Expert Systems with Applications 39 (2012) 869–880

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Expert Systems with Applications

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

Uncertain induced aggregation operators and its applicationin tourism management

José M. Merigó a,⇑, Anna M. Gil-Lafuente a, Onofre Martorell b

a Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spainb Department of Business Administration, University of Balearic Islands, Palma de Mallorca, Spain

a r t i c l e i n f o

Keywords:Interval numbersWeighted averageOWA operatorAggregation operatorsTourism managementMulti-person decision-making

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.07.085

⇑ Corresponding author. Tel.: +34 93 402 19 62; faxE-mail addresses: jmerigo@ub.edu (J.M. Merigó

Lafuente), onofre.martorell@uib.es (O. Martorell).

a b s t r a c t

We develop a new decision making approach for dealing with uncertain information and apply it in tour-ism management. We use a new aggregation operator that uses the uncertain weighted average (UWA)and the uncertain induced ordered weighted averaging (UIOWA) operator in the same formulation. Wecall it the uncertain induced ordered weighted averaging – weighted averaging (UIOWAWA) operator.We study some of the main advantages and properties of the new aggregation such as the uncertainarithmetic UIOWA (UA-UIOWA) and the uncertain arithmetic UWA (UAUWA). We study its applicabilityin a multi-person decision making problem concerning the selection of holiday trips. We see that depend-ing on the particular type of UIOWAWA operator used, the results may lead to different decisions.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The weighted average (WA) is one of the most common aggrega-tion operators found in the literature. It can be used in a wide rangeof problems including statistics, economics and engineering. An-other interesting aggregation operator is the ordered weightedaveraging (OWA) operator (Yager, 1988). The OWA operator pro-vides a parameterized family of aggregation operators that rangefrom the maximum to the minimum. For further reading on theOWA operator and some of its applications, refer to Ahn (2009),Beliakov, Pradera, and Calvo (2007), Chang and Wen (2010), Cheng,Wang, and Wu (2009), Kacprzyk and Zadrozny (2009), Liu, Cheng,Chen, and Chen (2010), Merigó (2010a, 2010b), Merigó, Casanovas,and Martínez (2010), Merigó and Gil-Lafuente (2008, 2010,2011a), Wei (2010a), Xu (2009, 2010), Xu and Da (2003), Xu andYager (2010), Yager (1993, 1998, 2009), and Yager and Kacprzyk(1997), Zhao, Xu, Ni, and Liu (2010), Zeng & Su (2011), Zhou andChen (2010). An interesting generalization of the OWA operator isthe induced OWA (IOWA) operator (Yager & Filev, 1999). Its mainadvantage is that it deals with complex reordering processes inthe aggregation by using order inducing variables. Since its intro-duction it has been studied by a lot of authors. For example, Merigóand Gil-Lafuente (2009) developed a generalization by using gener-alized and quasi-arithmetic means. Chiclana, Herrera-Viedma,Herrera, and Alonso (2004) and Xu and Da (2003) introduced ageometric version and applied it in group decision making. Wu, Li,

ll rights reserved.

: +34 93 403 98 82.), amgil@ub.edu (A.M. Gil-

Li, and Duan (2009) presented a continuous geometric version.Merigó and Casanovas (2009) applied it in a decision making prob-lem with Dempster–Shafer belief structure. Chen and Chen (2003)and Merigó and Casanovas (2010) studied the use of fuzzy numbersin the IOWA operator. For further reading, see Merigó (2011),Merigó, Gil-Lafuente, and Gil-Aluja (2011a, 2011b), Tan and Chen(2010), Wei (2010b), Wei, Zhao, and Lin (2010) and Yager (2003).

Usually, when using these approaches it is considered thatthe available information are exact numbers. However, thismay not be the real situation found in the specific problem con-sidered. Sometimes, the available information is vague or impre-cise and it is not possible to analyze it with exact numbers.Therefore, it is necessary to use another approach that is ableto assess the uncertainty such as the use of interval numbers.By using interval numbers we can consider a wide range of pos-sible results between the maximum and the minimum. Note thatin the literature, there are a lot of studies dealing with uncertaininformation represented in the form of interval numbers(Merigó, López-Jurado, Gracia, & Casanovas, 2009; Merigó &Wei, 2011; Wei, 2009; Xu & Da, 2002; Xu & Da, 2003; Xu &Yager, 2010).

Recently, some authors have tried to unify the WA and the OWAin the same formulation. It is worth noting the work developed byTorra (1997) with the introduction of the weighted OWA (WOWA)operator and the work of Xu and Da (2003) concerning the hybridaveraging (HA) operator. Both models arrived to a partial unifica-tion between the OWA and the WA because both concepts were in-cluded in the formulation as particular cases. However, as it hasbeen studied by Merigó (2008), these models seem to be a partialunification but not a real one because they can unify them but they

870 J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880

cannot consider how relevant these concepts are in the specificproblem considered. For example, in some problems we may preferto give more importance to the OWA operator because we believethat it is more relevant and vice versa. This problem is solved withthe ordered weighted averaging – weighted averaging (OWAWA)operator (Merigó, 2008, 2010b).

In this paper, we present a new approach to unify the IOWAoperator with the WA when the available information is uncertainand can be assessed with interval numbers. We call it the uncertaininduced ordered weighted averaging – weighted averaging (UIOW-AWA) operator. The main advantage of this approach is that it uni-fies the OWA and the WA taking into account the degree ofimportance that each concept has in the formulation and consider-ing that the information is given with interval numbers. Thus, weare able to consider situations where we give more or less impor-tance to the UIOWA and the UWA depending on our interests andthe problem analyzed. Furthermore, by using the UIOWAWA, weare able to use a complex reordering process in the OWA operatorin order to represent complex attitudinal characters. We also studydifferent properties of the UIOWAWA operator and further gener-alizations such as the mixture UIOWAWA (MUIOWAWA) operatorand the infinitary UIOWAWA operator. Moreover, we discuss sev-eral particular cases including the UWA, the UIOWA, the uncertainarithmetic UIOWA (UA-UIOWA) and the uncertain arithmetic UWA(UAUWA).

We also analyze the applicability of the new approach and wesee that it is possible to develop an astonishingly wide range ofapplications. For example, we can apply it in a lot of problemsregarding statistics, economics, engineering and decision theory.In this paper, we focus on a decision making problem concerningtourism management. We develop a multi-person decision makingproblem where a decision maker wants to select an optimal holi-day trip. The main advantage of the UIOWAWA in these problemsis that it is possible to consider the subjective probability (or thedegree of importance) and the attitudinal character of the decisionmaker at the same time.

This paper is organized as follows. In Section 2 we revise somebasic concepts and in Section 3 we suggest a new method for deal-ing with uncertain weights in the aggregation process. In Section 4we present the new aggregation operator. Section 5 analyzes theapplicability of the UIOWAWA operator including an applicationin tourism management and in Section 6 we summarize the mainresults of the paper.

2. Preliminaries

In this Section we briefly review the interval numbers, theUWA, the UOWA operator, the UIOWA operator and the OWAWAoperator.

2.1. The interval numbers

The interval numbers (Bachs, Merigó, López-Jurado, & Gracia-Ramos, 2008; Moore, 1966) are a very useful and simple techniquefor representing uncertainty. They have been used in a wide rangeof applications and can be defined as follows.

Definition 1. Let a = [a1, a2] = {x|a1 6 x 6 a2}, then, a is called aninterval number. Note that a is a real number if a1 = a2.

The interval numbers can be expressed in different forms. For

example, assume a 4-tuple [a1, a2, a3, a4], that is to say, a quadru-plet, and let a1 and a4 represent the minimum and the maximumof the interval number, respectively, and a2 and a3 represent theinterval that it is most possible to occur. Note thata1 6 a2 6 a3 6 a4. If a1 = a2 = a3 = a4, then the interval number is

an exact number. If a2 = a3, it is a triplet, and if a1 = a2 anda3 = a4, it is a simple 2-tuple interval number.

In the following, we review some basic operations. Let A and Bbe two triplets, where A = [a1, a2, a3] and B = [b1, b2, b3].

