**2. Preliminaries**

This section recalls some definitions about the weighted aggregation operators, m-polar fuzzy sets, and fuzzy soft sets to achieve our main aim, proposing new algorithms for solving group decision-making based on alternative m-polar fuzzy soft aggregation operators, in the next sections.

#### *2.1. Weighted Aggregation Operator*

The weighted minimum and the weighted maximum are two important aggregation operators dealing with objects having non-negative weights *ω*1, ··· , *ωK* such that ∑*Ki*=<sup>1</sup> *ωi* = 1. However, there is no unique solution to formulate them. For instance, Fagin and Wimmers [2] introduced the below formula to obtain the weighted minimum and the weighted maximum, respectively.

$$\min\_{\omega\_{1\prime},\cdots,\omega\_{K}}^{\odot}(\mathbf{x}\_{1\prime}\cdot\cdots\prime,\mathbf{x}\_{K}) = \sum\_{i=1}^{K} \left[\mathbf{i}\cdot(\omega\_{\sigma(i)} - \omega\_{\sigma(i+1)})\cdot\min(\mathbf{x}\_{\sigma(1)\prime}\cdot\cdots\cdot,\mathbf{x}\_{\sigma(i)})\right] \tag{1}$$

$$\max\_{\omega\_{\omega\_1}, \dots, \omega\_K}^{\odot} (\mathbf{x}\_1, \dots, \mathbf{x}\_K) = \sum\_{i=1}^{K} \left[ \mathbf{i} \cdot (\omega\_{\sigma(i)} - \omega\_{\sigma(i+1)}) \cdot \max(\mathbf{x}\_{\sigma(1)}, \dots, \mathbf{x}\_{\sigma(i)}) \right] \tag{2}$$

where *σ* is a permutation that orders the weights as follows: *ωσ*(1) ≥ *ωσ*(2) ≥ ··· ≥ *ωσ*(*K*) and *ωσ*(*k*+<sup>1</sup>) = 0.

The IOWA operator, introduced by Yager and Filev [6], and the IOWG operator, given by Xu and Da [7], are also some commonly used tools for aggregating weighted objects. The IOWA operator is defined by

$$IOWA(\langle \omega\_1, \mathbf{x}\_1 \rangle\_\prime \cdot \cdot \cdot, \langle \omega\_K, \mathbf{x}\_K \rangle) = \sum\_{j=1}^K w\_j \cdot y\_j \tag{3}$$

where *yj* is the value of *xi* that has the *j*th largest *ωi* and *ωi* in *<sup>ω</sup>i*, *xi* is referred to as the order inducing variable and *xi* as the argumen<sup>t</sup> variable. The weights *w*1, ··· , *wK* such that ∑*Ki*=<sup>1</sup> *ωi* = 1 are the associated weights to the IOWA operator that can be defined by a quantifier function *Q* : [0, 1] → [0, 1]. Here, the re-ordering step of *xi*s is carried out by the variable *ωi* rather than the value of *xi*, that is used to handle the re-ordering step in OWA operator, i.e., the collection *x*1, ··· , *xK* is re-ordered as max{*<sup>ω</sup>i*}, *y*1≥···≥ min{*<sup>ω</sup>i*}, *yK*.

An IOWG operator is defined by

$$I\text{OWG}(\langle\omega\_1, \mathbf{x}\_1\rangle, \dots, \langle\omega\_K, \mathbf{x}\_K\rangle) = \prod\_{j=1}^K y\_j^{w\_j} \tag{4}$$

where *yj* is the value of *xi* that has the *j*th largest *ωi* and *ωi* in *<sup>ω</sup>i*, *xi* is referred to as the order inducing variable and *xi* as the argumen<sup>t</sup> variable. Note that here also the re-ordering step is based on the inducing variable *ωi* and weights *w*1, ··· , *wK* such that ∑*Ki*=<sup>1</sup> *ωi* = 1 are the associated weights to the IOWG operator.

## *2.2. Fuzzy Sets*

Today, fuzzy sets are known as an effective tool for modeling vague data [32–35]. If *U* is a non-empty set of elements, then a fuzzy subset *X* of *U* is a set of ordered pairs (*<sup>u</sup>*, *μX*(*u*)) such that *u* ∈ *U* and *μX* : *U* → [0, 1] is a membership function where *μX*(*u*) shows the membership degree of element *u* in *X*. Any fuzzy set *R* in *U* × *U* is called a fuzzy relation on *U*. If *R* represents a fuzzy preference relation on *U*, then, for each pair (*<sup>u</sup>*, *v*) ∈ *U* × *U*, the value *μR*(*<sup>u</sup>*, *v*) shows the preference degree of *u* over *v*. Here, *μR*(*<sup>u</sup>*, *v*) = 0.5 indicates indifference between *u* and *v* (*u* ∼ *v*), while *μR*(*<sup>u</sup>*, *v*) ∈ (0.5, 1] shows *u* is preferred to *v* (*u v*). Moreover, generally, we have *μR*(*<sup>u</sup>*, *v*) + *μR*(*<sup>v</sup>*, *u*) = 1.

**Theorem 1** (Multiplicative transitivity)**.** *[36] If <sup>u</sup>*(.) *is a utility function on the set X* = {*<sup>x</sup>*1, ··· , *xn*} *such that the value u*(*xi*) = *ui shows the utility of alternative xi* ∈ *X, then the fuzzy preference relation R defined by μR*(*xi*, *xj*) = *rij* = *ui ui*+*uj , which is a fuzzy preference relation satisfying multiplicative transitivity condition, i.e., rji rijrkj rjk*= *rki rikfor all i*, *j*, *k* ∈ {1, ··· , *<sup>n</sup>*}*.*

To extend the traditional fuzzy sets dealing with unipolar data into the multi-polar information, the concept of m-polar fuzzy set is defined as below.

**Definition 1** (m-polar fuzzy set)**.** *[37] An m-polar fuzzy set (M-pFS) X on U is a mapping μ* : *U* → [0, 1]*m, where* [0, 1]*m refers to as the multiplication of* [0, 1] ×···× [0, 1] *m-times, such that μ*(*u*) = -(*μ*<sup>1</sup>(*u*), ··· , *<sup>μ</sup><sup>m</sup>*(*u*).*,* **0** = (0, 0, ··· , 0) *and* **1** = (1, 1, ··· , 1) *are the least and greatest elements, respectively, and μ<sup>c</sup>*(*u*)=(<sup>1</sup> − *<sup>μ</sup>*<sup>1</sup>(*u*), ··· , 1 − *μ<sup>m</sup>*(*u*)) *shows its complement. The set of all m-polar fuzzy sets over U is represented by <sup>m</sup>*(*U*)*.*

If {*μk*}*k* is a family of M-pFSs over the universe *U*, then for any *u* ∈ *U*:


#### *2.3. Fuzzy Soft Sets*

Theory of soft sets is presented based on the approximate descriptions of the set *U*. A soft set is characterized by a set-valued mapping *f* : *P* → 2*<sup>U</sup>* where *P* is a set of parameters and 2*<sup>U</sup>* shows the power set of *U*. By combining the definitions of fuzzy sets and soft sets a new concept called fuzzy soft set is proposed.

**Definition 2** (Fuzzy soft set)**.** *[12] A fuzzy soft set (FSS), denoted by fP or* (*f* , *<sup>P</sup>*)*, is a mapping f* : *P* → [0, 1]*<sup>U</sup> where for every p* ∈ *P, f*(*p*) *is a fuzzy subset of U with membership function fp* : *U* → [0, 1] *where* **0 ˜** *and* **1˜***, defined by* **0˜**(*p*)(*u*) = 0 *and* **1˜**(*p*)(*u*) = 1 ∀*u* ∈ *U and p* ∈ *P, is called the null fuzzy soft set and the absolute fuzzy soft set, respectively. Moreover, the complement of* (*f* , *<sup>P</sup>*)*, denoted by* (*f c*, *<sup>P</sup>*)*, is defined by f c* : *P* → [0, 1]*<sup>U</sup> where* ∀*p* ∈ *P, f cp*(*u*) = 1 − *fp*(*u*) *for all u* ∈ *U.*

If {(*fk*, *Pk*)}*k* is a family of FSSs over the universe *U*, then, for any *u* ∈ *U*:


#### **3. A New Weighted Aggregation Operator for M-pFSSs**

In this section, we introduce a new weighted aggregation operator, called M-pFSMWM operator, to improve the aggregating tools for multi-polar inputs with non-negative weights under fuzzy soft environment. The advantages of this new operator is also demonstrated by some theorems and properties. To this end, we first develop the new concept of m-polar fuzzy soft sets (M-pFSS) and then introduce the M-pFSMWM operator in the domain of m-polar fuzzy soft sets.

