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Projects: Critical Paths Dr. Ron Lembke Operations Management
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PERT & CPM Network techniques Developed in 1950’s CPM by DuPont for chemical plants PERT by U.S. Navy for Polaris missile Consider precedence relationships & interdependencies Each uses a different estimate of activity times
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Completion date? On schedule? Within budget? Probability of completing by...? Critical activities? Enough resources available? How can the project be finished early at the least cost? Questions Answered by PERT & CPM
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PERT & CPM Steps Identify activities Determine sequence Create network Determine activity times Find critical path Earliest & latest start times Earliest & latest finish times Slack
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Activity on Node (AoN) 2 4? Years Enroll Receive diploma Project: Obtain a college degree (B.S.) 1 month Attend class, study etc. 1 1 day 3
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Activity on Arc (AoA) 4,5 ? Years Enroll Receive diploma Project: Obtain a college degree (B.S.) 1 month Attend class, study, etc. 1 1 day 234
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AoA Nodes have meaning Graduating Senior Applicant Project: Obtain a college degree (B.S.) 1 Alum 234 Student
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Liberal Arts Sidebar Alum = ? Alumnus Alumna Alumni Alumnae Alumni
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Network Example You’re a project manager for Bechtel. Construct the network. ActivityPredecessors A-- BA CA DB EB FC GD HE, F
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Network Example - AON A CEFBDGHZ
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Network Example - AOA 2 4 5136879 A C F E B D H G
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AOA Diagrams 231 A C B D A precedes B and C, B and C precede D 241 A C B D 354 Add a phantom arc for clarity.
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Critical Path Analysis Provides activity information Earliest (ES) & latest (LS) start Earliest (EF) & latest (LF) finish Slack (S): Allowable delay Identifies critical path Longest path in network Shortest time project can be completed Any delay on activities delays project Activities have 0 slack
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Critical Path Analysis Example
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Network Solution A A E E D D B B C C F F G G 1 6 2 3 1 43
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Earliest Start & Finish Steps Begin at starting event & work forward ES = 0 for starting activities ES is earliest start EF = ES + Activity time EF is earliest finish ES = Maximum EF of all predecessors for non-starting activities
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Activity A Earliest Start Solution For starting activities, ES = 0. A A E E D D B B C C F F G G 1 6 2 3 1 43
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Earliest Start Solution A A E E D D B B C C F F G G 1 6 2 3 1 43
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Latest Start & Finish Steps Begin at ending event & work backward LF = Maximum EF for ending activities LF is latest finish; EF is earliest finish LS = LF - Activity time LS is latest start LF = Minimum LS of all successors for non-ending activities
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Earliest Start Solution A A E E D D B B C C F F G G 1 6 2 3 1 4 3
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Latest Finish Solution A A E E D D B B C C F F G G 1 6 2 3 1 43
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Compute Slack
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Critical Path A A E E D D B B C C F F G G 1 6 2 3 1 43
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New notation Compute ES, EF for each activity, Left to Right Compute, LF, LS, Right to Left C 7 LSLF ESEF
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Exhibit 6 A 21 E 5 D 2 B 5 C 7 F 8 G 2
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Exhibit 6 A 21 E 5 D 2 B 5 C 7 F 8 G 2 21282836 3638 2833 26282126 021 F cannot start until C and D are done. G cannot start until both E and F are done.
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Exhibit 6 A 21 E 5 D 2 B 5 C 7 F 8 G 2 2126 021 26283136 3638 21282836 21282836 3638 2833 26282126 021 E just has to be done in time for G to start at 36, so it has slack. D has to be done in time for F to go at 28, so it has no slack.
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Gantt Chart - ES 0510152025303540 A B C D E F G
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Solved Problem 2 A 1 B 4 C 3 D 7 E 6 F 2 H 9 I 4 G 7
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Solved Problem 2 A 1 0101 0101 B 4 1515 1515 C 3 6969 1414 D 7 2929 1818 E 6 511 F 2 911 810 H 9 918 817 I 4 1822 G 7 1118
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Summary Activity on Node representation Calculated –ES, EF for all activities –LS, LF for all activities (working backwards) –Slack for each activity Identified critical path(s)
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