Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets and their application in decision-making

Abstract

Although some statistical tools, such as mean and median, used for modelling a problem containing parameters or alternatives with multiple intuitionistic fuzzy values because these values are obtained in a specific period, decrease uncertainty, they lead to data loss. However, interval-valued intuitionistic fuzzy values can overcome such a concern. For this reason, the present study proposes the concept of interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets (d-sets) and presents several of its basic properties. Moreover, by using d-sets, we suggest a new soft decision-making method and apply it to a problem concerning the eligibility of candidates for two vacant positions in an online job advertisement. Since it is the first method proposed in relation to this structure (d-sets), it is impossible to compare this method with another in this sense. To deal with this difficulty, we introduce four new concepts, i.e. mean reduction, mean bireduction, mean bireduction-reduction, and mean reduction-bireduction. By using these concepts, we apply four state-of-the-art soft decision-making methods to the problem. We then compare the ranking performances of the proposed method with those of the four methods. Besides, we apply five methods to a real problem concerning performance-based value assignment to some filters used in image denoising and compare the ranking performances of these methods. Finally, we discuss d-sets and the proposed method for further research.

1 Introduction

The standard mathematical tools are incompetent at overcoming some problems containing uncertainties in many areas, such as engineering, physics, computer science, economics, social sciences, and medical sciences. To overcome this drawback, many new mathematical tools, such as fuzzy sets (Zadeh 1965), intuitionistic fuzzy sets (Atanassov 1986), interval-valued fuzzy sets (Gorzałczany 1987; Zadeh 1975), and soft sets (Molodtsov 1999), have been proposed. Moreover, some hybrid versions of these concepts, such as fuzzy soft sets (Maji et al. 2001), fuzzy parameterized soft sets (Çağman et al. 2011a), fuzzy parameterized fuzzy soft sets (fpfs-sets) (Çağman et al. 2010), fuzzy parameterized interval-valued fuzzy soft sets (Alkhazaleh et al. 2011), intuitionistic fuzzy parameterized soft sets (Deli and Çağman 2015), intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets (ifpifs-sets) (Karaaslan 2016), fuzzy parameterized intuitionistic fuzzy soft sets (fpifs-sets) (Sulukan et al. 2019), and interval-valued fuzzy parameterized intuitionistic fuzzy soft sets (ivfpifs-sets) (Kamacı 2019) have been introduced. So far, the studies have been conducted on these concepts in many fields, such as decision-making (Çağman and Enginoğlu 2010a, b, 2012; Çağman et al. 2011b; Enginoğlu 2012; Enginoğlu and Çağman n.d.; Enginoğlu and Memiş 2018; Enginoğlu et al. 2018; Hao et al. 2018; Joshi 2020; Maji et al. 2002; Razak and Mohamad 2011, 2013; Selvachandran et al. 2017), algebra (Çıtak and Çağman 2015, 2017; Sezgin 2016; Sezgin et al. 2019; Ullah et al. 2018), topology (Atmaca 2017; Enginoğlu et al. 2015; Riaz and Hashmi 2017; Şenel 2016; Thomas and John 2016; Zorlutuna and Atmaca 2016), analysis (Riaz et al. 2018; Şenel 2018), and the other (Garg and Arora 2020; Iqbal and Rizwan 2019; Maji et al. 2003; Niewiadomski 2013).

However, without losing data, the concepts mentioned above cannot directly model a problem in which values are assigned to a parameter or an alternative with multiple measurement results. Suppose that there are twenty speedometers which send ten signals in an hour in the Dardanelles Strait and every ten signals are stored as a measurement result. Furthermore, assume that a signal is accepted as a positive signal if the instant flow is sufficient to turn the impeller of a turbine or as a negative signal if insufficient. Let $a^{x}_n$ and $b^x_n$ denote the numbers of positive and negative signals transmitted by a speedometer x for nth measurement, respectively. Here, $a^x_n + b^x_n = 10$, for all unsigned integer number n. Then, for $(a^x_n)=(5,3,6,8,1)$ which shows the results of five measurements and the membership function defined by $\mu (x):=\frac{1}{10n}\sum _{i=1}^n{a^x_i}$, the membership degree of the speedometer x to the fuzzy set $\mu$ can be computed as 0.46. Therefore, the nonmembership degree of the speedometer x is computed as $\nu (x)=1-\mu (x)=0.54$. However, considering multi-values as a single value refers to data loss. On the other hand, for

$$\begin{aligned} \alpha (x):= & \left[ \frac{\min \limits _{n}a^x_n}{\max \limits _{n}a^x_n + \max \limits _{n}b^x_n} , \frac{\max \limits _{n}a^x_n}{\max \limits _{n}a^x_n + \max \limits _{n}b^x_n}\right] \\ \beta (x):= & \left[ \frac{\min \limits _{n}b^x_n}{\max \limits _{n}a^x_n + \max \limits _{n}b^x_n} , \frac{\max \limits _{n}b^x_n}{\max \limits _{n}a^x_n + \max \limits _{n}b^x_n}\right] \end{aligned}$$

$(a^x_n)=(5,3,6,8,1)$, and $(b^x_n)=(5,7,4,2,9)$, the membership/nonmembership degree of the speedometer x can be computed as $^{[0.06,0.47]}_{[0.12,0.53]}$ via the concept of interval-valued intuitionistic fuzzy sets (ivif-sets) provided in Atanassov (2020) and Atanassov and Gargov (1989) and which is being “data-friendly”. As a result, the values 0.46 and 0.54 denote that the positive and negative signal numbers for one hundred signals of speedometer x are 46 and 54, respectively. On the other hand, the value [0.06, 0.47] signifies that the positive signal numbers for one hundred signals of speedometer x range from 6 to 47. Similarly, the value [0.12, 0.53] suggests that the negative signal numbers for one hundred signals of speedometer x occur between 12 and 53. Therefore, since an interval-valued intuitionistic fuzzy value contains more information than fuzzy values, it is more convenient to model such a problem.

Recently, many researchers have focused on both theoretical and applied studies concerning the concept of ivif-sets. For example, Sotirov et al. (2018) have propounded an approach combining intuitionistic fuzzy logic via the intercriteria analysis method. Thereafter, Atanassov et al. (2019) have described a new intercriteria analysis method based on interval-valued intuitionistic fuzzy assessment. Moreover, Kim et al. (2018) have proposed a method by using ivif-sets for the assessment and evaluation of all the classes in question. Besides, Luo and Liang (2018) have suggested a novel similarity measure based on ivif-sets. They then have applied it to pattern recognition and medical diagnosis. Furthermore, Liu and Jiang (2020) have defined a new distance measure of ivif-sets and applied it in consideration of the well known ideal house selection.

The concept of ivif-sets has been used to overcome uncertainties, especially in multi-criteria decision-making problems. For example, Mishra and Rani (2018) have extended the scope of this concept to attend to a method called as a weighted aggregated sum product assessment and applied it to a decision-making problem. Moreover, Priyadharsini and Balasubramaniam (2019) have suggested a multi-criteria decision-making method using accuracy function and applied it to an investment company’s selection problem. In addition to the aforementioned studies, the concept of ivif-sets has also been used in a wide range of fields, including topology and algebra (Hemavathi et al. 2018; Park 2016, 2017; Senapati and Shum 2019).

However, more general forms are needed for mathematical modelling of some problems in the event that the parameters or alternatives (objects) have more serious uncertainties. Therefore, in this paper, we define the concept of the interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets (d-sets) by combining the concepts of interval-valued intuitionistic fuzzy parameterized soft sets (Deli and Karataş 2016) and interval-valued intuitionistic fuzzy soft sets (Jiang et al. 2010; Min 2008). Since this concept has a great modelling ability and provides new fields of study for researchers, it is worth conducting the study.

In Sect. 2 of this study, we present some of the basic definitions and properties required in the next sections of the paper. In Sect. 3, we define the concept of d-sets and investigate some of its basic properties. In Sect. 4, we suggest a new soft decision-making method by using d-sets. This method provides selecting optimal elements from the alternatives. In Sect. 5, we apply this method to a problem of the determination of eligible candidates for the positions. In Sect. 6, we define four new concepts, i.e. mean reduction, mean bireduction, mean bireduction-reduction, and mean reduction-bireduction. By using these concepts, we apply four soft decision-making methods constructed via ifpifs-sets, ivfpifs-sets, fpifs-sets, and fpfs-sets to the problem mentioned above. We then compare the ranking performance of the proposed method with those of the four methods. In Sect. 7, we apply five methods to a real problem concerning performance-based value assignment to some filters used in image denoising, so that we can order them in terms of performance. Moreover, we compare the ranking performances of these methods. Finally, we discuss d-sets and the proposed method for further research. This study is a part of the first author’s PhD dissertation.

2 Preliminaries

In this section, we first provide several well-known definitions. Throughout this paper, let Int([0, 1]) be the set of all closed classical subintervals of [0, 1].

Definition 1

Let $\gamma _1,\gamma _2 \in Int([0,1])$. For $\gamma _1:=[\gamma ^-_1,\gamma ^+_1]$ and $\gamma _2:=[\gamma ^-_2,\gamma ^+_2]$, if $\gamma ^-_1 \le \gamma ^-_2$ and $\gamma ^+_1 \le \gamma ^+_2$, then $\gamma _1$ is called a subinterval of $\gamma _2$ and is denoted by $\gamma _1 {{\tilde{\subseteq }}} \gamma _2$.

Definition 2

Let $\gamma _1,\gamma _2 \in Int([0,1])$. Then, $\gamma _1 {{\tilde{\le }}} \gamma _2\Leftrightarrow \gamma _1 {{\tilde{\subseteq }}} \gamma _2$.

Definition 3

Let $\gamma ,\gamma _1,\gamma _2 \in Int([0,1])$, $c \in {\mathbb {R}}^{+}$, $\gamma :=[\gamma ^-,\gamma ^+]$, $\gamma _1:=[\gamma ^-_1,\gamma ^+_1]$, and $\gamma _2:=[\gamma ^-_2,\gamma ^+_2]$. Then, $\gamma _1 + \gamma _2 := [\gamma ^-_1 + \gamma ^-_2,\gamma ^+_1 + \gamma ^+_2]$, $\gamma _1 - \gamma _2 := [\gamma ^-_1 - \gamma ^+_2,\gamma ^+_1 - \gamma ^-_2]$, $\gamma _1 \cdot \gamma _2 := [\gamma ^-_1 \cdot \gamma ^-_2,\gamma ^+_1 \cdot \gamma ^+_2]$, and $c \cdot \gamma := [c \cdot \gamma ^-,c \cdot \gamma ^+]$.

Secondly, we present the concept of interval-valued intuitionistic fuzzy sets (Atanassov and Gargov 1989) and some of its basic properties (Atanassov 1994; Atanassov and Gargov 1989).

Definition 4

(Atanassov and Gargov 1989) Let E be a universal set and $\kappa$ be a function from E to $Int([0,1]) \times Int([0,1])$. Then, the set $\left\{ {{^{\alpha (x)}_{\beta (x)}}}x : x \in E \right\}$ being the graphic of $\kappa$ is called an interval-valued intuitionistic fuzzy set (ivif-set) over E.

Here, for all $x \in E$, $\alpha (x):=[\alpha ^-(x),\alpha ^+(x)]$ and $\beta (x):=[\beta ^-(x),\beta ^+(x)]$ such that $\alpha ^+(x)+\beta ^+(x) \le 1$. Moreover, $\alpha$ and $\beta$ are called membership function and nonmembership function in an ivif-set, respectively.

Note 1

Since $[\alpha (x),\alpha (x)]:=\alpha (x)$, for all $x \in E$, we use ${{^{\alpha (x)}_{\beta (x)}}}x$ instead of ${{^{[\alpha (x),\alpha (x)]}_{[\beta (x),\beta (x)]}}}x$. Moreover, we do not display the elements $^{0}_{1}x$ in an ivif-set.

In the present paper, the set of all ivif-sets over E is denoted by IVIF(E). In IVIF(E), since the $graph(\kappa )$ and $\kappa$ generate each other uniquely, the notations are interchangeable. Therefore, as long as it does not cause any confusion, we denote an ivif-set $graph(\kappa )$ by $\kappa$.

Example 1

Let $E=\{x_1,x_2,x_3,x_4\}$ be a universal set. Then,

$$\begin{aligned} \kappa =\left\{ {^{[0.2,0.4]}_{[0.4,0.6]}}x_1,{^{[0,0.2]}_{[0.5,0.7]}}x_2,{^{[0.3,0.5]}_{[0.1,0.2]}}x_4 \right\} \end{aligned}$$

is an ivif-set over E.

In the present paper, for $\lambda ,\varepsilon \in Int([0,1])$, let $^{\lambda }_{\varepsilon }E$ denote an ivif-set $\kappa$ over E such that $\alpha (x)=\lambda$ and $\beta (x)=\varepsilon$, for all $x \in E$.

Definition 5

(Atanassov 1994) Let $\kappa \in IVIF(E)$. For all $x \in E$, if $\alpha (x)=0$ and $\beta (x)=1$, then $\kappa$ is called empty ivif-set and is denoted by $^{0}_{1}E$ or $0_{E}$.

Definition 6

(Atanassov 1994) Let $\kappa \in IVIF(E)$. For all $x \in E$, if $\alpha (x)=1$ and $\beta (x)=0$, then $\kappa$ is called universal ivif-set and is denoted by $^{1}_{0}E$ or $1_{E}$.

Definition 7

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2 \in IVIF(E)$. For all $x \in E$, if $\alpha _1(x) {{\tilde{\le }}} \alpha _2(x)$ and $\beta _2(x) {{\tilde{\le }}} \beta _1(x)$, then $\kappa _1$ is called a subset of $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\subseteq }}} \kappa _2$.

Proposition 1

Let $\kappa ,\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa {{\tilde{\subseteq }}} 1_{E}$, $0_{E} {{\tilde{\subseteq }}} \kappa$, $\kappa {{\tilde{\subseteq }}} \kappa$, and $[\kappa _1 {{\tilde{\subseteq }}} \kappa _2 \wedge \kappa _2 {{\tilde{\subseteq }}} \kappa _3] \Rightarrow \kappa _1 {{\tilde{\subseteq }}} \kappa _3$.

Definition 8

Let $\kappa _1,\kappa _2 \in IVIF(E)$. Then, $\kappa _1 {{\tilde{\le }}} \kappa _2\Leftrightarrow \kappa _1 {{\tilde{\subseteq }}} \kappa _2$.

Definition 9

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2 \in IVIF(E)$. If $\kappa _1 {{\tilde{\le }}} \kappa _2$ and $\kappa _2 {{\tilde{\le }}} \kappa _1$, then $\kappa _1$ and $\kappa _2$ are called equal ivif-sets and is denoted by $\kappa _1=\kappa _2$.

Proposition 2

Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $[\kappa _1=\kappa _2 \wedge \kappa _2=\kappa _3] \Rightarrow \kappa _1=\kappa _3$.

