# Linear Programming CPM – Critical Path Method Can normal task times

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```					         CPM – Critical Path Method
Can normal task times be reduced?

Is there an increase in direct costs?
• Overtime, etc…

Can there be a reduction in indirect costs?
• Less daily rental charges
• Bonus for early completion
• Avoid penalties for running late
• Avoid cost of late startup

CPM – Critical Path Method
Example:

Normal Time   Crash Time   Cost of Crashing
Job   Predecessors    (days)       (days)       per Day (\$)
A          -           10            7               4
B          -           5             4               2
C         B            3             2               2
D        A,C           4             3               3
E        A,C           5             3               3
F         D            6             3               5
G         E            5             2               1
H        F,G           5             4               4

CPM – Critical Path Method
Enumerative Approach:

Reduce job H by 1 day: Total Cost improves by \$5 - \$4 = \$1.

Reduce job A by 2 days: Total cost improves by \$10 - \$8 = \$2.

Reduce job A by an additional day, and job B by a day? Total cost
improves by \$5 - \$4 - \$2 = -\$1. Therefore do not take this action.

Reduce job A by an additional day, and job C by a day? Total cost
improves by \$5 - \$4 - \$2 = -\$1. Therefore do not take this action.

Evaluate combinations of reducing path 3-4-6 and 3-5-6 by one day.
D & E = \$5 - \$3 - \$3 = -\$1        F & E = \$5 - \$5 - \$3 = -\$3
D & G = \$5 - \$3 - \$1 = \$1         F & G = \$5 - \$5 - \$1 = -\$1
Therefore, reduce job D & G by 1 day: TC improves by \$5 - \$3 -\$1 = \$1.

Overall improvement: \$1 + \$2 + \$1 = \$4.
CPM – Critical Path Method
LP Approach:

Let tij – decision variable for time to complete task connecting
events i and j.
kij – normal completion time of task connecting events i and j.
lij – minimum completion time of task connecting events i and j.
Cij – incremental cost of reducing task connecting events i and j.

Model I: Given project must be complete by some time T, which tasks
should be reduced to minimize the total cost?

Min      Z   Cij (kij  tij )
i   j

s.t.      t j  ti  tij     for all jobs (i,j)
lij  tij  kij    for all jobs (i,j)
t n  t1  T
ti  0             for all i
CPM – Critical Path Method
LP Approach:

Model II: Given an additional budget of \$B for “crashing” tasks, what minimum
project completion time can be obtained while staying within your budget?

Min       Z  t n  t1

s.t.      t j  ti  tij                     for all jobs (i,j)
lij  tij  kij                    for all jobs (i,j)

 C
i   j
ij   (kij  tij )  B
for all i
ti  0
CPM – Critical Path Method
LP Approach:

Model III: Let \$F represent the indirect cost which is proportional to the
project duration. In other words, the indirect cost for the project is \$F times
the overall project duration, \$F*T. Find the minimum project cost.

Min       Z  F (tn  t1 )   Cij (kij  tij )
i   j

s.t.            t j  ti  tij            for all jobs (i,j)
lij  tij  kij           for all jobs (i,j)
ti  0                    for all i
PERT – Program Evaluation
and Review Technique
CPM dealt with tasks having durations with known certainty.

PERT deals with tasks having uncertain durations.

PERT assumes tasks durations can be estimated with the following:
1) A most probable time, m.
2) An optimistic time, a.
3) A pessimistic time, b.
PERT – Program Evaluation
and Review Technique
PERT usually assumes a Beta distribution for tasks durations.

a  4m  b
An estimate for the mean, m, is the following:   m
6

ba
2

An estimate for the variance,s2, is the following:   s 
2

 6 
PERT – Determining Expected
Project Duration
Consider the following network:

Optimistic Most Probable Pessimistic
Job   Predecessors Time (a)     Time (m)     Time (b)
A          -          2            5            8
B         A           6            9           12
C         A           6            7            8
D        B,C          1            4            7
E         A           8            8            8
F        D,E          5           14           17
G         C           3           12           21
H        F,G          3            6            9
I         H           5            8           11

Note: Example 3.7-4 in book.
PERT – Determining Expected
Project Duration
Calculate the Average, Standard Deviation and Variance

Standard
Job   Average Time   Deviation   Variance
A          5            1           1
B          9            1           1
C          7           1/3        1/9
D          4            1           1
E          8            0           0
F         13            2           4
G         12            3           9
H          6            1           1
I          8            1           1
PERT – Determining Expected
Project Duration
The network diagram with earliest and latest start times.

What jobs are on the critical path?
A-B-D-F-H-I
What is the expected project completion time, T?
E(T) = 5+9=4+13+6+8 = 45 days
What is the variance of T?
V(T) = 1+1+1+4+1+1 = 9            s(T) = 91/2 = 3 days
PERT – Determining Expected
Project Duration
Assuming all task durations are independent and identically
distributed, IID, then T has a normal distribution with mean E(T)
and variance V(T).

Given E(T) = 45 days and s(T) = 3 days:

What is the probability the project is completed within 45 days?

P(T <= 45) = .50

What is the probability the project is completed with 40 days?
40  45
P(T <= 40) =       P( Z            )  P( Z  1.67)  .05
3
What is the probability the project is completed with 49 days?
49  45
P(T <= 49) =    P( Z          )  P( Z  1.33)  1  .092  .908
3

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