A Dynamic Interval-Valued Intuitionistic Fuzzy Sets Applied to Pattern Recognition

Abstract

We present dynamic interval-valued intuitionistic fuzzy sets (DIVIFS), which can improve the recognition accuracy when they are applied to pattern recognition. By analyzing the degree of hesitancy, we propose some DIVIFS models from intuitionistic fuzzy sets (IFS) and interval-valued IFS (IVIFS). And then we present a novel ranking condition on the distance of IFS and IVIFS and introduce some distance measures of DIVIFS satisfying the ranking condition. Finally, a pattern recognition example applied to medical diagnosis decision making is given to demonstrate the application of DIVIFS and its distances. The simulation results show that the DIVIFS method is more comprehensive and flexible than the IFS method and the IVIFS method.

1. Introduction

Professor Zadeh’s paper on fuzzy sets [1] has influenced many researchers and has been applied to many application fields, such as pattern recognition [2], fuzzy reasoning [3], portfolio selection [4], and decision making [5]. In 1986, Atanassov [6] introduced membership function, nonmembership function, and hesitancy function and launched the concept of intuitionistic fuzzy sets (IFS), which generalized the FS theory. In the research field of IFS, Yager [7] discussed its cut set characteristics, some scholars (such as [8–13]) applied it to medical diagnosis, Dengfeng and Chuntian [14] and Li et al. [15] applied it to pattern recognition, and Xu [16–20] and Wei [21] applied it to decision making. In 1989, Atanassov and Gargov [22] proposed interval-valued intuitionistic fuzzy sets (IVIFS) based on the comparative analysis of interval-valued fuzzy sets and IFS. Thereafter, many researchers have investigated IVIFS: some researchers discussed the cut set characteristics of IVIFS [23] based on IFS, some applied it to pattern recognition [16, 19, 20, 24], others applied it to decision making analysis [18, 20, 25–29], and Zhang et al. [29] studied interval-valued intuitionistic fuzzy reasoning. However, there are few references related to the construction of IVIFS from IFS at present. Thus, in this paper, the main objective is to study the construction and application of IVIFS from IFS.

First, we introduce a series of definitions and construction methods of DIVIFS and propose some DIVIFS models according to IFS and IVIFS. Then, on the basis of the traditional distance measures, DIVIFS is applied to pattern recognition on medical diagnosis. The simulation results show that the method introduced in this paper is more comprehensive and flexible than the IFS method and the IVIFS method. Thus, this paper can provide valuable conclusion for the application research of IFS and IVIFS. The model of DIVIFS is also useful for the generalization from fuzzy reasoning [3] and intuitionistic fuzzy reasoning to interval-valued intuitionistic fuzzy reasoning [29].

2. Construction of DIVIFS

Definition 1. An IFS 𝐴 in universe 𝑋 is formulated as follows [6, 22]: 𝐴=𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋, (1) where 𝜇_𝐴∶𝑋→[0,1] and 𝜈_𝐴∶𝑋→[0,1] with the condition 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 for each 𝑥∈𝑋. 𝜇_𝐴(𝑥)∈[0,1] and 𝜈_𝐴(𝑥)∈[0,1] denote a degree of membership and a degree of nonmembership of 𝑥 to 𝐴, respectively. For each IFS in 𝑋, we call 𝜋_𝐴(𝑥)=1−𝜇_𝐴(𝑥)−𝜈_𝐴(𝑥) a degree of hesitancy of 𝑥 to 𝐴, 0≤𝜋_𝐴(𝑥)≤1 for each 𝑥∈𝑋.

Theorem 2. Let 𝐴 be an IFS as mentioned above; then 𝜇_𝐴(𝑥)−𝜇_𝐵(𝑥)+𝜈_𝐴(𝑥)−𝜈_𝐵(𝑥)+𝜋_𝐴(𝑥)−𝜋_𝐵(𝑥)=0. (2) Based on Definition 1, one has Theorem 2.

