Order 1328631: Project Management
tutorthammy
Week5.pdf
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Project Management Class – 5
Scheduling the Project
Learning Outcomes
Ø Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM)
Ø Establishing precedence relationships and activity sequence
Ø Developing Project Schedule
Ø Finding critical path, critical time and activity slack time
Ø Calculating probabilistic activity times, and probability of completing the project on time
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Introduction
Ø A schedule is a listing of a project's milestones, activities, and deliverables, usually with intended start and finish dates.
Ø The schedule can be prepared in several formats
1. Gantt charts 2. PERT network 3. CPM network
PERT and CPM Networks
Ø PERT and CPM developed independently in 1950’s Ø Program Evaluation and Review Technique (PERT) is a statistical tool
§ By a team from U.S. Navy, Booz-Allen Hamilton, and Lockheed Aircraft § Probabilistic activity durations
Ø Critical Path Method (CPM) § By Morgan Walker from Dupont De Nemours Inc. § Deterministic activity durations § Includes both time and cost estimates to allow time/cost trade-offs
Ø Both employ networks to schedule and display task sequences
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The Language of PERT/CPM
Ø Activity: A task or set of tasks required by the project and use resources and time.
Ø Event: An identifiable state resulting from completion of one or more activities
§ Consumes no resources or time § Predecessor activities must be completed
Ø Milestones: Identifiable and noteworthy events that mark significant progress, for example
§ Deliverables that meet deadlines can be milestones § The usage of time or resources can be milestones
The Language of PERT/CPM (Cont.)
Ø Network: A diagram of nodes (activities or events) and arrows (directional arcs) that illustrate the technological relationships of activities
Ø Path: A series of connected activities between two events
Ø Critical path: The set of activities on a path that, if delayed, will delay the completion date of the project
Ø Critical Time: The time required to complete all activities on the critical path
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The Importance of Activity Sequence
Ø The first step to calculate the project completion time is defining the sequence of activities.
Ø Activity “A” is predecessor of activity “B” (or “B” is successor activity for “A”) when working on “B” cannot be started unless “A” has been completed.
Ø All the following activities are predecessor for activity “Paint the walls”: § Clear the floor area near the wall and cover it § Remove pictures from the wall § Clean dirt, oil, stain from the wall § Fill and smooth any cracks or holes in the wall § Mask any surrounding areas where this paint is not wanted
Building the Network
Ø There are two ways of displaying a project network § Activities on arrows (AOA) network
ü The activities are shown as arrows and events as nodes ü Generally more difficult to draw but depicts the technical
relationships of the activities well § Activities on nodes (AON) network
ü Each task is shown as a node and the technological relationship is shown by the arrows
Ø AON network usually associated with CPM Ø AOA network usually associated with PERT
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Establishing Precedence Relationships
Precedence Relationships: Determining the sequence for undertaking activities.
Sample AON Network
S t a r t
FI N IS H
Task Predecessor A – B – C A D B E B F C, D G E
A
B
C
D
E
F
G
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Sample AOA Network
Task Predecessor A – B – C A D B E B F C, D G E
S t a r t
FI N IS H
1 3
2 4
A
B
C
E
D
F
G
Purchase and deliver equipment
Construct hospital
Develop information system
Install medical equipment
Train nurses and support staff
Select administration staff
Site selection and survey
Select medical equipment
Prepare final construction plans
Bring utilities to site
Interview applicants for nursing and support staff
Organizing and Site Preparation Physical Facilities and Infrastructure Level 1
Level 0
