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

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

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

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

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

AoA Nodes have meaning Graduating Senior Applicant Project: Obtain a college degree (B.S.) 1 Alum 234 Student

Liberal Arts Sidebar Alum = ? Alumnus Alumna Alumni Alumnae Alumni

Network Example You’re a project manager for Bechtel. Construct the network. ActivityPredecessors A-- BA CA DB EB FC GD HE, F

Network Example - AON A CEFBDGHZ

Network Example - AOA A C F E B D H G

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.

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 A A E E D D B B C C F F G G

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 For starting activities, ES = 0. A A E E D D B B C C F F G G

Earliest Start Solution A A E E D D B B C C F F G G

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 A A E E D D B B C C F F G G

Latest Finish Solution A A E E D D B B C C F F G G

Compute Slack

Critical Path A A E E D D B B C C F F G G

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

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

Exhibit 6 A 21 E 5 D 2 B 5 C 7 F 8 G F cannot start until C and D are done. G cannot start until both E and F are done.

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

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

Solved Problem 2 A B C D E F H I G

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)