Group-decision making with induced ordered weighted logarithmic aggregation operators

. This paper presents the induced generalized ordered weighted logarithmic aggregation (IGOWLA) operator, this operator is an extension of the generalized ordered weighted logarithmic aggregation (GOWLA) operator. It uses order-induced variables that modify the reordering process of the arguments included in the aggregation. The principal advantage of the introduced induced mechanism is the consideration of highly complex attitude from the decision makers. We study some families of the IGOWLA operator as measures for the characterization of the weighting vector. This paper presents the general formulation of the operator and some special cases, including the induced ordered weighted logarithmic geometric averaging (IOWLGA) operator and the induced ordered weighted logarithmic aggregation (IOWLA). Further generalizations using quasi-arithmetic mean are also proposed. Finally, an illustrative example of a group decision-making procedure using a multi-person analysis and the IGOWLA operator in the area of innovation management is analyzed. Innovation management.


Introduction
Aggregation operators are becoming very popular in the literature, especially in the areas of economics, statistics and engineering [1]. Currently, the literature presents an extensive amount of aggregation operators [2][3][4]. The ordered weighted average (OWA) operator [5] stands as one of the most disseminated aggregation operator in the field. The OWA operator proposes a parameterized family including the maximum, the minimum and the average. This classic operator has been widely applied from applications in expert systems, group decision making, neural networks, data base systems, to fuzzy systems [6,7].
Yager and Filev [8] introduced an extension to the OWA operator, the induced ordered weighted average (IOWA) operator. This extension allows a broad-er treatment and representation of complex information. The introduced mechanism applies a reordering process to the arguments, here, a set of orderinduced variables determines the order of the aggregation. This reordering mechanism of the IOWA operator has attracted much attention, motivating a broad diversity of applications [9,10] e.g. [11] develops new families of IOWA operators. In [12] dissimilarity functions are included in the analysis, [13] study the use of fuzzy numbers, [14] consider intuitionistic fuzzy information, and [15] analyze hesitant fuzzy sets and the Shapley framework. [16,17] develop induced aggregation operators with linguistic information, [18] with distance measures, [19] with heavy operators and moving averages, [20] with Bonferroni and heavy operators, [21] with prioritized operators and [22] with distances and multi-region operators.
An interesting generalization of the OWA operator results when applying quasi-arithmetic means in the aggregation process. The outcome is the quasiarithmetic ordered weighted aggregation (Quasi-OWA) operator [23]. This operator combines a wide range of mean operators, including the generalized mean, the OWA operator, the ordered weighted geometric (OWG) operator, and the ordered weighted quadratic averaging (OWG) operator, among others. Some of the most representative extensions of the Quasi-OWA operator are, e.g., the uncertain induced quasi-arithmetic OWA (Quasi-UIOWA) operator [24], the combined continuous quasi-arithmetic generalized Choquet integral aggregation operator [25] and the quasi intuitionistic fuzzy ordered weighted averaging operator [26], among others.
Zhou and Chen [27] propose a generalization of the ordered weighted geometric averaging (OWGA) operator based on an optimal model. The introduced operator is the generalized ordered weighted logarithmic aggregation (GOWLA) operator. This contribution includes a set of parameterized families, such as the step generalized ordered weighted logarithmic averaging (Step-GOWLA) operator, the window generalized ordered weighted logarithmic averaging (Window-GOWLA) operator, and the S-GOWLA, among others. A further generalization of the GOW-LA operator is that introduced by Zhou, Chen, and Liu [28] designated the generalized ordered weighted logarithmic proportional averaging (GOWLPA) operator. Some generalizations of this operator are the generalized hybrid logarithmic proportional averaging (GHLPA) operator and the quasi ordered weighted logarithmic partial averaging (Quasi-OWLPA) operator. Following the trend of developing aggregation operators based on optimal deviation models, Zhou, Chen, and Liu [28] introduce the generalized ordered weighted exponential proportional aggregation operator (GOWEPA), which is further generalized to develop the generalized hybrid exponential proportional averaging (GHEPA) operator and the generalized hybrid exponential proportional averaging-weighted average (GHEPAWA) operator. Recently, Zhou, Tao, Chen, and Liu [29] have introduced an additional generalization to the GOWLA designated the generalized ordered weighted logarithmic harmonic averaging (GOWLHA) operator, including the generalized hybrid logarithmic harmonic averaging (GHLHA) operator and the generalized hybrid logarithmic harmonic averaging weighted average (GHLHAWA) operator.
The aim of this paper is the introduction of the induced generalized ordered weighted logarithmic aggregation (IGOWLA) operator. The newly introduced operator is an extension of the optimal deviation model [27] adding the order-induced variables that change the previous ordering mechanism of the arguments. The introduction of this mechanism seeks a broader representation of the complexity in certain scenarios.
We study a series of properties and families of the operator such as the induced ordered weighted logarithmic geometric averaging (IOWLGA) operator, the induced ordered weighted logarithmic harmonic averaging (IOWLHA) operator, and the induced ordered weighted logarithmic aggregation (IOWLA) operator, among others. Furthermore, we present some extensions of the operator, first, using quasiarithmetic means, obtaining the quasi induced generalized ordered weighted logarithmic aggregation operator (Quasi-IGOWLA) operator; second, utilizing moving averages, we develop the induced generalized ordered weighted logarithmic moving average (IGOWLMA) operator.
This paper also proposes an illustrative example to show the main characteristics of the IGOWLA operator. The example includes a multi-person decisionmaking analysis in the field of innovation management. The application seeks to exemplify a strategic decision-making process where a series of experts need to assess and choose new productos from a portfolio of options. The case includes a highly complex attitudinal character from management. Results show a clear difference in the aggregation when applying order-induced variables instead of using traditional operators. The operator can be useful for other decision-making applications in business, such as human resource management, strategic decision making, marketing, etc. This paper is organized as follows. Section 2 presents basic concepts of the OWA, IOWA, Quasi-IOWA, and GOWLA operators. Section 3 presents the IGOWLA operator, the characterization of the weighting vector and families. Section 4 presents the extension of the Quasi-IGOWLA operator. In Section 5 proposes an illustrative application of a decisionmaking procedure utilizing the IGOWLA. Finally, Section 7 summarizes the concluding remarks of the paper.

