DECISION MODELING WITH DECISION MODELING WITH MICROSOFT EXCELMICROSOFT EXCEL

Chapter 14Chapter 14Chapter 14Chapter 14

Copyright 2001Copyright 2001Prentice HallPrentice Hall

Part 1Part 1

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IIII NTRODUCTIONNTRODUCTION

The task of managing major projects is an ancient The task of managing major projects is an ancient and honorable art.and honorable art.

In about 2600 B.C., the Egyptians build the Great In about 2600 B.C., the Egyptians build the Great Pyramid for King Khufu.Pyramid for King Khufu.

The Greek historian Herodotus claimed that 400,000 The Greek historian Herodotus claimed that 400,000 men worked for 20 years to build this structure.men worked for 20 years to build this structure.

Modern projects ranging from building a suburban Modern projects ranging from building a suburban shopping center to putting a man on the moon are shopping center to putting a man on the moon are amazingly large, complex, and costly.amazingly large, complex, and costly.

Completing such projects on time and within budget Completing such projects on time and within budget is not an easy task. is not an easy task.

Indeed, the complicated problems of scheduling Indeed, the complicated problems of scheduling such projects are often structured by the such projects are often structured by the interdependence of activities.interdependence of activities.

Typically, certain activities may not be initiated Typically, certain activities may not be initiated before others have been completed.before others have been completed.

Some key questions to be answered in this chapter Some key questions to be answered in this chapter are:are:

1.1. What is the expected project completion What is the expected project completion date?date?

2.2. What is the potential “variability” in this date? What is the potential “variability” in this date?

3.3. What are the scheduled start and completion What are the scheduled start and completion dates for each specific activity?dates for each specific activity?

4.4. What activities are What activities are criticalcritical in the sense that in the sense that they must be completed exactly as scheduled they must be completed exactly as scheduled in order to meet the target for overall project in order to meet the target for overall project completion?completion?

5.5. How long can How long can noncriticalnoncritical activities be delayed activities be delayed before a delay in the overall completion date before a delay in the overall completion date is incurred?is incurred?

6.6. How might resources be concentrated most How might resources be concentrated most effectively on activities in order to speed up effectively on activities in order to speed up project completion?project completion?

7.7. What controls can be exercised on the flows What controls can be exercised on the flows of expenditures for the various activities of expenditures for the various activities throughout the duration of the project in order throughout the duration of the project in order that the overall budget can be adhered to?that the overall budget can be adhered to?

To answer these questions, we will use the methods To answer these questions, we will use the methods PERT (PERT (PProgram rogram EEvaluation valuation RRevieweview T Technique) and echnique) and CPM (CPM (CCriticalritical P Path ath MMethod).ethod).

Both of these approaches to scheduling represents Both of these approaches to scheduling represents a project as a network.a project as a network.

When a project involves uncertain elements, the When a project involves uncertain elements, the representation of the project requires a stochastic representation of the project requires a stochastic network.network.

PERT was developed in the late 1950s by the Navy PERT was developed in the late 1950s by the Navy Special Projects Office in cooperation with the Special Projects Office in cooperation with the management consulting firm of Booz, Allen, and management consulting firm of Booz, Allen, and Hamilton.Hamilton.

The technique was used in the engineering and The technique was used in the engineering and development program of the Polaris missile.development program of the Polaris missile.

Many firms and government agencies today require Many firms and government agencies today require all contractors to use PERT.all contractors to use PERT.

CPM differs from PERT in the details of how time CPM differs from PERT in the details of how time and cost are treated.and cost are treated.

The implementation of PERT and CPM had an The implementation of PERT and CPM had an immediate impact on scheduling projects because it immediate impact on scheduling projects because it allowed the practice of “management by exception.”allowed the practice of “management by exception.”

CPM was developed in 1957 by J. E. Kelly of CPM was developed in 1957 by J. E. Kelly of Remington Rand and M. R. Walker of Du Pont.Remington Rand and M. R. Walker of Du Pont.

TYPICAL PROJECT: TYPICAL PROJECT: THE GLOBAL OIL CREDIT CARD OPERATIONTHE GLOBAL OIL CREDIT CARD OPERATION

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Moving the Global Oil credit card operation to Moving the Global Oil credit card operation to Des Moines, Iowa, from the home office in Dallas is Des Moines, Iowa, from the home office in Dallas is an important project.an important project.

