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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Weighted Score And Topsis Presentation 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