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Module G Chapter 8
Project Management – Critical Path Method (CPM)
Activity Immediate Activity Time
Predecessor (weeks)
a – 16
b – 14
c a 8
d a 5
e b 4
f b 6
g c 10
h d,e 15
Critical path and minimum project completion time can be found manually or by LP.
Earliest Times Latest Times Slack
Time
Act. (i,j) Pred. ESij EFij LSij LFij Sij
a (1,2) – 0 16 0 16 0 critical
b (1,3) – 0 14 3 17 3
c (2,4) a 16 24 18 26 2
d (2,5) a 16 21 16 21 0 critical
e (3,5) b 14 18 17 21 3
f (3,6) b 14 20 30 36 16
g (4,6) c 24 34 26 36 2
h (5,6) d,e 21 36 21 36 0 critical
Linear Programming Formulation
Let xi 0 be the earliest event time of node i, for i = 1, 2, 3, 4, 5, 6.
Then we have constraints:
x2 x1 16
x3 x1 14
x6 x5 15
and xi 0 , for i = 1, 2, 3, 4, 5, 6.
For the objective, we could just minimise x6, but if we want to ensure that all the xi s are
6
earliest event times we just minimise their sum: Z xi .
i 1
Project Evaluation and Review Technique (PERT)
Problem 18, pages 346-7, Stone River Textile Mill
Time Estimates (weeks)
Activity Predecessors a m b
a – 1 2 3
b – 2 5 8
c – 1 3 5
d a 4 10 25
e a 3 7 12
f b 10 15 25
g c 5 9 14
h d,e 2 3 7
i d,e,f 1 4 6
j d,e,f,g 2 5 10
k h,i,j 2 2 2
Text page 315: The activity times are assumed to have a beta distribution, which has a
minimum value (a), a maximum value (b) and a most likely value (m).
The mean and variance of a beta distribution can be estimated by simple formulae:
a 4m b ba
2
mean: t , variance: v
6 6
Act. (i,j) a m b t (mean) v (variance)
a (1,2) 1 2 3 2.000 0.111
b (1,3) 2 5 8 5.000 1.000
c (1,4) 1 3 5 3.000 0.444
d (2,5) 4 10 25 11.500 12.250
e (2,6) 3 7 12 7.167 2.250
f (3,7) 10 15 25 15.833 6.250
g (4,8) 5 9 14 9.167 2.250
dummy (5,6) 0 0 0 0.000 0.000
dummy (6,7) 0 0 0 0.000 0.000
h (6,9) 2 3 7 3.500 0.694
dummy (7,8) 0 0 0 0.000 0.000
i (7,9) 1 4 6 3.833 0.694
j (8,9) 2 5 10 5.000 1.778
k (9,10) 2 2 2 2.000 0.000
Find critical path and expected (mean) duration.
Use normal distribution approximation to calculate probabilities of duration being in
certain ranges.
Project crashing
Problem 27, page 352
Activity Time Activity Cost
(weeks) ($)
Activity (i,j) Predecessor Normal Crash Normal Crash
a (1,2) – 16 8 2000 4400
b (1,3) – 14 9 1000 1800
c (2,4) a 8 6 500 700
d (2,5) a 5 4 600 1300
e (3,5) b 4 2 1500 3000
f (3,6) b 6 4 800 1600
g (4,6) c 10 7 3000 4500
h (5,6) d,e 15 10 5000 8000
Activity (i,j) Pred. Max Crash Crash Cost
(wks) ($/wk)
a (1,2) – 8 300
b (1,3) – 5 160
c (2,4) a 2 100
d (2,5) a 1 700
e (3,5) b 2 750
f (3,6) b 2 400
g (4,6) c 3 500
h (5,6) d,e 5 600
Crash manually or by LP.