Global’s board of directors has set a firm deadline of Global’s board of directors has set a firm deadline of 22 weeks for the move and has put the Operations 22 weeks for the move and has put the Operations Analysis Group in charge of the move.Analysis Group in charge of the move.

The move is difficult to coordinate because it The move is difficult to coordinate because it involves many different divisions within the involves many different divisions within the company.company.

Real estate must select one of three available Real estate must select one of three available office sites.office sites.

Personnel has to determine which employees Personnel has to determine which employees from Dallas will move, how many new from Dallas will move, how many new employees to hire, and who will train them.employees to hire, and who will train them.

The systems group and the treasurer’s office The systems group and the treasurer’s office must organize and implement the operating must organize and implement the operating procedures and the financial arrangements for procedures and the financial arrangements for the new operation.the new operation.

The architects will have to design the interior The architects will have to design the interior space and oversee needed structural space and oversee needed structural improvements.improvements.

Office partitions, computer facilities, Office partitions, computer facilities, furnishings, and so on, must all be provided furnishings, and so on, must all be provided for the existing building.for the existing building.

A second complicating factor is the inter-A second complicating factor is the inter-dependence of activities (i.e., some parts of the dependence of activities (i.e., some parts of the project cannot be started until other parts are project cannot be started until other parts are completed).completed).Consider two obvious examples: Consider two obvious examples:

Global cannot construct the interior of an office Global cannot construct the interior of an office before it has been designed. before it has been designed.

Global cannot hire new employees until it has Global cannot hire new employees until it has determined its personnel requirements.determined its personnel requirements.

THE ACTIVITY LISTTHE ACTIVITY LIST PERT and CPM are specifically designed for projects PERT and CPM are specifically designed for projects of this sort.of this sort.

The first step is to define the activities in the project The first step is to define the activities in the project and to establish precedence relationships.and to establish precedence relationships.

The first The first activity listactivity list prepared for the move is shown prepared for the move is shown below.below.

This is the This is the most importantmost important part of any PERT or CPM part of any PERT or CPM project and must be a group effort so that no project and must be a group effort so that no important activities are overlooked.important activities are overlooked.

Note that the columns labeled Note that the columns labeled TimeTime and and ResourcesResources are indications of things to come.are indications of things to come.

Each activity (represented by a letter) is placed on a Each activity (represented by a letter) is placed on a separate line.separate line.

An activity’s immediate predecessors are recorded An activity’s immediate predecessors are recorded on the same line.on the same line.An immediate predecessor is an activity that must An immediate predecessor is an activity that must be completed prior to the start of the activity in be completed prior to the start of the activity in question (e.g., Global cannot start activity question (e.g., Global cannot start activity CC until until activity activity BB is completed). is completed).

THE GANTT CHARTTHE GANTT CHART Before discussing PERT and CPM, let’s look at Before discussing PERT and CPM, let’s look at another graphical approach, the Gantt chart another graphical approach, the Gantt chart (developed by Henry L. Gantt in 1918).(developed by Henry L. Gantt in 1918).

Activities

Time (weeks)

Activities

Time (weeks)

The horizontal axis is time while each activity is The horizontal axis is time while each activity is listed on the vertical axis.listed on the vertical axis.

The beginning of the bar represents The beginning of the bar represents the earliest the earliest possible starting possible starting time for the activity.time for the activity.

Each bar represents the Each bar represents the anticipated as well as anticipated as well as

actual duration.actual duration.

As each As each activity is activity is

completed, the completed, the appropriate bar appropriate bar

is shaded.is shaded.

At any point in time, it is clear which activities are on At any point in time, it is clear which activities are on schedule and which are not.schedule and which are not.

Activities

Time (weeks)

As of week 13, activities As of week 13, activities DD, , EE, and , and HH are behind are behind schedule.schedule. Activity Activity GG has has

actually been actually been completed and completed and

hence is ahead of hence is ahead of schedule.schedule.

The previous example shows how the Gantt chart is The previous example shows how the Gantt chart is mainly used as a record-keeping device for following mainly used as a record-keeping device for following the progression in time of the subtasks of a project.the progression in time of the subtasks of a project.

With the Gantt chart, we can see which tasks are on, With the Gantt chart, we can see which tasks are on, behind or ahead of schedule.behind or ahead of schedule.

