A novel approach for ranking in interval-valued information systems

Abstract

In decision making analysis, ranking decision with anti-noise capability is a very desirable issue. This paper focuses on analysing data in interval-valued information systems. A new approach ranking with weighted standardized cardinality (WSC) and inclusion indicator in interval-valued information systems is proposed. It is a novel generalization of ranking method based on dominance classes. The new approach overcomes the drawback of sensitivity to noise for raking using dominance classes-based method. In this paper, each object with all attribute values is regarded as an interval-valued fuzzy set (IVF-set). By defining WSC with controlling parameters and single interval inclusion indicator, two kinds of inclusion indicators of the IVF-set are constructed. The approach is based on the novel idea of raking with WSC and inclusion indicator rather than dominance classes. Experiments verify that the new approach is noise-resistant.

Keywords

Interval-valued information systems ranking decision weighted standardized cardinality inclusion indicator

1 Introduction

In many data analysis applications, information and knowledge are stored and represented in a table called an information system, which provides a convenient way to describe a finite set of objects within a universe by a finite set of attributes [19]. In decision making analysis, we often need to deal with data with various types value, such as single value, symbol value, set value, interval value, et al. [2 , 33]. To discover knowledge in information systems, the theory of rough sets proposed by Pawlak [19] has been studied deeply and applied in decision analysis, patten recognition, machine learning et al. [11 , 34]. In real world, it is important to order the objects according to their attributes in an information system. This kind of decision making is called ranking decision, which has been widely used in economy, management and engineering areas. To solve the ranking problem, Greco et al. [14, 15] present a dominance-based rough set approach (DRSA) which is an extension form of Pawlak’s rough set based on dominance relations. This approach is studied and developed in ranking problem and knowledge reduction in ordered information systems and interval-valued information systems [21 , 40]. Zhang and Xu [37] proposed a new method for ranking intuitionistic fuzzy values by using the similarity measure and the accuracy degree, and then they applied the proposed ranking method to multi-attribute decision making.

In fuzzy set theory, which is proposed by Zadeh in 1965 [35], the term cardinality was most commonly used in many areas. As an important characteristic of a set, the cardinality of a crisp set is the number of elements in the set and the cardinality of a fuzzy set is often focused on ordinary cardinal, real numbers or some generalization of cardinal numbers. In scalar approaches, the cardinalities of fuzzy sets were defined with a mapping to each fuzzy set, assigning a single ordinary cardinal number or a non-negative real number. The scalar cardinality was proposed by De Luca and Termini who named it as the power of a finite fuzzy set [7]. The power of a finite fuzzy set A is defined as the sum of the membership degrees of the fuzzy set A. The problem of counting fuzzy sets has generated a lot of literature since Zadeh’s initial conception. As a widely used concept in fuzzy areas, cardinality plays an important role in fuzzy databases and information systems. The relationship between fuzzy mappings and scalar cardinalities was studied in [10, 24]. In the researches on cardinalities, Wygralak developed an axiomatic approach to scalar cardinalities which covers all previous approaches [29]. Casasnovas and Torrens proposed the axiomatic theory of fuzzy cardinalities and investigated the scalar cardinality of finite fuzzy sets based on t-norms and t-conorms [5, 6]. Szmidt and Kacprzyk introduced the scalar cardinality of intuitionistic fuzzy sets [27]. Considering the power character of cardinality, we propose a weighted standardized cardinality approach for interval fuzzy sets and apply it to rank the objects in interval-valued fuzzy systems.

Inclusion measure is an important topic in fuzzy set theory. Different from binary discrimination, the researchers proposed some indicators giving the degree to which a fuzzy set is contained another one. The inclusion indicator has been developed by constructive and axiomatic approaches and has been applied to various fields such as approximate reasoning, data redundancy in fuzzy relational database, rough set theory, uncertainty analysis, and fuzzy formal context analysis [3 , 39]. Monotonicity is an important property of an inclusion indicator and Zhang studied the hybrid monotonic inclusion measure between two fuzzy sets [39]. In some sense, the monotonicity of an inclusion indicator can be used to measure the dominance degree between two fuzzy sets.

In decision making analysis, Zhang and Qiu proposed to order the objects based on a dominance relation [40] and Qian et al. generalized the approach for ranking all objects based on dominance classes and the entire dominance degree in interval ordered information systems [21]. However, the ranking approach based on dominance classes can be influenced by noise easily. To overcome the drawback, we propose the ranking approaches by two ways, one is an absolute ranking approach based on weighted standardized cardinality the other is a relative ranking method tie to inclusion indicator.

The remainder of this paper is organized as follows. In Section 2, we review some basic notions related to fuzzy sets, Atanassov’s fuzzy sets and interval-valued fuzzy sets, and then introduce the ranking method based on dominance classes and the entire dominance degree. In Section 3, we propose the weighted standardized cardinality of IVF-set, and give the ranking results of the objects according to the weighted standardized cardinality in an interval-valued information system. In Section 4, we construct inclusion indicators of IVF-sets by weighted standardized cardinality and single interval inclusion indicator respectively. The ranking result is obtained by the inclusion indicators. In Section 5, an illustrative example is presented. We then conclude this paper with a summary and directions for future research in Section 6.

2 Preliminaries

In this section we review some basic notions related to fuzzy sets (F-sets), Atanassov’s intuitionistic fuzzy sets (AIF-sets), interval valued fuzzy sets (IVF-sets) and ordered information systems.

2.1 F-sets, AIF-sets and IVF-sets

Let U be a finite non-empty set, an F-set A on U is defined as a set of ordered pairs

A = {〈x, μ_A (x) 〉|x ∈ U} , where μ_A : U ⟶ [0, 1] is the membership function of A and μ_A (x) denotes the grade of x belonging into A. A family of all F-sets in U will be denoted by $F (U)$ .

According to Zadeh’s seminal paper [35] introducing fuzzy sets, the intersection and union of two F-sets $A, B \in F (U)$ are defined as,

A ∩ B = {〈x, min (μ_A (x) , μ_B (x)) 〉|x ∈ U} ,

A ∪ B = {〈x, max (μ_A (x) , μ_B (x)) 〉|x ∈ U} .

The complement of F-set A is defined as,

thicksimA = {〈x, 1 - μ_A (x) 〉|x ∈ U} .

In intuitionistic fuzzy sets theory [1], an AIF-set A on U is defined as

A = {〈x, μ_A (x) , ν_A (x) 〉|x ∈ U} , where μ_A (x) and ν_A (x) are degrees of membership and nonmembership of x in A. Moreover they satisfy μ_A (x) , ν_A (x) ∈ [0, 1] and 0 ≤ μ_A (x) + ν_A (x) ≤1. π_A (x) =1 - μ_A (x) - ν_A (x) is called the degree of indeterminancy of x in A. A family of all AIF-sets in U will be denoted by $AIF (U)$ .

