9. pengambilan keputusan multi kriteria

7/30/2019 9. Pengambilan Keputusan Multi Kriteria

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PENGAMBILAN

KEPUTUSAN DENGAN

MULTI KRITERIA

OLEH:

DR. SITI AISJAH, SE., MS

PROGRAM PASCASARJANA MAGISTER MANAJEMEN

FAKULTAS EKONOMI UNIVERSITAS BRAWIJAYA


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POKOK BAHASAN

1. Program tujuan

2. Interpretasi Grafik dari Program Tujuan

3. Solusi Komputer untuk Masalah Program

tujuan dengan QM for Windows andExcel.

4. The Analytical Hierarchy Process (AHP)


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Overview

Pengambilan keputusan dengan beberapa kriteria,multiple criteria, untuk satu tujuan.

Tiga tehnik untuk memecahkan masalah: program tujuan(goal programming), the analytical Hierarchy process

(AHP) dan model penghitungan nilai (scoring). Program tujuan hampir sama dengan model program

linear dengan suatu fungsi tujuan, variabel-variabelkeputusan dan batasan-batasan.

Proses analisis bertingkat (analytical Hierarchy process-AHP) metode untuk membuat urutan alternatif keputusandan memilih yang terbaik pada saat pengambil keputusanmemiliki beberapa tujuan atau kriteria, untuk mengambilkeputusan tertentu.


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Overview

Model penghitungan skor (scorng)

serupa AHP, tetapi secara matematis

lebih sederhana.


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Goal Programming

Discussion Model Formulation

Maximize Z=$40x1 + 50x2Batasan:

1x1 + 2x2 40 jam kerja4x2 + 3x2 120 pon tanah liat

x1, x2 0Dimana x1 = jumlah mangkok yang diproduksi

x2 = jumlah cangkir yang diproduksi

Beberapa tujuan berdasarkan tingkat kepentingan:1. Untuk menghindari PHK, perusahaan menggunakan waktu tidak

kurang dari 40 jam per hari;2. Perusahaan mencapai tingkat keuntungan $1,600 per hari;3. Perusahaan lebih memilih untuk tidak menyimpan tanah liat lebih

dari 120 pon per hari;4. Perusahaan berusaha untuk meminimumkan jumlah waktu lembur.


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Goal Programming

Goal Constraint Requirements

Semua batasan tujuan merupakan persamaan

yang menyertakan variabel penyimpangan d-

dan d+.

Variabel penyimpangan positif (d+) merupakan

jumlah pada tingkat tujuan telah terlampaui.

Variabel penyimpangan negatif (d-) merupakan

jumlah pada tingkat tujuan tidak tercapai


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- Labor goals constraint (1, less than 40 hours labor; 4, minimumovertime):

minimize P1d1-, P4d1+

- Add profit goal constraint (2, achieve profit of $1,600):

minimize P1d1-, P2d2-, P4d1+- Add material goal constraint (3, avoid keeping more than 120

poundsof clay on hand):

minimize P1d1-, P2d2-, P3d3+, P4d1+

- Complete goal programming model:

minimize P1d1-, P2d2-, P3d3+, P4d1+subject to

x1 + 2x2 + d1- - d1+ = 40

40x1 + 50 x2 + d2 - - d2 + = 1,600

4x1 + 3x2 + d3 - - d3 + = 120

x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Goal Programming

Discussion Model Goal Constraints and

Objective Function


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Goal Programming

Alternative Forms of Discussion Model Goal Constraints

- Changing fourth-priority goal limits overtime to 10 hours instead of minimizingovertime:

d1- + d4 - - d4+ = 10; minimize P1d1 -, P2d2 -, P3d3 +, P4d4 +

- Addition of a fifth-priority goal- important to achieve the goal formugs:x1 + d5 - = 30 bowls; x2 + d6 - = 20 mugs; minimize P1d1 -, P2d2 -, P3d3 -,P4d4 -, 4P5d5 -, 5P5d6