Level 2
Relocation of a Hospital Owner: Project Manager
Owner: Mr/Mrs X Owner: Mr/Mrs Y
Owner: Mr/Mrs X1
Owner: Mr/Mrs X2
Owner: Mr/Mrs X3
Owner: Mr/Mrs X4
Owner: Mr/Mrs X5
Owner: Mr/Mrs X6
Owner: Mr/Mrs Y1
Owner: Mr/Mrs Y2
Owner: Mr/Mrs Y3
Owner: Mr/Mrs Y4
Owner: Mr/Mrs Y5
Finding the Critical Path and Critical Time Relocation of Hospital (WBS)
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Finding the Critical Path and Critical Time Relocation of Hospital (Activity Sequence)
Activity Immediate Predecessors Activity times
(wks) START 0 Organizing and Site Preparation
A. Selecting administrative staff start 12 B. Site selection and survey start 9 C. Select medical equipment A 10 D. Prepare final construction plans B 10 E. Bring utilities to site B 24 F. Interview applicants for nursing and support staff A 10
Physical Facilities and Infrastructure G. Purchase and deliver equipment C 35 H. Construct hospital D 40 I. Develop information system A 15 J. Install medical equipment E, G, H 4 K. Train nurses and support staff F, I, J 6
FINISH K 0
FinishStart
A
B
C
D
E
F
G
H
I
J
K A — B — C A D B E B F A G C H D I A J E,G,H K F,I,J
Immediate Predecessor
Finding the Critical Path and Critical Time Relocation of Hospital (Diagramming)
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Developing the Schedule
Earliest start time (ES) – the earliest finish time of the immediately preceding activities Earliest finish time (EF) – the earliest start time plus its estimated duration EF = ES + t Latest finish time (LF) – the latest start time of the activity that immediately follows. Latest start time (LS) – the latest finish time minus its estimated duration
LS = LF – t Activity Slack – the maximum length of time an activity can be delayed without delaying the entire project LF – EF or LS – ES
T U
T U
Developing the Schedule (Cont.)
Latest finish time
Latest start time
Activity
Duration
Earliest start time
Earliest finish time
0
2
12
14
A
12
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FinishStart
A
B
C
D
E
F
G
H
I
J
K
Path Time (wks)
A-I-K 33 A-F-K 28 A-C-G-J-K 67 B-D-H-J-K 69 B-E-J-K 43
Consider the sequence of activities (paths) between a project’s start and finish.
Finding the Critical Path and Critical Time Relocation of Hospital
K
6
C
10
G
35
J
4
H
40
B
9
D
10
E
24
I
15
FinishStart
A
12
F
10
0 Earliest start
time 12
Earliest finish time
0 9
9 33
9 19 19 59
22 5712 22
59 63
12 27
12 22 63 69
Earliest start time (ES) – the earliest finish time of the immediately preceding activities
EF = ES + t
Finding the Critical Path and Critical Time Relocation of Hospital (Cont.)
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K
6
C
10
G
35
J
4
H
40
B
9
D
10
E
24
I
15
FinishStart
A
12
F
10
0 9
9 33
9 19 19 59
22 5712 22
59 63
12 27
12 22 63 690 12
The Critical Path takes 69 weeks
Path Time (wks)
A-I-K 33 A-F-K 28 A-C-G-J-K 67 B-D-H-J-K 69 B-E-J-K 43
Finding the Critical Path and Critical Time Relocation of Hospital (Cont.)
Notes on Sample Project
Ø All activities, and thus all paths, must be completed to finish the project
Ø The shortest time for completion of the network is equal to the longest path through the network (in this case B-D-H-J-K)
Ø If any activity on this path is even slightly delayed, the project will be delayed
Ø Question: How long each activity can be delayed without delaying the entire project?
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K
6
C
10
G
35
J
4
H
40
B
9
D
10
E
24
I
15
FinishStart
A
12
F
10
0 9
9 33
9 19 19 59
22 5712 22
59 63
12 27
12 22 63 690 12
48 63
53 63
59 63
24 59
19 59
35 59
14 24
9 19
2 14
0 9
63 69
Latest start time Latest finish time
Latest finish time (LF) – the latest start time of the activity that immediately follows.LS = LF – t
Developing the Schedule Relocation of Hospital
Developing the Schedule: Activity Slack
K
6
C
10
G
35
J
4
H
40
B
9
D
10
E
24
I
15
FinishStart
A
12
F
10
0 9
9 33
9 19 19 59
22 5712 22
59 63
12 27
12 22 63 690 12
48 63
53 63
59 63
24 59
19 59
35 59
14 24
9 19
2 14
0 9
63 69
S = 0
S = 2
S = 26
S = 0
S = 36
S = 2
S = 2
S = 41 S = 0
S = 0 S = 0
S = LF – EF or LS – ES Question: How long each activity can be delayed without delaying the entire project?