Preliminaries
In the present section, we briefly review some of the principal contributions in the field of aggregation operators. Specifically, we describe the OWA operator, the induced OWA operator, the Quasi-IOWA operator and the GOWLA operator.

The OWA operator
The ordered weighted averaging operator introduced by Yager [5] proposes a family of aggregation operators that have been used in a plethora of applications (see, e.g., [7]). The OWA operator can be defined as follows: Definition 1. An OWA operator is a mapping : n OWA R R → , which has an associated n vector where j b is the jth largest of the arguments i a .
It has been demonstrated that the OWA operator is commutative, idempotent, bounded and monotonic [5]. Furthermore, we can obtain the ascending OWA or the descending OWA by generalizing the direction of the reordering process [30].

The induced OWA operator
The induced ordered weighted averaging operator, introduced by [8] presents an extension of the OWA operator. This extension allows a reordering process that is defined by order-induced variables rather than the traditional ordering constructed from the values of the arguments.

The Quasi-IOWA operator
The quasi-arithmetic induced ordered weighted aggregation (Quasi-IOWA) operator presents an extension of the Quasi-OWA operator. The main difference is the reordering process; in this case, orderinduced variables dictate the complex reordering of the arguments. The Quasi-IOWA operator can be defined as follows: Note that the Quasi-IOWA can also be viewed as a generalized form of the IOWA operator by using quasi-arithmetic means. The Quasi-IOWA has a wide variety of particular cases [31] including, e.g., the IGOWA operator, the IOWA operator, the IOWGA operator, the IOWQA operator, and the IOWHA operator.

The GOWLA operator
The generalized ordered weighted logarithmic aggregation (GOWLA) operator [27] introduces a parameterized family of aggregation operators including the step-GOWLA operator, the window-GOWLA operator, the S-GOWLA operator and the GOWHLA operator The GOWLA operator can be formulated as follows: ; therefore, 1 j a  following the notation in Zhou and Chen [27],

The induced GOWLA operator
This paper presents the induced GOWLA (IGOWLA) operator it is in fact an extension of the GOWLA operator introduced by [27]; the new formulation of the IGOWLA includes a previous reordering step, this means that the IGOWLA operator is not defined by the values and order of the arguments i a but by order-induced variables i u , that define the position of the arguments i a by the values of the i u [32]. This extension allows a generalized ordering process, where decision making can consider highly complex conditions. The IGOWLA operator can be defined as follows: where λ is a parameter such that  It is possible to differentiate the operator between the descending induced generalized OWA (DI-GOWA) operator, and the ascending induced generalized OWA (AIGOWA) operator. Regardless, the operators noted above are connected by the relationship of Note that n j j Ww =  .

