Answer to Question #169318 in Operations Research for Ra

Question

A company has three warehouses F₁, F₂ and F₃ which supply goods to four warehouses W₁, W₂, W₃ and W₄. The daily factory capacities of F₁, F₂ and F₃ are respectively, six units, one unit and ten units. The demand of the warehouses W₁, W₂, W₃ and W₄ are respectively, seven, five, three and two units. Unit transportation costs are as follows :

W₁ W₂ W₃ W₄

F₁ 2 3 11 7

F₂ 1 0 6 1

F₃ 5 8 15 9

To find an initial base feasible solution by the Vogel's approximation method.

Answer 1

Given Table is-

$W_1$ $W_2$ $W_3$ $W_4$

$f_1\\$ 2 3 11 7 6

$f_2$ 1 0 6 1 1

$f_3$ 5 8 15 9 10

7 5 3 2

Here ∑ai = ∑bj = $17$

(i.e) Total Availability =Total Requirement

∴The given problem is balanced transportation problem.

Hence there exists a feasible solution to the given problem.

First let us find the difference (penalty) between the first two smallest costs in each row and column and write them in brackets against the respective rows and columns

Choose the largest difference. Here the difference is 5 which corresponds to column D1 and D2. Choose either $W_1$ or $W_2$ arbitrarily. Here we take the column D1 . In this column choose the least cost. Here the least cost corresponds to $(f_1, W_1)$ . Allocate min (7, 6) = 6 units to this Cell.

Transportation schedule :

f₁→ W₁, f₁→W₂, f₂→W₂, f₂→W₄, f₃→W₃, f₃→W₄

This initial transportation cost-

$=(7\times2)+(6\times 3)+(5\times 0)+(1\times 1)+(3\times 15)+(10\times 9)\\=14+18+0+1+45+90\\=168$