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

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

with k objective functions; objective function values z i = f i (x) and objective vectors z = (z 1,…, z k )  R k l Feasible objective region Z  R k is image of S. Thus z  Z Problem

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

Background for Nautilus l 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 l Past experiences affect DMs’ hopes l 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

Background for Nautilus, cont l Kahneman and Tversky (1979): Prospect theory l Critique of expected utility theory as a descriptive model of decision making under risk l 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 l 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

Background for Nautilus, cont l 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 l Most preferred solution may not be found l Group decision making:  Negotiators easily anchor at starting Pareto optimal solution if it is advantageous for their interests

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

Z= f (S) The Idea of Nautilus

Nautilus Algorithm l Main underlying tool: achievement function based on a reference point q l Given the current values z h, two possibilities for preference information: –Rank relative importance of improving each current value: the higher rank r, the more important improvement is –Give points to each current objective value: the more points you allocate, the more improvement is desired  q i h =p i /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),

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

Z= f (S) Some Iterations of Nautilus

Implementation Ideas by Petri Eskelinenby Suvi Tarkkanen

Representing DM’s preferences: Challenges  Current preference expressing ways very rough  Converting objective improvement ranking to scalarizing function parameters: infinite number of possibilities l 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?

Background for the New Preference Model l 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 l We can assume: no interest to improve some objectives without improving others (all objectives are to be optimized) l 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

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

Expressing DM’s Preferences: Three Possibilities DM sets the values  1,  2, ,  k directly DM says that improvement of f i 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 f i by one unit should be accompanied by improvement of f j by  ij units. –One has to ensure that values  ij fully and consistently define values  i such that  j /  i =  ij for any i, j = 1,...,k, i  j

Expressing DM’s Preferences: Example 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!

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  Expressing DM’s Preferences: Example / Objectives and Parties (1)(2)(3)(4) (a)Xx (b)X x (c)xXX

The City 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. Expressing DM’s Preferences: Example / Negotiations 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...

Representing DM’s Preferences: Model Geometrical interpretation: find the farthest objective vector along the half-line q  t, t ≥ 0: max{t: q  t  Z} What if the objective vector found is not Pareto optimal? Improve objective functions starting from q as much as possible in the direction , inside the set Z q z2z2 z1z1 zkzk... z3z3 z = q  t, t ≥ 0 z0z0

Representing DM’s Preferences inside Nautilus Same scalarizing function q z2z2 z1z1 zkzk... z3z3 z = q  t, t ≥ 0 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 z0z0 z*z* z max z * is better than z 0 (along the line) z max is better than z * (Pareto domination)

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

Thank you! Industrial Optimization Group