Gau and Buehrer [12] defined vague sets. Bustince and Burillo [4] showed that the notion of vague set is the same as that of AIF-set.

The intersection and union of two AIF-sets $A, B \in AIF (U)$ are defined as,

$\begin{matrix} A \cap B & = & {〈 x, \min (μ_{A} (x), μ_{B} (x)), \\ \max (ν_{A} (x), ν_{B} (x)) 〉 | x \in U}, \end{matrix}$

$\begin{matrix} A \cup B & = & {〈 x, \max (μ_{A} (x), μ_{B} (x)), \\ \min (ν_{A} (x), ν_{B} (x)) 〉 | x \in U} . \end{matrix}$

The complement of AIF-set A is defined as,

thicksimA = {〈x, ν_A (x) , μ_A (x) 〉|x ∈ U} .

Another extension of fuzzy sets is interval-valued fuzzy sets theory introduced by Gorzalczany [13]. An IVF-set A on U is defined as

A = {〈x, [l_A (x) , r_A (x)] 〉|x ∈ U} , where l_A (x) and r_A (x) are the left (lower) and right (upper) ends of membership interval, and satisfy 0 ≤ l_A (x) ≤ r_A (x) ≤1. The greatest IVF-set is IVF universe set 1_U = {〈x, [1, 1] 〉|x ∈ U} and the least IVF-set is IVF empty set 0_U = {〈x, [0, 0] 〉|x ∈ U}. A family of all IVF-sets in U will be denoted by $IVF (U)$ .

Note that IVF-sets are called grey sets by Deng [8].

The intersection and union of two IVF-sets $A, B \in IVF (U)$ are defined as,

$\begin{matrix} A \cap B & = & {〈 x, [\min (l_{A} (x), l_{B} (x)), \\ \min (r_{A} (x), r_{B} (x))] 〉 | x \in U}, \end{matrix}$

$\begin{matrix} A \cup B & = & {〈 x, [\max (l_{A} (x), l_{B} (x)), \\ \max (r_{A} (x), r_{B} (x))] 〉 | x \in U} . \end{matrix}$

The complement of IVF-set A is defined as,

thicksimA = {〈x, [1 - r_A (x) , 1 - l_A (x)] 〉|x ∈ U} .

Obviously, an ordinary F-set A can be written as an AIF-set A = {〈x, μ_A (x) , 1 - μ_A (x) 〉|x ∈ U} , or an IVF-set A = {〈x, [μ_A (x) , μ_A (x)] 〉|x ∈ U} . An AIF-set A = {〈x, μ_A (x) , ν_A (x) 〉|x ∈ U} can be represented by means of an IVF-set A = {〈x, [μ_A (x) , 1 - ν_A (x)] 〉|x ∈ U} and vice versa. In Glad Deschrijver’s opinion [9], there exists an isomorphism between $AIF (U)$ and $IVF (U)$ , so IVF-sets theory is equivalent to AIF-sets theory, and the generalization of F-sets theory. All results presented in the follows can be straightforwardly transformed to AIF-set and F-set.

2.2 Ranking all objects by using the dominance relation in an interval-valued information system

In many data analysis applications, information and knowledge are stored and represented in a table and it provides a convenient way to describe a finite set of objects within a universe by a finite set of attributes [19].

An interval-valued information system (IVIS) is an order tuple S = (U, AT, V, f), where U is a finite non-empty set of objects, AT is a finite non-empty set of attributes, V = ⋃ _a∈ATV_a and V_a is a range of a ∈ AT, f : U × AT → V is called an information function [22] satisfying f (x, a) ∈ V_a and denoted by f (x, a) = [a^l (x) , a^r (x)] ⊆ [0, 1] , where a^l (x) and a^r (x) are the least and greatest grade of object x owning attribute a, respectively. Obviously, an ordinary fuzzy information system satisfies f (x, a) = a^l (x) = a^r (x) for all x ∈ U and a ∈ AT, we regard it as a special form of an IVIS.

In [21], Qian, Liang and Dang defined a concept of interval ordered information system, in which all attributes are criteria, that is, the range of each attribute is ordered according to their preference. In an IVIS S = (U, AT, V, f), for all objects x_i, x_j ∈ U and attribute a_k ∈ AT, x_j ≽ _{a
_k}x_i means that x_j is at least as good as x_i with respect to a_k. For any attribute subset A ⊆ AT, we can divide A into two subsets A_inc and A_dec, where A_inc is according to increasing preference and A_dec is according to decreasing preference, A = A_inc ∪ A_dec. For simplicity, we denote $[a_{k}^{l} (x_{i}), a_{k}^{r} (x_{i})]$ by [l_ik, r_ik]. For any object x_j ∈ U and attribute a_k ∈ A, if a_k ∈ A_inc then x_j ≽ _{a
_k}x_i ⇔ l_ik ≤ l_jk and r_ik ≤ r_jk, else if a_k ∈ A_dec then x_j ≽ _{a
_k}x_i ⇔ l_jk ≤ l_ik and r_jk ≤ r_ik. Furthemore, we define x_j ≽ _Ax_i ⇔ x_j ≽ _{a
_k}x_i, for any a_k ∈ A (A ⊆ AT). The dominance relation $R_{A}^{⩾}$ is defined by $\begin{matrix} R_{A}^{⩾} = {(x_{i}, x_{j}) \in U \times U | x_{j} ≽_{A} x_{i}} = \\ {(x_{i}, x_{j}) \in U \times U | l_{ik} \leq l_{jk}, r_{ik} \leq r_{jk} (\forall a_{k} \in A_{inc}); \\ l_{jk} \leq l_{ik}, r_{jk} \leq r_{ik} (\forall a_{k} \in A_{dec}); A = A_{inc} \cup A_{dec}} . \end{matrix}$ According to the dominance relation $R_{A}^{⩾}$ , the dominance class $[x_{i}]_{A}^{⩾}$ can be induced as follows $\begin{matrix} [x_{i}]_{A}^{⩾} = {x_{j} \in U | (x_{i}, x_{j}) \in R_{A}^{⩾}} = \\ {x_{j} \in U | l_{ik} \leq l_{jk}, r_{ik} \leq r_{jk} (\forall a_{k} \in A_{inc}); \\ l_{jk} \leq l_{ik}, r_{jk} \leq r_{ik} (\forall a_{k} \in A_{dec}); A = A_{inc} \cup A_{dec}} . \end{matrix}$

Remark 2.1 Since the complement of an interval [l, r] ⊆ [0, 1] is [1 - r, 1 - l]. In an IVIS S = (U, AT, V, f), we can use the complement $[l_{ik}^{'}, r_{ik}^{'}] = [1 - r_{ik}, 1 - l_{ik}]$ instead of the interval value [l_ik, r_ik] of an attribute a_k according to decreasing preference for all objects, then a_k will convert to an increasing preference attribute. Therefore, we only need to consider the case in which the IVIS with all attributes according to increasing preference.

