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Program Evaluation And Review Technique (PERT)Program Evaluation And Review Technique (PERT)Critical Path Method (CPM)

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Program Evaluation And Review Technique

• Untuk sebanyak mungkin mengurangi adanya penundaan, maupun gangguan produksi

• Mengkoordinasikan berbagai bagian suatu pekerjaan secara menyeluruh dan mempercepat selesainya proyek.

• Suatu pekerjaan yang terkendali dan teratur, karena

g q

jadwal dan anggaran dari suatu pekerjaan telah ditentukan terlebih dahulu sebelum dilaksanakan.

• Pencapaian suatu taraf tertentu dimana waktu merupakan dasar penting dari PERT dalam penyelesaian kegiatan-kegiatan bagi suatu proyek.

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Critical Path Method (metode jalur k iti )

• Diselesaikan secara tepat waktu serta tepat biaya.Metode perencanaan dan pengendalian proyek-proyek

• Prinsip pembentukan jaringan. • Jumlah waktu yang dibutuhkan dalam setiap tahap

kritis)

• Jumlah waktu yang dibutuhkan dalam setiap tahap suatu proyek dianggap diketahui dengan pasti,

• Hubungan antara sumber yang digunakan dan waktu yang diperlukan untuk menyelesaikan proyek.

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LAMA PENGERJAAN

NOMOR PENGERJAAN

TANGGAL SELESAI

TANGGAL MULAI

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Pekerjaan KelangsunganProyek

Data , Waktu, Biaya

Informasi Sasaran Arti Panah

PERT Perencanaan Dan

Pengendalian Proyek

Belum Pernah Dkerjakan,

Belum Diketahui Waktu Pengerjaan

Tercepat, Terlama Terlayak

Tepat Waktu, Sebab Dengan Penyingkatan Waktu Maka Biaya Proyek

Turut Mengecil,

Anak Panah Menunjukkan Tata Urutan (Hubungan Presidentil)

CPM Menjadwalkan D

Sudah Pernah Dik j k

Telah Diketahui Ol h E l t

Waktu P j

Tepat Biaya Tanda Panah Ad l h CPM Dan

Mengendalikan Aktivitas

Dikerjakan Oleh Evaluator Pengerjaan Waktu Yang Paling Tepat

DanLayak Untuk

Adalah Kegiatan

A Project A Project Suatu set pekerjaan yang dilakukan secara sekuensial

Tujuan (Goals)Menjamin suatu project

▪ mencapai tujuannya▪ Selesai tepat waktuSelesai tepat waktu▪ Sesuai Anggaran▪ Sesuai dengan sumber daya

Menyediakan mekasnisme monitoring

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Jalur Kritis / Critical Path:Jalur Kritis / Critical Path:Suatu aktivitas sekuensial yang menuju pada penyelesaian project.

Slack:Jumlah fleksibitas dalam menjadwalan aktivitas yang j y gtidak kritis.

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An Activity On Node (AON) Network Representationof the Klonepalm 2000 Computer Project

EImmediate EstimatedImmediate Estimated

A90

H28

E21

D20

B15

G14

F25

C5 Activity Predecessor Completion Time

A None 90B A 15C B 5D G 20E D 21F A 25

Activity Predecessor Completion TimeA None 90B A 15C B 5D G 20E D 21F A 25

NoneA

A

B

J45

I30

F A 25G C,F 14H D 28I A 30J D,I 45

F A 25G C,F 14H D 28I A 30J D,I 45

A

A

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Activity Description Immediate Predecessor

Time Estimate (days)Predecessor (days)

A Select teams 3

B Mail out invitations A 5

C Arrange accommodations 10

D Plan promotion B, C 3

E Print tickets B, C 5

F Sell tickets E 10

Seberapa cepatTurnamen dapatDisesaikan?

Aktivitas manakah Yang kritis?

