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Weighted Score And Topsis

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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 …

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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  • Transcript

    • 1. Multi-Criteria Decision Making MCDM Approaches
    • 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
    • 6. 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
    • 8. 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 ?
    • 9. 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 .
    • 10. 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
    • 11. 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.
    • 12. 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 )
    • 13. Example Using Weighted Scoring Method
      • Objective
        • Selecting a car
      • Criteria
        • Style, Reliability, Fuel-economy
      • Alternatives
        • Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
    • 14. 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
    • 15. 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
    • 16. 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.
    • 17. 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)
    • 18. 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
    • 19. 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
    • 20. 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
    • 21. 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 ] ½ i = 1, …, m
      • j
      • Similarly, the separation from the negative ideal alternative is:
      • S ' i = [  ( v j ' – v ij ) 2 ] ½ i = 1, …, m
      • j
    • 22. 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 ?
    • 23. 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
    • 24. 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)
    • 25. 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
    • 26. 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
    • 27. 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
    • 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} S i * = [  (v j * – v ij ) 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 S i *
       (v j * –v ij ) 2 S i * = [  (v j * – v ij ) 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}
      • S i ' = [  (v j ' – v ij ) 2 ] ½ for each row j
      Style Rel. Fuel Saturn Ford (.046 -.040 ) 2 (.244 -.164 ) 2 (.018) 2 (-.012) 2 Civic 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 S i '
       (v j ' –v ij ) 2 S i ' = [  (v j ' – v ij ) 2 ] ½ Saturn Ford 0.006904 0.083 0.001629 0.040 0.000361 0.019 Civic Mazda 0.002228 0.047
    • 34. Steps of TOPSIS
      • Step 5: Calculate the relative closeness to the ideal solution C i * = S ' i / (S i * +S ' i )
      S ' i /(S i * +S ' i ) C i * 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