Characterization of the weighting vector
When defining the IGOWLA operator, it is interesting to analyze the characterization of the weighting vector. Following the procedures developed by Yager [5,33] and the descriptions stated in [32] we can obtain the degree of orness or attitudinal Due to the induced properties [31], the attitudinal character of the IGOWLA operator can be described from two different perspectives. If we focus on the attitudinal character, then we can use the same measure as in the OWA operator [5] because we want to measure the complex attitude, which depends solely on the weighting vector. In this case, the formulation is as follows: Bal W = . Note that this measure is applicable to any induced aggregation operators [8,32].
Finally, the divergence measure of the weighting vector can be obtained by: Example 2. Following the arguments described in Example 1, the characterization of the weighting vector result is shown in Table 1:

IGOWLA operator families
A group of families of the IGOWLA operator can be described when analyzing the parameter . Table  2 presents some of the resulting cases of special interest: Note that this formulation can also be presented as the IOWLA operator: Example 3. Following the arguments described in Example 1, the results for each family of the IGOW-LA operator are shown in Table 3.

The Quasi-IOWLA operator
It is possible to generate an additional generalization of the general ordered weighted averaging operators by utilizing quasi-arithmetic means instead of the ordinary means (see, e.g., [32,35]). In the case of the IGOWLA operator, we suggest the use of a similar methodology to construct the Quasi-IOWLA operator.
Definition 7. A Quasi-WLA operator of dimension n is a mapping QWLA : ΩΩ n → with an associated weighting vector W of dimension n such that . This approach is equivalent for the Quasi-WLA and the Quasi-OWLA. Therefore, all these operators share the properties studied for the IGOWLA operator; specifically, it is bounded, idempotent and commutative. However, as shown in section 3, in some cases, it is not monotonic.
Observe that we can also distinguish between the descending Quasi-DIOWLA and the ascending Quasi-AIOWLA. The relationship found between the descending and the ascending operators is The wide range of operators that quasi-arithmetic means provide have proven to be effective when treating problems covering a wide range of complexities [36], including geometric aggregations, quadratic aggregations, and harmonic aggregations. Proposition 2. In case of any ties, replacing the tied arguments with the quasi-arithmetic logarithmic average operator is proposed [32].

Decision-making process
The IGOWLA operator is suitable for a wide range of applications in decision making processes (see, e.g., [37][38][39][40]). Here, a decision-making application in innovation management is proposed to show the variations and benefits of the newly introduced orderinduced mechanisms of the IGOWLA operator.
The main reason for selecting this topic is the presentation of information, which, in the case of innovation, has been stated to be imprecise and uncertain [41,42]. Therefore, there is the motivation to use the opinion of different decision makers or experts to find a suitable solution.
Strategic decision making in innovation management addresses diverse aspects that include not only imprecise information [43] but also a certain level of attitudinal character on the part of the decision makers, e.g., the possibility of different strategic outcomes [44], complexity and unfamiliar interactions [45], the lack of information [46], time, flexibility and control. In this sense, the use of inducing variables should aid in the complex decision-making procedure.
Innovation management considers a wide range of problems to be assessed, one of them is the correct selection new products to be developed, this from a portfolio of possible prototypes. The effectiveness with which an organization manages its new products portfolio is often a key determinant of competitive advantage [47]. Here, portfolio management deals with the allocation of the scarce resources of the business, namely: money, time, people, machinery, etc. to potential developments under uncertain conditions. The key concepts to analyze are quantity, quality, and organizational capability for new product development. The selected new products to be developed must correctly align with business objectives and balance several elements such as timespan and risk.
The process to follow in the selection of strategies in innovation management with the IGOWLA operator and the application when introducing a multiperson analysis can be summarized as follows: Step 1. Assuming that    based on its preferences.

, , , m A A A A =
Step 2. Based on the highly complex attitudinal character of the case, introduce a set of orderinducing variables ( ) hi mn u  corresponding to each alternative h and characteristic i . Include a ( ) 12 , , , n W w w w = weighting vector, make sure that this verctor satisfies the IGOWLA operator formulation, next, define a  value to be applied in the aggregation operation.
Step 3. In this case we propose the weighted average to aggregate the information provided by the decision-makers E and the vector X . The aggregated information results in the collective payoff matrix Step 4. Solve for the IGOWLA operator as described in Eq. 6. Please note that  value is typically set as 1; however, any of the families described in section 3.2 can be used, depending on the problem analyzed.
Step 5. After solving for the IGOWLA operator, set a ranking of the alternatives; compare the results of the specific problem and propose a decisionmaking approach.

