Projects: Critical Paths Dr. Ron Lembke Operations Management

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

Questions Answered by PERT & CPM 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?

PERT & CPM Steps Identify activities Determine sequence Create network Determine activity times Find critical path Earliest & latest start times Earliest & latest finish times Slack

3 1 2 Activity on Node (AoN) Project: Obtain a college degree (B.S.) Attend class, study etc. Receive diploma Enroll 3 1 2 1 month 4? Years 1 day

1 2 3 4 Activity on Arc (AoA) Project: Obtain a college degree (B.S.) Attend class, study, etc. Receive diploma Enroll 1 2 3 4 1 month 4,5 ? Years 1 day

1 2 3 4 AoA Nodes have meaning Project: Obtain a college degree (B.S.) GraduatingSenior Applicant Student Alum

Liberal Arts Sidebar Alum = ? Alumnus Alumna Alumni Alumnae Alumni

Network Example You’re a project manager for Bechtel. Construct the network. Activity Predecessors A -- B A C A D B E B F C G D H E, F

Network Example - AON D G B A E Z C H F

Network Example - AON F H C A E Z B G D

Network Example - AON E H C G A Z F B D

Network Example - AOA D G 3 6 8 B E A 1 2 5 H 7 9 C F 4

AOA Diagrams A precedes B and C, B and C precede D B A 1 2 3 D 4 C 3 B A 1 2 C 4 D 5 Add a phantom arc for clarity.

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

Critical Path Analysis Example

Network Solution B D E A G 2 6 3 1 1 C F 3 4

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

Activity A Earliest Start Solution D B C F G 1 6 2 3 4 For starting activities, ES = 0.

Earliest Start Solution D B C F G 1 6 2 3 4

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

Earliest Start Solution D B C F G 1 6 2 3 4

Latest Finish Solution D B C F G 1 6 2 3 4

Compute Slack

Critical Path A E D B C F G 1 6 2 3 4

New notation C 7 Compute ES, EF for each activity, Left to Right LS LF Compute ES, EF for each activity, Left to Right Compute, LF, LS, Right to Left

Exhibit 6 C 7 F 8 A 21 G 2 B 5 D 2 E 5

Exhibit 6 21 28 28 36 C 7 F 8 0 21 36 38 A 21 G 2 28 33 21 26 26 28 B 5 D 2 E 5 F cannot start until C and D are done. G cannot start until both E and F are done.

Exhibit 6 21 28 28 36 C 7 F 8 21 28 28 36 0 21 36 38 A 21 G 2 0 21 36 38 28 33 21 26 26 28 B 5 D 2 E 5 21 26 26 28 31 36 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.

Gantt Chart - ES A C F B D E G 0 5 10 15 20 25 30 35 40

Solved Problem 2 B 4 E 6 G 7 A 1 C 3 I 4 F 2 D 7 H 9

Solved Problem 2 B 4 E 6 G 7 C 3 A 1 I 4 F 2 D 7 H 9 1 5 5 11 11 18 1 5 5 11 B 4 E 6 11 18 1 5 5 11 G 7 11 18 0 1 1 4 18 22 A 1 C 3 I 4 0 1 6 9 8 10 18 22 F 2 1 8 9 11 8 17 D 7 H 9 2 9 9 18

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)