#### *3.1. m-Polar Fuzzy Soft Sets*

Motivated by m-polar fuzzy sets given in Definition 1, the notion of m-polar fuzzy soft set is developed to model data dealing with multi-polar or multi-feature attributes. Basic operations of m-polar fuzzy soft sets are also discussed in this section.

**Definition 3** (m-polar fuzzy soft set)**.** *Let U and P be two non-empty sets of alternatives and parameters, respectively. The pair* (*f* , *P*) *where f is the mapping f* : *P* → *m*(*U*) *such that for any p* ∈ *P the f*(*p*) *is an m-polar fuzzy subset of U can be defined as an m-polar fuzzy soft set (M-pFSS) over U. It means, for each p* ∈ *P and any u* ∈ *U, f*(*p*)(*u*) *is an m-tuple fp*(*u*)=(*f* 1*p* (*u*), *f* 2*p* (*u*), ··· , *f mp* (*u*)) *such that the f sp*(*u*)*, for s* = 1, 2, ··· , *m, represents the relation between object u* ∈ *U and feature s of parameter p.*

The set of all m-polar fuzzy soft sets is shown by *mfs*(*U*). Furthermore, an m-polar fuzzy soft set (*f* , *P*) is called a null M-pFSS, shown by **0˜**, or an absolute M-pFSS, shown by **1˜**, if for any *p* ∈ *P*, *f <sup>s</sup>*(*p*)(*u*) = 0 and *f <sup>s</sup>*(*p*)(*u*) = 1, respectively, for all *u* ∈ *U* and 1 ≤ *s* ≤ *m*. The complement of M-pFSS (*f* , *P*) is also an M-pFSS, shown by (*f c*, *<sup>P</sup>*), where for any *p* ∈ *P* and *u* ∈ *U*: *f sc*(*p*)(*u*) = 1 − *f <sup>s</sup>*(*p*)(*u*) for all *s* = 1, 2, ··· , *m*.

**Example 1.** *Let us suppose a person wants to rate four restaurants* {*<sup>x</sup>*1, *x*2, *x*3, *<sup>x</sup>*4} *according to the parameters* {*p*1 = *Location*, *p*2 = *Meal*, *p*3 = *Services*}*. Let he/she considers the different aspects of these parameters as follows: The location of the restaurants includes close to the main road, in green surroundings, and in shopping mall. The meal of the restaurants includes main course, appetizer (starter), and dessert. The services of the restaurants include parking lot, live music, and free Wi-Fi connectivity.*

*Assume that the person uses the linguistic variables "No" (0), "Yes" (1), "Very Poor" (0), "Poor" (0.1), "Medium Poor" (0.3), "Medium" (0.5), "Medium Good"(0.7), "Good" (0.9), "Very Good" (1) "Very Far" (0), "Far" (0.1), "Medium Far" (0.3), "Medium Close"(0.7), "Close" (0.9), and "Very Close" (1) shown in Table 1 for describing the performance of each alternative with respect to these parameters.*


**Table 1.** Linguistic variables for describing each alternative with respect to the parameters.

*Thus, a three-polar fuzzy soft set (that is shown in Table 2) can help he/she to explain his/her opinion about these four restaurants.*


**Table 2.** Tabular representation of M-pFSS (*f* , *<sup>P</sup>*).

*For example, if the person considers the meal of the restaurant x*1*, then the 3-tuple (1,0,0.5) means that the main course of the restaurant x*1 *is very good while the starter and the dessert are very poor and medium, respectively.*

**Definition 4.** *Let* {(*fk*, *Pk*)}*k be a family of M-pFSSs over the common universe U and parameter sets Pk. Then, for any u* ∈ *U:*


 *3.* ( ˜D*k fk*)(*a*)(*u*) = inf*k*{ *fk*(*a*)(*u*)} = (inf*k*{ *f* 1*k* (*a*)(*u*)}, ··· , inf*k*{ *f mk* (*a*)(*u*)})*, for all a* ∈ G*<sup>k</sup>*∈*<sup>K</sup>Pk.*

**Proposition 1.** *Let U and F be the universal sets of objects and parameters, respectively, and P*, *Q, and E are some subsets of F. Assume that* (*f* , *<sup>P</sup>*),(*g*, *Q*)*, and* (*h*, *E*) *are some m-polar fuzzy soft sets over U where fp*, *gq*, *he* ∈ *m*(*U*) *for all p* ∈ *P*, *q* ∈ *Q, and e* ∈ *E. Then:*


*4. (Associative)* (*f* , *<sup>P</sup>*)∨˜ I(*g*, *Q*)∨˜ (*h*, *E*)J= I(*f* , *<sup>P</sup>*)∨˜ (*g*, *Q*)J∨˜ (*h*, *E*) *and* (*f<sup>P</sup>*)∧˜I(*g*,*Q*)∧˜(*h*,*E*)J =I(*f<sup>P</sup>*)∧˜(*g*,*Q*)J∧˜(*h*,*<sup>E</sup>*)*.*

 , , *5. (Distributive)* (*f* , *<sup>P</sup>*)∨˜ I(*g*, *Q*)∧˜ (*h*, *E*)J = I(*f* , *<sup>P</sup>*)∨˜ (*g*, *Q*)J∧˜ I(*f* , *<sup>P</sup>*)∨˜ (*h*, *E*)J *and* (*f* , *<sup>P</sup>*)∧˜ I(*g*, *Q*)∨˜ (*h*, *E*)J = I(*f* , *<sup>P</sup>*)∧˜ (*g*, *Q*)J∨˜ I(*f* , *<sup>P</sup>*)∧˜ (*h*, *<sup>E</sup>*)J*.*

**Proof.** Trivial by Definitions 3 and 4.

**Proposition 2** (De Morgan Law)**.** *Let U and F be the universal sets of objects and parameters, respectively. Assume that* (*f* , *P*) *and* (*g*, *Q*) *are two m-polar fuzzy soft sets over U where P and Q are the subsets of F and fp*, *gq* ∈ *m*(*U*) *for all p* ∈ *P and q* ∈ *Q. Then:*

*1.* I(*f* , *<sup>P</sup>*)∨˜ (*g*, *Q*)J*<sup>c</sup>* = (*f c*, *<sup>P</sup>*)∧˜ (*g<sup>c</sup>*, *Q*)*. 2.* I(*f* , *<sup>P</sup>*)∧˜ (*g*, *Q*)J*<sup>c</sup>* = (*f c*, *<sup>P</sup>*)∨˜ (*g<sup>c</sup>*, *Q*)*.*

**Proof.** It is proved easily by Definitions 3 and 4.

#### *3.2. The M-pFSMWM Operator*

In this subsection, we develop the M-pFSMWM operator in the domain of M-pFSSs. This new aggregation operator is used to reach the process of consensus in group decision-making problems with weighted m-polar fuzzy soft inputs. Additionally, we show the M-pFSMWM is a well-defined operator having the behavioral properties.

Let *DK* = {(*fk*, *P*)| *fk* : *P* → *<sup>m</sup>*(*u*), *fk* ∈ *mfs*(*U*), *k* = 1, 2, ··· , *K*} be a collection of m-polar fuzzy soft sets over *U* and *P*, such that for all *k*: *fk*(*p*)(*u*)=(*f* 1*k* (*p*)(*u*), ··· , *f mk* (*p*)(*u*)) ∈ [0, 1]*m* for *p* ∈ *P* and *u* ∈ *U* where [0, 1]*m* refers to as the multiplication of [0, 1] ×···× [0, 1] m-times, with non-negative weights *ω*1, ··· , *ωK* ∈ [0, 1] where ∑*Kk*=<sup>1</sup> *ωk* = 1. In the following, we develop a new weighted aggregation operator for m-polar fuzzy soft sets (M-pFSMWM operator) based on the weighted minimum operator given in Equation (1) and M-pFS maximum defined in Definition 4.

