Chapter 14
A construction superintendent explained to his client the
massive cost overruns. He told of concrete trucks
arriving too soon and having to dump their loads; of
having to re-pour the foundation because the gravel
underlay had not been set; of having to cut many holes
through the concrete floor pad because the pipes had
been covered before being inspected. Everything
happened as scheduled, the poor man lamented. “You
should have used PERT,” the client admonished.
1
Project Planning
with PERT and CPM
The Importance of Time
in Planning
 Projects may have uncertainty.
 Projects may take a long time.
 Activities must be scheduled.
2
Basic Concepts of PERT
 PERT has an activity orientation. An
activity is an effort that consumes time and
resources.
 The central focus of all analysis and
procedures is the PERT Network.
 PERT ties together all project activities as
predecessors and successors.
 Events are project milestones and serve as
the logical glue (network nodes) for
connecting activities (network arcs).
3
Project Activities
in Constructing a House
4
PERT Network for
Constructing a House
5
Earliest Possible Event Times
(TE)
6
Finding TE Values
7
Latest Allowable Event Times
(TL)
8
9
Finding TL Values
10
Critical Paths
and Event Slack
11
Activity Scheduling
 Project activities need two scheduled times:
the starting time and the finishing time.
 PERT finds early and late values for each.
 All are relative to time 0 for the project start.
 An activity’s early starting time equals the
TE of its beginning event: ES = TE.
 An activity’s late finishing time equals the
TL for its ending event: LF = TL.
 From those, 2 times are computed using the
expected activity completion time t:
Early finishing time: EF = ES + t.
12
Late starting time: LS = LF - t.
Scheduling Parameters for
Constructing a House
 The following apply.
 Scheduled times must fall between ES and
LF and allow at least time t to complete.
13
Activity Slack
 The activity slack expresses the scheduling
leeway for the activity. It is computed by:
activity slack = LS - ES = LF - EF
 Critical activities have zero slack. These lie
on a critical path.
 Event slack and activity slack measure
different things and are only loosely related.
14
Using QuickQuant with PERT
 QuickQuant will construct a PERT network
and evaluate it using only tabular data input.
15
Time Cost Tradeoffs
 It is possible to speed up a project by crashing some
activities at extra cost. The following data apply.
16
 To reduce project time, speed up one or more activities on
the critical path.
Time-Cost Tradeoffs
 By successively reducing project time by 1 day at lowest
cost, these plans were found.
17
PERT Network for Plan 4
18
PERT Network for Plan 7
19
Probabilistic Analysis
 The activity completion times are really
random variables and probabilities apply.
 They are well represented by the modified
beta distribution.
 There are three time parameters:
most likely m optimistic a
pessimistic b
 The expected value or mean and variance are
computed from:
t =
20
a + 4m + b
6
s2
(b - a)2
=
62
Single Path Probabilities
 The normal distribution represents probabilities
for the duration T of a single path.
m
 ti
i on path
s2 
2
s
 i
i on path
 Using the above, the mean and variance are
computed. From them, probabilities may be found
that T lies in any specified interval.
 Probabilities for the duration of the a priori
critical path can be computed.
 WARNING: That will not represent the duration
of the project itself. The m computed above
understates expected project duration.
21
Probabilities for Time to
Construct a House
 The following data apply.
22
Probabilities for Time to
Construct a House
 The expected duration of the critical path
a-b-d-e-i-k-l is:
m = 5 + 2 + 12 + 10 + 1 + 3 + 9 = 42 days
 The variance for duration of that path is
s 2 = .444+.444+7.111+1.000+0+.250+.444
= 9.693
and the standard deviation is the square root
of the above
s = 3.11 days
 Using 40 days, z = (40 – 42)/3.11 = -.64
and Pr[T > 40] = .5 + .2389 = .7389
This applies to the critical path, not to the
23
project itself.
Single-Path Probabilities and
QuickQuant
 QuickQuant provides a detailed
probabilistic analysis of the critical path.
24
What’s Wrong with
Single-Path Probabilities?
 Single path probabilities do not properly
apply to the project itself because many
paths might end up to be the longest.
 The critical path is one of many possibilities.
 Durations of all these paths are not independent
random variables.
 To properly assess likelihoods for project
duration, build the house (on paper) many
times and see what happens.
 QuickQuant will do that. The procedure is
called Monte Carlo simulation.
25
Simulating House Construction
with PERT
 QuickQuant provided these simulation results.
26
Simulating House Construction
with PERT
 The second QuickQuant screen tells us about
unexpected longest paths.
27
 The a priori critical path was of longest duration
only 437 times out of 500. In some projects, it
may be longest as little as 1% of the time, or less.
