1. CSM 2006, Laxenburg, 28-30 August 2006 1 Hierarchical reference approach to multi-criteria analysis of discrete alternatives JANUSZ GRANAT National Institute of Telecommunications,Warsaw, and Warsaw University of Technology, Poland MAREK MAKOWSKI International Institute for Applied System Analysis, Laxenburg, Austria ANDRZEJ P. WIERZBICKI Center for Strategic Development of Science and Technology, Japan Advanced Institute of Science and Technology, Ichikawa, Japan, and National Institute of Telecommunications,Warsaw, Poland

2. CSM 2006, Laxenburg, 28-30 August 2006 2 Outline Motivation The limitation of the existing approaches Hierarchical criteria aggregations Applications Conclusions

3. CSM 2006, Laxenburg, 28-30 August 2006 3 The criteria for selection of energy technologies level 1level 2level 3 criteriacriteria/indicatorsindicators Economy Financial requirements Production cost Investment Fuel Price Resources Availability Generation potential Environment Global warming Total waste Social Employment Risk aversion

4. CSM 2006, Laxenburg, 28-30 August 2006 4 Hierarchical weighting 0,6 0,4 0,2 0,8 0,1 0,8 0,120,480,040,320,04

5. CSM 2006, Laxenburg, 28-30 August 2006 5 Bottom-up weighting 0,3 0,7 0,2 0,1 0,5 0,1 0,20,10,50,1

6. CSM 2006, Laxenburg, 28-30 August 2006 6 Compensatory versus noncompensatory criteria  Compensatory criteria – an improvement of a criterion can be rationally substantiated to compensate a deterioration of another criterion. e.g. operational costs and investment costs  Noncompensatory criteria are such that no rational substantiation exists for defining weighting coefficients. e.g. costs and loss of human life

7. CSM 2006, Laxenburg, 28-30 August 2006 7 Ranking „ranking” „classification” „partial ordering”

8. CSM 2006, Laxenburg, 28-30 August 2006 8 Subjective versus objective ranking Full objectivity is obviously – after Heisenberg and Quine – not attainable, but in many situations we must try to be as much objective as possible.

9. CSM 2006, Laxenburg, 28-30 August 2006 9 Objective ranking Weighting coefficients and/or aspiration and reservation levels should be determined in some objective or intersubjectively fair fashion. We shall consider three possible ways of achieving this goal:  neutral  statistical  voting

10. CSM 2006, Laxenburg, 28-30 August 2006 10 Neutral weights - objective weighting coefficients for compensatory criteria and weighting coefficients equal in size for all noncompensatory criteria aspirations/reservations - a neutral definition of reference points e.g. all aspiration levels equal to 67% of criteria ranges, all reservation levels equal to 33% of these ranges

11. CSM 2006, Laxenburg, 28-30 August 2006 11 Voting  A voting procedure between a group of decision makers.  Many voting procedures, see H.Nurmi (1999).  Voting results actually only in intersubjective aggregation.

12. CSM 2006, Laxenburg, 28-30 August 2006 12 Statistical Based on some meaningful statistics.  weights - it is very difficult to find statistical data to substantiate weighting coefficients  aspirations/reservations - the average score of all options, e.g.: q a i = q m i +(q max i –q m i )/2; q r i = q m i -(q m i –q min i )/2 q m i - is average value of the i-th criterion for all decision options q a i, q r i - aspiration and the reservation levels, respectively

13. CSM 2006, Laxenburg, 28-30 August 2006 13 Approaches to hierarchical criteria aggregation Compensatory aggregation on lower level, noncompensatory analysis on upper level. Noncompensatory aggregation both on lower and on upper level Noncompensatory aggregation with weighting coefficients as importance factors

14. CSM 2006, Laxenburg, 28-30 August 2006 14 Compensatory aggregation on lower level, noncompensatory analysis on upper level. q C = ∑ i є C w i q i for all C = A,…H qAqA qBqB q1q1 q2q2 q1q1 q3q3 q2q2

15. CSM 2006, Laxenburg, 28-30 August 2006 15 Noncompensatory aggregation both on lower and on upper level qAqA qBqB q1q1 q2q2 q1q1 q3q3 q2q2

16. CSM 2006, Laxenburg, 28-30 August 2006 16 Noncompensatory aggregation with weighting coefficients treated as importance factors The weights are interpreted as importance factors and are used for modification of neutral aspiration and reservation levels e.g.:

17. CSM 2006, Laxenburg, 28-30 August 2006 17 Noncompensatory aggregation with weighting coefficients treated as importance factors - weights b c d e a q2q2 q2q2

18. CSM 2006, Laxenburg, 28-30 August 2006 18 Noncompensatory aggregation with weighting coefficients treated as importance factors – neutral aspiration b c d e a q2q2 q2q2 (0.73, 0,73) (0.23, 0,23)

19. CSM 2006, Laxenburg, 28-30 August 2006 19 Noncompensatory aggregation with weighting coefficients treated as importance factors – weighted aspiration b c d e a q2q2 q2q2 (0.51, 0,22) (0.16, 0,07) w=(0.7, 0.3)

20. CSM 2006, Laxenburg, 28-30 August 2006 20 Preservation of Pareto optimality after hierarchical aggregation Theorem. In a hierarchical aggregation of criteria, suppose that the functions used to aggregate criteria in groups on the lower level are strictly monotone with respect to the partial orders defining the vector optimization problems on lower level. Then any decision option that is Pareto optimal in the space of aggregated criteria is also Pareto optimal in the original space of all lower level criteria (with respect to the overall partial order induced by the partial orders for all groups of criteria).

21. CSM 2006, Laxenburg, 28-30 August 2006 21 Electricity supply technologies - hierarchical weighting Economy (1, 0, 0) Environment (0, 1, 0) Social (0, 0, 1) Equal weights (0.5,0.5,0.5)

22. CSM 2006, Laxenburg, 28-30 August 2006 22 Compensatory aggregation on lower level, noncompensatory analysis on upper level Aspirations/reservations noncompensatory upper level compensatory lower level Equal weights (0.5,0.5,0.5)

23. CSM 2006, Laxenburg, 28-30 August 2006 23 Noncompensatory aggregation both on lower and on upper level compensatory upper level noncompensatory lower level noncompensatory upper level noncompensatory lower level noncompensatory upper level compensatory lower level

24. CSM 2006, Laxenburg, 28-30 August 2006 24 Conclusions (I)  Distinction between subjective and objective ranking  Distinction between compensatory and noncompensatory groups of criteria.  Approaches to hierarchical aggregation of criteria:  Compensatory aggregation on lower level, noncompensatory analysis on upper level;  Noncompensatory aggregation both on lower and on upper level;  Noncompensatory aggregation with weighting coefficients treated as importance factors.

25. CSM 2006, Laxenburg, 28-30 August 2006 25 Conclusions (II)  The discussion and a theorem on the preservation of Pareto optimality after hierarchical aggregation with strictly monotone aggregating functions.  The resulting approaches will be used on the problem of the selection of electricity supply technologies