An assignment problem is solved to minimize the total processing time of four jobs (1, 2, 3 and 4) on four different machines such that each job is processed exactly by one machine and each machine processes exactly one job. The minimum total processing time is found to be 500 minutes. Due to a change in design, the processing time of Job 4 on each machine has increased by 20 minutes. The revised minimum total processing time will be _____________ minutes (in integer).

The correct answer is 520.

Step-by-Step Explanation:

1. Understanding the Assignment Problem:
In a standard assignment problem involving four jobs (1, 2, 3, and 4) and four machines, the objective is to assign each job to exactly one machine such that the total processing time is minimized. Let $t_{ij}$ represent the processing time of Job $i$ on Machine $j$. The problem finds an optimal assignment where each job $i$ is matched with a unique machine $j$ to yield the minimum total processing time of 500 minutes.

2. Analyzing the Change:
According to the problem statement, the processing time of Job 4 on each machine has increased by 20 minutes. This means for all machines $j$ (where $j=\; 1,\; 2,\; 3,\; 4$):
New processing time $t_{4j}^{\text{new}}=\; t\_\{4j+\; 20$

3. Effect on the Optimal Assignment:
Since Job 4 must be processed by exactly one machine in any valid assignment, any feasible solution (including the optimal one) will contain exactly one processing time corresponding to Job 4. Since the processing time of Job 4 on every machine has increased by exactly 20 minutes, the total processing time of every feasible assignment configuration increases by exactly 20 minutes. This uniform change does not alter the relative differences between different assignment combinations, meaning the optimal assignment mapping remains the same.

4. Calculating the Revised Minimum Total Time:
The new minimum total processing time is simply the original minimum total processing time plus the constant increase in Job 4's processing time:
New minimum time = 500 + 20 = 520 minutes.