- Complete model with new goals for both overtime and production:minimize P1d1 -, P2d2 -, P3d3 -, P4d4 -, 4P5d5 -, 5P5d6 -subject to

x1 + 2x2 + d1- - d1+ = 40

40x1 + 50 x2 + d2 - - d2 + = 1,6004x1 + 3x2 + d3 - - d3 + = 120

d1 + + d4- - d4+ =10x1 + d5 - = 30x2 + d6- = 20

x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +, d4-, d4 +, d5 -, d6-, 0


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Graphical Interpretation of Goal

Programming (1 of 6)

minimize P1d1-, P2d2

-, P3d3+, P4d1

+

subject to

x1 + 2x2 + d1-

- d1+

= 4040x1 + 50 x2 + d2

- - d2+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Persamaan:

Gambar 9.1. Batasan Tujuan


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Graphical Interpretation of Goal Programming (2 of 6)


-, P3d3+, P4d1

+

subject to x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2- - d2

+ =

1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Figure 9.2 The first-priority goal: Minimize d1-


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-, P3d3+, P4d1

+


+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Figure 9.3 The second-priority goal: Minimize d2-


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-, P3d3+, P4d1

+


+ = 40

40x1 + 50 x2 + d2 - - d2 + = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Figure 9.4 The third-priority goal: Minimize d3-

G hi l I i f G l P i ( f 6)


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-, P3d3+, P4d1

+


+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Figure 9.5

The fourth-priority goal: (minimize d1-) and the solution


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- Goal programming solutions do not always achieve all goals and they are notoptimal, they achieve the best or most satisfactory solution possible.


-, P3d3+, P4d1

+

subject to

x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

x1 = 15 bowls

x2 = 20 mugs

d1- = 15 hours


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Computer Solution of Goal Programming Problems

QM for Windows (1 of 3)


-, P3d3+, P4d1

+


+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.1

Comp ter Sol tion of Goal Programming Problems


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-, P3d3+, P4d1

+

subject to

x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.2



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-, P3d3+, P4d1

+

subject to

x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.3


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Excel Spreadsheets (1 of 3)


-, P3d3+, P4d1

+

subject to

x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2 - - d2 + = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.4


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-, P3d3+, P4d1

+


+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.5



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Excel Spreadsheets (3 of 3)minimize P1d1

-, P2d2-, P3d3

+, P4d1+


+ = 40

40x1 + 50 x2 + d2-

- d2+

= 1,6004x1 + 3x2 + d3

- - d3+ = 120

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+ 0

Exhibit 9.6



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Altered Problem Excel Spreadsheets (1 of 5)

Exhibit 9.7


-, P3d3-, P4d4

-, 4P5d5-, 5P5d6

-


+ = 40

40x1

+ 50 x2

+ d2

- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

d1+ + d4- - d4+ =10

x1 + d5 - = 30

x2 + d6-= 20

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+, d4-, d4+, d5 -, d6-, 0



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Minimize

P1d1-, P2d2

-, P3d3-, P4d4

-, 4P5d5-, 5P5d6

-

subject to

x1 + 2x2 + d1-

- d1+

= 4040x1 + 50 x2 + d2

- - d2+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

d1+ + d4- - d4+ =10

x1 + d5 - = 30

x2 + d6-= 20

x1, x2, d1-

, d1+

, d2-

, d2+

, d3-

, d3+,

d4-, d4+

, d5 -, d6-, 0

Exhibit 9.8


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Minimize

P1d1-, P2d2

-, P3d3-, P4d4

-, 4P5d5-, 5P5d6

subject to

x1 + 2x2 + d1- - d1

+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

d1+ + d4- - d4+ =10

x1 + d5 - = 30

x2 + d6-= 20

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+, d4-, d4+, d5 -, d6-, 0

Exhibit 9.9



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-, P3d3-, P4d4

-, 4P5d5-, 5P5d6

-

subject to x1 + 2x2 + d1- - d1+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

