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Advanced Aspects of the Interactive NAUTILUS Method Enabling Gains without Losses

Advanced Aspects of the Interactive NAUTILUS Method Enabling Gains without Losses. Kaisa Miettinen kaisa.miettine@jyu.fi Dmitry Podkopaev University of Jyväskylä, Department of Mathematical Information Technology Francisco Ruiz Mariano Luque

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Advanced Aspects of the Interactive NAUTILUS Method Enabling Gains without Losses

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  1. Advanced Aspects of the Interactive NAUTILUS Method Enabling Gains without Losses Kaisa Miettinen kaisa.miettine@jyu.fi Dmitry Podkopaev University of Jyväskylä, Department of Mathematical Information Technology Francisco Ruiz Mariano Luque University of Malaga, Department of Applied Economics (Mathematics) Jyväskylä Malaga

  2. Contents • Some concepts • Interactive method Nautilus for nonlinear multiobjective optimization • Background • Algorithm • New approach to expressing preferences • Background • Example • Preference model • Conclusions

  3. Problem • with k objective functions;objective function valueszi = fi(x) and objective vectors z = (z1,…, zk) Rk • Feasible objective region Z Rk is image of S. Thus z  Z

  4. Concepts • Point x* S (and z  Z) is Pareto optimal (PO) if there exists no other point xS such that fi(x)  fi(x*) for all i=1,…,k and fj(x) <fj(x*) for somej • Ranges in the PO set: • Ideal objective vector • Nadir objective vector • Decision maker (DM) responsible for final solution • Goal: help DM in finding most preferred (PO) solution • We need preference information from DM

  5. Background for Nautilus • Typically methods deal with Pareto optimal solutions only, as no other solutions are expected to be interesting for the DM • Trading off necessitated: impairment in some objective(s) must be allowed in order to get a new solution • Past experiences affect DMs’ hopes • We do not react symmetrically to gains and losses • Requirement of trading off may hinder DM’s willingness to move from the current Pareto optimal solution

  6. Background for Nautilus, cont • Kahneman and Tversky (1979): Prospect theory • Critique of expected utility theory as a descriptive model of decision making under risk • Our attitudes to losses loom larger than gains • Pleasure of gaining some money seems to be lower than the dissatisfaction of losing the same amount of money • The past and present context of experience defines an adaptation level, or reference point, and stimuli are perceived in relation to this reference point • If we first see a very unsatisfactory solution, a somewhat better solution is more satisfactory than otherwise

  7. Background for Nautilus, cont • Typically low number of iterations is taken in interactive methods • Anchoring: solutions considered may fix our expectations (DM fixes thinking on some (possible irrelevant) information • Time available for solution process limited • Choice of starting point may play a significant role • Most preferred solution may not be found • Group decision making: • Negotiators easily anchor at starting Pareto optimal solution if it is advantageous for their interests

  8. The Idea of Nautilus • Learning-oriented interactive method • DM starts from the worst i.e. nadir objective vector and moves towards PO set • Improvement in each objective at each iteration • Gain in each objective at every iteration – no need for impairment • Only the final solution is Pareto optimal • Objective vector obtained dominates the previous one • DM can always go backwards if desired • The method allows the DM to approach the part of the PO set (s)he wishes

  9. Z=f(S) The Idea of Nautilus

  10. NautilusAlgorithm • Main underlying tool: achievement function based on a reference point q • Given the current values zh, two possibilities for preference information: • Rank relative importance of improving each current value: the higher rank r, the more important improvement is • Give 0-100 points to each current objective value: the more points you allocate, the more improvement is desired qih=pi/100, Miettinen, K., Eskelinen, P., Ruiz, F., Luque, M. (2010) NAUTILUS Method: An Interactive Technique in Multiobjective Optimization based on the Nadir Point, European Journal of Operational Research, 206(2), 426-434.

  11. Nautilus Algorithm, cont. • At the beginning, DM sets number of steps (iterations) to be taken itn(can be changed) and specifies preferences related to nadir obj. vector • ith= number of iterations left • With q=zh-1, minimize achievement function to get fh=f(xh). The next iteration point is • At the last iteration ith=1 and zh= fh • At each iteration, range of reachable obj.values shrinks • We calculate zh,lo and zh,up • zh,lois obtained by solving e-constraint problems • zh,up is obtained from the current obj.values • We also calculate distance to PO set

  12. Z=f(S) Some Iterations of Nautilus

  13. Implementation Ideas by Petri Eskelinen by Suvi Tarkkanen

  14. Representing DM’s preferences:Challenges • Current preference expressing ways very rough • Converting objective improvement ranking to scalarizing function parameters: infinite number of possibilities • Distributing percents / points among objectives: how to interpret the correspondence between the distribution and the selection rule? • Is there any straightforward and transparent way of expressing preferences and converting them into the algorithm?