Definition 10

Let $\kappa _1,\kappa _2 \in IVIF(E)$. If $\kappa _1 {{\tilde{\subseteq }}} \kappa _2$ and $\kappa _1 \ne \kappa _2$, then $\kappa _1$ is called a proper subset of $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\subsetneq }}} \kappa _2$.

Definition 11

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x \in E$, if $\alpha _3(x)=\sup \{\alpha _1(x),\alpha _2(x)\}$ and $\beta _3(x)=\inf \{\beta _1(x),\beta _2(x)\}$, then $\kappa _3$ is called union of $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\cup }}} \kappa _2$.

Proposition 3

(Atanassov 1994) Let $\kappa ,\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa {{\tilde{\cup }}} \kappa =\kappa$, $\kappa _1 {{\tilde{\cup }}} \kappa _2=\kappa _2 {{\tilde{\cup }}} \kappa _1$, and $(\kappa _1 {{\tilde{\cup }}} \kappa _2){{\tilde{\cup }}} \kappa _3=\kappa _1 {{\tilde{\cup }}} (\kappa _2 {{\tilde{\cup }}} \kappa _3)$.

Proposition 4

Let $\kappa ,\kappa _1,\kappa _2 \in IVIF(E)$. Then, $\kappa {{\tilde{\cup }}} 0_{E}=\kappa$, $\kappa {{\tilde{\cup }}} 1_{E}=1_{E}$, and $[\kappa _1 {{\tilde{\subseteq }}} \kappa _2 \Rightarrow \kappa _1 {{\tilde{\cup }}} \kappa _2=\kappa _2]$.

Definition 12

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x \in E$, if $\alpha _3(x)=\inf \{\alpha _1(x),\alpha _2(x)\}$ and $\beta _3(x)=\sup \{\beta _1(x),\beta _2(x)\}$, then $\kappa _3$ is called intersection of $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\cap }}} \kappa _2$.

Proposition 5

(Atanassov 1994) Let $\kappa ,\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa {{\tilde{\cap }}} \kappa =\kappa$, $\kappa _1 {{\tilde{\cap }}} \kappa _2=\kappa _2 {{\tilde{\cap }}} \kappa _1$, and $(\kappa _1 {{\tilde{\cap }}} \kappa _2) {{\tilde{\cap }}} \kappa _3=\kappa _1 {{\tilde{\cap }}} (\kappa _2 {{\tilde{\cap }}} \kappa _3)$.

Proposition 6

Let $\kappa ,\kappa _1,\kappa _2 \in IVIF(E)$. Then, $\kappa {{\tilde{\cap }}} 0_{E}=0_{E}$, $\kappa {{\tilde{\cap }}} 1_{E}=\kappa$, and $[\kappa _1 {{\tilde{\subseteq }}} \kappa _2 \Rightarrow \kappa _1 {{\tilde{\cap }}} \kappa _2=\kappa _1]$.

Proposition 7

(Atanassov 1994) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa _1 {{\tilde{\cup }}} (\kappa _2 {{\tilde{\cap }}} \kappa _3)=(\kappa _1 {{\tilde{\cup }}} \kappa _2) {{\tilde{\cap }}} (\kappa _1 {{\tilde{\cup }}} \kappa _3)$ and $\kappa _1 {{\tilde{\cap }}} (\kappa _2 {{\tilde{\cup }}} \kappa _3)=(\kappa _1 {{\tilde{\cap }}} \kappa _2) {{\tilde{\cup }}} (\kappa _1 {{\tilde{\cap }}} \kappa _3)$.

Definition 13

Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x\in E$, if $\alpha _3(x)=\inf \{\alpha _1(x),\beta _2(x)\}$ and $\beta _3(x)=\sup \{\beta _1(x),\alpha _2(x)\}$, then $\kappa _3$ is called difference between $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\setminus }}} \kappa _2$.

Proposition 8

Let $\kappa \in IVIF(E)$. Then, $\kappa {{\tilde{\setminus }}} 0_{E} =\kappa$ and $\kappa {{\tilde{\setminus }}} 1_{E}=0_{E}$.

Note 2

It must be noted that the difference is non-commutative and non-associative. For example, let $E=\{x\}$, $\kappa _1=\left\{ {{^{[0.1,0.3]}_{[0.2,0.4]}}}x \right\}$, $\kappa _2=\left\{ {{^{[0.4,0.5]}_{[0,0.1]}}}x \right\}$, and $\kappa _3=\left\{ {{^{[0.5,0.7]}_{[0.1,0.2]}}}x \right\}$. Since $\kappa _1 {{\tilde{\setminus }}} \kappa _2=\left\{ {{^{[0,0.1]}_{[0.4,0.5]}}}x \right\}$ and $\kappa _2 {{\tilde{\setminus }}} \kappa _1=\left\{ {{^{[0.2,0.4]}_{[0.1,0.3]}}}x \right\}$, then $\kappa _1 {{\tilde{\setminus }}} \kappa _2 \ne \kappa _2 {{\tilde{\setminus }}} \kappa _1$. Similarly, since $\kappa _1 {{\tilde{\setminus }}} (\kappa _2 {{\tilde{\setminus }}} \kappa _3)=\left\{ {{^{[0.1,0.3]}_{[0.2,0.4]}}}x \right\}$ and $(\kappa _1 {{\tilde{\setminus }}} \kappa _2) {{\tilde{\setminus }}} \kappa _3=\left\{ {{^{[0,0.1]}_{[0.5,0.7]}}}x \right\}$, then $\kappa _1 {{\tilde{\setminus }}} (\kappa _2 {{\tilde{\setminus }}} \kappa _3) \ne (\kappa _1 {{\tilde{\setminus }}} \kappa _2) {{\tilde{\setminus }}} \kappa _3$.

Definition 14

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2 \in IVIF(E)$. For all $x \in E$, if $\alpha _2(x)=\beta _1(x)$ and $\beta _2(x)=\alpha _1(x)$, then $\kappa _2$ is called complement of $\kappa _1$ and is denoted by $\kappa ^{{\tilde{c}}}_1$. It is clear that, $\kappa ^{{\tilde{c}}}_1=1_{E} {{\tilde{\setminus }}} \kappa _1$.

Proposition 9

Let $\kappa ,\kappa _1,\kappa _2 \in IVIF(E)$. Then, $(\kappa ^{\tilde{c}})^{{\tilde{c}}}=\kappa$, $0_{E}^{{\tilde{c}}}=1_{E}$, $\kappa _1 {{\tilde{\setminus }}} \kappa _2=\kappa _1 {{\tilde{\cap }}} \kappa _2^{{\tilde{c}}}$, and $[\kappa _1 {{\tilde{\subseteq }}} \kappa _2 \Rightarrow \kappa _2^{\tilde{c}} {{\tilde{\subseteq }}} \kappa _1^{{\tilde{c}}}]$.

Proposition 10

(Atanassov and Gargov 1989) Let $\kappa _1,\kappa _2 \in IVIF(E)$. Then, De Morgan’s laws are valid, i.e. $(\kappa _1 {{\tilde{\cup }}} \kappa _2)^{\tilde{c}}=\kappa ^{{\tilde{c}}}_1 {{\tilde{\cap }}} \kappa ^{{\tilde{c}}}_2$ and $(\kappa _1 {{\tilde{\cap }}} \kappa _2)^{{\tilde{c}}}=\kappa ^{{\tilde{c}}}_1 {{\tilde{\cup }}} \kappa ^{{\tilde{c}}}_2$.

Definition 15

Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x\in E$, if

$$\begin{aligned} \alpha _3(x)=\sup \{\inf \{\alpha _1(x),\beta _2(x)\},\inf \{\alpha _2(x),\beta _1(x)\}\} \end{aligned}$$

and

$$\begin{aligned} \beta _3(x)=\inf \{\sup \{\beta _1(x),\alpha _2(x)\},\sup \{\beta _2(x),\alpha _1(x)\}\} \end{aligned}$$

then $\kappa _3$ is called symmetric difference between $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\triangle }}} \kappa _2$.

Proposition 11

Let $\kappa ,\kappa _1,\kappa _2 \in IVIF(E)$. Then, $\kappa {{\tilde{\triangle }}} 0_{E}=\kappa$, $\kappa {{\tilde{\triangle }}} 1_{E}=\kappa ^{{\tilde{c}}}$, and $\kappa _1 {{\tilde{\triangle }}} \kappa _2=\kappa _2 {{\tilde{\triangle }}} \kappa _1$.

Note 3

It must be noted that the symmetric difference is non-associative. For example, let $E=\{x\}$, $\kappa _1=\left\{ {{^{[0.2,0.3]}_{[0.4,0.7]}}}x \right\}$, $\kappa _2=\left\{ {{^{[0.2,0.4]}_{[0.5,0.6]}}}x \right\}$, and $\kappa _3=\left\{ {{^{[0.1,0.4]}_{[0,0.5]}}}x \right\}$. Since $\kappa _1 {{\tilde{\triangle }}} (\kappa _2 {{\tilde{\triangle }}} \kappa _3)=\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}x \right\}$ and $(\kappa _1 {{\tilde{\triangle }}} \kappa _2) {{\tilde{\triangle }}} \kappa _3=\left\{ {{^{[0.1,0.4]}_{[0.2,0.5]}}}x \right\}$, then $\kappa _1 {{\tilde{\triangle }}} (\kappa _2 {{\tilde{\triangle }}} \kappa _3) \ne (\kappa _1 {{\tilde{\triangle }}} \kappa _2) {{\tilde{\triangle }}} \kappa _3$.

Definition 16

Let $\kappa _1,\kappa _2 \in IVIF(E)$. If $\kappa _1 {{\tilde{\cap }}} \kappa _2=0_{E}$, then $\kappa _1$ and $\kappa _2$ are called disjoint.

Definition 17

(Atanassov 1994) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x \in E$, if $\alpha _3(x)=[\alpha ^-_1(x) + \alpha ^-_2(x)-\alpha ^-_1(x) \alpha ^-_2(x),\alpha ^+_1(x) + \alpha ^+_2(x)-\alpha ^+_1(x) \alpha ^+_2(x)]$ and $\beta _3(x)=[\beta ^-_1(x ) \beta ^-_2(x),\beta ^+_1(x) \beta ^+_2(x)]$, then $\kappa _3$ is called sum of $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 \tilde{+} \kappa _2$.

Proposition 12

(Atanassov 1994) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa _1 \tilde{+} \kappa _2=\kappa _2 \tilde{+} \kappa _1$ and $(\kappa _1 \tilde{+} \kappa _2) \tilde{+} \kappa _3=\kappa _1 \tilde{+} (\kappa _2 \tilde{+} \kappa _3)$.

Proposition 13

Let $\kappa \in IVIF(E)$. Then, $\kappa \tilde{+} 0_{E}=\kappa$ and $\kappa \tilde{+} 1_{E}=1_{E}$.

Definition 18

(Atanassov 1994) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. For all $x \in E$, if $\alpha _3(x)=[\alpha ^-_1(x) \alpha ^-_2(x),\alpha ^+_1(x) \alpha ^+_2(x)]$ and $\beta _3(x)=[\beta ^-_1(x) + \beta ^-_2(x)-\beta ^-_1(x) \beta ^-_2(x),\beta ^+_1(x) + \beta ^+_2(x)-\beta ^+_1(x) \beta ^+_2(x)]$, then $\kappa _3$ is called product of $\kappa _1$ and $\kappa _2$ and is denoted by $\kappa _1 {{\tilde{\cdot }}} \kappa _2$.

Proposition 14

(Atanassov 1994) Let $\kappa _1,\kappa _2,\kappa _3 \in IVIF(E)$. Then, $\kappa _1 {{\tilde{\cdot }}} \kappa _2=\kappa _2 {{\tilde{\cdot }}} \kappa _1$ and $(\kappa _1 {{\tilde{\cdot }}} \kappa _2) {{\tilde{\cdot }}} \kappa _3=\kappa _1 {{\tilde{\cdot }}} (\kappa _2 {{\tilde{\cdot }}} \kappa _3)$.

Proposition 15

Let $\kappa \in IVIF(E)$. Then, $\kappa \, {{\tilde{\cdot }}} \, 0_{E}=0_{E}$ and $\kappa \, {{\tilde{\cdot }}} \, 1_{E}=\kappa$.

Proposition 16

(Atanassov 1994) Let $\kappa _1,\kappa _2 \in IVIF(E)$. Then, $(\kappa _1 \tilde{+} \kappa _2)^{{\tilde{c}}}=\kappa ^{{\tilde{c}}}_1 \, {{\tilde{\cdot }}} \, \kappa ^{{\tilde{c}}}_2$ and $(\kappa _1 {{\tilde{\cdot }}} \kappa _2)^{\tilde{c}}=\kappa ^{{\tilde{c}}}_1 \tilde{+} \kappa ^{{\tilde{c}}}_2$.

3 Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets

In this section, we first define the concept of interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets and introduce some of its basic properties. The primary purpose of the present section is to make a theoretical contribution to the conceptualization of soft sets (Molodtsov 1999) and ivif-sets (Atanassov and Gargov 1989).

Definition 19

Let U be a universal set, E be a parameter set, $\kappa \in IVIF(E)$, and f be a function from $\kappa$ to IVIF(U). Then, the set $\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x,f({{^{\alpha (x)}_{\beta (x)}}}x)\right) : x\in E \right\}$ being the graphic of f is called an interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft set (d-set) parameterized via E over U (or briefly over U).

Note 4

We do not display the elements $(^{0}_{1}x,0_{U})$ in a d-set. Here, $0_{U}$ is the empty ivif-set over U.

In the present paper, the set of all d-sets over U is denoted by $D_E(U)$. In $D_E(U)$, since the graph(f) and f generate each other uniquely, the notations are interchangeable. Therefore, as long as it does not cause any confusion, we denote a d-set graph(f) by f.

Example 2

Let $E=\{x_1,x_2,x_3,x_4\}$ be a parameter set, $U=\{u_1,u_2,u_3,u_4,u_5\}$ be a universal set, $\kappa =\left\{ {^{[0.2,0.5]}_{[0.3,0.4]}}x_1,{^{[0.3,0.4]}_{[0.1,0.3]}}x_2,{^{1}_{0}}x_4\right\}$, $f\left( {^{[0.2,0.5]}_{[0.3,0.4]}}x_1\right) =\left\{ {^{[0.1,0.3]}_{[0.2,0.6]}}u_2,{^{[0.8,0.9]}_{[0,0.1]}}u_3\right\}$, $f\left( {^{[0.3,0.4]}_{[0.1,0.3]}}x_2\right) =0_{U}$, $f({^{0}_{1}}x_3)=0_{U}$, and $f({^{1}_{0}}x_4)=\left\{ {^{0.7}_{0.2}}u_5\right\}$. Then, the d-set f over U is as follows:

$$\begin{aligned} f&=\left\{ \left( {^{[0.2,0.5]}_{[0.3,0.4]}}x_1,\right. \left\{ {^{[0.1,0.3]}_{[0.2,0.6]}}u_2,{^{[0.8,0.9]}_{[0,0.1]}}u_3\right\} \right) , \\&\left. \qquad\left( {^{[0.3,0.4]}_{[0.1,0.3]}}x_2,0_{U}\right) ,\left( {^{1}_{0}}x_4,\left\{ {^{0.7}_{0.2}}u_5\right\} \right) \right\} \end{aligned}$$

Since ignoring a portion of the values of parameters or alternatives of a d-set may be a necessary or facilitating way to the solution in some decision-making problems, the definition of the restriction of a d-set, unlike the simple restriction, should be as follows. Thus, the restriction of a d-set again belongs to the $D_E(U)$. For more detail, see (Enginoğlu 2012; Enginoğlu and Çağman n.d.).