1. A + B = [a1 + b1, a2 + b2, a3 + b3].2. A � B = [a1 � b3, a2 � b2, a3 � b1].3. A � k = [k � a1, k � a2, k � a3], for k > 0.4. A � B = [min(a1 � b1, a1 � b3, a3 � b1, a3 � b3), a2 � b2, max(a1 �

b1, a1 � b3, a3 � b1, a3 � b3)], for R.5. A � B = [min(a1 � b1, a1 � b3, a3 � b1,

a3 � b3), a2 � b2, max(a1 � b1, a1 � b3, a3 � b1, a3 � b3)], for R.

Sometimes, the ranking of the intervals is difficult because it isnot clear which interval number is higher, so we must establish anadditional criterion for ranking the interval numbers. For simplicity,we follow the following method throughout the paper. For 2-tuples,calculate the arithmetic mean of the interval, with (a1 + a2)/2. For3-tuples and above, calculate a weighted average that yields moreimportance to the central values. That is, for 3-tuples, (a1 +4a2 + a3)/6. For 4-tuples, we calculate: (a1 + 4a2 + 4a3 + a4)/10, andso on. In the case of a tie between the intervals, we select the intervalwith the lowest difference, i.e., (a2 � a1). For 3-tuples and aboveodd-tuples, we select the interval with the highest central value.Note that for 4-tuples and above even-tuples, we must calculatethe average of the central values following the initial criteria.

The main advantage of this method is that we can reduce theinterval number into a representative and exact number of theinterval. To understand the usefulness of this method, we presenta simple example.

Example 1. Assume we want to rank the following intervalnumbers: A = (38, 47, 57), B = (39, 45, 55) and C = (42, 46, 51). Ini-tially, it is not clear which is higher. Obviously, we can use a widerange of methods depending on the importance we want to give toeach tuple of the interval. We assume in this paper a ranking basedon (a1 + 4a2 + a3)/6. Thus, we assume that the central value is moreimportant than the extreme values. We convert the triplet to exactnumbers. Thus:

A ¼ ð38þ 4� 47þ 57Þ=4 ¼ 47:16:

B ¼ ð39þ 4� 45þ 55Þ=4 ¼ 45:66:

C ¼ ð42þ 4� 46þ 51Þ=4 ¼ 46:16:

With these results, we can reorder the interval numbers such thatA > C > B.

Note that other operations and ranking methods could be stud-ied (Bachs et al., 2008; Moore, 1966) but in this paper we focus onthose discussed above.

2.2. The uncertain weighted average

The uncertain weighted average (UWA) is an extension of theweighted average for situations in which the available informationis uncertain and can be assessed using interval numbers. It can bedefined as follows.

Definition 1. Let X be the set of interval numbers. An UWAoperator of dimension n is a mapping UWA: Xn ? X that has anassociated weighting vector W of dimension n with ~wi 2 ½0; 1� andPn

i¼1 ~wi ¼ 1 such that:

UWAð~a1; ~a2; . . . ; ~anÞ ¼Xn

i¼1

~wi~ai; ð1Þ

where ~ai is an interval number.

J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880 871

Note that in Section 3, we explain how to address uncertainweights ~wi so the sum of these weights is equal to 1. The same con-dition applies for all the other uncertain aggregation operators thatuse uncertain weights.

2.3. The uncertain OWA operator

The UOWA operator (Xu & Da, 2002) is an extension of theOWA operator that uses interval numbers in the arguments thatwill be aggregated. It provides a parameterized family of aggre-gation operators that include the uncertain maximum, theuncertain minimum and the UA, among others. It can be definedas follows.

Definition 2. Let X be the set of interval numbers. An UOWAoperator of dimension n is a mapping UOWA: Xn ? X that has anassociated weighting vector W of dimension n with

Pnj¼1 ~wj ¼ 1

and ~wj 2 ½0; 1� and such that:

UOWAð~a1; ~a2; . . . ; ~anÞ ¼Xn

j¼1

~wj~bj; ð2Þ

where ~bj is the jth largest of the ~ai, and each ~ai is an interval number.Sometimes it is not clear which interval number is higher in the

reordering process of the UOWA operator. In order to solve thisproblem, we recommend the method explained in Section 2.1 forranking interval numbers.

2.4. The uncertain induced OWA operator

The uncertain induced OWA operator (Xu, 2006) is an exten-sion of the OWA operator that uses the main characteristics oftwo well known aggregation operators: the induced OWA andthe uncertain OWA operator. Therefore, it uses interval numbersfor representing the uncertain information and a complex reor-dering process that it is based on order inducing variables. Itcan be defined as follows:

Definition 3. Let X be the set of interval numbers. An UIOWAoperator of dimension n is a mapping UIOWA: Xn �Xn ? X thathas an associated weighting vector W of dimension n withwj 2 ½0; 1� and

Pnj¼1 ~wj ¼ 1, such that:

UIOWAðhu1; ~a1i; . . . ; hun; ~aniÞ ¼Xn

j¼1

wjbj; ð3Þ

where bj is the ~ai value of the UIOWA pair hui; ~aii having the jth larg-est ui, ui is the order inducing variable and each ~ai is an intervalnumber.

From a generalized perspective of the reordering step it ispossible to distinguish between descending (DUIOWA) andascending (AUIOWA) orders. Note that in this case, it is not nec-essary to compare interval numbers because the reordering stepis developed with order inducing variables. The only case wherewe need to compare interval numbers is in the final result. Fordoing this, we recommend to use the criteria explained in Sec-tion 2.1.

2.5. The OWAWA operator

The ordered weighted averaging – weighted average (OWAWA)operator is a new model that unifies the OWA operator and theweighted average in the same formulation. Its main advantage isthat it can unify both concepts considering the degree of impor-tance that each one has in the aggregation process. It can be de-fined as follows.

Definition 4. An OWAWA operator of dimension n is a mappingOWAWA: Rn ? R that has an associated weighting vector W ofdimension n such that wj 2 [0, 1] and

Pnj¼1wj ¼ 1, according to the

following formula:

OWAWA ða1; . . . ; anÞ ¼Xn

j¼1

v jbj; ð4Þ

where bj is the jth largest of the ai, each argument ai has an associ-ated weight (WA) vi with

Pni¼1v i ¼ 1 and vi 2 [0, 1], v j ¼ bwjþ

ð1� bÞv j with b 2 [0, 1] and vj is the weight (WA) vi ordered accord-ing to bj, that is, according to the jth largest of the ai.

As we can see, if b = 1, we get the OWA operator and if b = 0, theWA. The OWAWA operator accomplishes similar properties thanthe usual aggregation operators. Note that we can distinguish be-tween descending and ascending orders, extend it by using mix-ture operators, and so on.

3. A method for dealing with uncertain weights in uncertainaggregation operators

In this section, we introduce a new method for dealing withuncertain weights in aggregation problems in which informationis uncertain. This method is a particular case of a more generalapproach. The main characteristics addressed here are asfollows:

� The weighting vector must sum to 1. If not, we must normalizeit to be consistent with usual aggregation methods.� In order to assess the weighting vector, we must have a method

for converting the interval numbers into exact numbers. Notethat we employ the method discussed in Section 2.1.� We always use interval weights (and interval arguments) to

provide the most complete information during the aggregationprocess. We only reduce the interval weights into exact num-bers to normalize the information and to obtain the final result,if necessary.� We assume that during the normalization process, the interval

weights can be less than or greater than the bounds becausethe weights were not normalized in the beginning due touncertainty.

The procedure that we suggest in this paper for intervalweights in uncertain aggregation operators can be summarizedas follows.

Step 1: Calculate the sum of all weights.Step 2: Convert the result into a representative exact number. Inthis paper, we use a = (a1 + 4a2 + a3)/6. The result obtained willbe the coefficient used for normalizing the initial weightingvector.Step 3: If a is 1, the initial weights are already normalized. If not,divide all initial weights ~wi or ~wj by a, that is, ~wi

a or~wj

a .Step 4: Use the weights ~wi

a or~wj

a in the uncertain aggregationprocess.Step 5: The result obtained in the aggregation process is thefinal uncertain aggregated result presented in the form of aninterval number.