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G Complete arrangements C 8

H Develop schedules G 3

I Practice D, H 2

J Conduct tournament F, I 3

Activity Expected Duration (weeks)

Immediate Predecessors

A 2

B 2

C 3 AD 2 B

A,2 C,3

E,1

Activities are represented by nodes:

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D 2 BE 1 C,D B,2 D,2

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Forward PassForward Pass:Calculate Earliest Start Times, Earliest Finish Times

Backward Pass:Calculate Latest Start Times, Latest Finish Times

SlackLatest Start Time – Earliest Start Time

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Activit Expecte Immediate Earliest Earliest Latest Latest Slacky d

Duration (weeks)

Predecessors

Start Time

Finish Time

Finish Time

Start Time

A 2

B 2

C 3 A

D 2 B

A,2

B,2

C,3

D,2

E,1

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D 2 B

E 1 C,D

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Activity

Expected

D ti

Immediate Predecess

Earliest Start Ti

Earliest Finish Ti

Latest Finish Ti

Latest Start Ti

Slack

Duration (weeks)

ors Time Time Time Time

A 2 0 2 2 0 0

B 2 0 2 3 1 1

C 3 A 2 5 5 2 0

D 2 B 2 4 5 3 1

E 1 C,D 5 6 6 5 0

Activities with 0 slack are on the critical path:

A,2

B,2

C,3

D,2

E,1

A,2

B,2

C,3

D,2

E,1

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0 1 2 3 4 5 6

Activity A Activity C Act. E

Act. EActivity DActivity BSlack

Time

p

Activity Duration (weeks)

ImmPred ES EF LF LS SLACK

A 5

B 4

C 3

D 2 A

E 6 B, C

F 3 D E

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F 3 D, E

G 7 E

H 5 F

I 4 F

J 2 G

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Activity

Description Imm Pred

Dur ES EF LS LF SLCK

A S l 3A Select teams 3

B Mail out invitations A 5C Arrange

accommodations10

D Plan promotion B, C 3E Print tickets B, C 5F Sell tickets E 10G Complete C 8G Complete

arrangementsC 8

H Develop schedules G 3I Practice D, H 2J Conduct tournament F, I 3

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Decision VariablesDecision Variables:

Objective Function: A,2

B 2

C,3

D 2

E,1

Constraints:

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B,2 D,2

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The terminal activityThe terminal activityThe single activity that identifies when the project is completed.If there is no natural terminal activity, add a dummy node with 0 duration:

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A, 1

B, 3

C, 1

D, 4

E, 2

Aktivitas digambarkan melelui tanda panahAktivitas digambarkan melelui tanda panah

Find the maximum cost flow

A,2

B,2

C,3

D,2E,1

Source: 1

Source: 1

Demand: 1

A,2

B,2

C,3

D,2

E,1

Interpretation: an arc has a flow of 1 if it is on the critical path

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Often there are penalties and bonuses for late Often there are penalties and bonuses for late or early completion of a project.

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Sebuah kontrak untuk menyelesaikan pekerjaanSebuah kontrak untuk menyelesaikan pekerjaandalam waktu 16 minggu. Terdapat bonus sebesar $12,000 untuk setiappekerjaan yang lebih awal per minggunya darijadwal. Penalti sebesar $15,000 per minggu keterlambatan. Kapankah waktu ideal dalam menyelesaikan proyekp y p ytsb?Aktivitas manakah yang harus diakselerasi dan

seberapa banyak?

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Activity Standard Minimum Extra Cost at Imm Maximum Incremental yDuration (weeks)

Duration (weeks)

Minimum Time ($000)

Pred Reduction Cost

A 5 3 8

B 4 2 14

C 3 1 16

D 2 1 7 A

E 6 3 21 B, C

F 3 2 4 D, E

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G 7 3 8 E

H 5 3 8 F

I 4 3 8 F

J 2 2 N/A G

The critical path method (CPM) is a deterministic The critical path method (CPM) is a deterministic approach to project planning.

Completion time depends only on the amount of money allocated to the activity.

Reducing an activity’s completion time is called “crashing”.

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There are two crucial time durations to consider for each activityfor each activity.

