multi criteria decision making
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Multi-Criteria Decision Making
MCDM Approaches
Membuat Keputusan Menggunakan Pelbagai Kriteria
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PENGENALAN
Zeleny (1982) dalam bukunya “Multiple Criteria Decision Making” mengatakan:
“Telah menjadi lebih susah untuk melihat dunia di sekeliling kita secara uni-dimensi dan menggunakan satu kriteria penilaian sahaja”
Sebenarnya kita selalu berdepan dengan keadaan untuk membuat keputusan berdasarkan banyak kriteria.
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Banyak masalah samada dipihak kerajaan, swasta atau individu akan melibatkatkan pelbagai objektif atau kriteria.
Contoh: bagaimana hendak mencari kawasan untuk loji nuklear, objektif terlibat mungkin merangkumi:
• Keselamatan (Safety)• Kesihatan (Health)• Alam Sekitar (Environment)• Kos (Cost)
PENGENALAN
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Contoh Masalah-masalah Pelbagai Kriteria
Dalam kajian kes R&D, (Moore et. al 1976), telah mengenalpasti objektif berikut:
• Keuntungan.• Pertumbuhan & kepelbagaian produk.• Peningkatan kadar milikan dalam pasaran. • Mempertahankan keupayaan teknikal.• Reputasi & Imej.• Penyelidikan yang menjangka persaiagan.
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Memilih calon isteri/ suami. Kriteria termasuklah:• Ugama (30%)• Cantik/ Tampan (20%)• Kekayaan (10%)• Keturunan / Family status (10%)• Pendidikan (20%)• Maskahwin/ Hantaran (10%)
% - Wajaran (weightages)
Contoh Masalah-masalah Pelbagai Kriteria
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Terdapat konflik dalam kriteria tersebut – semua kriteria kecuali Maskawin menggambarkan prinsip semakin tinggi nilai semakin baik.
Persoalannya bagaimana kita hendak ubahsuai @ “normalize” kriteria supaya menjadi sama dengan kriteria lain?
Supaya semua penilaian kita menjadi konsisten dan angka akhir akan memberikan satu skor yang bermakna.
Contoh Masalah-masalah Pelbagai Kriteria
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Dalam penilaian projek pun melibatkan proses yang sama:
1. “Problem Tree”2. “Objective Tree”3. Strategi4. Kenalpasti projek/ program5. Penilaian setiap projek/program6. Membuat keputusan - MCDM
Contoh Masalah-masalah Pelbagai Kriteria
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Contoh Matrik Keputusan Pelbagai Kriteria
Contoh MCDM
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Approaches For MCDM Several approaches for MCDM exist. We
will cover the following:
• Weighted score method.• TOPSIS method• Analytic Hierarchy Process (AHP) • Goal programming ?
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Weighted score method
Determine the criteria for the problem Determine the weight for each criteria.
The weight can be obtained via survey, AHP, etc.
Obtain the score of option i using each criteria j for all i and j
Compute the sum of the weighted score for each option .
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Weighted score method
In order for the sum to make sense all criteria scale must be consistent, i.e.,
More is better or less is better for all criteria
Example: In the wife selection problem, all criteria
(Religion, Beauty, Wealth, Family status, Family relationship, Education) more is better
If we consider other criteria (age, dowry) less is better
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Weighted score method
Let Sij score of option i using criterion j wj weight for criterion j Si score of option i is given as:
Si = wj Sij
j
The option with the best score is selected.
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Weighted Score Method The method can be modified by using U(Sij)
and then calculating the weighted utility score.
To use utility the condition of separability must hold.
Explain the meaning of separability:U(Si) = wj U(Sij)U(Si) U( wj Sij)
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Example Using Weighted Scoring Method
Objective• Selecting a car
Criteria• Style, Reliability, Fuel-economy
Alternatives• Civic Coupe, Saturn Coupe, Ford Escort,
Mazda Miata
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Weights and Scores Weight 0.3 0.4 0.3 Si
Style Reliability Fuel Eco.
Saturn
Ford
7 9 9
8 7 8
9 6 8
Civic
Mazda
6 7 8
8.4
7.6
7.5
7.0
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TOPSIS METHOD Technique of Order Preference by
Similarity to Ideal Solution This method considers three types of
attributes or criteria
• Qualitative benefit attributes/criteria• Quantitative benefit attributes• Cost attributes or criteria
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TOPSIS METHOD In this method two artificial alternatives are
hypothesized:
Ideal alternative: the one which has the best level for all attributes considered.
Negative ideal alternative: the one which has the worst attribute values.
TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative.
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Input to TOPSIS TOPSIS assumes that we have m alternatives
(options) and n attributes/criteria and we have the score of each option with respect to each criterion.
Let xij score of option i with respect to criterion j
We have a matrix X = (xij) mn matrix. Let J be the set of benefit attributes or criteria
(more is better) Let J' be the set of negative attributes or criteria
(less is better)
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Steps of TOPSIS Step 1: Construct normalized decision
matrix. This step transforms various attribute
dimensions into non-dimensional attributes, which allows comparisons across criteria.
