2Characteristics of a “project”: A project is unique (not routine),A project is composed of interrelated sub-projects/activities,It is associated woth a large investment.
3What is Project Management To schedule and control the progress and cost of a project.
4PERT/CPM: Input: Output: Activities in a project; Precedence relationships among tasks;Expected performance times of tasks.Output:The earliest finish time of the project;The critical path of the project;The required starting time and finish time of each task;Probabilities of finishing project on a certain date;...
5PERT/CPM is supposed to answer questions such as: How long does the project take?What are the bottle-neck tasks of the project?What is the time for a task ready to start?What is the probability that the project is finished by some date?How additional resources are allocated among the tasks?
6PERT Network: It is a directed network. Each activity is represented by a node.An arc from task X to task Y if task Y follows task X.A ‘start’ node and a ‘finish’ node are added to show project start and project finish.Every node must have at least one out-going arc except the ‘finish’ node.
7Example of Foundry Inc., p.523 ActivityImmediate PredecessorsA-BCDEFGD, EHF, G
11Performance Time t of an Activity t is calculated as follows:wherea=optimistic time,b=pessimistic time,m=most likely time.Note: t is also called the expected performance time of an activity.
12Variance of Activity Time t If a, m, and b are given for the optimistic, most likely, and pessimistic estimations of activity k, variance k2 is calculated by the formula
13Variance, a Measure of Variation Variance is a measure of variation of possible values around the expected value.The larger the variance, the more spread-out the random values.The square root of variance is called standard deviation.
14Example, Foundry Inc., p.525 A 1 2 3 B 4 C D 6 E 7 F 9 G 11 H Activity varianceA123B4CD6E7F9G11H
15Critical PathIt is the “longest” path in the PERT network from the start to the end.It determines the duration of the project.It is the bottle-neck of the project.
16Time and Timings of an Activity: t=estimated performance time;ES=Earliest starting time;LS=Latest starting time;EF=Earliest finish time;LF=Latest finish time;s=Slack time of a task.
17Uses of Time and Timings Earliest times (ES and EF) and latest times (LS and LF) show the timings of an activity’s “in/out” of project.ES and LS of an activity tell the time when the preparations for that activity must be done.For calculating the critical path.
18Computing Earliest Times Step 1. Mark “start” node: ES=EF=0.Step 2. Repeatedly do this until finishing all nodes:For a node whose immediate predecessors are all marked, mark it as below:ES = Latest EF of its immediate predecessors,EF = ES + tNote: EF=ES at the Finish node.
19Computing Latest Times: Step 1. Mark “Finish” node:LF = LS = EF of “Finish” node.Step 2. Repeatedly do this until finishing all nodes:For a node whose immediate children’s are all marked with LF and LS, mark it as below:LF = Earliest LS of its immediate children,LS = LF – tNote: LS=LF at Start node.
20Computing Slack TimesFor each activity:slack = LS – ES = LF – EF
21Foundry Inc. ExampleCalculate ES, EF, LS, LF, and slack for each activity of the Foundry Inc. example on its PERT network, given the data about the project as in the next slide.
22Example, Foundry Inc. 0.111 0.444 1.777 5 A 1 2 3 B 4 C D 6 E 7 F 9 G ActivityambtvarianceA1230.111B4CD60.444E7F91.777G115H
23Network for Foundry Inc. A2C2F3ES=EF=ES=EF=ES=EF=LS=LF=LS=LF=LS=LF=slack=slack=slack=StartE4H2FinishES=EF=ES=EF=ES=EF=ES=EF=LS=LF=LS=LF=LS=LF=LS=LF=slack=slack=B3D4G5ES=EF=ES=EF=ES=EF=LS=LF=LS=LF=LS=LF=slack=slack=slack=Network for Foundry Inc.
24Example of Hospital Project: Calculate ES, EF, LF, LS and slack of each activity in this project on its PERT network, given the data about the project as in the next slide.
25Example: A Hospital Project ActivityImmediate Predecessor(s)Performance time t (weeks)A—12B9C10DE24FG35H40I15JE, G, H4KF, I, G6
26A Hospital Project F 10 K 6 A 12 slack= I 15 slack= slack= slack= G 35 ES=EF=K6LS=LF=A12ES=EF=slack=ES=EF=LS=LF=I15LS=LF=slack=slack=ES=EF=LS=LF=slack=G35StartFinishC10ES=EF=ES=EF=ES=EF=ES=EF=LS=LF=LS=LF=slack=LS=LF=LS=LF=slack=D10H40ES=EF=ES=EF=B9J4LS=LF=LS=LF=ES=EF=ES=EF=slack=slack=LS=LF=LS=LF=slack=slack=E24ES=EF=LS=LF=A Hospital Projectslack=
27Slack and the Critical Path The slack of any activity on the critical path is zero.If an activity’s slack time is zero, then it is must be on the critical path.