Definition 3. An IVIFS 𝐴 in universe 𝑋 is stated as follows [22]: 𝐴=𝑥,𝑀_𝐴(𝑥),𝑁_𝐴(𝑥)∣𝑥∈𝑋, (3) where 𝑀_𝐴(𝑥)=[𝜇⁻_𝐴(𝑥),𝜇⁺_𝐴(𝑥)], 𝑁_𝐴(𝑥)=[𝜈⁻_𝐴(𝑥),𝜈⁺_𝐴(𝑥)], and 𝐻_𝐴(𝑥)=[𝜋⁻_𝐴(𝑥),𝜋⁺_𝐴(𝑥)] with the conditions 𝑀_𝐴(𝑥)⊆[0,1], 𝑁_𝐴(𝑥)⊆[0,1], and 𝐻_𝐴(𝑥)⊆[0,1]. The interval 𝑀_𝐴(𝑥), 𝑁_𝐴(𝑥), and 𝐻_𝐴(𝑥)denote a degree range of membership, a degree range of non-membership, and a degree range of hesitancy of 𝑥 to 𝐴, respectively. For each IVIFS in 𝑋, we call 𝜋⁻_𝐴(𝑥)=1−𝜇⁺_𝐴(𝑥)−𝜈⁺_𝐴(𝑥) lower boundary of 𝑥 to 𝐴 and 𝜋⁺_𝐴(𝑥)=1−𝜇⁻_𝐴(𝑥)−𝜈⁻_𝐴(𝑥) upper boundary of hesitancy of 𝑥 to 𝐴, respectively, where 0≤𝜋⁻_𝐴(𝑥)≤𝜋⁺_𝐴(𝑥)≤1 for each 𝑥∈𝑋.

Theorem 4. Let 𝐴 be an IVIFS as mentioned above; then 𝜇⁺_𝐴(𝑥)−𝜇⁺_𝐵(𝑥)+𝜈⁺_𝐴(𝑥)−𝜈⁺_𝐵(𝑥)+𝜋⁻_𝐴(𝑥)−𝜋⁻_𝐵(𝑥)𝜇=0,_−𝐴(𝑥)−𝜇⁻_𝐵(𝑥)+𝜈⁻_𝐴(𝑥)−𝜈⁻_𝐵(𝑥)+𝜋⁺_𝐴(𝑥)−𝜋⁺_𝐵(𝑥)=0. (4) Based on Definition 3, one has Theorem 4.

Definition 5. 𝑋 is a universe of discourse. A DIVIFS 𝐴 in 𝑋 is an object having the following form: 𝐴=𝑥,𝑀_𝐴(𝑥),𝑁_𝐴(𝑥)∣𝑥∈𝑋, (5) where 𝑀_𝐴(𝑥), 𝑁_𝐴(𝑥), and 𝐻_𝐴(𝑥) are the same as Definition 3. Let 𝜇⁺_𝐴(𝑥)=𝜇⁻_𝐴(𝑥)+𝛼_𝐴(𝑥), 𝜈⁺_𝐴(𝑥)=𝜈⁻_𝐴(𝑥)+𝛽_𝐴(𝑥). Obviously, we have 𝛼_𝐴(𝑥)+𝛽_𝐴(𝑥)=𝜋⁺_𝐴(𝑥)−𝜋⁻_𝐴(𝑥), 𝛼_𝐴(𝑥)≥0, 𝛽_𝐴(𝑥)≥0. Thus, DIVIFS is equivalent to IVIFS, and IFS is a special case of DIVIFS when 𝛼_𝐴(𝑥)=𝛽_𝐴(𝑥)=0.
From Definition 3, let all sample data be divided into three parts, 𝜇⁻_𝐴(𝑥) being the firm support party of event 𝐴, 𝜈⁻_𝐴(𝑥) representing the firm opposition party of event 𝐴, and 𝜋⁺_𝐴(𝑥) showing all the absent party. In the absent party, 𝜋⁻_𝐴(𝑥) is the firm absent party, and 𝜋⁺_𝐴(𝑥)−𝜋⁻_𝐴(𝑥) is the convertible absent party, in which each sample may become either the support party or the opposition party. If there is 𝛼_𝐴(𝑥) sample supporting event 𝐴 and 𝛽_𝐴(𝑥) sample opposing event 𝐴, we have DIVIFS based on Definition 5. If the proportion of the absent party converted to the support party is 𝜆_𝐴1(𝑥) and the proportion of the absent party converted to the opposition party is 1−𝜆_𝐴1(𝑥), then the model will become a DIVIFS with single variable, where 𝛼_𝐴(𝑥)=𝜆_𝐴1(𝑥)(𝜋⁺_𝐴(𝑥)−𝜋⁻_𝐴(𝑥)), 𝛽_𝐴(𝑥)=(1−𝜆_𝐴1(𝑥))(𝜋⁺_𝐴(𝑥)−𝜋⁻_𝐴(𝑥)). If the firm absent party is 𝜋⁻_𝐴(𝑥)=(1−𝜆_𝐴0(𝑥))𝜋⁺_𝐴(𝑥), then the convertible absent party is 𝜋⁺_𝐴(𝑥)−𝜋⁻_𝐴(𝑥)=𝜆_𝐴0(𝑥)𝜋⁺_𝐴(𝑥), and then we will get the other DIVIFS definition as follows.