Definition 20

Let $f, f_1 \in D_E(U)$ and $A \subseteq E$. Then, $A f_1$-restriction of f, denoted by $f_{A f_1}$, is defined by

$$\begin{aligned} \alpha _{A\kappa _1}(x):= & \left\{ \begin{array}{cl} \alpha (x),&{} x\in A\\ \alpha _1(x),&{} x \in E {\setminus } A \end{array}\right. \\ \beta _{A\kappa _1}(x):= & \left\{ \begin{array}{cl} \beta (x),&{} x\in A\\ \beta _1(x),&{} x \in E {\setminus } A \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} f_{A f_1}\left({{^{\alpha _{A\kappa _1}(x)}_{\beta _{A\kappa _1}(x)}}}x\right):=\left\{ \begin{array}{cl} f({{^{\alpha (x)}_{\beta (x)}}}x),&{} x\in A\\ f_1({{^{\alpha _{1}(x)}_{\beta _{1}(x)}}}x),&{} x \in E {\setminus } A \end{array}\right. \end{aligned}$$

Example 3

Let us consider the d-set f provided in Example 2, $A=\{x_1,x_3\}$, and $f_1\in D_E(U)$ such that

$$\begin{aligned} f_1&=\left\{ \left( {^{[0.1,0.6]}_{[0.2,0.3]}}x_2,\left\{ {^{[0.7,0.8]}_{[0,0.1]}}u_3,{^{[0.5,0.7]}_{[0,0.1]}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{[0.1,0.3]}_{[0.2,0.4]}}x_3,1_{U}\right) ,\\&\left. \qquad\left( {^{1}_{0}}x_4,\left\{ {^{[0.3,0.8]}_{[0.1,0.2]}}u_1,{^{[0.1,0.2]}_{[0,0.4]}}u_2\right\} \right) \right\} \end{aligned}$$

Then,

$$\begin{aligned} f_{Af_1}&=\left\{ \left( {^{[0.2,0.5]}_{[0.3,0.4]}}x_1,\left\{ {^{[0.1,0.3]}_{[0.2,0.6]}}u_2,{^{[0.8,0.9]}_{[0,0.1]}}u_3\right\} \right) , \right. \\&\qquad\left( {^{[0.1,0.6]}_{[0.2,0.3]}}x_2,\left\{ {^{[0.7,0.8]}_{[0,0.1]}}u_3,{^{[0.5,0.7]}_{[0,0.1]}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{1}_{0}}x_4,\left\{ {^{[0.3,0.8]}_{[0.1,0.2]}}u_1,{^{[0.1,0.2]}_{[0,0.4]}}u_2\right\} \right) \right\} \end{aligned}$$

Definition 21

Let $f \in D_E(U)$. If $\kappa =0_{E}$ and for all $x \in E$, $f({^{0}_{1}}x)=0_{U}$, then f is called empty d-set and is denoted by ${\tilde{0}}$.

Definition 22

Let $f \in D_E(U)$. If $\kappa =1_{E}$ and for all $x \in E$, $f({^{1}_{0}}x)=1_{U}$, then f is called universal d-set and is denoted by ${\tilde{1}}$.

Definition 23

Let $f_1,f_2 \in D_E(U)$. If $\kappa _1 {{\tilde{\subseteq }}} \kappa _2$ and for all $x \in E$, $f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\subseteq }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_1$ is called a subset of $f_2$ and is denoted by $f_1 {{\tilde{\subseteq }}} f_2$.

Proposition 17

Let $f,f_1,f_2,f_3 \in D_E(U)$. Then,

• (i) $f {{\tilde{\subseteq }}} {\tilde{1}}$

• (ii) ${\tilde{0}} {{\tilde{\subseteq }}} f$

• (iii) $f {{\tilde{\subseteq }}} f$

• (iv) $[f_1 {{\tilde{\subseteq }}} f_2 \wedge f_2 {{\tilde{\subseteq }}} f_3] \Rightarrow f_1 {{\tilde{\subseteq }}} f_3$

Remark 1

$f_1 {{\tilde{\subseteq }}} f_2$ does not imply that every element of $f_1$ is an element of $f_2$. For example, let $E=\{x_1,x_2,x_3\}$ be a parameter set, $U=\{u_1,u_2,u_3\}$ be a universal set,

$$\begin{aligned} f_1&=\left\{ \left( {^{[0.1,0.7]}_{[0.2,0.3]}}x_1,\left\{ {{^{[0.3,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0.3,0.4]}_{[0,0.1]}}}u_2,{{^{[0,0.2]}_{[0.5,0.6]}}}u_3 \right\} \right) ,\right. \\&\qquad\left( {^{[0.2,0.3]}_{[0.4,0.5]}}x_2,\left\{ {{^{[0.1,0.3]}_{[0.4,0.5]}}}u_1,{{^{[0,0.1]}_{[0.3,0.9]}}}u_2, {{^{[0.3,0.4]}_{[0.4,0.6]}}}u_3\right\} \right) ,\\&\left. \qquad\left( {^{[0.5,0.8]}_{[0.1,0.2]}}x_3,\left\{ {{^{[0.2,0.3]}_{[0.2,0.6]}}}u_1,{{^{[0.3,0.4]}_{[0.4,0.6]}}}u_2,{{^{[0.2,0.3]}_{[0.4,0.7]}}}u_3 \right\} \right) \right\} , \end{aligned}$$

and

$$\begin{aligned} f_2&=\left\{ \left( {^{[0.2,0.8]}_{[0.1,0.2]}}x_1,\left\{ {{^{[0.4,0.5]}_{[0,0.3]}}}u_1,{{^{[0.5,0.8]}_{[0,0.1]}}}u_2,{{^{[0.2,0.4]}_{[0.3,0.5]}}}u_3 \right\} \right) ,\right. \\&\qquad\left( {^{[0.3,0.4]}_{[0.2,0.5]}}x_2,\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0,0.1]}_{[0,0.9]}}}u_2, {{^{[0.4,0.5]}_{[0.3,0.4]}}}u_3\right\} \right) ,\\&\left. \qquad\left( {^{[0.7,0.8]}_{[0,0.1]}}x_3,\left\{ {{^{[0.3,0.4]}_{[0.1,0.5]}}}u_1, {{^{[0.6,0.8]}_{[0,0.1]}}}u_2,{{^{[0.4,0.5]}_{[0.1,0.3]}}}u_3\right\} \right) \right\} \end{aligned}$$

Since $\kappa _1 {{\tilde{\subseteq }}} \kappa _2$ and $f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\subseteq }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, for all $x \in E$, then $f_1 {{\tilde{\subseteq }}} f_2$.On the other hand, $f_1 \nsubseteq f_2$ because

$$\begin{aligned} \left( {^{[0.1,0.7]}_{[0.2,0.3]}}x_1,\left\{ {{^{[0.3,0.4]}_{[0.2,0.5]}}}u_1,\right. \right. \left. \left. {{^{[0.3,0.4]}_{[0,0.1]}}}u_2,{{^{[0,0.2]}_{[0.5,0.6]}}}u_3\right\} \right) \notin f_2 \end{aligned}$$

although it belong to $f_1$.

Definition 24

Let $f_1,f_2 \in D_E(U)$. If $\kappa _1=\kappa _2$ and for all $x \in E$, $f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x)=f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_1$ and $f_2$ are called equal d-sets and is denoted by $f_1=f_2$.

Proposition 18

Let $f_1,f_2,f_3 \in D_E(U)$. Then,

• (i) $[f_1=f_2 \wedge f_2=f_3] \Rightarrow f_1=f_3$

• (ii) $[f_1 {{\tilde{\subseteq }}} f_2 \wedge f_2 {{\tilde{\subseteq }}} f_1] \Leftrightarrow f_1=f_2$

Definition 25

Let $f_1,f_2 \in D_E(U)$. If $f_1 {{\tilde{\subseteq }}} f_2$ and $f_1 \ne f_2$, then $f_1$ is called a proper subset of $f_2$ and is denoted by $f_1 {{\tilde{\subsetneq }}} f_2$.

Definition 26

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 {{\tilde{\cup }}} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cup }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called union of $f_1$ and $f_2$ and is denoted by $f_1 {{\tilde{\cup }}} f_2$.

Definition 27

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 {{\tilde{\cap }}} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cap }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called intersection of $f_1$ and $f_2$ and is denoted by $f_1 {{\tilde{\cap }}} f_2$.

Proposition 19

Let $f,f_1,f_2,f_3 \in D_E(U)$. Then,

• (i) $f {{\tilde{\cup }}} f=f$ and $f {{\tilde{\cap }}} f=f$

• (ii) $f {{\tilde{\cup }}} {\tilde{0}}=f$ and $f {{\tilde{\cap }}} {\tilde{0}}={\tilde{0}}$

• (iii) $f {{\tilde{\cup }}} {\tilde{1}}={\tilde{1}}$ and $f {{\tilde{\cap }}} {\tilde{1}}=f$

• (iv) $f_1 {{\tilde{\cup }}} f_2=f_2 {{\tilde{\cup }}} f_1$ and $f_1 {{\tilde{\cap }}} f_2=f_2 {{\tilde{\cap }}} f_1$

• (v) $(f_1 {{\tilde{\cup }}} f_2) {{\tilde{\cup }}} f_3=f_1 {{\tilde{\cup }}} (f_2 {{\tilde{\cup }}} f_3)$

  $(f_1 {{\tilde{\cap }}} f_2) {{\tilde{\cap }}} f_3=f_1 {{\tilde{\cap }}} (f_2 {{\tilde{\cap }}} f_3)$

• (vi) $f_1 {{\tilde{\cup }}} (f_2 {{\tilde{\cap }}} f_3)=(f_1 {{\tilde{\cup }}} f_2) {{\tilde{\cap }}} (f_1 {{\tilde{\cup }}} f_3)$

  $f_1 {{\tilde{\cap }}} (f_2 {{\tilde{\cup }}} f_3)=(f_1 {{\tilde{\cap }}} f_2) {{\tilde{\cup }}} (f_1 {{\tilde{\cap }}} f_3)$

• (vii) $[f_1 {{\tilde{\subseteq }}} f_2 \Rightarrow f_1 {{\tilde{\cup }}} f_2=f_2]$

  $[f_1 {{\tilde{\subseteq }}} f_2 \Rightarrow f_1 {{\tilde{\cap }}} f_2=f_1]$

Definition 28

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 {{\tilde{\setminus }}} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\setminus }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called difference between $f_1$ and $f_2$ and is denoted by $f_1 {{\tilde{\setminus }}} f_2$.

Proposition 20

Let $f \in D_E(U)$. Then,

• (i) $f {{\tilde{\setminus }}} {\tilde{0}} =f$

• (ii) $f {{\tilde{\setminus }}} {\tilde{1}}={\tilde{0}}$

Note 5

It must be noted that the difference is non-commutative and non-associative.

Definition 29

Let $f_1,f_2 \in D_E(U)$. If $\kappa _2=\kappa ^{{\tilde{c}}}_1$ and for all $x\in E$, $f_2({{^{\beta _1(x)}_{\alpha _1(x)}}}x)=(f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x))^{\tilde{c}}$, then $f_2$ is called complement of $f_1$ and is denoted by $f^{{\tilde{c}}}_1$. That is, for all $x \in E$, $f_1^{\tilde{c}}({{^{\beta _1(x)}_{\alpha _1(x)}}}x)=(f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x))^{\tilde{c}}$. It is clear that, $f^{{\tilde{c}}}_1 = {\tilde{1}} {{\tilde{\setminus }}} f_1$.

Proposition 21

Let $f,f_1,f_2 \in D_E(U)$. Then,

• (i) $(f^ {{\tilde{c}}})^{{\tilde{c}}}=f$

• (ii) ${\tilde{0}}^{{\tilde{c}}}={\tilde{1}}$

• (iii) $f_1 {{\tilde{\setminus }}} f_2=f_1 {{\tilde{\cap }}} f_2^{{\tilde{c}}}$

• (iv) $f_1 {{\tilde{\subseteq }}} f_2 \Rightarrow f_2^{{\tilde{c}}} {{\tilde{\subseteq }}} f_1^{{\tilde{c}}}$

Proposition 22

Let $f_1,f_2 \in D_E(U)$. Then, the following De Morgan’s laws are valid.

• (i) $(f_1 {{\tilde{\cup }}} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 {{\tilde{\cap }}} f^{{\tilde{c}}}_2$

• (ii) $(f_1 {{\tilde{\cap }}} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 {{\tilde{\cup }}} f^{{\tilde{c}}}_2$

Definition 30

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 {{\tilde{\triangle }}} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\triangle }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called symmetric difference between $f_1$ and $f_2$ and is denoted by $f_1 {{\tilde{\triangle }}} f_2$.

Proposition 23

Let $f,f_1,f_2 \in D_E(U)$. Then,

• (i) $f {{\tilde{\triangle }}} {\tilde{0}}=f$

• (ii) $f {{\tilde{\triangle }}} {\tilde{1}}=f^ {{\tilde{c}}}$

• (iii) $f_1 {{\tilde{\triangle }}} f_2=f_2 {{\tilde{\triangle }}} f_1$

Note 6

It must be noted that the symmetric difference operation is non-associative.

Definition 31

Let $f_1,f_2 \in D_E(U)$. If $f_1 {{\tilde{\cap }}} f_2={\tilde{0}}$, then $f_1$ and $f_2$ are called disjoint.

