Multi-Criteria Decision Making MCDM Approaches
Introduction <ul><li>  Zeleny (1982) opens his book “Multiple Criteria Decision Making” with a statement: </li></ul><ul><l...
Introduction <ul><li>Many public sector problems and even private decision involve multiple objectives and goals. As an ex...
Examples of Multi-Criteria Problems <ul><li>In a case study on the management of R&D research (Moore et. al 1976) , the fo...
Examples of Multi-Criteria Problems <ul><li>In determining an electric route  for power transmission in a city, several ob...
Examples of Multi-Criteria Problems <ul><li>In selecting a major at KFUPM, several objectives can be considered. These obj...
Examples of Multi-Criteria Problems <ul><li>Wife selection problem . This problem is a good example of multi-criteria deci...
Approaches For MCDM <ul><li>Several approaches for MCDM exist. We will cover the following: </li></ul><ul><ul><li>Weighted...
Weighted score method <ul><li>Determine the  criteria  for the problem </li></ul><ul><li>Determine the  weight  for each c...
Weighted score method <ul><li>In order for the sum to make sense all criteria scale must be consistent, i.e.,  </li></ul><...
Weighted score method <ul><li>Let S ij  score of option i using criterion j </li></ul><ul><li>w j  weight for criterion j ...
Weighted Score Method <ul><li>The method can be modified by using U(S ij ) and then calculating the weighted utility score...
Example Using Weighted Scoring Method <ul><li>Objective </li></ul><ul><ul><li>Selecting a car </li></ul></ul><ul><li>Crite...
Weights and Scores <ul><li>Weight  0.3  0.4  0.3  S i </li></ul>Civic Mazda   6 7   8 8.4 7.6 7.5 7.0 Style Reliability Fu...
TOPSIS METHOD <ul><li>T echnique of  O rder  P reference by  S imilarity to  I deal  S olution </li></ul><ul><li>This meth...
TOPSIS METHOD <ul><li>In this method two artificial alternatives are hypothesized : </li></ul><ul><li>Ideal alternative : ...
Input to TOPSIS <ul><li>TOPSIS assumes that we have  m  alternatives (options) and  n   attributes/criteria and we have th...
Steps of TOPSIS <ul><li>Step 1:  Construct normalized  decision matrix.  </li></ul><ul><li>This step transforms various at...
Steps of TOPSIS <ul><li>Step 2:  Construct the weighted normalized decision matrix.  </li></ul><ul><li>Assume we have a se...
Steps of TOPSIS <ul><li>Step 3:  Determine the ideal and negative ideal solutions. </li></ul><ul><li>Ideal solution. </li>...
Steps of TOPSIS <ul><li>Step 4:   Calculate the separation measures for each alternative.  </li></ul><ul><li>The separatio...
Steps of TOPSIS <ul><li>Step 5:  Calculate the relative closeness to the ideal solution C i * </li></ul><ul><li>C i *   = ...
Applying TOPSIS Method to Example <ul><li>Weight  0.1  0.4  0.3  0.2 </li></ul>Civic Mazda   6 7   8 6 Cost Style Reliabil...
Applying TOPSIS to Example <ul><li>m = 4  alternatives (car models)  </li></ul><ul><li>n = 4   attributes/criteria </li></...
Steps of TOPSIS <ul><li>Step 1(a):  calculate (  x 2 ij  ) 1/2  for each column   </li></ul>Style Rel. Fuel Saturn Ford 4...
Steps of TOPSIS <ul><li>Step 1 (b):  divide each column by (  x 2 ij  ) 1/2  to get  r ij </li></ul>Style Rel. Fuel Satur...
Steps of TOPSIS <ul><li>Step 2 (b):  multiply each column by w j  to get  v ij .  </li></ul>Style Rel. Fuel Saturn Ford 0....
Steps of TOPSIS <ul><li>Step 3 (a):  determine ideal solution A*.  </li></ul><ul><li>A* = {0.059, 0.244, 0.162, 0.080} </l...
