Computers & Operations Research 39 (2012) 1673–1681

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Computers & Operations Research

0305-05

doi:10.1

n Corr

E-m

kaisa.m

rua@um

journal homepage: www.elsevier.com/locate/caor

A two-slope achievement scalarizing function for interactivemultiobjective optimization

Mariano Luque a,n, Kaisa Miettinen b, Ana B. Ruiz a, Francisco Ruiz a

a University of Malaga, Calle Ejido 6, 29071 Malaga, Spainb Department of Mathematical Information Technology, P.O. Box 35 (Agora), FI-40014 University of Jyvaskyla, Finland

a r t i c l e i n f o

Available online 10 October 2011

Keywords:

Multiobjective programming

Interactive methods

Reference point methods

Pareto optimality

48/$ - see front matter & 2011 Elsevier Ltd. A

016/j.cor.2011.10.002

esponding author. Tel.: þ34 952131173; fax:

ail addresses: mluque@uma.es (M. Luque),

iettinen@jyu.fi (K. Miettinen), abruiz@uma.es

a.es (F. Ruiz).

a b s t r a c t

The use of achievement (scalarizing) functions in interactive multiobjective optimization methods is

very popular, as indicated by the large number of algorithmic and applied scientific papers that use this

approach. Key parameters in this approach are the reference point, which expresses desirable objective

function values for the decision maker, and weights. The role of the weights can range from purely

normalizing to fully preferential parameters that indicate the relative importance given by the decision

maker to the achievement of each reference value. Technically, the influence of the weights in the

solution generated by the achievement scalarizing function is different, depending on whether the

reference point is achievable or not. Besides, from a psychological point of view, decision makers also

react in a different way, depending on the achievability of the reference point. For this reason, in this

work, we introduce the formulation of a new achievement scalarizing function with two different

weight vectors, one for achievable reference points, and the other one for unachievable reference

points. The new achievement scalarizing function is designed so that an appropriate weight vector is

used in each case, without having to carry out any a priori achievability test. It allows us to reflect the

decision maker’s preferences in a better way as a part of an interactive solution method, and this can

cause a quicker convergence of the method. The computational efficiency of this new formulation is

shown in several test examples using different reference points.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Many real life problems involve dealing with several criteria,which must be maximized or minimized simultaneously. Suchproblems are called multiobjective optimization problems whenboth the criteria and the constraints that determine the feasibleset of alternatives can be mathematically expressed by functions.Because the criteria, also known as objective functions, typicallyare conflicting, it is impossible to find a solution where all theobjectives can reach their individual optima simultaneously.Instead, we can identify compromise solutions, that is, so-calledPareto optimal or nondominated points, where none of theobjectives can get a better value without deteriorating at leastone of the other objectives.

Many methods have been developed for solving multiobjectiveoptimization problems during the years. They can be classified inthree classes according to the role of the decision maker (DM) inthe solution process (see, e.g., [6,11]). In the so-called a posteriori

ll rights reserved.

þ34 952132061.

(A.B. Ruiz),

methods a representation of nondominated points is first gener-ated and displayed to the DM who then is supposed to select thebest of them as the final solution. The difficulty here is that it maybe cognitively difficult for the DM to analyze all the providedsolutions. Alternatively, the DM can specify desires and hopesbefore the solution process in the so-called a priori methods. Thedrawback here is that it may be difficult for the DM to setexpectations on a realistic level before getting to know theproblem. Finally, the third group comprises of interactive methods.The idea behind these algorithms is the gradual incorporation ofthe DM’s preferences during the interactive and iterative solutionprocess.

Interactive multiobjective optimization methods have beenwidely studied and used in real applications (see, e.g., [11,16] andreferences therein). In them, a solution pattern is formulated andrepeated iteratively, and the DM takes actively part in thesolution process by specifying and refining his/her preferenceinformation. There are many interactive methods and basicallythey differ from each other in what kind of information is askedfor and shown to the DM at each iteration, as well as in the waythe solutions are calculated. Examples of different types ofpreference information asked from the DM include marginal ratesof substitution, surrogate values for trade-offs, classification of

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–16811674

objective functions and reference points. For further details, see[5,6,10,11,23] and references therein.

One of the ways to generate nondominated points in interactivemethods is the use of an achievement scalarizing function. Thepopularity of the achievement scalarizing functions [24] in theframework of interactive methods is unquestionable. It is normallyused within two different ways of specifying preference information.In reference point-based approaches (see [2,7,8,18,25]), the DM givesa reference value to each objective (these values constitute a so-calledreference point), while in classification-based approaches (see [1,14]),the DM classifies the objectives into different categories (objectives tobe improved, objectives than can be worsened, etc.). However,methods based on classification are closely related to referencepoint-based methods because a reference point can be formed oncea classification has been made [14]. Once a reference point is given,the achievement scalarizing function is optimized to find the non-dominated point that is, in some sense, closest to the reference point.For an overview of achievement scalarizing functions, see [13].

Regarding the interactive reference point-based methods, themain difference between them is the form of the weightingcoefficients used in the achievement scalarizing function. In[12,13,21], wide studies of the performance of some of thesemethods (including classification based procedures), and of dif-ferent weights, are presented and the following conclusion isreached: solutions obtained using different reference point-basedmethods (or classification-based ones) are, in fact, different. As aconsequence, a synchronous approach is proposed in [14] wheresome of these solutions are calculated at each iteration andshown to the DM, who chooses the most preferred one accordingto his/her preferences.

