1. Handling Preferences in the Pre-conflicting Phase of Decision Making Processes under Multiple Criteria Dmitry Podkopaev, Kaisa Miettinen Industrial Optimization Group www.mit.jyu.fi/optgroup/ Department of Mathematical Information Technology University of Jyväskylä, Finland ADT2011

2. Outline and motivation Are objectives conflicting or consistent? Problem statement and context Pre-conflicting phase of decision making Preference model Direction of simultaneous improvement of objectives Domination relation based on trade-off information Preference model characterization Application example Concluding remarks 2

3. Are objectives conflicting or consistent? Why multiple objectives can be calledconflicting? Because usually methods and research concentrate on Pareto optimal solutions: improvement in some objective function value is possible only when another objective(s) impairs 3

4. Are objectives conflicting or consistent? Why multiple objectives can be calledconsistent? Because the DM wishes to optimize all objectives simultaneously Simultaneous improvement of objectives may produce synergy effect Many utility functions are quasi-concave the utility increases faster in some directions of increasing objective values 4 Guerraggio A., Molho E. The origins of quasi concavity: a development between mathematics and economics. Historia Mathematica. 2004, 31, 62 75.

5. Problem statement and context Among elements of X (feasible solutions) evaluated by f =(f 1,f 2,...,f k ): X k (objective functions) with criteria of the type the more the better, find an element which is most preferred for the decision maker (DM). 5

6. Problem statement and context where Y = {f (x ): x X } is the set of outcomes such that the greater a component, the better for the DM. Find an outcome which is most preferred for the DM. 6

7. Problem statement and context 7 where Y = {f (x ): x X }. Assume that the outcome set cannot be presented or even sketched for the DM: X is defined implicitly (e.g. via constraints); f is hard to compute (e.g. simulation is needed); Y has a complex structure to comprehend (e.g. nonsmooth/stochastic; k is a big number). or

8. Pre-conflicting phase Assume that the outcome set cannot be presented or even sketched for the DM. Then interactive methods are used 8 The DM provides preference information to the method The method derives Pareto optimal solutions and shows to the DM The method learns about DMs preferences The DM learns about the problem (the outcome set) BEGINENDBEGIN Belton V., Branke J., Eskelinen P., Greco S., Molina J., Ruiz F., Slowinski R. Interactive multiobjective optimization from a learning perspective. In : Branke et al. (eds.), Multiobjective Optimization. Interactive and Evolutionary Approaches, Springer, 2008, 405-433.

9. Pre-conflicting phase In the beginning, before the structure of the outcome set is revealed, it is difficult to express preference in terms of: attainable objective function values; solution comparisons; trade-off information. We propose to ask the DM about a most promising direction of simultaneous improvement of objectives, starting from undesirable values of each objective. 9 y1y1 y2y2 ykyk... y3y3 empty objective space

10. Preference model: simultaneous improvement of objectives s k – starting point (a hypothetical outcome) If the problem arose from the desire to improve the existing solution, the outcome of that solution can serve as the starting point. The DM may provide worst imaginable values of objective functions to use them as the starting point components. The nadir point (calculated or estimated) 10 Deb K., Miettinen K., Chaudhuri S. Towards an Estimation of Nadir Objective Vector Using a Hybrid of Evolutionary and Local Search Approaches. IEEE Transactions on Evolutionary Computation. 2010, 14(6), 821 841.

11. Preference model: simultaneous improvement of objectives δ k – direction of improvement The DM may set the values δ 1, δ 2,…,δ k directly, provided that (s)he can operate with them. The DM says that the unitary increase of the i -th objective should be accompanied by increasing each other objective j i by θ j. Then δ i =1 and δ j = θ j, j i. The DM defines proportions of improvement freely, picking up pairs (i,j ), j i, and setting the desirable improvement ratio θ ij. It has to be ensured that values {θ ij } completely and consistently define k values {δ i }. 11

12. Preference model: simultaneous improvement of objectives s k – starting point δ k – direction of improvement The DM wants to improve s as much as possible, increasing the objective function values in proportions δ. 12 y1y1 y2y2 ykyk... y3y3 s y = s + t δ, t >0 max{ t : s +t δ Y, t >0 }

13. Preference model: domination relation and trade-off information β ij > 0 – bound on trade-off coefficient between i -th and j -th objectives: how much does DM agree to pay in terms of the i -th objective for unitary gain in j -th objective. 13 y*y* β ji 1 y dominates y * y is dominated by y *

14. Preference model: characterization 14 y2y2 y1y1 ykyk... y3y3 s Podkopaev D. Incorporating Explicit Tradeoff Information to Interactive Methods Based on the Chebyshev type Scalarizing Function. Reports of the Department of Mathematical Information Technology. Series B: Scientific Computing. No. B9/2010. University of Jyvaskyla, 2010.

