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PERT/CPM-1 Project Management with PERT/CPM

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PERT/CPM-2 PERT/CPM PERT : program evaluation and review technique CPM : critical path method Use a project network, Activity-on-Node (AON): –Nodes: activities, or tasks, to be performed –Arcs: show immediate predecessors to an activity –Times: duration times of activities are written next to the node

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PERT/CPM-3 Reliable Construction Company Example Activity list for the Reliable Construction Co. project Activity Activity Description Immediate Predecessors Estimated Duration AExcavate-2 weeks BLay the foundationA4 weeks CPut up the rough wallB 10 weeks DPut up the roofC6 weeks EInstall the exterior plumbingC4 weeks FInstall the interior plumbingE5 weeks GPut up the exterior sidingD7 weeks HDo the exterior paintingE,G9 weeks IDo the electrical workC7 weeks JPut up the wallboardF,I8 weeks KInstall the flooringJ4 weeks LDo the interior paintingJ5 weeks MInstall the exterior fixturesH2 weeks NInstall the interior fixturesK,L6 weeks

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PERT/CPM-4 START A B C ED G H M I F J K L N FINISH 0 2 4 10 6 7 4 7 8 5 4 5 6 0 9 2 Activity Code A. Excavate B. Foundation C. Rough wall D. Roof E. Exterior plumbing F. Interior plumbing G. Exterior siding H. Exterior painting I. Electrical work J. Wallboard K. Flooring L. Interior painting M. Exterior fixtures N. Interior fixtures The project network for the Reliable Construction Co. project

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PERT/CPM-5 Bake a Cake Example TaskImmediate predecessors Task Time A: Buy frosting ingredients—½ hr B: Clean up kitchen—1 hr C: Buy cake ingredients—½ hr D: Prepare frostingA,B¼ hr E: Prepare batter & bakeB,C2 hrs F: Frost cakeD,E½ hr 1/2 A B D E F 1 1/4 2 1/2 Start C 0 Finish

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PERT/CPM-6 Critical Path A path through a project network is a route from START to FINISH. The length of path is the sum of the task times (durations) of the nodes (activities) on the path. The critical path is the longest path. The project duration is the length of the longest path.

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PERT/CPM-7 The paths and lengths through Reliable’s project network PathLength START →A →B →C →D →G →H →M→FINISH START →A →B →C →E →H →M →FINISH START →A →B →C →E →F →J →K →N →FINISH START →A →B →C →E →F →J →L →N →FINISH START →A →B →C →I →J →K →N →FINISH START →A →B →C →I →J →L →N →FINISH 2 + 4 + 10 + 6 + 7 + 9 + 2 =40 weeks 2 + 4 + 10 + 4 + 9 + 2 =31 weeks 2 + 4 + 10 + 4 + 5 + 8 + 4 + 6 =43 weeks 2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 =44 weeks 2 + 4 + 10 + 7 + 8 + 4 + 6 =41 weeks 2 + 4 + 10 + 7 + 8 + 5 + 6 =42 weeks

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PERT/CPM-8 To Find the Critical Path and Slacks 1.Work from top to bottom in the network, calculating –ES = earliest start time for an activity EF = earliest finish time for an activity ES for activity i = largest EF of the immediate predecessors ES = 0 if no immediate predecessors EF = ES + activity duration time 2.Work from bottom to top in the network, calculating –LS = latest start time for an activity LF = latest finish time for an activity LS = LF – activity duration time LF for activity i = smallest LS of the immediate successors LF at Finish = EF at Finish if no immediate successors –Slack = LF - EF = LS - ES If slack is zero, the activity is on the critical path.