Definition 6. 𝑋 is a universe of discourse. DIVIFS 𝐴 in 𝑋 is an object having the following form 𝐴=𝑥,𝑀_𝐴(𝑥),𝑁_𝐴(𝑥)∣𝑥∈𝑋, (6) where 𝑀_𝐴(𝑥), 𝑁_𝐴(𝑥), and 𝐻_𝐴(𝑥) are the same as Definition 5. For each 𝜆_𝐴0(𝑥)∈[0,1] and for each 𝜆_𝐴1(𝑥)∈[0,1], we assume that 𝜇⁺_𝐴(𝑥)=𝜇⁻_𝐴(𝑥)+𝜆_𝐴0(𝑥)𝜆_𝐴1(𝑥)𝜋⁺_𝐴(𝑥), 𝜈⁺_𝐴(𝑥)=𝜈⁻_𝐴(𝑥)+𝜆_𝐴0(𝑥)(1−𝜆_𝐴1(𝑥))𝜋⁺_𝐴(𝑥) and that 𝜋⁻_𝐴(𝑥)=(1−𝜆_𝐴0(𝑥))𝜋⁺_𝐴(𝑥). Evidently, the DIVIFS model is also equivalent to IVIFS. If 𝜆_𝐴0(𝑥)=0, then DIVIFS is IFS. And if 𝜇⁻_𝐴(𝑥)=𝜇_𝐴(𝑥), 𝜈⁻_𝐴(𝑥)=𝜈_𝐴(𝑥), 𝜋⁺_𝐴(𝑥)=𝜋_𝐴(𝑥), then we have a special DIVIFS which is derived from IFS.

Definition 7. A DIVIFS derived from IFS is given as follows: 𝐴=𝑥,𝑀_𝐴(𝑥),𝑁_𝐴(𝑥)∣𝑥∈𝑋, (7) where 𝑀_𝐴(𝑥), 𝑁_𝐴(𝑥), and 𝐻_𝐴(𝑥) are the same as Definition 5 and 𝜇⁻_𝐴(𝑥)=𝜇_𝐴(𝑥), 𝜈⁻_𝐴(𝑥)=𝜈_𝐴(𝑥), 𝜋⁺_𝐴(𝑥)=𝜋_𝐴(𝑥), 𝜇⁺_𝐴(𝑥)=𝜇_𝐴(𝑥)+𝜆_𝐴0(𝑥)𝜆_𝐴1(𝑥)𝜋_𝐴(𝑥), 𝜈⁺_𝐴(𝑥)=𝜈_𝐴(𝑥)+𝜆_𝐴0(𝑥)(1−𝜆_𝐴1(𝑥))𝜋_𝐴(𝑥), 𝜋⁻_𝐴(𝑥)=(1−𝜆_𝐴0(𝑥))𝜋_𝐴(𝑥), for each 𝜆_𝐴0(𝑥)∈[0,1] and for each 𝜆_𝐴1(𝑥)∈[0,1]. This DIVIFS model is also an extension of the IFS model.

3. Conventional Distance Measures of IFS and IVIFS

Derived from the definitions of FS, IFS, and IVIFS, some distance measures are discussed (Figure 1). Let 𝑑=1−𝑆, Δ𝜇=𝜇_𝐴−𝜇_𝐵, Δ𝜈=𝜈_𝐴−𝜈_𝐵, and Δ𝜋=𝜋_𝐴−𝜋_𝐵. According to Definition 1 and Theorem 2, Δ𝜇+Δ𝜈+Δ𝜋=0.

(a)

(b)

(c)

(d)

Figure 1 

Distance between each patient and each disease on IFS calculated by Xu’s methods.