  15. Background for the New Preference Model • DM aims at improving all the objectives simultaneously there is no conflict at the beginning as perceived by DM • The conflict appears only when achieving the Pareto optimal set • We can assume: no interest to improve some objectives without improving others (all objectives are to be optimized) • There may be certain proportions in which the objectives should be improved to achieve the most intensive synergy effect • E.g. concave utility function grows faster in certain directions of simultaneous increase of objective function values

  16. Direction of Consistent Improvement of Objectives Starting point: q=(q1,,qk)  Z Direction of consistent improvement of objectives:  =(1,, k)  Rk, where i> 0 for alli DM wants to improve objective functions starting from q as much as possible, by decreasing the objective values in the proportions represented by 

  17. Expressing DM’s Preferences: Three Possibilities • DM sets the values 1, 2,, k directly • DM says that improvement of fi by one unit should be accompanied by improvement of each other objective j, j=1,...,k, by a value j.Then i := 1; j := j for all j=1,...,k, ji • DM defines for any chosen pairs of objectives i, j, ij:the improvement of fiby one unit should be accompanied by improvement of fj by ij units. • One has to ensure that values ijfully and consistently define values i such that j /i = ij for any i, j = 1,...,k, ij

  18. Expressing DM’s Preferences: Example City Municipality border Fresh Fishery Ltd. water pollution water pollution • 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 low dissolved oxygen (DO)level low dissolved oxygen (DO)level No information about possibilities before design starts!

  19. Expressing DM’s Preferences:Example / Objectives and Parties 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. The City Council, represented by two vice-mayors. Interest of parties in objectives 

  20. Expressing DM’s Preferences:Example / Negotiations • TheCity Council DM (c), on the right of the organizer, proposes to start from the following direction of improvement: 1 = 1,5 mg/L, 2 = 2 mg/L, 3 = 0,5 pp, 4 = 1 pp. • Association „ Citizens for clear water” (a) disagrees that 2 > 1 and insists that clear water at the city level is more important than at the municipality border. Thus (a) proposes to increase 1 to 3:1 = 3 mg/L, 2 = 2 mg/L, 3 = 0,5 pp, 4 = 1 pp. • The Fresh Fishery manager (b) indicates that comparing to 1 and 2 (DO levels), the value of 3 is disproportionally small. (b) reminds that Fishery is a co-investor and threatens to quit, if the following requirements will not be met: 3/1 0,5; 3/2 0,5; and 3/4 0,75. Thereby (b) proposes to set: 1 = 3 mg/L, 2 = 2 mg/L, 3 = 1,5 pp, 4 = 1 pp. • Association „ Citizens for clear water” (a) disagrees that 2 > 1 and insists that clear water at the city level is more important than at the municipality border. Thus (a) proposes to increase 1 to 3:1 = 3 mg/L, 2 = 2 mg/L, 3 = 0,5 pp, 4 = 1 pp. • The Fresh Fishery manager (b) indicates that comparing to 1 and 2 (DO levels), the value of 3 is disproportionally small. (b) reminds that Fishery is a co-investor and threatens to quit, if the following requirements will not be met: 3/1 0,5; 3/2 0,5; and 3/4 0,75. Thereby (b) proposes to set: 1 = 3 mg/L, 2 = 2 mg/L, 3 = 1,5 pp, 4 = 1 pp. • (c) proposes to decrease 1 to 2 mg/L and 3 to 1 pp, which does not violate conditions imposed by (a) and (b) • And so on...

  21. Representing DM’s Preferences: Model Improve objective functions starting from q as much as possible in the direction  , inside the set Z z2 Geometrical interpretation: find the farthest objective vector along the half-line qt, t ≥ 0: max{t: qt  Z} q z = qt, t ≥ 0 z0 z1 ... What if the objective vector found is not Pareto optimal? z3 zk

  22. Representing DM’s Preferencesinside Nautilus Improve objective functions starting from q as much as possible in the direction , inside set Z, or since there exists an objective vector dominating points on the line z2 Samescalarizingfunction z = qt, t ≥ 0 q • z* is better than z0(along the line) z0 z* zmax • zmax is better than z*(Pareto domination) z1 ... z3 zk

  23. Conclusions • We have described trade-off –free Nautilus providing new perspective to solving problems • We have developed new ways for preference information specification • Before the Pareto optimal set is reached, one can say that there is no conflict among objectives – they should all be optimized • DM’s preferences can be expressed as a direction of consistent improvement of objectives • Then the Chebyshev-type scalarizing function can be used as in the original Nautilus

  24. Thank you! Industrial Optimization Group http://www.mit.jyu.fi/optgroup kaisa.miettinen@jyu.fi http://www.mit.jyu.fi/miettine/engl.html

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