Although different achievement scalarizing functions have beendeveloped (for example, an additive achievement scalarizing func-tion in [22]), the most widely used ones so far are extensions of theL1-distance (Chebychev distance). When the reference point isunachievable, these achievement scalarizing functions minimizethe L1-distance between the reference point and the feasible set.In other words, the maximum (unwanted) deviation between thecoordinates of the reference point and the feasible set is minimized.On the other hand, when the reference point is achievable, theseachievement scalarizing functions minimize the maximum value ofthe negative differences between the coordinates of the referencepoint and the nondominated set, which is equivalent to maximizingthe minimum deviation between the coordinates of the referencepoint and the feasible set in the objective space.

Several studies support the idea that it is better to use differentvectors of weights, depending on the reference point given by theDM. In [21], it is shown that the effect of the weights on therelations between the reference values and the correspondingcomponents of the optimal solution are completely differentdepending on whether the reference point is achievable or not.Furthermore, in [3] some reference point-based methods arecompared, and experiments with real DMs are carried out todetermine which solutions they prefer. In fact, solutions obtainedby two different methods are compared: STOM [18] (SatisfyingTrade-Off Method), where the reference point is projected ontothe nondominated objective set, in the direction joining the idealpoint with the reference point, and GUESS [2] (a naıve approach)where the reference point is projected onto the nondominatedobjective set, in the direction joining the nadir vector with thereference point. The conclusion is that if the reference point isachievable, a higher percentage of DMs (55.2%) prefers the STOM

solution rather than the GUESS solution, while if the referencepoint is unachievable, a higher percentage of DMs (61%) prefersthe GUESS solution instead of the STOM solution.

Different methods use different vectors of weights, which arehardly ever controlled by the DM in any way. But studies reflect

that, when the reference point is achievable, the DMs tend toprefer the solutions obtained with certain weights, while whenthe reference point is unachievable, they often prefer others. Onthe other hand, the DM does not necessarily know in advancewhether the reference point (s)he gives is achievable or not.Therefore, in this paper we suggest an achievement scalarizingfunction which automatically chooses a vector of weights forachievable reference points, and a different one for unachievablereference points.

Another possible application of an achievement scalarizing func-tion which automatically adjusts itself according to the achievabilityof reference points can be found in [9], where it is shown that the useof preferential weights in achievement scalarizing functions allows usto obtain solutions that are more satisfactory to the DM, and speedsup the convergence of the algorithm. Several alternatives are providedfor considering such preferential weights. In one of them, it isnecessary to determine whether the reference point is achievable ornot, because different weights are used in each case. This implies tofirst solve a subproblem in order to check the achievability of thereference point before the actual achievement scalarizing functioncan be solved.

As said, in this paper, we propose a new achievement scalarizingfunction with two different weight vectors: one is automatically usedfor unachievable reference points and the other one for achievablereference points. The advantage of this function is that we do notneed to test whether the reference point is achievable or not beforeoptimizing the achievement scalarizing function. Instead, the optimi-zation process itself guarantees that the appropriate weight vectorwill be used in each case. This means that, for example, the previouslymentioned preferential weights problem, proposed in [9], can besolved in a single problem, or following [3], we can consider theSTOM weights for achievable reference points and the GUESS weightsfor unachievable reference points, again solving only a single problemper iteration of the interactive method.

To the authors’ knowledge, an achievement scalarizing functioncorresponding to the one proposed here cannot be found in theliterature. The most closely related function is described in [26],where an achievement scalarizing function with three differentvectors of weights is proposed for a double reference point approach(involving both desirable and acceptable reference values, that is,aspiration and reservation levels, respectively). This function isdefined in a branch-wise fashion, so that different weights are useddepending on the relative position of the objective vector withrespect to the reservation and the aspiration levels. Nevertheless, inthis case the achievability of the reference point is not the issue(both the reservation and the aspiration levels can be achievable ornot, without altering the corresponding weights), and the functiondefined implies an if–then formulation every time the achievementscalarizing function is evaluated.

The remainder of this paper is organized as follows. In Section 2,we introduce the main concepts and notations used. The newachievement scalarizing function is defined in Section 3, demon-strating the efficiency of the solutions obtained, and that anappropriate vector of weights is used in each case. In Section 4,the case of differentiable problems is analyzed. Some computationaltests show the performance of our new achievement scalarizingfunction for both differentiable and nondifferentiable cases inSection 5 and finally, some conclusions are drawn in Section 6.

2. Formulation and background concepts

We consider multiobjective optimization problems of the form

minimize ff 1ðxÞ,f 2ðxÞ, . . . ,f kðxÞg

subject to xAS ð1Þ

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–1681 1675

involving k ðZ2Þ conflicting objective functions f i : S-R to beminimized simultaneously. The decision variables x¼ ðx1,x2, . . . ,xnÞ

T belong to the nonempty compact feasible region S� Rn.Objective vectors in objective space Rk consist of objective values

fðxÞ ¼ ðf 1ðxÞ, f 2ðxÞ, . . . ,f kðxÞÞT and the image of the feasible region

is called feasible objective region Z ¼ fðSÞ.In multiobjective optimization, objective vectors are optimal if

none of their components can be improved without deterioratingat least one of the others. That is, a decision vector x0AS is said tobe efficient if there does not exist another xAS such thatf iðxÞr f iðx

0Þ for all i¼ 1, . . . ,k and f jðxÞo f jðx0Þ for at least one

index j. On the other hand, a decision vector x0AS is said to beweakly efficient if there does not exist another xAS such thatf iðxÞo f iðx

0Þ for all i¼ 1, . . . ,k. The corresponding objective vectorsfðxÞ are called (weakly) nondominated objective vectors. All thenondominated objective vectors form the nondominated objective

set. Note that the nondominated objective set is a subset of the setof weakly nondominated points.

