Project Scheduling with limited resources

7/29/2019 Project Scheduling with limited resources

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Network/ Project Scheduling withNetwork/ Project Scheduling with

Limited ResourcesLimited Resources


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Consider the following scheduling model displayingConsider the following scheduling model displaying

the precedence relationships:the precedence relationships:

An Example:An Example:

BeginBegin EndEndII

33

IIII

22 IVIV

11VV

22

VIIVII

11

IIIIII

11

IXIX

22

VIIIVIII

22

VIVI

44

Precedence relationships indicate which activitiesPrecedence relationships indicate which activities

must be completed before others can begin.must be completed before others can begin.

1. SCHEDULING WITH LIMITED RESOURCES1. SCHEDULING WITH LIMITED RESOURCES

(Workload Smoothing/ Resource Leveling)(Workload Smoothing/ Resource Leveling)


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This table shows the duration of each activity (inThis table shows the duration of each activity (in

weeks) and the resources required (number ofweeks) and the resources required (number of

people) to complete each activity.people) to complete each activity.

TIME REQUIRED TO NO. PEOPLE (PER WK)TIME REQUIRED TO NO. PEOPLE (PER WK)

ACTIVITY COMPLETE (WKS) REQUIRED TO COMPLETEACTIVITY COMPLETE (WKS) REQUIRED TO COMPLETE

II 33 66

IIII 22 33IIIIII 11 33

IVIV 11 33

VV 22 66

VIVI 44 55

VIIVII 11 33VIIIVIII 22 44

IXIX 22 33


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The proposed activity schedule which will achieveThe proposed activity schedule which will achieve

the overall completion time of 9 weeks is giventhe overall completion time of 9 weeks is given

below.below.

Note that this diagram respects precedenceNote that this diagram respects precedence

relationships and at the same time shows whenrelationships and at the same time shows when

each activity should start and how long it will take.each activity should start and how long it will take.

WeeksWeeks1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

II

IIIIIIIIII

IVIV

VVVIIVII

IXIX

VIIIVIII

VIVI


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Now consider the number of people per weekNow consider the number of people per week

required to implement the proposed schedule asrequired to implement the proposed schedule as

displayed in thedisplayed in thepersonnel loading chartpersonnel loading chartshownshown

below.below.

Weeks

Total people

scheduled

The proposedThe proposed

schedule makesschedule makes

an erratican erratic

utilization ofutilization ofpersonnel.personnel.

HeuristicHeuristic

programs canprograms can

be applied tobe applied toemployemploy

resources moreresources more

smoothly.smoothly.


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SlackSlack is the maximum amount of time an activity canis the maximum amount of time an activity can

be delayed without delaying overall projectbe delayed without delaying overall projectcompletion.completion.

WORKLOAD SMOOTHING HEURISTICWORKLOAD SMOOTHING HEURISTIC


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WeeksWeeks1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

II

IIIIIIIIII

IVIV

VVVIIVII

IXIX

VIIIVIII

VIVI

If the completion time of activity V were delayed,If the completion time of activity V were delayed,

then activity VI would also be delayed and thethen activity VI would also be delayed and the

project could not be completed by the end of theproject could not be completed by the end of the

ninth week.ninth week.

In contrast, the completion of activity VIII could beIn contrast, the completion of activity VIII could be

delayed by 3 weeks without delaying the completiondelayed by 3 weeks without delaying the completion

of the project. Activity VIII has a slack of 3 weeks.of the project. Activity VIII has a slack of 3 weeks.

Activity VActivity V

has no slackhas no slack

(0 weeks).(0 weeks).


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The following heuristic is given to provide aThe following heuristic is given to provide a

smoother workload across time:smoother workload across time:

1.1. Determine the maximum required resources inDetermine the maximum required resources in

the proposed schedule, saythe proposed schedule, say mm workers/week.workers/week.

2.2. In each week, impose a new upper limit ofIn each week, impose a new upper limit ofmm-1-1

for resource utilization, and, if possible, revisefor resource utilization, and, if possible, revise

the proposed schedule to satisfy thisthe proposed schedule to satisfy thisconstraint.constraint.

a.a. Beginning with the earliest week violatingBeginning with the earliest week violating

the constraint, consider the activitiesthe constraint, consider the activities

contributing to the overload and movecontributing to the overload and moveforward the one withforward the one with mostmostslack as little asslack as little as

possible until it contributes to nopossible until it contributes to no

overloading, but without delaying theoverloading, but without delaying the

completion of the entire project.completion of the entire project.


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NoteNote, do not move any activities with zero, do not move any activities with zero

slack. If there are ties, move forward theslack. If there are ties, move forward the

activity that contributesactivity that contributes leastleastto theto the

overload (i.e., requires the fewest people).overload (i.e., requires the fewest people).

b.b. The heuristic terminates when the currentThe heuristic terminates when the current

overload cannot be decreased.overload cannot be decreased.


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Consider the following proposed plan. The activityConsider the following proposed plan. The activity

label appearslabel appears belowbeloweach arrow while the number ofeach arrow while the number of

people required each week appearspeople required each week appears aboveabove the arrow.the arrow.

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek

Total personnelTotal personnel

New limitNew limit

12 9 6 15 13 9 5 5 512 9 6 15 13 9 5 5 5

66

II33

IIII

33

IIIIII

66

VV55

VIVI

33

IVIV33

VIIVII

33IXIX

44

VIIIVIII

Add together theAdd together the

activities intersectingactivities intersecting

a particular week toa particular week to

get the totalget the totalpersonal.personal. The distance from the end of theThe distance from the end of thearrow to 9 weeks is the slack time.arrow to 9 weeks is the slack time.