Additionally, it is possible to convert this final result into a rep-resentative exact number by setting a = (a1 + 4a2 + a3)/6. A generalway to do this that includes the previous method is by using aweighted average a ¼

Pni¼1wiai, where ai is the ith value of the

interval number, and wi is the weight given to each value of theinterval. Note that this weighted average is a general expressionthat can be used in Step 2.

872 J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880

Example 2. Assume the following interval weights for use in anaggregation process: W = (w1 = (0.2, 0.3, 0.4), w2 = (0.4, 0.5, 0.6),w3 = (0.3, 0.4, 0.5)).

Step 1: We calculate the sum of the weighting vector W

W ¼ ð0:2;0:3;0:4Þ þ ð0:4; 0:5; 0:6Þ þ ð0:3;0:4;0:5Þ¼ ð0:9;1:2;1:5Þ:

Step 2: We convert the interval weighting vector into a repre-sentative exact number

W ¼ 0:9þ 4� 1:2þ 1:56

¼ 1:2:

Step 3: As the sum is not 1, we normalize all initial weights bydividing them by 1.2

w1 ¼ð0:2;0:3;0:4Þ

1:2¼ ð0:166; 0:25;0:333Þ:

w2 ¼ð0:4;0:5;0:6Þ

1:2¼ ð0:333; 0:416;0:5Þ:

w3 ¼ð0:3;0:4;0:5Þ

1:2¼ ð0:25;0:333;0:416Þ:

Step 4: Now we use these weights in the uncertain aggregationprocess.To prove that these weights are consistent with usual aggrega-tions, we prove that the sum of the new weighting vector isequal to 1. Thus, we calculate the sum of the new weightingvector

W� ¼ ð0:166;0:25;0:333Þ þ ð0:333;0:416; 0:5Þþ ð0:25;0:333;0:416Þ ¼ ð0:75;1;1:25Þ:

Now we convert the interval obtained into a representative exactnumber, with (0.75 + 4 � 1 + 1.25)/6 = 1. As we can see, the sumof the weighting vector is equal to 1 in the particular case that weuse a representative exact number.

Moreover, it is worth noting that there are many other methodsthat could be considered depending on the assumptions we makein this analysis. Especially, when the results obtained in the previ-ous approach do not sum to 1 in the central value, we may prefer toconsider another approach.

For example, we could use a method where the normalizationprocess consists in normalizing the weights w2h using w2hPm

h¼1w2h

,

where w2h is the hth central value of the weighting vector, andthe w1h and w3h using w1hPm

h¼1

w1hþw3h2ð Þ and w3hPm

h¼1

w1hþw3h2ð Þ. Note that

according to this method, we always assume that the sum of thecentral values of the weighting vector is equal to 1.

These and other methods will be considered in more detail infuture research. Note that it is straightforward to implement thesemethods when using interval probabilities because the probabilitycan be treated as a weighted average.

4. The uncertain induced OWAWA operator

In this section we present the UIOWAWA operator. First, webriefly discuss some previous models that already used theOWA and the WA in the same formulation and see how theyshould be formulated with uncertain induced aggregation opera-tors. Second, we analyze the main aspects of the UIOWAWAoperator. Third, we analyze several measures for characterizingthe weighting vector and fourth, we study several families ofUIOWAWA operators.

4.1. Analyzing previous models dealing with the weighted average andOWA operator

Note that some previous models already considered the possi-bility of using OWA operators and WAs in the same formulation.The main models are the weighted OWA (WOWA) operator (Torra,1997; Torra & Narukawa, 2007) and the hybrid averaging (HA)operator (Xu & Da, 2003). Although they seem to be a good ap-proach, they are not so complete than the UIOWAWA because theycan unify OWAs and WAs in the same model but they can not takein consideration the degree of importance of each concept in theaggregation process. Moreover, in some particular cases we alsofind inconsistencies (Merigó, 2008). Note that in this case, wewould be talking about the uncertain induced hybrid averaging(UIHA) operator (Merigó & Casanovas, 2010; Merigó et al., 2009)and the uncertain induced weighted OWA (UIWOWA) operator.Other methods that could be considered are the concept of imme-diate probability (Engemann, Filev, & Yager, 1996; Merigó, 2010;Yager, Engemann, & Filev, 1995) applied to the WA that we couldcall in this case, the uncertain induced immediate WA (UII-WA).The UIWOWA operator can be defined as follows.

Definition 5. Let P and W be two weighting vectors of dimension n[P = (p1, p2, . . . , pn)], [W = (w1, w2, . . . , wn)], such that pi 2 [0, 1] andPn

i¼1pi ¼ 1, and wj 2 [0, 1] andPn

j¼1wj ¼ 1. In this case, a mappingUIWOWA: Xn �Xn ? X is an UIWOWA operator of dimension n if:

UIWOWAðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ ¼Xn

i¼1

xi~arðiÞ ð5Þ

where {r(1), . . . , r(n)} is a permutation of {1, . . . , n} such thatur(i�1) P ur(i) for all i = 2, . . . , n, and the weight xi is defined as:

xi ¼ w�Xj6i

prðjÞ

!�w�

Xj<i

prðjÞ

!; ð6Þ

with w⁄ a monotone increasing function that interpolates the points(i/n,

Pj6iwj) together with the point (0, 0). w⁄ is required to be a

straight line when the points can be interpolated in this way.The UIHA operator (Merigó, 2008; Merigó et al., 2009) can be

defined as follows.

Definition 6. An UIHA operator of dimension n is a mapping UIHA:Xn �Xn ? X that has an associated weighting vector W ofdimension n with wj 2 [0, 1] and

Pnj¼1wj ¼ 1, such that:

UIHAðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ ¼Xn

j¼1

wjbj; ð7Þ

where bj is the ai value (ai ¼ nxi~ai, i = 1, 2, . . . , n) of the IOWA pairhui; ~aii having the jth largest ui, ui is the order-inducing variable,x = (x1, x2, . . . , xn) is the weighting vector of the ~ai, withxi 2 [0, 1] and

Pni¼1xi ¼ 1.

As we have mentioned before, there are other methods thatcould be considered. Especially, we also want to consider the con-cept of immediate probability. Note that we give a definition wherewe extend the immediate probability to the case where we useWAs instead of probabilities. Thus, we can refer to this model asthe uncertain induced immediate weighted average (UII-WA). Itcan be defined as follows.

Definition 7. An UII-WA operator of dimension n is a mapping UII-WA: Xn �Xn ? X that has an associated weighting vector W ofdimension n with wj 2 [0, 1] and

Pnj¼1wj ¼ 1, such that:

UII�WAðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ ¼Xn

j¼1

v jbj; ð8Þ

J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880 873

where bj is the ai value of the UII-WA pair hui; ~aii having the jth larg-est ui, ui is the order-inducing variable, each ~ai has associated a WAvi, vj is the associated WA of bj, and v j ¼ ðwjv j=

Pnj¼1wjv jÞ.

As mentioned before, these approaches are useful in some par-ticular situations but they cannot consider the degree of impor-tance that the WA and the OWA have in the aggregation. Toovercome this limitation, we present the IOWAWA operator. Thisaggregation is more robust because it unifies the WA and theOWA considering the degree of importance that each concept hasin the formulation. It can be defined as follows.

4.2. The new approach

The uncertain induced OWAWA (UIOWAWA) operator is anaggregation operator that uses the WA and OWAs in the same for-mulation. It also uses order inducing variables in order to representthe reordering process from a general point of view. Moreover, theUIOWAWA also deals with an uncertain environment that cannotbe assessed with exact numbers but it is possible to use intervalnumbers. It can be defined as follows.

Definition 8. Let X be the set of interval numbers. An UIOW-AWA operator of dimension n is a mapping UIOWAWA:Xn �Xn ? X that has an associated weighting vector W ofdimension n such that wj 2 [0, 1] and

Pnj¼1wj ¼ 1, according to

the following formula:

UIOWAWA ðhu1; ~a1i; . . . ; hun; ~aniÞ ¼Xn

j¼1

v jbj; ð9Þ

where bj is the ~aj value of the UIOWAWA pair hui; ~aji having the jthlargest ui, ui is the order inducing variable, each argument ~aj has anassociated weight vi, both represented with intervals numbers, withPn

i¼1v i ¼ 1 and vi 2 [0, 1], v j ¼ bwj þ ð1� bÞv j with b 2 [0, 1] and vj

is the weight vi ordered according to bj, that is, according to the jthlargest of the ~aj.

