CH_08_PROJECT_MANAGEMENT

Analysis with Microsoft Project (9 of 13)

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8-51

Analysis with Microsoft Project (10 of 13)

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Analysis with Microsoft Project (11 of 13)

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Analysis with Microsoft Project (12 of 13)

Exhibit 8.14

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Analysis with Microsoft Project (13 of 13)

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Project Crashing and
Time-Cost Trade-Off Overview

■ Project duration can be reduced by assigning more resources to

project activities.

■ However, doing this increases project cost.

■ Decision is based on analysis of trade-off between time and cost.

■ Project crashing is a method for shortening project duration by
reducing one or more critical activities to a time less than normal

activity time.

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Project Crashing and Time-Cost Trade-Off
Example Problem (1 of 5)

Figure 8.19 The Project Network for Building a House 8-57

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Project Crashing and Time-Cost Trade-Off
Example Problem (2 of 5)

Crash cost & crash time have a linear relationship:

Total Crash Cost = $2000
Total Crash Time 5 weeks

= $400 / wk

Figure 8.20 8-58

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Project Crashing and Time-Cost Trade-Off
Example Problem (3 of 5)

Table 8.4

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Project Crashing and Time-Cost Trade-Off
Example Problem (4 of 5)

Figure 8.21 Network with Normal Activity Times and Weekly Crashing Costs

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Project Crashing and Time-Cost Trade-Off
Example Problem (5 of 5)

As activities are crashed, the critical path may change and
several paths may become critical.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Figure 8.22
Revised Network with
Activity 1 Crashed

8-61

Project Crashing and Time-Cost Trade-Off
Project Crashing with QM for Windows

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 8.16 8-62

Project Crashing and Time-Cost Trade-Off
General Relationship of Time and Cost (1 of 2)

■ Project crashing costs and indirect costs have an inverse
relationship.

■ Crashing costs are highest when the project is shortened.

■ Indirect costs increase as the project duration increases.

■ Optimal project time is at minimum point on the total
cost curve.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 8-63

Project Crashing and Time-Cost Trade-Off
General Relationship of Time and Cost (2 of 2)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Figure 8.23
The Time-Cost Trade-Off

8-64

The CPM/PERT Network
Formulating as a Linear Programming Model

The objective is to minimize the project duration (critical path time).

General linear programming model with AOA convention:

Minimize Z = Σxi
subject to: i

xj - xi ≥ tij for all activities i → j
xi, xj ≥ 0

Where:

xi = earliest event time of node i
xj = earliest event time of node j
tij = time of activity i → j

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The CPM/PERT Network
Example Problem Formulation and Data (1 of 2)

Figure 8.24

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The CPM/PERT Network
Example Problem Formulation and Data (2 of 2)

Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 8-67

subject to:

x2 - x1 ≥ 12
x3 - x2 ≥ 8
x4 - x2 ≥ 4
x4 - x3 ≥ 0
x5 - x4 ≥ 4
x6 - x4 ≥ 12
x6 - x5 ≥ 4
x7 - x6 ≥ 4
xi, xj ≥ 0

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The CPM/PERT Network
Example Problem Solution with Excel (1 of 4)

B6:B12

Exhibit 8.17

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The CPM/PERT Network
Example Problem Solution with Excel (2 of 4)

Exhibit 8.18 8-69

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The CPM/PERT Network
Example Problem Solution with Excel (3 of 4)

Exhibit 8.19

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The CPM/PERT Network
Example Problem Solution with Excel (4 of 4)

Exhibit 8.20

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Project Crashing with Linear Programming
Example Problem – Model Formulation

Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46
+ 200y56 + 7000y67

subject to:

y12 ≤ 5 y12 + x2 - x1 ≥ 12 x7 ≤ 30

y23 ≤ 3 y23 + x3 - x2 ≥ 8 xi, yij ≥ 0

y24 ≤ 1 y24 + x4 - x2 ≥ 4 Objective is to

y34 ≤ 0 y34 + x4 - x3 ≥ 0 minimize the
y45 ≤ 3 y45 + x5 - x4 ≥ 4 cost of crashing
y46 ≤ 3 y46 + x6 - x4 ≥ 12

y56 ≤ 3 y56 + x6 - x5 ≥ 4

y67 ≤ 1 x67 + x7 - x6 ≥ 4

xi = earliest event time of node I

xj = earliest event time of node j

yij = amount of time by which activity i → j is crashed

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Project Crashing with Linear Programming
Excel Solution (1 of 3)

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Project Crashing with Linear Programming
Excel Solution (2 of 3)

Exhibit 8.22

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Project Crashing with Linear Programming
Excel Solution (3 of 3)

Exhibit 8.23

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Example Problem
Problem Statement and Data (1 of 2)

Given this network and the data on the following slide, determine the
expected project completion time and variance, and the probability
that the project will be completed in 28 days or less.

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Example Problem
Problem Statement and Data (2 of 2)

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Example Problem Solution (1 of 4)

Step 1: Compute the expected activity times and variances.

t = a + 4m + b v =  b - a 2
6 

6 



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Example Problem Solution (2 of 4)

Step 2: Determine the earliest and latest activity times & slacks

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Example Problem Solution (3 of 4)

Step 3: Identify the critical path and compute expected
completion time and variance.

Critical path (activities with no slack): 1 → 3 → 5 → 7
Expected project completion time: tp = 9+5+6+4 = 24 days
Variance: vp = 4 + 4/9 + 4/9 + 1/9 = 5 (days)2

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Example Problem Solution (4 of 4)

Step 4: Determine the Probability That the Project Will be
Completed in 28 days or less (µ = 24, σ = √5)
Z = (x - µ)/σ = (28 -24)/√5 = 1.79

Corresponding probability from Table A.1, Appendix A, is .4633 and
P(x ≤ 28) = .4633 + .5 = .9633.

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