UNCERTAIN GROUP DECISION-MAKING WITH INDUCED AGGREGATION OPERATORS AND EUCLIDEAN DISTANCE

. In this paper, we present the induced uncertain Euclidean ordered weighted averaging distance (IUEOWAD) operator. It is an extension of the OWA operator that uses the main characteristics of the induced OWA (IOWA), the Euclidean distance and uncertain information represented by interval numbers. The main advantage of this operator is that it is able to consider complex attitudinal characters of the decision-maker by using order-inducing variables in the aggregation of the Euclidean distance. Moreover, it is able to deal with uncertain environments where the information is very imprecise and can be assessed with interval numbers. We study some of its main properties and particular cases such as the uncertain maximum distance, the uncertain minimum distance, the uncertain normalized Euclidean distance (UNED), the uncertain weighted Euclidean distance (UWED) and the uncertain Euclidean ordered weighted averaging distance (UEOWAD) operator. We also apply this aggregation operator to a group decision-making problem regarding the selection new artillery weapons under uncertainty.


Introduction
arguments that depends on order-inducing variables. The IOWAD generalizes the OWAD operator and provides a parameterized family of distance aggregation operators between the maximum and the minimum distance. Going a step further, Merigó and Casanovas (2011b) presented the induced Euclidean ordered weighted averaging distance (IEOWAD) operator, which uses the IOWA operator and the Euclidean distance in the same formulation. It generalizes the Euclidean ordered weighted averaging distance (EOWAD) operator and provides a parameterized family of distance aggregation operators between the maximum and the minimum distance based on a complex reordering process that reflects the complex attitudinal character of the decision-maker. The IEOWAD operator is very useful for decision-making problems because it can establish a comparison between an ideal, though unrealistic, alternative and available options in order to find the optimal choice. As such, the optimal choice is the alternative closest to the ideal one. The main advantage of the IEOWAD operator is that it is able to deal with complex attitudinal characters (or complex degrees of orness) in the decision process by using order-inducing variables. In doing so, we are able to deal with more complex problems that are closer to real-world situations. For further research on the use of the OWA and IOWA in distance measures, e.g. Karayiannis (2000), Merigó and Casanovas (2011c), Peng et al. (2013), Zeng (2013), Zeng and Su (2012).
Usually, when using the IEOWAD operator, it is assumed that the available information is clearly known and can be assessed with exact numbers. The aim of this paper is to present a new uncertain decision-making model by using the IEOWAD operator. To do so, we will develop an induced uncertain Euclidean ordered weighted averaging distance (IUEOWAD) operator, which is an extension of the IEOWAD operator with interval numbers. Therefore, it is an extension of the OWA operator that uses the main characteristics of the induced OWA (IOWA), the Euclidean distance and uncertain information represented by interval numbers. It is also a unified model that uses the uncertain IOWA (UIOWA) operator and the Euclidean distance in the same formulation. By using distance measures in the IUEOWAD operator, we can compare the real-world information with ideal information and see which alternative better fits with the interests of the decision-maker. For example, in human resource selection, we can establish an ideal candidate that would perfectly fit the company and compare it with the real-world alternatives that we have in the market and select the candidate with closest results to the ideal one. The main advantage of the IUEOWAD is that it can deal with uncertain environments represented in the form of interval numbers and with complex reordering processes of the information. In decision-making problems, this means that we are considering complex attitudinal characters of the decision-maker that reflect psychological aspects, pressure, utility, and a lot of other aspects. Furthermore, this new operator generalizes a wide range of uncertain distance measures and aggregation operators such as the uncertain maximum distance, the uncertain minimum distance, the uncertain normalized Euclidean distance (UNED), the uncertain weighted Euclidean distance (UWED) and the uncertain Euclidean ordered weighted averaging distance (UEOWAD) operator. We study some of its particular cases.
We study the applicability of the IUEOWAD operator and we see that it is very broad because all the previous studies that use the induced aggregation operators or the Euclidean distance can be revised with this new approach. For example, we can apply it to statistics, eco-nomics, engineering, physics and medicine. In this paper we focus on a group decision-making problem of a military unit purchasing new artillery weapons. Thus, we are able to consider a wide range of aggregation operators that could be used in the aggregation. We can see that, depending on the aggregation operator used, the results may lead to different decisions. Therefore, the decision maker is able to consider a wide range of scenarios and select the one that is in accordance with his interests. Moreover, by using several experts in the analysis, we obtain information that it is more robust because the opinion of several experts is always better than the opinion of one.
This paper is organized as follows. In Section 1 we briefly review some basic concepts. Section 2 presents the IUEOWAD operator and Section 3 analyzes different families of IUEOWAD operators. Section 4 briefly describes the decision-making process based on the developed operator and we give a numerical example in Section 5. The final section summarizes the main conclusions of the paper.

