BA 560 Management of Information System

PERT Program Evaluation and Review Technique

Estimation of Task Times In CPM, we assume that the task durations are known with certainty. This may not be realistic in many project settings.  How long does it take to design a switch?

 PERT tries to account for the uncertainty in task durations. Key question: What is the probability of completing project by given deadline? PERT

SEEM 3530

2

CPM vs. PERT CPM (critical path method) PERT (program evaluation and review technique) Both approaches work on a project network, which graphically portrays the activities of the project and their relationships.  CPM assumes that activity times are deterministic, while PERT views the time to complete a task as a random variable. PERT

SEEM 3530

3

Estimation of the duration of project activities (1) The deterministic approach (CPM), which

ignores uncertainty thus results in a point estimate (e.g. The duration of task 1 = 23 hours, etc.) (2) The stochastic approach (PERT), which considers the uncertain nature of project activities by estimating the expected duration of each activity and its corresponding variance. To analyse the past data to construct the probabilistic distribution of a task. PERT

SEEM 3530

4

Estimation of the activity duration Example: An activity was performed 40 times in the past, requiring a time between 10 to 70 hours. The figure below shows the frequency distribution.

PERT

SEEM 3530

5

Estimation of the activity duration

The probability distribution of the activity is approximated by a probability frequency distribution.

PERT

SEEM 3530

6

Estimation of the activity duration

In project scheduling, we usually use a beta distribution to represent the time needed for each activity.

PERT

SEEM 3530

7

Estimation of the activity duration  Three key values we use in the time estimate for each activity:

a = optimistic time, which means that there is little

chance that the activity can be completed before this time; m = most likely time, which will be required if the execution is normal; b = pessimistic time, which means that there is little chance that the activity will take longer.

PERT

SEEM 3530

8

Estimation of Mean and SD  The expected or mean time is given by: D= (a+4m+b)/6

The variance is: V = (b-a) 2/36  The standard deviation is (b - a)/6  For our example (Figure 7-3), we have a=10, b=70, m=35. Therefore D=36.6, and V2 =100. PERT

SEEM 3530

9

Estimation of Mean and SD Beta-distribution

a

m

b

a  4m  b t Expected task time: 6 2 ba ba 2 )  ( Standard deviation:   6

PERT

SEEM 3530

6

10

The PERT Approach The PERT (Program evaluation and review technique) approach addresses situations where uncertainties must be considered.

PERT

SEEM 3530

11

The PERT Approach (cont’d)  Now assume that the activity times are independent random variables.  Further, assume that there are n activities in the project, k of which are critical. Denote the activity times of the critical activities by the random variables di with mean E(di) and variances V(di), for i=1,2, …, k.  Then, the total project time (the total length of the critical path) is the random variable:  X= d1 + d2 +,…, +dk PERT

SEEM 3530

12

The PERT Approach (cont’d)  The mean project length, E(X), and its variance, V(X): E(X)= E(d1)+E(d2)+,…, +E(dk) V(X)= V(d1)+V(d2)+,…, +V(dk)

 Assumption:

 Activity times are independent random variables.  The project duration (=sum of times of activity on a critical path) is normally distributed.  Based on the Central Limit Theorem, which states that the distribution of the sum of independent random variables is approximately normal when the number of terms in the sum if sufficiently large.

PERT

SEEM 3530

13

The PERT Approach (cont’d)  Using a normal distribution, the probability of completing the project in not more than some given time T: X-E(X) T -E(X) T -E(X) P(X  T) = P( ------------  ------------- ) = P(Z  ----------) V(X)1/2 V(X)1/2 V(X)1/2 where Z is the standard normal deviate with mean 0 and variance 1. • The probability for P(Z < ), given any , can be found using normal distribution tables.

PERT

SEEM 3530

14

PERT

SEEM 3530

15

Example: Shopping Mall Renovation Activity A: Prepare initial design B: Identify new potential clients C: Develop prospectus for tenants D: Prepare final design E: Obtain planning permission F: Obtain finance from bank G: Select contractor H: Construction I: Finalize tenant contracts J: Tenants move in PERT

IP a 1 4 A 2 A 1 D 1 E 1 D 2 G, F 10 B, C, E 6 I, H 1

SEEM 3530

m 3 5 3 8 2 3 4 17 13 2

b 5 12 10 9 3 5 6 18 14 3 16

Example: Issues to Address 1. Schedule the project. 2. What is the probability of completing the project in 36 weeks?

PERT

SEEM 3530

17

Expected Activity Time and SD Act A B C D E F G H I J PERT

a 1 4 2 1 1 1 2 10 6 1

m 3 5 3 8 2 3 4 17 13 2

b 5 12 10 9 3 5 6 18 14 3

t 3 6 4 7 2 3 4 16 12 2 SEEM 3530

1 4 3  5 2 t 3 6 0.44 1.78 1.78   (124 ) 1.78 6 1.78 0.11 0.44 0.44 1.78 1.78 0.11 2

2

18

CPM with Expected Activity Times I,12

B,6

1

C,4

J,2

E,2

End

F,3 A,3

PERT

D,7

G,4

SEEM 3530

H,16

19

Critical Path and Expected Time 1. Critical path: A-D-E-F-H-J.

2. Expected Completion time: 33 weeks 3. What is the probability to complete the project within 36 weeks? -- Use the critical path to assess the probability

PERT

SEEM 3530

20

Probability Assessment Expected project completion time: Sum of the expected activity times along the critical path. Used to obtain probability of project  = 3+7+2+3+16+2 = 33 completion

Variance of project-completion time Sum of the variances along the critical path.

2 = 0.44+1.78+0.11+0.44+1.78+0.11= 4.66  = 2.15 PERT

SEEM 3530

21

Assessment by Normal Distribution P(X  36) = ? Assume X ~ N(33, 2.152)

Normal Distribution  = 2.15

-  36 - 33 T = = 1.4 z = .  2.15

Standardized Normal Distribution

 = 33 36 PERT

P(Z  1.4) = ?

 =1 z

X SEEM 3530

 = 0 1.4 z

Z 22

Obtain the Probability Standardized Normal Probability Table (Portion) Z

.00

.01

.02

P(Z