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Multi-Criteria Decision Making

Multi-Criteria Decision Making

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Multi-Criteria Decision Making

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  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 Sij score of option i using criterion j • wj weight for criterion j • Si score of option i is given as: Si =  wjSij j The option with the best score is selected.

  12. Weighted Score Method • The method can be modified by using U(Sij) and then calculating the weighted utility score. • To use utility the condition of separability must hold. • Explain the meaning of separability: U(Si) =  wj U(Sij) U(Si)  U( wjSij)

  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. Style Reliability Fuel Eco. 7 9 9 8 7 8 Saturn 9 6 8 Ford Weights and Scores Weight 0.3 0.4 0.3 Si 8.4 7.6 7.5 7.0 Civic 678 Mazda

  15. TOPSIS METHOD • Technique of Order Preference by Similarity to Ideal Solution • 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 xijscore of option i with respect to criterion j We have a matrix X = (xij) 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: rij = xij/ (x2ij) 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 wj for j = 1,…n. • Multiply each column of the normalized decision matrix by its associated weight. • An element of the new matrix is: vij = wj rij

  20. Steps of TOPSIS • Step 3: Determine the ideal and negative ideal solutions. • Ideal solution. A* = { v1*, …, vn*}, where vj*={ max (vij) if j  J ; min (vij) if j  J' } i i • Negative ideal solution. A' = { v1', …,vn' },where v' = {min (vij) if j  J ; max (vij) if j  J' } ii

  21. Steps of TOPSIS • Step 4: Calculate the separation measures for each alternative. • The separation from the ideal alternative is: Si *= [  (vj*– vij)2 ] ½ i = 1, …, m j • Similarly, the separation from the negative ideal alternative is: S'i = [  (vj' – vij)2 ] ½ i = 1, …, m j

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

  23. Style Reliability Fuel Eco. 7 9 9 8 8 7 8 7 Saturn 9 6 8 9 Ford Applying TOPSIS Method to Example Weight 0.1 0.4 0.3 0.2 Cost Civic 678 6 Mazda

  24. Applying TOPSIS to Example • m = 4 alternatives (car models) • n = 4 attributes/criteria • xij = score of option i with respect to criterion j X= {xij} 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 (x2ij )1/2 for each column Style Rel. Fuel Cost Civic 49 81 81 64 64 49 64 49 Saturn 81 36 64 81 Ford Mazda 36 49 64 36 xij2i 230 215 273 230 (x2)1/2 15.17 14.66 16.52 15.17

  26. Steps of TOPSIS • Step 1 (b): divide each column by (x2ij )1/2 to get rij Style Rel. Fuel Cost Civic 0.46 0.61 0.54 0.53 Saturn 0.53 0.48 0.48 0.46 Ford 0.59 0.41 0.48 0.59 0.40 0.480.48 0.40 Mazda

  27. Steps of TOPSIS • Step 2 (b): multiply each column by wj to get vij. Style Rel. Fuel Cost Civic 0.046 0.244 0.162 0.106 Saturn 0.053 0.192 0.144 0.092 Ford 0.059 0.164 0.144 0.118 0.040 0.1920.144 0.080 Mazda

  28. Steps of TOPSIS • Step 3 (a): determine ideal solution A*. A* = {0.059, 0.244, 0.162, 0.080} Style Rel. Fuel Cost Civic 0.046 0.2440.162 0.106 0.053 0.192 0.144 0.092 Saturn 0.059 0.164 0.144 0.118 Ford 0.040 0.1920.144 0.080 Mazda

  29. Steps of TOPSIS • Step 3 (a): find negative ideal solution A'. A' = {0.040, 0.164, 0.144, 0.118} Style Rel. Fuel Cost Civic 0.046 0.244 0.162 0.106 0.053 0.192 0.144 0.092 Saturn 0.059 0.1640.1440.118 Ford 0.040 0.1920.144 0.080 Mazda

  30. Steps of TOPSIS • Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080}Si*= [  (vj*– vij)2 ] ½for each row j Style Rel. Fuel Cost Civic (.046-.059)2 (.244-.244)2(0)2(.026)2 Saturn (.053-.059)2 (.192-.244)2(-.018)2(.012)2 Ford (.053-.059)2 (.164-.244)2(-.018)2 (.038)2 Mazda (.053-.059)2 (.192-.244)2(-.018)2 (.0)2

  31. Steps of TOPSIS • Step 4 (a): determine separation from ideal solution Si* (vj*–vij)2 Si* = [  (vj*– vij)2 ] ½ Civic 0.000845 0.029 Saturn 0.003208 0.057 Ford 0.008186 0.090 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} Si'= [  (vj'– vij)2 ] ½for each row j Style Rel. Fuel Cost Civic (.046-.040)2 (.244-.164)2(.018)2(-.012)2 Saturn (.053-.040)2 (.192-.164)2(0)2(-.026)2 Ford (.053-.040)2 (.164-.164)2(0)2 (0)2 Mazda (.053-.040)2 (.192-.164)2(0)2 (-.038)2

  33. Steps of TOPSIS • Step 4 (b): determine separation from negative ideal solution Si' Si' = [  (vj'– vij)2 ] ½ (vj'–vij)2 Civic 0.006904 0.083 Saturn 0.001629 0.040 Ford 0.000361 0.019 Mazda 0.002228 0.047

  34. Steps of TOPSIS • Step 5: Calculate the relative closeness to the ideal solution Ci*= S'i / (Si* +S'i ) S'i /(Si*+S'i) Ci* Civic 0.083/0.112 0.74  BEST Saturn 0.040/0.097 0.41 Ford 0.019/0.109 0.17 Mazda 0.047/0.105 0.45