Example 4

Let $E=\{x_1,x_2,x_3\}$ be a parameter set, $U=\{u_1,u_2,u_3,u_4\}$ be a universal set,

$$\begin{aligned} f_1&=\left\{ \left( {^{[0.1,0.5]}_{[0.2,0.3]}}x_1,\left\{ {{^{1}_{0}}}u_2,{{^{[0,0.1]}_{[0.5,0.6]}}}u_3, {{^{[0,0.1]}_{[0.5,0.7]}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{1}_{0}}x_2,\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0,0.1]}_{[0,0.5]}}}u_2, {{^{[0.4,0.5]}_{[0.3,0.4]}}}u_3,{{^{[0.5,0.6]}_{[0.1,0.3]}}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{[0.3,0.4]}_{[0,0.1]}}x_3,\left\{ {{^{[0.3,0.4]}_{[0.1,0.5]}}}u_1,{{^{[0,0.7]}_{[0.1,0.2]}}}u_3 \right\} \right) \right\} , \end{aligned}$$

and

$$\begin{aligned} f_2&=\left\{ \left( {^{[0.2,0.3]}_{[0,0.5]}}x_2,\left\{ {{^{[0.1,0.2]}_{[0.4,0.5]}}}u_1,{{^{[0.3,0.7]}_{[0.1,0.2]}}}u_2, {{^{[0.3,0.4]}_{[0.4,0.6]}}}u_3\right\} \right) ,\right. \\&\left. \qquad\left( {^{[0.3,0.8]}_{[0,0.1]}}x_3,\left\{ {{^{1}_{0}}}u_1,{{^{[0.2,0.3]}_{[0,0.4]}}}u_3,{{^{[0.1,0.4]}_{[0.5,0.6]}}}u_4 \right\} \right) \right\} \end{aligned}$$

Then,

$$\begin{aligned} f_1 {{\tilde{\cup }}} f_2&=\left\{ \left( {^{[0.1,0.5]}_{[0.2,0.3]}}x_1,\left\{ {{^{1}_{0}}}u_2,{{^{[0,0.1]}_{[0.5,0.6]}}}u_3, {{^{[0,0.1]}_{[0.5,0.7]}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{1}_{0}}x_2,\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0.3,0.7]}_{[0,0.2]}}}u_2, {{^{[0.4,0.5]}_{[0.3,0.4]}}}u_3,{{^{[0.5,0.6]}_{[0.1,0.3]}}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{[0.3,0.8]}_{[0,0.1]}}x_3,\left\{ {{^{1}_{0}}}u_1,{{^{[0.2,0.7]}_{[0,0.2]}}}u_3, {{^{[0.1,0.4]}_{[0.5,0.6]}}}u_4\right\} \right) \right\} ,\\ f_1 {{\tilde{\cap }}} f_2&=\left\{ \left( {^{[0.2,0.3]}_{[0,0.5]}}x_2,\left\{ {{^{[0.1,0.2]}_{[0.4,0.5]}}}u_1,{{^{[0,0.1]}_{[0.1,0.5]}}}u_2, {{^{[0.3,0.4]}_{[0.4,0.6]}}}u_3\right\} \right) ,\right. \\&\left. \qquad\left( {^{[0.3,0.4]}_{[0,0.1]}}x_3,\left\{ {{^{[0.3,0.4]}_{[0.1,0.5]}}}u_1,{{^{[0,0.3]}_{[0.1,0.4]}}}u_3,\right\} \right) \right\} ,\\ f_1 {{\tilde{\setminus }}} f_2&=\left\{ \left( {^{[0.1,0.5]}_{[0.2,0.3]}}x_1,\left\{ {{^{1}_{0}}}u_2,{{^{[0,0.1]}_{[0.5,0.6]}}}u_3, {{^{[0,0.1]}_{[0.5,0.7]}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{[0,0.5]}_{[0.2,0.3]}}x_2,\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0,0.1]}_{[0.3,0.7]}}}u_2, {{^{[0.4,0.5]}_{[0.3,0.4]}}}u_3,{{^{[0.5,0.6]}_{[0.1,0.3]}}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{[0,0.1]}_{[0.3,0.8]}}x_3,\left\{ {{^{[0,0.4]}_{[0.2,0.3]}}}u_3\right\} \right) \right\} ,\\ f_1^{{\tilde{c}}}&=\left\{ \left( {^{[0.2,0.3]}_{[0.1,0.5]}}x_1,\left\{ {{^{1}_{0}}}u_1,{{^{[0.5,0.6]}_{[0,0.1]}}}u_3, {{^{[0.5,0.7]}_{[0,0.1]}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{0}_{1}}x_2,\left\{ {{^{[0.2,0.5]}_{[0.2,0.4]}}}u_1,{{^{[0,0.5]}_{[0,0.1]}}}u_2, {{^{[0.3,0.4]}_{[0.4,0.5]}}}u_3,{{^{[0.1,0.3]}_{[0.5,0.6]}}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{[0,0.1]}_{[0.3,0.4]}}x_3,\left\{ {{^{[0.1,0.5]}_{[0.3,0.4]}}}u_1,{{^{1}_{0}}}u_2,{{^{[0.1,0.2]}_{[0,0.7]}}}u_3,{{^{1}_{0}}}u_4 \right\} \right) \right\} , \end{aligned}$$

and

$$\begin{aligned} f_1 {{\tilde{\triangle }}} f_2&=\left\{ \left( {^{[0.1,0.5]}_{[0.2,0.3]}}x_1,\left\{ {{^{1}_{0}}}u_2,{{^{[0,0.1]}_{[0.5,0.6]}}}u_3, {{^{[0,0.1]}_{[0.5,0.7]}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{[0,0.5]}_{[0.2,0.3]}}x_2,\left\{ {{^{[0.2,0.4]}_{[0.2,0.5]}}}u_1,{{^{[0,0.5]}_{[0.1,0.2]}}}u_2, {{^{[0.4,0.5]}_{[0.3,0.4]}}}u_3,{{^{[0.5,0.6]}_{[0.1,0.3]}}}u_4\right\} \right) ,\\&\left. \qquad\left( {^{[0,0.1]}_{[0.3,0.4]}}x_3,\left\{ {{^{[0.1,0.5]}_{[0.3,0.4]}}}u_1,{{^{[0.1,0.4]}_{[0,0.3]}}}u_3, {{^{[0.1,0.4]}_{[0.5,0.6]}}}u_4\right\} \right) \right\} \end{aligned}$$

Definition 32

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 \tilde{+} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) \tilde{+} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called sum of $f_1$ and $f_2$ and is denoted by $f_1 \tilde{+} f_2$.

Definition 33

Let $f_1,f_2,f_3 \in D_E(U)$. If $\kappa _3=\kappa _1 {{\tilde{\cdot }}} \kappa _2$ and for all $x\in E$, $f_3({{^{\alpha _3(x)}_{\beta _3(x)}}}x)=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cdot }}} f_2({{^{\alpha _2(x)}_{\beta _2(x)}}}x)$, then $f_3$ is called product of $f_1$ and $f_2$ and is denoted by $f_1 {{\tilde{\cdot }}} f_2$.

Proposition 24

Let $f,f_1,f_2,f_3 \in D_E(U)$. Then,

• (i) $f \tilde{+} {\tilde{0}}=f$ and $f \, {{\tilde{\cdot }}} \, {\tilde{0}}={\tilde{0}}$

• (ii) $f \tilde{+} {\tilde{1}}={\tilde{1}}$ and $f \, {{\tilde{\cdot }}} \, {\tilde{1}}=f$

• (iii) $f_1 \tilde{+} f_2=f_2 \tilde{+} f_1$ and $f_1 {{\tilde{\cdot }}} f_2=f_2 {{\tilde{\cdot }}} f_1$

• (iv) $(f_1 \tilde{+} f_2) \tilde{+} f_3=f_1 \tilde{+} (f_2 \tilde{+} f_3)$

  $(f_1 {{\tilde{\cdot }}} f_2) {{\tilde{\cdot }}} f_3=f_1 {{\tilde{\cdot }}}(f_2 {{\tilde{\cdot }}} f_3)$

• (v) $(f_1 \tilde{+} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 \, {{\tilde{\cdot }}} f^{{\tilde{c}}}_2$ and $(f_1 {{\tilde{\cdot }}} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 \tilde{+} f^{{\tilde{c}}}_2$

Definition 34

Let $f_1 \in D_{E_1}(U)$, $f_2 \in D_{E_2}(U)$, and $f_3 \in D_{E_1\times E_2}(U)$. For all $(x,y) \in E_1\times E_2$, if $\alpha _3(x,y)=$ $\inf \{\alpha _1(x),\alpha _2(y)\}$, $\beta _3(x,y)=\sup \{\beta _1(x),\beta _2(y)\}$, and $f_3({{^{\alpha _3(x,y)}_{\beta _3(x,y)}}}(x,y))=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cap }}} f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y)$, then $f_3$ is called and-product of $f_1$ and $f_2$ and is denoted by $f_1 \wedge f_2$.

Definition 35

Let $f_1 \in D_{E_1}(U)$, $f_2 \in D_{E_2}(U)$, and $f_3 \in D_{E_1\times E_2}(U)$. For all $(x,y) \in E_1\times E_2$, if $\alpha _3(x,y)=\sup \{\alpha _1(x),\alpha _2(y)\}$, $\beta _3(x,y)=\inf \{\beta _1(x),\beta _2(y)\}$, and $f_3({{^{\alpha _3(x,y)}_{\beta _3(x,y)}}}(x,y))=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cup }}} f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y)$, then $f_3$ is called or-product of $f_1$ and $f_2$ and is denoted by $f_1 \vee f_2$.

Definition 36

Let $f_1 \in D_{E_1}(U)$, $f_2 \in D_{E_2}(U)$, and $f_3 \in D_{E_1\times E_2}(U)$. For all $(x,y) \in E_1\times E_2$, if $\alpha _3(x,y)=\inf \{\alpha _1(x),\beta _2(y)\}$, $\beta _3(x,y)=\sup \{\beta _1(x),\alpha _2(y)\}$, and $f_3({{^{\alpha _3(x,y)}_{\beta _3(x,y)}}}(x,y))=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cap }}} f^{{\tilde{c}}}_2({{^{\beta _2(y)}_{\alpha _2(y)}}}y)$, then $f_3$ is called andnot-product of $f_1$ and $f_2$ and is denoted by $f_1 {{{\overline{\wedge }}}} f_2$.

Definition 37

Let $f_1 \in D_{E_1}(U)$, $f_2 \in D_{E_2}(U)$, and $f_3 \in D_{E_1\times E_2}(U)$. For all $(x,y) \in E_1\times E_2$, if $\alpha _3(x,y)=\sup \{\alpha _1(x),\beta _2(y)\}$, $\beta _3(x,y)=\inf \{\beta _1(x),\alpha _2(y)\}$, and $f_3({{^{\alpha _3(x,y)}_{\beta _3(x,y)}}}(x,y))=f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cup }}} f^{{\tilde{c}}}_2({{^{\beta _2(y)}_{\alpha _2(y)}}}y)$, then $f_3$ is called ornot-product of $f_1$ and $f_2$ and is denoted by $f_1 {{{\underline{\vee }}}} f_2$.

Proposition 25

Let $f_1 \in D_{E_1}(U)$, $f_2 \in D_{E_2}(U)$, and $f_3 \in D_{E_3}(U)$. Then,

• (i) $(f_1 \vee f_2) \vee f_3=f_1 \vee (f_2 \vee f_3)$

• (ii) $(f_1 \wedge f_2) \wedge f_3=f_1 \wedge (f_2 \wedge f_3)$

Proof

Let $E_{123}=E_1\times E_2 \times E_3$. Then, the proof of (i) is as follows:

$$\begin{aligned}&(f_1 \vee f_2) \vee f_3 \\&\quad = \left\{ \left( {{^{\sup \{\alpha _1(x),\alpha _2(y)\}}_{\inf \{\beta _1(x),\beta _2(y)\}}}}(x,y),f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cup }}} f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y)\right) : (x,y) \in E_{12} \right\} \\&\qquad\quad \vee \left\{ \left( {{^{\alpha _3(z)}_{\beta _3(z)}}}z,f_3({{^{\alpha _3(z)}_{\beta _3(z)}}}z)\right) : z\in E_3 \right\} \\&\quad = \left\{ \left( {^{\sup \{\sup \{\alpha _1(x),\alpha _2(y)\},\alpha _3(z)\}}_{\inf \{\inf \{\beta _1(x),\beta _2(y)\},\beta _3(z)\}}}(x,y,z),\right. \right. \\&\qquad\quad \left. \left. \left( f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) {{\tilde{\cup }}} f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y)\right) {{\tilde{\cup }}} f_3({{^{\alpha _3(z)}_{\beta _3(z)}}}z)\right) : (x,y,z) \in E_{123} \right\} \\&\quad = \left\{ \left( {^{\sup \{\alpha _1(x),\sup \{\alpha _2(y),\alpha _3(z)\}\}}_{\inf \{\beta _1(x),\inf \{\beta _2(y),\beta _3(z)\}\}}}(x,y,z), f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x) \right. \right. \\&\qquad\quad \left. \left. {{\tilde{\cup }}} \left( f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y) {{\tilde{\cup }}} f_3({{^{\alpha _3(z)}_{\beta _3(z)}}}z)\right) \right) : (x,y,z) \in E_{123} \right\} \\&\quad = \left\{ \left( {{^{\alpha _1(x)}_{\beta _1(x)}}}x,f_1({{^{\alpha _1(x)}_{\beta _1(x)}}}x)\right) : x\in E_1 \right\} \\&\qquad\quad \vee \left\{ \left( {{^{\sup \{\alpha _2(y),\alpha _3(z)\}}_{\inf \{\beta _2(y),\beta _3(z)\}}}}(y,z),f_2({{^{\alpha _2(y)}_{\beta _2(y)}}}y) {{\tilde{\cup }}} f_3({{^{\alpha _3(z)}_{\beta _3(z)}}}z)\right) : (y,z) \in E_{23} \right\} \\&\quad = f_1 \vee (f_2 \vee f_3) \end{aligned}$$

$\square$

Proposition 26

Let $f_1 \in D_{E_1}(U)$ and $f_2 \in D_{E_2}(U)$. Then, the following De Morgan’s laws are valid.

• (i) $(f_1 \vee f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 \wedge f^{{\tilde{c}}}_2$

• (ii) $(f_1 \wedge f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 \vee f^{{\tilde{c}}}_2$

• (iii) $(f_1 {{{\underline{\vee }}}} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 {{{\overline{\wedge }}}} f^{{\tilde{c}}}_2$

• (iv) $(f_1 {{{\overline{\wedge }}}} f_2)^{{\tilde{c}}}=f^{{\tilde{c}}}_1 {{{\underline{\vee }}}} f^{{\tilde{c}}}_2$

Note 7

It must be noted that the products mentioned above of d-sets are non-commutative and non-distributive. Moreover, andnot-product and ornot-product are non-associative.

4 The proposed soft decision-making method

In this section, we first define an aggregate ivif-set of a d-set.

Definition 38

(Huang et al. 2013) Let $\kappa \in IVIF(E)$ and $|E|=n$. Then, the average cardinality of $\kappa$, denoted by $|\kappa |_a$, is defined by

$$\begin{aligned} |\kappa |_a:=\frac{1}{2} \sum \limits _{i=1}^n \left( 1 + \frac{\alpha ^-(x_i)+\alpha ^+(x_i)}{2} - \frac{\beta ^-(x_i)+\beta ^+(x_i)}{2}\right) \end{aligned}$$

Definition 39

A function ${\mathcal {A}} : D_E(U) \rightarrow IVIF(U)$ defined by ${\mathcal {A}}(f)=f^*$ is called an aggregation operator over U and $f^*$ is called aggregate ivif-set of f. Here, $f^*=\left\{ {{^{\omega ^*(u)}_{\theta ^*(u)}}}u : u \in U \right\}$ such that $\omega ^*(u)=\frac{1}{|\kappa |_a} \sum \limits _{x \in E} \alpha (x) \omega _x(u)$, and $\theta ^*(u)=\frac{1}{|\kappa |_a} \sum \limits _{x \in E} \beta (x) \theta _x(u)$, for $f=\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x,\left\{ {{^{\omega _x(u)}_{\theta _x(u)}}}u : u\in U \right\} \right) : x\in E \right\}$.

