Advanced Aspects of the Interactive NAUTILUS Method Enabling Gains without Losses

Kaisa Miettinenkaisa.miettine@jyu.fi

Dmitry PodkopaevUniversity of Jyväskylä, Department of Mathematical Information Technology

Francisco Ruiz

Mariano LuqueUniversity of Malaga, Department of Applied Economics (Mathematics)

Jyväskylä

Malaga

Contents

Some concepts Interactive method

Nautilus for nonlinear multiobjective optimization

Background Algorithm

New approach to expressing preferences

Background Example Preference model

Conclusions

with k objective functions; objective function values zi = fi(x) and objective vectors z = (z1,…, zk) Rk

Feasible objective region Z Rk is image of S. Thus z Z

Problem

Concepts Point x* S (and z Z) is Pareto optimal (PO) if

there exists no other point xS such that

fi(x) fi(x*) for all i =1,…,k and fj(x) < fj(x*) for some j

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

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

Background for Nautilus, contKahneman 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 moneyThe 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

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

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

nadzz 0

lo,1zz

Z=f

(S)

lo,2z

1f

1z2z

2f

lo,3z

The Idea of Nautilus

Nautilus Algorithm 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, )(/1 **

inadi

hi

hi zzq

)/())(())((maxminimize **

1,...,1

inadi

k

i

hii

hii

hi

kizzqfqf

xx

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.

)(/1 **i

nadi

hi zzr

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,lo is obtained by solving e-constraint problems– zh,up is obtained from the current obj.values

We also calculate distance to PO set

hhhhhh ititit fzz )/1()/)1(( 1

nadzz 0

lo,1zz

Z=f

(S)

lo,2z

1f

1z2z

2f

lo,3z

Some Iterations of Nautilus

Implementation Ideas

by Petri Eskelinen by 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

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

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

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 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 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, ji

• DM defines for any chosen pairs of objectives i, j, ij:the improvement of fi by one unit should be accompanied by improvement of fj 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, ij

Expressing DM’s Preferences: Example

Fresh Fishery Ltd.

City

Municipality border

water pollution

water pollutionlow dissolved

oxygen (DO)level

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) X x

(b) X x

(c) x X X

• 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 qt, t ≥ 0:

max{t: qt 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

z2

z1

zk

...z3

z = qt

, t ≥

0

z0

Representing DM’s Preferencesinside Nautilus

Same scalarizing function

q

z2

z1

zk

...z3

z = qt

, 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

z0

z*zmax

• z* is better than z0

(along the line)

• zmax is better than z*

(Pareto domination)

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

Thank you!

Industrial Optimization Group http://www.mit.jyu.fi/optgroup

kaisa.miettinen@jyu.fihttp://www.mit.jyu.fi/miettine/engl.html