According to fuzzy sets [1], the distance measure of FS is composed of membership degree: 𝑑_Za(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖Δ𝜇^𝑝1/𝑝. (8)

In [6], Atanassov introduced a distance measure (9) of IFS from membership degree and non-membership degree: 𝑑_At(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖Δ𝜇^𝑝+|Δ𝜈|^𝑝1/𝑝. (9)

Considering the degree range of hesitancy, Szmidt and Kacprzyk presented a novel distance measure (10) according to Definition 1 and Theorem 2 in 2000 [30]: 𝑑_S,K(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖Δ𝜇^𝑝+|Δ𝜈|^𝑝+Δ𝜇+Δ𝜈^𝑝1/𝑝. (10)

In [14], Dengfeng and Chuntian proposed a novel distance model as follows: 𝑑_L,C(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖Δ𝜇−Δ𝜈^𝑝1/𝑝. (11)

At the same time, Grzegorzewski [31] and Hung and Yang [32] proposed a Hausdorff distance measure of IFS, respectively: 𝑑_P,H(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖maxΔ𝜇,|Δ𝜈|. (12)

In 2005, Wang and Xin [33] presented a new distance measure which is made up of 𝑑_At and 𝑑_P,H: 𝑑_W(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖Δ𝜇+|Δ𝜈|+2maxΔ𝜇,|Δ𝜈|4. (13)

In 2005, Liu [34] introduced some similar distance formulas as mentioned above.

In Szmidt and Kacprzyk [8–10], the ratio-based measure of similarity is defined as follows: 𝑑_RB()=𝑑()𝐴,𝐵𝐴,𝐵𝑑𝐴,𝐵^𝐶. (14) If 𝐵 is a set (𝜇_𝐵(𝑥),𝜈_𝐵(𝑥),𝜋_𝐵(𝑥)), Szmidt and Kacprzyk describe 𝐵^𝐶 to be a set (𝜈_𝐵(𝑥),𝜇_𝐵(𝑥),𝜋_𝐵(𝑥)). For example, if 𝐵 is (𝜇_𝐵(𝑥)=0.7, 𝜈_𝐵(𝑥)=0, 𝜋_𝐵(𝑥)=0.3), then 𝐵^𝐶 is (0, 0.7, 0.3), and so on. And 𝑑 denotes a kind of distance measure about 𝐴 and 𝐵.

In 2010, Xia and Xu [20] presented some weighted distance formulas, where 𝑘≥0 and 𝜆≥0 denote the weights: 𝑑_𝐷𝑘(𝐴,𝐵)=^𝑛_𝑖=1𝑤_𝑖(1−𝑘)Δ𝜇−𝑘Δ𝜈^𝑝1/𝑝,𝑑_{𝐹𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖(1−𝑘)Δ𝜇−𝑘Δ𝜈^𝑝+(1−𝜆)Δ𝜈−𝜆Δ𝜇^𝑝+(1−𝑘−𝜆)Δ𝜇+Δ𝜈^𝑝1/𝑝,𝑑𝑘+𝜆≤1,_{𝐺𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖𝑘Δ𝜇^𝑝+|𝜆Δ𝜈|^𝑝+𝑘Δ𝜇+𝜆Δ𝜈^𝑝1/𝑝,𝑑𝑘+𝜆≤1,_{𝐻𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖𝑘Δ𝜇^𝑝+𝜆Δ𝜇−(1−𝜆)Δ𝜈^𝑝+(𝑘−𝜆)Δ𝜇+(1−𝜆)Δ𝜈^𝑝1/𝑝,𝑑𝑘+𝜆≤1,_{𝐻∗𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖𝑘Δ𝜇^𝑝+𝑘𝜆Δ𝜇−(1−𝜆)Δ𝜈^𝑝+(1−𝜆)𝑘Δ𝜇+Δ𝜈^𝑝1/𝑝,𝑑𝑘+𝜆≤1,_{𝐽𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖(1−𝑘)Δ𝜇−𝑘Δ𝜈^𝑝+|𝜆Δ𝜈|^𝑝+(1−𝑘)Δ𝜇+(𝜆−𝑘)Δ𝜈^𝑝1/𝑝,𝑑𝑘+𝜆≤1,_{𝐽∗𝑘,𝜆}()=𝐴,𝐵0.5^𝑛_𝑖=1𝑤_𝑖(1−𝑘)Δ𝜇−𝑘𝜆Δ𝜈^𝑝+|𝜆Δ𝜈|^𝑝+(1−𝑘)Δ𝜇+𝜆Δ𝜈^𝑝1/𝑝,𝑘+𝜆≤1. (15)(16)(17)(18)(19)(20)(21)