Steps of TOPSIS <ul><li>Step 3 (a):  find negative ideal solution A ' .  </li></ul><ul><li>A '  = {0.040, 0.164, 0.144, 0....
Steps of TOPSIS <ul><li>Step 4 (a):  determine separation from ideal solution  A* = {0.059, 0.244, 0.162, 0.080}   S i *  ...
Steps of TOPSIS <ul><li>Step 4 (a):  determine separation from ideal solution  S i * </li></ul> (v j * –v ij ) 2 S i *  =...
Steps of TOPSIS <ul><li>Step 4 (b):  find separation from negative ideal solution A '  = {0.040, 0.164, 0.144, 0.118}   </...
Steps of TOPSIS <ul><li>Step 4 (b):  determine separation from negative ideal solution  S i ' </li></ul> (v j ' –v ij ) 2...
Steps of TOPSIS <ul><li>Step 5:  Calculate the relative closeness to the ideal solution  C i *   = S ' i  / (S i *  +S ' i...
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Weighted Score And Topsis

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  • Weighted Score And Topsis

    1. 1. Multi-Criteria Decision Making MCDM Approaches
    2. 2. Introduction <ul><li> Zeleny (1982) opens his book “Multiple Criteria Decision Making” with a statement: </li></ul><ul><li>“ 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” </li></ul>
    3. 3. Introduction <ul><li>Many public sector problems and even private decision involve multiple objectives and goals. As an example: </li></ul><ul><li>Locating a nuclear power plant involves objectives such as: </li></ul><ul><ul><li>Safety </li></ul></ul><ul><ul><li>Health </li></ul></ul><ul><ul><li>Environment </li></ul></ul><ul><ul><li>Cost </li></ul></ul>
    4. 4. Examples of Multi-Criteria Problems <ul><li>In a case study on the management of R&D research (Moore et. al 1976) , the following objectives have been identified: </li></ul><ul><ul><li>Profitability </li></ul></ul><ul><ul><li>Growth and diversity of the product line </li></ul></ul><ul><ul><li>Increased market share </li></ul></ul><ul><ul><li>Maintained technical capability </li></ul></ul><ul><ul><li>Firm reputation and image </li></ul></ul><ul><ul><li>Research that anticipates competition </li></ul></ul>
    5. 5. Examples of Multi-Criteria Problems <ul><li>In determining an electric route for power transmission in a city, several objectives could be considered: </li></ul><ul><ul><li>Cost </li></ul></ul><ul><ul><li>Health </li></ul></ul><ul><ul><li>Reliability </li></ul></ul><ul><ul><li>Importance of areas </li></ul></ul>
    6. 6. Examples of Multi-Criteria Problems <ul><li>In selecting a major at KFUPM, several objectives can be considered. These objectives or criteria include: </li></ul><ul><ul><li>Job market upon graduation </li></ul></ul><ul><ul><li>Job pay and opportunity to progress </li></ul></ul><ul><ul><li>Interest in the major </li></ul></ul><ul><ul><li>Likelihood of success in the major </li></ul></ul><ul><ul><li>Future job image </li></ul></ul><ul><ul><li>Parent wish </li></ul></ul>
    7. 7. Examples of Multi-Criteria Problems <ul><li>Wife selection problem . This problem is a good example of multi-criteria decision problem. Criteria include: </li></ul><ul><ul><li>Religion </li></ul></ul><ul><ul><li>Beauty </li></ul></ul><ul><ul><li>Wealth </li></ul></ul><ul><ul><li>Family status </li></ul></ul><ul><ul><li>Family relationship </li></ul></ul><ul><ul><li>Education </li></ul></ul>
    8. 8. Approaches For MCDM <ul><li>Several approaches for MCDM exist. We will cover the following: </li></ul><ul><ul><li>Weighted score method ( Section 5.1 in text book). </li></ul></ul><ul><ul><li>TOPSIS method </li></ul></ul><ul><ul><li>Analytic Hierarchy Process (AHP) </li></ul></ul><ul><ul><li>Goal programming ? </li></ul></ul>