Note that it is also possible to formulate the UIOWAWA opera-tor separating the part that strictly affects the UIOWA operator andthe UWAs.

Definition 9. Let X be the set of interval numbers. An UIOWAWAoperator is a mapping UIOWAWA: Xn �Xn ? X of dimension n, ifit has an associated weighting vector W, with

Pnj¼1 ~wj ¼ 1 and

~wj 2 ½0; 1� and a weighting vector V, withPn

i¼1 ~v i ¼ 1 and ~v i 2 ½0; 1�,both represented with intervals numbers such that:

UIOWAWA ðhu1; ~a1i; . . . ; hun; ~aniÞ

¼ bXn

j¼1

~wjbj þ ð1� bÞXn

i¼1

~v i~ai; ð10Þ

where bj is the ~ai value of the UIOWAWA pair hui; ~aii having the jthlargest ui, ui is the inducing variable, each argument ~ai is an intervalnumber and b 2 [0, 1].

Note that we use the methodology explained in Section 3 foraddressing uncertain weights. Moreover, if b = 1, we get the UIO-WA operator and if b = 0, the UWA. It is also worth noting thatwe can also consider the parameter b as an interval number. In thiscase, we also assume the same approach explained in Section 3 fordealing with uncertain weights. However, due to the fact that weonly have two results, the process is easier.

Example 3. Assume that we want to give a degree of importanceof (40%, 50%) to the UIOWA and a (50%, 70%) to the UWA. Followingthe methodology explained in Section 3, first we sum the degreesof importance: (0.4, 0.5) + (0.5, 0.7) = (0.9, 1.2). As we can see,(0.9 + 1.2)/2 = 1.05. Now, we normalize the initial weights with1.05. Thus:

b ¼ ð0:4;0:5Þ1:05

¼ ð0:38;0:47Þ:

1� b ¼ ð0:5;0:7Þ1:05

¼ ð0:47;0:66Þ:

As we can see, if we now convert these results into exact numbers,we get: b = (0.38 + 0.47)/2 = 0.43, and 1 � b = (0.47 + 0.66)/2 = 0.57.Obviously, b + (1 � b) = 1.

In the following, we are going to give a simple example on howto aggregate with the UIOWAWA operator. For simplicity, we as-sume that the weighting vector W and V and the parameter b areexact numbers.

Example 4. Assume the following arguments in an aggregationprocess: ([20, 30], [60, 70], [40, 50], [30, 40]). Assume the followingweighting vector W = (0.2, 0.2, 0.3, 0.3) and the following subjec-tive weighting vector V = (0.4, 0.3, 0.2, 0.1). Note that for simplicity,we have already converted the uncertain weights in usual exactweights. Assume the following order inducing variables:U = (8, 4, 7, 3). In this example, we propose that the subjectiveinformation has a degree of importance of 60% while the weightingvector W a degree of 40% (b = 0.4). If we want to aggregate thisinformation by using the UIOWAWA operator, we will get thefollowing. The aggregation can be solved either with Eqs. (9) and(10). With Eq. (10) we get the following.

UIOWAWA ¼ 0:4� ð0:2� ½20;30� þ 0:2� ½40;50� þ 0:3� ½60;70�þ 0:3� ½30;40�Þ þ 0:6� ð0:4� ½20;30� þ 0:3

� ½60;70� þ 0:2� ½40;50� þ 0:1� ½30;40�Þ¼ ½37:8;47:8�:

Note that different types of interval numbers could be used inthe aggregation such as 2-tuples, triplets, quadruplets, etc.

When using interval numbers we have the additional problemof how to reorder the arguments because now we are using inter-val numbers. For simplicity, we recommend the criteria explainedin Section 2.1. Note that in the reordering of the arguments of theUIOWAWA operator, this is not a problem because they are reor-dered according to the order inducing variables. Thus, the only casewhere we have to establish a ranking is in the final results.

Note that it is possible to distinguish between the descendingUIOWAWA (DUIOWAWA) and the ascending UIOWAWA (AUIOW-AWA) operator by using wj ¼ w�n�jþ1, where wj is the jth weight ofthe DUIOWAWA and w�n�jþ1 the jth weight of the AUIOWAWAoperator.

Additionally, if the weighting vector is not normalized, i.e.,W ¼

Pnj¼1wj – 1, or V ¼

Pnj¼1v i – 1, then, the UIOWAWA operator

can be reconstructed following the methodology explained in Sec-tion 3.

If B is a vector corresponding to the ordered arguments bj, weshall call this the ordered argument vector and WT is the transposeof the weighting vector, then, the UIOWAWA operator can be ex-pressed as:

UIOWAWA ðhu1; ~a1i; . . . ; hun; ~aniÞ ¼WT B: ð11Þ

The UIOWAWA is monotonic, commutative, bounded and idempo-tent. It is monotonic because if ~ai P ei, for all ~ai, then, UIOWAWA(hu1, ~a1i, . . . , hun ~ani) P UIOWAWA (hu1, e1i, . . . , hun, eni). It is com-mutative because any permutation of the arguments has the sameevaluation. That is, UIOWAWA (hu1, ~a1i, . . . , hun, ~ani) = UIOWAWA(hu1, e1i, . . . , hun, eni), where (hu1, e1i, . . . , hun, eni) is any permuta-tion of the arguments (hu1, e1i, . . . , hun, eni). It is bounded becausethe UIOWAWA aggregation is delimitated by the uncertain mini-mum and the uncertain maximum. That is, Min{~ai} 6 UIOWAWA

874 J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880

(hu1, ~a1i, . . . , hun, ~ani) 6Max{~ai}. It is idempotent because if ~ai = a,for all ~ai, then, UIOWAWA (hu1, ~a1i, . . . , hun, ~ani) = a.

It is also worth noting that in the UIOWAWA operator we find anew semi boundary condition when we only consider the boundsof the UIOWA or the UWA. This property can be proven as follows.

Theorem 1 (Semi boundary conditions). Assume f is the UIOWAWAoperator, then:

b�Minf~aig þ ð1� bÞ �Xn

i¼1

~wi~ai 6 f ðhu1; ~a1i; . . . ; hun; ~aniÞ

6 b�Maxf~aig þ ð1� bÞ �Xn

i¼1

~wi~ai ð12Þ

Proof. Let max{~ai} = ~c, and min{~ai} = ~d, then

f ðhu1; ~a1i; . . . ; hun; ~aniÞ ¼ ~bXn

j¼1

~wjbj þ ð1� ~bÞXn

i¼1

~v i~ai

6 ~bXn

j¼1

~wj~c þ ð1� ~bÞXn

i¼1

~v i~ai

¼ ~b~cXn

j¼1

~wj þ ð1� ~bÞXn

i¼1

~v i~ai; ð13Þ

f ðhu1; ~a1i; . . . ; hun; ~aniÞ ¼ ~bXn

j¼1

~wjbj þ ð1� ~bÞXn

i¼1

~v i~ai

6 ~bXn

j¼1

~wj~dþ ð1� ~bÞ

Xn

i¼1

~v i~ai

¼ ~b~dXn

j¼1

~wj þ ð1� ~bÞXn

i¼1

~v i~ai; ð14Þ

SincePn

j¼1 ~wj ¼ 1, we get

f ðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ 6 ~b~c þ ð1� ~bÞXn

i¼1

~v i~ai ð15Þ

f ðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞP ~b~dþ ð1� ~bÞXn

i¼1

~v i~ai: ð16Þ

Therefore,

b�Minf~aig þ ð1� ~bÞ �Xn

i¼1

~wi~ai 6 f ðhu1; ~a1i; . . . ; hun; ~aniÞ

6 ~b�Maxf~aig þ ð1� ~bÞ

�Xn

i¼1

~wi~ai: �

As we can see, if b = 1, we get the usual boundary conditions. Notethat a similar semi boundary condition could be analyzed from theUIOWA perspective. That is:

~b�Minf~aiig þ ð1� ~bÞ �Xn

j¼1

~wjbj 6 f ðhu1; ~a1i; . . . ; hun; ~aniÞ

6 ~b�Maxf~aig þ ð1� ~bÞ �Xn

j¼1

~wjbj ð17Þ

Note that if wi = 1/n for all i, the semi boundaries become the orlikeand the andlike S-UIOWA operator (Merigó, 2008; Merigó & Gil-Lafuente, 2009; Yager, 1993). In summary, we could refer to thesenew semi boundary conditions as the maximum – UWA

(Max-UWA) and the minimum – UWA (Min-UWA), and themaximum – UIOWA (Max-UIOWA) and the minimum – UIOWA(Min-UIOWA) operator.