It is important to note that in the context of the Gantt It is important to note that in the context of the Gantt chart, “on schedule” means “it has been completed chart, “on schedule” means “it has been completed no later than the earliest possible completion time.”no later than the earliest possible completion time.”

However, this is too simple a concept for whether an However, this is too simple a concept for whether an activity is on schedule.activity is on schedule.

The appropriate point of view should be whether The appropriate point of view should be whether the the overall projectoverall project is being delayed in terms of a target is being delayed in terms of a target completion date.completion date.

The Gantt chart also fails to reveal which activities The Gantt chart also fails to reveal which activities are are immediate predecessorsimmediate predecessors of other activities. of other activities.

We will now see that the network representation We will now see that the network representation contains the immediate predecessor information contains the immediate predecessor information that we need.that we need.

This information is of vital importance in This information is of vital importance in determining project completion time.determining project completion time.

THE NETWORK DIAGRAMTHE NETWORK DIAGRAM In a PERT In a PERT network diagramnetwork diagram, ,

each activity is represented by an arrow each activity is represented by an arrow that is called a that is called a branchbranch or or arcarc..

the beginning and end of each activity (an the beginning and end of each activity (an eventevent) is indicated by a circle that is called ) is indicated by a circle that is called a a nodenode..

When an When an activityactivity is completed, the is completed, the eventevent occurs. occurs.

Constructing the Network DiagramConstructing the Network Diagram The following The following network diagram shows activities network diagram shows activities AA through through CC. .

1111

2222

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4444

AA

BB CC

Note that the numbers assigned to Note that the numbers assigned to the nodes are arbitrary. They are the nodes are arbitrary. They are

simply used to identify events and simply used to identify events and do no imply anything about do no imply anything about

precedence relationships.precedence relationships.

In the network diagram, In the network diagram, each activity must start at each activity must start at

the node in which its the node in which its immediate predecessors immediate predecessors

ended.ended.For example, activity For example, activity CC starts at node because its starts at node because its immediate predecessor, activity immediate predecessor, activity BB, ended there., ended there.

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Now we need to add activity Now we need to add activity DD to the network. to the network. However, note that activities However, note that activities AA and and CC are both are both immediate predecessors to activity immediate predecessors to activity DD. .

Therefore, nodes and must be combined so that Therefore, nodes and must be combined so that activity activity DD can start from it. can start from it.

22 44

1111

3333

2222

4444

AA

BB

CC

5555

DDEE

Note that activity Note that activity EE, which , which has only has only DD as an immediate as an immediate predecessor, can be added predecessor, can be added

with no difficulty. with no difficulty.

However, adding activity However, adding activity FF creates new problems. creates new problems.

Since Since FF has has CC as an immediate predecessor, it would as an immediate predecessor, it would emanate from node . However, this would imply emanate from node . However, this would imply that that AA is also an immediate predecessor to is also an immediate predecessor to FF, which , which is incorrect.is incorrect.

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The Use of Dummy ActivitiesThe Use of Dummy Activities This dilemma is This dilemma is solved by introducing a solved by introducing a dummy activitydummy activity, which is , which is represented by a dashed line in the network represented by a dashed line in the network diagram.diagram.

The dummy activity is fictitious in the sense that it The dummy activity is fictitious in the sense that it requires no time or resources. requires no time or resources.

It merely provides a pedagogical device that enables It merely provides a pedagogical device that enables us to draw a network representation that correctly us to draw a network representation that correctly maintains the appropriate precedence relationships.maintains the appropriate precedence relationships.

1111

2222

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AA

BB

CC

7777DD EE

4444FF 6666

Here is the resulting network diagram with the Here is the resulting network diagram with the dummy activity.dummy activity.

The procedure is generalized as follows:The procedure is generalized as follows:

Suppose that we wish to add an activity Suppose that we wish to add an activity AA to the to the network starting at node network starting at node NN, but not all of the , but not all of the activities that enter node activities that enter node NN are immediate are immediate predecessors of activity predecessors of activity AA..

Create a new node Create a new node MM with a dummy activity running with a dummy activity running from node from node MM to node to node NN. .

Take those activities that are currently entering node Take those activities that are currently entering node NN and that are immediate predecessors of activity and that are immediate predecessors of activity AA and reroute them to enter node and reroute them to enter node MM..