Let us assume that for problem (1) the set of nondominatedobjective vectors contains more than one vector. Because it is oftenuseful to know the ranges of objective vectors in the nondominatedset, we calculate the ideal objective vector z% ¼ ðz%

1, . . . ,z%

k ÞT ARk by

minimizing each objective function individually in the feasibleregion, that is, z%

i ¼minxASf iðxÞ ¼minxAEf iðxÞ for all i¼ 1, . . . ,k,where E is the set of efficient solutions. This gives lower boundsfor the objectives. The upper bounds are set by the nadir objective

vector znad ¼ ðznad1 , . . . ,znad

k ÞT , which can be defined as

znadi ¼maxxAEf iðxÞ for all i¼ 1, . . . ,k. Usually, the nadir objective

vector is difficult to obtain. Its components can be approximatedusing a pay-off table but in general this kind of an estimation is notnecessarily too good (see, e.g., [11] and references therein.)

Furthermore, sometimes an utopian objective vector

z%% ¼ ðz%%

1 , . . . ,z%%

k ÞT is defined as a vector strictly better than the

ideal objective vector. Then, we set z%%

i ¼ z%

i �e for all i¼ 1, . . . ,k,where e40 is a small real number. In what follows, we assumethat the set of nondominated objective vectors is bounded andthat there are available estimations of the ranges of the non-dominated points.

All nondominated points can be regarded as equally desirable inthe mathematical sense and thus, a decision maker (DM) has toidentify the most preferred one among them. A DM is a person whocan express preference information related to the conflicting objec-tives and we assume that less is preferred to more in each objectivefor her/him. Here we assume that the DM specifies preferences inthe form of reference points. A reference point is an objective vectorq¼ ðq1, . . . ,qkÞ

T , where each qi (i¼ 1, . . . ,kÞ is a desirable objectivevalue for fi which has been provided by the DM.

Typically, when solving multiobjective optimization problems,the multiple objective functions and preferences specified by theDM are combined in real-valued scalarizing functions. Scalarizingfunctions can be optimized with appropriate single objectiveoptimization solvers and they are formulated so that they gen-erate (weakly) nondominated points for the original problem.

The main idea of interactive methods based on referencepoints [26] is the following. At each iteration h, the DM mustprovide desirable values or aspiration levels qi

h for every objectivefi (i¼ 1, . . . ,kÞ, and these levels define a reference pointqh ¼ ðqh

1, . . . ,qhkÞ

T reflecting her/his hopes. Next, a scalarizingfunction known as an achievement (scalarizing) function isminimized in order to find a solution that best satisfies the hopesexpressed. The DM can then give a new reference point and theiterative solution process continues until the DM has found themost preferred solution as the final solution and wants to stop.For simplicity, we will skip the iteration number h in the notationfrom now on and, thus, the reference point will be denoted byq¼ ðq1, . . . ,qkÞ

T .

In this paper, our starting point is the following achievementscalarizing function [24–26]

sðq,fðxÞ,lÞ ¼ maxi ¼ 1,...,k

fmiðf iðxÞ�qiÞgþrXk

i ¼ 1

ðf iðxÞ�qiÞ, ð2Þ

which must be minimized in S

minimize sðq,fðxÞ,lÞsubject to xAS: ð3Þ

The value mi40 is a weight assigned to the objective function fi,whose role can vary from a purely normalizing coefficient to apreferential parameter (see [21]). The optimal solution of problem(3) is denoted by xn and the corresponding objective vector byfn ¼ fðxnÞ. The parameter r40 is a so-called augmentation coeffi-cient. Problem (3) produces efficient solutions with bounded trade-offs between objectives (so-called properly efficient solutions), whichoften in practice are more useful than weakly efficient solutions (see,e.g., [11,26] for more details). It has also been shown that augmenta-tion terms may improve computational efficiency [15].

A possible drawback of the achievement scalarizing functionin (3) is that it is generally nondifferentiable even if the funct-ions in the original problem (1) are all differentiable. However,this can be overcome if we introduce a new real-valuedvariable and new constraints and use an equivalent differentiableformulation

minimize aþrXk

i ¼ 1

ðf iðxÞ�qiÞ

subject to miðf iðxÞ�qiÞra for all i¼ 1, . . . ,k,

xAS,aAR: ð4Þ

This achievement scalarizing function extends the idea of aminmax distance in the following sense: if the aspiration levelsare not simultaneously achievable, then it is the minmax distance.On the other hand, if these values are achievable, then the optimalvalues achieved improve the given reference levels ‘‘as much aspossible’’. To be more specific, we say that a reference point isachievable if qAZþRk

þ where Rkþ ¼ fuARk

j uiZ0 for i¼ 1, . . . ,kg.Otherwise, we say that it is unachievable. Given that if q isachievable, there exists any xn such that sðq,fðxnÞ,mÞr0 [1,26],we can easily judge the achievability of the reference point bystudying the sign of the optimal achievement scalarizing functionvalue. Overall, we can say that the achievement scalarizingfunction projects the reference point in the set of nondominatedpoints independently whether the reference point is achievableor not.