A further interesting issue is the problem of ties in the order-inducing variables. In order to solve this problem, we recommendto use the method explained by Yager and Filev (1999) where theyreplace the tied arguments by their average. Note also that differ-ent kinds of attributes may be used for the order-inducing vari-ables of the UIOWAWA operator with the only requirement ofhaving a linear ordering Yager and Filev (1999).

Some other generalizations can be developed following Mesiarand Pap (2008), Mesiar and Spirkova (2006), Spirkova (2009) andTorra and Narukawa (2010). Following Spirkova (2009), we canuse a generating function r for the order-inducing variables suchthat, r: I ? R, being I � R a closed interval I = [a, b]. We can also de-velop a generating function for the arguments of the UIOWAWAoperator that represents the internal formation of this informationsuch that, s: Xm ? X. Moreover, we also use a weighting function ffor the weighting vector. Thus, we obtain the mixture UIOWAWA(MUIOWAWA) operator as follows.

Definition 10. Let X be the set of interval numbers. A MUIOW-AWA operator of dimension n is a mapping MUIOWAWA:Xn �Xn ? X that has associated a vector of weighting functionsf, r: I ? ]0,1[, is some positive continuous function, s: Xm ? X,such that:

MUIOWAWA ðhrxðu1Þ; syð~a1Þi; . . . ; hrxðunÞ; syð~anÞiÞ

¼Pn

j¼1fjðsyðbjÞÞsyðbjÞPnj¼1fjðsyðbjÞÞ

; ð18Þ

where sy(bj) is the sy(~ai) value of the MUIOWAWA pair hrx(ui), sy(~ai)ihaving the jth largest rx(ui), ui is the order inducing variable, x and yindicates that each order-inducing variable and each argument isformed by using a different function where sy(bj) is the jth largestof the sy(~ai), ~ai is the argument variable, y indicates that each argu-ment is formed by using a different function and all the informationis provided using interval numbers.

Note that it is possible to use a more general expression of theprevious formula assuming that the generating function of theweighting vector does not depend on the arguments bj and can de-pend on a lot of other circumstances. Thus, we get the followingexpression where we also assume that the generating function isimplicitly used in the normalization process:

UIOWAWA� ðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ ¼Xn

j¼1

fjðv jÞsyðbjÞ: ð19Þ

Note that in the new formulation, we include all the availablefamilies of UIOWA operators and a lot of other situations. Obvi-ously, the UIOWAWA⁄ can be used in the whole UIOWA andUWA literature and in future extensions.

Another interesting result that we could analyze consists inusing infinitary aggregation operators (Mesiar & Pap, 2008). There-fore, we can represent the aggregation process with an unlimitednumber of arguments. Note that

P1j¼1v j ¼ 1. By using, the UIOWA-

WA operator we get the infinitary UIOWAWA (1-UIOWAWA)operator as follows.

1-UIOWAWA ðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ ¼X1j¼1

v jbj: ð20Þ

Note that the reordering process is very complex because wehave an unlimited number of arguments. For further reading forthe usual OWA, see Mesiar and Pap (2008).

J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880 875

4.3. Measures for characterizing the weighting vector

Several measures for characterizing the weighting vector V (Vand W) such as the orness measure, the entropy of dispersion,the divergence of W and the balance operator can be extended tothe UIOWAWA framework. Following a similar methodology as ithas been developed for the OWA operator (Merigó, 2008; Yager,1988) we can use these measures in the following way.

If we analyze the orness–andness measure to the UIOWAWAoperator, we get the following expressions. For the degree oforness:

aðVÞ ¼ ~bXn

j¼1

~w�jn� jn� 1

� �þ ð1� ~bÞ

Xn

j¼1

~v�jn� jn� 1

� �: ð21Þ

Note that we follow the methodology explained in Section 3 foraddressing uncertain weights. Additionally, it is also possible toconvert these interval weights into exact numbers in order to ob-tain the usual results (Merigó, 2008, 2010b; Yager, 1988). Note that~w�j and ~v�j are the ~wj and ~v j weights of the UIOWAWA aggregationordered according to the values of the order inducing variables ui.As we can see, if b = 1, we get the orness measure of the UIOWAoperator and if b = 0, we obtain the orness measure of the weightedaverage. It is straightforward to calculate the andness measure byusing the dual. That is, AndnessðVÞ ¼ 1� aðVÞ.

The entropy of dispersion (Yager, 1988) measures the amountof information being used in the aggregation. If we extend theentropy of dispersion to the UIOWAWA operator, we get thefollowing:

HðVÞ ¼ � ~bXn

j¼1

~wj lnð ~wjÞ þ ð1� ~bÞXn

i¼1

~v i lnð~v iÞ !

: ð22Þ

Note that ~v i is the ith weight of the UWA aggregation. As we cansee, if b = 1, we obtain the Yager entropy of dispersion and if b = 0,we get the classical Shannon entropy (Shannon, 1948). Note alsothat it is possible to consider other entropy measures in the UIOW-AWA operator.

The divergence of W (Yager, 2002) can also be extended withthe UIOWAWA operator. Thus, we get the following divergenceof V:

DivðVÞ ¼ ~bXn

j¼1

~wjn� jn� 1

� að ~WÞ� �2

!

þ ð1� ~bÞXn

j¼1

~v jn� jn� 1

� að~VÞ� �2

!: ð23Þ

Note that if b = 1, we get the UIOWA divergence and if b = 0, theUWA divergence. Note that for any step aggregation we getDiv�ðVÞ ¼ 0. If we only use a step aggregation in the UWA part,then we get the UIOWA divergence and vice versa. Note also that

Table 1Mixing families of UIOWA and UWA operators.

UIOWA

UIOWA Max Min UA

UWA UWA UIOWAWA Max-UWA Min-UWA UA-UWAMax UIOWA-Max Max Min-Max UA-MaxMin UIOWA-Min Max-Min Min UA-MinUA UIOWA-UA Max-UA Min-UA UAStep UIOWA-Step Max-Step Min-Step UA-StepOly. UIOWA-Oly. Max-Oly. Min-Oly. UA-Oly.Cent. UIOWA-Cent. Max-Cent. Min-Cent. UA-Cent.ME UIOWA-ME Max-ME Min-ME UA-MEBUM UIOWA-BUM Max-BUM Min-BUM UA-BUM

Etc.

Abbreviations: UA = uncertain average; Oly. = olympic; Cent. = centered; ME = maximal

we have presented the results of the divergence by using the or-ness measure but it is also possible to present them by using theandness measure, that is, the dual: orness = 1 � andness.

The balance operator (Yager, 1996) measures the balance of theweights against the tendency to the maximum or to the minimum.For the UIOWAWA operator it is formulated as follows:

BalðVÞ ¼ ~bg�1Xn

j¼1

gnþ 1� 2j

n� 1

� �~wj

!

þ ð1� ~bÞg�1Xn

j¼1

gnþ 1� 2j

n� 1

� �~v j

!: ð24Þ

Note that if g(b) = b, then, we get the usual balance operator ap-plied to the UIOWAWA operator as follows:

BalðVÞ ¼Xn

j¼1

nþ 1� 2jn� 1

� �v j: ð25Þ

If b = 1, we get the classic balance operator developed by Yager(1996) applied to the UIOWA operator and if b = 0, we obtain thebalance operator of the UWA.

