**2. Preliminaries**

This section reviews several basic concepts concerning the IFS and the induced aggregated distance methods.

**Definition 1.** *An IFS P in a set Z* = {*<sup>z</sup>*1, *z*2,..., *zn*} *is defined as in (1) [1]:*

$$P = \{ \langle z, (\mu\_P(z), v\_P(z)) \rangle | z \in Z \} \tag{1}$$

*where the function* 0 ≤ *μP*(*z*) ≤ 1 *and* 0 ≤ *vP*(*z*) ≤ 1 *denote as the degree of membership and the non-membership, respectively, and satisfy* 0 ≤ *μP*(*z*) + *vP*(*z*) ≤ 1*. For convenient calculation, the pair α* = (*μα*, *<sup>v</sup>α*) *is signed as an intuitionistic fuzzy number (IFN) [24], where μα*, *vα* ∈ [0, 1] *and μα* + *vα* ≤ 1*.*

**Definition 2.** *The intuitionistic fuzzy distance (IFD) between IFNs α*1 *and α*2 *is given by the following formula [9]:*

$$d\_{IFD}(a\_1, a\_2) = |a\_1 - a\_2| = \frac{1}{2}(|\mu\_{a\_1} - \mu\_{a\_2}| + |v\_{a\_1} - v\_{a\_2}|) \tag{2}$$

As one of the most widely used and effective extensions of the ordered weighted averaging (OWA) methods [25], the IOWA operator [26] aggregates information by its reordering rule, performed with the order-inducing variables to accommodate a more complicated attitude of decision-makers.

**Definition 3.** *An IOWA is defined as follows:*

$$\text{IOVA}(\langle u\_1, a\_1 \rangle, \dots, \langle u\_n, a\_n \rangle) = \sum\_{j=1}^n w\_j b\_j \tag{3}$$

*where W* = (*<sup>w</sup>*1, *w*2,..., *wn*)*<sup>T</sup> is the weight vector satisfying w*1 + ... + *wn* = 1 *and wj* ∈ [0, 1]*. bj is the reordered value of ai in the argument ui*, *ai having the jth largest order-inducing variable ui.*

Based on the work of IOWAD proposed by Merigó and Casanovas [11], Zeng and Su [12] introduced the IFIOWAD operator by combining the advantages of the induced aggregation and the IFD. For IFSs *A* = (*<sup>α</sup>*1,..., *<sup>α</sup>n*) and *B* = (*β*1,..., *βn*), it is formulated as follows:

**Definition 4.** *An IFIOWAD operator is defined by a weight vector W with* 0 ≤ *wj* ≤ 1 *and w*1 + ... + *wn* = 1*; and an order-inducing vector U* = (*<sup>u</sup>*1,..., *un*)*, such that:*

$$\text{IFIOWMAD}(\langle u\_1, a\_1, \beta\_1 \rangle, \dots, \langle u\_{\text{il}}, a\_{\text{il}\prime} \beta\_{\text{il}} \rangle) = \sum\_{j=1}^{n} w\_j d\_{\text{IFD}}(a\_{\sigma(j)}, \beta\_{\sigma(j)}) \tag{4}$$

*where dIFD*(*ασ*(*j*), *βσ*(*j*)) *is the reordering of dIFD*(*<sup>α</sup>j*, *βj*) *induced by the decreasing the order of uj, and dIFD*(*<sup>α</sup>j*, *βj*) *is the IF distance between IFNs αj and βj.*

Although the IFIOWAD operator is considered a useful and powerful measure tool, its inherent defect often leads to the loss of information and biased results that can be observed from the following example.

**Example 1.** *Let A* = {(0.3, 0.5),(0.5, 0.2),(0.7, 0.1),(0.4, 0.5)} *and B* = {(0.4, 0.6),(0.7, 0.4), (0.2, 0.7),(0.6, 0.2)} *be two sections of IFNs, and let the order-inducing variables be U* = (7, 8, 3, <sup>5</sup>)*. The main steps for the aggregation of the above arguments based on the IFIOWAD operator are shown as follows:*

1. Calculate the distances *dIFD*(*<sup>α</sup>j*, *βj*) (*j* = 1, 2, 3, 4) using Equation (2):

$$d\_{IFD}(\mathfrak{a}\_1, \mathfrak{z}\_1) = \frac{1}{2}(|0.3 - 0.4| + |0.5 - 0.6|) = 0.1,$$

