Project Scheduling

1
Project Scheduling

• Basic Approach

2
Introduction

• A project is a collection of tasks that must be
  completed in minimum time or at minimal cost.
• It is made up by a set of tasks or activities,
  some of which must be completed before others can
  be started.
• The ones that must be completed before a
  particular activity can be started are called
  predecessors.
• Immediate predecessors are the ones that must be
  done just prior to the commencement of a
  particular activity.
• A feasible scheduling is one that schedules the
  activities without violating any of the immediate
  predecessor scheduling constraints.

3
Possible Objectives

• Some objectives of project scheduling include
• Completing the project as early as possible by
  determining an earliest start and finish time for
  each of the activities
• Determining the likelihood a project will be
  completed within a certain time period
• Finding a minimum cost schedule that completes
  the project by a certain date
• Finding a minimum time to complete a project
  within budget restrictions
• Investigating the results of possible delays in
  one or more of an activitys completion time
• Evaluating the costs and benefits of reducing the
  time of performing one or more of the activities

4
Activities

• An activity could be
• Quite specific
• e.g. install light switch in third bathroom
• Less detailed
• e.g. install electrical for the house
• The degree of specificity used depends on the
  application and availability of data.
• Each activity has a set of immediate predecessor
  activities that must be completed immediately
  prior to starting the activity.

5
Activity Completion Times

• Associated with each activity is an estimated
  completion time. These time could be
• Deterministic
• The completion time is known with certainty
• Probabilistic
• The completion time varies according to some
  probability distribution with an estimated mean
  and standard deviation
• Determined by the amount spent to perform the
  activity.

6
Example

• KLONE COMPUTERS, INC.
• KLONE Computers manufactures computers.
• It is about to design, manufacture, and market
  the Klonepalm 2000 palmbook computer.
• In broad terms, the three major tasks to perform
  are to
• Design and manufacture the computer
• Train staff and vendor representatives on the
  features and use of the computer
• Advertise the computer

7
Detailed Activities
Activity Description A Prototype model
design B Purchase of materials Manufacturing
C Manufacture of prototype model
activities D Revision of design E Initial
production run
F Staff training Training activities G Staff
input on prototype models H Sales training
I Pre-production advertising Advertising
activities campaign
J Post-redesign advertising campaign
8
Precedence Relations
9
The PERT/CPM Approach for Project Scheduling

• PERT stands for Program Evaluation and Review
  Technique and CPM stands for critical path
  method.
• Both were methods for project scheduling
  developed independently in the late 1950s.
• The concepts have merged over the years so that
  now we simply call the approach the PERT/CPM
  approach.
• It uses a network representation (a set of nodes
  and a set of arcs) of the project.
• Nodes represent the activities and reflect their
  completion times
• Arcs reflect immediate predecessor relationships
  with arrows
• PERT/CPM is used for scheduling activities such
  that the projects completion time is minimized.

10
The PERT/CPM Network
A 90
11
OBJECTIVES

• Management at KLONE would like to schedule the
  activities to minimize the project completion
  time.
• Management wishes to know
• The earliest start and finish times for each
  activity that will allow the project to be
  completed in this minimal time.
• The latest start and finish times for each
  activity which will not alter this minimal time.
• Which activities must adhere to rigid schedules
  and which activities have slack in their
  schedules.

12
Earliest Start (ES) andEarliest Finish (EF)
Times

• The ES and EF times are determined by making a
  forward pass through the network as follows
• For all the activities which have no immediate
  predecessors
• The earliest start time (ES) 0
• The earliest finish time (EF) the activitys
  duration
• Then select a node for which EF of all its
  immediate predecessors has been determined.
• ES Max EF (of all its immediate predecessors)
• EF ES Activity Duration
• Repeat this process until all nodes have been
  evaluated

Minimum Project Completion Time is The maximum EF
in the project.
13
Earliest Start and Finish Times

• We enter these as (ES,EF) above each node.

170)
(149,
110)
(90,
(105,
105)
21
15
5
(149,
177)
90)
115)
(0,
(90,
(115,
129)
(129,
149)
28
90
25
14
20
194)
(149,
120)
(90,
45
30
Earliest Project completion time MAX(EF) 194
14
Latest Start (LS) andLatest Finish (LF) Times

• The LS and LF times are determined by making a
  backward pass through the network as follows
• For all the activities which are not predecessors
  for any other activity
• The latest finish time (LF) project completion
  time
• The latest start time (LS) LF Activity
  Duration
• Select a node which is the immediate predecessor
  for nodes whose LS times have all been determined
  
• LF Min LS (all nodes for which it is a
  predecessor)
• LS LF - Activity Duration
• Repeat this process until all nodes have been
  evaluated

15
Latest Start and Finish Times

• We enter these as (LS,LF) below each node.

