Mathematical Ranking (and Consensus Forming) Method

1. Mathematical Ranking (and Consensus Forming) Method Hiroaki Ishii Graduate School of Information Science and Technology Osaka University, Japan

2. Decision making based on Mathematical evaluation Data envelopment analysis（DEA) －－－ mathematical evaluation method for measuring the efficiency of decision-making units (DMU) on the basis of the observed data practiced in comparable DMUs, such as public departments (governments, universities, libraries, hospitals, etc), banking, etc. DEA is originated by Charnes et al. and extended by Banker et al. The basic DEA models are known as CCR and BCC named after the authors’ initials.

3. input Virtual Input= Weight Output weight Virtual output

5. DEFINITION (CCR EFFICIENCY) 1. is CCR-efficient if and there exist at least one optimal with 2. Otherwise, is CCR-inefficient. PRODUCTION POSSIBILITY SET P 1. The observed activities belong to P • If an activity (x, y) belongs to P, then activity (tx, ty) belongs to P • for any positive scalar t. • For an activity (x, y) in P, any semi-positive activity with is included in P. 4. Any semi-positive linear combination of activities in P belongs to P.

7. For ranking of alternatives, one of the most familiar methods is to compare the weighted sum of their votes, after determining suitable weights of each alternative. Borda initially proposed the “Method of Marks” more than two hundred years ago so as to obtain an agreement among different opinions. His method is surely a useful method evaluating consumers’ preferences of commodities in marketing, or in ranking social policies in political sciences, for instance. It is, however, difficult to determine suitable weight of each alternative a priori. In this context, Cook and Kress formulated the measure to automatically decide on the total rank order weight in order to hold the advantage using Data Envelopment Analysis model. Later, Green et al. evolved the measure so as to make it possible to decide on the total rank order of all candidates.

8. And so, the weight of first, second, and third place are set to be 5,3, and 1, respectively, where these are given a priori. The total Score of i-th candidate, , is given as follows: . Here, denote the number of i-th place votes earned by candidate i. Table 1 Voting data, MVP of CL (Central League) in JPB

9. In this example, if different weights are given to places, then the result of ranking becomes different. Then, an important issue is how to determine proper weights of first, second, and third place. Since all candidates want to be ranked first place, they wish each weight to be assigned so as to maximize their own composite score. So, Green’s method by using LP which can determine the value of weights, is a very useful method. However, their method has undesirable points from the viewpoint of application.

10. Rank ordering method by Green et alEach candidate j=1,2,…, m obtained number yj1 of vote as first place, yj2 as second place, yjk of k-th place the weight of k-th place (k=1,2,…,K). Assign the each weight so as to maximize the weighted sum to her/his vote．(1)

11. Total Ranking Method based on DEA Among m objects, n persons select till k ranks according to their preference Weight×captured votes s.t. Caseⅰ(Green et al.) Caseⅱ(Noguchi et al.)

12. ・・・ mean ・・・ ・・・ ・・・ ・・・ ・・・ ・・・ ・・・・ ・・・ ・・・ Geometric mean

13. Example Six DMU (A～F)

14. In case of many categories, we solve the following linear programming problems

15. s.t. ⅰ) ⅱ) Evaluation of DMU based on various data mDMU : S categories Efficiency of DEA Multiple choice (Changing weight order) Different type Candidates are chosen.

16. weight category ・・・ ・・・ ・・・ ・・・ mean ・・・ ・・・ ・・・ ・・・ ・・・ ・・・ ・・・・ ・・・ ・・・ Efficiency of DEA ・・・ S ・・・ 1 2 ・・・ ・・・ ・・・ ・・・・ ・・・

17. AHP

23. Example Application to Apparel Maker Selction Problem • Criteria Selection, candidate of maker, Hierarchy construction 2. Using DEA, making scores of makers from characteristics 3． Based on AHP, subjective evaluation with respect to Priority 4. Total Preference of maker Selection of maker

24. Hierarchy of Apparel Maker Selection Problem -1 Level １ Purpose Selection of apparel maker Applicable of AHP directly ＡＨＰ Level ２ Evaluation Points Sewing Design Assortment Not Applicable of AHP ＡＨＰ？ Level ３ Objects Maker A Maker B Maker C Maker D

25. Hierachy-2 Selection of Apparel maker Level １ Purpose Applicable of AHP directly ＡＨＰ Level ２ Evaluation Points Sewing Design Assortment Vote by staff (Piecewise comparison may be replaced by Voting Vote,First rank, second rank DEA Level ３ Objects Maker A Maker B Maker C Maker D Selection of Suit Brand (DEA+AHP)

26. Voting data (Input DATA) Sewing Design Assortment 2nd 1st 1st 2nd 1st 2nd 8 9 7 7 0 1 Maker A 2 6 7 7 8 2 Maker B 5 3 3 4 8 3 Maker C 3 2 1 1 13 10 Maker D

27. （obtained votes as the first rank)×weight for the first rank）+ （obtained votes as the second rank) x weight for the 2nd rank Linear sum for maker A maximize by two weughts wsa1 wsa2 Max 8*wsa1+9* wsa2 Subject to 8*wsa1+9* wsa2≦1, 6*wsa1+7* wsa2 ≦1, 5*wsa1+3* wsa2 ≦1, 1*wsa1+1* wsa2 ≦1, wsa1≧2* wsa2., wsa2≧2/{20*2(2+1)}. Sewing 1st 2nd Maker A 8 9 1.00 6 0.76 7 Maker B 0.52 Maker C 5 3 0.12 1 1 Maker D