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PERT/CPM-9 1.Work down the network calculating ES and EF (ES at Start = 0, EF at Start = 0) 2.Work backward up the network calculating LS and LF (LF and LS at Finish is EF and ES at Finish) D E F 2 ½ Start C 0 ½ Finish S = (ES, LS) F = (EF, LF) Slack = LS – ES = LF - EF A 0 S=() F=() Slack= B 1 ½ ¼ The complete project network showing ES, LS, EF and LF for each activity of the baking example

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PERT/CPM-10 Critical Path is Start →B→E →F →Finish Activity D has slack of 1¾ hours (Start of D could be delayed without affecting total project duration) Also, activities A and C have slack of 2 ¼ and ½ respectively. D E F 2 ½ Start C 0 ½ Finish S = (ES, LS) F = (EF, LF) Slack = LS – ES = LF - EF A 0 S=(3, 3) F=(3 ½, 3 ½) Slack=0 S=(0, ½) F=(½, 1) Slack= ½ S=(0, 0) F=(0, 0) Slack=0 S=(0, 2 ¼ ) F=(½, 2 ¾ ) Slack=2 ¼ B 1 S=(0, 0) F=(1,1) Slack=0 S=(1, 2 ¾) F=(1 ¼, 3) Slack=1 ¾ S=(1, 1) F=(3, 3) Slack=0 S=(3 ½, 3 ½) F=(3 ½, 3 ½) Slack=0 ½ ¼ The complete project network showing ES, LS, EF and LF for each activity of the baking example

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PERT/CPM-11 START A B C E D G H M I F J K L N FINISH 0 2 4 10 6 7 4 7 8 5 4 5 6 0 9 2 S= (0, 0) F= (0, 0) S= (38,42) F= (40,44) S= (33,34) F= (37,38) S= (33,33) F= (38,38) S= (38,38) F= (44,44) S= (25,25) F= (33,33) S= (16,18) F= (23,25) S= (16,20) F= (22,26) S= (22,26) F= (29,33) S= (29,33) F= (38,42) S= (20,20) F= (25,25) S= (16, 16) F= (20, 20) S= (6, 6) F= (16, 16) S= (2, 2) F= (6, 6) S= (0, 0) F= (2, 2) S= (44,44) F= (44,44) S = (ES, LS) F = (EF, LF) The complete project network showing ES, LS, EF and LF for each activity of the Reliable Construction Co. project

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PERT/CPM-12 Slack for Reliable’s activities ActivitySlack (LF - EF)On Critical Path? A0Yes B0 C0 D4No E0Yes F0 G4No H4 I2 J0Yes K1No L0Yes M4No N0Yes

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PERT/CPM-13 The spreadsheet used by MS project for entering the activity list for the Reliable Construction Co. project

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PERT/CPM-14 Incorporate Uncertain Activity Duration Times (Probabilistic) PERT Three-Estimate Approach m = most likely estimate of activity duration time o = optimistic estimate of activity duration time p = pessimistic estimate of activity duration time Assume Beta distribution of activity time Approximately:

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PERT/CPM-15 ActivityOptimistic Estimate o Most Likely Estimate m Pessimistic Estimate p A12321/9 B23 ½841 C6918104 D45 ½1061 E14 ½544/9 F441051 G56 ½1171 H581794 I37 ½971 J39981 K44440 L15 ½751 M12321/9 N55 ½964/9 Expected value and variance of the duration of each activity for Reliable’s project

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PERT/CPM-16 The paths and path lengths through Reliable’s project network when the duration of each activity equals its pessimistic estimate PathLength START →A →B →C →D →G →H →M→FINISH START →A →B →C →E →H →M →FINISH START →A →B →C →E →F →J →K →N →FINISH START →A →B →C →E →F →J →L →N →FINISH START →A →B →C →I →J →K →N →FINISH START →A →B →C →I →J →L →N →FINISH 3 + 8 + 18 + 10 + 11 + 17 + 3 =70 weeks 3 + 8 + 18 + 5 + 17 + 3 =54 weeks 3 + 8 + 18 + 5 + 10 + 9 + 4 + 9 =66 weeks 3 + 8 + 18 + 5 + 10 + 9 + 7 + 9 =69 weeks 3 + 8 + 18 + 9 + 9 + 4 + 9 =60 weeks 3 + 8 + 18 + 9 + 9 + 7 + 9 =63 weeks

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PERT/CPM-17 For a path (typically the critical path), find the mean length (time) µ p and the variance σ p 2 mean length of path µ p = sum of mean activity times on the path ( because E[X+Y] = E[X] + E[Y]) Variance of length of path σ p 2 = sum of variances of activity times on path (because we assume independence: Var [X+Y] = Var [X] + Var [Y] if X,Y independent ) Example: critical path Start →A →B →C →E →F →J →L →N →Finish µ p = 2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44 σ p 2 = 1/9 + 1 + 4 + 4/9 + 1 + 1 + 1 + 4/9 =9