Secondly, we suggest a soft decision-making method that assigns a performance-based value to the alternatives via this aggregate ivif-set. Thus, we can choose the optimal elements among the alternatives.

Algorithm Steps of the Proposed Method

Step 1.:

Construct a d-set f over U

Step 2.:

Obtain the aggregate ivif-set $f^*$ of f

Step 3.:

Obtain the values $s(u)=\omega ^*(u)-\theta ^*(u)$, for all $u \in U$

Step 4.:

Obtain the decision set $\{^{d(u_k)}u_k | u_k\in U\}$ such that

$$\begin{aligned} d(u_k)= & \left[ \frac{s^{-}(u_k) + |\min \limits _{i}s^{-}(u_i)|}{\max \limits _{i}s^{+}(u_i)+ |\min \limits _{i}s^{-}(u_i)|},\frac{s^{+}(u_k) + |\min \limits _{i}s^{-}(u_i)|}{\max \limits _{i}s^{+}(u_i) + |\min \limits _{i}s^{-}(u_i)|}\right] \end{aligned}$$

Step 5.:

Select the optimal elements among the alternatives via linear ordering relation provided in Xu and Yager (2006)

$$\begin{aligned}&\left[ \gamma ^{-}_1,\gamma ^{+}_1\right] \\&\quad \le _{_{XY}} \left[ \gamma ^{-}_2,\gamma ^{+}_2\right] \Leftrightarrow \left[ \left( \gamma ^{-}_1 + \gamma ^{+}_1 < \gamma ^{-}_2 + \gamma ^{+}_2 \right) \right. \\&\quad \quad \vee \left( \gamma ^{-}_1 + \gamma ^{+}_1 \right. = \left. \left. \gamma ^{-}_2 + \gamma ^{+}_2 \wedge \gamma ^{+}_1 - \gamma ^{-}_1 \le \gamma ^{+}_2 - \gamma ^{-}_2 \right) \right] \end{aligned}$$

Here, $s(u):=[s^{-}(u),s^{+}(u)]$, for all $u \in U$.

5 An illustrative example for the proposed method

In this section, we apply the proposed method to a problem concerning the eligibility of candidates for two vacant positions in a job advertisement. Assume that six candidates, denoted by $U=\{u_1,u_2,u_3,u_4,u_5,u_6\}$, have applied for two vacant positions announced by a company. Then, the human resources department (HR) of the company has firstly determined a parameter set $E=\{x_1,x_2,x_3,x_4\}$ such that $x_1=$ “knowledge of software”, $x_2=$ “knowledge of foreign language”, $x_3=$ “age”, and $x_4=$ “experience”.

Secondly, HR has obtained the ivif-value of the parameters by the membership and the nonmembership functions defined by

$$\begin{aligned} \alpha (x):= & \left[ \frac{\min \limits _{n}\mu ^x_{n}}{\max \limits _{n}\mu ^x_{n} + \max \limits _{n}\nu ^x_{n} + \min \left\{ \min \limits _{k \in I} \pi ^x_{k},\min \limits _{t \in J} \pi ^x_{t}\right\} } , \right. \left. \right. \\&\left. \frac{\max \limits _{n}\mu ^x_{n}}{\max \limits _{n}\mu ^x_{n} + \max \limits _{n}\nu ^x_{n} + \min \left\{ \min \limits _{k \in I} \pi ^x_{k},\min \limits _{t \in J} \pi ^x_{t}\right\} }\right] \end{aligned}$$

and

$$\begin{aligned} \beta (x):= & \left[ \frac{\min \limits _{n}\nu ^x_{n}}{\max \limits _{n}\mu ^x_{n} + \max \limits _{n}\nu ^x_{n} + \min \left\{ \min \limits _{k \in I} \pi ^x_{k},\min \limits _{t \in J} \pi ^x_{t}\right\} } , \right. \left. \right. \\&\left. \frac{\max \limits _{n}\nu ^x_{n}}{\max \limits _{n}\mu ^x_{n} + \max \limits _{n}\nu ^x_{n} + \min \left\{ \min \limits _{k \in I} \pi ^x_{k},\min \limits _{t \in J} \pi ^x_{t}\right\} }\right] \end{aligned}$$

such that $I=\left\{ k : \max \limits _{n}\mu ^x_{n}=\mu ^x_{k}\right\}$ and $J=\left\{ t : \max \limits _{n}\nu ^x_{n}=\nu ^x_{t}\right\}$. Here, ($\mu ^{x}_{n})$, $(\nu ^{x}_{n})$, and $(\pi ^{x}_{n})$ are ordered s-tuples which indicate the degrees of membership, nonmembership, and indeterminacy obtained according to the criteria determined by HR, for parameters.

For example, HR determines five software programs and, for the nth software programs, $\mu ^{x_1}_{n}$, $\nu ^{x_1}_{n}$, and $\pi ^{x_1}_{n}$ denote the numbers of employees who have a valid certificate, who do not know how to use the software, and who hold the knowledge of the software but have no valid certificate, respectively. If $(\mu ^{x_1}_{n})=(18,10,15,$ 16, 12), $(\nu ^{x_1}_{n})=(1,5,3,1,7)$, and $(\pi ^{x_1}_{n})=(1,5,$ 2, 3, 1), then the membership degree and the nonmembership degree of the parameter $x_1$ are [0.38, 0.69] and [0.04, 0.27], respectively. Similarly, the ivif-values of the other parameters can be constructed by HR. Thus, an ivif-set $\kappa$ over E can be given as follows:

$$\begin{aligned} \kappa =\left\{ {^{[0.38,0.69]}_{[0.04,0.27]}}x_1,{^{[0.53,0.6]}_{[0.34,0.37]}}x_2,{^{[0.05,0.25]}_{[0.32,0.45]}}x_3,{^{[0.4,0.52]}_{[0.26,0.38]}}x_4 \right\} \end{aligned}$$

Here, [0.38, 0.69] means that the positive effect of “knowledge of software” on success occurs between 38 and 69%. Moreover, [0.04, 0.27] means that the negative effect of “knowledge of software” on success ranges from 4 to 27%.

The application of the soft decision-making method proposed in Sect. 4 is as follows:

Step 1.:

The d-set f modelling the decision-making problem mentioned above is as follows:

$$\begin{aligned} f&=\left\{ \left( {^{[0.38,0.69]}_{[0.04,0.27]}}x_1,\left\{ {{^{[0.36,0.46]}_{[0.11,0.28]}}}u_1,{{^{[0,0.16]}_{[0.38,0.53]}}}u_2,\right. \right. \right. \\&\left.\qquad \left. {{^{[0.15,0.22]}_{[0,0.75]}}}u_3, {{^{[0.2,0.62]}_{[0.15,0.24]}}}u_4\right\} \right) ,\\&\qquad\left( {^{[0.53,0.6]}_{[0.34,0.37]}}x_2,\left\{ {{^{[0.24,0.32]}_{[0.34,0.65]}}}u_1,{{^{[0.28,0.35]}_{[0,0.62]}}} u_2,\right. \right. \\&\qquad \left. \left. {{^{[0.25,0.32]}_{[0.1,0.64]}}}u_3,{{^{[0.12,0.44]}_{[0.45,0.54]}}}u_4,{{^{1}_{0}}}u_5,{{^{[0,0.34]}_{[0,0.2]}}}u_6\right\} \right) ,\\&\qquad\left( {^{[0.05,0.25]}_{[0.32,0.45]}}x_3,\left\{ {^{[0.35,0.55]}_{[0.15,0.29]}}u_1,{{^{[0.26,0.35]}_{[0.43,0.55]}}}u_3,\right. \right. \\&\qquad\left. \left. {{^{[0.38,0.52]}_{[0.19,0.25]}}}u_4,{{^{[0.15,0.24]}_{[0.29,0.72]}}}u_6\right\} \right) ,\\&\qquad\left( {^{[0.4,0.52]}_{[0.26,0.38]}}x_4,\left\{ {{^{0}_{[0.52,0.58]}}}u_1, {{^{1}_{0}}}u_2,{{^{[0.18,0.28]}_{[0.27,0.7]}}}u_3,\right. \right. \\&\qquad\left. \left. \left. {{^{0}_{[0.1,1]}}}u_4,{{^{[0.35,0.4]}_{[0,0.1]}}}u_5,{{^{[0.28,0.48]}_{[0.33,0.34]}}}u_6\right\} \right) \right\} \end{aligned}$$

The ivif-value of the candidates for each parameter has been obtained by the membership function $\omega _x(u)$ and the nonmembership function $\theta _x(u)$ defined by

$$\begin{aligned}&\left[ \frac{\min \limits _{n}{(\mu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}{(\nu ^{x}_{u})}_{n} + \min \left\{ \min \limits _{k \in I} {(\pi ^{x}_{u})}_{k},\min \limits _{t \in J} {(\pi ^{x}_{u})}_{t}\right\} } ,\right. \\&\quad \left. \frac{\max \limits _{n}{(\mu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}{(\nu ^{x}_{u})}_{n} + \min \left\{ \min \limits _{k \in I} {(\pi ^{x}_{u})}_{k},\min \limits _{t \in J} {(\pi ^{x}_{u})}_{t}\right\} }\right] \end{aligned}$$

and

$$\begin{aligned}&\left[ \frac{\min \limits _{n}{(\nu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}{(\nu ^{x}_{u})}_{n} + \min \left\{ \min \limits _{k \in I} {(\pi ^{x}_{u})}_{k},\min \limits _{t \in J} {(\pi ^{x}_{u})}_{t}\right\} } , \right. \\&\quad \left. \frac{\max \limits _{n}{(\nu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}{(\nu ^{x}_{u})}_{n} + \min \left\{ \min \limits _{k \in I} {(\pi ^{x}_{u})}_{k},\min \limits _{t \in J} {(\pi ^{x}_{u})}_{t}\right\} }\right] \end{aligned}$$

respectively, such that $I=\left\{ k : \max \limits _{n}{(\mu ^{x}_{u})}_{n}={(\mu ^{x}_{u})}_{k}\right\}$ and $J=\left\{ t : \max \limits _{n}{(\nu ^{x}_{u})}_{n}={(\mu ^{x}_{u})}_{t}\right\}$. Here, $({(\mu ^{x}_{u})}_{n})$, $({(\nu ^{x}_{u})}_{n})$, and $({(\pi ^{x}_{u})}_{n})$ are ordered s-tuples which indicate the degrees of membership, nonmembership, and indeterminacy according to the parameters of the candidates.

For example, ten questions are asked to the candidates regarding each software program and they are asked to answer these questions using a three-level Likert scale, i.e. positive, negative, and indeterminant. Here, ${(\mu ^{x_1}_{u_5})}_{n}$, ${(\nu ^{x_1}_{u_5})}_{n}$, and ${(\pi ^{x_1}_{u_5})}_{n}$ denote the number of positive, negative, and indeterminant answers according to the $x_1$ parameter of the candidate $u_5$, respectively. If $({(\mu ^{x_1}_{u_5})}_{n})=(0,0,0,0,0)$, $({(\nu ^{x_1}_{u_5})}_{n})=(10,10,10,10,10)$, and $({(\pi ^{x_1}_{u_5})}_{n})=(0,0,0,0,0)$, then the membership degree and the nonmembership degree of the candidate $u_5$ according to the parameter $x_1$ are $[0,0]=0$ and $[1,1]=1$, respectively. Similarly, the ivif-values of the other candidates can be constructed.

Step 2.:

$f^*$ is as follows:

$$\begin{aligned}&\left\{ {^{[0.1253,0.2878]}_{[0.1349,0.2968]}}u_1,{^{[0.2440,0.3739]}_{[0.1491,0.3660]}}u_2,{^{[0.1221,0.2567]}_{[0.1076,0.4239]}}u_3,\right. \\&\quad \left. {^{[0.0706,0.3657]}_{[0.1094,0.3369]}}u_4,{^{[0.2981,0.3595]}_{[0.1602,0.3373]}}u_5,{^{[0.0532,0.2285]}_{[0.0973,0.3547]}}u_6 \right\} \end{aligned}$$

where

$$\begin{aligned} \omega ^*(u_1)&=\frac{[0.38,0.69] \cdot [0.36,0.46] + [0.53,0.6] \cdot [0.24,0.32]}{2.2475}\\&+\frac{[0.05,0.25] \cdot [0.35,0.55] + [0.4,0.52] \cdot [0,0]}{2.2475} \\ &=[0.1253,0.2878] \end{aligned}$$

and

$$\begin{aligned} \theta ^*(u_1)&=\frac{[0.04,0.27] \cdot [0.11,0.28] + [0.34,0.37] \cdot [0.34,0.65]}{2.2475} \\&+\frac{[0.32,0.45] \cdot [0.15,0.29] + [0.26,0.38] \cdot [0.52,0.58]}{2.2475}\\ &=[0.1349,0.2968] \end{aligned}$$

Step 3.:

For all $u \in U$, the values s(u) are as follows:

$$\begin{aligned} s(u_1)= & [-0.1715,0.1529], s(u_2)=[-0.1220,0.2248], \\ s(u_3)= & [-0.3018,0.1491], s(u_4)=[-0.2663,0.2563], \\ s(u_5)= & [-0.0392,0.1993], s(u_6)=[-0.3015,0.1313] \end{aligned}$$

Step 4.:

The decision set is as follows:

$$\begin{aligned}&\left\{ ^{[0.2334,0.8148]}u_1, ^{[0.3223,0.9436]}u_2, ^{[0,0.8079]}u_3, \right. \\&\,\,\,\left. ^{[0.0636,1]}u_4, ^{[0.4706,0.8980]}u_5, ^{[0.0005,0.7760]}u_6\right\} \end{aligned}$$

where $d(u_1)$ is calculated as follows:

$$\begin{aligned} d(u_1)=\left[ \frac{-0.1715 + |-0.3018|}{0.2563 + |-0.3018|},\frac{0.1529 + |-0.3018|}{0.2563 + |-0.3018|}\right] \end{aligned}$$

Step 5.:

According to the linear ordering relation $(\le _{_{XY}})$, the ranking order $u_6 \prec u_3 \prec u_1 \prec u_4 \prec u_2 \prec u_5$ is valid. The results show that $u_5$ and $u_2$ are more eligible for the vacant positions than the others. Thus, the candidates $u_5$ and $u_2$ are selected for the positions announced by the company.

6 Comparison results

In this section, we first provide the definitions of fuzzy sets (Zadeh 1965), intuitionistic fuzzy sets (Atanassov 1986), fuzzy parameterized fuzzy soft sets (Çağman et al. 2010), fuzzy parameterized intuitionistic fuzzy soft sets (Sulukan et al. 2019), and intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets (Karaaslan 2016) by taking into account the notations used throughout this paper.