Preliminaries
In this section, we briefly review some basic concepts about the interval numbers, the IOWA and the IEOWAD operator.

Interval numbers
The interval number (Moore 1966;Xu, Da 2002) is a very useful and simple technique for representing the uncertainty, which has been used in an astonishingly wide range of applications. The interval numbers can be expressed in different forms. For practical reasons, we use the notation introduced by Xu and Da (2002). According to this notation, an interval number a  can be noted as: , In the following, we are going to review some basic interval number operations as follows: be two interval numbers, then: Different approaches have been developed for dealing with interval number such as Zhou (2011), Liu et al. (2012), Pei (2013), Xu (2013), Xu and Cai (2012), Yue (2013). In order to measure deviation between interval numbers, Xu (2008) introduced a distance for each pair of interval numbers as following: is called the distance between a  and b  . Obviously, the smaller the value of ( ) , d a b    , the closer between a  and b  .

The Induced OWA Operator
The IOWA operator was introduced by Yager and Filev (1999) and it represents an extension of the OWA operator. The main difference is that the reordering step of the IOWA is carried out with order-inducing variables, rather than depending on the values of the arguments. The IOWA operator also includes the maximum, the minimum and the average operators, as special cases. It can be defined as follows: Definition 2. An IOWA operator of dimension n is a mapping IOWA: n n R R R × → that has an associated weighting vector W with [0,1] where j b is i a value of the IOWA pair , i i u a having the j th largest i u , i u is the order inducing variable and i a is the argument variable.

The IEOWAD operator
The IEOWAD (Merigó, Casanovas 2011b) operator is a distance measure that uses the IOWA operator in the normalization process of the Euclidean distance. Then, the reordering of the individual distances is developed with order inducing variables. For two sets  When using the IEOWAD operator, it is assumed that the available information is represented in the form of exact numbers. However, this may not be the real situation found in the decision-making problems. Sometimes the available information is vague or imprecise and it is not possible to analyze it with exact numbers. In this case, it is more suitable to use interval numbers to assess the uncertainty. In the following, we shall develop the uncertain induced Euclidean OWA distance (UIEOWAD).

Induced Uncertain Euclidean OWA Distance (IUEOWAD) operator
The induced uncertain Euclidean OWA distance (IUEOWAD) operator is an extension of the IEOWAD operator that uses uncertain information in the aggregation. The main difference between the IUEOWAD and IEOWAD operators is that the IUEOWAD operator addresses uncertain information represented using interval numbers, while the IEOWAD operator uses exact numbers. The reason for using this operator is that the uncertain factors that affect our decisions are sometimes not clearly known; thus, in order to assess the problem, we must use interval numbers in order to consider the range of uncertain results that could occur in the future. By using interval numbers, we obtain a more complete aggregation operator that considers the maximum and minimum results that could occur for a given problem. Moreover, by using the IUEOWAD, we obtain a generalization that includes a wide range of uncertain distance measures and uncertain aggregation operators such as the uncertain maximum distance, the uncertain minimum distance, the uncertain normalized Euclidean distance (UNED), the uncertain weighted Euclidean distance (UWED) and the uncertain Euclidean ordered weighted averaging distance (UEOWAD) operator. It can be defined as follows.
Similarly, we have Assume the following weighting vector (0.2,0.4,0.1,0.3) W = and the order-inducing variables (2,3,9,6) U = . If we calculate the distance between A  and B  using the UIEOWAD, then we have: From the above example, we can see that, compared with existing kinds of uncertain induced aggregation operators (Xu 2006;Merigó, Casanovas 2011d, the distance of two interval numbers adopted in the IUEOWAD operator can reduce computations for interval numbers operations. Second, final evaluation values of uncertain induced aggregation operators are interval numbers. There are many methods to compare and rank interval numbers, and different methods employed may generate different and inconsistent results. The results of our proposed operator are crisp numbers, which can avoid inconsistent results by using an interval ranking method to rank the alternatives' final evaluation values. From a generalized perspective of the reordering step, we can distinguish between the descending IUEOWAD (DIUEOWAD) operator and the ascending IUEOWAD (AIUEOWAD) operator by using w j = w* n-j+1 , where w j is the jth weight of the DUIEOWAD and w* n-j+1 the jth weight of the AUIEOWAD operator.
Note that if the weighting vector is not normalized, i.e.
Similar to the IEOWAD operator, the IUEOWAD operator is also commutative, monotonic, bounded and idempotent. A further interesting issue is the problem of ties in the reordering process of the order inducing variables. In order to solve this problem, we recommend following the policy explained by Yager and Filev (1999) about replacing the tied arguments by their average. Note that in this case, it would mean that we are replacing the tied arguments by their uncertain normalized Euclidean distance.
Another interesting issue to consider is the measures for characterizing the weighting vector W of the IUEOWAD operator such as the attitudinal character, the entropy of dispersion, the divergence of W and the balance operator (Yager 1988;Merigó, Casanovas 2011a). The entropy of dispersion is defined as follows: The balance operator can be defined as: And the divergence of W : The degree of orness can be defined as follows: ( )

