Topic 11
Basic Project
Scheduling
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Overview
No resource
Constraints
Project
Scheduling
Resource
Constraints
Critical Path
Method (CPM)
Program Evaluation
and Review Technique
(PERT)
Heuristic Resource
Leveling
Integer Programming
Formulations
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
Critical Path Method (CPM)

Think of unlimited machines in parallel
… and n jobs with precedence
constraints
Processing times pj as before

Objective to minimize makespan


July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
3
Critical Path Method

Forward procedure:



Starting at time zero, calculate the
earliest each job can be started
The completion time of the last job is the
makespan
Backward procedure

Starting at time equal to the makespan,
calculate the latest each job can be
started so that this makespan is realized
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
4
Forward Procedure
Step 1:
Set at time t = 0 for all jobs j with no predecessors,
Sj’=0 and set Cj’ = pj.
Step 2:
Compute for each job j
S 'j  max Ck' ,
all k  j
Cj’ = Sj’ + pj.
Step 3:
The optimal makespan is Cmax  max C1' , C2' ,..., Cn'
STOP

July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson

5
Backward Procedure
Step 1:
Set at time t = Cmax for all jobs j with no successors,
Cj’’= Cmax and set Sj’’ = Cmax - pj.
Step 2:
Compute for each job j
C 'j'  min S k'' ,
k all j
Sj’’ = Cj’’ - pj.
Step 3:
''
''
Verify that 0  min{ S1 ,..., S n }.
STOP
Lecture Notes for Planning and Scheduling
July 11, 2017
Prepared by Siggi Olafsson
6
Comments





The forward procedure gives the earliest
possible starting time for each job
The backwards procedures gives the latest
possible starting time for each job
If these are equal the job is a critical job.
If these are different the job is a slack job,
and the difference is the float.
A critical path is a chain of jobs starting at
time 0 and ending at Cmax.
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
7
Example
j
1
2
3
4
pj
5
6
9
12 7
2
5
4
6
7
8
12 10 6
9
10 11 12 13 14
10 9
7
8
7
5
7
10
1
6
9
3
11
5
July 11, 2017
12
14
13
8
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
8
Forward Procedure
5+6=11 11+12=23
2
4
5
1
23+10=33
7
33+9=42
10
43+8=51 51+5=56
14+12=26 26+10=36
6
9
5+9=14
3
12
11
5
14+7=21
July 11, 2017
Cmax  56
8
26+6=32
36+7=43
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
14
43+7=50
9
Backwards Procedure
24-12=12 34-10=24
2
4
14-9=5
1
10
36-10=26
26-12=14
56-5=51
43-7=36
6
9
3
12
11
5
35-10=26
July 11, 2017
43-9=34
7
51-8=43
8
43-7=36
51-8=43
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
56
14
56-5=51
10
Critical Path
2
4
7
10
1
6
9
3
11
5
July 11, 2017
12
14
13
8
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
11
Topic 12
Variable Processing
Times
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Time/Cost Trade-Offs





Assumed the processing times were
fixed
More money  shorter processing time
min
max
Start with linear costs p j  p j  p j
Processing time
Marginal cost
cj 
July 11, 2017
c aj  c bj
min
p max

p
j
j
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
Linear Costs
Resources (money)
c aj
cbj
p
July 11, 2017
min
j
p
max
j
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Processing
time
14
Solution Methods