Similarly, we have

$$d\_{IFD}(\mathfrak{a}\_2, \mathfrak{k}\_2) = 0.2,\\ d\_{IFD}(\mathfrak{a}\_3, \mathfrak{k}\_3) = 0.55,\\ d\_{IFD}(\mathfrak{a}\_4, \mathfrak{k}\_4) = 0.25.$$

2. Reordering the *dIFD*(*<sup>α</sup>j*, *βj*) (*j* = 1, 2, 3, 4) according to the decreasing values of the variable *uj* yields:

$$d\_{IFD}(\mathfrak{a}\_{\sigma(1)}, \mathfrak{f}\_{\sigma(1)}) = d\_{IFD}(\mathfrak{a}\_{2\prime}\mathfrak{f}\_{2}) = 0.2,\\ d\_{IFD}(\mathfrak{a}\_{\sigma(2)}, \mathfrak{f}\_{\sigma(2)}) = d\_{IFD}(\mathfrak{a}\_{1\prime}\mathfrak{f}\_{1}) = 0.1,$$

$$d\_{IFD}(\mathfrak{a}\_{\sigma(3)}, \mathfrak{f}\_{\sigma(3)}) = d\_{IFD}(\mathfrak{a}\_4, \mathfrak{f}\_4) = 0.25,\\ d\_{IFD}(\mathfrak{a}\_{\sigma(4)}, \mathfrak{f}\_{\sigma(4)}) = d\_{IFD}(\mathfrak{a}\_3, \mathfrak{f}\_3) = 0.55.$$

3. Let the associated weighting vector be *W* = (0.3, 0.4, 0.1, 0.2)*<sup>T</sup>*, then the aggregation result yields:

$$\text{IFIOWAND}(L', A, B) = 0.3 \times 0.2 + 0.4 \times 0.1 + 0.1 \times 0.25 + 0.2 \times 0.55 = 0.225.$$

If we adjust the values of the order-inducing variables to *U* = (8, 10, 1, <sup>6</sup>), then the aggregation result would be:

$$\text{IFIOWAD}(\mathcal{U}', A, B) = 0.3 \times 0.2 + 0.4 \times 0.1 + 0.1 \times 0.25 + 0.2 \times 0.55 = 0.225.$$

One can be observed that we ge<sup>t</sup> the same aggregated results for different values of the order-inducing variables. The reason is that the order-inducing variables in the IFIOWAD operator only play the induced role and are not integrated into the actual aggregation results, thus corresponding aggregation results cannot embody the variation caused by a change of order-inducing variables. In the next section we will develop a new method to overcome this drawback.

#### **3. The IFWIOWAD Operator**

To solve the feedback problem of the existing IFIOWAD operator, we propose an improved aggregation method, named the intuitionistic fuzzy weighted IOWA distance (IFWIOWAD) operator. It can be formulated as follows:

**Definition 5.** *Let A* = (*<sup>α</sup>*1, ... , *<sup>α</sup>n*) *and B* = (*β*1, ... , *βn*) *be two sets of IFNs. An IFWIOWAD operator is defined by W with* 0 ≤ *wj* ≤ 1 *and w*1 + ... + *wn* = 1*, and an order-inducing vector U* = (*<sup>u</sup>*1, ... , *un*)*, such that:*

$$\text{IFWIIONAND}(\langle u\_1, a\_1, \beta\_1 \rangle, \dots, \langle u\_n, a\_n, \beta\_n \rangle) = \sum\_{j=1}^n \alpha\_j d\_{IFD}(a\_{\sigma(j)}, \beta\_{\sigma(j)}) \tag{5}$$

*where j* (*j* = 1, ... , *n*) *is a moderated weight that relatively depends on weight wj* ∈ *W and order-inducing variable uj*∈ *U, defined as:*

$$\varpi\_{j} = \frac{w\_{j}u\_{\sigma(j)}}{\sum\_{j=1}^{n} w\_{j}u\_{\sigma(j)}} \tag{6}$$

