Download ASSUMPTIONS: • Activities are statistically independent (no

PERT
(PROGRAM EVALUATION AND REVIEW TECHNIQUE)
ASSUMPTIONS:
• Activities
are
statistically
independent
(no
corrective actions)
• The critical path is significantly longer than any
near critical path
• The critical path can analysed on the basis of the
central limit theorem (each activity gives little
contribution to the total variance)
LIUC - Master in Project Management
Enrico Cagno
©
• The duration of the single activity is described by
independent statistical variables
• Every single activity is described by a Beta
2
distribution with mean t and variance σ t
• The
estimate
of
the
distribution
parameters
considers three values a, m, b, corresponding to
optimistic, the most likely and the pessimistic
duration of the activity considered (5% percentile;
mode; 95% percentile)
LIUC - Master in Project Management
Enrico Cagno
©
• The estimate of the variance is given by:
2
σ
2
t
 (b − a ) 
=

6

N.B.: as a rule of a thumb for unimodal distributions
standard deviation corresponds to a 1/6 of the
variation range of the considered variable. This rule
stems from the fact that at least 89% of every
distribution can be found within 3 standard
deviations from the mean; for the normal distribution
the percentage grows over 99,7%
LIUC - Master in Project Management
Enrico Cagno
©
• The value t can be approximated by:
t=
( a + 4m + b )
LIUC - Master in Project Management
6
Enrico Cagno
©
BETA DISTRIBUTION
The distribution of the duration of any activity is
estimated by means of a beta distribution defined
by the function:

0

α
γ
f ( t ) =  k ⋅ ( t − a ) ⋅ (b − t )

0

−∞ < t < a
a ≤ t ≤ b; a , b ≥ 0
α , β > −1
As a matter of fact, the actual distribution is not
known
so
that
the
formulae
proposed
are
approximations.
LIUC - Master in Project Management
Enrico Cagno
©
Nevertheless
this
distribution
has
some
good
properties:
• two non-negative intercepts;
• continuous;
• unimodal;
• low probabilities of occurrence given to the
pessimistic (a) and optimistic (b) durations;
• high flexibility;
But…why not triangular distribution?
LIUC - Master in Project Management
Enrico Cagno
©
The project expected duration is given by:
N
TM =
∑t
i
i=1
and the corresponding variance by:
N
2
T
σ =
∑σ
i =1
2
ti
⇒ Normal distribution and central limit theorem
(n>15; al least n>6)
LIUC - Master in Project Management
Enrico Cagno
©
• The probability that the project completion time will
be shorter than a fixed one (Ts) will be
Ts − TM 

