**2. Preliminaries**

This section presents the basic concepts that have been used throughout the paper including OWA, IOWA, OWMA operators and Pythagorean membership grades, their formulations and their main characteristics.

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

The main advantage of the OWA operator developed by [3] is that the process of reordering the weights is based on the values of the attributes. Thus, it is possible to obtain the maximum and minimum results. The formulation is as given below.

**Definition 1.** *An OWA operator of dimension n is a mapping OWA* : *Rn* → *R that has an associated weighting vector* W *of n dimensions with wj* ∈ [0, 1] *and n* ∑ *j*=1 *wj* = 1, *such that:*

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

*where bj is the jth largest of ai*.

Instead of basing its reordering process on the values of the attributes, the induced OWA (IOWA) operator [10] uses an induced vector. This makes it possible to discriminate some arguments by using weights based on the information and knowledge of the decisionmaker. The formulation is as follows:

**Definition 2.** *An IOWA operator of n dimensions is an application IOWA* : *Rn* × *Rn* → *R that has an associated weighting vector W of n dimensions, where the sum of the weights is 1 and wj* ∈ [0, 1]*, where an induced set of ordering variables is included* (*ui*) *such that 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 bj is the ai value of the OWA pair ui*, *ai with the jth largest ui, ui is the order inducing variable and ai is the argument variable*.

The moving average is a method used to forecast the future for different variables by using historical data, which is why it is a common technique in economics and statistics [41]. The moving average can be defined as follows [42]:

**Definition 3.** *Given* {*ai*}*Ni*=1*, the moving average of n dimensions is defined as the sequence* {*si*}*<sup>N</sup>*−*n*+<sup>1</sup> *i*=1*, which is obtained by taking the arithmetic mean of the sequence of n terms, such that:*

$$s\_i = \frac{1}{n} \sum\_{j=i}^{i+n-1} a\_j. \tag{3}$$

*It is important to note that, in every case,n<N*.

Moreover, following the idea of moving averages and the OWA operator, the ordered weighted moving average (OWMA) and the induced ordered weighted moving average can be proposed [31].

**Definition 4.** *An OWMA operator of m dimensions is a mapping OWMA* : *Rm* → *R that has an associated weighting vector W of m dimensions with wj* ∈ [0, 1] *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1,*such that:*

$$
\langle OWMA(a\_{1+t}, a\_{2+t}, \dots, a\_{n+t}) \rangle = \sum\_{j=1+t}^{m+t} w\_j b\_{j\nu} \tag{4}
$$

*where bj is the jth largest argument of ai, m is the total number of arguments considered in the whole sample and t indicates the movement of the average from the initial analysis*.

**Definition 5.** *An IOWMA operator of m dimensions is a mapping IOWMA* : *R<sup>M</sup>* × *R<sup>M</sup>* → *R that has an associated weighting vector W of m dimensions with wj* ∈ [0, 1] *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1*, such that:*

$$IONMA(\langle u\_{1+t}, a\_{1+t} \rangle, \langle u\_{2+t}, a\_{2+t} \rangle, \dots, \langle u\_{n+t}, a\_{n+t} \rangle) = \sum\_{j=1+t}^{m+t} w\_j b\_{j\cdot} \tag{5}$$

*where bj is the ai value of the IOWMA pair ui, ai is the jth largest ui, ui is the order-inducing variable, ai is the argument variable, m is the total number of arguments considered in the whole sample and t indicates the movement of the average from the initial analysis*.

#### *2.2. Pythagorean Membership Grades Used in Multiple Criteria Decision-Making*

**Definition 6.** *MCDM considers a finite collection X of alternatives and a set of q criteria that we desire to satisfy. These criteria are referred to as cj for j* = 1 *to q. Each criterion cj is associated with an importance weight wj such that wj* ∈ [0, 1] *and* ∑ *wj* = 1*. Likewise, cj*(*x*) *indicates the degree of satisfaction of criterion cj by alternative x*.

