1. 1 Multi-Criteria Decision Making MCDM Approaches

2. 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. 3 Introduction w Many public sector problems and even private decision involve multiple objectives and goals. As an example: w Locating a nuclear power plant involves objectives such as: Safety Health Environment Cost

4. 4 Examples of Multi-Criteria Problems w 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. 5 Examples of Multi-Criteria Problems w In determining an electric route for power transmission in a city, several objectives could be considered: Cost Health Reliability Importance of areas

6. 6 Examples of Multi-Criteria Problems w 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. 7 Examples of Multi-Criteria Problems w 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. 8 Approaches For MCDM w 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. 9 Weighted score method w Determine the criteria for the problem w Determine the weight for each criteria. The weight can be obtained via survey, AHP, etc. w Obtain the score of option i using each criteria j for all i and j w Compute the sum of the weighted score for each option.

10. 10 Weighted score method w In order for the sum to make sense all criteria scale must be consistent, i.e., w More is better or less is better for all criteria Example: w In the wife selection problem, all criteria (Religion, Beauty, Wealth, Family status, Family relationship, Education) more is better w If we consider other criteria (age, dowry) less is better

11. 11 Weighted score method w Let S ij score of option i using criterion j w w j weight for criterion j w 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. 12 Weighted Score Method w The method can be modified by using U(S ij ) and then calculating the weighted utility score. w To use utility the condition of separability must hold. w Explain the meaning of separability: U(S i ) =  w j U(S ij ) U(S i )  U(  w j S ij )

13. 13 Example Using Weighted Scoring Method w Objective Selecting a car w Criteria Style, Reliability, Fuel-economy w Alternatives Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

14. 14 Weights and Scores Weight 0.3 0.4 0.3 S i StyleReliabilityFuel Eco. Saturn Ford 799799 878878 968968 Civic Mazda 67 867 8 8.4 7.6 7.5 7.0

15. 15 TOPSIS METHOD w Technique of Order Preference by Similarity to Ideal Solution w This method considers three types of attributes or criteria Qualitative benefit attributes/criteria Quantitative benefit attributes Cost attributes or criteria

16. 16 TOPSIS METHOD w In this method two artificial alternatives are hypothesized: w Ideal alternative: the one which has the best level for all attributes considered. w Negative ideal alternative: the one which has the worst attribute values. w TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative.

17. 17 Input to TOPSIS w 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. w Let x ij score of option i with respect to criterion j We have a matrix X = (x ij ) m  n matrix. w Let J be the set of benefit attributes or criteria (more is better) w Let J' be the set of negative attributes or criteria (less is better)

18. 18 Steps of TOPSIS w Step 1: Construct normalized decision matrix. w This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria. w Normalize scores or data as follows: r ij = x ij / (  x 2 ij ) for i = 1, …, m; j = 1, …, n i

19. 19 Steps of TOPSIS w Step 2: Construct the weighted normalized decision matrix. w Assume we have a set of weights for each criteria w j for j = 1,…n. w Multiply each column of the normalized decision matrix by its associated weight. w An element of the new matrix is: v ij = w j r ij

20. 20 Steps of TOPSIS w Step 3: Determine the ideal and negative ideal solutions. w Ideal solution. A* = { v 1 *, …, v n * }, where v j * ={ max (v ij ) if j  J ; min (v ij ) if j  J' } i i w Negative ideal solution. A' = { v 1 ', …, v n ' }, where v' = { min (v ij ) if j  J ; max (v ij ) if j  J' } i

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

22. 22 Steps of TOPSIS w 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. 23 Applying TOPSIS Method to Example Weight 0.1 0.4 0.3 0.2 StyleReliabilityFuel Eco. Saturn Ford 79987998 87878787 96899689 Civic Mazda 67 86 Cost

24. 24 Applying TOPSIS to Example w m = 4 alternatives (car models) w n = 4 attributes/criteria w x ij = score of option i with respect to criterion j X = {x ij } 4  4 score matrix. w J = set of benefit attributes: style, reliability, fuel economy (more is better) w J' = set of negative attributes: cost (less is better)

25. 25 Steps of TOPSIS w Step 1(a): calculate (  x 2 ij ) 1/2 for each column StyleRel.Fuel Saturn Ford 49818164 64496449 81366481 Civic Mazda Cost  x ij 2 i (  x 2 ) 1/2 36496436 230215273230 15.1714.6616.5215.17

26. 26 Steps of TOPSIS w Step 1 (b): divide each column by (  x 2 ij ) 1/2 to get r ij StyleRel.Fuel Saturn Ford 0.460.610.540.53 0.530.480.480.46 0.590.410.480.59 Civic Mazda 0.400.480.480.40 Cost

27. 27 Steps of TOPSIS w Step 2 (b): multiply each column by w j to get v ij. StyleRel.Fuel Saturn Ford 0.0460.2440.1620.106 0.0530.1920.1440.092 0.0590.1640.1440.118 Civic Mazda 0.0400.1920.1440.080 Cost

28. 28 Steps of TOPSIS w Step 3 (a): determine ideal solution A*. A* = {0.059, 0.244, 0.162, 0.080} StyleRel.Fuel Saturn Ford 0.0460.2440.1620.106 0.0530.1920.1440.092 0.0590.1640.1440.118 Civic Mazda 0.0400.1920.1440.080 Cost

29. 29 Steps of TOPSIS w Step 3 (a): find negative ideal solution A'. A' = {0.040, 0.164, 0.144, 0.118} StyleRel.Fuel Saturn Ford 0.0460.2440.1620.106 0.0530.1920.1440.092 0.0590.1640.1440.118 Civic Mazda 0.0400.1920.1440.080 Cost

30. 30 Steps of TOPSIS w 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 StyleRel.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 w 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 0.0008450.029 0.0032080.057 0.0081860.090 Civic Mazda 0.0033890.058

32. 32 Steps of TOPSIS w Step 4 (b): find separation from negative ideal solutionA' = {0.040, 0.164, 0.144, 0.118} S i ' = [  (v j '– v ij ) 2 ] ½ for each row j StyleRel.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 w 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 0.0069040.083 0.0016290.040 0.0003610.019 Civic Mazda 0.0022280.047

34. 34 Steps of TOPSIS w Step 5: Calculate the relative closeness to the ideal solution C i * = S' i / (S i * +S' i ) S' i /(S i * +S' i )Ci*Ci* Saturn Ford 0.083/0.1120.74  BEST 0.040/0.0970.41 0.019/0.1090.17 Civic Mazda 0.047/0.1050.45