P { T ≤ Ts } = P  z ≤

σT 

where z dove z represents the statistical variable
from a Standardized Normal Distribution.
LIUC - Master in Project Management
Enrico Cagno
©
EXAMPLE (AOA):
Act.
Pred.
a
m
b
t
σ t2
A
-
2
4
12
5
2.78
B
-
3
6
9
6
1.00
C
A
1
2
9
3
1.78
D
A
3
3
9
4
1.00
E
B
1
2
3
2
0.11
F
B
2
8
8
7
1.00
G
C
1
2
9
3
1.78
H
D, E
4
5
12
6
1.78
I
F
1
3
5
3
0.44
LIUC - Master in Project Management
Enrico Cagno
©
C (3)
2
5
A (5)
G (3)
D (4)
1
4
H (6)
7
E (2)
I (3)
B (6)
3
F (7)
C (3)
6
G (3)
A (5)
D (4)
H (6)
E (2)
B (6)
I (3)
F (7)
LIUC - Master in Project Management
Enrico Cagno
©
B-F-I (critical path)
T1 = 1 6
σ T2 = 2 .4 4
1
σ T = 1.5 6
1
P { B − F − I ≤ 1 5}
z =
15 − T1
15 − 16
=
= − 0 . 64
σ T1
1 . 56
Φ ( 0 .6 4 ) = 0 .2 4
Φ ( − 0 . 6 4 ) = 0 .5 − 0 . 2 4 = 0 . 2 6
LIUC - Master in Project Management
Enrico Cagno
©
A-D-H (near critical path)
T2 = 1 5
σ T2 = 5.5 6
2
σ T = 2 .3 6
2
P { A − D − H ≤ 1 5}
z =
15 − T 2
15 − 15
=
= 0
σ T2
2 . 36
Φ ( 0 ) = 0 .5
LIUC - Master in Project Management
Enrico Cagno
©
ex 1 (AON)
Activity
Predecessor
a
m
b
A
-
12
14
16
B
-
1
3
5
C
A, B
2
5
5
D
B
5
7
15
E
C, D
2
4
6
F
E
6
10
14
a) Determine the critical path.
b) Determine the duration of the project with a
probability of 90%.
c) Determine the probability of completing the project
within 31 months and 33 months.
LIUC - Master in Project Management
Enrico Cagno
©
a)
Activity Predecessor a
m
b
t
σt
σ2t
A
-
12 14 16
14
2/3 0.44
B
-
1
3
5
3
2/3 0.44
C
A, B
2
5
5
4.5
1/2 0.25
D
B
5
7 15
8
5/3
E
C, D
2
4
6
4
2/3 0.44
F
E
6 10 14
10
4/3
2.8
1.8
A
C
B
E
F
D
LIUC - Master in Project Management
Enrico Cagno
©
There are three paths in this tree: A-C-E-F, B-C-E-F e
B-D-E-F.
Path
A-C-E-F
B-C-E-F
B-D-E-F
T
σ2T
31.2 2.93
21.2 2.93
26 5.48
σT
1.7
1.7
2.3
The critical path in terms of expected duration is A-CE-F.
Nevertheless it’s clear that also path B-D-E-F should
be investigated as, even if its expected duration is 26,
its variance is twice bigger than the one of path A-CE-F.
LIUC - Master in Project Management
Enrico Cagno
©
b)
Duration of the project with a probability of 90%:
Path
D90
A-C-E-F
33.4
B-C-E-F
23.4
B-D-E-F
29
Path A-C-E-F is critical e there is a probability of 90%
to complete the project within 33,4 months.
c)
The completion probability with 31 months is near
50% (within 33 months is lower than 90%)
LIUC - Master in Project Management
Enrico Cagno
©
ex 2 (AON; succ. starts same day pred.)
Duration [weeks]
Activity Predecessor
Min
Mode
Max
A
-
10
22
28
B
A
4
4
10
C
B
4
6
14
D
B
1
2
3
E
C
1
5
9
F
D
7
8
9
G
E,F
2
2
2
H
G
5
7
11
I
G
3
6
9
J
H,I
5
7
15
LIUC - Master in Project Management
Enrico Cagno
©
1) Calculate the probability to complete the project
within 50 weeks;
2) Starting from the 55th week, for each week of delay
- for 4 weeks span – a fee of 2,5% of the contract
value will be paid. Estimate the risk to pay a fee.
LIUC - Master in Project Management
Enrico Cagno
©
1)
Activity
t
σ2
(min+4mode+max) / 6 [(max - min) / 6]2
A
21
324/36
B
5
1
C
7
100/36
D
2
4/36
E
5
64/36
F
8
4/36
G
2
0
H
7,33
1
I
6
1
J
8
100/36
The critical path is:
A,B,C,E,G,H,J
N.B. The successor activity stars the same day the
predecessor one ends.
LIUC - Master in Project Management
Enrico Cagno
©
0
21
21
21
26
5
A
0
26
21
C
26
33
7
B
21
33
26
38
5
E
33
33
38
38
40
40
47,33
2
7,33
G
38
H
40
40
47,33 55,33
47,33
8
J
26
28
28
2
D
28
36
40
8
30
47,33 55,33
6
I
F
30
46
38
41,33
47,33
Considering only critical path activities:
Tm =  Σ ti (critical) = 55,33
σ2 = Σ  σi2 (critical) = 660/36 = 18,33
σ = 4,28
P ( T ≤ 50 ) = P [ z ≤ (50 – Tm) / σ] = P ( z
≤ 1,245 ) = 10,66%
LIUC - Master in Project Management
Enrico Cagno
©
2)
Calculating
the
probability
that
the
project
is
completed beyond 55th, 56th, 57th and 58th week:
P ( T > 55 ) =
P ( z ≤ 0,07 )
= 52,8 %
P ( T > 56 ) = 1 - P ( z ≤ 0,16 ) = 43,6 %
P ( T > 57 ) = 1 - P ( z ≤0,39 ) = 34,8 %
P ( T > 58 ) = 1 - P ( z ≤0,62 ) = 26,8 %
P (55 < T ≤  56) = P (T ≤ 56) - P (T ≤ 55) =
= P (T > 55) - P (T > 56) =
= 52,8 - 43,6 = 9,2 %
P (56 < T ≤ 57) = 43,6 - 34,8 = 8,8 %
P (57 < T ≤ 58) = 34,8 - 26,8 = 8 %
P (T > 58 ) = 26,8 %
R = 0,092 ⋅ 2,5% + 0,088 ⋅ 5% + 0,080 ⋅ 7,5% +
0,268 ⋅ 10% = 3,95 %
LIUC - Master in Project Management
Enrico Cagno
©
PERT FLAWS:
• overlooks near critical paths;
• assumes independent activities;
• undervalues merging points;
• underestimates project duration.
For example:
Activity 1
10, 12, 14
Milestone A
Activity 2
10, 12, 14
Milestone B
Activity 3
10, 12, 14
⇒ little used (except PERT-like estimations)
LIUC - Master in Project Management
Enrico Cagno
©