Normalize scores or data as follows:
rij = xij/ (x2ij) for i = 1, …, m; j = 1, …, ni
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Steps of TOPSIS Step 2: Construct the weighted normalized
decision matrix. Assume we have a set of weights for each
criteria wj for j = 1,…n. Multiply each column of the normalized
decision matrix by its associated weight. An element of the new matrix is:
vij = wj rij
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Steps of TOPSIS Step 3: Determine the ideal and negative ideal
solutions.
Ideal solution. A* = { v1
* , …, vn
*}, where vj
* ={ max (vij) if j J ; min (vij) if j J' }
i i
Negative ideal solution.
A' = { v1' , …, vn' }, wherev' = { min (vij) if j J ; max (vij) if j J' }
i i
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Steps of TOPSIS
Step 4: Calculate the separation measures for each alternative.
The separation from the ideal alternative is: Si
* = [ (vj
*– vij)2 ] ½ i = 1, …, m j
Similarly, the separation from the negative ideal alternative is:
S'i = [ (vj' – vij)2 ] ½ i = 1, …, m j
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Steps of TOPSIS
Step 5: Calculate the relative closeness to the ideal solution Ci
*
Ci*
= S'i / (Si* +S'i ) , 0 Ci
* 1
Select the option with Ci* closest to 1.
WHY ?
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Applying TOPSIS Method to Example
Weight 0.1 0.4 0.3 0.2
Style Reliability Fuel Eco.
Saturn
Ford
7 9 9 8
8 7 8 7
9 6 8 9
Civic
Mazda
6 7 8 6
Cost
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Applying TOPSIS to Example m = 4 alternatives (car models) n = 4 attributes/criteria
xij = score of option i with respect to criterion j
X = {xij} 44 score matrix. J = set of benefit attributes: style, reliability, fuel
economy (more is better) J' = set of negative attributes: cost (less is better)
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Steps of TOPSIS
Step 1(a): calculate (x2ij )1/2 for each column
Style Rel. Fuel
Saturn
Ford
49 81 81 64
64 49 64 49
81 36 64 81
Civic
Mazda
Cost
xij2i
(x2)1/2
36 49 64 36
230 215 273 230
15.17 14.66 16.52 15.17
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Steps of TOPSIS
Step 1 (b): divide each column by (x2ij )1/2
to get rij
Style Rel. Fuel
Saturn
Ford
0.46 0.61 0.54 0.53
0.53 0.48 0.48 0.46
0.59 0.41 0.48 0.59
Civic
Mazda
0.40 0.48 0.48 0.40
Cost
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Steps of TOPSIS
Step 2 (b): multiply each column by wj to get vij.
Style Rel. Fuel
Saturn
Ford
0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
Civic
Mazda
0.040 0.192 0.144 0.080
Cost
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Steps of TOPSIS
Step 3 (a): determine ideal solution A*. A* = {0.059, 0.244, 0.162, 0.080}
Style Rel. Fuel
Saturn
Ford
0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
Civic
Mazda
0.040 0.192 0.144 0.080
Cost
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Steps of TOPSIS
Step 3 (a): find negative ideal solution A'. A' = {0.040, 0.164, 0.144, 0.118}
Style Rel. Fuel
Saturn
Ford
0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
Civic
Mazda
0.040 0.192 0.144 0.080
Cost
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Steps of TOPSIS
Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} Si
* = [ (vj
*– vij)2 ] ½ for each row j
Style Rel. Fuel
Saturn
Ford
(.046-.059)2 (.244-.244)2 (0)2 (.026)2 Civic
Mazda
Cost
(.053-.059)2 (.192-.244)2 (-.018)2 (.012)2
(.053-.059)2 (.164-.244)2 (-.018)2 (.038)2
(.053-.059)2 (.192-.244)2 (-.018)2 (.0)2
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Steps of TOPSIS
Step 4 (a): determine separation from ideal solution Si
*
(vj
*–vij)2 Si* = [ (vj
*– vij)2 ] ½
Saturn
Ford
0.000845 0.029
0.003208 0.057
0.008186 0.090
Civic
Mazda 0.003389 0.058
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Steps of TOPSIS Step 4 (b): find separation from negative ideal
solution A' = {0.040, 0.164, 0.144, 0.118} Si' = [ (vj'– vij)2 ] ½ for each row
j
Style Rel. Fuel
Saturn
Ford
(.046-.040)2 (.244-.164)2 (.018)2 (-.012)2Civic
Mazda
Cost
(.053-.040)2 (.192-.164)2 (0)2 (-.026)2
(.053-.040)2 (.164-.164)2 (0)2 (0)2
(.053-.040)2 (.192-.164)2 (0)2 (-.038)2
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Steps of TOPSIS
Step 4 (b): determine separation from negative ideal solution Si'
(vj'–vij)2 Si' = [ (vj'– vij)2 ] ½
Saturn
Ford
0.006904 0.083
0.001629 0.040
0.000361 0.019
Civic
Mazda 0.002228 0.047
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Steps of TOPSIS
Step 5: Calculate the relative closeness to the ideal solution Ci
* = S'i / (Si
* +S'i )
S'i /(Si
*+S'i) Ci*
Saturn
Ford
0.083/0.112 0.74 BEST
0.040/0.097 0.41
0.019/0.109 0.17
Civic
Mazda 0.047/0.105 0.45
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