4.4. Families of UIOWAWA operators

Different families of UIOWAWA operators are found by study-ing the weighting vector W and the coefficient . By doing this anal-ysis, we can see different forms of the weighting vector that can beused depending on the interests of the decision maker in the anal-ysis. In Table 1 we present a general overview of different particu-lar methods that could be implemented in the analysis.

If we focus on the coefficient b, in order to consider the degreeof importance that the UIOWA and the UWA has in the aggrega-tion, we get the following particular cases.

� If b = 1, we get the UIOWA operator.� If b = 0, we get the uncertain weighted average (UWA).� Note that when b increases, we are giving more importance to

the UIOWA operator and when b decreases, we give moreimportance to the UWA.

And if we look to the weighting vector V , we get for example,the following ones.

� The uncertain maximum (wp = 1 and wj = 0, for all j – p, andbp = Max{~ai}, and vp = 1 and vi = 0, for all i – p, and ap = Max{~ai}).� The uncertain minimum (wp = 1 and wj = 0, for all j – p, and

bp = Min{~ai}, and vp = 1 and vi = 0, for all i – p, and ap4 = Min{~ai}).� The uncertain maximum-UWA (wp = 1 and wj = 0, for all j – p,

and bp = Max{~ai}).� The uncertain minimum-UWA (wp = 1 and wj = 0, for all j – p,

and bp = Min{~ai}).

Step Olympic Centered ME BUM

Step-UWA Oly-UWA Cent-UWA ME-UWA BUM-UWAStep-Max Oly-Max Cent-Max ME-Max BUM-MaxStep-Min Oly-Min Cent-Min ME-Min BUM-MinStep-UA Oly-UA Cent-UA ME-UA BUM-UAStep Oly-Step Cent-Step ME-Step BUM-StepStep-Oly. Olympic Cent-Oly. ME-Oly. BUM-Oly.Step-Cent. Oly-Cent. Centered ME-Cent. BUM-Cent.Step-ME Oly-ME Cent-ME ME BUM-MEStep-BUM Oly-BUM Cent-BUM ME-BUM BUM

entropy; BUM = basic unit monotonic function.

876 J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880

� The uncertain maximum-UIOWA (vp = 1 and vi = 0, for all i – p,and ap = Max{~ai}).� The uncertain minimum-UIOWA (vp = 1 and vi = 0, for all i – p,

and ap = Min{~ai}).� The uncertain average (UA) (wj = 1/n, and vj = 1/n, for all ~ai).� The step-UIOWAWA (vk ¼ 1 and v j ¼ 0, for all j – k).� The general olympic-UIOWAWA operator (v j ¼ 0 for

j = 1, 2, . . . , k, n, n � 1, . . . , n � k + 1; and for all others v j� = 1/(n � 2k), where k < n/2).� The centered-UIOWAWA operator (if it is symmetric, strongly

decaying from the center to the maximum and the minimum,and inclusive).

Remark 1. It is worth noting that the UIOWAWA operatorincludes the uncertain arithmetic UWA (UAUWA) and the uncer-tain arithmetic UIOWA (UAUIOWA) operator. The UAUWA isobtained if wj = 1/n for all j, and can be formulated as follows:

UA� UWAðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ

¼ 1n

~b~ai þ ð1� ~bÞXn

i¼1

~v i~ai: ð26Þ

If vi = 1/n, for all i, then, we get the unification between the UA andthe UIOWA operator, that is, the UAUIOWA operator. It can be for-mulated as follows:

UA� UIOWAðhu1; ~a1i; hu2; ~a2i; . . . ; hun; ~aniÞ

¼ ~bXn

j¼1

~wjbj þ ð1� ~bÞ1n

~ai: ð27Þ

Theorem 1. If the interval numbers are reduced to the usual exactnumbers, then, the UIOWAWA operator becomes the IOWAWAoperator (Merigó, 2008).

Proof. Assume a triplet = (a1, a2, a3). If a1 = a2 = a3, then(a1, a2, a3) = a. Thus, we get the IOWAWA operator. h

Note that other families of UIOWAWA operators could be foundfollowing a similar methodology as it has been developed in a widerange of papers for the OWA operator and its extensions (Merigó &Casanovas, 2011a, 2011b, 2011c; Merigó & Gil-Lafuente, 2011b).Furthermore, we may find that the UIOWA uses one type of familyand the UWA another one. For example, the UIOWA uses a cen-tered aggregation and the UWA an olympic one. In Table 1, wehave briefly presented different families that use a different typeof weighting vector in the UIOWA and in the UWA.

5. Applicability of the UIOWAWA operator

In this section we discuss the applicability of the UIOWAWAoperator. First, we give a general overview of potential applicationsfor future research. Second, we present an application in multi-person decision making and third, we provide an illustrativeexample in a decision making process in tourism management.

5.1. Introduction

The UIOWAWA operator can be applied in a lot of applications.In summary, we arrive to the statement that all the studies that usethe UIOWA (note that the OWA is included as a particular case) orthe UWA, can be revised and extended by using this new approach.The reason is that the UIOWAWA includes the UIOWA and theUWA as particular cases. Therefore, we can always use the partic-ular type of UIOWAWA that adapts to the problem without consid-ering the rest of particular cases. For example, simply consideringthe UIOWA or the UWA.

In the following, we present some of the main research areaswhere we can use the UIOWA operator. Note that inside each field,there are a lot of potential applications that could be done. In gen-eral, we can use the UIOWAWA operator in the following areas:

� Statistics: The UIOWAWA is a fundamental instrument to revisethe majority of the statistical sciences such as descriptive statis-tics, regression, probability theory, hypothesis testing and infer-ence statistics.� Soft Computing and Fuzzy Set Theory: All aspects of fuzzy set

theory that use statistical techniques based on the UWA orthe UIOWA can be revised and extended with the UIOWAWAoperator. Specially, the theory of aggregation operators isstrongly affected by this new approach.� Decision Theory and Operational Research: The use of the

UIOWAWA operator can imply a lot of new improvements inthe current models.� Engineering: A wide range of applications can be developed in

different engineering areas such as electrical engineering,industrial engineering and mechanical engineering.� Economics and Business Administration: We can use it in sev-

eral problems concerning financial management, marketing,accounting, strategic management and human resourcemanagement.� Politics: We can use it when making political decisions concern-

ing international, national or local decisions.� Mathematics: The UIOWAWA operator can also be imple-

mented in several mathematical problems, especially thoseconcerning with mathematical statistics.� Physics: Physical statistics can be strongly revised by using the

UIOWAWA operator in order to obtain deeper formulations ofthe available studies.� Chemistry: A lot of statistical techniques used in chemistry can

be revised by using this new approach.� Biology, Medicine and Pharmacy: All the theories and tech-

niques concerning biostatistics and related topics can be revisedwith the UIOWAWA operator.� Other sciences: A lot of other applications could be developed in

many other sciences including communication studies, psychol-ogy, sociology and geography.

5.2. Applicability in multi-person decision making

In the following, we develop an application in a multi-persondecision-making problem. The main reason for using this approachis that we can assess the information by using the opinion of sev-eral persons. We focus on a decision making problem in tourismmanagement.

The decision-making process to follow with the UIOWAWAoperator in multi-person decision-making can be summarized asfollows. Note that in the literature, we may find a lot of other groupdecision-making models (Cabrerizo, Alonso, & Herrera-Viedma,2009; Wei, 2009, 2010; Wu et al., 2009).