15. Preference model: characterization 15 y2y2 y1y1 ykyk... y3y3 s If ŷ is non-dominated in Y, then return ŷ Otherwise continue moving along δ until it is dominated by any outcome. After that return an outcome y* dominating.

16. 16 Y Application example: NAUTILUS Miettinen K., Eskelinen P., Ruiz F., Luque M. NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point. European Journal of Operational Research. 2010, 206, 426-434.

17. Application example: NAUTILUS 17 Fresh Fishery Ltd. City Municipality border water pollution low dissolved oxygen (DO) level Invest to water treatment facilities in order to... increase the DO level at the City; increase the DO level at the municipality border. Undesirable effects: the return of investments at Fresh Fishery decreases; the city taxes grow. No information about possibilities before design starts!

18. Application example: NAUTILUS 18 Objectives: (1)Dissolved oxygen (DO) level at the city max; (2)DO level at the municipality boarder max; (3)The percent return of investments at Fresh Fishery max; (4)Increase of the city taxes min. Negotiation parties: (a)Association Citizens for clear water (b)Business Development Manager of the Fresh Fishery. (c)The City Council, represented by two vice-mayors. Interest of parties in objectives (1)(2)(3)(4) (a)Xx (b)X x (c)xXX

19. Application example: NAUTILUS 19 The City Council DM (c), on the right of the organizer, proposes to start from the following direction of improvement: δ 1 = 1.5 mg/L, δ 2 = 2 mg/L, δ 3 = 0.5 pp, δ 4 = 1 pp. Association Citizens for clear water (a) disagrees that δ 2 > δ 1 and insists that clear water at the city level is more important than at the municipality border. Thus (a) proposes to increase δ 1 to 3: δ 1 = 3 mg/L, δ 2 = 2 mg/L, δ 3 = 0.5 pp, δ 4 = 1 pp. The Fresh Fishery manager (b) indicates that comparing to δ 1 and δ 2 (DO levels), the value of δ 3 is disproportionally small. (b) reminds that Fishery is a co-investor and threatens to quit, if the following requirements will not be met: δ 3 / δ 1 0.5; δ 3 / δ 2 0.5; and δ 3 / δ 4 0.75. Thereby (b) proposes to set: δ 1 = 3 mg/L, δ 2 = 2 mg/L, δ 3 = 1.5 pp, δ 4 = 1 pp. Association Citizens for clear water (a) disagrees that δ 2 > δ 1 and insists that clear water at the city level is more important than at the municipality border. Thus (a) proposes to increase δ 1 to 3: δ 1 = 3 mg/L, δ 2 = 2 mg/L, δ 3 = 0.5 pp, δ 4 = 1 pp. The Fresh Fishery manager (b) indicates that comparing to δ 1 and δ 2 (DO levels), the value of δ 3 is disproportionally small. (b) reminds that Fishery is a co-investor and threatens to quit, if the following requirements will not be met: δ 3 / δ 1 0.5; δ 3 / δ 2 0.5; and δ 3 / δ 4 0.75. Thereby (b) proposes to set: δ 1 = 3 mg/L, δ 2 = 2 mg/L, δ 3 = 1,5 pp, δ 4 = 1 pp. (c) proposes to decrease δ 1 to 2 mg/L and δ 3 to 1 pp, which does not violate conditions imposed by (a) and (b) And so on...

20. Concluding remarks Asymmetric attitude of DM to gains and losses [1] Expressing preferences in terms of gains: a direction of simultaneous improvement of objectives Similar ideas have been exploited in negotiation support [2] Combining the direction of improvement with a domination relation scalarizing function model Reference point approach [3] : similar specification, different characterization Further discussion: applicability of the direction of simultaneous improvement concept in multiple criteria / group decision making; combining the direction of simultaneous improvement and bounds on trade-offs in one preference model explaining the direction of simultaneous improvement concept in terms of properties of gradient of a quasi-concave utility function 20 [1] Kahneman D., Tversky A. Prospect Theory: An Analysis of Decisions Under Risk. Econometrica, 1979, 47, 263-291 [2] Ehtamo H., Kettunen E., Hämäläinen R. Searching for joint gains in multi- party negotiations. European Journal of Operational Research, 2001, 130 (1), 54-69. [3] Wierzbicki A. The Use of Reference Objectives in Multiobjective Optimization. Lecture Notes in Economics and Mathematical Systems. Springer, 1980, 177, 468-486.