For simplicity, given two vectors z1 ¼ ðz11, . . . ,z1

k Þ and z2 ¼

ðz21, . . . ,z2

k Þ, let us denote

z1rz2 if z1i rz2

i for all i¼ 1, . . . ,k and

z1 �z2 in the opposite case ðthere exists some j with z1j 4z2

j Þ and

z1�z2 ¼ ðz11�z2

1, . . . ,z1k�z2

k Þ

3. A two-slope achievement scalarizing function

Let us introduce a new achievement scalarizing function withtwo different weight vectors, where either of them is used depend-ing on whether the reference point is achievable or not. The twodifferent weight vectors produce two different slopes in the projec-tion direction of the reference point onto the nondominatedobjective set. This allows us to reflect the DM’s preferences in abetter way by means of the consideration of appropriate weightvectors for each case (an achievable or an unachievable referencepoint).

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–16811676

The two-slope achievement scalarizing function is defined asfollows:

sðq,fðxÞ,mU ,mAÞ ¼ s0ðq,fðxÞ,mU ,mAÞþrXk

i ¼ 1

ðf iðxÞ�qiÞ, ð5Þ

with r40 and where s0 is given by

s0ðq,fðxÞ,mU ,mAÞ ¼ maxi ¼ 1,...,k

fmaxfmUi ðf iðxÞ�qiÞ,0gþminfmA

i ðf iðxÞ�qiÞ,0gg:

ð6Þ

Function (5) must be minimized over S

minimize sðq,fðxÞ,mU ,mAÞ

subject to xAS: ð7Þ

Next, we will prove that, when optimizing the achievementscalarizing function (5), the weight vector mU ¼ ðmU

1 , . . . ,mUk Þ is

used to project unachievable reference points onto the nondomi-nated objective set, while the weight vector mA ¼ ðmA

1 , . . . ,mAk Þ is

used to project achievable reference points. These weight vectorsare automatically chosen by the function, and thus, no a prioriachievability test on the reference point is needed. To this end, letus give an alternative formulation of function (5). Let x be afeasible solution for (1), and let fðxÞ be its corresponding objectivevector. Then, for each iAf1, . . . ,kg, we have: if f iðxÞrqi, then

maxfmUi ðf iðxÞ�qiÞ,0gþminfmA

i ðf iðxÞ�qiÞ,0g

¼ 0þmAi ðf iðxÞ�qiÞ ¼ mA

i ðf iðxÞ�qiÞ:

If f iðxÞ4qi, then

maxfmUi ðf iðxÞ�qiÞ,0gþminfmA

i ðf iðxÞ�qiÞ,0g

¼ mUi ðf iðxÞ�qiÞþ0¼ mU

i ðf iðxÞ�qiÞ

and therefore,

sðq,fðxÞ,mU ,mAÞ ¼ maxi A IA ðxÞj A IU ðxÞ

fmAi ðf iðxÞ�qiÞ,mU

j ðf jðxÞ�qjÞgþrXk

i ¼ 1

ðf iðxÞ�qiÞ,

ð8Þ

where IAðxÞ ¼ fi9f iðxÞrqig and IU

ðxÞ ¼ fj9f jðxÞ4qjg. Obviously,IAðxÞ \ IU

ðxÞ ¼ | and IAðxÞ [ IU

ðxÞ ¼ f1, . . . ,kg.Observe that if IU

ðxÞ ¼ |, which means that fðxÞrq, then

sðq,fðxÞ,mU ,mAÞ ¼ maxi ¼ 1,...,k

fmAi ðf iðxÞ�qiÞgþr

Xk

i ¼ 1

ðf iðxÞ�qiÞ ¼ sðq,fðxÞ,mAÞ

ð9Þ

and if IUðxÞa|, which means that fðxÞ�q, then

sðq,fðxÞ,mU ,mAÞ ¼ maxjA IU

ðxÞfmU

j ðf jðxÞ�qjÞgþrXk

i ¼ 1

ðf iðxÞ�qiÞ ¼ sðq,fðxÞ,mUÞ

ð10Þ

Therefore,

sðq,fðxÞ,mU ,mAÞ ¼sðq,fðxÞ,mAÞr0 if IU

ðxÞ ¼ | ðfðxÞrqÞ,

sðq,fðxÞ,mUÞ40 if IUðxÞa| ðfðxÞ�qÞ:

(ð11Þ

Given this formulation of the achievement scalarizing func-tion, we can now prove the following result:

Theorem 1. Let q be a reference point, and let mUi ,mA

i 40ði¼ 1, . . . ,kÞ, be components of two vectors of weights. Let s be the

achievement scalarizing function defined in (5). Then, vector xn is an

optimal solution of problem (7) if and only if xn is an optimal solution

of problem (3), taking mi ¼ mAi 40 ði¼ 1, . . . ,kÞ, if q is achievable, and

mi ¼ mUi 40 ði¼ 1, . . . ,kÞ, if q is unachievable.

Proof. Let us distinguish two cases:

1.