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Managerial Implications
Ø The primary attention of the project manager must be to activities on the critical path
Ø If anything delays one of these activities, the project will be late Ø Projects are easier to manage when there is project slack Ø Please verify the impact of the following delays in “relocation of
hospital” project completion time: § Project F delayed by 15 weeks § Project H delayed by 7 weeks § Project A by 5 weeks
The Gantt Chart
Ø Henry Gantt developed the Gantt chart around 1917 Ø It displays project activities as bars measured against a horizontal time
scale
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The Gantt Chart (Cont.)
Ø Gantt charts are easy to draw
Ø It can contain a considerable amount of information and is an excellent communication device about the state of a project.
Ø Problems arise when several tasks begin at the same time and have the same duration
§ Technical dependencies are harder to see on a Gantt chart
§ Can make it hard to find critical path
Ø PERT/CPM methods are often used as complements to Gantt charts
Precedence Diagramming
Ø One of the shortcoming of PERT/CPM network method is that it does not allow for leads and lags between two activities. § An activity can start as soon as its predecessor activities are completed. § What if a follow-on activity cannot begin until a certain amount of time after
its predecessor is completed. § For example, a successor may have to wait for paint to dry or cement to
harden.
Ø We may need to define the following relationship § Finish to start (F to S): Finish of Activity A to start of Activity B § Start to start (S to S): Start of Activity A to start of Activity B § Finish to finish (F to F): Finish of Activity A to finish of Activity B § Start to finish (S to F): Start of Activity A to finish of Activity B
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Precedence Diagramming Conventions
Calculating Probabilistic Activity Times
Ø In CPM analysis each activity time is treated as a random variable derived from a beta probability distribution.
Ø The most likely time estimate (m) is the mode of beta distribution. Ø Therefore, it is the time with the highest probability of occurrence.
a m bMean Time
a m b Mean Time
3σ 3σ
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Calculating Probabilistic Activity Times (Cont.)
Ø Estimate “a” is such that the actual duration of the task will be “a” (or lower) in less than 1 percent of the time. § Thus “a” is an optimistic estimate
Ø Estimate “b” is such that the actual finish time will be “b” (or greater) in less than 1 percent of the time § Thus “b” is a pessimistic estimate
Ø Estimate m is the most likely time
a m bMean Time
Ø The mean of the beta distribution for each activity time can be estimated by
te = a + 4m + b
6
Ø The variance of the beta distribution for each activity is
σ2 = b – a
6
2
Statistical Analysis
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Ø Suppose that the project team has arrived at the following time estimates for activity B (site selection and survey) of the Hospital project:
a = 7 weeks, m = 8 weeks, and b = 15 weeks
a. Calculate the expected time and variance for activity B.
b. Calculate the expected time and variance for the other activities in the project.
Statistical Analysis: Hospital Example
𝒕𝒆 = 𝟕 + 𝟒×𝟖 + 𝟏𝟓
𝟔 = 𝟓𝟒 𝟔 = 𝟗 𝐰𝐞𝐞𝐤𝐬 𝝈𝟐 =
𝟏𝟓 − 𝟕 𝟔
𝟐
= 𝟏.𝟕𝟖 𝐰𝐞𝐞𝐤𝐬
b. The following table shows the expected activity times and variances for this project. Time Estimates (week) Activity Statistics
Activity Optimistic (a) Most Likely (m) Pessimistic (b) Expected Time (te) Variance (σ2)
A 11 12 13 12 0.11 B 7 8 15 9 1.78 C 5 10 15 10 2.78 D 8 9 16 10 1.78 E 14 25 30 24 7.11 F 6 9 18 10 4.00 G 25 36 41 35 7.11 H 35 40 45 40 2.78 I 10 13 28 15 9.00 J 1 2 15 4 5.44 K 5 6 7 6 0.11
Statistical Analysis: Hospital Example (Cont.)
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Ø Central limit theorem: Sum of the independent, identically distributed random variables follows a normal distribution.