**Definition 5** (M-pFSMWM Operator)**.** *Let DK* = {(*fk*, *P*) ∈ *mfs*(*U*)|*k* = 1, 2, ··· , *K*} *be a collection of m-polar fuzzy soft sets over U and P with non-negative weights ω*1, ··· , *ωK* ∈ [0, 1] *such that* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Let value α where α* ∈ {1, 2, ··· , *K*} *be the required consensus degree. An M-pFSMWM operator of dimension K and at consensus degree α is a mapping M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*) : A*<sup>K</sup>*∈<sup>N</sup>(*mfs*(*U*))*<sup>K</sup>* → *mfs*(*U*) *that is defined by*

*M* − *pFSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)8 *f*1, ··· , *fK*9(*p*)(*u*) = *CK*,*<sup>α</sup>* max *l*=1 *α*∑*k*=1 *k* · (*ωσ*(*<sup>δ</sup>k* (*l*)) − *ωσ*(*<sup>δ</sup>k*+<sup>1</sup>(*l*)))· min{ *f* 1*<sup>σ</sup>*(*<sup>δ</sup>*1(*l*))(*p*)(*u*), ··· , *f* 1*<sup>σ</sup>*(*<sup>δ</sup>k* (*l*))(*p*)(*u*)}*<sup>δ</sup>k* (*l*)∈Δ*K*,*<sup>α</sup>*(*l*), ··· , *CK*,*<sup>α</sup>* max *l*=1 *α*∑*k*=1 *k* · (*ωσ*(*<sup>δ</sup>k* (*l*)) − *ωσ*(*<sup>δ</sup>k*+<sup>1</sup>(*l*)))· min{ *f m<sup>σ</sup>*(*<sup>δ</sup>*1(*l*))(*p*)(*u*), ··· , *f m<sup>σ</sup>*(*<sup>δ</sup>k* (*l*))(*p*)(*u*)}*<sup>δ</sup>k*(*l*)∈Δ*K*,*<sup>α</sup>*(*l*) (5)

*for u* ∈ *U and p* ∈ *P where the sum* ∑*<sup>α</sup>k*=<sup>1</sup>[...] *refers to as the weighted minimum over different choices α of K, σ is the permutation operator, CK*,*<sup>α</sup>* = *K*! *<sup>α</sup>*!(*<sup>K</sup>*−*<sup>α</sup>*)! *is the binomial coefficient, and* <sup>Δ</sup>*K*,*<sup>α</sup>*(*l*) *is an indexing set, where card*(<sup>Δ</sup>*K*,*<sup>α</sup>*(*l*)) = *α, including lth α-combination from a set of K elements. Thus,* {*<sup>δ</sup>*1(*l*)··· , *δα*(*l*)}*CK*,*<sup>α</sup> l*=1 *traverses all the α-combinations of the set* {1, 2, ··· , *K*} *and f sδk* (*l*)(*p*)(*u*) *represents the δkth element in lth α-combination of K for feature s; s* = 1, 2, ··· , *m.*

In the following, the various properties of M-pFSMWM operator including idempotency, boundedness, monotonicity, and commutativity (symmetry) are discussed.

**Theorem 2.** *Let* {(*fk*, *<sup>P</sup>*)}*Kk*=<sup>1</sup> *and* {(*gk*, *<sup>P</sup>*)}*Kk*=1*, for k* = 1, 2, ··· , *K, be two collections of some m-polar fuzzy soft sets over U and P with non-negative weights ωk that for all k: ωk* ∈ [0, 1] *and* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Let the required consensus degree α is given. Then, the M-pFSMWM operator has the following properties.*

*1. (Idempotency) If* (*fk*, *P*)=(*f* , *P*) *for all k, then*

$$M - pFSMWM^{(K,a,m)} \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle = (f, P)$$

*2. (Boundary Conditions)*

$$M - pFSMWM^{(K,a,m)} \langle \tilde{\mathbf{0}}, \dots, \tilde{\mathbf{0}} \rangle = \tilde{\mathbf{0}}$$

*and*

$$M - pFSMWM^{(K,\alpha,m)} \langle \mathbf{\tilde{1}}, \dots, \mathbf{\tilde{1}} \rangle = \mathbf{\tilde{1}}$$

*3. (Monotonicity) If* (*fk*, *P*)≤˜ (*gk*, *P*) *for all k, then*

$$M - pFSMWM^{(K,\mathfrak{a},m)} \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle \check{\leq} M - pFSMWM^{(K,\mathfrak{a},m)} \langle (\mathfrak{g}\_1, P), \dots, (\mathfrak{g}\_{K'}, P) \rangle$$

*4. (Boundedness)*

$$\min\_{k} \{ (f\_k, P) \}\_{k=1}^{K} \subseteq M - pFSMVM^{(K, \mathfrak{a}, m)} \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle \subseteq \max\_{k} \{ (f\_k, P) \}\_{k=1}^{K}$$

*5. (Commutativity or Symmetry)*

$$M - pFSMWM^{(K,\mathfrak{a},m)} \langle (f\_1, P), \dots, (f\_K, P) \rangle = M - pFSMWM^{(K,\mathfrak{a},m)} \langle (f\_{\sigma(1)}, P), \dots, (f\_{\sigma(K)}, P) \rangle$$

*where σ is any permutation of* {1, 2, ··· , *k*}*;*

**Proof.** 1. Let for all *k*: (*fk*, *P*)=(*f* , *<sup>P</sup>*). Thus, it is clear that the distinct *α*-combinations of *K* objects is reduced to the trivial case *K*-combination of *K* with *CK*,*<sup>K</sup>* = 1 and *ωk* = 1*K* for all *k*, i.e., the unweighted case. Thus,

$$\begin{aligned} (M - pFSMWM^{(K,\mathfrak{a},m)} \langle f\_1, \dots, f\_K \rangle)(p)(u) &= M - pFSMWM^{(K,K,m)} \langle f, \cdot, \cdot, f \rangle(p)(u) \\ &= (f^1(p)(u), \cdot, \cdot, f^m(p)(u)) = f(p)(u) \end{aligned}$$

since ∑*Kk*=<sup>1</sup> *k*.(*<sup>ω</sup><sup>k</sup>* − *<sup>ω</sup>k*+<sup>1</sup>). min{ *f* <sup>1</sup>(*p*)(*u*), ··· , *f* <sup>1</sup>(*p*)(*u*)} = ∑*<sup>K</sup>*−<sup>1</sup> *k*=1 *k*.( 1*K* − 1*K* ). *f* <sup>1</sup>(*p*)(*u*) + *K*. 1*K* . *f* <sup>1</sup>(*p*)(*u*).


**Theorem 3.** *Let* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *<sup>P</sup>*)*, where K* ≥ 2*, be some m-polar fuzzy soft sets over U and P such that for all k: fk*(*p*)(*u*)=(*f* 1*k* (*p*)(*u*), ··· , *f mk* (*p*)(*u*)) ∈ [0, 1]*m for p* ∈ *P and u* ∈ *U, with non-negative weights ω*1, ··· , *ωK* ∈ [0, 1] *where* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Then, the aggregated value M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)8(*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*)9 *is still an m-polar fuzzy soft set over U.*

**Proof.** Let *DK* = {(*fk*, *P*)| *fk*(*p*)(*u*)=(*f* 1*k* (*p*)(*u*), ··· , *f mk* (*p*)(*u*)) ∈ [0, <sup>1</sup>]*<sup>m</sup>*; *k* = 1, 2, ··· , *K*, *p* ∈ *P*} be a set of m-polar fuzzy soft arguments. Since, for each *k*, **0** ≤ *fk*(*p*)(*u*) ≤ **1**, then clearly by Theorem 2 we have **0** ≤ *M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)*f*1, ··· , *fK*(*p*)(*u*) ≤ **1**. This means that *M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)*f*1, ··· , *fK*(*p*)(*u*) ∈ [0, <sup>1</sup>]*<sup>m</sup>*. Now, define the function *F* : *P* → *m*(*U*) such that for any *p* ∈ *P* and *u* ∈ *U*, *<sup>F</sup>*(*p*)(*u*) = *M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)*f*1, ··· , *fK*(*p*)(*u*), given by Equation (5), which is an m-tuple of real numbers in the unit interval [0, 1]. This shows the *M* − *pFSMWM* operator of m-polar fuzzy soft sets is still an m-polar fuzzy soft set.

