Weighted Score And Topsis

Multi-Criteria Decision Making MCDM Approaches

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

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

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

In determining an electric route for power transmission in a city, several objectives could be considered:
Cost
Health
Reliability
Importance of areas

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

Wife selection problem . This problem is a good example of multi-criteria decision problem. Criteria include:
Religion
Beauty
Wealth
Family status
Family relationship
Education

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 ?

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 .

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

Let S ij score of option i using criterion j
w j weight for criterion j
S i score of option i is given as:
S i = w j S ij
j
The option with the best score is selected.

Weighted Score Method
The method can be modified by using U(S ij ) and then calculating the weighted utility score.
To use utility the condition of separability must hold.
Explain the meaning of separability:
U(S i ) = w j U(S ij )
U(S i ) U( w j S ij )

Example Using Weighted Scoring Method
Objective
Selecting a car
Criteria
Style, Reliability, Fuel-economy
Alternatives
Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

Weights and Scores
Weight 0.3 0.4 0.3 S i
Civic Mazda 6 7 8 8.4 7.6 7.5 7.0 Style Reliability Fuel Eco. Saturn Ford 7 9 9 8 7 8 9 6 8

TOPSIS METHOD
T echnique of O rder P reference by S imilarity to I deal S olution
This method considers three types of attributes or criteria
Qualitative benefit attributes/criteria
Quantitative benefit attributes
Cost attributes or criteria

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.

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 x ij score of option i with respect to criterion j
We have a matrix X = (x ij ) m n 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)

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:
r ij = x ij / ( x 2 ij ) for i = 1, , m; j = 1, , n
i

Steps of TOPSIS
Step 2: Construct the weighted normalized decision matrix.
Assume we have a set of weights for each criteria w j for j = 1,n.
Multiply each column of the normalized decision matrix by its associated weight.
An element of the new matrix is:
v ij = w j r ij

Steps of TOPSIS
Step 3: Determine the ideal and negative ideal solutions.
Ideal solution.
A* = { v 1 * , , v n * }, where
v j * ={ max (v ij ) if j J ; min (v ij ) if j J ' }
i i
Negative ideal solution.
A' = { v 1 ' , , v n ' }, where
v' = { min (v ij ) if j J ; max (v ij ) if j J ' }
i i

Steps of TOPSIS
Step 4: Calculate the separation measures for each alternative.
The separation from the ideal alternative is:
S i * = [ (v j * v ij ) 2 ] i = 1, , m
j
Similarly, the separation from the negative ideal alternative is:
S ' i = [ ( v j ' v ij ) 2 ] i = 1, , m
j

Steps of TOPSIS
Step 5: Calculate the relative closeness to the ideal solution C i *
C i * = S ' i / (S i * +S ' i ) , 0 C i * 1
Select the option with C i * closest to 1.
WHY ?

Applying TOPSIS Method to Example
Weight 0.1 0.4 0.3 0.2
Civic Mazda 6 7 8 6 Cost Style Reliability Fuel Eco. Saturn Ford 7 9 9 8 8 7 8 7 9 6 8 9

Applying TOPSIS to Example
m = 4 alternatives (car models)
n = 4 attributes/criteria
x ij = score of option i with respect to criterion j
X = {x ij } 4 4 score matrix.
J = set of benefit attributes: style, reliability, fuel economy (more is better)
J ' = set of negative attributes: cost (less is better)

Steps of TOPSIS
Step 1(a): calculate ( x 2 ij ) 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 x ij 2 i ( x 2 ) 1/2 36 49 64 36 230 215 273 230 15.17 14.66 16.52 15.17

Steps of TOPSIS
Step 1 (b): divide each column by ( x 2 ij ) 1/2 to get r ij
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

Steps of TOPSIS
Step 2 (b): multiply each column by w j to get v ij .
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

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

Weighted Score And Topsis

negative ideal

weighted score

negative ideal

fuel saturn

ideal solution

saturn ford

negative attributes

ideal alternative

Text of Weighted Score And Topsis

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