Illustrative Example
This paper proposes an illustrative example of the IGOWLA operator in a strategic decision-making process of portfolio management with multi-person inputs. Other business decision-making applications in the field of innovation management can be assessed e.g. knowledge management, project management, organization and structure, among others, please see [48].
Step 1. Let's assume that company Y is involved in the design of fast-moving consumer goods in the alimentary sector. The company must decide from its portfolio of new products and select one of five potential enhanced beverage concepts. Thus, we have: • A1 Super Sport: vitamin C with electrolytes • A2 High Energy: vitamin C with caffeine • A3 Fast Recover: vitamins B5, B6 and B12 • A4 0 Sugar Sport: vitamin C with electrolytes and no sugar • A5 AntiOx: manganese plus vitamin B3 This problem requires the inputs of several experts of the company to assure the relevance, appropriateness and a strategic alignment to the requirements of the business. The company sets 6 key factors to be analyzed in the selection process: • S1 Expected benefits • S2 Alignment to business • S3 Development costs • S4 Technical viability • S5 Risk • S6 Time to market The experts are divided in groups of 3 (Tables 4 -9). The first group (Table 4 and 5) has two engineering experts, the second includes two experts from marketing and sales (Table 6 and 7), and in the third group two financial experts (Table 8 and 9). The experts are asked to provide their opinion in a scale of 1 to 100, their opinions are bounded to the expected performance of each product based on the key factors selected by management. This case requires, firstly to generate a multi-aggregation process so the opinions of the groups can be aggregated. Secondly, we need to aggregate all the information into a sole collective payoff matrix. Once we obtain the matrix, we use the IGOWLA operator to generate the final results and aid the board of directors in the selection of the most suitable alternative for the elements that constitute the problem.
Step Step 3. For this case, we will consider the weighting vectors X , representing the different importance of each expert in the analysis. For the first group of experts we have . All the elements have been correctly defined, therefore we can obtain results by first aggregating the opinions of the three groups of experts using the weighted average; the results are shown in Tables 10, 11 and 12. Using this information, we now aggregate the three subgroups into a collective payoff matrix. The results are shown in Table 13.
Step 4. Solving for the IGOWLA operator families, we aggregate the collective information and obtain results. Table 14 show the final aggregations.
Step 5. The problem requires a visualization of the diverse decisions that can be generated. Therefore, we establish a ranking of the performance of each product. The preferred ordering of the alternatives is presented in Table 15. The ≻ symbol represents preferred to.

A A A A A
Results show that the elements have been ordered in different ways, depending directly on the operator utilized in the aggregation of the arguments. In this hypothetical case, which includes the diverse expert opinion of six persons and the highly complex attitudinal characteristics of the direction board, the exercise concludes that the concepts that should be firstly developed are products: 1 A (Super Sport) and 4 A (No Sugar Sport). Please note that the induced operators show a different ranking from the traditional ones, this indicates a clear difference when introducing order-induced mechanism to the reordering process. The aggregated results show no specific ties; therefore, the use of the proposed quasi-arithmetic means is not required in this case. Please also note that the multi-person process can be aggregated and presented in many other approaches; in this example it is assumed that the management board needed the information presented as represented in the example.

Conclusions
This paper presents the IGOWLA operator, it is a generalization of the GOWLA operator, therefore the introduced operator shares its main characteristics. The order-induced variables included in the formulation of the IGOWLA operators, allows an even wider representation of the possible highly complex attitude of decision makers in certain problems.
Diverse measures for characterizing the weighting vector have been analyzed; specifically, we have studied the degree of orness measure, the dispersion measure, the balance measure and the divergence measure. Note that some of these measures can be calculated from two different perspectives, depending on the attitudinal character or the numerical value of the weighting vector. Furthermore, we describe several families of the IGOWLA operator based on the parameter, including the IOWLGA operator, the IOWLHA operator, the IOWGA operator, the IOWLA operator, the IOWLQA operator, the IOWLC operator, and the maximum and minimum IGOWLA operators.
We introduce diverse generalizations of the IGOWLA operator. First, using the notion of quasiarithmetic means, we introduce the QWLA operator, the QOWLA operator, and the QIOWLA operator, therefore adding the option of considering geometric aggregations, quadratic aggregations and harmonic aggregations into the process.
The IGOWLA operator has been designed to aid group decision making, and could be used in several areas, such as economics, statistics and engineering problems. This paper proposes an illustrative example of a possible utilization of the IGOWLA operator. Here, a multi-expert for strategic decision-making process in the area of innovation management is exemplified. The case deals with the assessment of a decision in portfolio management of a company. The objective is the selection of new products to be developed based on diverse characteristics of the products and the alignment to the objectives and preferences of the studied case. This example seeks to show the components of the IGOWLA operator, i.e. the order-induced variables, the construction of scenarios including the generalized lambda vector, and the option of dealing with diverse expert opinions and the highly complex attitudinal characteristics of the aggregation elements.
Further developments and research need to be assessed. Firstly, deepen the mathematical characteristics of the logarithmic properties that build the IGOWLA operator. Secondly, new extensions should be developed e.g. to assess uncertain information, i.e., fuzzy numbers, linguistic variables and interval numbers, the inclusion of distance measures and the possibility of working with heavy aggregations, the new extensions allow the construction of complex formulations that could aid decision making problems in wider scenarios. Finally, new decision-making problems in diverse fields of knowledge should be considered for the application of the newly introduced tools.