d1+ + d4- - d4+ =10

x1 + d5 - = 30

x2 + d6-= 20

x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +, d4-, d4 +, d5 -, d6-, 0

Exhibit 9.10


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-, P3d3-, P4d4

-, 4P5d5-, 5P5d6

-


+ = 40

40x1 + 50 x2 + d2- - d2

+ = 1,600

4x1 + 3x2 + d3- - d3

+ = 120

d1+ + d4- - d4+ =10

x1 + d5 - = 30

x2

+ d6-= 20

x1, x2, d1-, d1

+, d2-, d2

+, d3-, d3

+, d4-, d4+, d5 -, d6-, 0

Exhibit 9.11


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The Analytical Hierarchy Process (AHP)

Overview

AHP is a method for ranking several decision alternatives and selecting the bestone when the decision maker has multiple objectives, or criteria, on which to base

the decision.

The decision maker makes a decision based on how the alternatives compare

according to several criteria.

The decision maker will select the alternative that best meets his or her decision

criteria.

AHP is a process for developing a numerical score to rank each decisionalternative based on how well the alternative meets the decision makers criteria.



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Pairwise Comparisons- In a pairwise comparison, two alternatives are compared according to a criterion and

one is preferred.

- A preference scale assigns numerical values to different levels of performance.

Table 9.1 Preference Scale for Pairwise Comparisions


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Pairwise Comparison Matrix

- A pairwise comparison matrix summarizes the pairwise comparisons for a criteria.

Income Level Infrastructure Transportation

A

B

C


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Developing Preferences Within Criteria (1 of 2)

- In synthesization, decision alternatives are prioritized with each criterion and then normalized:



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Developing Preferences Within Criteria (2 of 2)

Table 9.2 The Normalized Matrix with Row Averages

Table 9.3 Criteria Preference Matrix


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Ranking the Criteria

Pairwise comparison

matrix:

- Preference vector: Market

Income

Infrastructure

Transportation

Table 9.4 Normalized Matrix for Criteria with Row Averages


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Developing an Overall Ranking (1 of 2)

Market

Income

Infrastructure

Transportation

Table 9.3 Criteria Preference Matrix


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Developing an Overall Ranking (2 of 2)- Overall score:

Site A score = .1993(.5012) + .6535(.2819) + .0860(.1790) + .0612(.1561) = .3091

Site B score = .1993(.1185) + .6535(.0598) + .0860(.6850) + .0612(.6196) = .1595

Site C score = .1993(.3803) + .6535(.6583) + .0860(.1360) + .0612(.2243) = .5314

-AHP Ranking:


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Summary of Mathematical Steps

1. Develop a pairwise comparison matrix for each decision alternative for each criteria.

2. Synthesization

a. Sum the values of each column of the pairwise comparison matrices.

b. Divide each value in each column by the corresponding column sum.

c. Average the values in each row of the normalized matrices.

d. Combine the vectors of preferences for each criterion.

3. Develop a pairwise comparison matrix for the criteria.

4. Compute the normalized matrix.5. Develop the preference vector.

6. Compute an overall score for each decision alternative

7. Rank the decision alternatives.



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Exhibit 9.12

Th A l i l Hi h P (AHP)


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Exhibit 9.13

Th A l ti l Hi h P (AHP)


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Exhibit 9.14

Th A l ti l Hi h P (AHP)


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Exhibit 9.15


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Scoring Model

- Each decision alternative graded in terms of how well it satisfies the criterion

according to following formula:

Si = gijwj

where

wj = a weight between 0 and 1.00 assigned to criteria j; 1.00 important,

0 unimportant; sum of total weights equals one.

gij = a grade between 0 and 100 indicating how well alternative i satisfies criteria j;

100 indicates high satisfaction, 0 low satisfaction.

Si = total score for alternative i; high score is desired

S i M d l


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Scoring Model

Example Problem

- Mall selection with four alternatives and five criteria:

S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75

S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50

S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = 76.00

S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) = 77.75Mall 4 preferred because of highest score, followed by malls 3, 2, 1.