Now, make activity Now, make activity AA start at node start at node MM..

This network diagram shows the complete activity This network diagram shows the complete activity list and all of the precedence relationships.list and all of the precedence relationships.

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8888HH

GG JJ

II

Select Select sitesite

PlanPlan

PersonnelPersonnelrequirementsrequirements

SelectSelectpersonnelpersonnel

FinancialFinancialarrangementsarrangements

DesignDesign

ConstructConstruct

MoveMove

HireHire TrainTrain

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CC

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EE

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8888HH

GG JJ

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Some computer programs may have difficulty with Some computer programs may have difficulty with activities activities HH and and GG since they both start at node and since they both start at node and end at node . end at node .

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77 These activities These activities may be read by the may be read by the computer program computer program as one instead of as one instead of two.two.

A dummy activity can be used to cure this condition.A dummy activity can be used to cure this condition.

Select Select Site [3]Site [3]

Plan [5]Plan [5]PersonnelPersonnelRequirements [3]Requirements [3]

SelectSelect

Personnel [2]Personnel [2]

FinancialFinancialArrangements [5]Arrangements [5]

Design [4]Design [4]

Construct [8]Construct [8]

MoveMove[2][2]

Hire [4]Hire [4] Train [3]Train [3]

1111

2222

3333

5555

AA

BB

CC

7777

DD

EE

4444FF 6666

9999

HHGG

JJ

II8888

Expected activity Expected activity completion completion times.times.

An Activity-on-Nodes ExampleAn Activity-on-Nodes ExampleThe previous network was an example of an Activity-The previous network was an example of an Activity-on-Arc (or AOA) approach. on-Arc (or AOA) approach.

In the Activity-on-Nodes (AON) approach, the In the Activity-on-Nodes (AON) approach, the activities are associated with the nodes of the activities are associated with the nodes of the network while the arcs of the network display the network while the arcs of the network display the precedence relationships.precedence relationships.

The Global Oil network would be represented as The Global Oil network would be represented as follows:follows:

AAAA DDDD

BBBB

EEEE JJJJ

CCCC FFFF

IIIIHHHH

GGGG

The activity list and an appropriate network diagram The activity list and an appropriate network diagram are useful devices for representing the precedence are useful devices for representing the precedence relationships among the activities in a project.relationships among the activities in a project.

TTTTHE CRITICAL PATH - HE CRITICAL PATH - MEETING THE BOARD’S DEADLINEMEETING THE BOARD’S DEADLINE

Recall that the board has set a firm goal of 22 weeks Recall that the board has set a firm goal of 22 weeks for the overall project to be completed.for the overall project to be completed.

The time estimates must first be incorporated before The time estimates must first be incorporated before we can tell if this goal can be reached.we can tell if this goal can be reached.

The PERT-CPM procedure requires management to The PERT-CPM procedure requires management to produce an estimate of the expected time it will take produce an estimate of the expected time it will take to complete each activity on the activity list. to complete each activity on the activity list.

These time estimates are given below:These time estimates are given below:

If you added up all of the expected activity times in If you added up all of the expected activity times in the above table, the total working time required to the above table, the total working time required to complete all the individual activities would becomplete all the individual activities would be

THE CRITICAL PATH CALCULATIONTHE CRITICAL PATH CALCULATION

3 + 5 + 3 + 4 + 8 + 2 + 4 + 2 + 5 + 3 = 39 weeks 3 + 5 + 3 + 4 + 8 + 2 + 4 + 2 + 5 + 3 = 39 weeks

However, this does not take into account those However, this does not take into account those activities which can be performed simultaneously.activities which can be performed simultaneously.

For example, activities For example, activities AA (3 weeks) and (3 weeks) and BB (5 weeks) (5 weeks) can be initiated at the same time. Therefore, the can be initiated at the same time. Therefore, the total time to complete both activities would be 5 total time to complete both activities would be 5 weeks.weeks.

To obtain a prediction of the minimum calendar time To obtain a prediction of the minimum calendar time required for overall project duration, we must find required for overall project duration, we must find the the critical pathcritical path in the network. in the network.

A A pathpath is a sequence of connected activities that is a sequence of connected activities that leads from the starting node to the completion leads from the starting node to the completion node (e.g., path node (e.g., path B-C-D-E-JB-C-D-E-J).).