Definition 40

(Zadeh 1965) Let E be a universal set and $\mu$ be a function from E to [0, 1]. Then, the set $\left\{ {^{\mu (x)}}x : x \in E \right\}$ being the graphic of $\mu$ is called a fuzzy set (f-set) over E. Besides, the set of all f-sets over E is denoted by F(E).

Definition 41

(Atanassov 1986) Let E be a universal set and $\eta$ be a function from E to $[0,1] \times [0,1]$. Then, the set $\left\{ {{^{\mu (x)}_{\nu (x)}}}x : x \in E \right\}$ being the graphic of $\eta$ is called an intuitionistic fuzzy set (if-set) over E.

Here, for all $x \in E$, $0 \le \mu (x)+\nu (x) \le 1$. Moreover, $\mu$ and $\nu$ are called the membership function and the nonmembership function in an if-set, respectively, and $\pi (x)=1-\mu (x)-\nu (x)$ is called the degree of indeterminacy of the element $x \in E$. Further, the set of all if-sets over E is denoted by IF(E).

Moreover, each fuzzy set can be written as $\left\{ {{^{\mu (x)}_{1-\mu (x)}}}x : x \in E \right\}$.

Definition 42

(Karaaslan 2016) Let U be a universal set, E be a parameter set, $\eta \in IF(E)$, and g be a function from $\eta$ to IF(U). Then, the set $\left\{ \left( {{^{\mu (x)}_{\nu (x)}}}x,g({{^{\mu (x)}_{\nu (x)}}}x)\right) : x\in E \right\}$ being the graphic of g is called an intuitionistic fuzzy parameterized intuitionistic fuzzy soft set (ifpifs-set) parameterized via E over U (or briefly over U). Besides, the set of all ifpifs-sets over U is denoted by $IFPIFS_E(U)$.

Definition 43

(Çağman et al. 2010) Let U be a universal set, E be a parameter set, $\mu \in F(E)$, and h be a function from $\mu$ to F(U). Then, the set $\left\{ \left( {^{\mu (x)}}x,h({^{\mu (x)}}x)\right) : x\in E \right\}$ being the graphic of h is called an fuzzy parameterized fuzzy soft set (fpfs-set) parameterized via E over U (or briefly over U). Besides, the set of all fpfs-sets over U is denoted by $FPFS_E(U)$.

Definition 44

(Sulukan et al. 2019) Let U be a universal set, E be a parameter set, $\mu \in F(E)$, and p be a function from $\mu$ to IF(U). Then, the set $\left\{ \left( {{^{\mu (x)}}}x,p({{^{\mu (x)}}}x)\right) : x\in E \right\}$ being the graphic of p is called an fuzzy parameterized intuitionistic fuzzy soft set (fpifs-set) parameterized via E over U (or briefly over U). Besides, the set of all fpifs-sets over U is denoted by $FPIFS_E(U)$.

Since the proposed method in Sect. 4 is the first method proposed in relation to this structure (d-sets), it is impossible to compare this method with another in this sense. However, if the uncertainties in the modelled problem are decreased, it is possible to compare the method with the others in a substructure, such as ifpifs-sets, fpifs-sets, and fpfs-sets. For this reason, secondly, we define four new concepts, i.e. mean reduction, mean bireduction, mean bireduction-reduction, and mean reduction-bireduction.

Definition 45

Let $f\in D_E(U)$, that is $f:=\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x, \left\{ {{^{\omega (u)}_{\theta (u)}}}u : u\in U \right\} \right) : x\in E \right\}$. Then, the ifpifs-set

$$\begin{aligned} \left\{ \left( {{^{\frac{\alpha ^-(x) + \alpha ^+(x)}{2}}_{\frac{\beta ^-(x) + \beta ^+(x)}{2}}}}x, \left\{ {{^{\frac{\omega ^-(u) + \omega ^+(u)}{2}}_{\frac{\theta ^-(u) + \theta ^+(u)}{2}}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

is called mean reduction of f and is denoted by $f_{mr}$.

Definition 46

Let $f\in D_E(U)$, that is $f:=\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x, \left\{ {{^{\omega (u)}_{\theta (u)}}}u : u\in U \right\} \right) : x\in E \right\}$. Then, the fpfs-set

$$\begin{aligned}&\left\{ \left( {^{\frac{\alpha ^-(x) + \alpha ^+(x) - \beta ^-(x) - \beta ^+(x) + 2}{4}}}x, \left\{ {^{\frac{\omega ^-(u) + \omega ^+(u) - \theta ^-(u) - \theta ^+(u) + 2}{4}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

is called mean bireduction of f and is denoted by $f_{mb}$.

Definition 47

Let $f\in D_E(U)$, that is $f:=\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x, \left\{ {{^{\omega (u)}_{\theta (u)}}}u : u\in U \right\} \right) : x\in E \right\}$. Then, the fpifs-set

$$\begin{aligned}&\left\{ \left( {^{\frac{\alpha ^-(x) + \alpha ^+(x) - \beta ^-(x) - \beta ^+(x) + 2}{4}}}x, \left\{ {{^{\frac{\omega ^-(u) + \omega ^+(u)}{2}}_{\frac{\theta ^-(u) + \theta ^+(u)}{2}}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

is called mean bireduction-reduction of f and is denoted by $f_{mbr}$.

Definition 48

Let $f\in D_E(U)$, that is $f:=\left\{ \left( {{^{\alpha (x)}_{\beta (x)}}}x, \left\{ {{^{\omega (u)}_{\theta (u)}}}u : u\in U \right\} \right) : x\in E \right\}$. Then, the ifpfs-set

$$\begin{aligned}&\left\{ \left( {{^{\frac{\alpha ^-(x) + \alpha ^+(x)}{2}}_{\frac{\beta ^-(x) + \beta ^+(x)}{2}}}}x,\right. \right. \\&\quad \left. \left. \left\{ {^{\frac{\omega ^-(u) + \omega ^+(u) - \theta ^-(u) - \theta ^+(u) + 2}{4}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

is called mean reduction-bireduction of f and is denoted by $f_{mrb}$.

Example 5

$f_{mr}$, $f_{mbr}$, and $f_{mb}$ for the d-set f provided in Section 5 are as follows:

$$\begin{aligned} f_{mr}&=\left\{ \left( {^{0.53}_{0.16}}x_1,\left\{ {{^{0.41}_{0.2}}}u_1,{{^{0.08}_{0.46}}}u_2,{{^{0.19}_{0.38}}}u_3, {{^{0.41}_{0.2}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{0.57}_{0.36}}x_2,\left\{ {{^{0.28}_{0.5}}}u_1,{{^{0.32}_{0.31}}}u_2, {{^{0.29}_{0.37}}}u_3,{{^{0.28}_{0.5}}}u_4,{{^{1}_{0}}}u_5,{{^{0.17}_{0.1}}}u_6\right\} \right) ,\\&\qquad\left( {^{0.15}_{0.39}}x_3,\left\{ {{^{0.45}_{0.22}}}u_1,{{^{0.31}_{0.49}}}u_3,{{^{0.45}_{0.22}}}u_4,{{^{0.2}_{0.51}}}u_6\right\} \right) , \\&\left. \qquad\left( {^{0.46}_{0.32}}x_4,\left\{ {{^{0}_{0.55}}}u_1, {{^{1}_{0}}}u_2,{{^{0.23}_{0.49}}}u_3,{{^{0}_{0.55}}}u_4,{{^{0.38}_{0.05}}}u_5,{{^{0.38}_{0.34}}}u_6\right\} \right) \right\} \\ f_{mbr}&=\left\{ \left( {^{0.69}}x_1,\left\{ {{^{0.41}_{0.2}}}u_1,{{^{0.08}_{0.46}}}u_2,{{^{0.19}_{0.38}}}u_3, {{^{0.41}_{0.2}}}u_4\right\} \right) ,\right. \\&\qquad\left( {^{0.61}}x_2,\left\{ {{^{0.28}_{0.5}}}u_1,{{^{0.32}_{0.31}}}u_2, {{^{0.29}_{0.37}}}u_3,{{^{0.28}_{0.5}}}u_4,{{^{1}_{0}}}u_5,{{^{0.17}_{0.1}}}u_6\right\} \right) ,\\&\qquad\left( {^{0.38}}x_3,\left\{ {{^{0.45}_{0.22}}}u_1,{{^{0.31}_{0.49}}}u_3,{{^{0.45}_{0.22}}}u_4,{{^{0.2}_{0.51}}}u_6\right\} \right) , \\&\left. \qquad\left( {^{0.57}}x_4,\left\{ {{^{0}_{0.55}}}u_1, {{^{1}_{0}}}u_2,{{^{0.23}_{0.49}}}u_3,{{^{0}_{0.55}}}u_4,{{^{0.38}_{0.05}}}u_5,{{^{0.38}_{0.34}}}u_6\right\} \right) \right\} \end{aligned}$$

and

$$\begin{aligned} f_{mb}&=\left\{ \left( ^{0.69}x_1,\left\{ {^{ 0.61}}u_1,{^{0.31}}u_2,{^{0.41}}u_3,{^{0.61}}u_4\right\} \right) ,\right. \\&\qquad\left( ^{0.61}x_2,\left\{ {^{0.39}}u_1,{^{0.5}}u_2,{^{0.46}}u_3,{^{0.39}}u_4,{^{1}}u_5,{^{0.54}}u_6\right\} \right) ,\\&\qquad\left( ^{0.38}x_3,\left\{ {^{0.62}}u_1,{^{0.41}}u_3,{^{0.62}}u_4,{^{0.35}}u_6\right\} \right) ,\\&\left. \qquad\left( ^{0.57}x_4,\left\{ {^{0.22}}u_1,{^{1}}u_2,{^{0.37}}u_3,{^{0.22}}u_4,{^{0.66}}u_5,{^{0.52}}u_6\right\} \right) \right\} \end{aligned}$$

Thirdly, we present the soft decision-making methods in Çağman et al. (2010), Karaaslan (2016), Kamacı (2019), and Sulukan et al. (2019) by considering the notations used throughout this study. Moreover, to sort all the alternatives instead of selecting only one optimum alternative, we rearrange the last step of the method provided in Karaaslan (2016), faithfully to the original. Furthermore, just as the concepts of intuitionistic fuzzy sets and interval-valued fuzzy sets are equivalent (Atanassov and Gargov 1989), so are the concepts of ivfpifs (Kamacı 2019) and ifpifs (Karaaslan 2016). Therefore, we express the algorithm of the method, provided in Kamacı (2019), by using ifpifs instead of ivfpifs.

Algorithm Steps of Method 1 (Karaaslan 2016)

Step 1.:

Construct an ifpifs-set

$$\begin{aligned} g=\left\{ \left( {{^{\mu (x)}_{\nu (x)}}}x, \left\{ {{^{\rho _x(u)}_{\sigma _x(u)}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

over U

Step 2.:

Obtain the if-set $g^*=\left\{ {{^{\rho ^*(u)}_{\sigma ^*(u)}}}u : u \in U \right\}$ such that $\rho ^*(u)=\frac{1}{|E|} \sum \limits _{x \in E} \mu (x)\rho _x(u)$ and $\sigma ^*(u)=\frac{1}{|E|} \sum \limits _{x \in E} \nu (x)\sigma _x(u)$. Here, |E| is cardinality of E.

Step 3.:

For all $u \in U$, obtain the values

$$\begin{aligned} \xi (u)=\frac{\rho ^*(u)}{\rho ^*(u) + \sigma ^*(u)} \end{aligned}$$

Step 4.:

Obtain the decision set $\left\{ ^{d(u_k)}u_k | u_k\in U\right\}$ such that $d(u_k)=\frac{\xi (u_k)}{\max \limits _{i}\xi (u_i)}$

Algorithm Steps of Method 2 (Kamacı 2019)

Step 1.:

Construct an ifpifs-set

$$\begin{aligned} g=\left\{ \left( {{^{\mu (x)}_{\nu (x)}}}x, \left\{ {{^{\rho _x(u)}_{\sigma _x(u)}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

over U

Step 2.:

Obtain the ivif-set $\kappa _1=\left\{ {{^{\alpha _1(u)}_{\beta _1(u)}}}u : u \in E \right\}$ such that

$$\begin{aligned} \alpha _1(u)= & \left[ 1-\prod \limits _{x \in E}(1-\mu (x)\rho _x(u)),1\right. \\&\left. -\prod \limits _{x \in E}(1-(1-\nu (x))\rho _x(u))\right] \end{aligned}$$

and

$$\begin{aligned} \beta _1(u)=\left[ \prod \limits _{x \in E}\mu (x)\sigma _x(u),\prod \limits _{x \in E}(1-\nu (x))\sigma _x(u)\right] \end{aligned}$$

Step 3.:

Obtain the ivif-set $\kappa _2=\left\{ {{^{\alpha _2(u)}_{\beta _2(u)}}}u : u \in U \right\}$ such that

$$\begin{aligned} \alpha _2(u)=\left[ \prod \limits _{x \in E}\mu (x)\rho _x(u),\prod \limits _{x \in E}(1-\nu (x))\rho _x(u)\right] \end{aligned}$$

and

$$\begin{aligned} \beta _2(u)= & \left[ 1-\prod \limits _{x \in E}(1-\mu (x)\sigma _x(u)),1\right. \\&\left. -\prod \limits _{x \in E}(1-(1-\nu (x))\sigma _x(u))\right] \end{aligned}$$

Step 4.:

Obtain the decision set $\kappa _3$ such that

$$\begin{aligned} \kappa _3:=\kappa _1 \tilde{+} \kappa _2 \end{aligned}$$

Step 5.:

Select the optimal elements among the alternatives via the ordering relation (Tan 2011; Xu 2007)

$$\begin{aligned} {{^{\alpha }_{\beta }}} \preceq {{^{{{\tilde{\alpha }}}}_{{{\tilde{\beta }}}}}} \Leftrightarrow \left[ \left( s_1({{^{\alpha }_{\beta }}}) < s_1({{^{{{\tilde{\alpha }}}}_{{{\tilde{\beta }}}}}})\right) \vee \left( s_1({{^{\alpha }_{\beta }}}) = s_1({{^{{{\tilde{\alpha }}}}_{{{\tilde{\beta }}}}}}) \wedge s_2({{^{\alpha }_{\beta }}}) \le s_2({{^{{{\tilde{\alpha }}}}_{{{\tilde{\beta }}}}}})\right) \right] \end{aligned}$$

such that

$$\begin{aligned} s_1({{^{\alpha }_{\beta }}})=\frac{\alpha ^{-}-\beta ^{-}+\alpha ^{+}-\beta ^{+}}{2} \end{aligned}$$

and

$$\begin{aligned} s_2({{^{\alpha }_{\beta }}})=\frac{\alpha ^{-}+\beta ^{-}+\alpha ^{+}+\beta ^{+}}{2} \end{aligned}$$

Here, ${{^{\alpha }_{\beta }}}:={{^{[{\alpha ^{-},\alpha ^{+}}]}_{[{\beta ^{-},\beta ^{+}}]}}}$ and ${{^{{{\tilde{\alpha }}}}_{{{\tilde{\beta }}}}}}:={{^{[{{{\tilde{\alpha }}}^{-},{{\tilde{\alpha }}}^{+}}]}_{[{{{\tilde{\beta }}}^{-},{{\tilde{\beta }}}^{+}}]}}}$ are ivif-values.

