**2. Preliminaries**

In this section, we review some of the required basic concepts related to the OWA operator in this article. This operator is supported by criteria that are the bases of a decision that integrate the expectations of the decision makers in the evaluation that he or she makes of the set of actions to be taken [39]. Likewise, the OWA operator has the versatility to add data without losing its mathematical properties. Furthermore, according to the arguments, the qualifications can obtain evaluated alternatives. Thus, operators such as the HOWA, IOWA, PrOWA, PIOWA and IHOWA have been proposed and studied. Additionally, the OWA operator has allowed the development of several extensions that combine new parameters and interactions with other methods and some other extensions [39]. Among these, the BON-OWA, BON-HOWA, BON-IOWA and BON-PrOWA will also be studied to fulfil the purpose of the research. Hence, each of the definitions of the operators mentioned above is presented below.

#### *2.1. OWA Operator and Its Main Extensions*

The OWA operator was introduced by Yager [18], and its main feature is that it is possible to obtain the maximum and minimum values according to the operator's rearrangemen<sup>t</sup> weight. The purpose of this operator is to obtain a single representative value from the aggregation of a series of data that reflect the predetermined optimism/pessimism parameters. It is defined as follows:

**Definition 1.** *An OWA operator of dimension n is a mapping of OWA* : *Rn* → *R with a weight vector W of dimension n with* ∑*n i*=1 *wi* = 1 *and wi* ∈ [0, 1] *such that:*

$$OWA(a\_1, a\_2, \dots, a\_n) = \sum\_{j=1}^n w\_j b\_j \tag{1}$$

*where bj is the jth element and the largest of the collection a*1, *a*2,..., *an*.

The fundamental characteristic of the OWA operator is that the rearrangemen<sup>t</sup> of the elements or arguments allows argumen<sup>t</sup> *aj* not to be associated with weight *wj* weight if all *wj*s are associated with the position in the order for aggregation.

**Definition 2.** *As introduced by Merigo and Gil-Lafuente [21], an IOWA operator of dimension n is an application IOWA* : *Rn* → *R that has an associated weight vector W of dimension n where the sum of the weights is 1, wj* ∈ [0, 1]*, and an induced set of variables of order are included (ui). The formula is*

$$IOWA(\langle u\_1, a\_1 \rangle, \langle u\_2, a\_2 \rangle, \dots, \langle u\_n, a\_n \rangle) = \sum\_{j=1}^n w\_j b\_{j\prime} \tag{2}$$

*where* (*b*1, *b*2,..., *bn*) *is simply* (*<sup>a</sup>*1, *a*2,..., *an*) *reordered descending or ascending according to the values of ui . bj is the ai value of the OWA pair* < *ui*, *ai* > *having the jth largest ui . ui is the order inducing variable, and ai is the argument variable. These operators take argument pairs, called OWA pairs, in which a component is used to induce an order on the second components that are then added.*

Among the extensions of the OWA operator that focus on the weight vector is the heavy OWA (HOWA) operator [23]. In this extension, the weight vector is not ∑*n j*=1 *wj* = 1 but is unbounded; therefore, the weighting vector can be 1 ≤ ∑*n j*=1 *wj* ≤ *n*. The definition is as follows:

**Definition 3.** *An HOWA operator is a mapping HOWA* : *Rn* → *R that is associated with a weight vector w, where wj* ∈ [0, 1] *and* 1 ≤ ∑*n j*=1 *wj* ≤ *n, such that*

$$HOWA\left(a\_1, a\_2, \dots, a\_n\right) = \sum\_{j=1}^n w\_j b\_{j\prime} \tag{3}$$

*where bj is the jth largest element of collection ai. It is also important to note that in some cases, it is possible that the weight vector is* − ∞ ≤ ∑*n j*=1 *wj* ≤ <sup>∞</sup>*, making it possible to under- or overestimate the results according to the expectations of the decision maker. It is important to note that Yager (2002) also developed a characteristic of the HOWA operator, which is called the beta value. This beta value can be defined as β*(*W*) = (|*W*| − <sup>1</sup>)/(*n* − 1)*. Note that if β* = 1*, we obtain the total operator, and if β* = 0*, we obtain the usual OWA operator.*