*where* (*σ*(1), ... , *σ*(*n*)) *is any possible permutation of* (1, ... , *<sup>n</sup>*)*, and clearly satisfies <sup>u</sup>σ*(*j*−<sup>1</sup>) ≥ *<sup>u</sup>σ*(*j*) *for j* > 1*. The distance dIFD*(*ασ*(*j*), *βσ*(*j*)) (*j* = 1, . . . , *n*) *is the reordering of dIFD*(*<sup>α</sup>j*, *βj*) *induced by <sup>u</sup>σ*(*j*)*.*

**Example 2.** *Assume the same collections of IFNS and order-inducing variables as defined in Example 1. Then the aggregation process by the IFWIOWAD is illustrated as follows:*

1. Record the order-inducing variables:

$$
\mu\_{\sigma(1)} = \mu\_{\mathfrak{J}} = 8, \mu\_{\sigma(2)} = \mathfrak{u}\_1 = 7, \mathfrak{u}\_{\sigma(3)} = \mathfrak{u}\_4 = 5, \mathfrak{u}\_{\sigma(4)} = \mathfrak{u}\_2 = 3.
$$

2. Calculate the moderated weight *j* using Equation (6):

$$\varpi\_1 = \frac{w\_1 \mu\_{\sigma(1)}}{\sum\_{j=1}^4 w\_j \mu\_{\sigma(j)}} = \frac{0.3 \times 8}{0.3 \times 8 + 0.4 \times 7 + 0.1 \times 5 + 0.2 \times 3} = 0.381...$$

Similarly,

$$\alpha\_2 = 0.445, \alpha\_3 = 0.079, \alpha\_4 = 0.095.$$

3. Compute the distance between *αi* and *βi* using Equation (2) (note that we can ge<sup>t</sup> these distances directly from Example 1:

$$d\_{IFD}(\mathfrak{a}\_1, \mathfrak{z}\_1) = 0.1,\\ d\_{IFD}(\mathfrak{a}\_2, \mathfrak{z}\_2) = 0.2,\\ d\_{IFD}(\mathfrak{a}\_3, \mathfrak{z}\_3) = 0.55,\\ d\_{IFD}(\mathfrak{a}\_4, \mathfrak{z}\_4) = 0.25.$$

4. Rank *dIFD*(*<sup>α</sup>j*, *βj*) (*j* = 1, 2, 3, 4) according to associated value of *<sup>u</sup>σ*(*j*):

$$d\_{IFD}(\mathfrak{a}\_{\sigma(1)}, \mathfrak{f}\_{\sigma(1)}) = d\_{IFD}(\mathfrak{a}\_2, \mathfrak{f}\_2) = 0.2,\\ d\_{IFD}(\mathfrak{a}\_{\sigma(2)}, \mathfrak{f}\_{\sigma(2)}) = d\_{IFD}(\mathfrak{a}\_1, \mathfrak{f}\_1) = 0.1,$$

$$d\_{IFD}(\mathfrak{a}\_{\sigma(3)}, \mathfrak{f}\_{\sigma(3)}) = d\_{IFD}(\mathfrak{a}\_4, \mathfrak{f}\_4) = 0.25,\\ d\_{IFD}(\mathfrak{a}\_{\sigma(4)}, \mathfrak{f}\_{\sigma(4)}) = d\_{IFD}(\mathfrak{a}\_3, \mathfrak{f}\_3) = 0.55.$$

Employ the IFWIOWAD operator defined in Equation (5) to obtain the aggregation result:

$$\text{IFWIOWAND}(IL, A, B) = 0.381 \times 0.2 + 0.445 \times 0.1 + 0.079 \times 0.25 + 0.095 \times 0.55 = 0.1927.$$

It is easy to see that we ge<sup>t</sup> a different aggregation value compared to the IFIOWAD operator. In addition, the variables *uj*(*j* = 1, ... , *n*) in the IFWIOWAD operator play dual functions, one is to induce the collection of arguments while the other moderates the weights that can overcome the drawback of the IFIOWAD operator caused by the limited role of the order-inducing variables.

