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Chapter 13. Project Management. Characteristics 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. What is Project Management.
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Chapter 13 Project Management
Characteristics 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.
What is Project Management • To schedule and control the progress and cost of a project.
PERT/CPM: • Input: • 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; • ...
PERT/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?
PERT 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.
Performance Time t of an Activity • t is calculated as follows: where a=optimistic time, b=pessimistic time, m=most likely time. • Note: t is also called the expected performance time of an activity.
Variance 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
Variance, 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.
Critical Path • It 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.
Time 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.
Uses 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.
Computing 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 + t • Note: EF=ES at the Finish node.
Computing 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 – t • Note: LS=LF at Start node.
Computing Slack Times • For each activity: • slack = LS – ES = LF – EF
Foundry Inc. Example • Calculate 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.
F 3 A 2 C 2 EF= EF= EF= ES= ES= ES= LF= LF= LF= LS= LS= LS= slack= slack= slack= Start E 4 H 2 Finish EF= EF= ES=EF= ES= ES= ES=EF= LF= LF= LS=LF= LS= LS= LS=LF= slack= slack= D 4 G 5 B 3 EF= EF= EF= ES= ES= ES= LF= LF= LF= LS= LS= LS= slack= slack= slack= Network for Foundry Inc.
Example 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.
F 10 EF= ES= K 6 LF= LS= A 12 EF= ES= slack= EF= ES= LF= LS= I 15 LF= slack= LS= slack= EF= ES= LF= LS= slack= G 35 Start Finish C 10 EF= ES= ES=EF= ES=EF= EF= LF= ES= LS= slack= LS=LF= LS=LF= LF= LS= slack= H 40 D 10 EF= EF= ES= ES= B 9 J 4 LF= LF= LS= LS= EF= ES= EF= ES= slack= slack= LF= LS= LF= LS= slack= slack= E 24 EF= ES= LF= LS= A Hospital Project slack=
Slack 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.
Critical Path, Examples • What is the critical path in the Foundry Inc. example? • What is the critical path in the Hospital project example?
Calculate 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.
C 4 EF= ES= LF= LS= A 2 slack= EF= ES= LF= LS= slack= D 3 Finish Start EF= ES= ES=EF= ES=EF= LF= LS= LS=LF= LS=LF= slack= E 2 B 7 EF= ES= EF= ES= LF= LS= LF= LS= slack= slack= Calculate the critical path
Example: Draw diagram and find critical path Activity Predecessor t A - 5 B - 3 C - 6 D B 4 E A 8 F C 12 G A,D 7 H E,G 6 I G 5
Example: Draw diagram and find critical path Activity Predecessor t A - 3 B - 4 C A 6 D B 5 E A,B 8 F C 2 G D,E,F 4 H E,F 5
Steps for Solving 13-1&2 • Calculate 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.
Probabilities in PERT • Since 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.
Probabilistic Information for Management • The expected project finish time and the variance of project finish time; • Probability the project is finished by a certain date.
Project 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)
Example, Foundry Inc. Critical path: A-C-E-G-H Variance of T, T2 = Project completion time, T =
Solved Problem 13-1&2, p.547-548Project completion time and variance Critical path: B-D-E-G Project completion time, T = Variance of T, T2 =
Probability Analysis • To 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).
The 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)
Notes (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)
Notes (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)
Example 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, and • So, P(project is finished within 16 weeks) = P(z<=Z) = P(z<=0.57) = 0.71566.
Examples 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 project • is actually completed within 246 days? • is actually completed within 240 days? • is actually completed within 256 days? • is not completed by the 256th day?
A 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?
Data of One-more-example: Activity Predecessor a m b A - 1 2 3 B - 5.5 7 8.5 C A 3.5 4 4.5 D A 2 3 4 E B,D 0 2 4
Example (cont.) Activity Predecessor t v A - 2 0.111 B - 7 0.250 C A 4 0.028 D A 3 0.111 E B,D 2 0.444
C 4 EF= ES= LF= LS= A 2 slack= EF= ES= LF= LS= slack= D 3 Finish Start EF= ES= ES=EF= ES=EF= LF= LS= LS=LF= LS=LF= slack= E 2 B 7 EF= ES= EF= ES= LF= LS= LF= LS= slack= slack= Calculate the critical path
Solving on QM • Critical 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.