**Remark 1.** *According to Definition 5 and Theorem 3, weights in M-pFSMWM operator first re-order the position of arguments, means in the re-ordered list the first object has the biggest weight. Then, the aggregated value is computed based on the weighting vector ω* = (*<sup>ω</sup>*1, ··· , *<sup>ω</sup>K*)*<sup>T</sup> related to the m-polar fuzzy soft sets* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *<sup>P</sup>*)*. Thus, the weights reflect positions and importance degrees of input arguments in the aggregated value in comparison with IOWA and IOWG operators where weights only show the position of arguments. Moreover, if α shows the consensus degree, then the α-combinations of set* {1, 2, ··· , *K*} *express different scenarios of agreement among K decision makers where the decision of first, second, ... , and last α individuals are checked one by one. Further, by choosing l* = 1, 2, ··· , *CK*,*α, all possible choices of agreement between K experts at consensus degree α are considered. Thus, the concept of fuzzy majority, expressed by linguistic variables such as "half plus one", "more than 75%", "most", and "almost all" can be taken into account by choosing K*2+ 1 ≤ *α* ≤ *K if K is an even number and K*+1 2≤ *α* ≤ *K if K is an odd number.*

From Theorems 2 and 3, the *M* − *pFSMWM* operator degenerates to some special aggregation operators as follows.

**Theorem 4.** *Let* {(*fk*, *<sup>P</sup>*)}*Kk*=<sup>1</sup> *be a set of m-polar fuzzy soft sets over U with non-negative weights ωk that, for all k, ωk* ∈ [0, 1] *and* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Then, the M* − *pFSMWM operator degenerates to some special aggregation operators as follows.*

$$\begin{aligned} 1. \quad &If \,\omega = (0, \cdot, \cdot, \underbrace{1 \ldots \cdot}\_{j - th}, \cdot, 0)^T \, \text{i.e., } \omega\_{\overline{j}} = 1 \,\text{for } k = j \text{ and } \omega\_{\overline{k}} = 0 \,\text{for } k \neq j \text{, then} \\\\ &M - pFSMWM^{(\text{K}, a, m)} \, \langle f\_1, \cdot, \cdot, f\_{\overline{k}} \rangle (p)(u) = f\_{\overline{j}}(p)(u) \end{aligned}$$

*2. When K* = *α, we have*

$$M - p\text{FSMWM}^{(\text{K},\mu,u)}(f\_1 \cdots \cdot, f\_{\text{K}})(p)(u) = \langle \min\_{\omega\_1, \dots, \omega\_{\text{K}}}^{\odot} \{ f\_k^1(p)(u) \}\_{k=1}^{\text{K}} \cdot \cdots \cdot \min\_{\omega\_1, \dots, \omega\_{\text{K}}}^{\odot} \{ f\_k^m(p)(u) \}\_{k=1}^{\text{K}} \rangle$$

*which is called the M-pFS weighted minimum operator.* 111

*3. When K* = *α: if ω* = ( *K* , *K* , ··· , *K* )*T, then*

$$M - pFSMWM^{(K,\mu,m)} \langle f\_1 \cdots \cdot, f\_K \rangle (p)(u) = \langle \min\_k \{ f\_k^1(p)(u) \}\_{k=1}^K \cdot \cdots \cdot, \min\_k \{ f\_k^m(p)(u) \}\_{k=1}^K \rangle$$

*which is the M-pFS minimum operator.*

*4. When K* = *α: if fσ*(1)≥··· ˜ ≥˜ *fσ*(*K*)*; and ωσ*(1) = 1 *and ωσ*(*k*) = 0 *for all k* = 1*, then*

$$M - pFSMWM^{(K,\mu,m)} \langle f\_1 \cdots \cdot, f\_K \rangle (p)(\mu) = \langle \max\_k \{ f\_k^1(p)(\mu) \}\_{k=1'}^K \cdot \cdots \cdot, \max\_k \{ f\_k^m(p)(\mu) \}\_{k=1}^K \rangle$$

*which is the M-pFS maximum operator.*

*5. When K* = *α: If fσ*(1)≥··· ˜ ≥˜ *fσ*(*K*)*; and ωσ*(*K*) = 1 *and ωσ*(*k*) = 0 *for all k* = *K, then*

$$M - pFSMWM^{(K,\mu,m)} \langle f\_1 \cdots, f\_K \rangle (p)(\mu) = \langle \min\_k \{ f\_k^1(p)(\mu) \}\_{k=1}^K \cdots, \min\_k \{ f\_k^m(p)(\mu) \}\_{k=1}^K \rangle$$


$$\begin{aligned} M - pFSMWM^{(\mathcal{K}, \mathfrak{a}, \mathfrak{m})} (f\_1, \dots, f\_{\mathcal{K}})(p)(u) &= M - pFSMWM^{(\mathcal{K}, \mathcal{K}, \mathfrak{m})} \langle f\_1, \dots, f\_{\mathcal{K}} \rangle(p)(u) \\ &= \left\langle \min\_{\omega\_1, \dots, \omega\_K}^{\odot} \{ f\_1^1(p)(u), \dots, f\_K^1(p)(u) \}, \dots, \right\rangle \\ &\quad \min\_{\omega\_1, \dots, \omega\_K}^{\odot} \{ f\_1^m(p)(u), \dots, f\_K^m(p)(u) \} \end{aligned}$$


#### **4. Application of M-pFSMWM Operator in Group Decision-Making**

In this section, the M-pFSMWM operator is applied to handle group decision-making problems with weighted m-polar fuzzy soft inputs.

In a group decision-making problem with m-polar fuzzy soft information, let *U* = {*<sup>u</sup>*1, *u*2, ··· , *uN*} and *P* = {*p*1, *p*2, ··· , *pM*} be the finite sets of alternatives and parameters, respectively, where *λ* = (*<sup>λ</sup>p*1 , *<sup>λ</sup>p*2 , ··· , *<sup>λ</sup>pM* )*T* is the weighting vector for the parameter set *P* such that ∀*y*: *<sup>λ</sup>py* ∈ [0, 1] and <sup>∑</sup>*My*=<sup>1</sup> *<sup>λ</sup>py* = 1. Additionally, let each *py* be a multi-polar parameter with *m* different aspects or features such that *<sup>λ</sup>py* = (*λ*1*py* , *λ*2*py* , ··· , *λmpy* )*T* is the weighting vector for the parameter *py* ∈ *P* where ∀*s*: *λspy* ∈ [0, 1] and ∑*ms*=<sup>1</sup> *λspy* = 1. Suppose that *DK* = { *f*1, *f*2, ··· , *fK*} is the set of decision makers and *ω* = (*<sup>ω</sup>*1, *ω*2, ··· , *<sup>ω</sup>K*)*<sup>T</sup>* is the weighting vector of *fk* where, for all *k*: *ωk* ∈ [0, 1] and ∑*Kk*=<sup>1</sup> *ωk* = 1. Assume that each decision maker *fk* applies an m-polar fuzzy soft set to present the linguistic evaluation about alternatives such that *fk*(*py*)(*ui*)=(*f* 1*k* (*py*)(*ui*), ··· , *f mk* (*py*)(*ui*)) ∈ [0, 1]*m* and each *f sk* (*py*)(*ui*) shows the satisfaction degree of alternative *ui* about feature *s* of attribute *py*. Moreover, let the required consensus degree *α* mean an alternative may be selected if it is acceptable for at least *α* individuals.