Ø The sum of the expected activity times on each path is the mean of a normal distribution
Ø Because the activity times are independent, the variance of the time distribution for critical path:
Analyzing Probabilities
𝑻𝑬 = ∑ 𝐄𝐱𝐩𝐞𝐜𝐭𝐞𝐝 𝐚𝐜𝐭𝐢𝐯𝐢𝐭𝐲 𝐭𝐢𝐦𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐜𝐫𝐢𝐭𝐢𝐜𝐚𝐥 𝐩𝐚𝐭𝐡
� � = 𝐌𝐞𝐚𝐧 𝐨𝐟 𝐧𝐨𝐫𝐦𝐚𝐥 𝐝𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧
𝝈𝒑𝟐 = O 𝐕𝐚𝐫𝐢𝐚𝐧𝐜𝐞 𝐨𝐟 𝐚𝐜𝐭𝐢𝐯𝐢𝐭𝐢𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐜𝐫𝐢𝐭𝐢𝐜𝐚𝐥 𝐩𝐚𝐭𝐡 �
�
Analyzing Probabilities (Cont.)
Ø To analyze probabilities of completing a project by a certain date T Ø Focus on the critical path Ø Use the z-transformation formula
𝑻𝑬 𝑻
𝒑 𝒕 ≤ 𝑻 = 𝒑 𝒛 ≤ 𝑻 − 𝑻𝑬 𝝈𝒑
𝝁 = 𝟎 𝒛
Equivalent to
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Ø Calculate the probability that Hospital will become operational in 72 weeks, using the critical path.
Ø The critical path B–D–H–J–K has a length of 69 weeks. TE = 9 + 10 + 40 + 4 + 6 = 69 weeks
Ø From the previous calculations, we obtain the variance of path B–D– H–J–K:
σ2 = 1.78 + 1.78 + 2.78 + 5.44 + 0.11 = 11.89 weeks
Analyzing Probabilities Critical Path Analysis
𝒑 𝒕 ≤ 𝟕𝟐 = 𝒑 𝒛 ≤ 𝟕𝟐 − 𝟔𝟗 𝟏𝟏.𝟗𝟖�
= 𝒑 𝒛 ≤ 𝟎.𝟖𝟕 ≅ 𝟎.𝟖𝟏
Ø Calculate the probability that Hospital will become operational in 72 weeks, using near critical path A–C–G–J–K.
Ø The critical path A–C–G–J–K has a length of 67 weeks. TE = 12 + 10 + 35 + 4 + 6 = 67 weeks
Ø From the previous calculations, we obtain the variance of path A–C– G–J–K:
σ2 = 0.11 + 2.78 + 7.11 + 5.44 + 0.11 = 15.55 weeks
Analyzing Probabilities Near Critical Path Analysis
𝒑 𝒕 ≤ 𝟕𝟐 = 𝒑 𝒛 ≤ 𝟕𝟐 − 𝟔𝟕 𝟏𝟓.𝟓𝟓�
= 𝒑 𝒛 ≤ 𝟏.𝟐𝟕 ≅ 𝟎.𝟗𝟎
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The Probability of Completing the Project on Time
Ø Can the project be completed in X days/weeks/months? Ø Completing a project by a specified time requires that all the paths in
the network be completed by the specified time Ø Determining the probability that a project is completed by a specified
time requires § Calculating the probability that every single path is finished by the
specified time § Multiplying these probabilities together § This requires the assumption that the paths are statistically independent
The Probability of Completing the Project on Time Relocation of Hospital
Ø Calculate the probability that Hospital will become operational in 72 weeks. § Identify all paths from start to finish § Find the probability of each path finishes in 72 weeks § Multiply all probabilities together
Path TE σ2 σp Z Probability A-I-K A-F-K
A-C-G-J-K B-D-H-J-K B-E-J-K
33 28 67 69 43
9.22 4.22 15.55 11.89 14.44
𝒛 = 𝟕𝟐 − 𝑻𝑬 𝝈𝒑
3.03 2.05 3.94 3.45 3.80
12.87 21.46 1.27 0.87 7.63
1 1
0.90 0.81 1
Probability that Hospital will become operational in 72 weeks = 1*1*0.9*0.81*1 = 0.73