Moreover, if the values of the order-inducing variables are changed to *U* = (8, 10, 1, <sup>6</sup>), then we can recalculate the moderated weights:

$$\varpi\_1 = \frac{w\_1 u\_{\sigma(1)}}{\sum\_{j=1}^4 w\_j u\_{\sigma(j)}} = \frac{0.3 \times 10}{0.3 \times 10 + 0.4 \times 8 + 0.1 \times 6 + 0.2 \times 1} = 0.429.1$$

Similarly,

> 2 = 0.456, 3 = 0.086, 4 = 0.029.

Thus, the aggregation of the IFWIOWAD operator will yield the following result:

$$\text{IFWIONAID}(\mathcal{U}, \mathcal{U}, B) = 0.429 \times 0.2 + 0.456 \times 0.1 + 0.086 \times 0.25 + 0.029 \times 0.55 = 0.16885.$$

As can be seen, in comparison to the IFIOWAD operator, the aggregation result of the IFWIOWAD is changed based on the adjustment of the values of *uj*(*j* = 1, ... , *<sup>n</sup>*), thus it can accommodate the variation caused by a change of order-inducing variables and yield better results.

Depending on the operational laws defined for the IFNs, one can drive some properties of the IFWIOWAD operator that are illustrated by the following theorems.

**Theorem 1.** *(Commutativity—distance measures). Let F be the IFWIOWAD operator, then* 

$$\bar{F}(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_{n\_r}, u\_{n\_r} \beta\_{n\_r} \rangle) = \bar{F}(\langle u\_1, \beta\_1, u\_1 \rangle, \dots, \langle u\_{n\_r} \beta\_{n\_r}, u\_{n\_r} \rangle) \tag{7}$$

**Theorem 2.** *(Commutativity—IOWA aggregation). Let* (*<sup>u</sup>*1,*s*1, *<sup>t</sup>*1,...,*un*,*sn*, *tn*) *is any possible permutation of argument vector* (*<sup>u</sup>*1, *α*1, *β*1,...,*un*, *αn*, *βn*)*, then*

$$\widetilde{F}(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle) = \widetilde{F}(\langle u\_1, s\_1, t\_1 \rangle, \dots, \langle u\_n, s\_n, t\_n \rangle) \tag{8}$$

**Theorem 3.** *(Monotonicity). If* |*αi* − *βi*| ≤ HH*α i* − *β i*HH *for all i, then*

$$\widetilde{F}(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle) \le \widetilde{F}(\langle u\_1, s\_1, t\_1 \rangle, \dots, \langle u\_n, s\_n, t\_n \rangle) \tag{9}$$

**Theorem 4.** *(Boundedness). Let min i* (|*<sup>α</sup>i* − *βi*|) = *d and maxi* ((|*<sup>α</sup>i* − *βi*|)) = *D, then*

$$d \le \widetilde{F}(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle) \le D \tag{10}$$

**Theorem 5.** *(Idempotency). If all d i* = |*αi* − *βi*| = *d for all i, then* 

$$\widetilde{F}(\langle u\_1, a\_1, \beta\_1 \rangle, \dots, \langle u\_n, a\_n, \beta\_n \rangle) = \widetilde{d} \tag{11}$$

It is straightforward to prove these theorems and therefore omitted for sake of brevity. Moreover, some particular cases of the IFWIOWAD operator can be explored by analyzing the order-inducing values and the weight vector. For example,

• If *U* = (*<sup>u</sup>*, 0, ··· , 0) (*u* = 0), then

$$\text{IFWIOWAD}(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle) = d\_{\text{IFD}}(a\_{\sigma(1)}, \beta\_{\sigma(1)}) \tag{12}$$

• If *U* = (0, ··· , 0, *u*) (*u* = 0), then

$$\text{IFWIOWAD}(\langle u\_1, a\_1, \beta\_1 \rangle, \dots, \langle u\_n, a\_{n\prime} \beta\_n \rangle) = d\_{IFD}(a\_{\sigma(n)\prime} \beta\_{\sigma(n)}) \tag{13}$$

• If *wj* = 0 and *wk* = 1, for all *j* = *k*, then

$$\text{WIEOVD}(\langle u\_1, p\_1, q\_1 \rangle, \dots, \langle u\_{n\prime}, p\_{n\prime} q\_n \rangle) = d\_{\text{IFD}}(u\_{\sigma(k)\prime} \beta\_{\sigma(k)}) \tag{14}$$

Especially, if *Dk* = max*i* {|*<sup>α</sup>i* − *βi*|}, then we ge<sup>t</sup> the intuitionistic fuzzy maximum distance; if *Dk* = min*i*{|*<sup>α</sup>i* − *βi*|}, the intuitionistic fuzzy minimum distance.

