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

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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

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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 Kelangsungan

Proyek 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 PWaktu j Tepat Biaya Tanda Panah Ad l h

CPM

Dan Mengendalikan

Aktivitas

Dikerjakan Oleh Evaluator Pengerjaan Waktu Yang Paling Tepat Dan Layak 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

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Jalur Kritis / Critical Path:

Suatu aktivitas sekuensial yang menuju pada penyelesaian project.

Slack:

Jumlah fleksibitas dalam menjadwalan aktivitas yang j y g tidak kritis.

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

E Immediate Estimated Immediate Estimated A 90 H 28 E 21 D 20 B 15 G 14 F 25 C

5 Activity Predecessor Completion Time

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

Activity Predecessor Completion Time

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

None

A

B

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

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 cepat Turnamen dapat Disesaikan? Aktivitas manakah Yang kritis? 11 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 A D 2 B A,2 C,3 E,1

Activities are represented by nodes:

D 2 B

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Forward Pass

Forward Pass:

Calculate Earliest Start Times, Earliest Finish Times

Backward Pass:

Calculate Latest Start Times, Latest Finish Times

Slack

Latest Start Time – Earliest Start Time

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Activit Expecte Immediate Earliest Earliest Latest Latest Slack y d

Duration (weeks)

Predecess

ors 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 D 2 B E 1 C,D

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Activit y Expecte d 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 15 0 1 2 3 4 5 6

Activity A Activity C Act. E

Act. E Activity D Activity B Slack Time p

Activity Duration _(weeks) _PredImm ES EF LF LS SLACK

A 5 B 4 C 3 D 2 A E 6 B, C F 3 D E F 3 D, E G 7 E H 5 F I 4 F J 2 G

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Acti vity Description Imm Pred Dur ES EF LS LF SLCK A S l 3 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 G Complete C 8 G Complete arrangements C 8 H Develop schedules G 3 I Practice D, H 2 J Conduct tournament F, I 3 17

Decision Variables

Decision Variables:

Objective Function:

A,2

B 2 C,3 D 2 E,1

Constraints:

B,2 D,2

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The

terminal activity

The

terminal activity

The single activity that identifies when the project is completed.

If there is no natural terminal activity, add a dummy node with 0 duration: 19 A, 1 B, 3 C, 1 D, 4 E, 2

Aktivitas digambarkan melelui tanda panah

Find the maximum cost flow

A,2 B,2 C,3 D,2 E,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

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Often there are penalties and bonuses for late

or early completion of a project.

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Sebuah kontrak untuk menyelesaikan pekerjaan Sebuah kontrak untuk menyelesaikan pekerjaan

dalam waktu 16 minggu.

Terdapat bonus sebesar $12,000 untuk setiap

pekerjaan yang lebih awal per minggunya dari jadwal.

Penalti sebesar $15,000 per minggu keterlambatan. Kapankah waktu ideal dalam menyelesaikan proyekp y p y

tsb?

Aktivitas manakah yang harus diakselerasidan seberapa banyak?

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Activityy Standard Minimum Extra Cost at Imm Maximum Incremental Duration (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 23 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

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

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There are two crucial time durations to consider

for each activity

for 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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18 _{…and save more on}completion time

Linearity assumption

16 14 12 10 8 Add to the normal cost... Add more to the

normal cost...

Crashing CC = $4400 CT = 12 days

completion time

_{Add 25% to the}

normal cost

Save 25% on

completion time

Cost ($100) 6 4 2 5 10 15 20 25 30 35 40 45 CT = 12 days

Marginal Cost

=

Additional Cost to get Max. Time Reduction_{Maximum 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.

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campaign to plan.