28. Voting Data Analysis from Sewing point of view A C D maximize B w1 0.08 0.08 0.1063 0.1063 mean 0.04 0.04 0.017 0.017 w2 Maker A 1.0000 1.0000 1.000 1.000 1.0000 0.757 Maker B 0.76 0.76 0.757 0.758 Maker C 0.52 0.52 0.583 0.551 0.583 Maker D 0.12 0.12 0.123 0.123 0.1215

29. Geometric mean of Preference rate Sewing Design Assortment Maker A 1.0000 0.9761 0.0597 Maker B 0.758 1.0000 0.1641 Maker C 0.551 0.4183 0.4160 Maker D 0.1215 0.3941 1.0000

30. Piecewise Comparison Sewing Design Assortment Priority Sewing 0.6495 1 5 3 Design 1/5 1 3 0.2295 0.121 Assortment 1/3 1/3 1

31. A maker：0.650*1.0000+0.223*0.9761+0.121*0.0597=0.8749 B maker：0.650*0.758+0.223*1.0000+0.121*0.1641=0.7356 C maker：0.650*0.551+0.223*0.4183+0.121*0.4160=0.5018 D maker：0.650*0.1215+0.223*0.3941+0.121*1.0000=0.2869 Selection of A maker is the most preferable

32. Conjoint analysis is a scaling method developed in mathematical psychology by American psychologist Luce and Turkey in 1964. A model of consumer’s preference formation in common use is the simple additive model. In this model, we think that each possible level of an attribute has a “part worth” to a level of an attribute, and the sum of the part worthies of its attributes is the “total worth” to a consumer of a product. Generally, conjoint analysis introduces a part worth value of each attribute of each product based upon some goodness fit criterion from preference rank ordinal data. The rank ordinal data in that case is a result of people’s selection by individuals.

33. Real estate Factors Scores z (ranking y) Styles Forms Totally Japanese Semi-European style Totally European Detached house Condominium A B C D E F 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 6(1) 5(2) 4(3) 3(4) 2(5) 1(6) Part worth value Example of Conjoint Analysis

34. Factor Analysis by Conjoint analysis

35. Ordinal scale of consumer’s preference for products 0-1 design matrix to indicate each level of products Order preserving transformation of y An additive conjoint model Average vector of z The part worth values to be estimated

36. minimize maximize To be minimized Optimal b Fitting criterion for conjoint analysis Quadratic fractional programming Numerator Difference between estimation and actual data denominator Variation of actual data

37. Parametric quadratic function with  • Theorem Let If then Further, the minimizer of solution of is also an optimal

38. is convex , Since Let b, be , substitute into Optimality conditions From (1), (2), obtain the optimal part worth value

39. Method B

40. SingleObjective Many Multi-purpose Problem Actually Multi-purpose More Applicable RESEARCH PURPOSE Marketing Consumer Preference Conjoint Analysis Conjoint Analysis combined with DEA

41. Voting Data multiple total evaluation Weighting Total Ordering Ordinal Data from total evaluation Factor—part worth value DEA・Conjoint Analysis • DEA･･･ • Conjoint Analysis･･

42. DEA Total evaluation Multipleevaluation Conjoint Analysis Conjoint Analysis combined with DEA 〈Example〉Voting Data

43. Preferable Weighting For each alternative DEA model Choose k preferences among m alternatives with ranking Objective function constraint Weight of each rank Captured votes Ranking by Cross-valuation

44. DEA－Ｃｏｎｊｏｉｎｔ Analysis Ｅｖａｌｕａｔｉｏｎ Method ｆｏｒ Voting Data Application Ａｐｐｌｉｃａｔｉｏｎ to development plan ｏｆ ｎｅｗ medicine From activation values of 40 samples, we find promising Compounds from various aspect.

45. (single objective case )Mother compound • Artificial example

46. Sample Activation value 1 7.6 2 7.1 3 6.6 4 6.3 5 5.9 6 5.1 7 5.1 8 4.9 9 6.4 10 5.4 The combination of substituents expected as the new medicines

47. Activation value Z 1 1 0 0 0 0 1 0 0 1 0 7.6 2 0 1 0 0 0 0 1 0 1 0 7.1 3 0 0 1 0 0 0 1 0 0 1 6.6 4 1 0 0 0 0 0 1 0 0 1 6.3 5 0 0 0 0 1 0 1 0 0 1 5.9 6 0 1 0 0 0 0 0 1 0 1 5.1 7 1 0 0 0 0 0 0 1 0 1 5.1 8 0 0 0 1 0 0 0 1 0 1 4.9 9 0 0 0 0 1 0 0 1 0 1 6.4 10 0 0 0 1 0 0 1 0 0 1 5.4 b Application to the design matrix

48. : : : Calculation Result Estimate a new medicine with combination of substituents of high part worth value Activation value (estimation) 8.54

49. Theme Theme Extending Distance Measure to Construct Joint Ballot Model Single Ballot Choosing “special and the best” objects Joint Ballot Choosing “the most favorable pairs” constructing a model to reflect voter’s favorable pairs when they place objects in the order

50. Consensus Formation Distance Function Minimizing “distance” which indicates degree of disagreement of voters Cook&Kress’s Relative Distance Position j Forward Indicator Vector Position j Backward Indicator Vector Relative Distance “degree of differences” are expressed