Chapter 7, Problem 10RP

Explanation of Solution

Optimal solution:

• Using u-v method, the first step is to check whether the solution is optimal or not.
• The table is shown below:

4	2	4
12	8	4

• The first step is to find the initial feasible solution.
• It is found using Vogels App method.
• In order to find the feasible solution, the first step is to find the difference between the two least cost cells which have not been allocated for both rows and columns.
• The next step is to select the largest of the differences of both the rows and columns and if there is a tie, the one with the maximum quantity is considered.
• The next step is to assign the largest quantity within the demand in that row or column where the cost is minimum.
• Then the last step is to eliminate the entire row or column.
• The cost value of cells is represented by${\text{c}}_{\text{ij}}$.
• Hence, ${\text{c}}_{\text{11}}$ is 10, ${\text{c}}_{\text{12}}$is 5, ${\text{c}}_{\text{22}}$is 5 and ${\text{c}}_{\text{23}}$ is 10.
• In this method, the value ${\text{u}}_{\text{i}}$and ${\text{v}}_{\text{j}}$ belongs to rows and columns respectively.
• The formula to find ${\text{u}}_{\text{i}}$ and ${\text{v}}_{\text{j}}$ is as follows:

  ${\text{u}}_{\text{i}}{\text{+ v}}_{\text{j}}{\text{= c}}_{\text{ij }}\text{= m + n – 1 = 3 + 2 – 1 = 4}$

• Here, $\text{m}$is the total number of rows and $\text{n}$is the total number of columns and ${\text{c}}_{\text{ij}}$is the cost value