After each expert prepares a linguistic or numerical judgment of alternatives based on the parameters *py*, the first stage is to reach consensus among a fuzzy majority or a partial agreemen<sup>t</sup> of them. This step is handled through the proposed aggregation operator M-pFSMWM by Equation (5) of the previous Section 3. The second stage of a MAGDM problem aims to find the best option with respect to the collective view. Thus, a ranking procedure is needed to derive the optimum choice. In the following subsections, we first define a fuzzy soft preference relationship over the universe *U* based on the collective view obtained by M-pFSMWM operator and then, propose a new score value function for ranking the preference order of objects.

#### *4.1. A Fuzzy Soft Preference Relationship*

The aim of this section is to define an square matrix *N* × *N* based on a fuzzy soft preference relationship over the set of alternatives *U*.

Let the m-tuple *M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)*f*1, ··· , *fK*(*py*)(*ui*)=(*u*1*iy*, ··· , *umiy*) present the performance of alternative *ui* ∈ *U* based on parameter *py* ∈ *P* and collective view obtained by M-pFSMWM operator. We define a fuzzy soft preference relationship on *U* by the mapping *R* ˜ : *P* → [0, 1]*<sup>U</sup>*×*<sup>U</sup>* where for each *py* ∈ *P*, *<sup>R</sup>*˜(*py*) is a fuzzy preference relationship on *U* which is characterized by the membership function *R* ˜(*py*) : *U* × *U* → [0, 1]. For any (*ui*, *uj*) ∈ *U* × *U* we define *R* ˜(*py*)(*ui*, *uj*) = *<sup>r</sup>*˜*ij*(*py*) such that

$$\bar{r}\_{ij}(p\_y) = \frac{\sum\_{s=1}^{m} \lambda\_{p\_y}^{s} u\_{iy}^{s}}{\sum\_{s=1}^{m} \lambda\_{p\_y}^{s} u\_{iy}^{s} + \sum\_{s=1}^{m} \lambda\_{p\_y}^{s} u\_{jy}^{s}} \tag{6}$$

**Definition 6.** *Suppose that U* = {*<sup>u</sup>*1, *u*2, ··· , *uN*} *is the set of alternatives and py* ∈ *P* = {*p*1, *p*2, ··· , *pM*} *is a parameter including m different aspects* {*p*<sup>1</sup>*y*, *p*2*y*, ··· , *pmy* }*. Let <sup>λ</sup>py* = (*λ*1*py* , ··· , *λmpy* )*T* ∈ [0, 1]*m show the weighting vector for parameter py* ∈ *P where* ∑*ms*=<sup>1</sup> *λspy* = 1*. If <sup>R</sup>*˜(*py*) *is the fuzzy preference relationship on U defined by* (6)*, then the N* × *N matrix R* ˜(*py*)=[*r*˜*ij*(*py*)]*<sup>N</sup>*×*<sup>N</sup> defined by*

$$
\bar{R}(p\_{\mathcal{Y}}) = [\bar{r}\_{ij}(p\_{\mathcal{Y}})]\_{N \times N} = \begin{bmatrix}
\bar{r}\_{11}(p\_{\mathcal{Y}}) & \bar{r}\_{12}(p\_{\mathcal{Y}}) & \cdots & \bar{r}\_{1N}(p\_{\mathcal{Y}}) \\
\bar{r}\_{21}(p\_{\mathcal{Y}}) & \bar{r}\_{22}(p\_{\mathcal{Y}}) & \cdots & \bar{r}\_{2N}(p\_{\mathcal{Y}}) \\
\vdots & \vdots & & \vdots \\
\bar{r}\_{N1}(p\_{\mathcal{Y}}) & \bar{r}\_{N2}(p\_{\mathcal{Y}}) & \cdots & \bar{r}\_{NN}(p\_{\mathcal{Y}})
\end{bmatrix} \tag{7}
$$

*where <sup>r</sup>*˜*ij*(*py*) ∈ [0, 1] *interprets the degree of preference of the alternative ui over the alternative uj with respect to the parameter py* ∈ *P. Moreover, <sup>r</sup>*˜*ii*(*py*) = 0.5 *for all* 1 ≤ *i* ≤ *N.*

In Definition 6, *<sup>r</sup>*˜*ij*(*py*) = 0.5 shows indifference between *ui* and *uj* based on the parameter *py* ∈ *P*, which is represented by *ui*<sup>∼</sup>*pyuj*, while *<sup>r</sup>*˜*ij*(*py*) ∈ (0.5, 1] shows *ui* is preferred to *uj* based on the parameter *py* ∈ *P* at degree *<sup>r</sup>*˜*ij*(*py*), i.e., *uipyuj*. Moreover, the fuzzy soft preference relationship *R* ˜ : *P* → [0, 1]*<sup>U</sup>*×*<sup>U</sup>* can be represented by matrix *R*˜ = [*r***˜***ijy*]*<sup>N</sup>*×*<sup>N</sup>* where each entry *<sup>r</sup>***˜***ijy* = [*r*˜*ij*(*py*)]*<sup>N</sup>*×*<sup>N</sup>* is in fact the *N* × *N* matrix *R* ˜(*py*). Hence, we have

$$\tilde{R} = [\tilde{r}\_{\vec{i}\vec{j}}]\_{N \times N} = \begin{bmatrix} [\tilde{r}\_{\vec{i}\vec{j}}(p\_1)]\_{N \times N} & [\tilde{r}\_{\vec{i}\vec{j}}(p\_2)]\_{N \times N} & \cdots & [\tilde{r}\_{\vec{i}\vec{j}}(p\_M)]\_{N \times N} \end{bmatrix}$$

**Proposition 3.** *The fuzzy preference relationship R* ˜(*py*) *clearly satisfies the following statements:*

*1. <sup>r</sup>*˜*ij*(*py*) + *<sup>r</sup>*˜*ji*(*py*) = 1 *2.* (*r* ˜ *ji*(*py*) *r* ˜ *ij*(*py*))(*<sup>r</sup>* ˜*kj*(*py*) *r* ˜*jk* (*py*)) = *r* ˜*ki*(*py*) *r* ˜*ik* (*py*) *3. If <sup>r</sup>*˜*ij*(*py*) ≥ 0.5 *and <sup>r</sup>*˜*jk*(*py*) ≥ 0.5*, then <sup>r</sup>*˜*ik*(*py*) ≥ max{*r*˜*ij*(*py*),*r*˜*jk*(*py*)}*.*

$$\text{for all } i, j, k = 1, 2, \cdot \cdot \cdot \text{, } N \text{ and } y = 1, 2, \cdot \cdot \cdot \text{, } M.$$

**Proof.** Item 1 is easily checked by Equation (6). Parts 2 and 3 are obtained by using *<sup>r</sup>*˜*ij*(*py*) = ∑*ms*=<sup>1</sup> *λspy usiy* ∑*ms*=<sup>1</sup> *λspy <sup>u</sup>siy*+∑*ms*=<sup>1</sup> *λspy usjy* = 1 1+ <sup>∑</sup>*ms*=<sup>1</sup> *λspy usjy* <sup>∑</sup>*ms*=<sup>1</sup> *λspy usiy* and *r* ˜*ji*(*py*) *r* ˜*ij*(*py*) = 1 *r* ˜*ij*(*py*) − 1 ( see also Theorem 1).

Proposition 3 says that the fuzzy preference relationship *R* ˜(*py*) satisfies the reciprocity and the restricted max-max transitivity. This means that, if *uipyuj* and *ujpyuk*, then *uipyuk*.

