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Operations Management Session 27: Project Management Scheduling the Project From Action Plan and WBS to Gantt chart and project network. Gantt Chart Project Network Activity-on-arrow Activity-on-node CPM and PERT Risk analysis involves determining the likelihood that a project can be completed on time Statistics Simulation Session 27 Operations Management 2 Scheduling the Project Session 27 Operations Management 3 History Late 1950s Critical Path Method (CPM) Dupont De Nemours Inc. developed the method Deterministic activity durations Program Evaluation and Review Technique (PERT) Session 27 U.S. Navy, Booz-Allen Hamilton, and Lockeheed Aircraft Probabilistic activity durations Operations Management 4 Activity The Language of PERT/CPM Task or set of tasks Takes time and needs resources Precedence Relationships The immediate predecessor activities Event Completion of one or more activities (to allow the next activity or activities to start) Zero duration, zero resource Milestones Significant events – showing completion of a significant portion of the project Session 27 Operations Management 5 The Language of PERT/CPM Network Diagram of nodes connected by directed arcs Shows technological relationships among activities Path A set of connected activities such that each activity on both sides is connected to one and only one other activity (with exception!) . Critical Path A path where a delay in any of its activities will delay the project The longest path on the network The shortest time to complete the project Critical Time The total time to complete all activities on the critical path Session 27 Operations Management 6 Two Types of Network Diagrams Activity-on-Arrow Network (Arrow Diagramming Method) Easier to show events and milestones More compatible with network theory techniques Sometimes requires dummy (artificial) activities Activity-on-Node Network (Precedence Diagramming Method) Easier representation No dummy activity Session 27 Operations Management 7 Activity on Arrow Network An Activity is an arc with two nodes at its beginning and its end a b c d Session 27 Operations Management 8 AoA: Activity Predecessors A list of immediate predecessors is needed. Task Predecessor a -- b a c b Task Predecessor a --- b -- c b d a Session 27 a b c a d b c Operations Management 9 AoA: Activity Predecessors Task Predecessor a --- b a c a d a Task Predecessor a --- b -- c -- d a,b,c Session 27 b a c d a d b c Operations Management 10 AoA May Need Dummy Activity Two activities have the same starting and ending nodes A single activity connects to two or more nodes Task Predecessor a --- b a c a d b,c b d a c Try this: a,b c and a,d e Session 27 Operations Management 11 AoA: A Power Plant Construction Project Task Description Predecessor a Design & engineering --- b Select site a c Select vendor a d Select personnel a e Prepare site b f Manufacture generator c g Prepare operation manual c h Install generator e,f i Train operators d,g j Obtain license h,i Session 27 Operations Management 12 AoA: A Power Plant Construction Project b a e c d Session 27 f g h i Operations Management j a:b:a c:a d:a e: b f:c g:c h:e,f i: d,g j: h,i 13 Draw AoN Network a:b:a c:a d:a e: b f:c g:c h:e,f i: d,g j: h,i Session 27 Operations Management 14 Draw AOA Activity Predecessor Duration a 6 b 2 c 5 d a 4 e a 3 f c 8 g b,e,f 9 h c 7 i b,e,f 4 j h 9 k d,i 6 Session 27 Operations Management 15 Transform into AON 2 b=2 1 3 Session 27 d=4 4 h=9 6 g=9 5 Operations Management 7 a:b:c:d:a e: a f:c g:b,e,f h:c i: b,e,f j: h k:d,i 16 Draw AoN Network a:b:c:d:a e: a f:c g:b,e,f h:c i: b,e,f j: h k:d,i Session 27 Operations Management 17 Critical Path and Critical time The critical path is the shortest time in which a project can be completed If a critical activity is delayed, the entire project will be delayed. There may be more than one critical path. Brute force approach to finding critical path: 1. identify all possible paths from start to finish 2. sum up duration of activities on each path 3. largest total indicates critical path Session 27 Operations Management 18 Critical Path Method: The Network 4 6 A1 A3 S 4 3 A4 A6 3 2 A2 A5 Session 27 E Find the Critical Path. Operations Management 19 Critical Path Method: Paths 4 6 A1 A3 S 3 A2 How many path? E 4 3 A4 A6 2 10 11 8 A5 Critical Path is the longest path. It is the shortest time to complete the project Session 27 Operations Management 20 Forward Path; Earliest Starts 6 4 4 A1 00 4 A3 4 10 10 3 4 0 S A4 4 0 4 3 0 A2 0 Session 27 8 8 5 A6 8 11 A5 3 3 2 E 11 11 3 5 Operations Management 21 Forward Path Max = 30 10 30 20 35 35 35 5 Session 27 Operations Management 22 Backward Path; Latest Starts 0 4 4 A1 00 00 S 03 5 4 6 11 5 4 4 4 0 3 4 6 2 6 6 A2 Session 27 3 10 8 8 3 E 11 10 8 8 A5 3 0 4 A4 4 3 11 A3 4 11 8 8 5 3 1111 A6 8 11 8 5 Operations