Other a parameterized family of the IFWIOWAD operator can be described by similar methods, as applied in references [27–31].

#### **4. The IFWIOWAWAD Operator**

From the examples illustrated in the Section 3, we can see that the proposed IFWIOWAD operator can effectively eliminate the defects of the existing methods. However, further analysis indicates that the IFWIOWAD operator also has some shortcomings; i.e., it cannot integrate the weight of integrated arguments—and thus the importance of the integrated date cannot be reflected in the aggregation process. Recently, Merigó [32] presented a unification of the OWA and the IOWA operators, and termed it the induced ordered weighted averaging–weighted average (IOWAWA) operator. The prominent feature of the IOWAWA operator is that it unifies the IOWA operator and weighted average (WA) in the same formula, and allows each of the two concepts to be assigned a degree of importance in the aggregation. The IOWAWA operator has been receiving increasing attention to date. For example, Zeng et al. [33] explored the usefulness of the IOWAWA in the intuitionistic fuzzy situation. Merigó et al. [34] studied the application of the IOWAWA in entrepreneurial fuzzy group decision-making problems. Merigó et al. [35] presented some new IOWAWA–based methods to compute variance and covariance. Zeng et al. [36] proposed some aggregation operators based on the IOWAWA method in Pythagorean fuzzy environment. Motivated the idea of the IOWAWA operator, in this section we present the IFWIOWAWAD operator that comprises a unified model that employs the main advantages of IFWIOWAD operator and the weighted average (WA) methods. Thus, it can perform the importance of attributes and complex attitude of experts in the decision-making framework.

**Definition 6.** *Let A* = (*<sup>α</sup>*1, ... , *<sup>α</sup>n*) *and B* = (*β*1, ... , *βn*) *be two sets of IFNs defined in set Z* = {*<sup>z</sup>*1, *z*2,..., *zn*} *and δi be the weight of the element zi*(*<sup>i</sup>* = 1, ... , *<sup>n</sup>*)*, satisfying δ*1 + ... + *δn* = 1 *and δi* ∈ [0, 1]*. Then, the IFWIOWAWAD is termed intuitionistic fuzzy weighted IOWA weighted average distance operator and defined as*

$$\text{IFWIOWAWAD}(\langle u\_1, a\_1, \beta\_1 \rangle, \dots, \langle u\_{n\_r} a\_{n\_r} \beta\_n \rangle) = \sum\_{j=1}^n \tilde{w}\_j d\_{IFD}(a\_{\sigma(j)}, \beta\_{\sigma(j)}) \tag{15}$$

*where dIFD*(*ασ*(*j*), *βσ*(*j*)) *is the argument value of dIFD*(*<sup>α</sup>j*, *βj*) *reordered by the order-inducing variable <sup>u</sup>σ*(*j*) *such that <sup>u</sup>σ*(*j*−<sup>1</sup>) ≥ *<sup>u</sup>σ*(*j*) *for* 1 < *j* ≤ *n. The combined weight of w is defined as follows:* 

$$
\tilde{w}\_{\hat{\jmath}} = \lambda \varpi\_{\hat{\jmath}} + (1 - \lambda)\delta\_{\sigma(\hat{\jmath})} \tag{16}
$$

*where λ* ∈ [0, 1]*, W* = (*<sup>w</sup>*1,..., *wn*)*<sup>T</sup> is the associated weighting vector that simply satisfies the condition* 0 ≤ *wj* ≤ 1 *and w*1 + ... + *wn* = 1*. j is defined by Equation (6), that is*

$$\mathcal{O}\_{\dot{j}} = \frac{w\_{\dot{j}} u\_{\sigma(\dot{j})}}{\sum\_{j=1}^{n} w\_{\dot{j}} u\_{\sigma(\dot{j})}} \tag{17}$$

