Part Five: Problem Set I Key

Transcription

1 Mat 114 Part Five: Problem Set I Key 1. Consider the following order-requirement digraph, where all indicated weights are in hours. a) What is the critical path time? For this we need to backtrack. The final vertex, T7 is assigned the label 3 Backtracking from there we assign T6 the label 5 (2+3) and T5 the label 4 (1+3) Backtracking from T5 gives T2 the label 6 (2+4); backtracking from T6 gives T4 the label 7 (2+5). To label T3 we backtrack from T5 and get 8 (4+4) and backtrack from T6 to get 9 (4+5). T3 gets the larger number and so is assigned the label 9. Finally we backtrack to assign T1 a label which is 12. (The larger of 9 from T2, 12 from T3 and 10 from T4). So the backtracking labels are: T1:12, T2:6, T3:9, T4:7, T5:4, T6:5, T7:3. The critical path time is the label assigned to the first vertex, T1. So this is 12 hours. b) What is the critical path? This is the path(s) from start to finish of length 12 which is T1 T3 T6 T7 c) If task T2 is delayed by some time, will this necessarily alter the critical path time? Not necessarily T2 is not on the critical path. A close inspection shows that a 3 hour delay has no affect on the job completion time. d) If task T3 is delayed by some time, will this necessarily alter the critical path time? Yes, definitely T3 is on the critical path. Any delay here delays the whole project.

2 2. Consider the following order-requirement digraph where all indicated weights are given in minutes. a) What is the total project time? Adding up the task times gives 4+2+1=3+3=2 = 15 minutes b) Using this information alone, what is the minimum time the project could take given (i) 1 processor, (ii) 2 processors, (iii) 3processors and (iv) 4 processors. Simply divide by the given number and round up to the nearest integer. So (i) 15 minutes, (ii) 8 minutes, (iii) 5 minutes and (iv) 4 minutes. c) What is the critical path time? Backtracking gives T6:2, T5:5, T4:5, T2:7, T3:6, T1: 11 So the critical path time is 11 minutes. d) Redo part b) in light of the information in c) We now have to use the numbers in part b) but also need to remember that the absolute minimum time is the critical path time of 11 minutes. So we get (i) 15 minutes, (ii) 11 minutes, (iii) 11 minutes and (iv) 11 minutes.

3 3. Consider the order-requirement digraph given in Problem 1. a) Given the priority list T7, T1, T4, T6, T2, T3, T5 use the list-processing algorithm to schedule the tasks on 2 processors. Write the solution using the task labels and idle times (if any) for each processor, separated by At time 0, the first task T7 is not ready. Next on the list is t1 which is ready and so the first processor works on this. Nothing else is ready and so the second processor is idle. After 3 hours, the first processor completes T1 and now T2, T3 and T4 are ready. T4 is the first of these on the list and so the first processor does this; T2 is before T3 and so the second processor works on this. These are both complete two hours later. At this point only T3 is ready and the first processor works on this while the second processor is idle. Four hours later, T3 is done and T5 and T6 are ready. T6 is earlier on the list and so the first processor gets this; the second processor works on T5. An hour later T5 is done and the second processor is idle but the only remaining task T7 is not ready and the second processor is idle. After one more hour the first processor completes T6 and immediately works on T7 leaving the second processor idle until the job is complete. Completed in 14 hours So Processor 1: T1, T4, T3, T6, T7 Processor 2: 3, T2, 4, T5, 4 b) Repeat part b) but using 3 processors. Processor 1: T1, T4, 2, T6, T7 Processor 2: 3,T2,2, T5,4 Processor 3: 3, T3, 5 Completed in 12 hours cannot do better since this is the critical path time.

4 4. Consider the order-requirement digraph given in Problem 1. a) Use the backtracking numbers found in part a of Problem One to come up with the critical path list for this problem. Break ties by using the processor with the smaller number. (So T2 would come before T3 or T5 in the list.) The backtracking numbers are T1:12, T2:6, T3:9, T4:7, T5:4, T6:5, T7:3. Arranging these in descending order gives T1:12, T3:9, T4:7, T2:6, T6:5, T5:4, T7:3. So the critical path list is T1, T3, T4, T2, T6, T5, T7. b) Use the critical path scheduling method to schedule the 7 tasks on 2 processors. Write the So Processor 1 T1, T3, T6, T7 Processor 2 3, T4, T2, T5, 5 Note that this is the best solution. The job has been completed in the critical path time with the fewest number of processors possible. 5. Consider the order-requirement digraph given in Problem 2. a) Use the critical path scheduling method to schedule the 7 tasks on 2 processors. Write the As with problem 4, we need backtracking numbers. We found these in problem 2c): T6:2, T5:5, T4:5, T2:7, T3:6, T1: 11. Arranging these in descending order gives T1:11, T2:7, T3:6, T4:5, T5:5, T6:2. So the critical path list is T1, T2, T3, T4, T5, T6. So Processor 1: T1, T2, T5, T6 Processor 2: 4, T3, T4, 3

5 b) Use the critical path scheduling method to schedule the 7 tasks on 3 processors. Write the Processor 1: T1, T2, T5, T6 Processor 2: 4, T3, 6 Processor 3: 4, T4, 4 c) What is the time savings for using the extra processor? There are no time savings. The solution in b) was optimal. The job was done in the critical path time with the minimum number of processors. Using the third processor just involves more expense for nothing more than slack time! d) Is the company working on this project likely to hire the third processor? What factors would affect their decision? No! The decision can rest on interplay between the extra expense for the third processor and the time savings. For example, there could be a penalty cost for missing a contract deadline. In this case it may be worthwhile to pay for an extra crew.