One important problem in intelligent decision-making is to rank the objects. In [40], Zhang and Qiu defined a concept of dominance degree for ranking all objects in classical ordered information systems. In [21], Qian, Liang and Dang developed the method in interval orderd information systems. In their studies, the following measure $D_{A} (x_{i}, x_{j}) = \frac{| \sim [x_{i}]_{A}^{⩾} \cup [x_{j}]_{A}^{⩾} |}{| U |}$ (1)is a dominance degree between objects x_i and x_j with respect to the dominance relation $R_{A}^{⩾}$ , where | · | is the count operation. From the dominance degree, one can calculate the entire dominance degree of each object in U according to the following formula $D_{A} (x_{i}) = \frac{1}{| U | - 1} \sum_{j \neq i} D_{A} (x_{i}, x_{j}), x_{i}, x_{j} \in U .$ (2) Equations.(1) and (2) show that the bigger the value of D_A (x_i) is, the better object x_i is. So the entire dominance degree can be used to rank objects in an ordered information system.

Example 2.1. An IVIS is listed in Table 1, where U = {x₁, x₂, x₃, x₄, x₅, x₆} and A = {a₁, a₂, a₃, a₄, a₅}. Rank all objects in U according to the entire dominance degree D_A (x_i).

Firstly, we obtain the dominance classes of all objects as follows $[x_{1}]_{A}^{⩾} = {x_{1}}, [x_{2}]_{A}^{⩾} = {x_{2}, x_{4}, x_{6}},$ $[x_{3}]_{A}^{⩾} = {x_{1}, x_{3}, x_{4}, x_{6}}, [x_{4}]_{A}^{⩾} = {x_{4}}, [x_{5}]_{A}^{⩾} = {x_{5}},$ $[x_{6}]_{A}^{⩾} = {x_{6}} .$ Then, we get the dominance degree of all objects by Equation (1), and list them in a matrix as $D_{A} = (\begin{matrix} 1 & 0.83 & 1 & 0.83 & 0.83 & 0.83 \\ 0.5 & 1 & 0.83 & 0.67 & 0.5 & 0.67 \\ 0.5 & 0.67 & 1 & 0.67 & 0.33 & 0.5 \\ 0.83 & 1 & 1 & 1 & 0.83 & 0.83 \\ 0.83 & 0.83 & 0.83 & 0.83 & 1 & 0.83 \\ 0.83 & 1 & 1 & 0.83 & 0.83 & 1 \end{matrix}) .$ Thus, we calculate the entire dominance degree of each object by Equation (2) as $\begin{matrix} D_{A} (x_{1}) = 0.864, & D_{A} (x_{2}) = 0.634, & D_{A} (x_{3}) = 0.534, \\ D_{A} (x_{4}) = 0.898, & D_{A} (x_{5}) = 0.830, & D_{A} (x_{6}) = 0.898 . \end{matrix}$

According to the values above, we can get the ranking result as $≻ x_{4} x_{6} ≽ x_{1} ≽ x_{5} ≽ x_{2} ≽ x_{3} .$

In Qian’s methods, one can rank all objects in an ordering information system from the perspective of dominance classification. However, this method is so strict that it will be influenced by noise easily. From Example 2.1, we obtained the ranking result by the entire dominance degree Equation (2). Considering the object x₅, its interval-values are greater than all the others’ in U with respect to all attributes in AT except a₄, but the ranking result shows that x₅ is not the best one. Obviously, ranking by dominance relation approach in a multi-attribute information system, some few attributes values (which may be seen as noise) can influence the whole ranking results, and the true best object can not be picked out. To avoid such problem and get an objective ranking result, we will make ranking by cardinality function and inclusion measure from a global point.

3 Ranking by hybrid weighted cardinality in an IVIS

In mathematics, the cardinality of a set is a measure of the “number of elements of the set”. There are two approaches to cardinality—-one compares sets directly using bijections and injections, and the other is to use cardinal numbers. The cardinality of a fuzzy set is also called its power, when no confusion with other notions of power is possible. As a notion of power, the cardinality can be used to measure a fuzzy set, and compare the superiority and inferiority of fuzzy sets.

Wygralak [29] developed an axiomatic theory of scalar cardinality of fuzzy sets which covers all previous approaches. In the sense of Wygralak’s theories, we would like to introduce a notion of the cardinality of IVF-sets and discuss its applications for ranking in an IVIS.

Let [a, b]/u denote an IVF-set such that ([a, b]/u) (u) = [a, b] and ([a, b]/u) (v) = [0, 0] for each v ≠ u, where [a, b] ⊆ [0, 1] and u, v ∈ U. So each IVF-set A is a union of singletons as A = ⋃ _u∈UA (u)/u.

Definition 3.1. A mapping card: $IVF (U) ⟶ [0, \infty)$ is called a scalar cardinality of IVF-sets, if the following postulates are satisfied: (P1)Coincidence: for all u ∈ U, card([1, 1]/u) =1; (P2)Monotonicity: for all [c, a] , [c, b] , [a, d] , [b, d] ⊆ [0, 1] and u, v ∈ U, a≤ b ⇒ card([a, d]/u) ≤ card ([b, d]/v), a≤ b ⇒ card([c, a]/u) ≤ card ([c, b]/v); (P3) Additivity:for all $A, B \in IVF (U)$ , supp(A)∩supp(B) =∅⇒ card (A ∪ B) = card (A) +card (B), where, supp (A) = {u|u∈ U and A (u) ≠[0, 0]}, for all $A \in IVF (U)$ .

In an IVIS S = (U, AT, V, f), for any x_i ∈ U, all attributes values of x_i with respect to (w.r.t.) AT form an IVF-set on AT and are called object-attribute IVF-set, denoted by ${\tilde{X}}_{i}$ , where ${\tilde{X}}_{i} (a_{k}) = [l_{ik}, r_{ik}],$ for any a_k ∈ AT. As the same, for any a_k ∈ AT, the attribute values of all objects w.r.t. a_k also form an IVF-set on U and are called attribute-object IVF-set, denoted by ${\tilde{A}}_{k}$ , where ${\tilde{A}}_{k} (x_{i}) = [l_{ik}, r_{ik}],$ for any x_i ∈ U. Obviously, for all x_i ∈ U, a_k ∈ AT, ${\tilde{X}}_{i} (a_{k}) = {\tilde{A}}_{k} (x_{i})$ . In this paper, we only need to consider the object-attribute IVF-sets for ranking objects. In many real life cases, the significance of attributes in an IVIS are often not equal to each other. Considering the weighting of each attribute, we will introduce the weighted standardized cardinality of IVF-sets by average means.

