1

Project Planning, Scheduling and Control

Project – a set of partially ordered, interrelated activities that must be

completed to achieve a goal.

2

Network Models

• PERT – Program Evaluation and Review Technique– probabilistic features

• CPM – Critical Path Method– cost/time trade-offs

project scheduler

3

Objectives

• Planning, scheduling, and control of complex projects

• Find critical activities to manage resources (management by exception)

• Determine flexibility of non-critical activities (slack)

• Estimate earliest completion time of project• Determine time – cost trade-offs

4

ServiceIndustry

DistributionIndustry

ProducingIndustry

Business and Industry – a taxonomy

Rawmaterials

ContinuousProcessing

DiscreteProducts

MiningDrillingFarming

Construction Manufacturing ChemicalsFoodRefinery

Batch MassProcessing Production

5

Production Systems

• Job shops• Flow shops• Batch production• Mass production• Cellular manufacturing• Project Shop• Continuous Processing

Gosh. Can you tell us more about these?

6

Project Shop

• single product in fixed location• material and labor brought to the site• usually job shop/flow shop associated• functionalized production system• examples include construction and

shipbuilding

7

The Elements of Project Scheduling

• Project Definition. Statement of project, goals, and resources required.

• Activity Definitions. Content and requirements of each activity.

• Project Scheduling. Specification of starting and ending times of all activities.

• Project Monitoring. Keeping track of the progress of the project.

8

Definitions• Activity – an effort (task) that requires resources and

takes a certain amount of time.• Event – a specific accomplishment or milestone (the

start or finish of an activity).• Project – a collection of activities and events leading

to a definable goal.• Network – a graphical representation of a project

depicting the precedence relationships among the activities and events.

• Critical Activity – an activity that if delayed will hold up the scheduled completion of a project.

• Critical Path – the sequence of critical activities that forms a continuous path from the start of a project to its completion.

9

Framework for Analysis

• Analyze project in terms of activities and events

• Determine sequence (precedence) of activities (develop network)

• Assign estimates of time, cost, and resources to all activities

• Identify the critical path• monitor, evaluate, and control progress of

project

10

Network Representation

Projects may be represented as networks with:

• Arrows representing activities.

• Nodes representing completion of a set of activities (milestones).

• Pseudo activities may be required to satisfy precedence relationships.

11

Network Development

1 2 3

events(nodes)

activities(arcs)

Activities have durationand may have precedence.

Define activities in terms oftheir beginning and ending events.e.g. Activity 1-2 must precede Activity 2-3

12

Network Development (continued)

1

2

3

4

Event 1 is start of projectActivities 1-2, 1-3, and1-4 have no predecessorsand may start simultaneously

13

Network Development (continued)

n-2

n-3

n

n-1

Event n is the end of theproject. Activities (n-3 – n,(n-2) – n, and (n-1) - nmust be completed for theproject to be completed.

14

Network Development (continued)

6

7

8

9

Activities 6-7, 6-8, and6-9 cannot start until activity 5-6has been completed.

5

burst event

15

Network Development (continued)

8

5

6

7Activities 5-8, 6-8, and7-8 must be completedbefore activity 8-9 may begin.

9

merge event

16

Network Development (continued)

8

5

6

7

Activities 5-8, 6-8, and 7-8 must be completedbefore activity 8-9, 8-10, or 8-11 may begin.

9

10

11

Gosh! Acombinedmerge andburst event.Are theserare or what?