Management 23 11 Backward Path 30 30 30 Min = 35 35 45 30 5 Session 27 Operations Management 24 Activity Slack Slack, or float: The amount of time a noncritical task can be delayed without delaying the project Slack—LFT – EFT or LST – EST EST—Earliest Start Time Largest EFT of all predecessors EFT—Earliest Finish Time EST + duration for this task LFT—Latest Finish Time Smallest LST of following tasks LST—Latest Start Time LFT – duration for this task Session 27 Operations Management 25 Computing Slack Times EST EFT Task = duration slack = xxxx LST Session 27 LFT Operations Management 26 Critical Path, Slacks 0 4 4 5 11 11 A3 A1 0 6 4 4 4 S 4 E 10 8 11 8 A4 3 3 6 4 6 A2 0 Session 27 2 3 11 A6 8 8 8 11 A5 3 3 5 Operations Management 27 Slack Times Example Task Pred. Dur. Task Pred. Dur. a -- 4 g c,d 1 b -- 3 h e 4 c a 3 i f 5 d a 2 j e,g 6 e b 6 k h,i 1 f b 4 For each task, compute ES, EF, LF, LS, slack Session 27 Operations Management 28 Slack Times Example c=3 slack= LST Task=dur slack=xxx EST g=1 slack= a=4 slack= EFT j=6 slack= d=2 slack= Finish Start e=6 slack= h=4 slack= b=3 slack= Session 27 LFT k=1 slack= f=4 i=5 slack= slack= Operations Management 29 Activity Times in PERT Optimistic (a) Activity duration to be ≤ a has 1% probability. ≥ a has 99% probability Pessimistic (b) Activity duration to be ≥ b has 1% probability ≤ b has 99% probability Most likely (m) The mode of the distribution All possible task durations (or task costs) can be represented by statistical distributions Session 27 Operations Management 30 Beta Distribution: The Probability Distribution of Activity Times Session 27 Operations Management 31 Activity Expected Time and Variance Mean, “expected time” TE = (a + 4m + b)/6 Standard deviation, = (b-a)/6 Variance 2 = [(b-a)/6]2 Session 27 Operations Management 32 95% & 90% Levels If we replace 99% with 95% or 90% levels Activity duration to be ≤ a has 5% probability Activity duration to be ≥ b has 5% probability (b a ) 3 .3 Activity duration to be ≤ a has 10% probability Activity duration to be ≥ b has 10% probability (b a ) 2 .6 Session 27 Operations Management 33 Probability of Completing the Critical Path on Time We assume the various activities are statistically independent of each other Individual variances (and mean) of the activities on a path can then be summed to find the variance (mean) of the path Determine the mean and standard deviation of the critical path Compute the probability of critical path being ≤ a Session 27 Operations Management 34 The Probability of Completing the Critical Path on Time Z DCP m CP 2 CP DCP = the desired completion date of the critical path mCP= the sum of the TE for the activities on the critical path 2CP = the sum of the variances of the activities on the critical path Given Z, the probability of having the standard normal variable being ≤ Z is the probability of completing the project in a time ≤ D Session 27 Ardavan Asef-Vaziri Operations Management 35 Selecting Risk and Finding D Select the probability of meeting the completion date and solve for the desired date, D DCP mCP Z CP Using the probability, you can compute Z and then solve for D 5/25/2017 Session 27 Operations Management 36 Probability of Completing a Project on Time Find all paths in the network Compute mean and standard deviation of each path Compute the probability of completing each path in ≤ the given time Calculate the probability that the entire project is completed within the specified time by multiplying these probabilities together Session 27 Operations Management 37 Critical Path Method: Paths Suppose all activities have beta distribution 4,1 A1 6,2 A3 E 3,1 4,1 S A4 3,0.5 A2 A6 2,0.5 A5 10 11 8 The first number is the mean; the second is standard deviation. Session 27 Operations Management 38 Probability of Completing CP in 12 days What is the probability of competing the critical path in a maximum of 12 days? DCP = the desired completion date of the critical path mCP= 4+4+3 = 11 2CP = 12+12+12 = 3 CP = 1.73 Z DCP mCP CP 12 11 0.58 1.73 Z= 0.58 P(z≤0.58) = 0.72 Session 27 Operations Management 39 Selecting Risk and Finding CP Time With a probability of 90%, in how many days will the CP be completed? From Standard Normal Table Z 90% = 1.28 DCP mCP Z CP mCP 11, CP 1.73, Z %90 1.28 DCP 11 1.28 1.73 11 2.21 13.21 Session 27 Operations Management 40 Probability of Completing The Project in 12 days The probability of completing the critical path in not more than 12 days was 0.72. We need to compute this probability for blue path and green path too, and then multiply these probabilities mCP= 6+4 = 10 2CP = 12+22 = 5 Z= (12-10)/2.24 = 0.89 P(z≤0.89) = 0.81 CP = 2.24 mCP= 3+2+3 = 8 2CP = 0.52+0.52+12 = 1.22 Z= (12-8)/1.1 = 3.63 P(z≤3.63) ≈ 1 CP = 1.1 The probability of competing the Project in not more than 12 days is 0.72×0.81×1 = 0.58 Session 27 Operations Management 41