$$C(\mathbf{x}) = \left(\sum\_{j=1}^{q} w\_j c\_j(\mathbf{x})\right),\tag{6}$$

*Let us now consider the situation in which the values of cj*(*x*) *are Pythagorean membership grades* [15]. *Here, each cj*(*x*) = (*a*(*x*), *b*(*x*))*, where a*(*x*) *and b*(*x*) ∈ [0, 1] *and aj*(*x*)<sup>2</sup> + *bj*(*x*)<sup>2</sup> ≤ 1.

$$\mathcal{C}(\mathbf{x}) = \left(\sum\_{j=1}^{q} w\_j a\_j(\mathbf{x}), \sum\_{j=1}^{q} w\_j b\_j(\mathbf{x})\right),\tag{7}$$

*where aj*(*x*) *and bj*(*x*) *indicate the degree of satisfaction of criterion cj by alternative x*.

Additionally, this formulation is completed via the following function, which is based on fuzzy rules [15].

$$F(r, \theta) = \frac{1}{2} + r\left(\frac{1}{2} - \frac{2\theta}{\pi}\right). \tag{8}$$

#### **3. Pythagorean Membership Grade Aggregation Operators**

This section presents new operators that combine the IOWA and OWMA operators with Pythagorean membership grades.

#### *3.1. Extensions of the Pythagorean OWA*

Extensions of the Pythagorean OWA are new propositions that combine the characteristics of IOWA and OWMA operators and Pythagorean membership grades. These new formulations are important because, when imprecise and ambiguous information is present in the problem, this needs to be analyzed, Pythagorean and OWA operators have proven to be useful [43–46]. Due to this, expanding the formulations by using more complex situations, such as those that can be analyzed with the IOWA and OWMA operators, presents a good opportunity to generate new results by considering a new reordering process based on induced values or problems that use time series. The new formulations are as follows. Since we present a new formulation, the notation *Bj*(*x*) = (*B*(*x*), *<sup>B</sup>*(*x*)) to *Bj*(*x*)<sup>2</sup> + *<sup>B</sup> j*(*x*)<sup>2</sup> ≤ 1 corresponds to the notation *aj*(*x*)<sup>2</sup> + *bj*(*x*)<sup>2</sup> ≤ 1 to distinguish the new contribution.

**Proposition 1.** *A Pythagorean membership grade induced OWA operator (PMGIOWA) is an extension of the OWA operator. Thus, an PMGIOWA operator is a map Rn* → *R that is associated with a weight vector w, with wj* ∈ [0, 1] *and n* ∑ *j*=1 *wj* = 1*. Additionally, each Bj*(*x*) = (*B*(*x*), *<sup>B</sup>*(*x*))*, where <sup>B</sup>*(*x*) *and <sup>B</sup>*(*x*) ∈ [0, 1] *and Bj*(*x*)<sup>2</sup> + *<sup>B</sup> j*(*x*)<sup>2</sup> ≤ 1*, such that:*

$$PMGIOWA(\mathbf{x}) = \left(\sum\_{j=1}^{q} w\_j B\_j(\mathbf{x}), \sum\_{j=1}^{q} w\_j B'\_j(\mathbf{x})\right),\tag{9}$$

*where Bj*(*x*) *and <sup>B</sup> j*(*x*) *indicate the degree of satisfaction of criterion Bj*(*x*) *by alternative x. Thus, Bj*(*x*) *is the ai value of the OWA pair ui*, *ai with the jth largest ui, ui is the order-inducing variable and ai is the argument variable*.

**Proposition 2.** *A Pythagorean membership grade OWMA operator (PMGOWMA) is a mapping PMGOWMA* : *R M* × *R M* → *R that has an associated weighting vector W of m dimensions with wj* ∈ [0, 1] *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1*. Additionally, each Bj*(*x*) = (*B*(*x*), *B* (*x*))*, where <sup>B</sup>*(*x*) *and B* (*x*) ∈ [0, 1] *and Bj*(*x*) 2 + *B j*(*x*) 2 ≤ 1, such that:

$$PMGOWA(\mathbf{x}) = \left(\sum\_{j=1+t}^{m+t} w\_j B\_j(\mathbf{x})\_\prime \sum\_{j=1+t}^{m+t} w\_j B'\_j(\mathbf{x})\right),\tag{10}$$

*where Bj*(*x*) *and B j*(*x*) *indicate the degree of satisfaction of criterion Bj*(*x*) *by alternative x. Thus, Bj*(*x*) *is the jth largest argument of ai, m is the total number of arguments considered in the whole sample and t indicates the movement of the average from the initial analysis*.