Normal completion time (NT)

Crash completion time (CT)

NT is achieved when a usual or normal cost (NC)is spent to complete the activity.

CT is achieved when a maximum crash cost (CC)is spent to complete the activity.

The Linearity Assumption

[Normal Time - Crash Time][Normal Time]

= [Crash Cost - Normal Cost][Normal Cost]

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Time2018

NormalNC = $2000NT = 20 days

A demonstration of the

Linearity assumption…and save oncompletion time…and save more on

Total Cost = $2600Job time = 18 days

161412108

Add to thenormal cost...

Add more to thenormal cost...

CrashingCC = $4400CT = 12 days

completion time Add 25% to thenormal cost

Save 25% on completion time

Cost ($100)

642

5 10 15 20 25 30 35 40 45

CT = 12 days

Marginal Cost = Additional Cost to get Max. Time ReductionMaximum Time reduction

= (4400 - 2000)/(20 - 12) = $300 per day

Meetings a Deadline at Minimum Cost

Let D be the deadline date to complete a project.

If D cannot be met using normal times, additional resources must be spent on crashing activities.

The objective is to meet the deadline D at minimal additional cost.

Tom Larkin’s political campaign problem illustrates the concept.

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TOM LARKIN’S POLITICAL CAMPAIGNTom Larkin has 26 weeks of mayoral election campaign to plan.

The campaign consists of the following activities

Immediate Normal Schedule Reduced ScheduleActivity Predecessor Time Cost Time Cost

A. Hire campain staff None 4 2.0K 2 5.0KB. Prepare position paper None 6 3.0 3 9C. Recruit volunteers A 4 4.5 2 10

Immediate Normal Schedule Reduced ScheduleActivity Predecessor Time Cost Time Cost

A. Hire campain staff None 4 2.0K 2 5.0KB. Prepare position paper None 6 3.0 3 9C. Recruit volunteers A 4 4.5 2 10D. Raise funds A,B 6 2.5 4 10E. File candidacy papers D 2 0.5 1 1F. Prepare campaign material E 13 13.0 8 25G. Locate/staff headquarters E 1 1.5 1 1.5H. Run personal campaign C,G 20 6.0 10 23.5I. Run media campaign F 9 7.0 5 16

D. Raise funds A,B 6 2.5 4 10E. File candidacy papers D 2 0.5 1 1F. Prepare campaign material E 13 13.0 8 25G. Locate/staff headquarters E 1 1.5 1 1.5H. Run personal campaign C,G 20 6.0 10 23.5I. Run media campaign F 9 7.0 5 16

NETWORK PRESENTATION

A C

F

G

I

H

FINISH

To meet the deadline date of 26 weeks some activities must be crashed.

DBE F

WINQSB CPM schedule with normal times.Project completion (normal) time = 36 weeks

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A c t iv i ty N T N C ($ ) C T C C T M ($ )A 4 2 0 0 0 2 5 0 0 0 2 $ 1 5 0 0

A c t iv i ty N T N C ($ ) C T C C T M ($ )A 4 2 0 0 0 2 5 0 0 0 2 $ 1 5 0 0

Mayoral Campaign Crash Schedule

A 4 2 0 0 0 2 5 0 0 0 2 $ 1 ,5 0 0B 6 3 0 0 0 3 9 0 0 0 3 2 0 0 0C 4 4 5 0 0 2 1 0 0 0 0 2 2 7 5 0D 6 2 5 0 0 4 1 0 0 0 0 2 3 7 5 0E 2 5 0 0 1 1 0 0 0 1 5 0 0F 1 3 1 3 0 0 0 8 2 5 0 0 0 5 2 4 0 0G 1 1 5 0 0 1 1 5 0 0 ** * ***H 2 0 6 0 0 0 1 0 2 3 5 0 0 1 0 1 7 5 0I 9 7 0 0 0 5 1 6 0 0 0 4 2 2 5 0