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PERT/CPM-18 Find the probability the project is completed in 47 weeks using an assumption of a Normal distribution

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PERT/CPM-19 Time-Cost Trade-offs If one can expedite the project, use money/resources to reduce task times, what is the best way to allocate money? For an activity, could pay extra to reduce time (crash) How much would it cost to reduce total project duration from 44 weeks to 40 weeks? Which activities should be “crashed”? Could calculate crash cost and crash time for all possible paths – but can also apply LP! Crash cost Normal cost Crash time Normal time Activity duration time $ assume linear

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PERT/CPM-20 Time-cost trade-off data for the activities of Reliable’s project Activity TimeCostMaximum Reduction in Time Crash Cost per Week Saved NormalCrashNormalCrash A2 weeks1 week$180,000$280,0001 week$100,000 B4 weeks2 weeks$320,000$420,0002 weeks$50,000 C10 weeks7 weeks$620,000$860,0003 weeks$80,000 D6 weeks4 weeks$260,000$340,0002 weeks$40,000 E4 weeks3 weeks$410,000$570,0001 week$160,000 F5 weeks3 weeks$180,000$260,0002 weeks$40,000 G7 weeks4 weeks$900,000$1,020,0003 weeks$40,000 H9 weeks6 weeks$200,000$380,0003 weeks$60,000 I7 weeks5 weeks$210,000$270,0002 weeks$30,000 J8 weeks6 weeks$430,000$490,0002 weeks$30,000 K4 weeks3 weeks$160,000$200,0001 week$40,000 L5 weeks3 weeks$250,000$350,0002 weeks$50,000 M2 weeks1 week$100,000$200,0001 week$100,000 N6 weeks3 weeks$330,000$510,0003 weeks$60,000 If do all tasks normal, 44 weeks, cost is 4.55 million. If do all tasks crash, 28 weeks, cost is 6.15 million.

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PERT/CPM-21 Marginal Cost Analysis For small networks, may reduce the project 1 week at a time, and observe the changes. Activity to Crash Crash CostLength of Path ABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN 403143444142 J$ 30,000403142434041 J$ 30,000403141423940 F$ 40,000403140413940 F$ 40,000403139403940

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PERT/CPM-22 Linear Programming to Make Crashing Decisions Let Z = total cost of crashing on any activity x j = reduction in the duration of activity j due to crashing j = A,B,C,…,N x j ≤ maximum reduction time = normal time – crash time y FINISH = project duration, time at which FINISH node is reached y j = start time of activity j y j ≥ y i + normal time i – x i i is an immediate predecessor of j F I J yJyJ y J ≥ y F + 5 - x F y J ≥ y I + 7 - x I yF5yF5 yI7yI7

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PERT/CPM-23 Linear Programming Model

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PERT/CPM-24 The schedule of cumulative project costs when all activities begin at their earliest start times or at their latest start times

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PERT/CPM-25 Time-cost trade-off data for the activities of the baking project ActivityNormal TimeCrash TimeNormal CostCrash Cost A: Buy frosting0.50.2505 B: Clean kitchen1.00.25010 C: Buy cake0.50.2505 D: Prepare frosting0.25 00 E: Prepare batter bake2.01.505 F: Frost cake0.50.2505

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PERT/CPM-26 If all are normal: ADF 1 ¼ BDF 1 ¾ BEF 3 ½ CEF 3 Total time is 3 ½. If all are crashed: ADF ¾ BDF ¾ BEF 2 CEF 2 Total time is 2. B D E F Start C A N C 1/2, 1/4 N C 1/2, 1/4 N C 2, 1 1/2 N C 1/2, 1/4 N C 1, 1/4 N C 1/4, 1/4 Finish The project network showing Normal Time and Crash Time for each activity of the baking project N: Normal time C: Crash time

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PERT/CPM-27 Could solve the crash LP for finish times between 3.5 and 2 to evaluate alternatives 25 16.67 11.67 8.33 5 2.5 022.252.52.7533.253.5 T= y Finish Cost