The campaign consists of the following activities

Immediate Normal Schedule Reduced Schedule

Activity Predecessor Time Cost Time Cost

A. Hire campain staff None 4 2.0K 2 5.0K

B. Prepare position paper None 6 3.0 3 9

C. Recruit volunteers A 4 4.5 2 10

Immediate Normal Schedule Reduced Schedule

Activity Predecessor Time Cost Time Cost

A. Hire campain staff None 4 2.0K 2 5.0K

B. Prepare position paper None 6 3.0 3 9

C. Recruit volunteers A 4 4.5 2 10

D. Raise funds A,B 6 2.5 4 10

E. File candidacy papers D 2 0.5 1 1

F. Prepare campaign material E 13 13.0 8 25

G. Locate/staff headquarters E 1 1.5 1 1.5

H. Run personal campaign C,G 20 6.0 10 23.5

I. Run media campaign F 9 7.0 5 16

D. Raise funds A,B 6 2.5 4 10

E. File candidacy papers D 2 0.5 1 1

F. Prepare campaign material E 13 13.0 8 25

G. Locate/staff headquarters E 1 1.5 1 1.5

H. Run personal campaign C,G 20 6.0 10 23.5

I. 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.

D

B E

F

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

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A c tiv ity 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 tiv ity N T N C ($ ) C T C C T M ($ ) AA 44 2 0 0 02 0 0 0 22 5 0 0 05 0 0 0 2 $ 1 5 0 02 $ 1 ,5 0 0 B 6 3 0 0 0 3 9 0 0 0 3 2 0 0 0 C 4 4 5 0 0 2 1 0 0 0 0 2 2 7 5 0 D 6 2 5 0 0 4 1 0 0 0 0 2 3 7 5 0 E 2 5 0 0 1 1 0 0 0 1 5 0 0 F 1 3 1 3 0 0 0 8 2 5 0 0 0 5 2 4 0 0 G 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 0 I 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 0 B 6 3 0 0 0 3 9 0 0 0 3 2 0 0 0 C 4 4 5 0 0 2 1 0 0 0 0 2 2 7 5 0 D 6 2 5 0 0 4 1 0 0 0 0 2 3 7 5 0 E 2 5 0 0 1 1 0 0 0 1 5 0 0 F 1 3 1 3 0 0 0 8 2 5 0 0 0 5 2 4 0 0 G 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 0 I 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 better

solved by a linear programming model.

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Linear Programming Approach

VariablesV

X_j = start time for activity j.

Y_j= the amount of crash in activity j. Objective Function

Minimize 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

≥ J H F E D C B A 2000Y 2750Y 3750Y 500Y 2400Y 17500Y 2250Y Min1500Y + + + + + + +

Minimize total crashing costs

26 ) FIN ( X ST ≤

Meet the deadline _XX₍(FIN_FIN₎) _XX ₍₂₀(9 Y_Y)₎ H I I − + ≥ − + ≥ 5 Y 1 Y 2 Y 2 Y 3 Y 2 Y E D C B A ≤ ≤ ≤ ≤ ≤ ≤ Maximum time-reduction constraints ) Y 6 ( X X ) Y 2 ( X X ) Y 2 ( X X ) Y 4 ( X X 1 X X ) Y 13 ( X X D D E E E F E E G C C H G H F F I − + ≥ − + ≥ − + ≥ − + ≥ + ≥ − + ≥ Activity can start only after all the predecessors are completed. 10 Y 5 Y H F ≤ ≤ ) Y 4 ( X X ) Y 4 ( X X ) Y 6 ( X X A A C A A D B B D − + ≥ − + ≥ − + ≥ A C F G I H FINISH

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WINQSB Crashing Optimal Solution

Crashing costs Most 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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The objective function becomes a constraint

Minimize X(FIN)

1500 Y_A+ 2000 Y_B+ 2750 Y_C+ 3750 Y_D+ 500 Y_E+ 2400 Y_F +1750 Y_H+ 2250 Y_J

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

This constraint becomes the objective function

X(FIN) 26

≤

( )

1500 Y_A+ 2000 Y_B+ 2750 Y_C+ 3750 Y_D+ 500 Y_E+ 2400 Y_F +1750 Y_H+ 2250 Y_J

≤

75,000 - 40,000 = 35,000

The rest of other crashing model constraints

remain the same.

WINQSB Crashing Analysis with a Budget of

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

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weeks, but there are weekly operating expenses

of $100.

The Objective Function becomes

Minimize

1500 Y_A_A+ 2000 Y_B_B+ 2750 Y_C_C+ 3750 Y_D_D+ 500 Y_E_E+ 2400 Y_F_F +1750 Y_H+ 2250 Y_J+ 100X(FIN)