**Definition 7.** *Let R* ˜(*py*)=[*r*˜*ij*(*py*)]*<sup>N</sup>*×*<sup>N</sup> be the N* × *N matrix defined in Definition 6. Then, the N* × *N matrix A* ˜ = [*a***˜***ij*]*<sup>N</sup>*×*<sup>N</sup> defined by*

$$A = [\tilde{\mathfrak{a}}\_{ij}]\_{N \times N} = \begin{bmatrix} \tilde{\mathfrak{a}}\_{11} & \tilde{\mathfrak{a}}\_{12} & \cdots & \tilde{\mathfrak{a}}\_{1N} \\ \tilde{\mathfrak{a}}\_{21} & \tilde{\mathfrak{a}}\_{22} & \cdots & \tilde{\mathfrak{a}}\_{2N} \\ \vdots & \vdots & & \vdots \\ \tilde{\mathfrak{a}}\_{N1} & \tilde{\mathfrak{a}}\_{N2} & \cdots & \tilde{\mathfrak{a}}\_{NN} \end{bmatrix} \tag{8}$$

*where is an M-polar fuzzy preference relationship on U characterized by the membership function A* ˜ : *U* × *U* → [0, 1]*<sup>M</sup> and, for* 1 ≤ *i*, *j* ≤ *N, a***˜***ij* = (*a*˜1*ij*, *a*˜2*ij*, ··· , *a*˜*Mij* ) ∈ [0, 1]*<sup>M</sup> shows the degrees of preference of the alternative ui over the alternative uj with respect to the parameter set P. For each y: a*˜*yij* = 1 *if <sup>r</sup>*˜*ij*(*py*) > 0.5*, a* ˜*y ij* = 0.5 *if <sup>r</sup>*˜*ij*(*py*) = 0.5*, and a*˜*yij* = 0 *if <sup>r</sup>*˜*ij*(*py*) < 0.5*. Clearly, for all i, each entry <sup>a</sup>***˜***ii* = (0.5, 0.5, ··· , 0.5) : ;< = *M*−*times .*

For all *i*, *j*: *A* ˜(*ui*, *uj*) = *a***˜***ij* = (0.5, 0.5, ··· , 0.5) : ;< = *M*−*times* shows indifference between *ui* and *uj* based on parameter set *P*, which is represented by *ui*∼*uj*, while *A* ˜(*ui*, *uj*) = *a***˜***ij* = (1, 1, ··· , 1) : ;< = *M*−*times* shows *ui* is preferredto*uj*(*uiuj*)forallparameters.

 **Proposition 4.** *The following statements hold for M-polar fuzzy preference relation A.* ˜


*for all i*, *j*, *k* ∈ {1, 2, ··· , *<sup>N</sup>*}*.*

#### **Proof.** It is obtained easily by Proposition 3.

#### *4.2. An Approach to Group Decision-Making Based on M-pFSMWM Operator*

The second stage of a group decision-making is to reach the process of selection based on an overall performance of alternatives in terms of the crisp or partial agreemen<sup>t</sup> among the experts. In this section, we introduce a new procedure for solving group decision-making problems based on the M-pFSMWM operator and the pairwise comparisons of alternatives that are obtained by the *N* × *N* matrix *A* ˜ defined in Equation (8).

By using Definitions 6 and 7, a new overall score value function *S* : *U* → *R* over the universe *U* is defined as below.

**Definition 8.** *The mapping S* : *U* → *R defined by*

$$S\_i = \sum\_{j=1, j \neq i}^{N} \sum\_{y=1}^{M} \lambda\_{p\_y} \cdot \vec{a}\_{ij}^y \tag{9}$$

*for i* = 1, ··· , *N is called the score value function over U where <sup>λ</sup>py shows the importance degree of parameter py* ∈ *P and a***˜***ij* = (*a*˜1*ij*, *a*˜2*ij*, ··· , *a*˜*Mij* ) ∈ [0, 1] *is the entry in ith row and jth column of matrix A.* ˜

In the following, we apply the M-pFSMWM operator for solving MAGDM problems based on m-polar fuzzy soft information (Algorithm 1).The proposed procedure uses Equation (9) for ranking the preference order of objects. We also clarify the idea of the proposed method in Algorithm 1 by the given flowchart in Figure 1.

**Figure 1.** Flowchart of the proposed Algorithm 1.

**Algorithm 1.** Finding the optimum solution in MAGDM problems based on M-pFSMWM operator for M-pFSSs.

**Input :**K different m-polar fuzzy soft sets (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*) over the set *U* such that |*U*| = *N* and |*P*| = *M* where m is the number of different aspects of each parameter and *K* shows the number of decision makers. Weighting vectors *ω* = (*<sup>ω</sup>*1, *ω*2, ··· , *<sup>ω</sup>K*)*<sup>T</sup>*, *λ* = (*<sup>λ</sup>p*1 , *<sup>λ</sup>p*2 , ··· , *<sup>λ</sup>pM* )*T*, and *<sup>λ</sup>py* = (*λ*1*py* , *λ*2*py* , ··· , *λmpy* )*T* for *y* = 1, ··· , *M*. Consensus degree *α* where *α* ≤ *K*. **Output:**Optimum solution. **begin** Step 1. Calculate *L* = *CK*,*<sup>α</sup>* = *K*! *<sup>α</sup>*!(*<sup>K</sup>*−*<sup>α</sup>*)! . Step 2. **for** *l* = 1, 2, . . . , *L* **do** Find the *l*th *α*-combination of the set {1, 2, ··· , *K*} presented by <sup>Δ</sup>*K*,*<sup>α</sup>*(*l*) where |<sup>Δ</sup>*K*,*<sup>α</sup>*(*l*)| = *α*. **end** Step 3. **for** *i* = 1, 2, . . . , *N* **do for** *y* = 1, 2, . . . , *M* **do for** *s* = 1, 2, . . . , *m* **do** Compute the m-tuple *M* − *<sup>p</sup>FSMWM*(*<sup>K</sup>*,*α*,*<sup>m</sup>*)*f*1, ··· , *fK*(*py*)(*ui*)=(*u*1*iy*, ··· , *umiy*) by using Equation (5) to derive matrix *C* ¯ = [*c***¯***iy*]*<sup>N</sup>*×*<sup>M</sup>* such that *<sup>c</sup>***¯***iy* = (*u*1*iy*, *u*2*iy*, ··· , *<sup>u</sup>miy*). **end end end** Step 4. **for** *y* = 1, 2, . . . , *M* **do for** *i* = 1, 2, . . . , *N* **do for** *j* = 1, 2, . . . , *N* **do** Utilize Equations (6) and (7) to compute matrix *R* ˜(*py*)=[*r*˜*ij*(*py*)]*<sup>N</sup>*×*N*. **end end end** Step 5. Regarding fuzzy relation matrices *R* ˜(*py*) and by Equation (8), construct the collective overall preference matrix *A* ˜ = [*a***˜***ij*]*<sup>N</sup>*×*N*. **if** *A is a diagonal matrix* ˜ **then** There is no optimal option over *U*; **else** Go to the Step 6. **end** Step 6. **for** *i* = 1, 2, . . . , *N* **do** Using Equation (9) to calculate the overall score value *Si*. **end** Step 7. Rank the alternatives *ui* based on *Si* and then select the best one(s). **end**

**Remark 2** (Analysing Algorithm 1)**.** *Let K decision makers evaluate N number of alternatives based on M number of parameters where the m-polar fuzzy soft sets are applied to present their linguistic evaluations of the alternatives. According to Algorithm 1, we first utilize the M-pFSMWM operator to obtain a collective view of decision makers. The M-pFSMWM operator allows us to have not only partial agreement within a group, such as "almost all", "most", "more than half" etc., but also different choices for a partial agreement at the consensus degree α.*