Families of IUEOWAD Operators
By choosing a different manifestation of the weighting vector in the IUEOWAD operator, we are able to obtain different types of distance aggregation operators. For example, we can obtain the uncertain maximum distance, the uncertain minimum distance, the uncertain normalized Euclidean distance (UNED), the uncertain weighted Euclidean distance (UWED) and the uncertain Euclidean ordered weighted averaging distance (UEOWAD) operator. The main advantage of using these families is that they can be very useful for the decision-maker in some specific situations. However, each family is just one particular case. Therefore, they can only be used in some particular cases, but they cannot be seen as a general model that can be used in all possible frameworks. Thus, the best way to assess all these particular cases is by using a general formulation such as the IUEOWAD operator that includes them all. Note that the particular case to be used will depend on the interests of the decision-maker in the specific problem at hand. Remark 1. The uncertain maximum distance is obtained by setting 1 1 w = , 0 , and the uncertain minimum distance by

Group decision-making with the IUEOWAD operator
In this section, we will propose a procedure for group decision-making problems with interval preference information by application of IUEOWAD operator. The main motivation for using this model is that the representation of the information is very complex and we need to use interval numbers and the opinion of several decision makers (experts) in order to correctly assess the problem. Moreover, the attitudinal character of the decision maker is very complex and we need to use order-inducing variables in order to assess it. This can be useful in a lot of situations, for example, when the board of directors of a company wants to take a decision. Obviously, the attitudinal character of the board of directors is very complex because it involves the decision of different persons and their interests may be different. Consider a group decision-making problem with uncertain preference information. Let

{ }
a preference value, which takes the form of interval argument, given by the decision maker k e E ∈ , for the alternative i x X ∈ with respect to the attribute i s S ∈ .
In general, there are benefit attributes and cost attributes in the multiple attribute decision-making problems. In order to measure all attributes in dimensionless units and facilitate inter-attribute comparisons, we introduce the following Eqs (10) and (11) (Xu 2011) to normalize each decision matrix , for benefit attribute j s ; (10) Then the process to follow in decision making based on the UIEOWAD operator can be summarized as follows (Zeng, Su 2011): Step 1. Normalize each decision matrix ( ) k A to ( ) k R by Eqs (10) and (11).
Step 2. Fix the ideal levels of each attribute to form the ideal strategy (see Table 1), where I is the ideal strategy expressed by a fuzzy subset, i s is the i th characteristic to consider. Step 3. Calculate the order-inducing variables ( ) ij m n u × to be used in the decision matrix for each alternative i and attribute j . Calculate also the weighting vector of the UIEOWAD operator.
Step 4. Use the uncertain weighted averaging (UWA) operator (Xu 2008) to aggregate the information of the decision-makers E by using the weighting vector V . The result is the uncertain unified decision matrix Step 5. Calculate the distance between the ideal values and the aggregated results by using the UIEOWAD operator. Note that it is possible to consider a wide range of IUEOWAD operators, such as those described in Sections 4.
Step 6. Adopt decisions according to the results found in the previous steps. Select the alternative/s that provides the best result/s. Moreover, establish an ordering or a ranking of the alternatives from the most to the least preferred alternative to enable consideration of more than one selection.