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To complete the project, the activities To complete the project, the activities on all pathson all paths must be completed (all paths must be traversed).must be completed (all paths must be traversed).

The task is to analyze the total amount of calendar The task is to analyze the total amount of calendar time required for all paths to be traversed and to find time required for all paths to be traversed and to find the the longestlongest path from start to finish. path from start to finish.

The longest path through the network is called the The longest path through the network is called the critical pathcritical path and will determine the overall project and will determine the overall project duration (because no other path will be longer).duration (because no other path will be longer).

The activities on the critical path are called The activities on the critical path are called critical critical activitiesactivities of the project since if they are delayed, of the project since if they are delayed, then the entire project will be delayed.then the entire project will be delayed.

It is this subset of activities that must be kept on It is this subset of activities that must be kept on schedule.schedule.

Earliest Start and Earliest Finish TimesEarliest Start and Earliest Finish Times Now let’s Now let’s look at the steps employed in finding a critical path. look at the steps employed in finding a critical path. Fundamental in this process is the Fundamental in this process is the earliest start time earliest start time for each activity. for each activity.To illustrate this idea, consider activity To illustrate this idea, consider activity DD, “design , “design facility.” Now assume that the project starts at time facility.” Now assume that the project starts at time zero and ask yourself:zero and ask yourself:

““What is the earliest time at which activity What is the earliest time at which activity DD can start?”can start?”

Clearly, it cannot start until activity Clearly, it cannot start until activity AA is complete (3 is complete (3 weeks). However, it also cannot start before the weeks). However, it also cannot start before the dummy activity is complete (0 weeks). dummy activity is complete (0 weeks).

Since the dummy cannot start until Since the dummy cannot start until BB and and CC are are complete (a total of 8 weeks), we see that complete (a total of 8 weeks), we see that DD cannot cannot start until 8 weeks have passed.start until 8 weeks have passed.

1111

2222

3333

5555

AA

BB

CC

7777

DD

EE

4444FF 6666

9999

HHGG

JJ

II8888

In this calculation, it is crucial to note that activities In this calculation, it is crucial to note that activities AA and and BB both start at time 0. both start at time 0.

Select Select Site [3]Site [3]

Plan [5]Plan [5]PersonnelPersonnelRequirements [3]Requirements [3]

SelectSelect

Personnel [2]Personnel [2]

FinancialFinancialArrangements [5]Arrangements [5]

Design [4]Design [4]

Construct [8]Construct [8]

MoveMove[2][2]

Hire [4]Hire [4] Train [3]Train [3]

AA is complete after 3 weeks. is complete after 3 weeks.BB requires another 2 weeks requires another 2 weeks

BB is complete after 5 weeks and C can start is complete after 5 weeks and C can start

C C is complete after 5 + 3 = 8 weeksis complete after 5 + 3 = 8 weeks

Thus, after 8 weeks, both Thus, after 8 weeks, both AA and and CC are complete and are complete and DD can start. In other words, can start. In other words,

earliest start for activity earliest start for activity DD = 8 weeks = 8 weeks

Another important concept is Another important concept is earliest finish timeearliest finish time for for each activity. If we let each activity. If we let

ESES = earliest start time for a given activity = earliest start time for a given activity

EFEF = earliest finish time for a given activity = earliest finish time for a given activity

tt = expected activity time for a given activity= expected activity time for a given activity

Then, for a given activity, the relation between Then, for a given activity, the relation between earliest start time and earliest finish time is earliest start time and earliest finish time is

EF = ES + EF = ES + tt

Consider the following rule:Consider the following rule:

Earliest Start Time RuleEarliest Start Time RuleThe ES time for an activity leaving a particular node The ES time for an activity leaving a particular node

is the is the largestlargest of the EF times for all activities of the EF times for all activities entering the node.entering the node.

Looking at nodes , , , and of the previous Looking at nodes , , , and of the previous network, the result is shown in brackets below network, the result is shown in brackets below [ES, EF][ES, EF]. .

11 22 33 44

The ES rule says that The ES rule says that the ES for activity the ES for activity DD

is equal to the larger is equal to the larger EF for all activities EF for all activities

entering node 2.entering node 2.

Continuing to each Continuing to each node in a node in a forward forward passpass through the through the

entire network, the entire network, the values values [ES, EF][ES, EF] are are then computed for then computed for

each activity.each activity.