Let us suppose that q is achievable. Therefore

there exists xAS such that fðxÞrq:

Then, relation (11) implies

sðq,fðxÞ,mU ,mAÞ ¼ sðq,fðxÞ,mAÞr0 and

sðq,fðxnÞ,mU ,mAÞ ¼ sðq,fðxnÞ,mAÞr0)

fðxnÞrq

Consequently

minxAS

sðq,fðxÞ,mU ,mAÞ ¼ minx A S

fðxÞr q

sðq,fðxÞ,mU ,mAÞ

¼ minx A S

fðxÞr q

sðq,fðxÞ,mAÞ ¼minxAS

sðq,fðxÞ,mAÞ:

Thus, the theorem is proved for the case when q is achievable.

2. Let us now suppose that q is unachievable. Therefore

for all xAS, fðxÞ�q:

Given relation (11), we have

for all xAS, sðq,fðxÞ,mU ,mAÞ ¼ sðq,fðxÞ,mUÞ40 and

minxAS

sðq,fðxÞ,mU ,mAÞ ¼minxAS

sðq,fðxÞ,mUÞ:

This completes the proof of the theorem. &

Combining Theorem 1 and the basic properties of the achieve-ment scalarizing function (2) (see, e.g. [11,25,26]), we can deducethat if r40 and if xn is an optimal solution of problem (7), thenxn is a properly efficient solution of (1). If r¼ 0, then xn is aweakly efficient solution of (1), and it is efficient if it is the uniqueoptimal solution of (7).

Therefore, the achievement scalarizing function (5) allows usto define a priori two vectors of weights: one for achievablereference points, and another one for unachievable referencepoints. Theorem 1 proves that the appropriate vector will be usedin each case, without having to carry out any achievability test.This can be useful to overcome certain difficulties when solvingmultiobjective problems with a reference point approach. Manyof the achievement scalarizing functions that can be found in theliterature just differ in the vector of weights used. But, as shownin [12,21], the corresponding solutions can be significantlydifferent. Moreover, it is claimed in [3] that the solution preferredby the DM can be different, depending on whether the referencepoint is achievable or not. Namely, when real DMs were asked tocompare the solutions obtained using the STOM and the GUESSmethods, and when the reference point was achievable, mostDMs preferred the STOM solution, obtained by solving problem(3) with the weight vector m¼ ð1=q�znnÞ, while, when thereference point was unachievable, most DMs preferred the GUESSsolution, obtained by solving problem (3) with the weight vectorm¼ ð1=znad�qÞ. Therefore, using our new achievement scalarizingfunction, we can utilize these findings by considering the follow-ing two weight vectors in (5):

mA ¼1

q�znn, mU ¼

1

znad�q: ð12Þ

Let us give a graphical idea of the performance of our two-slope achievement scalarizing function with an unachievable andan achievable reference point. Given a reference point q, let usdenote by fS the STOM solution, by fG the GUESS solution, and byfn the solution obtained with the two-slope achievement scalar-izing function (7). In Fig. 1 we can see that for the same reference

q

Z = f (S)

f1

f2

μU1f G = f*

f Sznad

f S = f*

z**z**

f G

znadZ = f (S)

f1

f2

q

μA1

Fig. 1. Graphical idea of the two-slope achievement scalarizing function.

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–1681 1677

point and with only two objective functions, the solutionsobtained, that is, fS and fG, can be substantially different. It canbe seen that using mU and mA defined in (12), when the referencepoint is achievable, we have fn ¼ fS and when it is unachievable,we have fn ¼ fG.

Another possible application of the two-slope achievementscalarizing function can be found in [9], where preferentialweights are used. Namely, after the DM has specified her/hisreference point, (s)he assigns objective functions to classes in anincreasing order of importance for achieving the correspondingaspiration levels. This importance evaluation allows us to allocatethe k objective functions into index sets Jr which represent theimportance levels r¼ 1, . . . ,s, where 1rsrk. If rot, then achiev-ing the aspiration levels of objective functions in the index set Jr isless important than achieving aspiration levels of the objectives inJt. One objective function can only belong to one index set butseveral objectives can be assigned to the same index set Jr. Thismeans that achieving their aspiration levels is equally important.In order to take into account these DM’s preferences, differentproblems have to be solved depending on whether the referencepoint is achievable or not. In [9], an achievability test has to becarried out first, and then the appropriate problem can be solved.But we can use the two-slope achievement scalarizing function,instead, by setting

mAi ¼

1

rðznad�znnÞ, mU

i ¼r

znad�znn, ð13Þ

for iA Jr and r¼ 1, . . . ,s.

4. Differentiable case

As it was mentioned in Section 2, when problem (1) isdifferentiable, we can use a differentiable scalarized problem(4), which is equivalent to problem (3). In this way, we do notlose the differentiability properties of the original problem, andwe can use a suitable single objective solver. In a similar way, letus look for a differentiable problem equivalent to (7). Taking intoaccount Theorem 1 and formulation (4), it is evident that if q isunachievable, then problem (7) is equivalent to the followingproblem:

minimize aþrXk

i ¼ 1

ðf iðxÞ�qiÞ

subject to mUi ðf iðxÞ�qiÞra ði¼ 1, . . . ,kÞ,

xAS, ð14Þ

while if q is achievable, then problem (7) is equivalent to thefollowing problem:

minimize bþrXk

i ¼ 1

ðf iðxÞ�qiÞ

subject to mAj ðf jðxÞ�qjÞrb ðj¼ 1, . . . ,kÞ,

xAS: ð15Þ

Therefore, solving (7) is equivalent to the following procedure:

1.