Grades for Alternative (0 to 100)

Decision Criteria

Weight

(0 to 1.00) Mall 1 Mall 2 Mall3 Mall4

School proximityMedian income

Vehicular traffic

Mall quality, size

Other shopping

0.300.25

0.25

0.10

0.10

4075

60

90

80

6080

90

100

30

9065

79

80

50

6090

85

90

70

Scoring Model


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Scoring Model

Excel Solution

Exhibit 9.16


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Goal Programming Example Problem

Problem Statement

- Public relations firm survey interviewer staffing requirements determination.- One person can conduct 80 telephone interviews or 40 personal interviews per

day.

- $50/ day for telephone interviewer; $70 for personal interviewer.

- Goals (in priority order):

a. At least 3,000 total interviews

b. Interviewer conducts only one type of interview each day. Maintain daily

budget of $2,500.

c. At least 1,000 interviews should be by telephone.

- Formulate a goal programming model to determine number of interviewers to

hire in order to satisfy the goals, and then solve the problem.


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Solution (1 of 2)

Step 1: Model Formulation


-, P3d3-

subject to

80x1 + 40x2 + d1- - d1+ = 3,000 interviews

50x1 + 70x2 + d2- - d2

+ = $2,500 budget

80x1 + d3- - d3

+ = 1,000 telephone interviews

where

x1 = number of telephone interviews

x2 = number of personal interviews



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Solution (2 of 2)

Step 2: The QM for Windows Solution



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AHP Ranking Problem Statement

- Purchasing decision, three model alternatives, three decision criteria.

- Pairwise comparison matrices:

- Prioritized decision criteria:

Gear Action

Bike X Y Z

X

YZ

1

37

1/3

14

1/7

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Price

Bike X Y Z

X

YZ

1

1/31/6

3

11/2

6

21

Weight/Durability

Bike X Y Z

X

YZ

1

1/31

3

12

1

1/21

Criteria Price Gears Weight

Price

GearsWeight

1

1/31/5

3

11/2

5

21



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AHP Ranking Problem Solution ( 1 of 4)

Step 1: Develop normalized matrices and preference vectors for all the pairwise comparison

matrices for criteria.

Price

Bike X Y Z Row Averages

X

YZ

0.6667

0.22220.1111

0.6667

0.22220.1111

0.6667

0.22220.1111

0.6667

0.22220.1111

1.0000

Gear Action


X

YZ

0.0909

0.27270.6364

0.0625

0.18750.7500

0.1026

0.17950.7179

0.0853

0.21320.7014

1.0000


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Step 1 continued: (Develop normalized matrices and preference vectors for all the pairwise

comparison matrices for criteria.)

Weight/Durability


X

YZ

0.4286

0.14290.4286

0.5000

0.16670.3333

0.4000

0.20000.4000

0.4429

0.16980.3873

1.0000

Criteria

Bike Price Gears WeightX

YZ

0.6667

0.22220.1111

0.0853

0.21320.7014

0.4429

0.16980.3873



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Step 2: Rank the criteria.

Price

Gears

Weight

0.1222

0.2299

0.6479

Criteria Price Gears Weight Row Averages

Price

Gears

Weight

0.6522

0.2174

0.1304

0.6667

0.2222

0.1111

0.6250

0.2500

0.1250

0.6479

0.2299

0.1222

1.0000



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AHP Ranking Problem Solution (4 of 4)

Step 3: Develop an overall ranking.

Bike X

Bike Y

Bike Z

Bike X score = .6667(.6479) + .0853(.2299) + .4429(.1222) = .5057

Bike Y score = .2222(.6479) + .2132(.2299) + .1698(.1222) = .2138

Bike Z score = .1111(.6479) + .7014(.2299) + .3873(.1222) = .2806

Overall ranking of bikes: X first followed by Z and Y (sum of scores equal 1.0000).

1222.0

2299.0

6479.0

3837.07014.01111.0

1698.02132.02222.0

4429.00853.06667.0

9. pengambilan keputusan multi kriteria

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