Pert

Activity Duration
1. “Guess” the activity duration,
• especially smaller activities.
2. Use Published materials:
– Crew A= 1 Foreman, 2 Masons, 1 Labor
– Crew A can install 200 12 Blocks/day

Spring 2008, PERT 1
King Saud University Dr. Khalid Al-Gahtani

Activity Duration
3. Use companies historical data:
– similar to method#2.
– the jobs must be similar.
– Field input helps to modify historical data.
4. Use the estimated labor costs to determine the
activity duration. Ex)
– Labor cost = $2000.
– Worker = $100/day
– task will take 20 worker days.
– If a 4 workers crew is used, t= 5 days.
– Estimate must be accurate.

Spring 2008, PERT 2

Example of using labor cost
• A masonry facade consisting of 3,800 ft2,
• total cost per worker hour = $31.5,
• total estimated cost of labor = $10,500,
• assuming 8-hour work/days and a crew of
6 workers;

Spring 2008, PERT 3

How many days should be allowed to complete this task?

(i) da ys sho uld be a llo w ed to co m p lete this task is:
2
V = 3,800 ft
R ate = $31.5/hr
T otal la bo r co st = $10,500
$ 10,500
N u m ber o f ho urs requ ired = = 333.33 hrs
$31.5/hr
3 3 3 .3 3 h rs
D uratio n = = 6.94 7 days
8 h rs/d ay.w orkers 6 w o rkers

Spring 2008, PERT 4

What is the production rate that the crew must attain to
keep the project on schedule and within the budget?

(ii) T he pro ductio n rate that the crew m ust attain to keep the pro ject
o n sc hedu le a nd w ithin the budget is:
2
3,800 ft 2
T heo retica l P ro ductio n R ate = = 547.2 ft /day
6.94 days
2
3,800 ft 2
A ctua l P ro ductio n R ate = = 542.8 ft /day
7 days

Spring 2008, PERT 5

Example of using Historical Data
# o f 12 B lo ck s B lo cks o n this
t (days) B lo cks/da y
o n P ro ject pro ject
8,000 16 500 22
12,000 22 546 20.2
9,000 19 474 23.2
7,000 15 467 23.6
13,000 20 , 650 16.6
14,0000 31 452 24.4

n
21.7
n
2.76
xi ( xi X)
2

i 1 i 1

n n 1

Spring 2008, PERT 6

Example of using Historical Data
= 21.7 days, = 2.76 days
- H o w ma ny da ys in o rder to be 85% ? t 85 % :
at 85%  x= 1.04 days (T able A ppendix 9-1b)
t-μ
x=  t 85 % = + (x× )
σ
= 21.7 + (1.04 × 2.76) = 24.6 25 days
,

- W hat % o f tim es < 28 da ys (% o f (t < 28 days)):
t-μ 28 - 21.7
x= = = 2.28
σ 2.76
% o f (t < 28 days) = 0.9887 = 98.87%
 at x= 2.28 from T able Appendix 9 -1b
Spring 2008, PERT 7

Program Evaluation and Review
Technique [PERT]
• Activity time is very much probabilities.
• three activity durations should be
estimated for each activity:
– Optimistic duration =a =4
– Pessimistic duration =b =7
– Most Likely duration =m =6

Spring 2008, PERT 8

• Optimistic Duration (a): Very favorable
conditions. Very low probability of being
completed within this duration.
(say p = 5%, shortest time)
• Pessimistic Duration (b): Activity
performed under very unfavorable
conditions. Again, very low probabilities
(say p= 5%)
• Most Likely Duration (m): Usually closet to
the actual durations. Very high probability.
Spring 2008, PERT 9

If this activity is performed a large number of times and
record of the actual durations is maintained, a plot of
frequencies of such durations will give the beta-curve (an
unsymmetrical curve).
Spring 2008, PERT 10

Example I

0.5

t e (5.8) m (6)

Since m > te ( 6 > 5.8 )
The person making activity estimates was pessimistic


Example II

0.5

a (4) m (6) t e (7) b (18)

Here, since te > m, the person making this estimate was
optimistic.