Algorithm Steps of Method 3 (Sulukan et al. 2019)

Step 1.:

Construct an fpifs-set over U

$$\begin{aligned} p=\left\{ \left( {{^{\mu (x)}}}x, \left\{ {{^{\rho _x(u)}_{\sigma _x(u)}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

Step 2.:

Obtain the values

$$\begin{aligned} \omega (u)=\frac{1}{|E|} \sum \limits _{x \in E} \mu (x)(\rho _x(u)-\sigma _x(u)) \end{aligned}$$

for all $u\in U$. Here, |E| is cardinality of E.

Step 3.:

Obtain the decision set $\left\{ ^{d(u_k)}u_k | u_k\in U\right\}$ such that $d(u_k)=\frac{{\omega (u_k)} + |\min \limits _{i}{\omega (u_i)}|}{\max \limits _{i}{\omega (u_i)} + |\min \limits _{i}{\omega (u_i)}|}$

Algorithm Steps of Method 4 (Çağman et al. 2010)

Step 1.:

Construct an fpfs-set over U

$$\begin{aligned} h=\left\{ \left( {^{\mu (x)}}x, \left\{ {{^{\nu _x(u)}}}u : u\in U \right\} \right) : x\in E \right\} \end{aligned}$$

Step 2.:

Obtain the f-set $h^*=\left\{ {{^{\nu ^*(u)}}}u : u \in U \right\}$ such that $\nu ^*(u)=\frac{1}{|E|} \sum \limits _{x \in E} \mu (x)\nu _x(u)$. Here, |E| is cardinality of E.

Step 3.:

Obtain the decision set $\left\{ ^{d(u_k)}u_k | u_k\in U\right\}$ such that $d(u_k)=\frac{\nu ^*(u_k)}{\max \limits _{i}\nu ^*(u_i)}$

Fourthly, we apply the proposed method and Method 1, 2, 3, and 4 to f, $f_{mr}$, $f_{mr}$, $f_{mbr}$, and $f_{mb}$ provided in Example 5, respectively. The decision sets and the ranking orders of the methods within their own structures are provided in Tables 1 and 2, respectively. The proposed method, Method 1, and Method 2 decide that the candidates $u_5$ and $u_2$ are eligible for the vacant positions. Thus, the candidates $u_5$ and $u_2$ are selected for the positions announced by the company. On the other hand, while Method 3 and 4 suggest the candidate $u_2$ for one of the two positions, it fails to decide between $u_1$ and $u_4$ for the other position. Moreover, these five methods propose that the candidates $u_6$ and $u_3$ are ineligible for the vacant positions. Furthermore, although the performances of the candidates $u_1$ and $u_4$ in the application of Method 1, 2, 3, and 4 are the same, the proposed method is capable of sorting them. Therefore, the proposed method has been successfully applied to the problem involving further uncertainties.

Table 1 The decision sets of the proposed method and Method 1, 2, 3, and 4

Table 2 The ranking orders of the five methods within their own structures

7 An application of the proposed method to a performance-based value assignment problem

In this section, we apply the proposed method and four state-of-the-art methods mentioned in the previous section to the performance-based value assignment problem for seven known filters used in image denoising, namely based on pixel density filter (BPDF) (Erkan and Gökrem 2018), modified decision based unsymmetrical trimmed median filter (MDBUTMF) (Esakkirajan et al. 2011), decision based algorithm (DBA) (Pattnaik et al. 2012), noise adaptive fuzzy switching median filter (NAFSMF) (Toh and Isa 2010), different applied median filter (DAMF) (Erkan et al. 2018), a new adaptive weighted mean filter (AWMF) (Tang et al. 2016), and adaptive Riesz mean filter (ARmF) (Enginoğlu et al. 2019). Hereinafter, let $U=\{u_1,u_2,u_3,u_4,u_5,u_6,u_7\}$ be an alternative set such that $u_1=$ “BPDF”, $u_2=$ “MDBUTMF”, $u_3=$ “DBA”, $u_4=$ “NAFSMF”, $u_5=$ “DAMF”, $u_6=$ “AWMF”, and $u_7=$ “ARmF”. Moreover, let $E=\{x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9\}$ be a parameter set determined by a decision-maker such that $x_1=$ “noise density 10%”, $x_2=$ “noise density 20%”, $x_3=$ “noise density 30%”, $x_4=$ “noise density 40%”, $x_5=$ “noise density 50%”, $x_6=$ “noise density 60%”, $x_7=$ “noise density 70%”, $x_8=$ “noise density 80%”, and $x_9=$ “noise density 90%”.

We consider four traditional images, i.e. “Cameraman”, “Lena”, “Jet Plane”, and “Baboon”, for the convenience of the experts who are asked to produce a suitable ranking order of the aforementioned filters. Thus, we can compare the results of the methods with the experts’ judgements. To this end, we present the results of the filters in Enginoğlu et al. (2019) by Structural Similarity (SSIM) (Wang et al. 2004) for the images at noise densities ranging from 10 to 90%, in Table 3. Further, let bold numbers in a table point out the best scores therein. Let $({(\mu ^{x}_{u})}_{n})$ be ordered s-tuples such that ${(\mu ^{x}_{u})}_{n}$ corresponds the SSIM results by nth image for filter u and noise density x. Moreover, the first, second, third, and fourth image are the Cameraman, Lena, Jet Plane, and Baboon, respectively.

Table 3 The SSIM results of the filters for the Cameraman, Lena, Jet Plane, and Baboon images

Secondly, we construct the d-set f via the membership function $\omega _x(u)$ and the nonmembership function $\theta _x(u)$ defined by

$$\begin{aligned} \left[ \frac{\min \limits _{n}{(\mu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}, \frac{\max \limits _{n}{(\mu ^{x}_{u})}_{n}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}\right] \end{aligned}$$

and

$$\begin{aligned} \left[ \frac{\min \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}, \frac{\max \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}{\max \limits _{n}{(\mu ^{x}_{u})}_{n} + \max \limits _{n}\{1-{(\mu ^{x}_{u})}_{n}\}}\right] \end{aligned}$$

For example, $({(\mu ^{x_1}_{u_3})}_{n})$ denote the SSIM results of four traditional images by DBA at noise density 10%, namely $({(\mu ^{x_1}_{u_3})}_{n})=(0.9938,0.9867,0.9875,0.9827)$. Since

$$\begin{aligned} \omega _{x_1}(u_3)= & \left[ \frac{0.9827}{0.9938 + 0.0173}, \frac{0.9938}{0.9938 + 0.0173}\right] \\= & [0.9719,0.9829] \end{aligned}$$

and

$$\begin{aligned} \theta _{x_1}(u_3)= & \left[ \frac{0.0062}{0.9938 + 0.0173}, \frac{0.0173}{0.9938 + 0.0173}\right] \\= & [0.0061,0.0171] \end{aligned}$$

then the membership degree and the nonmembership degree of the filter $u_3$ according to the parameter $x_1$ are [0.9719, 0.9829] and [0.0061, 0.0171], respectively. Similarly, the ivif-values of the other filters can be constructed. Suppose that the noise removal performances of the filters are more significant in high noise density, in which noisy pixels outnumber uncorrupted pixels, then performance-based success would be more important in the presence of high noise densities than of other densities. For example, let

$$\begin{aligned} \kappa &=\left\{ {^{[0,0.01]}_{[0.9,0.95]}}x_1, {^{[0,0.05]}_{[0.85,0.9]}}x_2, {^{[0,0.1]}_{[0.8,0.85]}}x_3, {^{[0.05,0.35]}_{[0.25,0.5]}}x_4, {^{[0.2,0.45]}_{[0.2,0.45]}}x_5,\right. \\&\qquad\left. {^{[0.25,0.5]}_{[0.05,0.35]}}x_6, {^{[0.8,0.85]}_{[0,0.1]}}x_7,{^{[0.85,0.9]}_{[0,0.05]}}x_8,{^{[0.9,0.95]}_{[0,0.01]}}x_9 \right\} \end{aligned}$$

Therefore, the d-set f, the ifpifs-set $f_{mr}$, the fpifs-set $f_{mbr}$, and the fpfs-set $f_{mb}$, respectively are as follows:

$$\begin{aligned}&\left\{ \left( {^{[0,0.01]}_{[0.9,0.95]}}x_1,\left\{ {{^{[0.9682,0.9796]}_{[0.0089,0.0204]}}}u_1,{{^{[0.9513,0.9706]}_{[0.0101,0.0294]}}}u_2, {{^{[0.9719,0.9829]}_{[0.0061,0.0171]}}}u_3,\right. \right. \right. \\&\quad {{^{[0.9391,0.9619]}_{[0.0153,0.0381]}}}u_4, {{^{[0.9807,0.9884]}_{[0.0040,0.0116]}}}u_5, {{^{[0.9569,0.9721]}_{[0.0126,0.0279]}}}u_6, \\&\quad \left. \left. {{^{[0.9844,0.9907]}_{[0.0031,0.0093]}}}u_7\right\} \right) , \left( {^{[0,0.05]}_{[0.85,0.9]}}x_2,\left\{ {{^{[0.9233,0.9511]}_{[0.0211,0.0489]}}}u_1, {{^{[0.8908,0.9132]}_{[0.0644,0.0868]}}}u_2,\right. \right. \\&\quad {{^{[0.9355,0.9603]}_{[0.0149,0.0397]}}}u_3,{{^{[0.8767,0.9229]}_{[0.0308,0.0771]}}}u_4, {{^{[0.9577,0.9742]}_{[0.0092,0.0258]}}}u_5, \\&\quad \left. \left. {{^{[0.9374,0.9608]}_{[0.0157,0.0392]}}}u_6,{{^{[0.9683,0.9808]}_{[0.0066,0.0192]}}}u_7\right\} \right) ,\\&\quad \left( {^{[0,0.1]}_{[0.8,0.85]}}x_3, \left\{ {{^{[0.8659,0.9133]}_{[0.0392,0.0867]}}}u_1,\right. \right. \\&\quad {{^{[0.6996,0.7804]}_{[0.1388,0.2196]}}}u_2,{{^{[0.8870,0.9296]}_{[0.0278,0.0704]}}}u_3, {{^{[0.8145,0.8839]}_{[0.0468,0.1161]}}}u_4, \\&\quad \left. \left. {{^{[0.9304,0.9571]}_{[0.0163,0.0429]}}}u_5,{{^{[0.9179,0.9491]}_{[0.0196,0.0509]}}}u_6,{{^{[0.9489,0.9688]}_{[0.0113,0.0312]}}}u_7\right\} \right) , \left( {^{[0.05,0.35]}_{[0.25,0.5]}}x_4,\right. \\&\quad \left\{ {{^{[0.7929,0.8642]}_{[0.0645,0.1358]}}}u_1,{{^{[0.7238,0.7769]}_{[0.1700,0.2231]}}}u_2, {{^{[0.8202,0.8877]}_{[0.0448,0.1123]}}}u_3, \right. \\&\quad {{^{[0.7486,0.8439]}_{[0.0607,0.1561]}}}u_4,{{^{[0.8977,0.9368]}_{[0.0241,0.0632]}}}u_5,{{^{[0.8952,0.9355]}_{[0.0242,0.0645]}}}u_6,\\&\left. \left. \quad {{^{[0.9234,0.9531]}_{[0.0171,0.0469]}}}u_7\right\} \right) ,\left( {^{[0.2,0.45]}_{[0.2,0.45]}}x_5,\left\{ {{^{[0.6972,0.8008]}_{[0.0956,0.1992]}}}u_1, {{^{[0.7469,0.8199]}_{[0.1072,0.1801]}}}u_2, \right. \right. \\&\quad {{^{[0.7364,0.8331]}_{[0.0703,0.1669]}}}u_3,{{^{[0.6769,0.8019]}_{[0.0732,0.1981]}}}u_4,{{^{[0.8561,0.9110]}_{[0.0342,0.0890]}}}u_5, {{^{[0.8629,0.9157]}_{[0.0315,0.0843]}}}u_6,\\&\left. \left. \quad {{^{[0.8871,0.9309]}_{[0.0253,0.0691]}}}u_7\right\} \right) ,\left( {^{[0.25,0.5]}_{[0.05,0.35]}}x_6, \left\{ {{^{[0.5878,0.7252]}_{[0.1375,0.2748]}}}u_1, \right. \right. \\&\quad {{^{[0.6749,0.8007]}_{[0.0736,0.1993]}}}u_2,{{^{[0.6382,0.7688]}_{[0.1006,0.2312]}}}u_3,{{^{[0.6085,0.7598]}_{[0.0888,0.2402]}}}u_4, {{^{[0.8088,0.8808]}_{[0.0471,0.1192]}}}u_5,\\&\left. \left. \quad {{^{[0.8201,0.8887]}_{[0.0428,0.1113]}}}u_6,{{^{[0.8393,0.9009]}_{[0.0375,0.0991]}}}u_7\right\} \right) , \left( {^{[0.8,0.85]}_{[0,0.1]}}x_7, \right. \\&\quad \left\{ {{^{[0.4609,0.6355]}_{[0.1898,0.3645]}}}u_1,{{^{[0.6023,0.7573]}_{[0.0877,0.2427]}}}u_2, {{^{[0.5269,0.6917]}_{[0.1434,0.3083]}}}u_3,\right. \\&\left. \left. \quad {{^{[0.5330,0.7130]}_{[0.1069,0.2870]}}}u_4,{{^{[0.7412,0.8387]}_{[0.0638,0.1613]}}}u_5, {{^{[0.7532,0.8469]}_{[0.0594,0.1531]}}}u_6,{{^{[0.7683,0.8566]}_{[0.0552,0.1434]}}}u_7\right\} \right) , \\&\quad \left( {^{[0.85,0.9]}_{[0,0.05]}}x_8,\left\{ {{^{[0.2962,0.5184]}_{[0.2594,0.4816]}}}u_1,{{^{[0.5040,0.6711]}_{[0.1619,0.3289]}}}u_2, {{^{[0.3951,0.6017]}_{[0.1917,0.3983]}}}u_3,\right. \right. \\&\quad {{^{[0.4465,0.6576]}_{[0.1314,0.3424]}}}u_4,{{^{[0.6428,0.7770]}_{[0.0889,0.2230]}}}u_5, {{^{[0.6525,0.7834]}_{[0.0856,0.2166]}}}u_6, \\&\quad \left. \left. {{^{[0.6624,0.7901]}_{[0.0821,0.2099]}}}u_7\right\} \right) ,\left( {^{[0.9,0.95]}_{[0,0.01]}}x_9,\left\{ {{^{[0.0745,0.3577]}_{[0.3591,0.6423]}}}u_1, {{^{[0.2702,0.3912]}_{[0.4877,0.6088]}}}u_2,\right. \right. \\&\quad {{^{[0.2683,0.5063]}_{[0.2556,0.4937]}}}u_3,{{^{[0.3420,0.5658]}_{[0.2105,0.4342]}}}u_4, {{^{[0.4815,0.6717]}_{[0.1382,0.3283]}}}u_5, \\&\quad \left. \left. \left. {{^{[0.4883,0.6771]}_{[0.1342,0.3229]}}}u_6,{{^{[0.4926,0.6804]}_{[0.1319,0.3196]}}}u_7\right\} \right) \right\} \end{aligned}$$