The IFWIOWAWAD operator can also be explicitly illustrated in terms of the two underlying rules of aggregation (i.e., WA and IOWA). Thus, the IFWIOWAWAD can be separated into a linear combination of the IF weighted distance (IFWD) [15] and the IFWIOWAD:

$$\begin{aligned} \text{IFWIOWAWAD}(\langle u\_1, \alpha\_1, \beta\_1 \rangle, \dots, \langle u\_n, \alpha\_n, \beta\_n \rangle) &= \\ \lambda \sum\_{j=1}^n \omega\_j d\_{IFD}(\alpha\_{\sigma(j)}, \beta\_{\sigma(j)}) + (1 - \lambda) \sum\_{i=1}^n \delta\_i d\_{IFD}(\alpha\_i, \beta\_i) \end{aligned} \tag{18}$$

**Example 3.** *(Continuing from Example 2). Let the weighting vector δ* = (*<sup>δ</sup>*1, *δ*2, *δ*3, *<sup>δ</sup>*4)*<sup>T</sup>* = (0.2, 0.3, 0.15, 0.35)*<sup>T</sup> and λ* = 0.6, *then with the help of Example 2, the rest steps using the IFWIOWAWAD operator are given as follows:*

1. Compute the combined weight *wj* (*j* = 1, 2, 3, 4) using Equation (16):

$$\begin{aligned} \tilde{w}\_1 &= \lambda \sigma\_1 + (1 - \lambda)\delta\_{\sigma(1)} = 0.6 \times 0.429 + (1 - 0.6) \times 0.3 = 0.3774, \\ \tilde{w}\_2 &= \lambda \sigma\_2 + (1 - \lambda)\delta\_{\sigma(2)} = 0.6 \times 0.456 + (1 - 0.6) \times 0.2 = 0.3536, \\ \tilde{w}\_3 &= \lambda \sigma\_3 + (1 - \lambda)\delta\_{\sigma(3)} = 0.6 \times 0.086 + (1 - 0.6) \times 0.35 = 0.1916, \\ \tilde{w}\_4 &= \lambda \sigma\_4 + (1 - \lambda)\delta\_{\sigma(4)} = 0.6 \times 0.029 + (1 - 0.6) \times 0.15 = 0.0774. \end{aligned}$$

2. Employ the IFWIOWAWAD operator defined in Equation (15) to perform the aggregation as follows:

> IFWIOWAWAD(*<sup>U</sup>*, *A*, *B*)= 0.3774 × 0.2 + 0.3536 × 0.1 + 0.1916 × 0.25 + 0.0774 × 0.55 = 0.20131

The aggregation of IFWIOWAWAD can also be performed using Equation (19) as following:

IFWIOWAWAD(*<sup>U</sup>*, *A*, *B*) = 0.6 × IFWIOWAD + 0.4 × IFWD = 0.6 × 0.16885 + 0.4 × (0.2 × 0.1 + 0.3 × 0.2 + 0.15 × 0.55 + 0.35 × 0.25)= 0.20131

Evidently, we ge<sup>t</sup> the same aggregate values for both methods. Moreover, we can see that, contrary to the IFWIOWAD operator, the IFWIOWAWAD operator cannot only consider the attitudinal character represented by the order induced variable, but also take into account the importance of the argumen<sup>t</sup> based on the weighted average method.

In the following results, we show some of the most important properties of the IFWIOWAWAD operator.

**Proposition 1.** *The IFWIOWAWAD is commutative if it follows (let ϕ be the IFWIOWAWAD operator for a simple notation):*

$$\varrho\left(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle\right) = \varrho\left(\langle u\_1, \beta\_1, u\_1 \rangle, \dots, \langle u\_n, \beta\_n, u\_n \rangle\right) \tag{19}$$

*or*

$$\varphi(\langle u\_1, u\_1, \beta\_1 \rangle, \dots, \langle u\_n, u\_n, \beta\_n \rangle) = \varphi(\langle u\_1, s\_1, t\_1 \rangle, \dots, \langle u\_n, s\_n, t\_n \rangle) \tag{20}$$

*where* (*<sup>u</sup>*1,*s*1, *<sup>t</sup>*1,...,*un*,*sn*, *tn*) *is a possible permutation of the argument vector* (*<sup>u</sup>*1, *α*1, *β*1,...,*un*, *αn*, *βn*)*.*

**Proposition 2.** *If* |*αi* − *βi*| ≤ |*s*1 − *t*1| *for all i, it follows that:*

$$\left|\varphi\left(\langle u\_1, u\_1, \beta\_1 \rangle, \ldots, \langle u\_n, u\_n, \beta\_n \rangle\right) \right| \le \left\langle \psi\left(u\_1, s\_1, t\_1\right), \ldots, \left\langle u\_n, s\_n, t\_n \right\rangle \right\rangle \tag{21}$$

Then the IFWIOWAWAD is monotonic.

