Question

Three companies \( C_1, C_2 \) and \( C_3 \) submit bids for three jobs \( J_1, J_2 \) and \( J_3 \). The costs involved per unit are given in the table below: \[ \begin{array}{c|ccc} & J_1 & J_2 & J_3 \\ \hline C_1 & 10 & 12 & 8 \\ C_2 & 9 & 15 & 10 \\ C_3 & 15 & 10 & 9 \\ \end{array} \]

Hint:

For assignment problems, use the Hungarian algorithm to minimize the total cost by assigning jobs to companies optimally.

Answer 1

Correct Answer: 27

We are tasked with finding the optimal assignment that minimizes the total cost. This is a typical assignment problem that can be solved using the Hungarian Method or by directly inspecting the minimum cost for each job assignment.

We will follow the steps below to determine the minimum cost:

• Step 1: Consider the costs associated with each job:
• For \( J_1 \), the minimum cost is \( 9 \) from \( C_2 \).
• For \( J_2 \), the minimum cost is \( 10 \) from \( C_3 \).
• For \( J_3 \), the minimum cost is \( 8 \) from \( C_1 \).

Step 2: Add up the minimum costs for the optimal assignment:

\[ 9 + 10 + 8 = 27 \]

Final Answer:

The cost of the optimal assignment is \( \boxed{27} \).