17

Dummy activityA CA DB D

W R O N G

7

5

6

9

10

A

B

C

D

75

6

9

10

A

B D

C

8

dummy has no resources and no duration

18

Project Networks

• Collection of nodes and arcs• Depicted graphically• Events are uniquely numbered• Arcs are labeled according to their beginning and

ending events– Ending events always have higher numbers than beginning

events

• Two activities cannot have the same beginning and ending events

• Activity lengths have no significance

19

Our Very Own Exampleproduct development

activity description precedence

A design promotion campaign -B initial pricing -C product design -D promotion cost analysis AE manufacture prototype CF test and redesign EG final pricing B,D,FH market test G

20

product development

1

2

3 4

6 75

A

B

C

D

E

F

G H

21

Notation

• i-j = an activity of a project• di-j = the duration of activity i-j• Ei = the earliest time event i can occur• ESi-j = the earliest start time of activity i-j• EFi-j = the earliest finish time of activity i-j• LSi-j = the latest start time of activity i-j• LFi-j = the latest finish time of activity i-j• Li = the latest time event i can occur

22

Our Very Own Exampleproduct development

activity precedence duration (days)

A (1-2) - 17B (1-5) - 7C (1-3) - 33D (2-5) A 6E (3-4) C 40F (4-5) E 7G (5-6) B,D,F 12H (6-7) G 48

23

product development – forward pass

1

2

3 4

6 75

A(17)

B(7)

C(33)

D(6)

E(40)

F(7)

G(12) H(48)

E1 = 0

ES1-2 = 0ES1-5 = 0ES1-3 = 0

EF1-2 = 17EF1-5 = 7EF1-3 = 33

E2 = 17E5 = 7E3 = 33

ES5-6 = 80EF5-6 = 92E6 = 92

ES2-5 = 17ES3-4 = 33

EF2-5 = 23EF3-4 = 73

E4 = 73

ES4-5 = 73EF4-5 = 80

E5 = 80

ES6-7 = 92EF6-7 = 140E7 = 140

24

product development – backward pass

1

2

3 4

6 75

A(17)

B(7)

C(33)

D(6)

E(40)

F(7)

G(12) H(48)

L1 = 0

LF1-2 = 74LF1-5 = 80LF1-3 = 33

L2 = 74L3 = 33

L6 = 92 LF5-6 = 92LS5-6 = 80

LF2-5 = 80 LS2-5 = 74

LF3-4 = 73 LS3-4 = 33

L4 = 73

LF4-5 = 80LS4-5 = 73

L5 = 80

L7 = 140LF6-7 = 140LS6-7 = 92

LS1-2 = 57LS1-5 = 73LS1-3 = 0

25

Activity Slack

Si-j = LSi-j – ESi-j Si-j = LFi-j – EFi-jor

Activity LS ES Slack

1-2 57 0 571-5 73 0 731-3 0 0 02-5 74 17 573-4 33 33 04-5 73 73 05-6 80 80 06-7 92 92 0

criticalactivities

26

Critical Path Method

An analytical tool that provides a schedule that completes the project in minimum time subject to the precedence constraints. In addition, CPM provides:

• Starting and ending times for each activity

• Identification of the critical activities (i.e., the ones whose delay necessarily delay the project).

• Identification of the non-critical activities, and the amount of slack time available when scheduling these activities.

27

critical path

1

2

3 4

6 75

A(17)

B(7)

C(33)

D(6)

E(40)

F(7)

G(12) H(48)

ES1-3 = 0LS1-3 = 0

ES5-6 = 80LS5-6 = 80

ES3-4 = 33 LS3-4 = 33

ES4-5 = 73LS4-5 = 73

ES6-7 = 92LS6-7 = 92

ES1-5 = 0LS1-5 = 73

ES2-5 = 17 LS2-5 = 74

ES1-2 = 0LS1-2 = 57

28

Critical Path Activities

• focus management attention• increase resources• eliminate delays• eliminate critical activities• overlap critical activities• break activity into smaller tasks• outsource or subcontract

29

Critical Path by LP

1

Min

. :

, pairs

n

ni

j i ij

E

subj to

E E d i j

earliest start times

1

1

Min

. :

, pairs

n

n ii

j i ij

nL L

subj to

L L d i j

latest start times

30

Activity Durations

a b

uniform

triangular

beta

31

More Activity Durations

let a = optimistic time b = pessimistic time m = most likely time

2

2

2 12

a b b a

2 2 22

3 18

a m b a b m ab am bm

2

24

6 18

a m b b a

uniform:

triangular:

beta:

32

activity durationsproduct development

activity a m b

A (1-2) 6 18 24 17 9 3B (1-5) 6 6 12 7 1 1C (1-3) 24 30 54 33 25 5D (2-5) 6 6 6 6 0 0E (3-4) 24 36 72 40 64 8F (4-5) 6 6 12 7 1 1G (5-6) 6 12 18 12 4 2H (6-7) 36 48 60 48 16 4

2

beta

note: based upon a 6 day workweek

33

critical path analysisproduct development

activity a m b

C (1-3) 24 30 54 33 25 5E (3-4) 24 36 72 40 64 8F (4-5) 6 6 12 7 1 1G (5-6) 6 12 18 12 4 2H (6-7) 36 48 60 48 16 4

sum 140 110

2

beta

From the Central Limit Theorem, project completiontime is normally distributed with a mean of 140 daysand a standard deviation of = 10.5 days.110

34

Probability Statements

Probability project will be completed by day 150 isgiven by:

150 140Pr 150 Pr Pr .95 .829

10.5

TT z

Probability project will be completed after day 130 is given by:

130 140Pr 130 Pr Pr .95 .171

10.5

TT z

35

Resource Constraints

Activity ES Duration staffing

1-2 0 17 51-5 0 7 71-3 0 33 102-5 17 6 43-4 33 40 64-5 73 7 35-6 80 12 56-7 92 48 6

36

Resource Profile – early start schedule

0 10 20 30 40 50 60 70 80

30

25

20

15

10

5

1-2

1-5

1-33-4

2-5

4-5 5-6

This doesn’t work.We need too manypeople at the startof the project!

37

Late Start Staffing

Activity ES Duration staffing

1-2 57 17 51-5 73 7 71-3 0 33 102-5 74 6 43-4 33 40 64-5 73 7 35-6 80 12 56-7 92 48 6

38

Resource Profile – late start schedule

0 10 20 30 40 50 60 70 80

30

25

20

15

10

5

1-21-5

1-33-4

2-5

4-5 5-6

Boss. Let’s go withthe late start schedule. Then we can layoffsome folks.

39

Time Costing Methods

• Suppose that projects can be expedited by reducing the time required for critical activities. Doing so results in an increase in some costs and a decrease in others. The goal is to determine the optimal number of days to schedule the project to minimize total cost.

• Assume that there is a linear time/cost relationship for each activity.

40

Time-Cost Trade-offs

timecrashtime

normaltime

crashcost

normalcost

,n ni j i jd c

,c ci j i jd c

41

Heuristic Crashing

c ni j i j

i j n ci j i j

c ck

d d

= $ / day

time costactivity normal crash normal crash k

C (1-3) 33 25 10 20 1.25E (3-4) 40 31 22 35 1.44F (4-5) 7 5 8 12 2.0G (5-6) 12 9 17 30 4.33H (6-7) 48 40 30 48 2.25

42

An LP approach

let yi-j = number of time units activity i-j is crashed K = indirect cost per day

,

min

. :

0

0 1,2,...,

i j i j nall i j

nj i i j i j

n ci j i j i j

i

k y K E

subt to

E E y d i j

y d d i j

E i n

43

The End

Backups Follow

44

Forward Pass

set Ei = 0i=1; j=2

set ESi-j = Ei EFi-j = Ei + di-j

Ej = max {Ej , EFi-j}

If i-j is an activity

set j = j + 1j <= n

i = i + 1j = 2

j > n

i < n

stop

i = n

If i-j not an activity

45

Backward Pass

set Li = En i=1; j=n

set LFi-j = Li LSi-j = Li - di-j

Lj = min {Lj , LFi-j}

If i-j is an activity

set i = i + 1i < n

j = j - 1i = 1

i = n

j > 0

stop

j = 0

If i-j not an activity