Let Δ𝜇⁻=𝜇⁻_𝐴−𝜇⁻_𝐵, Δ𝜇⁺=𝜇⁺_𝐴−𝜇⁺_𝐵, Δ𝜈⁻=𝜈⁻_𝐴−𝜈⁻_𝐵, Δ𝜈⁺=𝜈⁺_𝐴−𝜈⁺_𝐵, Δ𝜋⁻=𝜋⁻_𝐴−𝜋⁻_𝐵, Δ𝜋⁺=𝜋⁺_𝐴−𝜋⁺_𝐵. According to Definition 3 and Theorem 4, Δ𝜇⁻+Δ𝜈⁻+Δ𝜋⁺=0, Δ𝜇⁺+Δ𝜈⁺+Δ𝜋⁻=0. Similar to the distance measures of IFS, some scholars define some distance measures made up of Δ𝜇⁻, Δ𝜇⁺, Δ𝜈⁻, Δ𝜈⁺, Δ𝜋⁻, Δ𝜋⁺.

4. Construction of Distance Measures on DIVIFS

First, we will define a novel essential ranking property of the distance measure on IFS and IVIFS.

Definition 8. Assume that 𝐴 and 𝐵 are IFSs, and then we have the following:  𝐴=𝐵 if and only if 𝜇_𝐴(𝑥)=𝜇_𝐵(𝑥), 𝜈_𝐴(𝑥)=𝜈_𝐵(𝑥), 𝜋_𝐴(𝑥)=𝜋_𝐵(𝑥). 𝐴⊆𝐵 if and only if 𝜇_𝐴(𝑥)≤𝜇_𝐵(𝑥), 𝜈_𝐴(𝑥)≥𝜈_𝐵(𝑥). 𝐴⊂𝐵 if and only if 𝜇_𝐴(𝑥)≤𝜇_𝐵(𝑥) and 𝜈_𝐴(𝑥)>𝜈_𝐵(𝑥), or 𝜇_𝐴(𝑥)<𝜇_𝐵(𝑥) and 𝜈_𝐴(𝑥)≥𝜈_𝐵(𝑥).

Definition 9. Assume that 𝐴 and 𝐵 are IVIFSs, and then we obtain the following:  𝐴=𝐵 if and only if 𝜇⁻_𝐴(𝑥)=𝜇⁻_𝐵(𝑥), 𝜇⁺_𝐴(𝑥)=𝜇⁺_𝐵(𝑥), 𝑣⁻_𝐴(𝑥)=𝑣⁻_𝐵(𝑥), 𝑣⁺_𝐴(𝑥)=𝑣⁺_𝐵(𝑥), 𝜋⁻_𝐴(𝑥)=𝜋⁻_𝐵(𝑥), 𝜋⁺_𝐴(𝑥)=𝜋⁺_𝐵(𝑥). 𝐴⊆𝐵 if and only if 𝜇⁻_𝐴(𝑥)≤𝜇⁻_𝐵(𝑥), 𝜇⁺_𝐴(𝑥)≤𝜇⁺_𝐵(𝑥), 𝑣⁻_𝐴(𝑥)≥𝑣⁻_𝐵(𝑥), 𝑣⁺_𝐴(𝑥)≥𝑣⁺_𝐵(𝑥). 𝐴⊂𝐵 if and only if 𝐴⊆𝐵 and 𝐴≠𝐵.
In general, a distance measure should conform to the following Theorem 10 according to its properties.

Theorem 10. 𝐴, 𝐵, and 𝐶 are IFSs; the distances among them must satisfy the following:(P1)𝑑(𝐴,𝐵)≥0 and 𝑑(𝐴,𝐵)=0 if and only if 𝐴=𝐵.(P2)𝑑(𝐴,𝐵)=𝑑(𝐵,𝐴).(P3) If 𝐶⊆𝐵⊆𝐴, then 𝑑(𝐴,𝐶)≥𝑑(𝐴,𝐵) and 𝑑(𝐴,𝐶≥𝑑(𝐵,𝐶). If 𝐶⊂𝐵⊂𝐴, then 𝑑(𝐴,𝐶)>𝑑(𝐴,𝐵) and 𝑑(𝐴,𝐶)>𝑑(𝐵,𝐶).(P4) If 𝐷⊆𝐶⊆𝐵⊆𝐴, then 𝑑(𝐴,𝐷)≥𝑑(𝐵,𝐶), and 𝑑(𝐴,𝐷)=𝑑(𝐵,𝐶) if and only if 𝐴=𝐵 and 𝐶=𝐷.  If 𝐷⊂𝐶⊂𝐵⊂𝐴, then 𝑑(𝐴,𝐷)>𝑑(𝐵,𝐶).
In fact, property (P4) is derived from property (P3). According to (P3), one has the following: If 𝐷⊆𝐶⊆𝐵⊆𝐴, then 𝑑(𝐴,𝐷)≥𝑑(𝐴,𝐶)≥𝑑(𝐵,𝐶), 𝑑(𝐴,𝐷)≥𝑑(𝐵,𝐷)≥𝑑(𝐵,𝐶)→𝑑(𝐴,𝐷)≥𝑑(𝐵,𝐶). If 𝐷⊂𝐶⊂𝐵⊂𝐴, then 𝑑(𝐴,𝐷)>𝑑(𝐴,𝐶)>𝑑(𝐵,𝐶), 𝑑(𝐴,𝐷)>𝑑(𝐵,𝐷)>𝑑(𝐵,𝐶)→𝑑(𝐴,𝐷)>𝑑(𝐵,𝐶).
Obviously, for all the distance formulas above, (P1), (P2), and (P3) can be obtained. However, most of them do not satisfy the ranking property (P4).