In the following theorem we propose a kind of scalar cardinalities in the sense of Definition 3.1.

Theorem 3.1. In an IVIS S = (U, AT, V, f), U = {x₁, x₂, . . . , x_n} , AT = {a₁, a₂, . . . , a_m}, a function SC: $IVF (U) ⟶ [0, 1]$ is a scalar cardinality of IVF-sets if

$SC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} f_{k} (l_{ik}, r_{ik}),$ where f_k : [0, 1] × [0, 1] ⟶ [0, 1] is a binary function for which (C1) f_k (0, 0) =0, f_k (1, 1) =1 ; (C2) a ≤ b ⇒ f_k (a, d) ≤ f_k (b, d) and f_k (c, a) ≤ f_k (c, b), for all [c, a] , [c, b] , [a, d] , [b, d] ⊆ [0, 1] , k = 1, 2, . . . , m.

Proof. Suppose that f_k (k = 1, 2, . . . , m) fulfill (C1)-(C2) and $SC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} f_{k} (l_{ik}, r_{ik})$ for each object-attribute IVF-set ${\tilde{X}}_{i} \in IVF (U)$ . We will prove that the function SC satisfies postulates (P1)–(P3) in Definition 3.1.

(1) (P1) is fulfilled following postulate (1) immediately.

(2) In the IVIS (U, AT, V, f), for x_i ∈ U, a_k, a_t ∈ AT, if ${\tilde{X}}_{i} (a_{k}) = [a, d], {\tilde{X}}_{i} (a_{t}) = [b, d]$ , then SC $({\tilde{X}}_{i} (a_{k}) / a_{k}) = SC ([a, d] / a_{k}) = f_{k} (a, d) \leq f_{k} (b, d) = SC ([b, d] / a_{t}) = SC ({\tilde{X}}_{i} (a_{t}) / a_{t})$ for a ≤ b. Similarly, if ${\tilde{X}}_{i} (a_{k}) = [c, a], {\tilde{X}}_{i} (a_{t}) = [c, b]$ , then $SC ({\tilde{X}}_{i} (a_{k}) / a_{k}) = SC ([c, a] / a_{k}) = f_{k} (c, a) \leq f_{k} (c, b) = SC ([c, b] /$ $a_{k}) = SC ({\tilde{X}}_{i} (a_{t}) / a_{t})$ for a ≤ b, i.e. (P2) is satisfied.

(3) Since f_k (0, 0) =0, then for any x_i, x_j ∈ U, $SC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} f_{k} (l_{ik}, r_{ik}) = \sum_{a_{k} \in supp ({\tilde{X}}_{i})} f_{k} (l_{ik}, r_{ik})$ and $SC ({\tilde{X}}_{j}) = \sum_{k = 1}^{m} f_{k} (l_{jk}, r_{jk}) = \sum_{a_{k} \in supp ({\tilde{X}}_{j})} f_{k} (l_{jk}, r_{jk})$ . So $supp ({\tilde{X}}_{i}) \cap supp ({\tilde{X}}_{j}) = \emptyset$ ⇒∀a_k ∈ AT ([l_ik, r_ik] = [0, 0] or [l_jk, r_jk] = [0, 0]) $\Rightarrow SC ({\tilde{X}}_{i} ⋃ {\tilde{X}}_{j}) = \sum_{k = 1}^{m} f_{k} (\max (l_{ik}, l_{jk}), \max (r_{ik}, r_{jk}))$ $= \sum_{a_{k} \in supp ({\tilde{X}}_{i})} f_{k} (l_{ik}, r_{ik}) + \sum_{a_{k} \in supp ({\tilde{X}}_{j})} f_{k} (l_{jk}, r_{jk}) =$ $SC ({\tilde{X}}_{i}) + SC ({\tilde{X}}_{j})$ . Thus, (P3) is satisfied.

Summarizing all above, we conclude that the function SC is a scalar cardinality of IVF-sets.□

Definition 3.2 In an IVIS S = (U, AT, V, f), U = {x₁, x₂, . . . , x_n} , AT = {a₁, a₂, . . . , a_m} , weighted standardized cardinality with respect to a weighting vector w, WSC is defined as

$WSC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} w_{k} f_{k} (l_{ik}, r_{ik}),$ where $w = (w_{1}, w_{2}, . . ., w_{m}), w_{k} \in [0, 1], \sum_{k = 1}^{m} w_{k} = 1$ and f_k : [0, 1] × [0, 1] ⟶ [0, 1] is a binary function satisfies the conditions (C1) and (C2) in Theorem 3.1.

WSC in Definition 3.2 is a weighted average and standardized form which developed from Theorem 3.1. It not only reflects the cardinality of an IVF-set ${\tilde{X}}_{i}$ with respect to object x_i but also considers the weighting of each attribute in AT. Then, we will construct the function f in WSC which will be used in ranking all objects in an IVIS.

Theorem 3.2. For real numbers α ∈ [0, 1] , p ∈ (0, ∞) and any [x, y] ⊆ [0, 1], a binary function family defined as $f_{p}^{α} (x, y) = {\begin{matrix} 2^{p - 1} ((1 - α) x + α y)^{p}, \\ 0 \leq (1 - α) x + α y \leq 0.5; \\ 1 - 2^{p - 1} (1 - (1 - α) x - α y)^{p}, \\ 0.5 < (1 - α) x + α y \leq 1 . \end{matrix}$ (3)Then for a given IVIS S = (U, AT, V, f), U = {x₁, x₂, . . . , x_n} , AT = {a₁, a₂, . . . , a_m} $WSC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} w_{k} f_{p}^{α_{k}} (l_{ik}, r_{ik})$ (4)is a weighted standardized cardinality, where $w_{k} \in [0, 1], \sum_{k = 1}^{m} w_{k} = 1, α_{k} \in [0, 1], k = 1, 2, . . ., m$ .

Proof. We only need to prove that the binary function $f_{p}^{α_{k}}$ in Equation (3) satisfies the conditions (C1) and (C2) in Definition 3.2.

(C1) In an IVIS S = (U, AT, V, f), it is easy to verify that $f_{p}^{α_{k}} (l_{ik}, r_{ik}) \in [0, 1]$ and $f_{p}^{α_{k}} (0, 0) = 0$ , $f_{p}^{α_{k}} (1, 1) = 1$ for any i ∈ {1, 2, . . . , n} , k ∈ {1, 2, . . . , m} and p ∈ (0, ∞).