**Proposition 3.** *A Pythagorean membership grade induced OWMA operator (PMGIOWMA) is a map PMGIOWMA* : *R M* × *R M* → *R that that has an associated weighting vector W of m dimensions with wj* ∈ [0, 1] *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1*. Additionally, each Bj*(*x*) = (*B*(*x*), *B* (*x*))*, where <sup>B</sup>*(*x*) *and B* (*x*) ∈ [0, 1] *and Bj*(*x*) 2 + *B j*(*x*) 2 ≤ 1, *such that:*

$$PMIGOWMA(\mathbf{x}) = \left(\sum\_{j=1+t}^{m+t} w\_j B\_j(\mathbf{x}), \sum\_{j=1+t}^{m+t} w\_j B'\_j(\mathbf{x})\right),\tag{11}$$

*where Bj*(*x*) *and B j*(*x*) *indicate the degree of satisfaction of criterion Bj*(*x*) *by alternative x. Thus, Bj*(*x*) *is the ai value of the OWA pair ui*, *ai with the jth largest ui*, *ui is the order-inducing variable and ai is the argument variable. Likewise, m is the total number of arguments considered in the whole sample and t indicates the movement of the average from the initial analysis*.

The Pythagorean membership grade has the property that the sum of the squares must be less than 1 [15]. We now prove that the new operators meet that condition.

**Theorem 1.** *If for i* = 1, 2, ... , *q*, *we have Bi*, *B j* ∈ [0, 1] *and wi* ∈ [0, 1] *with* (*ui*, *ai*) 2 + (*u i*, *a i*) 2 ≤ 1 *and* ∑*i wi* = 1*, then* (∑*i wi*(*ui*, *ai*))<sup>2</sup> + (∑*i wi*(*u i*, *a i*))<sup>2</sup> ≤ 1.

**Proof.**

(∑*i wi*(*ui*, *ai*))<sup>2</sup> + (∑*i wi*(*u i*, *a i*))<sup>2</sup> = ∑*i w*<sup>2</sup> *i* (*ui*, *ai*) 2 2 + ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wi*(*ui*, *ai*)*wj uj*, *aj* + ∑*i w*<sup>2</sup> *i* (*u i*, *a i*) 2 + ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wi*(*u i*, *a i*)*wj u j*, *a j* . (∑*i wi*(*ui*, *ai*))<sup>2</sup> + (∑*i wi*(*u i*, *a i*))<sup>2</sup> = ∑*i w*<sup>2</sup> *i* (*ui*, *ai*) 2 + (*u i*, *a i*) 2 + ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wiwj* (*ui*, *ai*) *uj*, *aj* + (*u i*, *a i*) *u j*, *a <sup>j</sup>*. (∑*i wi*(*ui*, *ai*))<sup>2</sup> + (∑*i wi*(*u i*, *a i*))<sup>2</sup> ≤ ∑*i w*<sup>2</sup> *i* + ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wiwj* (*ui*, *ai*) *uj*, *aj* +(*u i*, *a i*) *u j*, *a <sup>j</sup>*.

**Theorem 2.** *For a moving average, if for i* = 1, 2, ... , *q, we have Bi*, *B i* ∈ [0, 1] *and wi* ∈ [0, 1] *with Bi*+*t* 2 + *B* <sup>2</sup> *i*+*t* ≤ 1 *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1*, then* (∑*i wiBi*+*t*) 2 + (∑*i wiB <sup>i</sup>*+*t*) 2 ≤ 1.