*To this end, Algorithm 1 starts with finding the subsets* <sup>Δ</sup>*K*,*<sup>α</sup>*(*l*) ⊆ {1, 2, ··· , *K*} *where l* = 1, 2, ··· , *CK*,*α. This helps us to check all possible cases of agreement between K decision makers at consensus level α. In fact, the value of α shows the number of possible iterations of Algorithm 1 (*1 ≤ *α* ≤ *K). By repeating Steps 1 and 2 for different value α until α* ≤ *K, the aggregated value moves from the minimum value to the maximum value. This guaranties Algorithm 1 is convergent (please also see Theorems 2 and 4). Then, at Step 3, matrix C* ¯ *is driven. Each entry of C* ¯ *shows the performance of alternative ui based on parameter py and the collective view of experts at degree α. In Step 4, the fuzzy preference relations R* ˜(*py*) *(for py* ∈ *P) give a comparison of objects based on the collective view of decision makers and each parameter py. The information of matrices R* ˜(*py*) *are then converted to the M-polar fuzzy soft preference relation A* ˜ = [*a***˜***ij*]*<sup>N</sup>*×*N, in Step 5, for providing comparison results where a* **˜** *ij* = (*a*˜1*ij*, *a*˜2*ij*, ··· , *a*˜*Mij* ) ∈ [0, 1]*<sup>M</sup> defined by a*˜*yij* = 0.5 *if <sup>r</sup>*˜*ij*(*py*) = 0.5*, a*˜*yij* = 1 *if <sup>r</sup>*˜*ij*(*py*) > 0.5*, otherwise a* ˜*y ij* = 0*. Moreover, if A* ˜ *is an upper triangle matrix such that a***˜***ij* = (0.5, 0.5, ··· , 0.5)*; a***˜***ij* = (1, 1, ··· , 1) *for i* < *j; and a***˜***ij* = (0, 0, ··· , 0) *for i* > *j, then we have the following descending chain u*1 *u*2 ··· *uN on U. If A* ˜ *is a lower triangle matrix such that a***˜***ij* = (0.5, 0.5, ··· , 0.5)*; a***˜***ij* = (1, 1, ··· , 1) *for i* > *j; and a***˜***ij* = (0, 0, ··· , 0) *for i* < *j, then we have the ascending chain uN uN*−1 ··· *u*1 *on U. However, if A* ˜ *is a diagonal matrix, then there is no optimal option on U. In the last step of Algorithm 1, the best option is*

*selected based on its rank in the resultant preference order. For MAGDM problems with benefit criteria, means more is better, the alternative with the highest score can be selected as the best option. However, for the problems dealing with cost criteria the counter condition should be considered.*

#### **5. The M-pFSIOWA and M-pFSIOWG Operators**

To compare different m-polar fuzzy soft weighted aggregation operators with the proposed operator in Section 3.2, in this section, we develop the m-polar fuzzy soft induced ordered weighted average (M-pFSIOWA) operator and m-polar fuzzy soft induced ordered weighted geometric (M-pFSIOWG) operator, which are the extensions of IOWA and IOWG operators, respectively. The re-ordering step of M-pFSIOWA and M-pFSIOWG operators are defined based on the weights of arguments *ω* = ( *ω*1, ··· *<sup>ω</sup>K*)*<sup>T</sup>*. Since the M-pFSIOWA and M-pFSIOWG operators are defined in the domain of M-pFSSs, these new families of IOWA and IOWG operators give more general methods for aggregating data than traditional IOWA and IOWG operators.

Motivated by development of OWA operator and OWG operator for FSs [38,39] and IFSSs [21], the extensions of these two aggregation operators for M-pFSSs are defined as below.

**Definition 9.** *1. The M-pFSIOWA operator of dimension k is the mapping M* − *pFSIOWA* : A *<sup>K</sup>*∈<sup>N</sup>(*mfs*(*U*))*<sup>K</sup>* → *mfs*(*U*) *such that for an associated weighting vector w* = ( *w*1, *w*2, ··· , *wK*)*T, where wj* ∈ [0, 1] *and* ∑*<sup>K</sup> j*=1*wj* = 1*, is defined as below:*

$$M - pFSIOWA\langle f\_1, \dots, f\_K \rangle (p\_y)(u\_i) = \left\langle \sum\_{j=1}^K w\_j \cdot F\_{jyi}^1 \cdot \dots \cdot \sum\_{j=1}^K w\_j \cdot F\_{jyi}^m \right\rangle \tag{10}$$

*2. The M-pFSIOWG operator of dimension k is the mapping M* − *pFSIOWG* : A *<sup>K</sup>*∈<sup>N</sup>(*mfs*(*U*))*<sup>K</sup>* → *mfs*(*U*) *such that for an associated weighting vector w* = ( *w*1, *w*2, ··· , *wK*)*T, where wj* ∈ [0, 1] *and* ∑*<sup>K</sup> j*=1 *wj* = 1*, can be defined by*

$$M - pFSIOWG\langle f\_1, \dots, f\_K \rangle (p\_y)(u\_i) = \left\langle \prod\_{j=1}^K (F\_{jyi}^1)^{w\_j} \cdot \dots \cdot \prod\_{j=1}^K (F\_{jyi}^m)^{w\_j} \right\rangle \tag{11}$$

*where Fsjyi is the kth value f s k* (*py*)(*ui*) *having the jth largest <sup>ω</sup>j of the weighting vector ω* = (*<sup>ω</sup>*1, *ω*2, ··· , *<sup>ω</sup>K*)*<sup>T</sup> for M-pFSSs* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *<sup>P</sup>*)*.*

The main steps of these operations are the re-ordering step according to the weighting vector *ω* = ( *ω*1, *ω*2, ··· , *<sup>ω</sup>K*)*<sup>T</sup>* and then determining the associated weighting vector *w* = ( *w*1, *w*2, ··· , *wK*)*<sup>T</sup>* to the aggregation operators M-pFSIOWA and M-pFSIOWG. Here, for each 1 ≤ *s* ≤ *m*, 1 ≤ *y* ≤ *M*, and 1 ≤ *i* ≤ *N*, the collection: *f s* 1 (*py*)(*ui*), ··· , *f s K*(*py*)(*ui*) is re-ordered as max{*<sup>ω</sup>k*}, *Fs* 1*yi* ≥ ··· ≥ min{*<sup>ω</sup>k*}, *FsKyi* where the weighting vector *ω* = ( *ω*1, *ω*2, ··· , *<sup>ω</sup>K*)*<sup>T</sup>* shows the weights of different decision makers.

**Theorem 5.** *Let* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *<sup>P</sup>*)*, where K* ≥ 2*, is some m-polar fuzzy soft set over U and P such that for all* 1 ≤ *k* ≤ *K: fk*(*py*)(*ui*)=( *f* 1 *k* (*py*)(*ui*), ··· , *f m k* (*py*)(*ui*)) ∈ [0, 1] *m for py* ∈ *P and ui* ∈ *U, with non-negative weights ω*1, ··· , *ωK* ∈ [0, 1] *where* ∑*<sup>K</sup> k*=1 *ωk* = 1*. Then, the aggregated value M* − *pFSIOWA*8 (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*) 9 *and M* − *pFSIOWG*8 (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*) 9 *are still an m-polar fuzzy soft set over U.*

**Proof.** Define the function *F* : *P* → *m*(*U*) such that for any *py* ∈ *P* and *ui* ∈ *U*, *<sup>F</sup>*(*py*)(*ui*) = *M* − *pFSIOWAf*1, ··· , *fK*(*py*)(*ui*) or *<sup>F</sup>*(*py*)(*ui*) = *M* − *pFSIOWGf*1, ··· , *fK*(*py*)(*ui*). Then, the assertion is trivial from *f s k* (*py*)(*ui*) ∈ [0, 1], *wk* ∈ [0, 1], ∑*<sup>K</sup> k*=1 *wk* = 1, and convexity of [0, 1].

The following properties are inherited to M-pFSIOWA and M-pFSIOWG operators from IOWA operator and IOWG operator, respectively.

**Theorem 6.** *Let* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*) *be some m-polar fuzzy soft sets over U and P with non-negative weights ω*1, ··· , *ωK* ∈ [0, 1] *where* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Let w* = (*<sup>w</sup>*1, ··· , *wK*)*<sup>T</sup> be the associated weighting vector to the M-pFSIOWA and M-pFSIOWG operators. Then,*

*1. (Idempotency) If* (*fk*, *P*)=(*f* , *P*) ∀*k, then*

$$M - pFSIOWA \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle = (f, P)$$

*and*

$$\langle M - pFSIOWGG \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle = (f, P)$$

*2. (Monotonicity) If* (*fk*, *P*)≤˜ (*gk*, *P*) ∀*k, then*

$$M - pFSIOWA \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle \xleftarrow{\sim} M - pFSIOWA \langle (g\_1, P), \dots, (g\_{K'}, P) \rangle$$

*and*

$$M - pFSIOWG\langle (f\_1, P), \dots, (f\_K, P) \rangle \stackrel{\sim}{\leq} M - pFSIOWG\langle (\mathcal{g}\_1, P), \dots, (\mathcal{g}\_K, P) \rangle$$

*3. (Boundedness)*

$$\min\_{k} \{ (f\_k, P) \} \tilde{\le} M - pFSIOWA \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle \tilde{\le} \max\_{k} \{ (f\_{k'} P) \}$$

*and*

$$\min\_{k} \{ (f\_k, P) \} \tilde{\le} M - pFSIOWG \langle (f\_1, P), \dots, (f\_{k'}, P) \rangle \tilde{\le} \max\_{k} \{ (f\_{k'}, P) \}$$

*4. (Commutativity or Symmetry)*

$$M - pFSIOWA \langle (f\_1, P), \dots, (f\_{K'}, P) \rangle = M - pFSIOWA \langle (f\_{\sigma(1)}, P), \dots, (f\_{\sigma(K)}, P) \rangle$$

*and*

$$M - pFSIOWG\langle (f\_1, P), \dots, (f\_{K'}, P) \rangle = M - pFSIOWG\langle (f\_{\sigma(1)}, P), \dots, (f\_{\sigma(K)}, P) \rangle$$

*where σ is any permutation of* {1, 2, ··· , *k*}*.*

We can also obtain some spacial cases of M-pFSIOWA and M-pFSIOWG operators by using different choices for *w*.