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PERT/CPM-28 Lasagna Dinner Example TaskTasks that must precede Time A: Buy the mozzarella cheese30 mins B: Slice the mozzarellaA5 mins C: Beat 2 eggs2 mins D: Mix eggs and ricotta cheeseC3 mins E: Cut up onions and mushrooms7 mins F: Cook the tomato sauceE25 mins G: Boil large quantity of water15 mins H: Boil the lasagna noodlesG10 mins I: Drain the lasagna noodlesH2 mins J: Assemble all the ingredientsI, F, D, B10 mins K: Preheat the oven15 mins L: Bake the lasagnaJ, K30 mins

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PERT/CPM-29 Construct project network G H I F J K L 25 15 10 2 30 10 A 30 B 5 C 2 E 7 D 3 Start Finish A→B →J →L 75 * C→D →J →L 45 E→F →J →L 73 G→H →I →J →L 45

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PERT/CPM-30 G H I F J K L 25 15 10 2 30 10 A 30 ES=0 EF=30 B 5 ES=30 EF=35 ES=35 EF=45 C 2 ES=0 EF=2 E 7 ES=0 EF=7 ES=0 EF=15 ES=15 EF=25 D 3 ES=2 EF=5 ES=7 EF=32 ES=25 EF=27 ES=0 EF=15 ES=45 EF=75 EF = ES + activity time (or duration) if no predecessors, ES = 0; otherwise ES = max (EF) (immediate predecessors) work forward through network Start Finish

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PERT/CPM-31 G H I F J K L 25 15 10 2 30 10 A 30 S=(0,0) F=(30,30) Slack=0 B 5 S=(30,30) F=(35,35) Slack=0 S=(35,35) F=(45,45) Slack=0 C 2 S=(0,30) F=(2,32) slack=30 E 7 S=(0,3) F=(7,10) Slack=3 S=(0,10) F=(15,25) Slack=10 S=(15,25) F=(25,35) Slack=10 D 3 S=(2,32) F=(5,35) slack=30 S=(7,10) F=(32,35) Slack=3 S=(25,33) F=(27,35) Slack=8 S=(0,30) F=(15,45) Slack=30 S=(45,45) F=(75,75) Slack=0 S = (ES, LS) F = (EF, LF) Slack = LF-EF=LS-ES Critical path has zero slack: A→B →J →L The complete project network showing ES, LS, EF, LF and Slack for each activity of the Lasagna dinner example Start Finish

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PERT/CPM-32 Because of a phone call, you will delayed by 6 minutes to cut onions and mushrooms (Task E). By how much will dinner be delayed? slack is 3, delay of 6 minutes will delay dinner by 6-3=3 If you use your food processor instead to reduce cutting time from 7 minutes to 2 minutes, will dinner still be delayed? ES=0 LS=8 LS=2 LF=10 2 slack = 8, so phone of 6 won’t delay dinner E

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PERT/CPM-33 All of the critical path, ES, LS, EF, LF, are based on estimates of the activity times. How can you incorporate uncertainty into planning? PERT 3 – estimate approach Most Likely Estimate (m) most probable event Optimistic Estimate (σ) if everything goes perfectly Pessimistic Estimate (p) if everything goes wrong

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PERT/CPM-34 What is the probability of meeting your deadline? Assume the distribution for activity time is a Beta distribution µ - 3 σ µ + 3 σ µ ± 3 σ interval captures 99.73% of distribution density f(t) t: time

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PERT/CPM-35 Consider Critical Path A B J L μ= 30 Buy mozzarella cheese σ 2 = (6 2/3) 2 μ= 5 Slice cheese σ 2 = (11/3) 2 μ= 10 Assemble σ 2 = (2) 2 = 4 μ= 30 Bake σ 2 = (3 2/3) 2 o=10 m=30 p=50 o=3 m=4 p=11 o=20 m=30 p=40

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PERT/CPM-36 Could calculate pessimistic length: 50 + 11 + 14 + 40 = 115 Longest pessimistic path may not be the critical mean path The mean length is sum of means: 30 + 5 + 10 + 30 = 75 = µ p Assume all task times are independent, variance for path is sum of variances:

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PERT/CPM-37 Assume distribution of path time is normal (central limit theorem if lots of tasks on a path) 67 75 83

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