PERT/CPM Evaluation for Home
Construction Example (Figure 14-26)
The activities on the
critical path are given in
O
P
Q
column
R (indicated
by
the word Yes). In this
case they are a, b, d, e, I,
k, and l.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
R
1
PERT/CPM EVALUATION RESULTS
2
3 PROBLEM: Home Construction Example
4
5
On
Activity - Event Table
6
Critical
Events
TEend - TEbeg
7
Activities
Name
1
2
3
4
5
6
7
8
9
10
11 Duration Activities
Name
Path?
8
a
Excavating
-1
1
5
a
Excavating
5
Yes
9
b
Pour Foundation
-1
1
2
b
Pour Foundation
2
Yes
10
c
Outside plumbing
-1
1
6
c
Outside plumbing
24
No
11
d
Framing
-1
1
12
d
Framing
12
Yes
12
e
Inside plumbing
-1
1
10
e
Inside plumbing
10
Yes
13
f
Wiring
-1
1
9
f
Wiring
11
No
14
g
Roofing
-1
1
5
g
Roofing
5
No
15
h
Brickwork
-1
1
9
h
Brickwork
17
No
16
i
Plumbing inspection
-1
1
1
i
Plumbing inspection
1
Yes
17
j
Shingling
-1
1
2
j
Shingling
6
No
18
k
Cover walls
-1
1
3
k
Cover walls
3
Yes
19
l
Interior finish
-1
1
9
l
Interior finish
9
Yes
20
m
Exterior finish
-1
1
7
m
Exterior finish
7
No
21
n
Landscaping
-1
1
8
n
Landscaping
11
No
22
D1
Dummy 1
-1
1
0
D1
Dummy 1
0
No
23
N
24
Project
Solution (Earliest Event Times, TEj)
26 =M26
25
1
2
3
4
5
6
7
8
9
10
11
Time
26
0
5
7
19
29
24
24
30
31
33
42
42
Sum of Earliest Event Times =
244
27
O
P
Q
R
R
28
8 =A8 =IF(ISBLANK(B8),"",B8)
=SUMPRODUCT($C$26:$M$26,C8:M8) =IF(R38=1,"Yes","No")
26 =SUM(C26:M26)
29
9 =A9 =IF(ISBLANK(B9),"",B9)
=SUMPRODUCT($C$26:$M$26,C9:M9) =IF(R39=1,"Yes","No")
30
10 =A10 =IF(ISBLANK(B10),"",B10) =SUMPRODUCT($C$26:$M$26,C10:M10) =IF(R40=1,"Yes","No")
31
The length of the
critical path is in
cell N26. It is 42
in this example.
The earliest
event times are
given in cells
C25:M26
28
1. Enter the
problem name in
cell B3.
PERT/CPM Evaluation for Home
Construction Example (Figure 14-26)
3. Enter data in the
O
P
Q
activity-event
table. R
Put a -1 in the column
where the activity
On
starts and a +1 where
Critical
TE
- TE
Activities
Path?
theName
activity
ends.
All
a
Excavating
5
Yes
b
Pour
Foundation
2in the Yes
other
entries
c
Outside plumbing
24
No
d
Framing
12 For Yes
row are blank.
e
Inside plumbing
10
Yes
example, cell11C8 hasNo a
f
Wiring
g
Roofing
5
No
-1 because activity
A
h
Brickwork
17
No
i
Plumbing inspection
1
Yes
starts with event
1 No
j
Shingling
6
k
Cover
wallscell D8 has
3
and
a +1Yes
l
Interior finish
9
Yes
m
Exterior
finish
7
because
activity
A No
n
Landscaping
11
No
ends
2. No
D1
Dummy
1 with event
0
A
B
C
D
E
F
G
H
I
J
K
L
M
N
1
PERT/CPM EVALUATION RESULTS
2
3 PROBLEM: Home Construction Example
4
5
Activity - Event Table
6
Events
7
Activities
Name
1
2
3
4
5
6
7
8
9
10
11 Duration
end
beg
8
a
Excavating
-1
1
5
9
b
Pour Foundation
-1
1
2
10
c
Outside plumbing
-1
1
6
11
d
Framing
-1
1
12
12
e
Inside plumbing
-1
1
10
13
f
Wiring
-1
1
9
14
g
Roofing
-1
1
5
15
h
Brickwork
-1
1
9
16
i
Plumbing inspection
-1
1
1
17
j
Shingling
-1
1
2
18
k
Cover walls
-1
1
3
19
l
Interior finish
-1
1
9
20
m
Exterior finish
-1
1
7
21
n
Landscaping
-1
1
8
22
D1
Dummy 1
-1
1
0
23
24
Project
Solution (Earliest Event Times, TEj)
N
25
1
2
3
4
5
6
7
8
9
10
11
Time
26
=M26
26
0
5
7
19
29
24
24
30
31
33
42
42
Sum of Earliest Event Times =
244
27
O
P
Q
R
R
28
8 =A8 =IF(ISBLANK(B8),"",B8)
=SUMPRODUCT($C$26:$M$26,C8:M8) =IF(R38=1,"Yes","No")
26 =SUM(C26:M26)
29
9 =A9 =IF(ISBLANK(B9),"",B9)
=SUMPRODUCT($C$26:$M$26,C9:M9) =IF(R39=1,"Yes","No")
30
10 =A10 =IF(ISBLANK(B10),"",B10) =SUMPRODUCT($C$26:$M$26,C10:M10) =IF(R40=1,"Yes","No")
31
2. Expand the table to accommodate the proper number of activities and
events by inserting or deleting rows and columns. If rows are inserted, make
29
sure to copy the formulas in columns O, P, Q, and R into these new rows.