(C2) For any α ∈ [0, 1] , p ∈ (0, ∞), we can deduce the partial derivative about first parameter of $f_{p}^{α}$ in Equation (3) as $\frac{\partial f_{p}^{α}}{\partial x} = 2^{p - 1} p (1 - α) ((1 - α) x + α y)^{p - 1} \geq 0 (0 < (1 - α) x + α y \leq 0.5)$ and $\frac{\partial f_{p}^{α}}{\partial x} =$ 2^p-1p (1 - α) (1 - (1 - α) x - αy) ^p-1≥ 0 (0.5 <(1 - α) x + αy < 1). Synchronously, $f_{p}^{α} (x, y) \in [0, 0.5] (0 \leq (1 - α) x + α y \leq 0.5)$ and $f_{p}^{α} (x, y) \in (0.5, 1] (0 \leq (0.5 < (1 - α) x + α y \leq 1)$ . Thus $f_{p}^{α} (x, y)$ is a left monotonicity increasing function. Since $\frac{\partial f_{p}^{α}}{\partial y} = 2^{p - 1} p α ((1 - α) x + α y)^{p - 1} \geq 0 (0 < (1 - α) x + α y \leq 0.5)$ and $\frac{\partial f_{p}^{α}}{\partial y} = 2^{p - 1} p α (1 - (1 - α) x - α y)^{p - 1} \geq 0 (0.5 < (1 - α) x + α y < 1)$ . Therefore, $f_{p}^{α} (x, y)$ is also a right monotonicity increasing function. Thus, (C2) is fulfilled.

Hence, WSC( ${\tilde{X}}_{i}$ ) in Equation (4) is a weighted standardized cardinality. □

In Equation (3), p and α are controlling parameters. Figure. 1 shows the surfaces of binary function $f_{p}^{α} (x, y)$ with 0 ≤ x ≤ y ≤ 1 when p and α are assigned different values.

Remark 3.1. In Theorem 3.2, α_k ∈ [0, 1] reflects an optimism degree with respect to attribute a_k. For an interval value [l_ik, r_ik], (1 - α_k) l_ik + α_kr_ik is an estimation value instead of the interval value [l_ik, r_ik]. When α_k = 1 the value reaches r_ik is the upper end of interval [l_ik, r_ik], the attribute a_k is the most optimistic, when α_k = 0 the value equals l_ik is the lower end of interval [l_ik, r_ik], the attribute a_k is the most pessimistic. In an IVIS, one can evaluate the significance and optimism degree of all attributes, and give the weighting value w_k and the optimism degree α_k of each attribute a_k, then calculate the weighted standardized cardinality of each object by Equation (4) for ranking.

Example 3.1. (Continued from Example 2.1) According to Equations.(3) and (4) in Theorem 3.2, let p = 1, α_k = 0.5, w_k = 0.2 (k = 1, 2, . . . , 5), we can obtain the weighted standardized cardinality of each object as follows $\begin{matrix} WSC ({\tilde{X}}_{1}) = 0.51, & WSC ({\tilde{X}}_{2}) = 0.35, \\ WSC ({\tilde{X}}_{3}) = 0.26, & WSC ({\tilde{X}}_{4}) = 0.53, \\ WSC ({\tilde{X}}_{5}) = 0.80, & WSC ({\tilde{X}}_{6}) = 0.59 . \end{matrix}$

Therefore, according to the weighted standardized cardinality, we can get the ranking result as $x_{5} ≽ x_{6} ≽ x_{4} ≽ x_{1} ≽ x_{2} ≽ x_{3} .$

From Example 3.1, a complete rank of objects can be obtained by using the weighted standardized cardinality. In the ranking result, objects x₄ and x₆ which are put into the same place according to the entire dominance degree in Example 2.1 have been further ranked, and x₅ has become the best one to correspond with human perceptions. Simultaneously, the property of rank preservation is still remained, that is to say, the ordering of other objects (i.e. x₁, x₂ and x₃) is unaltered. Obviously, as a ranking approach, the weighted standardized cardinality in an IVIS is effective. This approach only depends on the interval-values but not dominance relation which is disturbed by noise easily. Analogously, another ranking approach depending on inclusion indicator will be proposed in next section.

4 Ranking all objects by inclusion indicator

In classical set theory, a binary discrimination between two sets is one being or not being a subset of another. To relax the rigidity, inclusion indicator was proposed and many researches about inclusion indicators were discussed in the theories of F-sets, IVF-sets and AIF-sets.

4.1 Inclusion indicators of IVF-sets

Now let us consider the problem of measuring the inclusion degree of IVF-sets. In [3], Bunstince introduced an axiomatic definition for the inclusion indicator of IVF-sets and applied it to approximate reasoning. In [36], Zeng and Guo extended the inclusion measure and introduced an axiomatic definition of inclusion indicator of IVF-sets which is different from Bustince’s.

Definition 4.1. A real function Inc: $IVF (U) \times IVF (U) ⟶ [0, 1]$ is called the inclusion indicator of IVF-sets, if Inc satisfies the following properties: (I1) Inc (1_U, 0_U) =0; (I2) Inc (A, B) =1 iff A ⊆ B, $\forall A, B \subseteq IVF (U)$ ; (I3) For all $A, B, C \subseteq IVF (U)$ , if A ⊆ B ⊆ C, then Inc (C, A) ≤ Inc (B, A),Inc (C, A) ≤ Inc (C, B).

In an IVIS S = (U, AT, V, f), an inclusion indictor between two object-attribute IVF-sets can reflect one object being or not being dominance than the other. By this opinion, we will make ranking in an IVIS by inclusion indictors of IVF-sets.

Definition 4.2. For U = {x₁, x₂, . . . , x_n} and AT = {a₁, a₂, . . . , a_m}, S = (U, AT, V, f, w, a) is called a hybrid weighted regular IVIS, if S is an IVIS satisfying: (1) For any x_i ∈ U, there exists a_k ∈ AT such that l_ik < 1 or r_ik > 0; (2) w_k, α_k ∈ (0, 1) for all k ∈ {1, 2, . . . , m}, where w = (w₁, w₂, . . . , w_m) is an attribute weighting vector satisfying $\sum_{k = 1}^{m} w_{k} = 1$ , a = (α₁, α₂, . . . , α_m) is an attribute optimism degree vector.

Theorem 4.1. Let S = (U, AT, V, f, w, a) be a hybrid weighted regular IVIS, WSC is a weighted standardized cardinality defined in Theorem 3.2, then for all i, j ∈ {1, 2, . . . , n} $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \frac{WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j})}{WSC ({\tilde{X}}_{i})}$ (5)is an inclusion indicator of object-attribute IVF-sets in S.

