1. Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Project Management Operations Management - 5 th Edition Chapter 9 Roberta Russell & Bernard W. Taylor, III

2. Copyright 2006 John Wiley & Sons, Inc.9-2 Lecture Outline   Review of Deterministic Setting   Probabilistic Activity Times   Project Crashing and Time-Cost Trade-off

3. Copyright 2006 John Wiley & Sons, Inc.9-3 Project Scheduling  Steps Define activities Define activities Sequence activities Sequence activities Estimate time Estimate time Develop schedule Develop schedule  Techniques Gantt chart Gantt chart CPM CPM PERT PERT Microsoft Project Microsoft Project

4. Copyright 2006 John Wiley & Sons, Inc.9-4 Gantt Chart   Graph or bar chart with a bar for each project activity that shows passage of time  Provides visual display of project schedule   Slack amount of time an activity can be delayed without delaying the project

5. Copyright 2006 John Wiley & Sons, Inc.9-5 |||||||||| Activity Design house and obtain financing Lay foundation Order and receive materials Build house Select paint Select carpet Finish work 0246810 MonthMonth 13579135791357913579 Example of Gantt Chart

6. Copyright 2006 John Wiley & Sons, Inc.9-6 CPM/PERT  Critical Path Method (CPM) DuPont & Remington-Rand (1956) DuPont & Remington-Rand (1956) Deterministic task times Deterministic task times Activity-on-node network construction Activity-on-node network construction  Project Evaluation and Review Technique (PERT) US Navy, Booz, Allen & Hamilton US Navy, Booz, Allen & Hamilton Multiple task time estimates Multiple task time estimates Activity-on-arrow network construction Activity-on-arrow network construction

7. Copyright 2006 John Wiley & Sons, Inc.9-7 AON Network for House Building Project 1 3 2 2 4 3 3 1 5 1 6 1 7 1Start Design house and obtain financing Order and receive materials Select paint Select carpet Lay foundation Build house Finish work

8. Copyright 2006 John Wiley & Sons, Inc.9-8 1 3 2 2 4 3 3 1 5 1 6 1 7 1Start Critical Path  Critical path Longest path through a network Longest path through a network Minimum project completion time Minimum project completion time A:1-2-4-7 3 + 2 + 3 + 1 = 9 months B:1-2-5-6-7 3 + 2 + 1 + 1 + 1 = 8 months C:1-3-4-7 3 + 1 + 3 + 1 = 8 months D:1-3-5-6-7 3 + 1 + 1 + 1 + 1 = 7 months

9. Copyright 2006 John Wiley & Sons, Inc.9-9 Mode Configuration 103 303 Activity number Activity duration Earliest start Latest start Earliest finish Latest finish

10. Copyright 2006 John Wiley & Sons, Inc.9-10 Probabilistic Time Estimates   Beta distribution a probability distribution traditionally used in CPM/PERT a = optimistic estimate m = most likely time estimate b = pessimistic time estimate where Mean (expected time): t = a + 4 m + b 6 Variance:  2 = b - a 6 2

11. Copyright 2006 John Wiley & Sons, Inc.9-11 Why Use the Beta Distribution?  Mean and variance can be estimated with 3 time estimates  Continuous, but no predetermined shape  Tradition

12. Copyright 2006 John Wiley & Sons, Inc.9-12 Examples of Beta Distributions P(time) Time amtbamtb m = t Time Time ba

13. Copyright 2006 John Wiley & Sons, Inc.9-13 Project Network with Probabilistic Time Estimates: Example 1 9,12,15 2 7,8,12 4 11,12,20 3 3,4,5 5 2,4,5 6 2,4,6 7 3,4,7 Start Design house and obtain financing Order and receive materials Select paint Select carpet Lay foundation Build house Finish work

14. Copyright 2006 John Wiley & Sons, Inc.9-14 Activity Time Estimates 1 91215121.00 1 91215121.00 278128.50.69 278128.50.69 334540.11 334540.11 4 11122013.172.25 4 11122013.172.25 5 2453.830.25 5 2453.830.25 624640.44 624640.44 73474.330.44 73474.330.44 TIME ESTIMATES (WKS)MEAN TIMEVARIANCE ACTIVITY ambt б 2

15. Copyright 2006 John Wiley & Sons, Inc.9-15 Activity Early, Late Times, and Slack ACTIVITY t б  ESEFLSLFS 1121.000120120 1121.000120120 28.50.691220.51220.50 28.50.691220.51220.50 340.11121616.520.54.5 340.11121616.520.54.5 4 13.172.2520.533.720.533.70 4 13.172.2520.533.720.533.70 5 3.830.2520.524.325.829.75.4 5 3.830.2520.524.325.829.75.4 640.4424.328.329.733.75.4 640.4424.328.329.733.75.4 74.330.4433.73833.7380 74.330.4433.73833.7380