PERT/CPM Evaluation for Home
Construction Example (Figure 14-26 )
4. Enter the
activity
O
P in
durations
column N (cells
N8:N22 in this
example).
A
B
C
D
E
F
G
H
I
J
K
L
M
N
Q
R
1
PERT/CPM EVALUATION RESULTS
2
3 PROBLEM: Home Construction Example
4
5
On
Activity - Event Table
6
Critical
Events
TEend - TEbeg
7
Activities
Name
1
2
3
4
5
6
7
8
9
10
11 Duration Activities
Name
Path?
8
a
Excavating
-1
1
5
a
Excavating
5
Yes
9
b
Pour Foundation
-1
1
2
b
Pour Foundation
2
Yes
10
c
Outside plumbing
-1
1
6
c
Outside plumbing
24
No
11
d
Framing
-1
1
12
d
Framing
12
Yes
12
e
Inside plumbing
-1
1
10
e
Inside plumbing
10
Yes
13
f
Wiring
-1
1
9
f
Wiring
11
No
14
g
Roofing
-1
1
5
g
Roofing
5
No
15
h
Brickwork
-1
1
9
h
Brickwork
17
No
16
i
Plumbing inspection
-1
1
1
i
Plumbing inspection
1
Yes
17
j
Shingling
-1
1
2
j
Shingling
6
No
18
k
Cover walls
-1
1
3
k
Cover walls
3
Yes
19
l
Interior finish
-1
1
9
l
Interior finish
9
Yes
20
m
Exterior finish
-1
1
7
m
Exterior finish
7
No
21
n
Landscaping
-1
1
8
n
Landscaping
11
No
22
D1
Dummy 1
-1
1
0
D1
Dummy 1
0
No
23
N
24
Project
Solution (Earliest Event Times, TEj)
26
=M26
25
1
2
3
4
5
6
7
8
9
10
11
Time
26
0
5
7
19
29
24
24
30
31
33
42
42
Sum of Earliest Event Times =
244
27
R
O
P
Q
R
28
26 =SUM(C26:M26)
8 =A8 =IF(ISBLANK(B8),"",B8)
=SUMPRODUCT($C$26:$M$26,C8:M8) =IF(R38=1,"Yes","No")
29
9 =A9 =IF(ISBLANK(B9),"",B9)
=SUMPRODUCT($C$26:$M$26,C9:M9) =IF(R39=1,"Yes","No")
30
10 =A10 =IF(ISBLANK(B10),"",B10) =SUMPRODUCT($C$26:$M$26,C10:M10) =IF(R40=1,"Yes","No")
31
5. Click on Tools and then Solver to get the Solver Parameters dialog box shown next.
30
1. Enter the
project time, cell
R26, in the
Target Cell line,
either with or
without the $
signs.
Solver Paramters Dialog Box
(Figure 14-27)
NOTE: Normally all these entries appear in the Solver Parameter dialog
box so you only need to click on the Solve button. However, you
should always check to make sure the entries are correct for the
problem you are solving.
2. The Target
Cell is to be
minimized so
click on Min in
the Equal To
line.
3. Enter the
decision
variables in the
By Changing
Cells line,
C26:M26.
4. The constraints are entered in the Subject to Constraints box by using the
Add Constraints dialog box shown next (obtained by clicking on the Add
button). If a constraint needs to be changed, click on the Change button. The
31
Change and Add Constraint dialog box function in the same manner.
CEMCO:
The Add Constraints Dialog Box
1. Enter Q8:Q22 (or $Q$8::$Q$22) in
the Cell Reference line. These are
the differences between the earliest
times at the ending and beginning
nodes for each activity.
Normally, all
these entries
already appear.
You will need to
use this dialog
box only if you
need to add a
constraint.
3. Enter the activity
durations N8:N22 in
the Constraint line
(or =$N$8:$N$22).
4. Click the OK
button.
If you need to
change a
constraint, the
Change
Constraint
dialog box
functions just
like this one.