Proof. In the hybrid weighted regular IVIS S = (U, AT, V, f, w, a), for all i, j ∈ {1, 2, . . . , n}, from Theorem 3.2, Equation (5) can be denoted as $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \frac{\sum_{k = 1}^{m} w_{k} f_{p}^{α_{k}} (l_{ik} \land l_{jk}, r_{ik} \land r_{jk})}{\sum_{k = 1}^{m} w_{k} f_{p}^{α_{k}} (l_{ik}, r_{ik})},$ (6)where “∧” is the minimum operator.

(I1) Since S is regular, for all i, j ∈ {1, 2, . . . , n}, ${\tilde{X}}_{i}$ is not a universal IVF-set and ${\tilde{X}}_{j}$ is not an empty IVF-set. By logical reasoning theory, Equation (5) satisfies condition (I1) in Definition 4.1.

(I2) It is obvious that $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = 1$ , when ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j}$ . Conversely, if $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \frac{WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j})}{WSC ({\tilde{X}}_{i})} = 1$ , then $WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j}) = WSC ({\tilde{X}}_{i})$ .

By Equation (4), we have

$WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j}) = \sum_{k = 1}^{m} w_{k} f_{p}^{α_{k}} (l_{ik} \land l_{jk}, r_{ik} \land r_{jk})$ ,

$WSC ({\tilde{X}}_{i}) = \sum_{k = 1}^{m} w_{k} f_{p}^{α_{k}} (l_{ik}, r_{ik})$ . Since l_ik ∧ l_jk ≤ l_ik, r_ik ∧ r_jk ≤ r_ik, and from the proof of Theorem 3.2 we know that $f_{p}^{α_{k}}$ is left and right monotonicity increasing, so $WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j}) \leq WSC ({\tilde{X}}_{i})$ . S is regular, i.e. 0 < α_k < 1, so $f_{p}^{α_{k}}$ is left and right strict monotonicity increasing. Assume that ${\tilde{X}}_{i} ⊈ {\tilde{X}}_{j}$ , then there exists k₀ ∈ {1, 2, . . . , m} which satisfies l_{jk
₀} < l_{ik
₀} or r_{jk
₀} < r_{ik
₀}, and then $f_{p}^{α_{k_{0}}} (l_{{ik}_{0}} \land l_{{jk}_{0}}, r_{{ik}_{0}} \land r_{{jk}_{0}}) < f_{p}^{α_{k_{0}}} (l_{{ik}_{0}}, r_{{ik}_{0}})$ . It follows that $WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j}) < WSC ({\tilde{X}}_{i})$ , which contradict with $WSC ({\tilde{X}}_{i} \cap {\tilde{X}}_{j}) = WSC ({\tilde{X}}_{i})$ . Thus, ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j}$ .

(I3) In the IVIS, for all i, j, s ∈ {1, 2, . . . , n}, if ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j} \subseteq {\tilde{X}}_{s}$ , then ${\tilde{X}}_{s} \cap {\tilde{X}}_{i} = {\tilde{X}}_{i}$ , ${\tilde{X}}_{j} \cap {\tilde{X}}_{i} = {\tilde{X}}_{i}$ , so $WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{i}) = WSC ({\tilde{X}}_{j} \cap {\tilde{X}}_{i})$ . By the monotonicity increasing property of $f_{p}^{α_{k}}$ , we have that $WSC ({\tilde{X}}_{s}) \geq WSC ({\tilde{X}}_{j})$ , Therefore, $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) = \frac{WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{i})}{WSC ({\tilde{X}}_{s})} \leq \frac{WSC ({\tilde{X}}_{j} \cap {\tilde{X}}_{i})}{WSC ({\tilde{X}}_{j})} = Inc ({\tilde{X}}_{j}, {\tilde{X}}_{i})$ . ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j}$ implies that ${\tilde{X}}_{s} \cap {\tilde{X}}_{i} \leq {\tilde{X}}_{s} \cap {\tilde{X}}_{j}$ , which means that $WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{i}) \leq WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{j})$ . It follows that $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) = \frac{WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{i})}{WSC ({\tilde{X}}_{s})} \leq \frac{WSC ({\tilde{X}}_{s} \cap {\tilde{X}}_{j})}{WSC ({\tilde{X}}_{s})} = Inc ({\tilde{X}}_{s}, {\tilde{X}}_{j})$ .

Summarizing (I1),(I2) and (I3) above, we conclude that Inc denoted by Equation (5) is an inclusion indicator of object-attribute IVF-sets in S. □

Remark 4.1. In an IVIS with all attributes according to increasing preference, the inclusion indicator of object-attribute IVF-sets $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j})$ can show the degree of x_j dominating x_i in some sense. Then Equation (2) the entire dominance degree of each object in U is revised as follows $D_{A} (x_{i}) = \frac{1}{| U | - 1} \sum_{j \neq i} (1 - Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j})), x_{i}, x_{j} \in U .$ (7)Then one can make ranking in an IVIS according to the entire dominance degree by Equation (7).

Example 4.1. (Continued from Example 2.1) According to Equations.(3), (4) and (5), let p = 1, α_k = 0.5, w_k = 0.2 (k = 1, 2, . . . , 5), we can obtain the inclusion degree of all objects, and list them in a matrix as follows ${Inc}_{A} = (\begin{matrix} 1 & 0.6667 & 0.5098 & 0.8824 & 0.9804 & 0.9804 \\ 0.9714 & 1 & 0.7429 & 1 & 0.9714 & 1 \\ 1 & 1 & 1 & 1 & 0.9615 & 1 \\ 0.8491 & 0.6604 & 0.4906 & 1 & 0.9623 & 0.9434 \\ 0.6250 & 0.4250 & 0.3125 & 0.6375 & 1 & 0.7000 \\ 0.8475 & 0.5932 & 0.4407 & 0.8475 & 0.9492 & 1 \end{matrix}) .$ Thus, we calculate the entire dominance degree of each object by Equation (7) as

$\begin{matrix} D_{A} (x_{1}) = 0.1961, & D_{A} (x_{2}) = 0.0629, \\ D_{A} (x_{3}) = 0.0077, & D_{A} (x_{4}) = 0.2189, \\ D_{A} (x_{5}) = 0.4600, & D_{A} (x_{6}) = 0.2644 . \end{matrix}$ Therefore, according to the entire dominance degree, we can get the ranking result as $x_{5} ≽ x_{6} ≽ x_{4} ≽ x_{1} ≽ x_{2} ≽ x_{3} .$

4.2 Single interval inclusion indicator

In Definition 4.1, if IVF-sets A and B are singletons, i.e. A = A (u)/u and B = B (u)/u, where A (u) , B (u) ⊆ [0, 1] and A (v) = B (v) = [0, 0] for each v ≠ u, then Inc (A, B) will be simplified to the single interval inclusion indicator between A (u) and B (u). We will present the equivalent definition as follows.