1111

2222

3333

B [0,5]B [0,5] C [5,8]C [5,8]

D [8,12]D [8,12]

4444F [8,10]F [8,10]

I [5,10]I [5,10]

A [0,3]A [0,3]

The network with the ES and EF values is given The network with the ES and EF values is given below:below:

1111

2222

3333

5555A [0,3]A [0,3]

B [0,5]B [0,5]C [5,8]C [5,8]

7777

D [8,12]D [8,12]

E [12,20]E [12,20]

4444

F [8,10]F [8,10] 6666

9999

HH[12,12][12,12]

J [20,23]J [20,23]

I [5,10]I [5,10]

8888G

[10,14]G

[10,14]

Note that the earliest finish time for Note that the earliest finish time for JJ is 23 weeks. is 23 weeks. Therefore, the earliest completion time for the entire Therefore, the earliest completion time for the entire project is 23 weeks. We have now answered the 1project is 23 weeks. We have now answered the 1stst question.question.

Latest Start and Latest Finish TimesLatest Start and Latest Finish Times We now We now proceed with a proceed with a backward passbackward pass calculation in order to calculation in order to identify possible start and completion dates, the identify possible start and completion dates, the activities on the critical path, and how long activities on the critical path, and how long noncritical activities may be delayed without noncritical activities may be delayed without affecting the overall completion date (answering the affecting the overall completion date (answering the 33rdrd, 4, 4thth, and 5, and 5thth questions). questions).

Now that we have the target completion date of 23 Now that we have the target completion date of 23 weeks, we can work backward from this date, weeks, we can work backward from this date, determining the determining the latestlatest date each activity can finish date each activity can finish without delaying the entire project.without delaying the entire project.

The backward pass begins at the completion node, The backward pass begins at the completion node, node . Then, trace back through the network node . Then, trace back through the network computing the computing the latest start timelatest start time and and latest finish timelatest finish time for each activity.for each activity.

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The relation between these quantities isThe relation between these quantities is

LSLS = latest start time for a given activity = latest start time for a given activity

LFLF = latest finish time for a given activity = latest finish time for a given activity

LS = LF - LS = LF - tt

The general rule isThe general rule is

Latest Finish Time RuleLatest Finish Time RuleThe LF time for an activity entering a particular node The LF time for an activity entering a particular node

is the is the smallestsmallest of the LS times for all activities of the LS times for all activities leaving that node.leaving that node.

The complete network with the The complete network with the LSLS and and LFLF values values (given below the (given below the [ES,EF][ES,EF] values) is shown below: values) is shown below:

1111

2222

3333

5555A [0,3]A [0,3]

B [0,5]

B [0,5]

C [

5,8]

C [

5,8]

7777

D [8,12]D [8,12]

E [12,20]

E [12,20]

4444F [8,10]F [8,10] 6666

9999

HH[12,12][12,12]

J [20,23]J [20,23]

I [5,10]I [5,10]

8888G

[10,14]G

[10,14]

[5,8][5,8]

[0,5][0,5]

[8,12][8,12]

[12,20]

[12,20][14,16][14,16]

[5,8

][5

,8]

[18,23][18,23]

[20,23][20,23]

[18,20][18,20]

[16,20][16,20]

Slack and the Critical PathSlack and the Critical Path The next step of the The next step of the algorithm is to identify the amount of slack, or free algorithm is to identify the amount of slack, or free time, associated with each activity.time, associated with each activity.SlackSlack is the amount of time an activity can be is the amount of time an activity can be delayed without affecting the completion date for the delayed without affecting the completion date for the overall project.overall project.

SlackSlack is the same concept covered in linear is the same concept covered in linear programming and is the extra time that could be programming and is the extra time that could be spent on that path without affecting the length of the spent on that path without affecting the length of the critical path.critical path.

For each activity, the slack is computed as:For each activity, the slack is computed as:

Slack = LS – ES = LF – EFSlack = LS – ES = LF – EFFor example, the slack for activity For example, the slack for activity GG is: is:

Slack for Slack for GG = LS for = LS for GG – ES for – ES for GG= 16 – 10 = 6 weeks= 16 – 10 = 6 weeks

Every activity on the critical path should have a Every activity on the critical path should have a slack of 0. This means that the slack of 0. This means that the activity cannot be activity cannot be delayed without affecting the entire projectdelayed without affecting the entire project. .