28Critical Path, Examples What is the critical path in the Foundry Inc. example?What is the critical path in the Hospital project example?
29Calculate the Critical Path: Step 1. Mark earliest times (ES, EF) on all nodes, forward;Step 2. Mark latest times (LF, LS) on all nodes, backward;Step 3. Calculate slack of each activity;Step 4. Identify the critical path that contain the activities with zero slack.
30Calculate the critical path 4ES=EF=LS=LF=A2slack=ES=EF=LS=LF=slack=D3FinishStartES=EF=ES=EF=ES=EF=LS=LF=LS=LF=LS=LF=slack=E2B7ES=EF=ES=EF=LS=LF=LS=LF=slack=slack=Calculate the critical path
31Example: Draw diagram and find critical path Activity Predecessor tABCD B 4E A 8F CG A,D 7H E,G 6I G 5
32Example: Draw diagram and find critical path Activity Predecessor tABC A 6D B 5E A,B 8F C 2G D,E,F 4H E,F 5
33Solved Problem 13-1&2, p.547-548 Calculate the Critical Path ActivityambImmediate predecessorA123-B4C56D8910EC, DFG
34Steps for Solving 13-1&2Calculate activity performance time t for each activity;Draw the PERT network;Calculate ES, EF, LS, LF and slack of each activity on PERT network;Identify the critical path.
35Probabilities in PERTSince the performance time t of an activity is from estimations, its actual performance time may deviate from t;And the actual project completion time may vary, therefore.
36Probabilistic Information for Management The expected project finish time and the variance of project finish time;Probability the project is finished by a certain date.
37Project Completion Time and its Variance The expected project completion time T:T = earliest completion time of the project.The variance of T, T2 :T2 = (variances of activities on the critical path)
38Example, Foundry Inc. 0.111 0.444 1.777 5 A 1 2 3 B 4 C D 6 E 7 F 9 G ActivityambtvarianceA1230.111B4CD60.444E7F91.777G115HCritical path: A-C-E-G-HVariance of T, T2 =Project completion time, T =
39Solved Problem 13-1&2, p.547-548 Project completion time and variance ActivityambtvarianceA1230.111B4C56D8910EFGCritical path: B-D-E-GProject completion time, T =Variance of T, T2 =
40Probability AnalysisTo find probability of completing project within a particular time x:1. Find the critical path, expected project completion time T and its variance T2 .3. Find probability from a normal distribution table (as on page 698).
41The Idea of the Approach The table on p.698 gives the probability P(z<=Z) where z is a random variable with standard normal distribution, i.e. zN(0,1); Z is a specific value.P(project finishes within x days)
42Notes (1) P(project is finished within x days) = P(z<=Z) P(project is not finished within x days)= 1P(project finishes within x days)= 1P(z<=Z)
43Notes (2) If x<T, then Z is a negative number. But the table on p.698 is only for positive Z values.For example, Z= 1.5, per to the symmetry feature of the normal curve,P(z<=1.5) = P(z>=1.5) = 1P(z<=1.5)
44Example of Foundry Inc. p.530-531 Project completion time T=15 weeks.Variance of project time, T2=3.111.We want to find the probability that project is finished within 16 weeks. Here, x=16, andSo, P(project is finished within 16 weeks)= P(z<=Z) = P(z<=0.57) =
45Examples of probability analysis If a project’s expected completing time is T=246 days with its variance T2=25, then what is the probability that the projectis actually completed within 246 days?is actually completed within 240 days?is actually completed within 256 days?is not completed by the 256th day?
46A Comprehensive Example Given the data of a project as in the next slide, answer the following questions:What is PERT network like for this project?What is the critical path?Activity E will be subcontracted out. What is earliest time it can be started? What is time it must start so that it will not delay the project?What is probability that the project can be finished within 10 weeks?What is the probability that the project is not yet finished after 12 weeks?
47Data of One-more-example: Activity Predecessor a m bABC AD AE B,D
48Example (cont.) Activity Predecessor t v A - 2 0.111 B - 7 0.250 C AD AE B,D
49Calculate the critical path 4ES=EF=LS=LF=A2slack=ES=EF=LS=LF=slack=D3FinishStartES=EF=ES=EF=ES=EF=LS=LF=LS=LF=LS=LF=slack=E2B7ES=EF=ES=EF=LS=LF=LS=LF=slack=slack=Calculate the critical path
50Solving on QMCritical path, ES, LS, EF, LF, and slack can be calculated by QM for Windows. We need to enter activities’ times and immediate predecessors.But QM does not provide the network.