**1. Introduction**

The Organization for Economic Co-operation and Development (OECD) has recently recognized open governmen<sup>t</sup> initiatives as critical drivers of citizens' trust and key aspects of the modernization, anticorruption, civic freedom, innovation, financial managemen<sup>t</sup> and human resource managemen<sup>t</sup> of the public sector of a country [1]. Moreover, a culture of transparency, participation and accountability that conforms to open governmen<sup>t</sup> yields opportunities for economic growth, as it promotes the creation of businesses, jobs and cost-effective public policies [2]. Nonetheless, the design, creation and implementation of effective open governmen<sup>t</sup> strategies pose a series of challenges for countries, including their alignment with national plans, strategic visions, public governance and technological resources [3–5].

Transparency and access to information are key issues for the establishment of open governments. Governmental transparency is the ability to determine what is happening inside the governmen<sup>t</sup> [6]. Moreover, transparency fosters the accountability of actions and offers information to citizens regarding governmental decisions [7], thereby dissuading corruption and promoting efficiency, democracy and legitimacy [8]. In this sense, information is an asset, and while some administrations may use it as a trigger for best practices, others may have a radically different opinion based on their own political, administrative,

**Citation:** Perez-Arellano, L.A.; Blanco-Mesa, F.; Leon-Castro, E.; Alfaro-Garcia, V. Bonferroni Prioritized Aggregation Operators Applied to Government Transparency. *Mathematics* **2021**, *9*, 24. https://dx.doi.org/10.3390/ math9010024

Received: 2 December 2020 Accepted: 21 December 2020 Published: 24 December 2020

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institutional and demographic contexts [9,10]. These variations based on country contexts constitute the difference between freedom of information laws, their design and operations and the challenges they have for their nations, e.g., Canada and the United Kingdom or the open governmen<sup>t</sup> of the People's Republic of China [11].

In Mexico, access to information is a citizen's right composed of three elements: normative design, institutional design, and procedures for access to public information and transparency obligations [12]. The National Institute of Transparency (INAI) is a specialized public institution that regulates transparency at the national level, including access to information, personal data protection and the development of methodologies to assess transparency [13]. Additionally, the ranking of transparency websites is measured through five components: institutional arrangements, open data, vertical collaboration, horizontal collaboration, and interface [14]. The main difficulties with this formula are that it takes an average of the results that depend on the state; some of the components are more important than others. Because the calculation is made with the same weights for each subindex for all the states, there is no real evaluation of transparency depending on the specific characteristics and problems of each state.

Recent developments in information technologies have opened the path for assessing decision-making in systemic environments. Expert and intelligent systems have proven effective in subjective, uncertain and highly complex scenarios [15,16]. In this context, to address some of the abovementioned challenges, a combination of several intelligent systems such as the Bonferroni means [17] and the ordered weighted averaging (OWA) operator [18] will be used. A special focus will be placed on the following extensions: (a) the Bonferroni ordered weighted averaging (BON-OWA) operator [19] allows adding information and making multiple comparisons between input arguments and capturing their interrelation to present information, (b) the induced ordered weighted averaging (IOWA) operator [20,21] uses induced variables in the reordering step instead of the traditional reordering based on the value of the arguments of the OWA operator, (c) the prioritized ordered weighted averaging (PrOWA) operator [22] introduces a mechanism for assigning specific weights to the participants in a group decision-making problem, and, finally, (d) the heavy ordered weighted averaging (HOWA) operator [23] features a nonbounded weighting vector that allows the over- or underestimation of results according to the expectation and knowledge of the decision maker.

According to Blanco-Mesa, León-Castro and Merigó [24], aggregation operators allow joining different pieces of information provided by several sources [25], ensuring the inclusion of all the fusion information [26,27] and combining several values into a single value [15,28]. Since the proposal of the BON-OWA operator, several new methodological contributions have been made, among which those developed by Blanco-Mesa, such as (1) the Bonferroni means with distance measures applied to entrepreneurship and human resource managemen<sup>t</sup> [29,30], (2) the Bonferroni induced operator and heavy operator applied to enterprise risk managemen<sup>t</sup> and sale forecasting [31,32], (3) the Bonferroni OWA variance used in strategic analysis in enterprise risk managemen<sup>t</sup> [33], and (4) the Bonferroni covariance OWA used in research and development investment problems [34], stand out as addressing decision-making problems in business management. Recently, a paper has been published that proposed measuring transparency with another aggregation method called the prioritized induced ordered weighted average weighted average (PIOWAWA) operator. This operator considers the degree of importance, reordering and weight factors given to the information in the same formulation by the decision maker and is assessed using a Colombian transparency case [35]. Additionally, formulations have become widespread, and extensions have been proposed with other operators, such as the induced OWA operator (IOWA) [20,21], the heavy OWA operator (HOWA) [23], the OWAWA operator [36] and immediate weights (IWs) [37].

Following the above ideas, it is interesting to explore other operators that can be combined with the Bonferroni means. In that sense, one of the operators that can be extended is the prioritized OWA operator [38]. This operator is characterized by balancing the impact that a decision maker has on decision problems where he or she does not have the same position in the final decision, i.e., this operator assigns an additional impact to some decision makers and less to others. In the case of this research, it is very useful in problems calculating and evaluating the importance of each component because of their interrelationship, their interdependence and the importance that various agents have in this evaluation process.

The objective of this paper is to present a new extension of the BON-OWA operator using the extensions described above in a single formulation. The introduced operator is the Bonferroni prioritized induced heavy OWA (BON-PrIHOWA) operator. The main advantage of this operator is the consideration of a group decision-making problem in a single formulation including a nonlimited to zero weighting vector and an induced weighting vector capable of assigning weights according to the highly complex conditions of the analyzed phenomena. These features allow the analysis of a changing classification according to the additional information provided and the consideration of new scenarios for accurate results. The newly introduced BON-PrIHOWA is used as a method for ranking the transparency portals for the 32 states in Mexico based on experts.

The remainder of this document is organized as follows. In Section 2, we present some of the basic aggregation operators. Section 3 presents the new proposed operator, the BON-PrIHOWA operator. In Section 4, the evaluation of the characteristics of the transparency websites in Mexico based on different experts and aggregation operators are included. Finally, in Section 5, the conclusions of the document are presented.