16. Copyright 2006 John Wiley & Sons, Inc.9-16 Critical Path Earliest, Latest, and Slack 10 12 30 2 20.5 2 1220.5 3 1216 1 16.520.5 5 24.3 1 25.8 29.7 4 20.533.7 3 20.533.7 6 24.328.3 1 29.733.7 7 38 1 33.738 Start Design house and obtain financing Select paint Lay foundation Select carpet Build house Finish work Order and receive materials

17. Copyright 2006 John Wiley & Sons, Inc.9-17  2 = б 1 2 + б 2 2 + б 4 2 + б 7 2  2 = 1.00 + 0.694 + 2.25 + 0.444 = 4.389 weeks Total project variance

18. Copyright 2006 John Wiley & Sons, Inc.9-18 Probabilistic Network Analysis Determine probability that project is completed within specified time where  = t p = project mean time  =project standard deviation x =proposed project time Z =number of standard deviations x is from mean Z =Z =Z =Z = x -  

19. Copyright 2006 John Wiley & Sons, Inc.9-19 Key Assumptions  Activity times are statistically independent Allows summation of individual expected activity times and variances to get the overall projects expected time and variance Allows summation of individual expected activity times and variances to get the overall projects expected time and variance  Network mean and variance normally distributed Book cites CLT (is it correct?) Book cites CLT (is it correct?)

20. Copyright 2006 John Wiley & Sons, Inc.9-20 Normal Distribution Of Project Time  = t p Timex ZZ Probability

21. Copyright 2006 John Wiley & Sons, Inc.9-21 House Example – 40 Weeks What is the probability that the project is completed within 40 weeks?  2 = 4.389 weeks  = 4.389  = 2.095 weeks Z = = = 0.9547 x -   40 - 38 2.095 From Table A.1, (appendix A) a Z score of 0.9547 corresponds to a probability of ~0.3289. Thus P(40) = 0.3289 + 0.5000 = 0.8289  = 38 Time (weeks) x = 40 P( x  40 weeks)

22. Copyright 2006 John Wiley & Sons, Inc.9-22 House Example – 35 Weeks  = 38 Time (weeks) x = 35 P( x  35 weeks) What is the probability that the project is completed within 35 weeks?  2 = 4.389 weeks  = 4.389  = 2.095 weeks Z = = = -1.43 x -   35 - 38 2.095 From Table A.1 (appendix A) a Z score of -1.43 corresponds to a probability of 0.4236. Thus P(35) = 0.5000 - 0.4236 = 0.0764

23. Copyright 2006 John Wiley & Sons, Inc.9-23 Project Crashing   Crashing reducing project time by expending additional resources   Crash time an amount of time an activity is reduced   Crash cost cost of reducing activity time  Goal reduce project duration at minimum cost

24. Copyright 2006 John Wiley & Sons, Inc.9-24 1 12 2 8 4 3 4 5 4 6 4 7 4 Project Crashing: Example

25. Copyright 2006 John Wiley & Sons, Inc.9-25 Project Crashing: Example (cont.) $7,000 – $6,000 – $5,000 – $4,000 – $3,000 – $2,000 – $1,000 – – ||||||| 02468101214Weeks Normal activity Normal time Normal cost Crash time Crashed activity Crash cost Slope = crash cost per week

26. Copyright 2006 John Wiley & Sons, Inc.9-26 Normal Activity and Crash Data TOTAL NORMALCRASHALLOWABLECRASH TIMETIMENORMALCRASHCRASH TIMECOST PER ACTIVITY(WEEKS)(WEEKS)COSTCOST(WEEKS)WEEK 1127$3,000$5,0005$400 2852,0003,5003500 3434,0007,00013,000 412950,00071,00037,000 5415001,1003200 6415001,1003200 74315,00022,00017,000 $75,000$110,700

27. Copyright 2006 John Wiley & Sons, Inc.9-27 1 12 2 8 3 4 5 4 6 4 7 4 $400 $500 $3000 $7000 $200 $7000 12 4 Project Duration: 36 weeks FROM … 1 7 2 8 3 4 5 4 6 4 7 4 $400 $500 $3000 $7000 $200 $7000 12 4 Project Duration: 31 weeks Additional Cost: $2000 TO…

28. Copyright 2006 John Wiley & Sons, Inc.9-28  Crashing costs increase as project duration decreases  Indirect costs increase as project duration increases  Reduce project length as long as crashing costs are less than indirect costs Time-Cost Relationship

29. Copyright 2006 John Wiley & Sons, Inc.9-29 Time-Cost Tradeoff Cost ($) Project duration CrashingTime Minimum cost = optimal project time Total project cost Indirect cost Direct cost