**Proof.**

$$\begin{split} \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}\right)^{2} + \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}'\right)^{2} &= \left(\sum\_{i} w\_{i}^{2} \boldsymbol{B}\_{i+1}^{2}\right)^{2} + \sum\_{1 \neq j, i < j} 2w\_{i} \boldsymbol{B}\_{i+1} \boldsymbol{w}\_{j} \boldsymbol{B}\_{j+1} + \left(\sum\_{i} w\_{i}^{2} \boldsymbol{B}\_{j+1}'\right) + \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}'\right)^{2} \\ \sum\_{i \neq j, i < j} 2w\_{i} \boldsymbol{B}\_{i+1}' \boldsymbol{w}\_{j} \boldsymbol{B}\_{j+1}' &= \\ \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}\right)^{2} + \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}'\right)^{2} &= \sum\_{i} w\_{i}^{2} \left(\boldsymbol{B}\_{i+1}^{2} + \boldsymbol{B}\_{i+1}^{2}\right) + \sum\_{1 \neq j, i < j} 2w\_{i} w\_{j} \left(\boldsymbol{B}\_{i+1} \boldsymbol{B}\_{j+1} + \boldsymbol{B}\_{i+1}' \boldsymbol{B}\_{j+1}'\right) \cdot \\ \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}\right)^{2} + \left(\sum\_{i} w\_{i} \boldsymbol{B}\_{i+1}'\right)^{2} &\leq \sum\_{i} w\_{i}^{2} + \sum\_{1 \neq j, i < j} 2w\_{i} w\_{j} \left(\boldsymbol{B}\_{i+1} \boldsymbol{B}\_{j+1} + \boldsymbol{B}\_{i+1}' \boldsymbol{B}\_{j+1}'\right) \cdot \\ \square$$

**Theorem 3.** *For an induced moving average, if for i* = 1, 2, ... , *q, we have Bi*, *<sup>B</sup> i* ∈ [0, 1] *and wi* ∈ [0, 1] *with* (*ui*+*t*, *ai*+*t*)<sup>2</sup> + (*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*)<sup>2</sup> ≤ 1 *and m*+*t* ∑ *j*=1+*<sup>t</sup> wj* = 1, *then* (∑*i wi*(*ui*+*t*, *ai*+*t*))<sup>2</sup> + (∑*i wi*(*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*))<sup>2</sup> ≤ 1.

**Proof.**

(∑*i wi*(*ui*+*t*, *ai*+*t*))<sup>2</sup> + (∑*i wi*(*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*))<sup>2</sup> = ∑*i w*2*i* (*ui*+*t*, *ai*+*t*)<sup>2</sup><sup>2</sup><sup>+</sup> ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wi*(*ui*+*t*, *ai*+*<sup>t</sup>*)*wj uj*+*t*, *aj*+*t* + ∑*i w*2*i* (*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*)<sup>2</sup><sup>+</sup> ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wi*(*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*<sup>t</sup>*)*wj <sup>u</sup> <sup>j</sup>*+*t*, *<sup>a</sup> <sup>j</sup>*+*t* (∑*i wi*(*ui*+*t*, *ai*+*t*))<sup>2</sup> + (∑*i wi*(*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*))<sup>2</sup> = ∑*i w*2*i* (*ui*+*t*, *ai*+*t*)<sup>2</sup> + (*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*)<sup>2</sup><sup>+</sup> ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wiwj* (*ui*+*t*, *ai*+*<sup>t</sup>*) *uj*+*t*, *aj*+*t* + *<sup>u</sup> <sup>j</sup>*+*t*, *<sup>a</sup> <sup>j</sup>*+*<sup>t</sup> <sup>u</sup> <sup>j</sup>*+*t*, *<sup>a</sup> <sup>j</sup>*+*t* (∑*i wi*(*ui*+*t*, *ai*+*t*))<sup>2</sup> + (∑*i wi*(*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*t*))<sup>2</sup> ≤ ∑*i w*2*i* + ∑1=*j*,*i*<*<sup>j</sup>* <sup>2</sup>*wiwj* (*ui*+*t*, *ai*+*<sup>t</sup>*) *uj*+*t*, *aj*+*t* + (*<sup>u</sup> <sup>i</sup>*+*t*, *<sup>a</sup> <sup>i</sup>*+*<sup>t</sup>*) *<sup>u</sup> <sup>j</sup>*+*t*, *<sup>a</sup> <sup>j</sup>*+*t*