    9. 9. Weighted score method <ul><li>Determine the criteria for the problem </li></ul><ul><li>Determine the weight for each criteria. The weight can be obtained via survey, AHP, etc. </li></ul><ul><li>Obtain the score of option i using each criteria j for all i and j </li></ul><ul><li>Compute the sum of the weighted score for each option . </li></ul>
    10. 10. Weighted score method <ul><li>In order for the sum to make sense all criteria scale must be consistent, i.e., </li></ul><ul><li>More is better or less is better for all criteria </li></ul><ul><li>Example: </li></ul><ul><li>In the wife selection problem , all criteria (Religion, Beauty, Wealth, Family status, Family relationship, Education) more is better </li></ul><ul><li>If we consider other criteria (age, dowry) less is better </li></ul>
    11. 11. Weighted score method <ul><li>Let S ij score of option i using criterion j </li></ul><ul><li>w j weight for criterion j </li></ul><ul><li>S i score of option i is given as: </li></ul><ul><li>S i =  w j S ij </li></ul><ul><li>j </li></ul><ul><li> The option with the best score is selected. </li></ul>
    12. 12. Weighted Score Method <ul><li>The method can be modified by using U(S ij ) and then calculating the weighted utility score. </li></ul><ul><li>To use utility the condition of separability must hold. </li></ul><ul><li>Explain the meaning of separability: </li></ul><ul><li>U(S i ) =  w j U(S ij ) </li></ul><ul><li>U(S i )  U(  w j S ij ) </li></ul>
    13. 13. Example Using Weighted Scoring Method <ul><li>Objective </li></ul><ul><ul><li>Selecting a car </li></ul></ul><ul><li>Criteria </li></ul><ul><ul><li>Style, Reliability, Fuel-economy </li></ul></ul><ul><li>Alternatives </li></ul><ul><ul><li>Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata </li></ul></ul>
    14. 14. Weights and Scores <ul><li>Weight 0.3 0.4 0.3 S i </li></ul>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. 15. TOPSIS METHOD <ul><li>T echnique of O rder P reference by S imilarity to I deal S olution </li></ul><ul><li>This method considers three types of attributes or criteria </li></ul><ul><ul><li>Qualitative benefit attributes/criteria </li></ul></ul><ul><ul><li>Quantitative benefit attributes </li></ul></ul><ul><ul><li>Cost attributes or criteria </li></ul></ul>
    16. 16. TOPSIS METHOD <ul><li>In this method two artificial alternatives are hypothesized : </li></ul><ul><li>Ideal alternative : the one which has the best level for all attributes considered. </li></ul><ul><li>Negative ideal alternative : the one which has the worst attribute values. </li></ul><ul><li>TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative. </li></ul>
    17. 17. Input to TOPSIS <ul><li>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. </li></ul><ul><li>Let x ij score of option i with respect to criterion j </li></ul><ul><li>We have a matrix X = (x ij ) m  n matrix. </li></ul><ul><li>Let J be the set of benefit attributes or criteria (more is better) </li></ul><ul><li>Let J ' be the set of negative attributes or criteria (less is better) </li></ul>
    18. 18. Steps of TOPSIS <ul><li>Step 1: Construct normalized decision matrix. </li></ul><ul><li>This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria. </li></ul><ul><li>Normalize scores or data as follows: </li></ul><ul><li>r ij = x ij / (  x 2 ij ) for i = 1, …, m; j = 1, …, n </li></ul><ul><li>i </li></ul>
    19. 19. Steps of TOPSIS <ul><li>Step 2: Construct the weighted normalized decision matrix. </li></ul><ul><li>Assume we have a set of weights for each criteria w j for j = 1,…n. </li></ul><ul><li>Multiply each column of the normalized decision matrix by its associated weight. </li></ul><ul><li>An element of the new matrix is: </li></ul><ul><li>v ij = w j r ij </li></ul>