21
15
5
194)
(173,
110)
(95,
115)
(110,
28
90
20
25
14
194)
(166,
(115,
115)
(90,
129)
(0,
149)
(129,
90)
45
30
(149,
194)
149)
(119,
16
Slack Times

• Activity start time and completion time may be
  delayed by deliberate reasons as well as by
  unforeseen reasons.
• Some of these delays may affect the overall
  completion date.
• The effects of these delays can be determined by
  the slack time, for each activity.

Slack time for an activity LS-ES or LF-EF
17
The Critical Path

• The activities with 0 slack time form at least
  one critical path of connected activities, each
  of which is an immediate predecessor for another
  activity on the path from the beginning (time
  0) to the end (the completion time of the
  project).
• Critical activities must be rigidly scheduled.
• Any delay in a critical activity will delay the
  entire project.
• The critical path is the longest in the network

Sum of the completion times of activities on a
critical path Project completion time
18
Slack Time Calculations

• Slack time LS - ES

Critical Path A ?F ?G ?D ? J
19
The Critical Path

20
Possible Delays

• There could be a delay in just one activity.
• Any delay more than the slack time for the
  activity will delay the entire project by the
  difference between the activity delay and the
  slack time
• There could be delays in more than one activity.
• If activities are on different paths or on the
  same path but separated by a critical activity,
  each of the delays is evaluated separately. The
  project delay max (these delays corresponding
  slack).
• Activities on the same path which are not
  separated by a critical activity share the slack.
  Both will have the same value for the slack and
  any combined delays in these activities that
  exceed this common slack results in a project
  delay equal to (total activity delay) (common
  slack).
• Usually with multiple delays the model is simply
  re-solved!

21
Examples of Activity Delays

• Activity G is delayed 5 days
• G is on the critical path (has 0 slack) so the
  project will be delayed 5 days.
• Activity E is delayed 15 days
• E has 24 days of slack so the project will not be
  delayed
• Activity B is delayed 15 days
• B has 5 days of slack so the project will be
  delayed 10 days
• Activity E is delayed 30 days and Activity I is
  delayed 30 days
• E and I are on different paths. E has 24 days of
  slack which could cause a 30-24 6 day delay I
  has 29 days of slack which could cause 30-29 1
  day delay. The project is delayed by the
  MAX(6,1) 6 days.
• Activity B is delayed 4 days and Activity E is
  delayed 4 days
• B and E are on the same path but are separated by
  critical activities (G and D). This is the same
  as the case above. B has 5 days slack so
  delaying it 4 days would not delay the project E
  has 24 days of slack so a 4 day delay will not
  delay the project Net effect No delay.
• Activity B is delayed 4 days and Activity C is
  delayed 4 days
• B and C are on the same path with no critical
  activity in between. They share the same 5 days
  of slack. So sense both are delayed 4 days for a
  total of 8 days, the project is delayed 8 5 3
  days.

22
A Linear Programming Approach to PERT/CPM

• Variables
• Xi The start time of the activities for iA,
  B, C, ,J
• X(FIN) Finish time of the project
• Objective function
• Complete the project in minimum time.
• Constraints
• For each arc a constraint
  states that the start time of M must not occur
  before the finish time of its immediate
  predecessor, L.

23
The Linear Program
Minimize X(FIN) ST

• X(FIN) ³ XE 21
• X(FIN) ³ XH 28
• X(FIN) ³ XJ 45
• XD ³ XG 14
• XE ³ XD 20 XG ³ XC 5
• XH ³ XD 20 XG ³ XF 25
• XJ ³ XD 20 XI ³ XD 90
• XJ ³ XI 30 XF ³ XA 90
• XC ³ XB 15
• XD ³ XG 14
• XB ³ XA 90

Example ofConstraints
Start Time for C Cs Duration
and
Start Time for F Fs Duration
All Xs ³ 0
24
Using the PERT-CPM Template
25
Using the PERT-CPM Template
26
(No Transcript)
27
(No Transcript)
28
Review

• Objectives of Project Scheduling
• Precedence Relation Chart showing immediate
  predecessors
• How to construct a PERT/CPM network
• Forward Pass for finding the earliest
  start/finish times (ES,EF)
• Backwards Pass for finding the latest
  start/finish times (LS,LF)
• Calculation and analysis of slack
• Finding the critical path
• Linear programming formulation
• Use of PERT-CPM template