**Theorem 7.** *Let* (*f*1, *<sup>P</sup>*), ··· ,(*fK*, *P*) *be some m-polar fuzzy soft sets over U and P with non-negative weights ω*1, ··· , *ωK* ∈ [0, 1] *where* ∑*Kk*=<sup>1</sup> *ωk* = 1*. Let w* = (*<sup>w</sup>*1, ··· , *wK*)*<sup>T</sup> be the associated weighting vector to the M-pFSIOWA and M-pFSIOWG operators. Then, the M-pFSIOWA operator and M-pFSIOWG operator degenerate to some special aggregation operators as follows.*

*1. If w* = ( 1*K* , ··· , 1*K* )*T, then*

$$M - pFSIOWA \langle f\_1, \dots, f\_K \rangle (p\_\mathcal{Y})(u\_i) = \langle \frac{1}{K} \sum\_{j=1}^K F\_{jyi}^1 \cdot \dots \cdot \frac{1}{K} \sum\_{j=1}^K F\_{jyi}^m \rangle\_i$$

*which we call the m-polar fuzzy soft arithmetic average operator, and*

$$M - pFSIO\mathcal{W}G\langle f\_1, \dots, f\_K \rangle (p\_y)(u\_i) = \langle \prod\_{j=1}^K (F^1\_{jyi})^\dagger \rangle \cdot \dots \cdot \prod\_{j=1}^K (F^m\_{jyi})^\dagger \cdot \rangle$$

*which we call the m-polar fuzzy soft geometric average operator. If w* = (1, 0, ··· , 0)*T, then*

$$M - pFSIOWA\langle f\_1, \dots, f\_K \rangle (p\_y)(\mu\_i) = \langle \max\_j \{ F^1\_{jyi} \}\_{j=1}^K, \dots, \max\_j \{ F^m\_{jyi} \}\_{j=1}^K \rangle$$

*and*

*2.*

$$M - pFSOWG\langle f\_1, \dots, f\_K \rangle (p\_y)(\mu\_i) = \langle \max\_j \{ F\_{jyi}^1 \}\_{j=1}^K \cdot \cdot \cdot \cdot \max\_j \{ F\_{jyi}^m \}\_{j=1}^K \rangle$$

*3. If w* = (0, ··· , 0, 1)*T, then*

$$M - p\text{FSOWA}\langle f\_1, \dots, f\_K \rangle (p\_\mathcal{Y})(u\_i) = \langle \min\_j \{ F\_{j\text{jir}}^1 \}\_{j=1}^K, \dots, \min\_j \{ F\_{j\text{j}i}^m \}\_{j=1}^K \rangle$$

*and*

$$M - pFSIOWG\langle f\_1, \dots, f\_K \rangle (p\_y)(\mu\_i) = \langle \min\_j \{ F^1\_{jyi} \}\_{j=1}^K \cdot \cdot \cdot \cdot \min\_j \{ F^m\_{jyi} \}\_{j=1}^K \rangle$$

#### *Application of M-pFSIOWA and M-pFSIOWG Operators in Group Decision-Making*

In this section, similar to Algorithm 1, we apply the M-pFSIOWA operator and M-pFSIOWG operator to propose a procedure for solving MAGDM problems with M-pFS inputs as the following Algorithm 2.

```
Algorithm 2. Finding the optimum solution in MAGDM problems based on M-pFSIOWA or M-pFSIOWG operators for M-pFSSs.
     Input :K different m-polar fuzzy soft sets (f1, P), ··· ,(fK, P) over the set U such that |U| = N and
               |P| = M where m is the number of different aspects of each parameter and K shows the number
               of decision makers. Weighting vectors ω = (ω1, ··· , ωK)T related to the m-polar fuzzy soft sets
               (f1, P), ··· ,(fK, P). Weighting vectors w = (w1, ··· , wK)T related to the M-pFSIOWA or
               M-pFSIOWG operators. λ = (λp1 , λp2 , ··· , λpM )T, and λpy = (λ1py , λ2py , ··· , λmpy )T for
               y = 1, ··· , M.
      Output :Optimum solution.
     begin
          Step 1. for i = 1, 2, . . . , N do
              for y = 1, 2, . . . , M do
                  for s = 1, 2, . . . , m do
                       Using the associated eighting vectors w = (w1, ··· , wK)T to compute the m-tuple
                        M − pFSIOWAf1, ··· , fK(py)(ui)=(u1iy, ··· , umiy) or
                        M − pFSIOWGf1, ··· , fK(py)(ui)=(u1iy, ··· , umiy) by Equation (10) or Equation (11)
                        and derive matrix C
                                             ¯ = [c¯iy]N×M such that c¯iy = (u1iy, u2iy, ··· , umiy).
                  end
              end
          end
          Step 2. for y = 1, 2, . . . , M do
              for i = 1, 2, . . . , N do
                  for j = 1, 2, . . . , N do
                       Utilize Equation (6) and Equation (7) to compute matrix R
                                                                                      ˜(py)=[r˜ij(py)]N×N based on
                        the matrix C
                                     ¯ = [c¯iy]N×M obtained in Step 1.
                  end
              end
          end
          Step 3. Using Equation (8) to construct the collective overall preference matrix A
                                                                                                  ˜ = [a˜ij]N×N
           according to the matrices R
                                        ˜(py)=[r˜ij(py)]N×N computed in Step 2.
          if A is a diagonal matrix
             ˜
                                    then
              There is no optimal option over U;
          else
              Go to the Step 4.
          end
          Step 4. for i = 1, 2, . . . , N do
              Calculate the overall score value Si based on the resultant matrix A
                                                                                       ˜
                                                                                        from Step 3 and Equation (9).
          end
          Step 5. Rank the alternatives ui based on Si and then select the best one(s).
     end
```
**Remark 3.** *Note that Algorithm 2 starts with computing matrix C* ¯ *based on M-pFSIOWA or M-pFSIOWG operators rather than the M-pFSMWM operator used in Algorithm 1. In Steps 2 and 3, the entries of the resultant matrix C* ¯ *are used to compute the matrices R* ˜(*py*) *and A*˜*. Then, Algorithm 2 is followed similarly with Algorithm 1.*

*Here, by repeating Step 1 for different iteration value α until α* ≤ *K, the aggregated value computed by M-pFSIOWA operator or M-pFSIOWG operator moves between the minimum value and the maximum value (please see Theorems 6 and 7). This guarantees Algorithm 2 is also convergent.*

## **6. Illustrative Example**

The general method for solving the MAGDM problems involves two main phases: (i) aggregation or consensus stage; and (ii) selection stage. The proposed methods in Equations (5), (10), and (11) can help us to aggregate different decision makers' judgments about the alternatives to obtain a collective decision matrix *C* ¯ . Further, Equations (8) and (9) provide an overall preference matrix *A* ˜ and the overall score values *Si* for the alternatives, respectively, to cover the selection phase. In the following, we compare the proposed procedures in Algorithms 1 and 2 for MAGDM problems with m-polar fuzzy soft information by some numerical examples.
