1. © J. Christopher Beck 20051 Lecture 5: Project Planning 2

2. © J. Christopher Beck 2005 2 Outline Time/Cost Tradeoffs Linear and non-linear Adding Workforce Constraints Slides borrowed from Twente & Iowa See Pinedo CD

3. © J. Christopher Beck 2005 3 Time/Cost Trade-Offs What if you could spend money to reduce the job duration More money  shorter processing time Run machine at higher speed

4. © J. Christopher Beck 2005 4 Linear Costs Money Processing time Marginal cost

5. © J. Christopher Beck 2005 5 Problem Spend money to reduce processing times so as to minimize: “Overhead” cost Cost per activity

6. © J. Christopher Beck 2005 6 Solution Methods Objective: minimum cost of project Time/Cost Trade-off Heuristic Good schedules Works also for non-linear costs Linear programming formulation Optimal schedules Non-linear version not easily solved

7. © J. Christopher Beck 2005 7 Source (dummy) node Sink node Minimal cut set Cut set Sources, Sinks, & Cuts

8. © J. Christopher Beck 2005 8 Step 1: Set all processing times at their maximum Determine all critical paths Construct the graph G cp of critical paths Time/Cost Trade-off Heuristic

9. © J. Christopher Beck 2005 9 Step 2: Determine all minimum cut sets in G cp Consider those sets where all processing times are larger than their minimum If no such set STOP; otherwise continue to Step 3 Time/Cost Trade-off Heuristic

10. © J. Christopher Beck 2005 10 Time/Cost Trade-Off Heuristic Step 3: For each minimum cut set: Compute the cost of reducing all processing times by one time unit. Take the minimum cut set with the lowest cost If this is less than the overhead per time unit go on to Step 4; otherwise STOP

11. © J. Christopher Beck 2005 11 Time/Cost Trade-Off Heuristic Step 4: Reduce all processing times in the minimum cut set by one time unit Determine the new set of critical paths Revise graph G cp and go back to Step 2

12. © J. Christopher Beck 2005 12 Example 4.4.2 Overhead: c o = 6 (cost of project per time unit)

13. © J. Christopher Beck 2005 13 Step 1: Maximum Processing Times, Find G cp 1 2 3 69 58 47 1110121413

14. © J. Christopher Beck 2005 14 Step 1: Maximum Processing Times, Find G cp 1 2 3 69 58 47 1110121413 Cost = overhead + job costs = c o * C max + Σc a j = 6 * 56 + 350 = 686

15. © J. Christopher Beck 2005 15 1 3 69 111214 c 1 =7 c 3 =4 c 6 =3c 9 =4 c 11 =2 c 12 =2 c 14 =8 Minimum cut set with lowest cost Cut sets: {1},{3},{6},{9}, {11},{12},{14}. Step 2 & 3: Min. Cut Sets in G cp & Lowest Cost

16. © J. Christopher Beck 2005 16 Step 4 & 1: Reduce Processing Time for Each Job by 1 1 2 3 69 58 47 1110121413 Cost = overhead + processing = c 0 * C max + Σjob costs = 6 * 55 + 352 = 682

17. © J. Christopher Beck 2005 17 Step 2 & 3: Min. Cut Sets in G cp & Lowest Cost 1 3 69 111214 c 1 =7 c 3 =4 c 6 =3c 9 =4 c 11 =2 c 12 =2 c 14 =8 13 Cut sets: {1},{3},{6},{9}, {11},{12,13},{14}. c 13 =4 Minimum cut set with lowest cost

18. © J. Christopher Beck 2005 18 1 3 69 111214 c 1 =7 c 3 =4 c 6 =3c 9 =4 c 11 =2 c 12 =2 c 14 =8 13 c 13 = 4 Next 3 iterations reduce processing time from 7 to 4 Cost = overhead + processing = c o * C max + Σjob costs = 6 * 52 + 355 = 667 Next 3 Iterations

19. © J. Christopher Beck 2005 19 1 3 69 111214 c 1 =7 c 3 =4 c 6 =3c 9 =4 c 11 =2 c 12 =2 c 14 =8 13 Reduce processing time next on job 6 c 13 = 4 Step 1,2, & 3 Q: why not 12?

20. © J. Christopher Beck 2005 20 After More Iterations … 1 3 69 111214 c 1 =7 c 3 =4 c 6 =3c 9 =4 c 11 =2 c 12 =2 c 14 =8 13 c 13 = 4 247 10 c 2 =2 c 4 =3 c 7 =4 c 10 =5

21. © J. Christopher Beck 2005 21 Linear Programming Formulation The heuristic does not guarantee optimum See example 4.4.3 Here total cost is linear so use LP Want to minimize

22. © J. Christopher Beck 2005 22 Minimize subject to earliest start time of job k processing time of job k Linear Program

23. © J. Christopher Beck 2005 23 Can Also Have Non-linear Costs Arbitrary function c j (p j ) → cost of setting job j to processing time p j LP doesn’t work! See Section 4.5 A question I like: Given processing times and c j (p j ), which algorithm would you use (heuristic or LP)?

24. © J. Christopher Beck 2005 24 What If Jobs Require Resources? Back to fixed durations Without resources → easy With resources → hard Resource Constraint Project Scheduling Problem (RCPSP)

25. © J. Christopher Beck 2005 25 32145678 3 2 1 4 5 6 1 2 3 4 5 6 RCPSP Example Resource requirements

26. © J. Christopher Beck 2005 26 32145678 3 2 1 4 5 6 910 1 2 34 56 What if R 1 = 4?

27. © J. Christopher Beck 2005 27 RCPSP n: jobs j=1,…,n N: resources i=1,…,N R k :availability of resource k p j :duration of job j R kj : requirement of job j for resource k P j :(immediate) predecessors of job j Minimize C max

28. © J. Christopher Beck 2005 28 RCPSP Example