Even thought „t‟ has a Beta distribution, T N( , 2)

T or


Variance
A ct ivit y ( A ) A ct ivit y (B )

a = 4 2

m = 6 6

b = 8 10
4 24 8 2 4(6) 10
te = 6 6
6 6

8 4 10 2
te ( A )
1 . 25 te ( B )
2 .5
3 .2 3 .2


Variance
• 2 (te) = [(b-a) /3.2]2
– (5%-95% Assumption)
• 2
(te) = Uncertainty about the activity
durations, where:
– If (b-a) is a large figure, greater uncertainty.
– If (b-a) is small amount, less uncertainty.


Project Duration (Te)
• Determine te for each activity
• Determine Slacks and Project Duration
(Te) by forward and backward passes as in
a CPM network.
• P (the project will be finished as time Te)
– or p(Te) = 0.5, Since
• p(te) = 0.5 i = 1, 2, 3, …, n


• Te‟s follow a normal distributions, and not beta-
distributions as activity durations te‟s do.

Te = t e*

Example 1
A B C D

a = 4 3 2 4
m = 6 8 4 5
b = 8 9 7 6
2 2 2 2 2 2
(t e ) = [(b-a)/3.2 ] (1.25 ) (1.875 ) (1.5625 ) (0.625 )
te 6 7.33 4.17 5

P ro ject D uratio ns T e = 6 + 7.33 + 4.17 + 5
o r, T e = 22.50 da ys

2 2 2 2
(T e ) = 1 .2 5 1 .8 7 5 1 .5 6 3 0 .6 2 5

= (T e ) = 2.81 3

TE TE

TE 2 TE

TE 3 TE

18.0 19.5 21.0 2 4.0 25.5 27.0
22.5
TE


Central Limit Theorem
• if number of CA > 4, the distribution of T is
approximately normal with mean T and variance
Vt given by:
– T = te1+ te2 + ……, tem (sum of the means)
– Vt = vt1 + vt2 + ……+ vtm (sum of the variances)
• The distributions of the sum of activity times will
BE NORMAL regardless of the shape of the
distribution of actual activity performance times.


PERT Computations
1)
(i) C o m pu te:
- E xpected activity d u ratio n ( t e )
a 4m b
te = w here
6
a = O ptim ist ic d uratio n
m = m o st lik e ly du rat io n
b = pessim ist ic du ratio n
- S tand ard d ev iatio n o f an act iv ity ( te)
b a
(t e ) =
3 .2


PERT Computations
( i)
2) C o m pute:
2
- V ar ia nce o f an act ivit y ( te)
2
2 b a
(t e ) =
3 .2
3)
( ii) D o C P M ana lysis, using ‘t e ’ as activit y tim e s.
*
4)
( iii) Ident ify C r it ica l A na lys is. t e
( iv) P ro ject T im e (T e )
5)
*
Te = te


PERT Computations
(6 ) C o m p u te:
i)
2
- P ro ject V aria nce (T e )
2 *
(T e ) = ( te )
- F o r m u lt ip le crit ica l p aths, co nsid er
the h ig he st to tal o f v ar ia nces .
- P ro ject Standard D ev iatio n (T e )
2 *
(T e ) = (te ) or
2
b a
(T e ) =
3 .2


Probability of Meeting a schedule Date
STEPS
i) Set the variance (Vt) of the initial event to zero,
ii) Ignore other scheduled dates (if any),
iii) Compute the mean duration of the longest path
(critical path) to the scheduled date event,
iv) Compute the total variance to the scheduled date
event using the longest path. For multiple critical paths,
consider the highest total of variances.
v) P (T ≤ Ts ) = P (Z ≤ z)
Ts -
z= Table 9.1 given z  find P
t

Example 2:
D uratio n E stim ates
activit y D epend o n a m b
A 1 1 7
B 1 4 7
C 2 2 8
D A 1 1 1
E B 2 5 14
F C 2 5 8
G E 3 6 15

Compute the followings:
1. Project mean duration and variance.
2. Probability of completing the project three days earlier than expected.
3. Probability of completing the project three days later than expected.
4. The date for the terminal event that meets a probability of being
finished with the project at or less than 84% of the time.
5. Probability of completing activity E by day 9.

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