Example 11. Let 𝐴=(0.5,0.4,0.1), 𝐵=(0.5,0.5,0), 𝐶=(0.4,0.4,0.2), and 𝐷=(0.4,0.5,0.1). For IFS 𝐴, we obtain 𝜇_𝐴(𝑥)=0.5, 𝜈_𝐴(𝑥)=0.4, and 𝜋_𝐴(𝑥)=0.1. In the following, we denote 𝜇_𝐴=𝜇_𝐴(𝑥)=0.5, 𝜈_𝐴=𝜈_𝐴(𝑥)=0.4, and 𝜋_𝐴=𝜋_𝐴(𝑥)=0.1.
According to Definition 3, if 𝜇⁻(𝑥)=𝜇⁺(𝑥)=𝜇(𝑥), 𝜈⁻(𝑥)=𝜈⁺(𝑥)=𝜈(𝑥), 𝜋⁻(𝑥)=𝜋⁺(𝑥)=𝜋(𝑥), then IVIFS is equivalent to IFS. Correspondingly, we can define some IVIFSs as follows, respectively: ⟨[],[,⟨[],[,⟨[],[,⟨[],[.𝐴=𝑥,0.5,0.50.4,0.4]⟩𝐵=𝑥,0.5,0.50.5,0.5]⟩𝐶=𝑥,0.4,0.40.4,0.4]⟩𝐷=𝑥,0.4,0.40.5,0.5]⟩ (22) If 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} is a universe of discourse, we define 𝐴=𝑥₁,[],[,𝑥0.5,0.50.4,0.4]…,_𝑛,[],[;𝑥0.5,0.50.4,0.4]𝐵=₁,[],[,𝑥0.5,0.50.5,0.5]…,_𝑛,[],[;𝑥0.5,0.50.5,0.5]𝐶=₁,[],[,𝑥0.4,0.40.4,0.4]…,_𝑛,[],[;𝑥0.4,0.40.4,0.4]𝐷=₁,[],[,𝑥0.4,0.40.5,0.5]…,_𝑛,[0.4,0.4],[0.5,0.5]. (23)
Thus we have 𝜇_𝐴(𝑥)=𝜇_𝐵(𝑥)>𝜇_𝐶(𝑥)=𝜇_𝐷(𝑥), 𝜈_𝐴(𝑥)=𝜈_𝐶(𝑥)<𝜈_𝐵(𝑥)=𝜈_𝐷(𝑥). According to Definitions 8 and 9, we have 𝐷⊂𝐶⊂𝐵⊂𝐴. Therefore, we obtain the necessary condition for distance measure on IFS and IVIFS: 𝑑(𝐴,𝐷)>𝑑(𝐵,𝐶).

Obviously, according to 𝐴, 𝐵, 𝐶, and 𝐷, we obtain Δ𝜇(𝐴,𝐷)=𝜇_𝐴−𝜇_𝐷()=0.1,Δ𝜈𝐴,𝐷=𝜈_𝐴−𝜈_𝐷()=−0.1,Δ𝜋𝐴,𝐷=𝜋_𝐴−𝜋_𝐷()=0,Δ𝜇𝐵,𝐶=𝜇_𝐵−𝜇_𝐶()=0.1,Δ𝜈𝐵,𝐶=𝜈_𝐵−𝜈_𝐶()=0.1,Δ𝜋𝐵,𝐶=𝜋_𝐵−𝜋_𝐶=0.2. (24)