Step 1: Let A = {a1, a2, . . . , an} be a set of finite alternatives,S = {s1, s2, . . . , sn} a set of finite states of nature (or attributes),forming the payoff matrix (ãhi)m�n. Let E = {e1, e2, . . . , ep} be afinite set of decision-makers. Let X = (x1, x2, . . ., xp) be theweighting vector of the decision-makers such that

Ppk¼1xk ¼ 1

and xk 2 [0, 1]. Note that the weights X are given in the formof interval numbers and we assess them following the method-ology discussed in Section 3. Each decision-maker provides hisown payoff matrix (ãhi

(k))m�n also represented by using intervalnumbers.Step 2: Calculate the order-inducing variables (uhi)m�n to beused in the payoff matrix for each alternative h and state of

J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880 877

nature i. Moreover, calculate the weighting vectorV ¼ ~b� ~W þ ð1� ~bÞ � ~V to be used in the UIOWAWA aggrega-tion. Note that W = (w1, w2, . . . , wn) such that

Pnj¼1wj ¼ 1 and

wj 2 [0, 1] and V = (v1, v2, . . . , vp) such thatPn

i¼1v i ¼ 1 andvi 2 [0, 1]. Both weighting vectors are given by using intervalnumbers.Step 3: Use the UWA to aggregate the information of the deci-sion-makers E by using the weighting vector X. The result isthe collective payoff matrix (ãhi)m�n. Thus, ~ahi ¼

Ppk¼1xk~ak

hi. Notethat it is also possible to use other aggregation operators suchas the UOWA and the UIOWA operator.Step 4: Calculate the aggregated results by using the UIOWAWAoperator explained in Eq. (9) and consider different families ofUIOWAWA operators by using some of cases discussed in Sec-tion 4.Step 5: Adoption of decisions according to the results found inthe previous steps. Select the alternative/s that provides thebest result/s. Moreover, establish an ordering or a ranking ofthe alternatives from the most to the less preferred alternative.

Note that this process can be summarized using the followingaggregation operator that we call the multi-person – UIOWAWA(MP-UIOWAWA) operator.

Definition 11. Let X be the set of interval numbers. A MP-UIOWAWA operator is a mapping MP-UIOWAWA: (Xn)p �Xn ? Xthat has a weighting vector X of dimension p with

Ppk¼1xk ¼ 1 and

xk 2 [0, 1] and a weighting vector W of dimension n withPnj¼1 ~wj ¼ 1 and ~wj 2 ½0; 1�, such that:

MP� UIOWAWA ðhu1; ð~a11; . . . ; ~ap

1Þi; . . . ; hun; ð~a1n; . . . ; ~ap

nÞiÞ ¼Xn

j¼1

v jbj;

ð28Þ

where bj is the ai value of the MP-UIOWAWA pair hui,~aii having thejth largest ui, ui is the order-inducing variable, ~ai ¼

Ppk¼1xk~ak

i , ~aki is

the argument variable provided by each person (or expert), eachargument ~ai has an associated weight (WA) ~v i with

Pni¼1 ~v i ¼ 1

and ~v i 2 ½0; 1�, v j ¼ ~b~wj þ ð1� ~bÞ~v j with ~b 2 ½0; 1� and ~v j is theweight (WA) ~v i ordered according to bj, that is, according to thejth largest ui.

Note that the MP-UIOWAWA operator accomplishes similarproperties than those explained in Section 3 such as the generaliza-tion with mixture operators, the use of infinitary aggregations, andso on.

The MP-UIOWAWA operator includes a wide range of particularcases following the methodology explained in Section 4 such as:

� The multi-person – UIOWA (MP-UIOWA) operator.� The multi-person – UWA (MP-UWA) operator.� The multi-person – UOWA (MP-UOWA) operator.� The multi-person – uncertain arithmetic mean (MP-UAM)

operator.� The multi-person – uncertain arithmetic UWA (MP-UA-UWA).� The multi-person – uncertain arithmetic UIOWA (MP-UA-

UIOWA).

5.3. Illustrative example in tourism management

In the following, we develop a simple numerical example of thenew approach. We focus on a multi-person decision-making prob-lem regarding the selection of holiday trips (selection of destina-tions) in tourism management. For further research in tourismmanagement, see for example (Martorell, 2006; Martorell, Mulet,& Roselló, 2008; Weaver & Lawton, 2010).

Step 1: Assume that a group of persons is planning their vaca-tion and they decide to make a trip to another country. Aftera general evaluation of different alternatives, they consider sixalternatives (destinations):� A1 = Trip to Mallorca (Spain).� A2 = Trip to Nice (France).� A3 = Trip to Cancun (Mexico).� A4 = Trip to El Cairo (Egypt).� A5 = Trip to New York (USA).� A6 = Trip to Hong Kong (China).

In order to assess these alternatives, there are six friends thatconstitute three couples. Thus, each couple provides its own deci-sion having three general opinions to be considered in the decisionprocess. After careful analysis, each couple evaluates 7 generalcharacteristics that are supposed to be the most influential in theirdecision:

� C1 = Price of the trip.� C2 = Tourist activities.� C3 = Sun and beach (weather attractiveness).� C4 = Willingness for doing the trip.� C5 = Facilities of the place (Bars and restaurants close to the

hotel, etc.).� C6 = Peace and stability.� C7 = Other variables.

Each couple evaluates each characteristic from 0 to 1, being 1the best result. Note that the best result means that this character-istic perfectly adapts to the interests of the couple. As they are notreally sure how to evaluate these alternatives, they use intervalnumbers in the analysis. The expected results for each coupledepending on the alternative A and the characteristic C are shownin Tables 2–4.

Step 2: In this example, the couples consider that they are a bitoptimistic. Thus, the weighting vector of the UIOWA isW = (0.3, 0.2, 0.1, 0.1, 0.1, 0.1, 0.1) with a degree of importanceof 30%; and the weighting vector UWA (70%): V =(0.2, 0.1, 0.1, 0.3, 0.1, 0.1, 0.1). Note that the weighting vector Xthat represents the importance of each expert in the analysisis: X = (0.4, 0.3, 0.3). That is, the opinion of the first couple is abit more influential than the other two. In order to consider areordering process not only based on the values of the argu-ments, they use order inducing variables that represent a reor-dering process adapted to the interests of the 6 persons that areconsidering this vocational trip. In this example, we assumethat the order inducing variables are: U = (24, 17, 14,28, 16, 12, 11).Step 3: With the available information, we can aggregate it inorder to make a decision. First, we aggregate the informationof the three couples in order to obtain a collective result foreach alternative and characteristic. These results are shown inTable 5.Step 4: Next, we aggregate this information by using theUIOWAWA operator. In order to provide a complete analysis ofthe different potential results that may occur depending on theinterests of the decision makers, we present a wide range of par-ticular cases of UIOWAWA operators. However, in case of doubt,we assume that these 6 persons will select the best result foundin the aggregation with the usual formulation of the UIOWAWAoperator. In this example, we consider the Max-UWA, the Min-UWA, the uncertain average (UA), the UWA, the UOWA, theUIOWA, the UA – UWA (UA-UWA), the UA – UOWA (UA-UOWA),the UA – UIOWA (UA-UIOWA), the UOWAWA and the UIOWAWAoperator. The results are presented in Tables 6 and 7.

Table 3Payoff matrix – couple 2.

C1 C2 C3 C4 C5 C6 C7

A1 (0.6, 0.7) (0.7, 0.8) (0.6, 0.7) (0.7, 0.8) (0.8, 0.9) (0.5, 0.6) (0.6, 0.7)A2 (0.7, 0.8) (0.6, 0.7) (0.7, 0.8) (0.4, 0.5) (0.4, 0.5) (0.7, 0.8) (0.5, 0.6)A3 (0.3, 0.4) (0.4,0.5) (0.7, 0.8) (0.6, 0.7) (0.5, 0.6) (0.7, 0.8) (0.6, 0.7)A4 (0.3, 0.4) (0.7, 0.8) (0.5, 0.6) (0.5, 0.6) (0.4,0.5) (0.5, 0.6) (0.8, 0.9)A5 (0.5, 0.6) (0.8, 0.9) (0.6, 0.7) (0.7, 0.8) (0.8, 0.9) (0.6, 0.7) (0.6, 0.7)A6 (0.3, 0.4) (0.6, 0.7) (0.5, 0.6) (0.6, 0.7) (0.6, 0.7) (0.6, 0.7) (0.5, 0.6)

Table 2Payoff matrix – couple 1.