A 4 2 0 0 0 2 5 0 0 0 2 $ 1 ,5 0 0B 6 3 0 0 0 3 9 0 0 0 3 2 0 0 0C 4 4 5 0 0 2 1 0 0 0 0 2 2 7 5 0D 6 2 5 0 0 4 1 0 0 0 0 2 3 7 5 0E 2 5 0 0 1 1 0 0 0 1 5 0 0F 1 3 1 3 0 0 0 8 2 5 0 0 0 5 2 4 0 0G 1 1 5 0 0 1 1 5 0 0 *** ** *H 2 0 6 0 0 0 1 0 2 3 5 0 0 1 0 1 7 5 0I 9 7 0 0 0 5 1 6 0 0 0 4 2 2 5 0

• Heuristic Approach– Three observations lead to the heuristic.

• The project time is reduced only by critical activities.• The maximum time reduction for each activity is limited.• The amount of time a critical activity can be reduced before another.

path becomes critical is limited.

– Small crashing problems with small number of critical paths can be solved by this heuristic approach.

– Problems with large number of critical paths are bettersolved by a linear programming model.

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Linear Programming ApproachVariablesVXj = start time for activity j.Yj = the amount of crash in activity j.

Objective FunctionMinimize the total additional funds spent on crashing activities.

ConstraintsConstraints▪ The project must be completed by the deadline date D▪ No activity can be reduced more than its Max. time reduction▪ Start time of an activity Finish time of immediate predecessor

≥

JHFEDCBA Y2250Y17500Y2400Y500Y37502750Y2000YMin1500Y +++++++Minimize total crashing costs

26)FIN(XST

≤Meet the deadline )Y20(X)FIN(X)Y9(X)FIN(X

H

II

−+≥−+≥

5Y1Y2Y2Y3 Y2 Y

E

D

C

B

A

≤≤≤≤≤≤

Maximum time-reductionconstraints

)Y6(XX)Y2(XX)Y2(XX)Y4(XX

1XX)Y13(XX

DDE

EEF

EEG

CCH

GH

FFI

−+≥−+≥−+≥−+≥

+≥−+≥ Activity can

start only afterall the predecessorsare completed.

10Y5Y

H

F

≤≤

)Y4(XX)Y4(XX)Y6(XX

AAC

AAD

BBD

−+≥−+≥−+≥A

D

C

BE F

G

I

H

FINISH

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WINQSB Crashing Optimal Solution

Crashing costsMost of the activities become critical !!

Deadline

Other Cases of Project Crashing

Operating Optimally within a given budget▪ When a budget is given, minimizing crashing costs is a constraint,

not an objective.▪ In this case the objective is to minimize the completion time.

Incorporating Time-Dependent Overhead CostsIncorporating Time Dependent Overhead Costs▪ When the project carries a cost per time unit during its duration, this cost is

relevant and must be figured into the model.

▪ In this case the objective is to minimize the total crashing cost+ total overhead cost

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TOM LARKIN - ContinuedThe budget is $75,000.

The objective function becomes a constraintMinimize X(FIN)

1500 YA+ 2000 YB + 2750 YC + 3750 YD + 500 YE + 2400 YF +1750 YH + 2250 YJ

1500 Y 2000 Y 2750 Y 3750 Y 500 Y 2400 Y

This constraint becomes the objective functionX(FIN) 26≤

( )

1500 YA+ 2000 YB + 2750 YC + 3750 YD + 500 YE + 2400 YF +1750 YH + 2250 YJ 75,000 - 40,000 = 35,000≤

The rest of other crashing model constraints remain the same.

WINQSB Crashing Analysis with a Budget of $75000

Project completion time Overall crashing costNormal time is 13 weeks Normal time is 17 weeks

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Administrative Costs of $100 per week.The campaign must be completed within 26weeks, but there are weekly operating expensesof $100.The Objective Function becomesMinimize

1500 YA+ 2000 YB + 2750 YC + 3750 YD + 500 YE + 2400 YFA B C D E F +1750 YH + 2250 YJ + 100X(FIN)

The other crashing model constraints remain the same.