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P R T

I.K. Gunarta

Department of Industrial EngineeringITS

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The technique is based on theassumption that an activitys duration

follows a probability distributioninstead of being a single value.

The probabilistic information about the

activities is translated into probabilisticinformation about the project.

Project Evaluation and Review Technique (PERT)

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PERT

Three time estimates are required to

compute the parameters of an activitys

duration distribution:

pessimistic time (tp ) - the time the activity wouldtake if things did not go well

most likely time (tm ) - the consensus best estimateof the activitys duration

optimistic time (to ) - the time the activity wouldtake if things did go well

. . . more

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From these three time estimates aboutan activity, two probability distribution

parameters are calculated: the mean(te ) and the variance (Vt ).

te = ( to + 4tm + tp ) / 6

Vt = [ ( tp - to ) / 6 ]2

PERT

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Steps in PERT Analysis

Draw the network.

Analyze the paths through the network

and find the critical path. The length of the critical path is the

mean of the project duration

probability distribution which isassumed to be normal.

. . . more

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Steps in PERT Analysis

The standard deviation of the projectduration probability distribution is

computed by adding the variances ofthe critical activities (all of the activitiesthat make up the critical path) andtaking the square root of that sum

Probability computations can now bemade using the normal distributiontable.

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Contoh

Aktivitas A M B Te [(b a)/6]2

A 17 29 47 30 25

B 6 12 24 13 9C 16 19 28 20 4

D 13 16 19 16 1

E 2 5 14 6 4

F 2 5 8 5 1

Berapa probabilitas proyek selesai dalam waktu kurang dari 67 hari?

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Network

0 A

B

C

D

E

F

30

30

30

30

13

43

50

20

43

16

59

50

6

56

59

5

64

6459

59

59

43

53

4330

5333

300

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Possible Project Duration

Ts = 67Te = 64

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Solusi

V Path = VA + VB + VD + VF

V Path = 25 + 9 + 1 + 1

path = 6

Z = (67 64)/

= + 0.5

P = 0.69

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Immed. Optimistic Most Likely Pessimistic

Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.)

A -- 4 6 8

B -- 1 4.5 5C A 3 3 3D A 4 5 6E A 0.5 1 1.5F B,C 3 4 5G B,C 1 1.5 5

H E,F 5 6 7I E,F 2 5 8J D,H 2.5 2.75 4.5K G,I 3 5 7

PERT Example

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PERT Example

ACTIVITY PREDECESSOR OPTIMISTIC

TIME

MOST

LIKELY

TIME

PESSIMISTIC

TIME

A - 4 6 8

B - 1 4,5 5

C A 3 3 3

D A 4 5 6

E A 0,5 1 1,5

F B, C 3 4 5

G B, C 1 1,5 5

H E, F 5 6 7

I E, F 2 5 8

J D, H 2,5 2,75 4,5

K G, I 3 5 7

Probability the project will be completed within 24 days?

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PERT Network

A

D

C

B

F

E

G

I

H

K

J

PERT Example

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Activity Expected Time and Variance

Activity Expected Time Variance

A 6 4/9

B 4 4/9C 3 0D 5 1/9E 1 1/36

F 4 1/9G 2 4/9H 6 1/9I 5 1J 3 1/9K 5 4/9

PERT Example

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Earliest/Latest Times

Activity ES EF LS LF Slack A 0 6 0 6 0 *critical

B 0 4 5 9 5

C 6 9 6 9 0 *D 6 11 15 20 9E 6 7 12 13 6F 9 13 9 13 0 *G 9 11 16 18 7

H 13 19 14 20 1I 13 18 13 18 0 *J 19 22 20 23 1K 18 23 18 23 0 *

PERT Example

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Probability the project will be completedwithin 24 hours

Vpath = VA + VC + VF + VI + VK= 4/9 + 0 + 1/9 + 1 + 4/9

= 2

path = 1.414

z = (24 - 23)/ = (24-23)/1.414 = .71

PERT Example

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Probability the project will be completedwithin 24 hours

From the Standard Normal Distributiontable:

P(z < .71) = .5 + .2612 = .7612

23 24

.5000

.2612

PERT Example