Slack = 3 wksSlack = 3 wks

The maximum required resource is 15 in period 4, soThe maximum required resource is 15 in period 4, so

impose a new upper limit of 14 in each week.impose a new upper limit of 14 in each week.

14 14 14 14 14 14 14 14 1414 14 14 14 14 14 14 14 14


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Applying the HeuristicApplying the Heuristic Now, move forward onlyNow, move forward only

those activities with positive slack. The movablethose activities with positive slack. The movable

activities contributing to the overload are IV, VIII,activities contributing to the overload are IV, VIII,

and IX. IV has the most slack.and IX. IV has the most slack.

33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 15 13 9 5 5 512 9 6 15 13 9 5 5 5

14 14 14 14 14 14 14 14 1414 14 14 14 14 14 14 14 14

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII


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33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 15 13 9 5 5 512 9 6 15 13 9 5 5 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII

Moving IV forward by 1 periodMoving IV forward by 1 period

reduces the utilization in week 4reduces the utilization in week 4

by 3 units but increases week 5by 3 units but increases week 5to 16 units.to 16 units.

12 1612 16


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33

IVIV

The best choice isThe best choice is

to move IV forwardto move IV forward

by 2 periods so thatby 2 periods so thatno upper limit willno upper limit will

be violated.be violated.

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 15 13 9 5 5 512 9 6 15 13 9 5 5 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII


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33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 12 13 12 5 5 512 9 6 12 13 12 5 5 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII

12 12 12 12 12 12 12 12 1212 12 12 12 12 12 12 12 12

The new upper limit is now 12 in each week.The new upper limit is now 12 in each week.


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1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 12 13 12 5 5 512 9 6 12 13 12 5 5 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33

IVIV

33VIIVII33

IXIX

44VIIIVIII

Now the only overload based on a limit of 12 isNow the only overload based on a limit of 12 is

caused by activities VIII and IX in week 5.caused by activities VIII and IX in week 5.


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1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 9 10 12 8 8 512 9 6 9 10 12 8 8 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33

IVIV

33VIIVII33

IXIX

44VIIIVIII

Activity IX has the most slack, and it must beActivity IX has the most slack, and it must be

advanced 3 weeks to begin in week 7.advanced 3 weeks to begin in week 7.

11 11 11 11 11 11 11 11 1111 11 11 11 11 11 11 11 11

The new upper limit is now 11 in each week.The new upper limit is now 11 in each week.


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1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

12 9 6 9 10 12 8 8 512 9 6 9 10 12 8 8 5

66

II33

IIII

33

IIIIII

66

VV

55

VIVI

33

IVIV

33VIIVII33

IXIX

44VIIIVIII

11 11 11 11 11 11 11 11 1111 11 11 11 11 11 11 11 11

Now there are violations in weeks 1 and 6.Now there are violations in weeks 1 and 6.

According to the algorithm, first move activity IIIAccording to the algorithm, first move activity III

forward 2 weeks and then activity IV forward 1 week.forward 2 weeks and then activity IV forward 1 week.


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33

IIIIII 33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

9 9 9 9 10 9 11 8 59 9 9 9 10 9 11 8 5

66

II33

IIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII

Now there are violations in weeks 1 and 6.Now there are violations in weeks 1 and 6.

According to the algorithm, first move activity IIIAccording to the algorithm, first move activity III

forward 2 weeks and then activity IV forward 1 week.forward 2 weeks and then activity IV forward 1 week.

The new upper limit is now 10 in each week.The new upper limit is now 10 in each week.10 10 10 10 10 10 10 10 1010 10 10 10 10 10 10 10 10

N th l i l ti i i k 7 S


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Now the only violation is in week 7. So, moveNow the only violation is in week 7. So, move

activity IV forward 2 weeks.activity IV forward 2 weeks.

10 10 10 10 10 10 10 10 1010 10 10 10 10 10 10 10 10

33

IIIIII 33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

9 9 9 9 10 9 11 8 59 9 9 9 10 9 11 8 5

66

II33

IIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII

H i i T iH i ti T i t Th l i h i blTh l ith i bl t


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Heuristic TerminatesHeuristic Terminates The algorithm is unable toThe algorithm is unable to

improve beyond the fifth proposal. Hence, thisimprove beyond the fifth proposal. Hence, this

schedule is the heuristic solution. The maximumschedule is the heuristic solution. The maximum

utilization is now 10 weeks and the minimum is 8.utilization is now 10 weeks and the minimum is 8.

33

IIIIII 33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek


New limitNew limit

9 9 9 9 10 9 8 8 89 9 9 9 10 9 8 8 8

66

II33

IIII

66

VV

55

VIVI

33VIIVII33

IXIX

44VIIIVIII

F thi d l ti l l ti i th tF thi d l ti l l ti i th t


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For this model, an optimal solution is one thatFor this model, an optimal solution is one that

minimizes the maximum utilizationminimizes the maximum utilization of personnel. Theof personnel. The

optimal schedule according to thisoptimal schedule according to this minimaxminimaxcriteriacriteria

is shown below.is shown below.

33

IIIIII33

IVIV

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9WeekWeek

Total personnelTotal personnel 9 9 9 9 9 9 9 8 89 9 9 9 9 9 9 8 8

66

II33

IIII

66

VV

55

VIVI

33

VIIVII33

IXIX

44

VIIIVIII


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End of SessionEnd of Session

Documents

Project Scheduling with limited resources