**Definition 4.** *The prioritized OWA (PrOWA) operator developed by Yager [40] is an aggregation operator that is useful when problem-solving decision makers do not have the same standing in the final decision. Thus, this operator allocates an additional impact to some decision makers and less to others. This operator can be defined as follows (Yager 2008, 2009a). A prioritized OWA (PrOWA) of dimension n is a mapping PrOWA* : *Rn* → *R that has an associated vk that is the corresponding weight of the jth criterion in the ith category.where Ci*(*x*) = *ai* ∈ [0, 1] *is the degree of satisfaction with criterion Ciby alternative x*.

$$V\_k \in [0, 1] \text{ and } \sum\_{k=1}^{n} V\_k = 1,\tag{4}$$

*where aind*(*k*) *is the kth largest element of collection Ci*(*x*).

$$\mathbb{C}\_{(x)} = \sum\_{i=1}^{q} \sum\_{l=1}^{n\_i} w\_{lj} \mathbb{C}\_{ij}(x),\tag{5}$$

*which allows us to obtain ind(j). We calculate this number using the subscript of the associated Ci.*

$$
\tilde{R}\_K = \sum\_{i=1}^k r\_{ind(i)'} \tag{6}
$$

$$r\_i = \frac{T\_i}{\sum\_{j=1}^{n} T\_j} \tag{7}$$

$$
v\_k = f\left(\mathbb{X}\_K\right) - f\left(\mathbb{X}\_{K-1}\right),\tag{8}$$

$$C\_{(x)} = \sum\_{i=1}^{n} v\_k \cdot a\_{ind(k)}.\tag{9}$$

$$T\_1 = 1, \ T\_i = \mathbb{C}\_{i-1} T\_{i-1} \quad \text{for } i = 2 \text{ to } n,\tag{10}$$

*where bj is the jth element that has the largest value of ui ; ui is the induced order of variables; v*ˆ*ij is the corresponding weight of the jth criterion in the ith category for each i* = 1, ... , *q and j* = 1, ... , *ii ; and Cij*(*x*) *measures the satisfaction of the jth criterion in the ith group by alternative x* ∈ *X for each i* = 1, . . . , *q and j* = 1, . . . , *ii*.

**Definition 5.** *A prioritized induced OWA (PIOWA) of dimension n is a mapping PIOWA* : *Rnx Rn* → *R that has an associated weight vector w of dimension n*, *where wj* ∈ [0, 1] *and* ∑*nj*=<sup>1</sup> *wj* = 1*, such that*

$$PIOWA(\langle u\_1, a\_1 \rangle, \langle u\_2, a\_2 \rangle, \dots, \langle u\_n, a\_n \rangle) = \sum\_{i=1}^q \sum\_{h=1}^{n\_i} b\_j \mathfrak{d}\_{ij} \mathbb{C}\_{ij}(\mathbf{x}), \tag{11}$$

*where bj is the jth element that has the largest value of ui ; ui is the induced order of variables; v*ˆ*ij is the corresponding weight of the jth criterion in the ith category for each i* = 1, ... , *q and j* = 1, ... , *ii ; and Cij*(*x*) *measure the satisfaction of the jth criterion in the ith group by alternative x* ∈ *X for each i* = 1, . . . , *q and j* = 1, . . . , *ii*.

Another extension takes the reordering process of the IOWA operator and the unbounded weighting vector of the HOWA operator. This operator is called the induced heavy OWA (IHOWA) operator. The definition is as follows (Merigó and Casanovas 2011).

**Definition 6.** *An IHOWA operator of dimension n is a mapping IHOWA* : *Rn* × *Rn* → *R that has an associated weighting vector W of dimension n with wj* ∈ [0, 1] *and* 1 ≤ ∑*nj*=<sup>1</sup> *wj* ≤ *n such that*

$$IHOWA(\langle u\_1, a\_1 \rangle, \langle u\_2, a\_2 \rangle, \dots, \langle u\_n, a\_n \rangle) = \sum\_{j=1}^n w\_j b\_{j\prime} \tag{12}$$

*where bj is the ai of the IHOWA pair* < *ui*, *ai* > *having the jth largest ui*. *ui is the order inducing variable, and ai is the argument variable.*