32
2. Enter >= as the sign because the differences in
ending and beginning earliest times cannot be less
than the activity durations, given next in Step 3.
PERT/CPM Evaluation for Home
Construction Example (Figure 14-26)
6. To find which activities are on the critical path (in column R) copy the portion of
the Sensitivity Report titled "Constraints" and paste it into this spreadsheet so that
theA word Constraints
is in
cell
N35
(for
this
example),
as shown
next.
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
1
PERT/CPM EVALUATION RESULTS
2
3 PROBLEM: Home Construction Example
4
5
On
Activity - Event Table
6
Critical
Events
TEend - TEbeg
7
Activities
Name
1
2
3
4
5
6
7
8
9
10
11 Duration Activities
Name
Path?
8
a
Excavating
-1
1
5
a
Excavating
5
Yes
9
b
Pour Foundation
-1
1
2
b
Pour Foundation
2
Yes
10
c
Outside plumbing
-1
1
6
c
Outside plumbing
24
No
11
d
Framing
-1
1
12
d
Framing
12
Yes
12
e
Inside plumbing
-1
1
10
e
Inside plumbing
10
Yes
13
f
Wiring
-1
1
9
f
Wiring
11
No
14
g
Roofing
-1
1
5
g
Roofing
5
No
15
h
Brickwork
-1
1
9
h
Brickwork
17
No
16
i
Plumbing inspection
-1
1
1
i
Plumbing inspection
1
Yes
17
j
Shingling
-1
1
2
j
Shingling
6
No
18
k
Cover walls
-1
1
3
k
Cover walls
3
Yes
19
l
Interior finish
-1
1
9
l
Interior finish
9
Yes
20
m
Exterior finish
-1
1
7
m
Exterior finish
7
No
21
n
Landscaping
-1
1
8
n
Landscaping
11
No
22
D1
Dummy 1
-1
1
0
D1
Dummy 1
0
No
23
N
24
Project
Solution (Earliest Event Times, TEj)
26
=M26
25
1
2
3
4
5
6
7
8
9
10
11
Time
26
0
5
7
19
29
24
24
30
31
33
42
42
Sum of Earliest Event Times =
244
27
R
O
P
Q
R
28
26 =SUM(C26:M26)
8 =A8 =IF(ISBLANK(B8),"",B8)
=SUMPRODUCT($C$26:$M$26,C8:M8) =IF(R38=1,"Yes","No")
29
9 =A9 =IF(ISBLANK(B9),"",B9)
=SUMPRODUCT($C$26:$M$26,C9:M9) =IF(R39=1,"Yes","No")
30
10 =A10 =IF(ISBLANK(B10),"",B10) =SUMPRODUCT($C$26:$M$26,C10:M10) =IF(R40=1,"Yes","No")
31
7. If all the earliest event times are desired, run Solver once more using the sum of the
33
earliest event times (cell R26) as the target cell.
Solver’s Sensitivity Report
To get Solver’s Sensitivity Report, highlight
Sensitivity Report in the Report box of the Solver
Results dialog box before clicking the OK button.
34
Portion of Sensitivity Report
Showing Critical Path (Figure 14-28)
Column R shows the critical path: a 1 indicates the corresponding
activity is on the critical path and a 0 that it is not.
N
O
35 Constraints
36
37
Cell
38
$Q$8
39
$Q$9
40
$Q$10
41
$Q$11
42
$Q$12
43
$Q$13
44
$Q$14
45
$Q$15
46
$Q$16
47
$Q$17
48
$Q$18
49
$Q$19
50
$Q$20
51
$Q$21
52
$Q$22
35
P
Name
Excavating TEend - TEbeg
Pour Foundation TEend - TEbeg
Outside plumbing TEend - TEbeg
Framing TEend - TEbeg
Inside plumbing TEend - TEbeg
Wiring TEend - TEbeg
Roofing TEend - TEbeg
Brickwork TEend - TEbeg
Plumbing inspection TEend - TEbeg
Shingling TEend - TEbeg
Cover walls TEend - TEbeg
Interior finish TEend - TEbeg
Exterior finish TEend - TEbeg
Landscaping TEend - TEbeg
Dummy 1 TEend - TEbeg
Q
R
Final
Value
Shadow
Price
5
2
24
12
10
11
8
20
1
3
3
9
7
8
0
S
1
1
0
1
1
0
0
0
1
0
1
1
0
0
0
T
U
Constraint Allowable Allowable
R.H. Side Increase Decrease
5
1E+30
5
2
1E+30
7
6
18
1E+30
12
1E+30
11
10
1E+30
2
9
2
1E+30
5
3
1E+30
9
11
1E+30
1
1E+30
2
2
1
1E+30
3
1
3
9
1
3
7
3
1
8
3
1
0
3
1