Therefore, any activity with a slack of 0 is a Therefore, any activity with a slack of 0 is a critical critical activityactivity and is on the and is on the critical pathcritical path..

The critical path activities are those with 0 slack.The critical path activities are those with 0 slack.

Spreadsheet Approach for the NetworkSpreadsheet Approach for the Network The The spreadsheet solution of this problem is most easily spreadsheet solution of this problem is most easily done with an activity-on-the-node (AON) approach. done with an activity-on-the-node (AON) approach.

=MAX(E3)=MAX(E3)=MAX(E2,E4)=MAX(E2,E4)=MAX(E5)=MAX(E5)=MAX(E4)=MAX(E4)=MAX(E7)=MAX(E7)=MAX(E7)=MAX(E7)=MAX(E3)=MAX(E3)=MAX(E6,E8,E9)=MAX(E6,E8,E9)

=D2+C2=D2+C2=G2-C2=G2-C2=MIN(F5)=MIN(F5)=MIN(F4,F10)=MIN(F4,F10)=MIN(F5,F7)=MIN(F5,F7)=MIN(F6)=MIN(F6)=MIN(F11)=MIN(F11)=MIN(F8,F9)=MIN(F8,F9)=MIN(F11)=MIN(F11)=MIN(F11)=MIN(F11)=E13=E13=E13=E13

=F2-D2=F2-D2=IF(H2=0,”YES”,”NO”)=IF(H2=0,”YES”,”NO”)

=MAX(E2:E11)=MAX(E2:E11)

We have now answered the following questions:We have now answered the following questions:

1.1. What is the expected project completion What is the expected project completion date?date?

3.3. What are the scheduled start and completion What are the scheduled start and completion dates for each specific activity?dates for each specific activity?

AnswerAnswer:: 23 weeks23 weeks

AnswerAnswer:: An activity may be scheduled to An activity may be scheduled to start at any date between “earliest start at any date between “earliest start” and “latest start.” The start” and “latest start.” The scheduled completion date will be scheduled completion date will be “start date + expected activity time.” “start date + expected activity time.”

4.4. What activities are What activities are criticalcritical in the sense that in the sense that they must be completed exactly as scheduled they must be completed exactly as scheduled in order to meet the target for overall project in order to meet the target for overall project completion?completion?

5.5. How long can How long can noncriticalnoncritical activities be delayed activities be delayed before a delay in the overall completion date before a delay in the overall completion date is incurred?is incurred?

AnswerAnswer:: The activities on the critical path The activities on the critical path are: are: BB, , CC, , DD, , EE, , JJ..

AnswerAnswer:: Any activity may be started as late Any activity may be started as late as the “latest start” date without as the “latest start” date without delaying the overall project delaying the overall project completion.completion.

The remaining questions will be answered later.The remaining questions will be answered later.

It is clear from the critical path analysis that we have It is clear from the critical path analysis that we have a problem. The board of directors wants to start a problem. The board of directors wants to start operating in Des Moines in 22 weeks, and with the operating in Des Moines in 22 weeks, and with the current plan 23 weeks are required.current plan 23 weeks are required.

There are two basic approaches to reducing the time There are two basic approaches to reducing the time required to complete a project:required to complete a project:

WAYS OF REDUCING PROJECT DURATIONWAYS OF REDUCING PROJECT DURATION

1.1. A A strategic analysisstrategic analysis: Here the analyst asks: : Here the analyst asks: “Does this project have to be done the way it “Does this project have to be done the way it is currently diagrammed?”is currently diagrammed?”

In particular, “Do all of the activities on the In particular, “Do all of the activities on the critical path have to be done in the specified critical path have to be done in the specified order?” Can we make arrangements to order?” Can we make arrangements to accomplish some of these activities in a accomplish some of these activities in a different way not on the critical path?different way not on the critical path?

2.2. A A tactical approachtactical approach: In this approach, the : In this approach, the analyst assumes that the current diagram is analyst assumes that the current diagram is appropriate and works at reducing the time of appropriate and works at reducing the time of certain activities on the critical path by certain activities on the critical path by devoting more resources to them.devoting more resources to them.