Solve problem (14). If the optimal value of a is strictly positive(which implies that q is unachievable), then the optimalsolution of (14) is the optimal solution to problem (7).

2.

If the optimal value of a is non-positive (which implies that qis achievable), then solve problem (15) and the optimalsolution of (15) is the optimal solution to problem (7).

This procedure maintains the differentiability of problem (1),but in some cases (when the reference point is achievable)we have to solve two problems. Because the main aim of thetwo-slope achievement scalarizing function is to use a singleoptimization problem to solve the problem, we want to find adifferentiable formulation equivalent to problem (7). To this end,let us consider the following problem:

minimize aþdðbþrXk

i ¼ 1

ðf iðxÞ�qiÞÞ

subject to mUi ðf iðxÞ�qiÞra ði¼ 1, . . . ,kÞ,

mAj ðf jðxÞ�qjÞrb ðj¼ 1, . . . ,kÞ,

xAS,aZ0,bAR, ð16Þ

where d and r are positive scalars. Although problems (7) and(16) are not exactly equivalent, the following theorem provesthat, under mild conditions, we can make their optimal solutionsbe as close as we wish. Therefore, in practice, the differentiableformulation (16) can be used when the original problem isdifferentiable.

Theorem 2. Let us consider problem (1), and let us suppose that all

objective functions are bounded, that is, there exists a positive

number M such that 9f iðxÞ9oM , for all i¼ 1, . . . ,k and xAS. Let

mUi ,mA

i 40 ði¼ 1, . . . ,kÞ, be components of two vectors of weights such

thatP

i ¼ 1,...,kmUi ¼

Pi ¼ 1,...,kmA

i ¼ 1. Let ðxðdÞ,aðdÞ,bðdÞÞ be the opti-

mal solution of problem (16). Then, the following results hold:

(i)

If q is achievable, then ðxðdÞ,bðdÞÞ is an optimal solution to

problem (15) with r¼ r.

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–16811678

(ii)

If q is unachievable, then for every e40, there exists a value

d40 such that

9aðdÞ�an9re,

where ðxn,anÞ is an optimal solution for problem (14) with

r¼ dr.

Proof. Let us prove each item separately.

(i)

First, let us suppose that q is achievable. Let us prove that, inthis case, aðdÞ ¼ 0. To this end, let us suppose that aðdÞ40,and let ðxn,bn

Þ be an optimal solution of problem (15). Giventhat q is achievable, we have fðxnÞrq, and therefore,ðxn,0,bn

Þ is a feasible solution for (16). Thus, we have

d � ðbðdÞþrXk

i ¼ 1

ðf iðxðdÞÞ�qiÞÞoaðdÞþd

� ðbðdÞþrXk

i ¼ 1

ðf iðxðdÞÞ�qiÞÞrd � ðbnþr

Xk

i ¼ 1

ðf iðxnÞ�qiÞÞ

and therefore

bðdÞþrXk

i ¼ 1

ðf iðxðdÞÞ�qiÞobnþr

Xk

i ¼ 1

ðf iðxnÞ�qiÞ,

which contradicts the fact that ðxn,bnÞ is an optimal solution

of (15). Thus, we have aðdÞ ¼ 0 and bðdÞr0, and this impliesthat ðxðdÞ,bðdÞÞ is an optimal solution of the followingproblem:

minimize dðbþrXk

i ¼ 1

ðf iðxÞ�qiÞÞ

subject to mAj ðf jðxÞ�qjÞrb ðj¼ 1, . . . ,kÞ,

xAS ð17Þ

which, obviously, is equivalent to (15) with r¼ r. Therefore,ðxðdÞ,bðdÞÞ is an optimal solution of problem (15) with r¼ r.

(ii)

Let us now suppose that q is unachievable. Then, we have

aðdÞ40, bðdÞ40, an40:

Let us define

bn¼ max

i ¼ 1,...,kfmA

i ðf iðxnÞ�qiÞg40

and

M¼maxfM , maxi ¼ 1,...,k

9qi9g:

Let us consider

d¼mine

4rkM,e

4Mmin

i

mUi

mAi

( )( ): ð18Þ

Let j0 be the index such that

maxi ¼ 1,...,k

fmAi ðf iðx

nÞ�qiÞg ¼ mAj0ðf j0ðxnÞ�qj0

Þ ¼ bn: ð19Þ

We also have

mUj0ðf j0ðxnÞ�qj0

Þran: ð20Þ

Dividing (20) by the second equality of (19) we can obtain

mUj0

mAj0

ran

bn) bnr

mAj0

mUj0

an: ð21Þ

From (18) and (21), we have

dre

4Mmini

mUi

mAi

( )

) dbnre

4Mmin

i

mUi

mAi

( )bnr

e4M

mini

mUi

mAi

( )�mA

j0

mUj0

an

re

4M

mUj0

mAj0

mAj0

mUj0

an ¼e

4Man

and thus

anþdbnranþe

4Man ¼ 1þ

e4M

� �an: ð22Þ

Given that ðxn,anÞ is an optimal solution of (14) with r¼ dr,we have

anþdrXk

i ¼ 1

ðf iðxnÞ�qiÞraðdÞþdr

Xk

i ¼ 1

ðf iðxðdÞÞ�qiÞ: ð23Þ

Similarly, since ðxðdÞ,aðdÞ,bðdÞÞ is an optimal solution of (16),the following relation holds:

aðdÞþd bðdÞþrXk

i ¼ 1

ðf iðxðdÞÞ�qiÞ

!

