A law office currently has five jobs to be scheduled, shown in the order in which they arrived.

Job

Processing Time (hours)

Due (hours)

V

18

24

W

20

35

X

43

46

Y

27

35

Z

55

63

Calculate Average Flow Time (Total flow time/# of jobs), Average Lateness (Total lateness / # of jobs) and Average Number of Jobs (Total flow time / Total processing time) by sequencing the tasks using Shortest Processing Time rule.

Question

A law office currently has five jobs to be scheduled, shown in the order in which they arrived.

Job

Processing Time (hours)

Due (hours)

V

18

24

W

20

35

X

43

46

Y

27

35

Z

55

63

Calculate Average Flow Time (Total flow time/# of jobs), Average Lateness (Total lateness / # of jobs) and Average Number of Jobs (Total flow time / Total processing time) by sequencing the tasks using Shortest Processing Time rule.

BlackTom AI · Accepted Answer

Question restated: A law office has five jobs V, W, X, Y, Z with processing times 18, 20, 43, 27, 55 hours and due dates 24, 35, 46, 35, 63 hours respectively. We are asked to compute Average Flow Time (AFT), Average Lateness (AL), and Average Number of Jobs (ANJ) after sequencing the tasks using the Shortest Processing Time (SPT) rule.

Option list to evaluate: 
- AFT = 50, AL = 8, ANJ = 4.26 
- AFT = 8.5, AL = 4.26, ANJ = 1.96 
- AFT = 26.3, AL = 12, ANJ = 3.24 
- AFT = 78.4, AL = 39, ANJ = 2.405 
- AFT = 63.6, AL = 27.6, ANJ = 2.16

Step 1: Apply the Shortest Processing Time sequencing rule. Order jobs by processing time from smallest to largest: V (18), W (20), Y (27), X (43), Z (55).
- This ordering is obtained by sorting the given processing times: 18, 20, 27, 43, 55.

Step 2: Compute completion (flow) times under the SPT order. Cumulative processing yields:
- V completes at 18 hours
- W completes at 18 + 20 = 38 hours
- Y completes at 38 + 27 = 65 hours
- X completes at 65 + 43 = 108 hours
- Z completes at 108 + 55 = 163 hours

Step 3: Calculate Total Flow Time and AFT. Total flow time is the sum of completion times: 18 + 38 + 65 + 108 + 163 = 392 hours. With 5 jobs, AFT = 392 / 5 = 78.4 hours.

Step 4: Calculate Lateness for each job, using lateness defined as max(0, completion time − due date) (i.e., lateness cannot be negative, common in scheduling problems when measuring delay). The due dates are: V 24, W 35, Y 35, X 46, Z 63. Compute individual latenesses:
- V: max(0, 18 − 24) = 0
- W: max(0, 38 − 35) = 3
- Y: max(0, 65 − 35) = 30
- X: max(0, 108 − 46) = 62
- Z: max(0, 163 − 63) = 100
Sum of latenesses = 0 + 3 + 30 + 62 + 100 = 195. Therefore AL = 195 / 5 = 39.

Step 5: Compute ANJ (Average Number of Jobs in the system) as Total Flow Time divided by Total Processing Time. Total processing time is 18 + 20 + 27 + 43 + 55 = 163. Thus ANJ = 392 / 163 ≈ 2.405.

Now, evaluate each option against the calculations:
- Option 1: AFT 50, AL 8, ANJ 4.26. The AFT 50 is inconsistent with the calculated 78.4; AL 8 is far from 39, and ANJ 4.26 is inconsistent with 2.405. This option does not align with the SPT results.
- Option 2: AFT 8.5, AL 4.26, ANJ 1.96. All three figures are far from the computed values; AFT 8.5 is far too low given the total processing time and schedule, so this is not correct.
- Option 3: AFT 26.3, AL 12, ANJ 3.24. Again, the AFT and AL do not match the computed results, while ANJ 3.24 is closer but not equal to 2.405, so this option is incorrect.
- Option 4: AFT 78.4, AL 39, ANJ 2.405. All three figures match the calculations when using the SPT order and the lateness definition described; AFT aligns with 78.4, AL with 39, and ANJ with approximately 2.405.
- Option 5: AFT 63.6, AL 27.6, ANJ 2.16. This set is inconsistent with the derived completion times and lateness sums; neither AFT nor AL nor ANJ match the calculated values.

Summary: The SPT sequence yields the completion times 18, 38, 65, 108, and 163 hours, producing a total flow time of 392 hours and an AFT of 78.4. Using nonnegative lateness, the lateness values are 0, 3, 30, 62, and 100, summing to 195, giving an AL of 39. The total processing time is 163, so ANJ is 392/163 ≈ 2.405. Therefore, among the provided options, the one that exactly matches these results is the fourth option, though this analysis presents the numerical steps and rationale leading to that conclusion without stating it directly as the final choice.

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