$$\begin{aligned}&\left\{ \left( {^{0.005}_{0.925}}x_1,\left\{ {{^{0.9739}_{0.0146}}}u_1,{{^{0.9609}_{0.0198}}}u_2,{{^{0.9774}_{0.0116}}}u_3,{{^{0.9505}_{0.0267}}}u_4,\right. \right. \right. \\&\left. \left. \quad {{^{0.9846}_{0.0078}}}u_5,{{^{0.9645}_{0.0202}}}u_6,{{^{0.9875}_{0.0062}}}u_7\right\} \right) ,\\&\quad \left( {^{0.025}_{0.875}}x_2,\left\{ {{^{0.9372}_{0.0350}}}u_1, {{^{0.9020}_{0.0756}}}u_2,{{^{0.9479}_{0.0273}}}u_3, \right. \right. \\&\quad \left. \left. {{^{0.8998}_{0.0539}}}u_4,{{^{0.9660}_{0.0175}}}u_5,{{^{0.9491}_{0.0274}}}u_6,{{^{0.9746}_{0.0129}}}u_7\right\} \right) ,\\&\quad \left( {^{0.05}_{0.825}}x_3, \left\{ {{^{0.8896}_{0.0630}}}u_1,{{^{0.7400}_{0.1792}}}u_2,{{^{0.9083}_{0.0491}}}u_3,{{^{0.8492}_{0.0815}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.9437}_{0.0296}}}u_5, {{^{0.9335}_{0.0352}}}u_6,{{^{0.9589}_{0.0212}}}u_7\right\} \right) , \\&\quad \left( {^{0.2}_{0.375}}x_4,\left\{ {{^{0.8286}_{0.1001}}}u_1,{{^{0.7504}_{0.1965}}}u_2,{{^{0.8540}_{0.0785}}}u_3,{{^{0.7963}_{0.1084}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.9172}_{0.0437}}}u_5,{{^{0.9154}_{0.0443}}}u_6,{{^{0.9382}_{0.0320}}}u_7\right\} \right) ,\\&\quad \left( {^{0.325}_{0.325}}x_5,\left\{ {{^{0.7490}_{0.1474}}}u_1, {{^{0.7834}_{0.1436}}}u_2,{{^{0.7847}_{0.1186}}}u_3, \right. \right. \\&\quad \left. \left. {{^{0.7394}_{0.1356}}}u_4,{{^{0.8836}_{0.0616}}}u_5,{{^{0.8893}_{0.0579}}}u_6,{{^{0.9090}_{0.0472}}}u_7\right\} \right) ,\\&\quad \left( {^{0.375}_{0.2}}x_6, \left\{ {{^{0.6565}_{0.2062}}}u_1,{{^{0.7378}_{0.1365}}}u_2,{{^{0.7035}_{0.1659}}}u_3,{{^{0.6841}_{0.1645}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.8448}_{0.0831}}}u_5, {{^{0.8544}_{0.0770}}}u_6,{{^{0.8701}_{0.0683}}}u_7\right\} \right) , \\&\quad \left( {^{0.825}_{0.05}}x_7,\left\{ {{^{0.5482}_{0.2772}}}u_1,{{^{0.6798}_{0.1652}}}u_2,{{^{0.6093}_{0.2258}}}u_3,{{^{0.6230}_{0.1969}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.7900}_{0.1125}}}u_5,{{^{0.8001}_{0.1062}}}u_6,{{^{0.8125}_{0.0993}}}u_7\right\} \right) ,\\&\quad \left( {^{0.875}_{0.025}}x_8,\left\{ {{^{0.4073}_{0.3705}}}u_1, {{^{0.5875}_{0.2454}}}u_2,{{^{0.4984}_{0.2950}}}u_3, \right. \right. \\&\quad \left. \left. {{^{0.5521}_{0.2369}}}u_4,{{^{0.7099}_{0.1560}}}u_5,{{^{0.7180}_{0.1511}}}u_6,{{^{0.7263}_{0.1460}}}u_7\right\} \right) ,\\&\left. \quad \left( {^{0.925}_{0.005}}x_9, \left\{ {{^{0.2161}_{0.5007}}}u_1,{{^{0.3307}_{0.5483}}}u_2,{{^{0.3873}_{0.3747}}}u_3,{{^{0.4539}_{0.3224}}}u_4,\right. \right. \right. \\&\left. \left. \left. \quad {{^{0.5766}_{0.2332}}}u_5, {{^{0.5827}_{0.2286}}}u_6,{{^{0.5865}_{0.2258}}}u_7\right\} \right) \right\} \end{aligned}$$

$$\begin{aligned}&\left\{ \left( {^{0.04}}x_1,\left\{ {{^{0.9739}_{0.0146}}}u_1,{{^{0.9609}_{0.0198}}}u_2,{{^{0.9774}_{0.0116}}}u_3,{{^{0.9505}_{0.0267}}}u_4,\right. \right. \right. \\&\left. \left. \quad {{^{0.9846}_{0.0078}}}u_5,{{^{0.9645}_{0.0202}}}u_6,{{^{0.9875}_{0.0062}}}u_7\right\} \right) ,\\&\quad \left( {^{0.075}}x_2,\left\{ {{^{0.9372}_{0.0350}}}u_1, {{^{0.9020}_{0.0756}}}u_2,{{^{0.9479}_{0.0273}}}u_3,\right. \right. \\&\quad \left. \left. {{^{0.8998}_{0.0539}}}u_4,{{^{0.9660}_{0.0175}}}u_5,{{^{0.9491}_{0.0274}}}u_6,{{^{0.9746}_{0.0129}}}u_7\right\} \right) ,\\&\quad \left( {^{0.1125}}x_3, \left\{ {{^{0.8896}_{0.0630}}}u_1, {{^{0.7400}_{0.1792}}}u_2,{{^{0.9083}_{0.0491}}}u_3,{{^{0.8492}_{0.0815}}}u_4, \right. \right. \\&\quad \left. \left. {{^{0.9437}_{0.0296}}}u_5, {{^{0.9335}_{0.0352}}}u_6,{{^{0.9589}_{0.0212}}}u_7\right\} \right) , \\&\quad \left( {^{0.4125}}x_4,\left\{ {{^{0.8286}_{0.1001}}}u_1,{{^{0.7504}_{0.1965}}}u_2,{{^{0.8540}_{0.0785}}}u_3,{{^{0.7963}_{0.1084}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.9172}_{0.0437}}}u_5,{{^{0.9154}_{0.0443}}}u_6,{{^{0.9382}_{0.0320}}}u_7\right\} \right) ,\\&\quad \left( {^{0.5}}x_5, \left\{ {{^{0.7490}_{0.1474}}}u_1, {{^{0.7834}_{0.1436}}}u_2,{{^{0.7847}_{0.1186}}}u_3, \right. \right. \\&\quad \left. \left. {{^{0.7394}_{0.1356}}}u_4,{{^{0.8836}_{0.0616}}}u_5,{{^{0.8893}_{0.0579}}}u_6,{{^{0.9090}_{0.0472}}}u_7\right\} \right) ,\\&\quad \left( {^{0.5875}}x_6, \left\{ {{^{0.6565}_{0.2062}}}u_1,{{^{0.7378}_{0.1365}}}u_2,{{^{0.7035}_{0.1659}}}u_3,{{^{0.6841}_{0.1645}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.8448}_{0.0831}}}u_5, {{^{0.8544}_{0.0770}}}u_6,{{^{0.8701}_{0.0683}}}u_7\right\} \right) , \\&\quad \left( {^{0.8875}}x_7,\left\{ {{^{0.5482}_{0.2772}}}u_1,{{^{0.6798}_{0.1652}}}u_2,{{^{0.6093}_{0.2258}}}u_3,{{^{0.6230}_{0.1969}}}u_4,\right. \right. \\&\left. \left. \quad {{^{0.7900}_{0.1125}}}u_5,{{^{0.8001}_{0.1062}}}u_6,{{^{0.8125}_{0.0993}}}u_7\right\} \right) ,\\&\quad \left( {^{0.925}}x_8,\left\{ {{^{0.4073}_{0.3705}}}u_1, {{^{0.5875}_{0.2454}}}u_2,{{^{0.4984}_{0.2950}}}u_3, \right. \right. \\&\quad \left. \left. {{^{0.5521}_{0.2369}}}u_4,{{^{0.7099}_{0.1560}}}u_5,{{^{0.7180}_{0.1511}}}u_6,{{^{0.7263}_{0.1460}}}u_7\right\} \right) ,\\&\quad \left( {^{0.96}}x_9, \left\{ {{^{0.2161}_{0.5007}}}u_1,{{^{0.3307}_{0.5483}}}u_2,{{^{0.3873}_{0.3747}}}u_3,{{^{0.4539}_{0.3224}}}u_4,\right. \right. \\&\left. \left. \left. \quad {{^{0.5766}_{0.2332}}}u_5, {{^{0.5827}_{0.2286}}}u_6,{{^{0.5865}_{0.2258}}}u_7\right\} \right) \right\} \end{aligned}$$

$$\begin{aligned}&\left\{ \left( ^{0.04}x_1,\left\{ {^{0.9796}}u_1,{^{0.9706}}u_2,{^{0.9829}}u_3,{^{0.9619}}u_4,\right. \right. \right. \\&\quad\left. \left. {^{0.9884}}u_5,{^{0.9721}}u_6,{^{0.9907}}u_7\right\} \right) ,\\&\quad \left( ^{0.075}x_2,\left\{ {^{0.9511}}u_1,{^{0.9132}}u_2,{^{0.9603}}u_3, \right. \right. \\&\quad \left. \left. {^{0.9229}}u_4,{^{0.9742}}u_5,{^{0.9608}}u_6,{^{0.9808}}u_7\right\} \right) ,\\&\quad \left( ^{0.1125}x_3,\left\{ {^{0.9133}}u_1,{^{0.7804}}u_2,{^{0.9296}}u_3, {^{0.8839}}u_4,\right. \right. \\&\left. \left. \quad {^{0.9571}}u_5,{^{0.9491}}u_6,{^{0.9688}}u_7\right\} \right) , \\&\quad \left( ^{0.4125}x_4,\left\{ {^{0.8642}}u_1,{^{0.7769}}u_2,{^{0.8877}}u_3,{^{0.8439}}u_4,\right. \right. \\&\left. \left. \quad {^{0.9368}}u_5,{^{0.9355}}u_6,{^{0.9531}}u_7\right\} \right) ,\\&\quad \left( ^{0.5}x_5,\left\{ {^{0.8008}}u_1,{^{0.8199}}u_2,{^{0.8331}}u_3, \right. \right. \\&\quad \left. \left. {^{0.8019}}u_4,{^{0.9110}}u_5,{^{0.9157}}u_6,{^{0.9309}}u_7\right\} \right) ,\\&\quad \left( ^{0.5875}x_6,\left\{ {^{0.7252}}u_1,{^{0.8007}}u_2,{^{0.7688}}u_3, {^{0.7598}}u_4,\right. \right. \\&\left. \left. \quad {^{0.8808}}u_5,{^{0.8887}}u_6,{^{0.9009}}u_7\right\} \right) , \\&\quad \left( ^{08875}x_7,\left\{ {^{0.6355}}u_1,{^{0.7573}}u_2,{^{0.6917}}u_3,{^{0.7130}}u_4,\right. \right. \\&\left. \left. \quad {^{0.8387}}u_5,{^{0.8469}}u_6,{^{0.8566}}u_7\right\} \right) ,\\&\quad \left( ^{0.925}x_8,\left\{ {^{0.5184}}u_1,{^{0.6711}}u_2,{^{0.6017}}u_3, \right. \right. \\&\quad \left. \left. {^{0.6576}}u_4,{^{0.7770}}u_5,{^{0.7834}}u_6,{^{0.7901}}u_7\right\} \right) ,\\&\quad \left( ^{0.96}x_9,\left\{ {^{0.3577}}u_1,{^{0.3912}}u_2,{^{0.5063}}u_3, {^{0.5658}}u_4,\right. \right. \\&\left. \left. \left. \quad {^{0.6717}}u_5,{^{0.6771}}u_6,{^{0.6804}}u_7\right\} \right) \right\} \end{aligned}$$

Thirdly, we apply the proposed method and Method 1, 2, 3, and 4 to f, $f_{mr}$, $f_{mr}$, $f_{mbr}$, and $f_{mb}$ provided in this section, respectively. In Tables 4 and 5, we present the decision sets and the ranking orders of the filters for the five methods within their own structures, respectively. Based on the values in Table 3, the proposed method, Method 3, and Method 4 produce the same ranking order as proposed by the experts, including the authors herein. In other words, the proposed method has been able to produce a valid ranking of the seven filters in view of the four images, which suggests that the method is also applicable to a larger number of images. On the other hand, Method 1 and 2 are generally observed to yield a ranking different from the one created by the experts although they say, “ARmF outperforms the other filters”. The above discussion shows that the proposed method can be successfully applied to performance based-value assignment problems so that alternatives can be ordered in terms of performance.

Table 4 The decision sets of filters for the proposed method and Method 1, 2, 3, and 4

Table 5 The ranking orders of the filters for the five methods within their own structures

8 Conclusion

In this paper, we defined the concept of d-sets. We then suggested a new soft decision-making method via the aggregation operator and gave an application of this method to a problem of the determination of eligible candidates in the recruitment process of a company. Moreover, we provided an real application of this method to evaluate the performances of seven filters used in image denoising. To compare this method with another method, we defined four new concepts, i.e. mean reduction, mean bireduction, mean bireduction-reduction, and mean reduction-bireduction. By using these concepts, we applied the proposed method and the four state-of-the-art soft decision-making methods to the aforesaid problems. The results showed that the proposed method was successfully applied to the problems involving further uncertainties.

In the future, effective soft decision-making methods based on group decision-making can be developed by using and/or/andnot/ornot-products of d-sets. Thus, it will be possible to compare such soft decision-making methods constructed by the same structure with the method proposed in this paper. By doing so, the decision-making performances of the methods can be evaluated in a more consistent and down-to-earth manner. Besides, to obtain ivif-values of alternatives or parameters with multiple measurement results, the different membership/nonmembership functions can be defined and compared with the results provided in this study. Moreover, it is necessary and worthwhile to conduct theoretical and applied studies in various fields, such as algebra and topology, and on varied topics, e.g., similarity and distance measurement, by making use of the d-sets. Furthermore, to overcome decision-making problems containing a large number of data and multiple measurement results, defining the matrix representations of d-sets have an enormous significance. Therefore, we have been recently studying the concept of d-matrices that we believe will use and improve d-sets’ skills in modelling.