test optimality using the Stepping Stone Method.