    20. 20. Steps of TOPSIS <ul><li>Step 3: Determine the ideal and negative ideal solutions. </li></ul><ul><li>Ideal solution. </li></ul><ul><li>A* = { v 1 * , …, v n * }, where </li></ul><ul><li>v j * ={ max (v ij ) if j  J ; min (v ij ) if j  J ' } </li></ul><ul><li> i i </li></ul><ul><li>Negative ideal solution. </li></ul><ul><li>A' = { v 1 ' , …, v n ' }, where </li></ul><ul><li>v' = { min (v ij ) if j  J ; max (v ij ) if j  J ' } </li></ul><ul><li>i i </li></ul>
    21. 21. Steps of TOPSIS <ul><li>Step 4: Calculate the separation measures for each alternative. </li></ul><ul><li>The separation from the ideal alternative is: </li></ul><ul><li>S i * = [  (v j * – v ij ) 2 ] ½ i = 1, …, m </li></ul><ul><li>j </li></ul><ul><li>Similarly, the separation from the negative ideal alternative is: </li></ul><ul><li>S ' i = [  ( v j ' – v ij ) 2 ] ½ i = 1, …, m </li></ul><ul><li>j </li></ul>
    22. 22. Steps of TOPSIS <ul><li>Step 5: Calculate the relative closeness to the ideal solution C i * </li></ul><ul><li>C i * = S ' i / (S i * +S ' i ) , 0  C i *  1 </li></ul><ul><li>Select the option with C i * closest to 1. </li></ul><ul><li>WHY ? </li></ul>
    23. 23. Applying TOPSIS Method to Example <ul><li>Weight 0.1 0.4 0.3 0.2 </li></ul>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. 24. Applying TOPSIS to Example <ul><li>m = 4 alternatives (car models) </li></ul><ul><li>n = 4 attributes/criteria </li></ul><ul><li>x ij = score of option i with respect to criterion j </li></ul><ul><li>X = {x ij } 4  4 score matrix. </li></ul><ul><li>J = set of benefit attributes: style, reliability, fuel economy (more is better) </li></ul><ul><li>J ' = set of negative attributes: cost (less is better) </li></ul>
    25. 25. Steps of TOPSIS <ul><li>Step 1(a): calculate (  x 2 ij ) 1/2 for each column </li></ul>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. 26. Steps of TOPSIS <ul><li>Step 1 (b): divide each column by (  x 2 ij ) 1/2 to get r ij </li></ul>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. 27. Steps of TOPSIS <ul><li>Step 2 (b): multiply each column by w j to get v ij . </li></ul>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. 28. Steps of TOPSIS <ul><li>Step 3 (a): determine ideal solution A*. </li></ul><ul><li>A* = {0.059, 0.244, 0.162, 0.080} </li></ul>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. 29. Steps of TOPSIS <ul><li>Step 3 (a): find negative ideal solution A ' . </li></ul><ul><li>A ' = {0.040, 0.164, 0.144, 0.118} </li></ul>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. 30. Steps of TOPSIS <ul><li>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 </li></ul>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. 31. Steps of TOPSIS <ul><li>Step 4 (a): determine separation from ideal solution S i * </li></ul> (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. 32. Steps of TOPSIS <ul><li>Step 4 (b): find separation from negative ideal solution A ' = {0.040, 0.164, 0.144, 0.118} </li></ul><ul><li>S i ' = [  (v j ' – v ij ) 2 ] ½ for each row j </li></ul>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. 33. Steps of TOPSIS <ul><li>Step 4 (b): determine separation from negative ideal solution S i ' </li></ul> (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. 34. Steps of TOPSIS <ul><li>Step 5: Calculate the relative closeness to the ideal solution C i * = S ' i / (S i * +S ' i ) </li></ul>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
    1. ¿Le ha llamado la atención una diapositiva en particular?

      Recortar diapositivas es una manera útil de recopilar información importante para consultarla más tarde.

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