Five jobs (J1, J2, J3, J4 and J5) need to be processed in a factory. Each job can be assigned to any of the five different machines (M1, M2, M3, M4 and M5). The time durations taken (in minutes) by the machines for each of the jobs, are given in the table. However, each job is assigned to a specific machine in such a way that the total processing time is minimum. The total processing time is _________ minutes.

	M1	M2	M3	M4	M5
J1	40	30	50	50	58
J2	26	38	60	26	3840
J3	40	34	28	24	30
J4	28	40	40	32	48
J5	28	32	38	22	44

To find the minimum total processing time, we can formulate this as an assignment problem. We need to assign each of the five jobs ($J_1$, $J_2$, $J_3$, $J_4$, and $J_5$) to a unique machine ($M_1$, $M_2$, $M_3$, $M_4$, and $M_5$) such that the sum of the processing times is minimized.

Let's write down the processing times for each job on each machine from the given table (noting that the entry for $J_2$ under $M_5$ is 38 minutes):

• J1: M1 = 40, M2 = 30, M3 = 50, M4 = 50, M5 = 58
• J2: M1 = 26, M2 = 38, M3 = 60, M4 = 26, M5 = 38
• J3: M1 = 40, M2 = 34, M3 = 28, M4 = 24, M5 = 30
• J4: M1 = 28, M2 = 40, M3 = 40, M4 = 32, M5 = 48
• J5: M1 = 28, M2 = 32, M3 = 38, M4 = 22, M5 = 44

Let's evaluate the optimal assignment step-by-step:

1. Assign job J5 to machine M4, which takes the minimum possible time for J5:
Time = 22 minutes.

2. Assign job J1 to machine M2, which takes the minimum possible time for J1:
Time = 30 minutes.

3. Assign job J4 to machine M1:
Time = 28 minutes.

4. Assign job J3 to machine M3:
Time = 28 minutes.

5. Assign job J2 to machine M5:
Time = 38 minutes.

Now, let's verify if all jobs and machines are uniquely paired:

• Jobs: J1, J2, J3, J4, J5 (All 5 assigned)
• Machines: M2, M5, M3, M1, M4 (All 5 assigned)

Calculating the total processing time:
$\text{Total Time}=30+38+28+28+22=146$ minutes.

Thus, the minimum total processing time is 146 minutes.