ranþd bnþr

Xk

i ¼ 1

ðf iðxnÞ�qiÞ

!: ð24Þ

From (22) to (24), we can deduce

anþdrXk

i ¼ 1

ðf iðxnÞ�qiÞraðdÞþdr

Xk

i ¼ 1

ðf iðxðdÞÞ�qiÞ

raðdÞþd bðdÞþrXk

i ¼ 1

ðf iðxðdÞÞ�qiÞ

!

ranþd bnþr

Xk

i ¼ 1

ðf iðxnÞ�qiÞ

!

r 1þe

4M

� �anþdr

Xk

i ¼ 1

ðf iðxnÞ�qiÞ:

Therefore

anraðdÞþdrXk

i ¼ 1

ðf iðxðdÞÞ�f iðxnÞÞr 1þ

e4M

� �an

) aðdÞ�anþdrXk

i ¼ 1

ðf iðxðdÞÞ�f iðxnÞÞ

r 1þe

4M

� �an�an ¼

e4M

an

and thus

aðdÞ�anre

4Manþdr

Xk

i ¼ 1ðf iðxðdÞÞ�f iðx

nÞÞ: ð25Þ

On the one hand

an ¼ maxi ¼ 1,...,k

fmUi ðf iðx

nÞ�qiÞgr2M ð26Þ

and on the other hand, from (18), we have

drXk

i ¼ 1

ðf iðxðdÞÞ�f iðxnÞÞr

e4rkM

rkðMþMÞ ¼ e=2: ð27Þ

From (25) to (27), we deduce

aðdÞ�anre

4M2Mþe=2¼ e:

Similarly

an�aðdÞrdrXk

i ¼ 1

ðf iðxðdÞÞ�f iðxnÞÞ

re

4rkMrkðMþMÞ ¼ e=2oe:

TableGener

Test

prob

[15]

[15]

[15]

[15]

[15]

[15]

[15]

[15]

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–1681 1679

Therefore, we have

9aðdÞ�an9re

and this completes the proof. &

It must be noted that the bounding condition on the values ofthe objective functions is usually fulfilled in real problems, and ifthe bound M is hard to derive, the users can always give a largeenough value. On the other hand, given that any strictly positivevalue of r suffices in (14) to obtain properly efficient solutions,condition r¼ dr is not restrictive in practice. Therefore, Theorem2 shows that solving problem (16) yields, in practice, optimalsolutions for problem (7).

5. Computational tests

In order to evaluate the behavior and potential of the two-slope achievement scalarizing function proposed, we have carriedout a series of computational tests. For these tests, we have usedthe 12 test problems described in [15] (referred to as [15]-1 to[15]-12), the test problem used in [4], and the test problemproposed in [9].

The aim of these tests is to show the usefulness of the two-slope achievement scalarizing function in a concrete case whendifferent weights are considered for achievable and unachievablereference points. Namely, we will use the vectors of weights givenin expression (12) and we will compare the two-slope schemewith an equivalent scheme. However, it is important to keep inmind that the two-slope achievement scalarizing function can beused for any pair of weight vectors, like for example, thepreferential weights described in [9].

Therefore, for each of the test problems considered, tenreference points have been randomly generated between itscorresponding ideal and nadir points (occasionally, some morereference points have been generated in order to obtain a similarnumber of achievable and unachievable reference points). For

1al case—computational tests.

lem

Two-slope

evals

Minmax

evals

Difference ach.

values

% Achiev. ref.

points

-1 2623 3463 5� 10�9 40

-2 15,513 21,568 2:2� 10�7 40

-3 1623 1946 0 30

-5 25,584 41,083 5:5� 10�7 60

-6 25,015 38,352 3:5� 10�7 50

-8 343,450 541,679 1:3� 10�7 60

-10 453,454 681,942 3:5� 10�5 50

-12 44,908 71,429 7:5� 10�9 50

Table 2Differentiable case—computational tests.

Test problem Two-slope evals Minmax evals

[15]-4 8.2 11.9

[15]-7 9.2 13.8

[15]-9 5.9 14.7

[15]-11 10.5 12.6

[4] 13.3 16.2

[9] 6.4 8.7

each reference point, we have obtained the corresponding effi-cient solution, following two schemes:

�

Two-slope scheme: minimizing the two-slope achievementscalarizing function (5), using the weights given in expression(12). � Minmax scheme: first, the GUESS solution is generated. If the

reference point is unachievable, then the process ends. Other-wise, the STOM solution is obtained.

According to the results obtained in Sections 3 and 4, thesetwo schemes can be regarded as equivalent. Therefore, in order tostudy which of them is computationally more efficient, we willcompare them in terms of the number of function evaluationsrequired to reach an optimal solution in each test problem. Wehave divided these tests in two groups. The convex problems havebeen solved using a differentiable scheme, while the rest havebeen solved using the general (non-differentiable) scheme.