Definition 4.3. A real function $D : I ([0, 1]) \times I ([0, 1]) ⟶ [0, 1]$ is called the single interval inclusion indicator, if $D$ satisfies: (D1) $D ([1, 1], [0, 0]) = 0$ ; (D2) $D ([a_{1}, a_{2}], [b_{1}, b_{2}]) = 1 iff a_{1} \leq b_{1}$ and a₂ ≤ b₂ for all $[a_{1}, a_{2}], [b_{1}, b_{2}] \in I ([0, 1])$ ; (D3) For all $[a_{1}, a_{2}], [b_{1}, b_{2}], [c_{1}, c_{2}] \in I ([0, 1])$ , if a₁ ≤ b₁ ≤ c₁ and a₂ ≤ b₂ ≤ c₂, then

$D ([c_{1}, c_{2}], [a_{1}, a_{2}]) \leq D ([b_{1}, b_{2}], [a_{1}, a_{2}])$ ,

$D ([c_{1}, c_{2}], [a_{1}, a_{2}]) \leq D ([c_{1}, c_{2}], [b_{1}, b_{2}])$ , where $I ([0, 1])$ is a family of all intervals in [0, 1].

Theorem 4.2. In a hybrid weighted regular IVIS S = (U, AT, V, f, w, a), U = {x₁, x₂, . . . , x_n} , AT = {a₁, a₂, . . . , a_m}, $D$ is a single interval inclusion indicator, for any x_i, x_j ∈ U and a_k ∈ AT, let $D_{ij}^{k} = D ([l_{ik}, r_{ik}], [l_{jk}, r_{jk}])$ , then $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \sum_{k = 1}^{m} w_{k} D_{ij}^{k}$ is an inclusion indicator between object-attribute IVF-sets ${\tilde{X}}_{i}$ and ${\tilde{X}}_{j}$ .

Proof. We only need to prove that if $D_{ij}^{k}$ satisfies (D1), (D2) and (D3) in Definition 4.3 then $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j})$ fulfils (I1), (I2) and (I3) in Definition 4.1.

(I1) In a regular IVIS S, the proof of postulate (I1) is similar to the proof of (I1) in Theorem 4.1.

(I2) For any x_i, x_j ∈ U, if ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j}$ , then l_ik ≤ l_jk and r_ik ≤ r_jk for all a_k ∈ AT, so $D_{ij}^{k} = 1 (k = 1, . . . . m)$ , thus $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \sum_{k = 1}^{m} w_{k} D_{ij}^{k} = \sum_{k = 1}^{m} w_{k} = 1$ . Conversely, if $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j}) = \sum_{k = 1}^{m} w_{k} D_{ij}^{k} = 1$ , then $D_{ij}^{k} = 1$ for all a_k ∈ AT follows $\sum_{k = 1}^{m} w_{k} = 1 (w_{k} > 0)$ . By postulate (D2), we have l_ik ≤ l_jk and r_ik ≤ r_jk for all a_k ∈ AT, i.e. ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j}$ .

(I3)In the IVIS, for all i, j, s ∈ {1, 2, . . . , n}, if ${\tilde{X}}_{i} \subseteq {\tilde{X}}_{j} \subseteq {\tilde{X}}_{s}$ , then l_ik ≤ l_jk ≤ l_sk and r_ik ≤ r_jk ≤ r_sk for all a_k ∈ AT. By postulate (D2), we have $D_{si}^{k} \leq D_{ji}^{k}$ and $D_{si}^{k} \leq D_{sj}^{k}$ , for all a_k ∈ AT. So $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) = \sum_{k = 1}^{m} w_{k} D_{si}^{k} \leq \sum_{k = 1}^{m} w_{k} D_{ji}^{k} = Inc ({\tilde{X}}_{j}, {\tilde{X}}_{i})$ and $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) = \sum_{k = 1}^{m}$ $w_{k} D_{si}^{k} \leq \sum_{k = 1}^{m} w_{k} D_{sj}^{k} = Inc ({\tilde{X}}_{s}, {\tilde{X}}_{j})$ , i.e. $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) \leq Inc ({\tilde{X}}_{j}, {\tilde{X}}_{i})$ and $Inc ({\tilde{X}}_{s}, {\tilde{X}}_{i}) \leq Inc ({\tilde{X}}_{s}, {\tilde{X}}_{j})$ .

Thus, we complete the proof of Theorem 4.2. □

In Theorem 4.2, the inclusion indicator $Inc ({\tilde{X}}_{i}, {\tilde{X}}_{j})$ constructed by single interval inclusion can also be used to make ranking.

Example 4.2. (Continued from Example 2.1) According to Equations.(3), (4), and (5) and Theorem 4.2, let p = 1, α_k = 0.5, w_k = 0.2 (k = 1, 2, . . . , 5), we can obtain the inclusion degree of all objects, and list them in a matrix as follows ${Inc}_{A} = (\begin{matrix} 1 & 0.6996 & 0.5349 & 0.8993 & 0.9818 & 0.9846 \\ 0.9750 & 1 & 0.7440 & 1 & 0.9714 & 1 \\ 1 & 1 & 1 & 1 & 0.9667 & 1 \\ 0.8621 & 0.6612 & 0.4904 & 1 & 0.9636 & 0.9500 \\ 0.6657 & 0.4407 & 0.3302 & 0.6620 & 1 & 0.7300 \\ 0.8466 & 0.6098 & 0.4457 & 0.8551 & 0.9538 & 1 \end{matrix}) .$ Thus, we calculate the entire dominance degree of each object by Equation (7) as

$\begin{matrix} D_{A} (x_{1}) = 0.1799, & D_{A} (x_{2}) = 0.0619, \\ D_{A} (x_{3}) = 0.0067, & D_{A} (x_{4}) = 0.2145, \\ D_{A} (x_{5}) = 0.4361, & D_{A} (x_{6}) = 0.2578 . \end{matrix}$

Therefore, according to the entire dominance degree according to a single interval inclusion indicator, we can get the ranking result as $x_{5} ≽ x_{6} ≽ x_{4} ≽ x_{1} ≽ x_{2} ≽ x_{3} .$

Example 4.1 and Example 4.2 show that the entire dominance degree based on inclusion indicator can not only get a complete rank of objects but also overcome the interference of noise.