The current expected times assume a certain The current expected times assume a certain allocation of resources. For example, the 8 allocation of resources. For example, the 8 weeks for construction (activity weeks for construction (activity EE) assumes a ) assumes a regular 8-hour workday. The contractor can regular 8-hour workday. The contractor can complete the job more rapidly by working complete the job more rapidly by working overtime, but at increased costs.overtime, but at increased costs.

The tactical approach takes into consideration CPM The tactical approach takes into consideration CPM models, which will be discussed later. For now, let’s models, which will be discussed later. For now, let’s focus on the so-called strategic questions.focus on the so-called strategic questions.

A Strategic AnalysisA Strategic Analysis This analysis is analogous to This analysis is analogous to “What if?” analysis done with spreadsheets. “What if?” analysis done with spreadsheets.

After reviewing the network, it is discovered that the After reviewing the network, it is discovered that the current network assumes that activity current network assumes that activity JJ, the training , the training of new employees, must be carried out in the new of new employees, must be carried out in the new building (after building (after EE is complete), and after records and is complete), and after records and key personnel have been moved (after key personnel have been moved (after HH is is complete).complete).

Perhaps these requirements can be changed. First, Perhaps these requirements can be changed. First, J J can be accomplished independently of can be accomplished independently of HH..

Moreover, an alternative training facility can be Moreover, an alternative training facility can be secured by arranging to use surplus classroom secured by arranging to use surplus classroom space in Des Moines at a minimal cost. The new space in Des Moines at a minimal cost. The new employees can be trained and ready to start the employees can be trained and ready to start the moment that construction ends.moment that construction ends.

On the other hand, a new activity will have to be On the other hand, a new activity will have to be added to the activity list: added to the activity list:

secure a training facility (activity secure a training facility (activity KK).).

All of these changes to the network may have All of these changes to the network may have created a new critical path with a still unsatisfactory created a new critical path with a still unsatisfactory minimum time (i.e., one greater than 22 weeks).minimum time (i.e., one greater than 22 weeks).

Spreadsheet Output for the Redefined NetworkSpreadsheet Output for the Redefined Network This This redefined activity list is shown in the form of the redefined activity list is shown in the form of the activity-on-the-arc (AOA) diagram.activity-on-the-arc (AOA) diagram.

1111

2222

3333

5555AA

BB

CC

7777

DDEE

4444FF

9999

HHGG

JJ

II

8888

6666KK

Select Select Site [3]Site [3]

Plan [5]Plan [5]

PersonnelPersonnelRequirements [3]Requirements [3]

SelectSelect

Personnel [2]Personnel [2]

FinancialFinancialArrangements [5]Arrangements [5]

Design [4]Design [4]

Construct [8]

Construct [8]Move [2]Move [2]Hire [4]

Hire [4]

Train [3]Train [3]

Secu

re S

ecure

Train

ing

[3]

Train

ing

[3]

Here is the same network using the Activity-on-the-Here is the same network using the Activity-on-the-Node (AON) approach. Node (AON) approach.

AAAA DDDD

BBBB

EEEE

JJJJ

CCCC FFFF

IIII HHHH

GGGG

KKKK

The modified spreadsheet is shown below:The modified spreadsheet is shown below:

=MAX(E3)=MAX(E3)=MAX(E2,E4)=MAX(E2,E4)=MAX(E5)=MAX(E5)=MAX(E4)=MAX(E4)=MAX(E4)=MAX(E4)=MAX(E7)=MAX(E7)=MAX(E3)=MAX(E3)=MAX(E8,E12)=MAX(E8,E12)=MAX(E7)=MAX(E7)

=D2+C2=D2+C2=G2-C2=G2-C2=MIN(F5)=MIN(F5)=MIN(F4,F10)=MIN(F4,F10)=MIN(F5,F7)=MIN(F5,F7)=MIN(F6)=MIN(F6)=E14=E14=MIN(F8,F9,F12)=MIN(F8,F9,F12)=MIN(F11)=MIN(F11)=E14=E14=E14=E14=E14=E14=MIN(F11)=MIN(F11)

=MAX(E2:E12)=MAX(E2:E12)

=F2-D2=F2-D2=IF(H2=0,”YES”,”NO”)=IF(H2=0,”YES”,”NO”)

Note that the redefined project completion time is 20 Note that the redefined project completion time is 20 weeks and the new critical path is weeks and the new critical path is BB--CC--DD--EE..

End of Part 1End of Part 1Please continue to Part 2Please continue to Part 2