Illustrative example
In this section, we utilize a practical group decision-making problem to illustrate the application of the developed approaches. A military unit is planning to purchase new artillery weapons and there are four feasible artillery weapons (alternatives) ( 1,2,3,4) i x i = to be selected (adapted from Xu 2011). When making a decision, the attributes considered are as follows: (1) 1 s , assault fire capability indices (m); (2) 2 s , reaction capability indices (evaluated using 1-5 scale); (3) 3 s , mobility indices (m); (4) 4 s , survival ability indices (evaluated using 0-1 scale); and (5) 5 s , cost ($). Among these five attributes, ( 1,2,3,4) j s j = are of benefit type and 5 s is of cost type. An expert group that consists of three experts ( 1,2,3)  Tables 2-4).  Since the attributes ( 1,2,3,4,5) j s j = have different dimension units, then we utilize (10) and (11) to transform the decision matrices ( ) k A into the normalized decision matrices ( ) k R (see Tables 5-7). According to the objectives of the experts, they establish a consensus ideal levels that an artillery weapon should have. The normalized results are shown in Table 8. To analyze the attitudinal character of the group of experts, we consider that they use order-inducing variables shown in Table 9, which represents the complex attitudinal character in the decision process. Note that in this example, the decision-maker assumes a different attitudinal character for each alternative because the results given by each alternative are not equal. The main advantage of using order inducing variables is that we can represent complex decision processes that include psychological factors such as time pressure, personal affects to each alternative and other related aspects. With this information, we can make an aggregation to make a decision. First, we aggregate the information of the three experts to obtain a collective uncertain decision matrix. We use the uncertain weighted averaging (UWA) operator to obtain this matrix while assuming that (0.2,0.5,0.3) V = . The results are shown in Table 10. With this information, we can use the proposed decision-making method based on the IUEOWAD operator to get the best alternative(s). In order to provide a complete analysis of the different potential results that may occur depending on the interests of the decision makers, we present a wide range of particular cases of IUEOWAD operators such as the uncertain maximum distance, the uncertain minimum distance, the UNED, the UWED, the UEOWAD, the IUEOWAD, the Step-IUEOWAD ( 2) k = and the Olympic-UIEOWAD. Note that for the UEOWAD and the UIEOWAD operators we assume the following vector (0.1,0.2,0.2,0.2,0.3) W = , formed as a consensus agreement between the three experts where they evaluate the ordered argument that should have more importance in the analysis. The aggregated results obtained by using the previous particular cases of IUEOWAD operators are shown in Tables 11. The optimal choice would be the alternative closest to the ideal. As we can see, for most of the cases 1 x is the optimal choice excepting for an extreme pessimistic situation where 2 x is optimal. A further interesting issue is to establish an ordering of the alternatives. This is very useful when we want to consider more than one alternative. The results are shown in Table 12.
As we can see, we get different orderings depending on the aggregation operator used. Note that the main advantage of using the IUEOWAD operator is that we can consider complex reordering processes in an uncertain framework that can be assessed with interval numbers. Another main advantage of the IUEOWAD operator is that it gives a complete view of the decision-making problem by considering different particular types of IUEOWAD operators. Thus, depending on the interests of the decision maker, he will select a different type of IUEOWAD operator that will lead him to different results and decisions. In addition, our approach is more flexible than the existing uncertain induced aggregation operators because it can provide the decision makers more choices as the ideal levels are assigned different values according to the features of the specific situations.

Conclusions
We have introduced the IUEOWAD operator as an aggregation operator that uses the main characteristics of the IOWA operator, the Euclidean distance and uncertain information represented in the form of interval numbers. The main advantage of this operator is that it provides more complete information because it represents the information in a more complete way considering the maximum and the minimum results that can occur. Moreover, it includes many different types of uncertain distance measures and aggregation operators, such as the UNED, the UWED and the UEOWAD operator.
We have also presented an application of the new approach to a group decision-making problem concerning the selection of artillery weapons. We have seen that the IUEOWAD is very useful because it represents very well the uncertain information by using interval numbers. The main advantage of the IUEOWAD operator in decision-making is that it shows a lot of different scenarios that could happen depending on the particular type of IUEOWAD operator used in the problem. Thus, the decision maker gets a more complete view of the decision problem and is able to select the alternative that is closest to his interests. Moreover, by using order-inducing variables, it is possible to consider different scenarios according to complex attitudinal characters.
In future research we expect to develop further extensions by adding new characteristics in the problem such as the use of probabilistic aggregations. We will also consider other decision making applications such as human resource management and product management. Shouzhen ZENG graduated from the Tianjin University and obtained the Master degree in applied mathematics. At present, he is studying his in-service doctor of applied statistics in Zhejiang Gongshang University. Currently he is a full-time Lecture in Zhejiang Wanli University. He has published more than 40 papers in journals, books and conference proceedings including journals such as Statistics Research, Economic Research, Knowledge-based Systems and Group Decision and Negotiation. His main research interests are aggregation operators, decision making, comprehensive evaluation and uncertainty.
Xiaojia YE graduated from the Zhejiang Gongshang University and obtained her Master's degree in Statistics. At present, she is studying her in-service doctor of applied statistics in Zhejiang Gongshang University. Her main research interests are aggregation operators, decision making, comprehensive evaluation and uncertainty.