**Proposition 3.** *The IFWIOWAWAD is bounded if it follows that:*

$$\min\_{i} (|\boldsymbol{\mu}\_{i} - \boldsymbol{\beta}\_{i}|) \le \boldsymbol{\varrho}(\langle \boldsymbol{\mu}\_{1}, \boldsymbol{\mu}\_{1}, \boldsymbol{\beta}\_{1} \rangle, \dots, \langle \boldsymbol{\mu}\_{n}, \boldsymbol{\mu}\_{n}, \boldsymbol{\beta}\_{n} \rangle) \le \max\_{i} (|\boldsymbol{\mu}\_{i} - \boldsymbol{\beta}\_{i}|) \tag{22}$$

**Proposition 4.** *If all d i* = |*αi* − *βi*| = *d for i* ∈ [1, *<sup>n</sup>*]*, it follows that:*

$$\widetilde{F}(\langle u\_1, \alpha\_1, \beta\_1 \rangle, \dots, \langle u\_n, \alpha\_n, \beta\_n \rangle) = \widetilde{d} \tag{23}$$

Then the IFWIOWAWAD operator is idempotent.

By selecting different values for the weights and parameters in the IFWIOWAWAD operator, we can derive some special intuitionistic fuzzy distance operators. For example:


Equivalently, many other special cases can be derived by analyzing the weighting vectors *W*, *V* and the order inducing variable vector *U* in a similar way (see [33–36]).

#### **5. A MADM Model Based on the IFWIOWAWAD Operator**

A framework of the MADM model based on the IFWIOWAWAD is presented in this section. The main process for the model is structured as follows:

**Step 1.** Each decision maker *ek* provides their opinions and thus forms the individual decision matrix, constructed as in (24):

$$D\_k = \begin{array}{c} \mathbb{C}\_1 & \cdots & \mathbb{C}\_n \\ A\_1 & \begin{pmatrix} a\_{11}^{(k)} & \cdots & a\_{1n}^{(k)} \\ \vdots & \ddots & \vdots \\ a\_{m1}^{(k)} & \cdots & a\_{mn}^{(k)} \end{pmatrix} \\ \end{array} \tag{24}$$

where *Ai* and *Cj* indicate the alternative *i*(*i* = 1, ... , *m*) and the attribute *j*(*j* = 1, ... , *<sup>n</sup>*), respectively. Meanwhile the IFNs *α*(*k*) *ij* = (*μ*(*k*) *ij* , *v*(*k*) *ij* ) represents the preference for *Ai* with respect to the attribute *Cj*.

**Step 2.** Employ the IF weighted average (IFWA) operator [24] to convert individual opinions of each decision makers into a group decision matrix *D* = -*<sup>α</sup>ij*.*m*×*n*, where

$$\mathfrak{a}\_{ij} = \text{IFWA}\left(\mathfrak{a}\_{ij}^{(1)}, \dots, \mathfrak{a}\_{ij}^{(t)}\right) \\ \mathfrak{i} = 1, \dots, m, j = 1, \dots, n. \tag{25}$$

**Step 3.** Construct the ideal alternative *I* and determine the order-inducing variables and weights used for the IFWIOWAWAD operator.

**Step 4.** Compute the weighted distance between the ideal alternative *I* and each *Ai*(*i* = 1, ... , 5) using the IFWIOWAWAD operator.

**Step 5.** Establish a ranking for the alternative *Ai*(*i* = 1, ... , 5) in accordance with the IFWIOWAWAD(*<sup>I</sup>*, *Ai*) obtained in step 4. The alternative with the smallest distance will be selected as the best.