5.1. General case

For these problems, the computational implementation hasbeen carried out in the Cþþ language using the LGO Solver Systemfor Continuous Global Optimization [19,20], for the solution of thesingle objective problems. This solver uses Branch and Bound þLocal Search for some problems, and Global Adaptive RandomSearch þ Local Search for others. In this case, the two-slopescheme consists of solving problem (7).

In the general (non-differentiable) case, the results obtainedare shown in Table 1. The first column contains the test problemreference. The next two columns show the number of functionevaluations for the two-slope scheme and for the minmaxscheme, respectively, required to solve each problem. It must bepointed out that these numbers are the average number ofevaluations for all the problems solved using the differentreference points generated. Finally, the last two columns containthe average differences of the optimal values of the achievementscalarizing functions used in both schemes (that is, it gives anidea of the accuracy of the solutions obtained), and the percen-tage of reference points that happened to be achievable.

We can observe that the optimal solutions of both schemes arevery similar since the differences between the optimal values(Difference ach. values) are very small (the greatest being3:5� 10�5). However, the average number of function evaluationswhen using the two-slope scheme is significantly smaller thanthat of the minmax scheme (for example, 2623–3463 in the firstproblem). In fact, the evaluations have been reduced by about31%, on the average. In addition, the percentage of achievablereference points is never greater than 60%. In the minmaxscheme, having an achievable reference point implies solvingtwo optimization problems instead of one. So, it is important tohave a similar number of achievable and unachievable referencepoints if we do not want the results to be biased.

Difference ach. values % Achiev. ref. points

3:2� 10�7 60

0 40

0 60

9:6� 10�6 50

2� 10�8 40

0 50

M. Luque et al. / Computers & Operations Research 39 (2012) 1673–16811680

To conclude, we can say that, in the general case, whendifferent weights are used for achievable and unachievablereference points, the two-slope achievement scalarizing functionrequires fewer function evaluations to reach the same optimalsolution in comparison to the minmax scheme.

5.2. Differentiable case

This implementation has also been carried out in Cþþ lan-guage, and we have considered the convex test problems since wehave used the NAG (Numerical Algorithms Group) library [17] tosolve the single objective problems. Given that now we have adifferentiable formulation, the two-slope scheme consists ofsolving problem (16).

Table 2 shows the results in the differentiable case with thesame columns as described for the general case.

First, one can note that the numbers of function evaluations inTable 2 are significantly lower in magnitude than those of Table 1.The reason of this behavior is that the LGO library used to solvethe single objective problems is based on a heuristic approachwhich requires, in general, a high number of function evaluations.On the other hand, the NAG library uses a sequential quadraticprogramming (SQP) method to solve the intermediate problemsof the differentiable case and the SQP method usually requiresfew a evaluations to find an optimal solution in comparison withany global solver, because derivative information is used.

Despite of the magnitudes of numbers, in this table we canobserve the same types of results as in Table 1. That is, theoptimal solutions found by both schemes are really close (thehighest difference being 9:6� 10�6, and some of them beingpractically 0), while the number of function evaluations requiredby the two-slope scheme is, again, significantly lower than thatneeded by the minmax scheme in all the test problems (theaverage reduction percentage being again about 31%).

Therefore, we can conclude that, when we wish to usedifferent weights for achievable and unachievable referencepoints, the use of the two-slope achievement scalarizing functionproduces the same optimal solutions, but requiring less functionevaluations, for both differentiable and non-differentiable pro-blems. This demonstrates the usefulness and the potential of ournew achievement scalarizing function from a computational pointof view.

6. Conclusions

Reference point based methods work by optimizing a so-calledachievement scalarizing function, which depends on the referencepoint, given by the DM, and on a vector of weights, whose role canvary from purely normalizing parameters to fully preferentialones. Some empirical studies have concluded that the DMs prefersolutions obtained with certain weights for achievable referencepoints, and those obtained with other weights for unachievableones. Even for purely normalizing weights, the interpretation ofthe effect of the weights also depends on the achievability of thereference point. Therefore, it is a sensible policy to use differentvectors of weights, depending on whether the reference point isachievable or not. To date, it was necessary to solve two problemsin order to do this (while the first problem solved acts as anachievability test). In this paper, we have developed a two-slopeachievement scalarizing function, which uses two vectors ofweights simultaneously. We have proved that, in practice, oneof the vectors of weights is used for achievable reference points,and the other one for unachievable ones. Therefore, it suffices tosolve a single optimization problem to get the solution we wish.

For differentiable problems, we have also developed a formu-lation that maintains the original differentiability of the problem,and whose optimal solution, under mild conditions, is as close aswe wish to the optimal solution of the two-slope achievementscalarizing function. Therefore, this formulation allows us to usesolvers designed for differentiable problems in this case.

Finally, a series of computational tests show that the solutionsobtained using the new two-slope achievement scalarizing func-tion are very accurate, while the number of function evaluationsexperiments a significant decrease. These results prove theusefulness of this formulation, if we wish to use different weightsfor different reference points.

A future search direction is to investigate the use of our newachievement scalarizing function in the framework of interactivemultiobjective optimization methods, as well as its hybridizationwith evolutionary algorithms. Naturally, testing with real-lifeproblems is in order as well.

Acknowledgments

This research was partly supported by the Andalusian RegionalMinistry of Innovation, Science and Enterprises (PAI group SEJ-445 and P09-FQM-5001), by the Ministry of Science and Innova-tion of Spain (Research Project MTM2010-14992) and by theAcademy of Finland (grant number 128495).

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