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Experience Mazda Zoom Zoom Lifestyle and Culture by Visiting and joining the Official Mazda Community at http://www.MazdaCommunity.org for additional insight into the Zoom Zoom Lifestyle and special offers for Mazda Community Members. If you live in Arizona, check out CardinaleWay Mazda's eCommerce website at http://www.Cardinale-Way-Mazda.com
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2
Introduction
Zeleny (1982) opens his book “Multiple Criteria Decision Making” with a statement:
“It has become more and more difficult to see the world around us in a unidimensional way and to use only a single criterion when judging what we see”
3
Introduction Many public sector problems and even
private decision involve multiple objectives and goals. As an example:
Locating a nuclear power plant involves objectives such as:
• Safety• Health• Environment• Cost
4
Examples of Multi-Criteria Problems
In a case study on the management of R&D research (Moore et. al 1976), the following objectives have been identified:
• Profitability• Growth and diversity of the product line• Increased market share• Maintained technical capability• Firm reputation and image• Research that anticipates competition
5
Examples of Multi-Criteria Problems
In determining an electric route for power transmission in a city, several objectives could be considered:
• Cost• Health• Reliability• Importance of areas
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Examples of Multi-Criteria Problems
In selecting a major at KFUPM, several objectives can be considered. These objectives or criteria include:
• Job market upon graduation• Job pay and opportunity to progress• Interest in the major• Likelihood of success in the major• Future job image• Parent wish
7
Examples of Multi-Criteria Problems
Wife selection problem. This problem is a good example of multi-criteria decision problem. Criteria include:• Religion• Beauty• Wealth• Family status• Family relationship• Education
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Approaches For MCDM Several approaches for MCDM exist. We
will cover the following:
• Weighted score method ( Section 5.1 in text book).
• 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, …, n
i
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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
22
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 ?
23
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
24
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)
25
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
26
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
27
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
28
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
29
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
30
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
31
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
32
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
33
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