5 Experiments

In Example 3.1, Example 4.1 and Example 4.2, we only give one case, i.e. the controlling parameter p = 1, every optimism degree α_i gets the median of [0,1] and the weighting value w_i are equal (i = 1, 2, 3, 4, 5). To demonstrate the effectiveness of the proposed ranking approaches, we perform experiments on the hybrid weighted IVIS which is shown in Table 1.

For each ranking approach, we divide the controlling parameter p into three instances: (1) 0 < p < 1 (let p = 0.5); (2) p = 1; (3) 1 < p (let p = 2).

The weighting vector w and optimism degree vector a will be combined to four cases:

Case 1. The weighting value w_i are equal and the optimism degree α_i are equal (i = 1, 2, 3, 4, 5) (let w₁ = w₂ = w₃ = w₄ = w₅ = 0.20 and α₁ = α₂ = α₃ = α₄ = α₅ = 0.50);

Case 2. The weighting value w_i are equal and the optimism degree α_i are not equal (i = 1, 2, 3, 4, 5) (let w₁ = w₂ = w₃ = w₄ = w₅ = 0.20 and α₁ = 0.71, α₂ = 0.58, α₃ = 0.50, α₄ = 0.45, α₅ = 0.41);

Case 3. The weighting value w_i are not equal and the optimism degree α_i are equal (i = 1, 2, 3, 4, 5) (let w₁ = 0.30, w₂ = 0.25, w₃ = 0.20, w₄ = 0.15, w₅ = 0.10 and α₁ = α₂ = α₃ = α₄ = α₅ = 0.50);

Case 4. The weighting value w_i are not equal and the optimism degree α_i are not equal (i = 1, 2, 3, 4, 5) (let w₁ = 0.30, w₂ = 0.25, w₃ = 0.20, w₄ = 0.15, w₅ = 0.10 and α₁ = 0.71, α₂ = 0.58, α₃ = 0.50, α₄ = 0.45, α₅ = 0.41).

We calculate the weighted standardized cardinality WSC and entire dominance degree $D_{A}^{I}$ and $D_{A}^{II}$ of each object, which are list in Tables 2–4, where $D_{A}^{I}$ denotes the entire dominance degree with respect to inclusion indicator and $D_{A}^{II}$ denotes the entire dominance degree with respect to single interval inclusion indicator. Table 5 shows the ranking results in all cases.

This case study examines how the scalar cardinality and inclusion indicator are used in measuring the dominance of objects in an IVIS. The results show that the two ranking approaches can both obtain complete ranking results of the objects and suppress the interference of noise. In the case study, a function family $f_{p}^{α}$ defined by Equation (3) is introduced into a hybrid weighted IVIS for computing the scalar cardinality, the inclusion indicator and the single interval inclusion indicator. The ranking results may be different if the weight factors w, α and controlling parameter p are changed.

Example 5.1. In the IVIS water-quality data of small central water supply shown in Table 6, we select 9 second grade indices(Aerobic bacteria count, As, Pb, Cr-sexavalence, pH, Total hardness, Anion synthetic detergent, Total α radioactivity and Total β radioactivity) from 4 first grade indices (Microorganism indices, Toxicological indices, General chemical indices and Radioactive indices), and use a₁thicksima₉ in stead of the 9 second grade indices,respectively. x₁thicksimx₁₀ are 10 water-samples from different small central water supply, which detected data in three months are normalized to interval-values in [0,1].

The dominance classes of all objects are $\begin{matrix} [x_{1}]_{A}^{⩾} = {x_{1}}, & [x_{2}]_{A}^{⩾} = {x_{2}}, & [x_{3}]_{A}^{⩾} = {x_{3}}, \\ [x_{4}]_{A}^{⩾} = {x_{4}}, & [x_{5}]_{A}^{⩾} = {x_{5}}, & [x_{6}]_{A}^{⩾} = {x_{6}}, \\ [x_{7}]_{A}^{⩾} = {x_{7}}, & [x_{8}]_{A}^{⩾} = {x_{8}}, & [x_{9}]_{A}^{⩾} = {x_{9}}, \\ [x_{10}]_{A}^{⩾} = {x_{10}} . \end{matrix}$

According to Equations.(1) and (2), the entire dominance degree of each object can be computed that D_A (x₁) = D_A (x₂) = D_A (x₃) = D_A (x₄) = D_A (x₅) = D_A (x₆) = D_A (x₇) = D_A (x₈) = D_A (x₉) = D_A (x₁₀) =0.9. In Table 6, there are some data especially smaller or larger than the other data according to the same second grade index. For instance, l_2,2 = 0.01, l_1,7 = 0.07, r_2,9 = 0.85, l_4,8 = 0.05, r_5,4 = 0.98, l_7,9 = 0.07 etc. The abnormality of these data may be due to recording error or some technological reasons and will influent the information system as noise. Obviously, under the influence of noise, all the water-samples can not be ranked by the traditional approach based on dominance classes. To overcome this drawback, we calculate the weighted standardized cardinality WSC and entire dominance degree $D_{A}^{I}$ and $D_{A}^{II}$ of each water-sample in Table 7 as follows (Here, let p = 1, α_k = 0.5, w_k = 1/9(k = 1, 2, . . . , 9)).

Therefore, we can get that the ranking result according to the weighted standardized cardinality WSC and entire dominance degree $D_{A}^{I}$ are the same that x₃ ≽ x₉ ≽ x₄ ≽ x₇ ≽ x₁ ≽ x₅ ≽ x₁₀ ≽ x₆ ≽ x₂ ≽ x₈, and the ranking result according to the entire dominance degree $D_{A}^{II}$ is x₃ ≽ x₄ ≽ x₉ ≽ x₇ ≽ x₁ ≽ x₅ ≽ x₁₀ ≽ x₂ ≽ x₆ ≽ x₈. Although local differences occur in these two ranking results, their overall trends are consistent.

6 Conclusions

Weighted standardized cardinality and inclusion indicator of IVF-sets are two kinds of measuring functions, the former can reflect the power of an IVF-set and the latter can measure the degree of an IVF-set being a subset of another. In our study, they are used to compare the dominance of objects from absolute and relative points of view, respectively. The novelties of the approach are mainly represented from two aspects: completely ranking and overcoming the interference of noise with the objectivity of ranking result. The hybrid weighted IVIS is a generalization of classical information system, so the ranking methods based on weighted standardized cardinality and inclusion indicator are also used in classical information system. The controlling parameter p in the function family $f_{p}^{α}$ provide us with multiple choices for weighted standardized cardinality and inclusion indicator in practical application.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 61170107, 61300121), the Natural Science Foundation of Hebei Province of PR China (Nos. A2014205157,A2013208175), the Science Foundation of Hebei Education Department of PR China (No. Q2012093), and Training Program for Leading Talents of Innovation Teams in the Universities of Hebei Province (LJRC022).

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