Project Planning - Technical University of Denmark · Activity-on-arc (AOA): Each activity is presented as a an arc. A node is used to separate an activity from each of its immediate

1Jesper Larsen

Department of Management Engineering / Operations Research

Project PlanningJesper Larsen

[email protected]

Department of Management Engineering

Technical University of Denmark

2Jesper Larsen


Project ManagementProject Management is a set of techniques thathelps management manage large-scale projects;projects that typically require coordination ofnumerous tasks throughout the organisation.

PERT = Program Evaluation and ReviewTechnique

CPM = Critical Path Method

The methods were developed independently but arenow often used interchangeably and are combinedin one acronym: PERT/CPM.

3Jesper Larsen


ExampleThe Reliable Construction Company has just madethe winning bid of a $ 5.4 million contract toconstruct a new plant.

There are the following constraints:

a penalty of $ 300000, if Reliable has notcompleted construction by the deadline of 47weeks from now, and

a bonus for speedy construction of $ 150000 tobe paid if the plant is ready within 40 weeks.

4Jesper Larsen


QuestionsHow can the project be displayed graphically tobetter visualize the flow of the activities?

What is the total time required to complete theproject if no delays occur?

When do the individual activities need to startand finish (at the latest) to meet this projectcompletion time?

When can the individual activities start andfinish (at the earliest) if no delays occur?

5Jesper Larsen


More questionsWhich are the critical bottleneck activities whereany delays must be avoided to prevent delayingproject completion?

For the other activities, how much delay can betolerated without delaying the projectcompletion?

If extra money is spent to expedite the project,what is the least expensive way of attempting tomeet the target completion time?

6Jesper Larsen


Example

7Jesper Larsen


Project Networks – VisualizationProject networks consists of a number of nodes anda number of arcs. There are two alternatives forpresenting project networks.

Activity-on-arc (AOA): Each activity ispresented as a an arc. A node is used toseparate an activity from each of its immediatepredecessors.

Activity-on-node (AON): Each activity isrepresented by a node. The arcs are used toshow the precedence relationships.

8Jesper Larsen


AON vs. AOAAON are considerably easier to construct thanAOA.

AON are easier to understand than AOA forinexperienced users.

AON are easier to revise than AOA when thereare changes in the network.

9Jesper Larsen


AON for the example

10Jesper Larsen


A path through the networkA path through a project network is one of theroutes following the arcs from the START node tothe FINISH node. The length of a path is the sum ofthe (estimated) durations of the activities on thepath.St→A→B→C→D→G→H→M→Fi 40St→A→B→C→E→H→M→Fi 31St→A→B→C→E→F→J→K→N→Fi 43St→A→B→C→E→F→J→L→N→Fi 44St→A→B→C→I→J→K→N→Fi 41St→A→B→C→I→J→L→N→Fi 42

11Jesper Larsen


The Critical PathThe estimated project duration equals thelength of the longest path through the projectnetwork. This longest path is called the criticalpath.

In the case of the Reliable Constructioncompany the critical path isSt→A→B→C→E→F→J→L→N→Fi with alength of 44 weeks.

12Jesper Larsen


Scheduling individual activitiesThe PERT/CPM scheduling procedure beginsby addressing when the activities need to startand finish at the earliest if no delays occur.

Having no delays means 1) actual durationequals estimated duration and 2) each activitybegins as soon as all its immediatepredecessors are finished.

ES = Earliest start for a particular activityEF = Earliest finish for a particular activity

where EF = ES + duration

13Jesper Larsen


Earliest Start Time RuleThe earliest start time of an activity is equal to thelargest of the earliest finish times of its immediatepredecessors, or

ES = max { EF of immediate predecessors }

14Jesper Larsen


Generating earliest start and finish times

15Jesper Larsen


Earliest start and finish times

16Jesper Larsen


Finding latest start and finish timesThe latest start time for an activity is the latestpossible time that it can start without delaying thecompletion of the project (so the finish node still isreached at its earliest finish time), assuming nosubsequent delays in the project. The latest finishtime has the corresponding definition with respectto finishing the activity.

LS = latest start time for a particular activity

LF = latest finish time for a particular activity

where LS = LF - duration

17Jesper Larsen


Generating latest start and finish times

18Jesper Larsen


Latest Finish Time RuleThe latest finish time of an activity is equal to thesmallest of the latest start times of its immediatesuccessors, or

LF = min { LS of the immediate successors }

19Jesper Larsen


Latest start and finish times

20Jesper Larsen


All information in one graph

21Jesper Larsen


Identifying slackThe slack for an activity is the difference betweenits latest finish time and its earliest finish time.

Slack Activities0 A, B, C, E, F, J, L, N> 0 D, G, H, I, K, M

Each activity with zero slack is on a critical paththrough the project network such that any delayalong this path will delay project completion.

22Jesper Larsen


Time-Cost Trade-offs for activitiesCrashing an activity refers to taking (costly)measures to reduce the duration of an activitybelow its normal time.

Crashing the project refers to crashing anumber of activities in order to reduce theduration of the project below its normal value.

The CPM method of time-cost trade-offs isconcerned with determining how much to crasheach activity in order to reduce the projectduration.

23Jesper Larsen


Crashing an activityThe data necessary for determining how muchto crash a particular activity are given by thetime-cost graph for the activity.

A linear correlation between the points areassumed.

normal timecrash time

crash cost

normal cost

Normal

Crash

24Jesper Larsen


Data for crashing

25Jesper Larsen


Considering Time-Cost trade-offsWhat is the least expensive way of crashing someactivities to reduce the estimated project duration tothe specified level?

One way: look at marginal costs.

26Jesper Larsen


Using LP to make crashing decisionsRestatement of the problem: Let Z be the total costof crashing activities. The problem then is tominimize Z, subject to the constraint that projectduration must be less than or equal to the timedesired by the project manager.

27Jesper Larsen


Modelling using LPWhat are our decision variables? xj = reductionin the duration of activity j due to crashing thisactivity.

How will our objective function look like? Theobjective function is to minimize the total cost ofcrashing activities:min 100000xA + 50000xB + . . . + 60000xN .

To impose the constraint that the project mustbe finished in less than or equal to a certainnumber of weeks we impose another variable:yFi.

28Jesper Larsen


Modelling using LP (cont)To help the LP assign the appropriate value toyFi given the values of xA, xB, . . . it is convenientto introduce: yj = start time of activity j.

NOW since the start time of each activity isdirectly related to to the start time and durationof each of its immediate predecessors we get:

start time of this activity ≥ (start time -duration) for this immediate predecessor

29Jesper Larsen


Putting it all together1. min 100000xA + 50000xB + . . . + 60000xN

2. Maximum reduction constraints:xA ≤ 1, xB ≤ 2, . . . , xN ≤ 3.

3. Non-negativity constraints:xj ≥ 0, yj ≥ 0, yFi ≥ 0

4. Start time constraints: For each arc we get arelationship between the two end points:yC ≥ yB + 4 − xB,

yD ≥ yC + 10 − xCetc.

5. Project duration constraints: yFi ≤ 40

Documents

Project Planning - Technical University of Denmark · Activity-on-arc (AOA): Each activity is presented as a an arc. A node is used to separate an activity from each of its immediate