Objective: minimum cost of project
Time/Cost Trade-Off Heuristic



Good schedules
Works also for non-linear costs
Linear programming formulation


Optimal schedules
Non-linear version not easily solved
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
15
Sources, Sinks, and Cuts
Cut set
Sink node
Source (dummy) node
July 11, 2017
Minimal cut set
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
16
Time/Cost Trade-Off Heuristic
Step 1:
Set all processing times at their maximum
pj  p
max
j
Determine all critical paths with these processing times
Construct the graph Gcp of critical paths
Continue to Step 2
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
17
Time/Cost Trade-Off Heuristic
Step 2:
Determine all minimum cut sets in Gcp
Consider those sets where all processing times are larger
than their minimum
pj  p
min
j
, j  Gcp
If no such set STOP; otherwise continue to Step 2
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
18
Time/Cost Trade-Off Heuristic
Step 3:
For each minimum cut set:
Compute the cost of reducing all processing times by one
time unit.
Take the minimum cut set with the lowest cost
If this is less than the overhead per time unit go on to Step
4; otherwise STOP
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
19
Time/Cost Trade-Off Heuristic
Step 4:
Reduce all processing times in the minimum cut set by
one time units
Determine the new set of critical paths
Revise graph Gcp and go back to Step 2
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
20
Example
j
1
2
3
4
5
pj
5
6
9
12 7
12 10 6
10 9
7
8
7
5
pj
3
5
7
9
9
7
6
5
5
2
5
6
7
8
8
3
9
10 11 12 13 14
5
cja 20 25 20 15 30 40 35 25 30 20 25 35 20 10
cj
7
July 11, 2017
2
4
3
4
3
4
4
4
5
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
2
4
8
21
Maximum Processing Times
2
4
7
10
1
6
9
3
11
5
July 11, 2017
12
14
13
8
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
22
Maximum Processing Times
Cmax  56
2
4
7
10
1
6
9
3
11
5
July 11, 2017
12
14
13
8
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
23
Critical Path Subgraph (Gcp)
C1=7
C6=3
6
1
3
C3=4
12
C14=8
14
11
C11=2
Cut sets: {1},{3},{6},{9},
{11},{12},{14}.
July 11, 2017
C12=2
C9=4
9
Minimum cut
set with lowest cost
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
24
Critical Path Subgraph (Gcp)
C1=7
C6=3
6
1
3
C3=4
12
11
C14=8
14
13
C11=2
Cut sets: {1},{3},{6},{9},
{11},{12,13},{14}.
July 11, 2017
C12=2
C9=4
9
C13=4
Minimum cut
set with lowest cost
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
25
Critical Path Subgraph (Gcp)
C1=7
C6=3
6
1
3
C3=4
12
11
C14=8
14
13
C11=2
Reduce processing time
next on job 6
July 11, 2017
C12=2
C9=4
9
C13=14
Reduce processing
time until = 4
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
26
Critical Path Subgraph (Gcp)
C2=2
2
C4=3
4
C7=4
7
C1=7
C6=3
6
1
3
C3=4
July 11, 2017
C9=4
9
C10=5
10
C12=2
12
11
C11=2
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
C14=8
14
13
C13=4
27
Linear Programming Formulation


The heuristic does not guaranteed optimum
Here total cost is linear

n
c0Cmax   c bj  c j p max
 pj
j

j 1

Want to minimize
n
c0Cmax   c j p j .
j 1
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
28
Linear Program
Minimize
subject to
n
c0Cmax   c j p j .
j 1
xk  p j  x j  0, j  k  A
pj  p
max
j
, j
p j  p min
j , j
x j  0, j
Cmax  x j  p j  0, j
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
29
Topic 13
PERT
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Program Evaluation and
Review Technique (PERT)




Assumed processing times deterministic
Processing time of j random with mean mj and
variance sj2.
Want to determine the expected makespan
Assume we have
pja = most optimistic processing time
pjm = most likely processing time (mode)
pjb = most pessimistic processing time
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
31
Expected Makespan




Estimate expected processing time
p aj  4 p mj  p bj
mj 
6
Apply CPM with expected processing times
Let Jcp be a critical path
Estimate expected makespan
Eˆ (Cmax ) 
July 11, 2017
m
jJ cp
j
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
32
Distribution of Makespan


Estimate the variance of processing times
b
a
p

p
j
s 2j  j
6
and the variance of the makespan
Vˆ (Cmax ) 

2
s
 j
jJ cp
Assume it is normally distributed
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
33
Discussion

Potential problems with PERT:

Always underestimates project duration


Non-critical paths ignored



critical path probability
critical activity probability
Activities are not always independent


other paths may delay the project
same raw material, weather conditions, etc.
Estimates by be inaccurate
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
34
Discussion

No resource constraints:



Critical Path Method (CPM)
Simple deterministic
Time/cost trade-offs



Linear cost (heuristic or exact)
Non-linear cost (heuristic)
Accounting for randomness (PERT)
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
35
Topic 14
Adding Resource
Constraints
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Resource Constraints





Renewable resources
Very hard problem
No LP
Can develop an IP
Let job n+1 be dummy job (sink) and
1 if job j is completed at time t
x jt  
0 otherwise.
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
37
Makespan

Let H bound the makespan, e.g.
n
H   pj
j 1

Completion time of job j is
and the makespan
H
t  x
t 1
jt
H
t  x
t 1
July 11, 2017
n 1,t
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
38
IP Formulation
H
t  x
Minimize
Subject to
t 1
H
t  x
t 1
n 1,t
H
jt
 pk   txkt  0
t 1
 t  p j 1 
 Rij  x ju   R j



j 1 
u t

n
H
x
t 1
July 11, 2017
jt
1
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
39
In Practice



The IP cannot be solved
(Almost) always resource constraints
Heuristic:
Resource constraint  Precedence constraint

Example: pouring foundation and sidewalk




Both require same cement mixer
Delaying foundation delays building
Precedence constraint: pour foundation first
Not a logical constraint
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
40
Optimality of Heuristic




Say n jobs need the same resource
Could otherwise all be done simultaneously
Add (artificial) precedence constraints
Have n! possibilities
2
3
4
5
6
2

6
24
120
720
Will we select the optimal sequence?
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
41
Decision Support

Resource leveling



Solve with no resource constraints
Plot the resource use as a function of time
If infeasible suggest precedence
constraints




Longest job
Minimum slack
User adds constraints
Start over
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
42
Example: One Resource
2
Uses resource
3
9
6
6
6
11
Resource Profile
Capacity
Time
10
July 11, 2017
20
Lecture Notes for Planning and Scheduling
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43