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Program Evaluation and Review Technique (PERT)

4. PROGRAM EVALUATION AND REVIEW TECHNIQUE (PERT)

 

PERT was originally developed in 1958 to 1959 as part of the Polaris Fleet Ballistic Missile Program of the United States' Navy.

 

The primary difference . between PERT and CPM is that PERT takes explicit account of the uncertainty in the activity duration estimates. CPM is activity oriented whereas PERT is event oriented. CPM gives emphasis on time and cost whereas PERT is primarily concerned with time.

 

In PERT, the probability distribution is specified by three estimates of the activity duration most likely duration (t111), an optimistic duration (t0) and a pessimistic duration (t p). This type of activity duration is assumed to follow the beta distribution with and

 

Mean = to + 4 t m + 1 p/ 6

Variance = ( T p – t 0 /6 )2

 

The network construction phase of PERT is identical to that of CPM. Furthermore, once mean and variance are computed for each activity, the critical path determination is identical to CPM. The earliest and latest event times for the network are random variables. Once the critical path is determined, probability statements may be made about the total project duration and about the slack at any event.

Example 3. A project consists of the following activities and different time estimates :

 

Activity           t 0          t m                            tp

1-2                   3          5                      8

1 - 3                  2          4                        8

1 -4                  6          8                       12

2-5                   5          9                      12

3-5                   3          5                        9

4 -6                  3          6                       10

5-6                   2          4                       8

 

(a) Draw the network.

(b) Determine the expected time and variance for each activity.

(c) Find the critical path and the project variance.

(d) What is the probability that the project will be completed by 22 days?

 

Solution. (a) Using the given information the resulting network is drawn as follows

Expected Time = t0 + 4tm + tp / 6 = tij

T12 = 5.17             t25 = 8.83              t=4.33

T13 =4.33              t35 = 5.33

T14 = 8.33             t46 =6.17

 

Varience = (tp – t0/ 6)2

 

σ212 = 0.694       σ =225 = 1.361   σ256 = 1

σ213 = 1               σ235 = 1

σ214 = 1               σ246 = 1.361

 

(c) Set                                   Es1 = Es1 + t12 =5.17

ES3 = ES1 = t13 = 4.33

ES4 = Max { ES3 = t35, ES2 + t25}

=Max {9.66,14} = 14

ES6 = Max { ES5 =t56, ES4+ t46}

=Max{18.33,14.5} = 18.33

Set                                                                         LF6 = ES6 =18.33

Then                                                                      LF5= LF6 – t56 = 14

LF4 =LF6 – t46 = 12.16

LF3 =LF5 – t35 = 8.67

LF2 =LF5 – t25 = 5.17

LF1 = Min { LF3 – t13, LF2 – t12, LF4 – t14}

=MIN. {4.34 , 0 , 3.83} = 0

 

 

 

 

 

Hence the critical path is (1) - (2)-(5)-(6)

 

Project variance = σ 2 122 252 56 = 0.694 + 1 .361 + 1 = 3.055.

 

(d) Here mean project length is 18.33.

Set                               z =x – 18 .33/√3 .055-~ n (0,1)

x = 22 , z = 2.1

For

 

Then the required probability

= p ( X ≤ 22)

= P ( Z ≤ 2.1)

= 0.5 +0.4821

= 0 .9821

=> there is 98.21% chance that the project will be completed by 22 days.

Example 4. A PERT network consists of I 0 activities. The precedence relationships and expected time and variance of activity times, in days, are given below :

 

Activity                                      a      b      c     d      e      f      g      h      i         j

Immediate                                -     a       a    -      b       c   d       d        e,f,g    h

predecessor (s)

Expected activity                     4     2     6     2     3       9     5     7      1       10

time

Variance of activity time        1      1      2     1    1      5      1     8        1      16

 

Construct an arrow diagram. Find the critical path based on expected times. Based on this

critical path find the probability of completing the project in 25 days.

 

 

 

Solution. The resulting network is given in Fig. 8.4.

Set                                           ES 1 = 0

ES 2 = ES1 + t12 = 4

ES 3 = ES2 + t23 = 6

ES 4 = ES2 + t24 = 10

ES 5 = ES1 + t15 = 2

 

ES6 = Max. {ES3 +  t 36 ES4 + t 46, ES5 + t56} = 19

ES7 = ES5 + s57 = 9

ES8 = Max. {ES6 + t 68, ES7 + t78} = 20

Example 4. A PERT network consists of I 0 activities. The precedence relationships and expected time and variance of activity times, in days, are given below :

 

Activity                                      a      b      c     d      e      f      g      h      i         j

Immediate                                -     a       a    -      b       c   d       d        e,f,g    h

predecessor (s)

Expected activity                     4     2     6     2     3       9     5     7      1       10

time

Variance of activity time        1      1      2     1    1      5      1     8        1      16

 

Construct an arrow diagram. Find the critical path based on expected times. Based on this

critical path find the probability of completing the project in 25 days.

 

 

 

Solution. The resulting network is given in Fig. 8.4.

