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Transcript of 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:
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- Safety
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- Health
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- Environment
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- 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:
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- Profitability
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- Growth and diversity of the product line
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- Increased market share
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- Maintained technical capability
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- Firm reputation and image
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- Research that anticipates competition
- Examples of Multi-Criteria Problems
- In determining an electric route for power transmission in a city, several objectives could be considered:
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- Cost
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- Health
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- Reliability
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- Importance of areas
- Examples of Multi-Criteria Problems
- In selecting a major at KFUPM, several objectives can be considered. These objectives or criteria include:
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- Job market upon graduation
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- Job pay and opportunity to progress
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- Interest in the major
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- Likelihood of success in the major
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- Future job image
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- Parent wish
- Examples of Multi-Criteria Problems
- Wife selection problem . This problem is a good example of multi-criteria decision problem. Criteria include:
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- Religion
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- Beauty
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- Wealth
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- Family status
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- Family relationship
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- Education
- Approaches For MCDM
- Several approaches for MCDM exist. We will cover the following:
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- Weighted score method ( Section 5.1 in text book).
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- TOPSIS method
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- Analytic Hierarchy Process (AHP)
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- 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 .
- 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
- Weighted score method
- 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
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- Selecting a car
- Criteria
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- Style, Reliability, Fuel-economy
- Alternatives
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- Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
- Weights and Scores
- Weight 0.3 0.4 0.3 S i
- 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
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- Qualitative benefit attributes/criteria
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- Quantitative benefit attributes
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- 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
- 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
- Steps of TOPSIS
- Step 1 (b): divide each column by ( x 2 ij ) 1/2 to get r ij
- Steps of TOPSIS
- Step 2 (b): multiply each column by w j to get v ij .
- Steps of TOPSIS
- Step 3 (a): determine ideal solution A*.
- A* = {0.059, 0.244, 0.162, 0.080}
- Steps of TOPSIS
- Step 3 (a): find negative ideal solution A ' .
- A ' = {0.040, 0.164, 0.144, 0.118}
- Steps of TOPSIS
- Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} S i * = [ (v j * v ij ) 2 ] for each row j
- Steps of TOPSIS
- Step 4 (a): determine separation from ideal solution S i *
- Steps of TOPSIS
- Step 4 (b): find separation from negative ideal solution A ' = {0.040, 0.164, 0.144, 0.118}
- S i ' = [ (v j ' v ij ) 2 ] for each row j
- Steps of TOPSIS
- Step 4 (b): determine separation from negative ideal solution S i '
- Steps of TOPSIS
- Step 5: Calculate the relative closeness to the ideal solution C i * = S ' i / (S i * +S ' i )