#### **6. An Example of Investment Selection**

Decision-making related to the selection of a suitable investment from finite feasible alternatives constitutes one of the most common and important activities in various business fields. The complexity of the assessment and selection process for investment projects necessitates a complex method: i.e., the multiple attribute decision-making (MADM) technique provides an efficient tool for decision makers to solve problems based on an evaluation or preference information given by multiple experts. In the past, many authors have proposed different MADM approaches for solving the selection of investment problems [37–41]. Previous findings have shown that the applications of the induced aggregation distance operators are very heartening and widely used in the decision-making process. This paper presents the application of the proposed model in the process of selecting investments in which a group of decision makers (or experts) are invited for the selection of a suitable strategy (adapted from Ref. [32]). Based on the market research and preliminary screening, there are five companies (alternatives) to be considered as potential investment options, namely a chemical company (*A*1), a food company (*A*2), a car company (*A*3), a furniture company (*A*4) and a computer company (*A*5). The main situations of company for investment are evaluated by the world economic growth rate: *C*1 = High growth rate, *C*2 = Medium growth rate, *C*3 = Low growth rate, *C*4 = Growth rate near 0 and *C*5 = Negative growth rate. The assessment of the alternatives with respect to each attribute given by three decision makers, are given in Tables 2–4. For example, the decision maker *e*1 called ten experts together to assess the situations (attributes) for these five companies. As for the *C*1 of the company *A*1, if six experts consider *C*1 strong while three experts consider *C*1 low and one expert do not judge whether *C*1 is strong or not, then the evaluation of company *A*1 relative to *C*1 can be represented by IFN (0.6,0.3) by using the statistical approach.


**Table 2.** Decision matrix *D*1 .



**Table 4.** Decision matrix *D*3.


In this problem, the weighting vector of the three experts is assumed to *V* = (0.3, 0.4, 0.3)*<sup>T</sup>* while, the collective results performed by the IFWA operator are listed in Table 5.

**Table 5.** Collective decision matrix *D* .


The order-inducing variables and the ideal alternative determined by the group of experts are shown in Tables 6 and 7, respectively.

**Table 6.** Order-inducing variables.



The weights *δi* for the attributes are given as 0.1, 0.25, 0.2, 0.35, 0.1 while the ordered weights, *wj* are assumed to be 0.15, 0.25, 0.2, 0.1, 0.3. Table 8 shows the aggregated results performed by the IFWIOWAWAD operator ( *λ* = 0.4).

**Table 8.** Aggregate results and ranking rendered by the IFWIOWAWAD operator.


Thus, *A*2 appears to be the best choice as it is closest to the ideal alternative while, the ranking of the five alternatives is *A*2 *A*5 *A*3 *A*4 *A*1.

To conduct a comparative analysis, we employ the IFIOWAWAD and IFOWAWAD operators in identical decision information to further explore the effectiveness of the order-inducing variables on the aggregation results. The results are shown in Table 9.

Thus, the rankings of the alternatives obtained by the IFIOWAWAD and IFOWAWAD operators are *A*2 *A*5 *A*3 *A*1 *A*4 and *A*5 *A*2 *A*3 *A*4 *A*1, respectively. From Tables 8 and 9, it is clear that the orderings of the alternatives may change if a different distance operator is used. It should be pointed out that the order-inducing variables in the IFIOWAWAD operator only perform a single induced function during the aggregation process. The IFOWAWAD operator integrates the importance of attributes and ordered weights into the formula to evaluate the IFS information, but fails to account for the attitudinal characters as it cannot infuse the order-inducing variables. However, the IFWIOWAWAD not only integrates both of the weights, but also captures the variation in the order-inducing variables, and thus achieves a more scientific and accurate result in comparison with other approaches.

**Table 9.** Aggregate results driven by the IFIOWAWAD and the IFOWAWAD operators.


Moreover, it is possible to conduct a sensitive analysis to explore the robustness of the ranking of the alternative with regards to the parameter *λ*, *λ* ∈ [0, 1]. The computation results are illustrated in Table 10.

**Table 10.** Ranking rendered by the IFWIOWAWAD operator with different values of *λ*

.


As can be seen, the ranking of alternatives may be different based on the different values of *λ*. Thus, the decision maker can select suitable values of *λ* to meet their interests or actual needs at hand. Therefore, this model is rather flexible as it provides more choices to decision makers for the selection of aggregation schemes by adjusting different values of the parameters.
