Session 27 Operations Management2 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 Management3 Scheduling the Project
Session 27 Operations Management4 History Late 1950s Critical Path Method (CPM) Dupont De Nemours Inc. developed the method Deterministic activity durations Program Evaluation and Review Technique (PERT) U.S. Navy, Booz-Allen Hamilton, and Lockeheed Aircraft Probabilistic activity durations
Session 27 Operations Management5 The Language of PERT/CPM Activity 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 Management6 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 Management7 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 Management8 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 Management9 AoA: Activity Predecessors A list of immediate predecessors is needed. TaskPredecessor abcabc -- a b abc TaskPredecessor abcdabcd --- -- b a a b d c
Session 27 Operations Management10 AoA: Activity Predecessors TaskPredecessor abcdabcd --- a a b d c TaskPredecessor abcdabcd --- -- a,b,c a b d c
Session 27 Operations Management11 AoA May Need Dummy Activity Two activities have the same starting and ending nodes A single activity connects to two or more nodes TaskPredecessor abcdabcd --- a b,c a b d c Try this: a,b c and a,d e
Session 27 Operations Management12 AoA: A Power Plant Construction Project TaskDescriptionPredecessor abcdefghijabcdefghij Design & engineering Select site Select vendor Select personnel Prepare site Manufacture generator Prepare operation manual Install generator Train operators Obtain license --- a b c e,f d,g h,i
Session 27 Operations Management13 a b cf d g e h i j AoA: A Power Plant Construction Project 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 Management16 Transform into AON 12 3 4 5 6 7 a = 6 b = 2g = 9 h = 9 k = 6 d = 4 c = 5 e = 3 i = 4 f = 8 j = 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
Session 27 Operations Management17 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 Management18 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 Management19 Critical Path Method: The Network 4 3 6 4 2 3 E A1 A3 A4 A6 A5 A2 S Find the Critical Path.
Session 27 Operations Management20 Critical Path Method: Paths A1A3A4A6A5A2 4 3 6 4 2 3 ES 10 Critical Path is the longest path. It is the shortest time to complete the project 118 How many path?
Session 27 Operations Management25 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 Management30 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 Management31 Beta Distribution: The Probability Distribution of Activity Times
Session 27 Operations Management32 Activity Expected Time and Variance Mean, “expected time” T E = (a + 4m + b)/6 Standard deviation, = (b-a)/6 Variance 2 = [(b-a)/6] 2
Session 27 Operations Management33 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 Activity duration to be ≤ a has 10% probability Activity duration to be ≥ b has 10% probability
Session 27 Operations Management34 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 Management35 Ardavan Asef-Vaziri The Probability of Completing the Critical Path on Time D CP = the desired completion date of the critical path CP = the sum of the T E for the activities on the critical path 2 CP = 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 Operations Management36 5/2/2015 Selecting Risk and Finding D Select the probability of meeting the completion date and solve for the desired date, D Using the probability, you can compute Z and then solve for D
Session 27 Operations Management37 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 Management38 Critical Path Method: Paths A1A3A4A6A5A2 4,1 3,0.5 6,2 4,1 2,0.5 3,1 ES 10 The first number is the mean; the second is standard deviation. 118 Suppose all activities have beta distribution
Session 27 Operations Management39 Probability of Completing CP in 12 days D CP = the desired completion date of the critical path CP = 4+4+3 = 11 2 CP = 1 2 + 1 2 + 1 2 = 3 CP = 1.73 Z= 0.58 P(z≤0.58) = 0.72 What is the probability of competing the critical path in a maximum of 12 days?
Session 27 Operations Management40 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
Session 27 Operations Management41 Probability of Completing The Project in 12 days CP = 6+4 = 10 2 CP = 1 2 + 2 2 = 5 CP = 2.24 Z= (12-10)/2.24 = 0.89 P(z≤0.89) = 0.81 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 CP = 3+2+3 = 8 2 CP = 0.5 2 + 0.5 2 + 1 2 = 1.22 CP = 1.1 Z= (12-8)/1.1 = 3.63 P(z≤3.63) ≈ 1 The probability of competing the Project in not more than 12 days is 0.72×0.81×1 = 0.58
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