Set                                           ES 1 = 0

ES 2 = ES1 + t12 = 4

ES 3 = ES2 + t23 = 6

ES 4 = ES2 + t24 = 10

ES 5 = ES1 + t15 = 2

 

ES6 = Max. {ES3 +  t 36 ES4 + t 46, ES5 + t56} = 19

ES7 = ES5 + s57 = 9

ES8 = Max. {ES6 + t 68, ES7 + t78} = 20

Set  ES6 =       LF 8 = 20

 

LF 7 = LF8 –t78 = 10

 

LF 6 = LF8 – t 68 = 19

 

LF5 = Min. {LF7 – t57, LF6 –t56 } = 3

 

LF 4 = LF6 – t 46 = 1 0

 

LF 3 = LF6 – t36 = 16

 

LF2 = Min. {LF3 – t 23, LF4 – t 24} = 4

 

LF1 = Min. {LF2 – t12 , LF5 - t5}

 

Hence the critical path is a – c-  f-  i on which ES = LF

Total expected time = 4 + 6 + 9 + 1 = 20

Project variance = 1 + 2 + 5 + 1 = 9

 

Z =x- 20/3 ~ N (0,1)

 

For x = 25, z = 1.67

Then the required probability = P(X ≤ 25)

= P(z ≤ 1.67)

= 0.5 + Φ (1.67)

= 0.5 + Q.4525

= 0.9525

::::> There is 95.25% chance that the project will be completed by 25 days.

 

 

1. For a small project of 12 activities, the details are given below :

Activity               Dependence                         Duration (9days)

 

A                                  -                                              9

B                                  -                                              4

C                                  -                                              7

D                                 B,C                                          8

E                                  A                                            7

F                                  C                                             5

G                                  E                                             10

H                                 E                                              8

I                              D,F,H                                          6

J                                    E                                             9

K                                  I ,J                                          10

L                                    G                                           2

 

 

(a) Draw a network for the following project:

 

Activities       1-2       1-3       1-4      2-5       2-6       3-6        5-7     6-7      4-7

Time (days)    8          12        4          9         3         9            3           6          5

 

 

(b) Determine total slack time all activities and identify  the identify the critical path

(c) Calculate  total  floats and free floats of  each activities  .

 

3. Consider the following informations :

Job               1-2       2-3      2-4       3-4       3-5       3-6       4-5       5-6

Time (days) 10        9          7          6          9          10        6          7

 

(a) Draw the network

(b) Find the critical path.

(c) Calculate total floats and free floats of each activities.

 

Draw the network  network using the given precedence conditions. Calculate the critical path and floats (total  and free))

 

Activity              A      B         C         D         E          F          G         H         I           J

Immediate         -         A         A         A         D         D         E          F,G      C,H    B

Predecessor (s)  1        4          2          2          3          3          2          1          3          2

Duration             1       4          2          2          3          3         2          1         3         2

 

 

5. A project consists of eight activities with the following time estimates:

Activity                                   Immediate                   Time (days)

Predecessor                 t 0      t m      t p

A                                             –                                 1      1      7

B                                             –                                 1      4     7

c                                              -                                     2      2     7

D                                             A                                   2      2     8

E                                              B                                  2      5    14

F                                              C                                    2     5     8

G                                             D, E                               3      6    15

H                                             F, G                               1      2     3

 

(a) Draw PERT network.

(b) Find the expected time for each activity.

(c) Determine the critical path.

(d) What is the probability that the project will be completed in (i) 22 days, (ii) 1 8 days ?

(e) What project duration will have 95% chance of completion

 

6. Consider the following project :

Activity                       Time estimates (in weeks)                   Predecessor

t    t m          tp

A                                        3      6        9                                     None

B                                        2       4      8                                     None

C                                        2      5       6                                         A

D                                        2      3       10                                       B

E                                        1       3        11                                      B

F                                        4       6        8                                       C,D

G                                       1       5       15                                       E

 

Find the critical path and its standard deviation. What is the probability that the project will be

completed by 1 8 weeks ?

 

7. A project has the following activities and other characteristics

Activity                     Preceding                     Time estimates(in weeks)

Activity                            t o         t m         t p

A                                 -                                       4         7          16

B                                  -                                       1         5          15

C                                 A                                     6        12          30

D                                 A                                      2         5           8

E                                  C                                    5         11         17

F                                  D                                     3          6          15

G                                 B                                    3           9          27

H                                E, F                                1          4           27

I                                   G                                    4         19         28

 

(a) Draw the PERT diagram

(b) Identify the critical path.

(c)  A find the probability that the project whose time estimates are given below

8 . A small project is composed of eight activities whose time estimates are given below

 

Time estimates

 

Activity                    Optimistic                  Most likely                  Pessimistic

 

0-1                              2                                 3                                   10

0-2                               4                                 5                                    6

1-2                                   0                                 3                                    0

1-3                                  9                                  7                                    8

1-4                                  1                                  7                                    8

2-5                                  3                                  5                                    19

3-4                                  0                                  0                                    0

4-5                               1                                     3                                   5

 

(a) Draw the PERT diagram .

(b) Identify  the critical  path .

©  Find the probability that  the project is completed time estimates are given below.

 

3. (b) CP : 1 – 2- 3 – 4 -5 -6

 

TF :      0, 0, 8, 0, 3, 9, 0, 0

FF :      0, 0, 8, 0, 3, 9, 0, 0

 

4. A-D-E-G-H-1, Project length = 12 months

 

5. (c) B-E-G-H (or, 1-3-5-6-7)

 

Variance =  82/ 9, Mean project length = 1 9

 

(d) P (X ≤ 22) = 0.8389 or 83.89%

P (X ≤ 1 8) = 0.3707 or 37.07%

 

(e) 23.97 or 24 days.

 

6. CP : A-C-F, expected duration = 1 6 weeks

Standard deviation = 1 .374.

P (X ≤ 1 8} = 0.928.

 

7. (b) A-C-E-H (i.e., 1-2-4-6-7-8)

(c) P (X ≤ 36) = 0.4207.

 

10. CP : A-C-F-H, expected duration = 17 weeks

Project variance = 1 .568.