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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)
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
Figure 9.2 The first-priority goal: Minimize d1-
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Graphical Interpretation of Goal Programming (3 of 6)
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
Figure 9.3 The second-priority goal: Minimize d2-
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Graphical Interpretation of Goal Programming (4 of 6)
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
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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Graphical Interpretation of Goal Programming (5 of 6)
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
Figure 9.5
The fourth-priority goal: (minimize d1-) and the solution
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Graphical Interpretation of Goal Programming (6 of 6)
- Goal programming solutions do not always achieve all goals and they are notoptimal, they achieve the best or most satisfactory solution possible.
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
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)
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
Exhibit 9.1
Comp ter Sol tion of Goal Programming Problems
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Computer Solution of Goal Programming Problems
QM for Windows (2 of 3)
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
Exhibit 9.2
Computer Solution of Goal Programming Problems
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Computer Solution of Goal Programming Problems
QM for Windows (3 of 3)
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
Exhibit 9.3
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Computer Solution of Goal Programming Problems
Excel Spreadsheets (1 of 3)
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
Exhibit 9.4
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Computer Solution of Goal Programming Problems
Excel Spreadsheets (2 of 3)
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
Exhibit 9.5
Computer Solution of Goal Programming Problems
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Computer Solution of Goal Programming Problems
Excel Spreadsheets (3 of 3)minimize P1d1
-, P2d2-, P3d3
+, P4d1+
subject to x1 + 2x2 + d1- - d1
+ = 40
40x1 + 50 x2 + d2-
- d2+
= 1,6004x1 + 3x2 + d3
- - d3+ = 120
x1, x2, d1-, d1
+, d2-, d2
+, d3-, d3
+ 0
Exhibit 9.6
Computer Solution of Goal Programming Problems
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Computer Solution of Goal Programming Problems
Altered Problem Excel Spreadsheets (1 of 5)
Exhibit 9.7
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
Computer Solution of Goal Programming Problems
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Computer Solution of Goal Programming Problems
Altered Problem Excel Spreadsheets (2 of 5)
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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Computer Solution of Goal Programming Problems
Altered Problem Excel Spreadsheets (3 of 5)
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
Computer Solution of Goal Programming Problems
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Computer Solution of Goal Programming Problems
Altered Problem Excel Spreadsheets (4 of 5)
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 9.10
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Computer Solution of Goal Programming Problems
Altered Problem Excel Spreadsheets (5 of 5)
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 9.11
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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.
The Analytical Hierarchy Process (AHP)
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The Analytical Hierarchy Process (AHP)
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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The Analytical Hierarchy Process (AHP)
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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The Analytical Hierarchy Process (AHP)
Developing Preferences Within Criteria (1 of 2)
- In synthesization, decision alternatives are prioritized with each criterion and then normalized:
The Analytical Hierarchy Process (AHP)
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The Analytical Hierarchy Process (AHP)
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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The Analytical Hierarchy Process (AHP)
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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The Analytical Hierarchy Process (AHP)
Developing an Overall Ranking (1 of 2)
Market
Income
Infrastructure
Transportation
Table 9.3 Criteria Preference Matrix
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The Analytical Hierarchy Process (AHP)
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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The Analytical Hierarchy Process (AHP)
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.
The Analytical Hierarchy Process (AHP)
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The Analytical Hierarchy Process (AHP)
Excel Spreadsheets (1 of 4)
Exhibit 9.12
Th A l i l Hi h P (AHP)
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The Analytical Hierarchy Process (AHP)
Excel Spreadsheets (2 of 4)
Exhibit 9.13
Th A l ti l Hi h P (AHP)
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The Analytical Hierarchy Process (AHP)
Excel Spreadsheets (3 of 4)
Exhibit 9.14
Th A l ti l Hi h P (AHP)
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The Analytical Hierarchy Process (AHP)
Excel Spreadsheets (4 of 4)
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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Goal Programming Example Problem
Solution (1 of 2)
Step 1: Model Formulation
minimize P1d1-, P2d2
-, 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
Goal Programming Example Problem
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Goal Programming Example Problem
Solution (2 of 2)
Step 2: The QM for Windows Solution
Goal Programming Example Problem
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Goal Programming Example Problem
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
1/41
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
Goal Programming Example Problem
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Goal Programming Example Problem
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
Bike X Y Z Row Averages
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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Goal Programming Example Problem
AHP Ranking Problem Solution ( 2 of 4)
Step 1 continued: (Develop normalized matrices and preference vectors for all the pairwise
comparison matrices for criteria.)
Weight/Durability
Bike X Y Z Row Averages
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
Goal Programming Example Problem
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Goal Programming Example Problem
AHP Ranking Problem Solution ( 3 of 4)
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
Goal Programming Example Problem
7/30/2019 9. Pengambilan Keputusan Multi Kriteria
49/49
Goal Programming Example Problem
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