C1 C2 C3 C4 C5 C6 C7

A1 (0.8, 0.9) (0.6, 0.7) (0.7, 0.8) (0.6, 0.7) (0.7, 0.8) (0.6, 0.7) (0.5, 0.6)A2 (0.7, 0.8) (0.5, 0.6) (0.7, 0.8) (0.5, 0.6) (0.8, 0.9) (0.8, 0.9) (0.4, 0.5)A3 (0.2, 0.3) (0.4, 0.5) (0.8, 0.9) (0.6, 0.7) (0.6, 0.7) (0.7, 0.8) (0.8, 0.9)A4 (0.4, 0.5) (0.8, 0.9) (0.6, 0.7) (0.6, 0.7) (0.5, 0.6) (0.6, 0.7) (0.8, 0.9)A5 (0.5, 0.6) (0.8, 0.9) (0.4, 0.5) (0.7, 0.8) (0.8, 0.9) (0.6, 0.7) (0.6, 0.7)A6 (0.3, 0.4) (0.7, 0.8) (0.4, 0.5) (0.6, 0.7) (0.6, 0.7) (0.7, 0.8) (0.4, 0.5)

Table 4Payoff matrix – couple 3.

C1 C2 C3 C4 C5 C6 C7

A1 (0.6, 0.7) (0.6, 0.7) (0.5, 0.6) (0.5, 0.6) (0.6, 0.7) (0.6, 0.7) (0.5, 0.6)A2 (0.7, 0.8) (0.6, 0.7) (0.7, 0.8) (0.5, 0.6) (0.8, 0.9) (0.6, 0.7) (0.5, 0.6)A3 (0.5, 0.6) (0.4, 0.5) (0.8, 0.9) (0.6, 0.7) (0.6, 0.7) (0.7, 0.8) (0.6, 0.7)A4 (0.4, 0.5) (0.7, 0.8) (0.5, 0.6) (0.6, 0.7) (0.8, 0.9) (0.5, 0.6) (0.8, 0.9)A5 (0.5, 0.6) (0.8, 0.9) (0.5, 0.6) (0.7, 0.8) (0.7, 0.8) (0.6, 0.7) (0.6, 0.7)A6 (0.3, 0.4) (0.7, 0.8) (0.6, 0.7) (0.5, 0.6) (0.6, 0.7) (0.7, 0.8) (0.3, 0.4)

Table 5Payoff matrix – collective results.

C1 C2 C3 C4 C5 C6 C7

A1 (0.68, 0.78) (0.63, 0.73) (0.61, 0.71) (0.6, 0.7) (0.7, 0.8) (0.57, 0.67) (0.53, 0.63)A2 (0.7, 0.8) (056, 0.66) (0.7, 0.8) (0.47, 0.57) (0.68, 0.78) (0.71, 0.81) (0.46, 0.56)A3 (0.32, 0.42) (0.4, 0.5) (0.77, 0.87) (0.6, 0.7) (0.57, 0.67) (0.7, 0.8) (0.68, 0.78)A4 (0.37, 0.47) (0.74, 0.84) (0.54, 0.64) (0.57, 0.67) (0.56, 0.66) (0.54, 0.64) (0.8, 0.9)A5 (0.5, 0.6) (0.8, 0.9) (0.49, 0.59) (0.7, 0.8) (0.77, 0.87) (0.6, 0.7) (0.6, 0.7)A6 (0.3, 0.4) (0.67, 0.77) (0.49, 0.59) (0.57, 0.67) (0.6, 0.7) (0.67, 0.77) (0.4, 0.5)

Table 6Aggregated results 1.

Max-UWA Min-UWA UA UWA UOWA UIOWA

A1 (0.7, 0.8) (0.53, 0.63) (0.61, 0.71) (0.62, 0.72) (0.64, 0.74) (0.62, 0.72)A2 (0.71, 0.81) (0.46, 0.56) (0.61, 0.71) (0.59, 0.69) (0.64, 0.74) (0.59, 0.69)A3 (0.77, 0.87) (0.32, 0.42) (0.57, 0.67) (0.55, 0.65) (0.62, 0.72) (0.55, 0.65)A4 (0.8, 0.9) (0.37, 0.47) (0.58, 0.68) (0.56, 0.66) (0.64, 0.74) (0.56, 0.66)A5 (0.8, 0.9) (0.49, 0.59) (0.63, 0.73) (0.63, 0.73) (0.68, 0.78) (0.63, 0.73)A6 (0.67, 0.77) (0.3, 0.4) (0.52, 0.62) (0.51, 0.61) (0.57, 0.67) (0.51, 0.61)

Table 7Aggregated results 2.

UA-UWA UA-UOWA UA-UIOWA UOWAWA UIOWAWA

A1 (0.61, 0.71) (0.62, 0.72) (0.61, 0.71) (0.62, 0.72) (0.62, 0.72)A2 (0.59, 0.69) (0.61, 0.71) (0.60, 0.70) (0.60, 0.70) (0.59, 0.69)A3 (0.56, 0.66) (0.59, 0.69) (0.57, 0.67) (0.57, 0.67) (0.55, 0.65)A4 (0.57, 0.67) (0.60, 0.70) (0.58, 0.68) (058, 0.68) (0.56, 0.66)A5 (0.63, 0.73) (0.65, 0.75) (0.63, 0.73) (0.65, 0.75) (0.63, 0.73)A6 (0.51, 0.61) (0.54, 0.64) (0.52, 0.62) (0.53, 0.63) (0.51, 0.61)

Table 8Ordering of the holiday trips (destinations).

Ordering Ordering

Max-UWA A4 = A5 A3 A2 A1 A6 UA-UWA A5 A1 A2 A4 A3 A6

Min-UWA A1 A5 A2 A4 A3 A6 UA-UOWA A5 A1 A2 A4 A3 A6

UA A5 A1 A2 A4 A3 A6 UA-UIOWA A5 A1 A2 A4 A3 A6

UWA A5 A1 A2 A4 A3 A6 UOWAWA A5 A1 A2 A4 A3 A6

UOWA A5 A4 A1=A2 A3 A6 UIOWAWA A5 A1 A2 A4 A3 A6

UIOWA A5 A1 A2 A4 A3 A6

878 J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880

J.M. Merigó et al. / Expert Systems with Applications 39 (2012) 869–880 879

Step 5: In order to provide more complete information of thedecision process we have to establish a ranking of the holidaytrips. This is useful because apart from the optimal choice wecan also see which method goes second, third and so on. Theordering of the alternatives is presented in Table 8. Note alsothat ‘‘ ’’ means ‘‘preferred to’’ and ‘‘=’’ means ‘‘equal to’’.

As we can see, depending on the aggregation operator used, theresults may be different leading to different rankings of the holidaytrips. Note that in this example the optimal choice is A5 because weassume that the final decision is only affected by the results ob-tained in the UIOWAWA operator.

6. Conclusions

We have presented a new decision making process based on anew aggregation operator called the UIOWAWA operator and wehave applied it in a tourism management problem. The UIOWAWAoperator is a new aggregation operator that unifies the UIOWAoperator with the UWA when the available information is uncertainand can be assessed with interval numbers. The main advantage ofthis operator is that it provides more complete information becauseit represents the information in a more complete way consideringthe maximum and the minimum results that can occur. Moreover,it also considers the degree of importance that each concept has inthe analysis allowing the aggregation to range from the UWA to theUIOWA operator. We have also studied several properties and par-ticular cases of this new approach.

We have analyzed the applicability of the new approach and wehave seen that it is very broad because it provides a more generalformulation of the UWA and the UIOWA operator. In this paper, wehave focused on an application in tourism management based onthe use of a multi-person decision making process where a groupof decision makers are looking for an optimal holiday trip. We haveseen that depending on the aggregation operator used the resultsmay lead to different decisions.

In future research, we expect to develop further developmentsby using other types of information such as fuzzy numbers, linguis-tic variables and expertons. We will also add other characteristicsin order to obtain a more complete formulation such as the use ofgeneralized and quasi-arithmetic means, distance measures andunified aggregation operators. Finally, we will also develop differ-ent types of applications especially in decision theory but also inother fields such as statistics, engineering, business and economics.

Acknowledgement

Support from the Projects JC2009-00189 and A/023879/09 fromthe Spanish Ministry of Science and Innovation is gratefullyacknowledged.

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