From formulas (8)–(14), we have 𝑑_Za(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.1^𝑝1/𝑝=𝑑_Za(),𝑑𝐵,𝐶_At(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.1^𝑝+0.1^𝑝1/𝑝=𝑑_At(),𝑑𝐵,𝐶_S,K(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.1^𝑝+0.1^𝑝1/𝑝<^𝑛_𝑖=1𝑤_𝑖0.1^𝑝+0.1^𝑝+0.2^𝑝1/𝑝=𝑑_S,K(),𝑑𝐵,𝐶_L,C(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.2^𝑝1/𝑝>0=𝑑_L,C(),𝑑𝐵,𝐶_P,H(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖max(0.1,0.1)=𝑑_P,H(),𝑑𝐵,𝐶_W(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.1+0.1+2max(0.1,0.1)4=𝑑_W(),𝑑𝐵,𝐶_RB(𝐴,𝐷)=+∞>1=𝑑_RB(𝐵,𝐶). (25) Only (11) and (14) satisfy property (P4).

Considering (15), we get 𝑑_𝐷𝑘(𝐴,𝐷)=^𝑛_𝑖=1𝑤_𝑖0.1^𝑝1/𝑝≥^𝑛_𝑖=1𝑤_𝑖(|1−2𝑘|0.1)^𝑝1/𝑝=𝑑_𝐷𝑘(𝐵,𝐶). (26) If 𝑘≠0 and 𝑘≠1, then we have 𝑑_𝐷𝑘(𝐴,𝐷)>𝑑_𝐷𝑘(𝐵,𝐶); otherwise, we have 𝑑_𝐷𝑘(𝐴,𝐷)=𝑑_𝐷𝑘(𝐵,𝐶).

From (16), we get 𝑑_{𝐹𝑘,𝜆}(𝐴,𝐷)=0.5^𝑛_𝑖=1𝑤_𝑖0.1^𝑝+0.1^𝑝1/𝑝,𝑑_{𝐹𝑘,𝜆}()=𝐵,𝐶0.5^𝑛_𝑖=1𝑤_𝑖0.1^𝑝(|1−2𝑘|^𝑝+|1−2𝜆|^𝑝+|1−𝑘−𝜆|^𝑝)^1/𝑝,𝑘+𝜆≤1. (27) If 𝑝=1, then we have 𝑘+𝜆<1/2→𝑑_𝐷𝑘(𝐴,𝐷)<𝑑_𝐷𝑘(𝐵,𝐶); 𝑘+𝜆≥1/2→𝑑_𝐷𝑘(𝐴,𝐷)≥𝑑_𝐷𝑘(𝐵,𝐶).

From (17), we get 𝑑_{𝐺𝑘,𝜆}()=𝐴,𝐷0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1𝜆|^𝑝+|0.1(𝑘−𝜆)|^𝑝1/𝑝≤0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1𝜆|^𝑝+|0.1(𝑘+𝜆)|^𝑝1/𝑝=𝑑_{𝐺𝑘,𝜆}(𝐵,𝐶). (28) Obviously, we get 𝑑_𝐷𝑘(𝐴,𝐷)≤𝑑_𝐷𝑘(𝐵,𝐶) for each 𝑘≥0, for each 𝜆≥0, and for each 𝑃>0.

For (18), we have 𝑑_{𝐻𝑘,𝜆}()=𝐴,𝐷0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1|^𝑝+|0.1(1−𝑘)|^𝑝1/𝑝,𝑑_{𝐻𝑘,𝜆}()=𝐵,𝐶0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1(1−2𝜆)|^𝑝+|0.1(1+𝑘−2𝜆)|^𝑝1/𝑝. (29) If 𝑝=1 and 𝑘+𝜆≤1, then we have 0≤𝜆<1/3, 2𝜆<𝑘→𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)<𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶); otherwise, 𝑑_𝐷𝑘(𝐴,𝐷)≥𝑑_𝐷𝑘(𝐵,𝐶).

For example, let 𝑝=1, 𝜆=0.2, and 𝑘=0.5, and then 𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)=0.5∑^𝑛_𝑖=1𝑤_𝑖0.2<0.5∑^𝑛_𝑖=1𝑤_𝑖0.22=𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶).

Similar to the result from (18), for (20), we have 𝑑_{𝐽𝑘,𝜆}()=𝐴,𝐷0.5^𝑛_𝑖=1𝑤_𝑖|0.1|^𝑝+|0.1𝜆|^𝑝+|0.1(1−𝜆)|^𝑝1/𝑝,𝑑_{𝐽𝑘,𝜆}()=𝐵,𝐶0.5^𝑛_𝑖=1𝑤_𝑖|0.1(1−2𝑘)|^𝑝+|0.1𝜆|^𝑝+|0.1(1+𝜆−2𝑘)|^𝑝1/𝑝. (30) If 𝑝=1 and 𝑘+𝜆≤1, then we have 0≤𝑘<1/3, 2𝑘<𝜆→𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)<𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶); otherwise, 𝑑_𝐷𝑘(𝐴,𝐷)≥𝑑_𝐷𝑘(𝐵,𝐶).

For example, let 𝑝=1, 𝑘=0.2, and 𝜆=0.5, and then 𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)=0.5∑^𝑛_𝑖=1𝑤_𝑖0.2<0.5∑^𝑛_𝑖=1𝑤_𝑖0.22=𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶).

For (19), we obtain 𝑑_𝐻^∗_𝑘,𝜆()=𝐴,𝐷0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1(1+𝑘𝜆−𝜆)|^𝑝+|0.1(1−𝜆)(1−𝑘)|^𝑝1/𝑝,𝑑_𝐻^∗_𝑘,𝜆()=𝐵,𝐶0.5^𝑛_𝑖=1𝑤_𝑖|0.1𝑘|^𝑝+|0.1(1−𝑘𝜆−𝜆)|^𝑝+|0.1(1−𝜆)(1+𝑘)|^𝑝1/𝑝,𝑘+𝜆≤1. (31)

If 𝑝=1 and 𝑘+𝜆≤1, then we have 0≤𝜆<1/2→𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)<𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶); otherwise, 𝑑_𝐷𝑘(𝐴,𝐷)≥𝑑_𝐷𝑘(𝐵,𝐶).

Similarly, we have the same results for (21), which will be as follows: if 𝑝=1 and 𝑘+𝜆≤1, then 0≤𝑘<1/2→𝑑_{𝐻𝑘,𝜆}(𝐴,𝐷)<𝑑_{𝐻𝑘,𝜆}(𝐵,𝐶); otherwise, 𝑑_𝐷𝑘(𝐴,𝐷)≥𝑑_𝐷𝑘(𝐵,𝐶).

From the analysis to the formulas from Xia and Xu above, only (15) basically meets the requirement of (P4); the others only meet the requirement of (P4) in part. Similarly, most of the conventional distance measures of IVIFS do not satisfy property (P4). Therefore, it is necessary to find a distance measure satisfying the necessary ranking property. Considering the component elements of the distance formula of IFS and IVIFS, we can define some appropriate distance measures of DIVIFS (Figure 2).

(a)

(b)

(c)

(d)

Figure 2 

Distance between each patient and each disease on DIVIFS.

According to Definition 7 and Example 11, we have 𝜇⁻_𝐴(𝑥)=0.5, 𝑣⁻_𝐴(𝑥)=0.4, 𝜋⁺_𝐴(𝑥)=0.1, 𝜇⁺_𝐴(𝑥)=0.5+𝜆₀𝜆₁0.1, 𝑣⁺_𝐴(𝑥)=0.4+𝜆₀(1−𝜆₁)0.1, 𝜋⁻_𝐴(𝑥)=(1−𝜆₀)0.1, 𝜇⁻_𝐵(𝑥)=0.5, 𝑣⁻_𝐵(𝑥)=0.5, 𝜋⁻_𝐵(𝑥)=0, 𝜇⁺_𝐵(𝑥)=0.5, 𝑣⁺_𝐵(𝑥)=0.5, 𝜋⁺_𝐵(𝑥)=0, 𝜇⁻_𝐶(𝑥)=0.4, 𝑣⁻_𝐶(𝑥)=0.4, 𝜋⁺_𝐶(𝑥)=0.2, 𝜇⁺_𝐶(𝑥)=0.4+𝜆₀𝜆₁0.2, 𝑣⁺_𝐶(𝑥)=0.4+𝜆₀(1−𝜆₁)0.2, 𝜋⁻_𝐶(𝑥)=(1−𝜆₀)0.2, 𝜇⁻_𝐷(𝑥)=0.4, 𝑣⁻_𝐷(𝑥)=0.5, 𝜋⁺_𝐷(𝑥)=0.1, 𝜇⁺_𝐷(𝑥)=0.4+𝜆₀𝜆₁0.1, 𝑣⁺_𝐷(𝑥)=0.5+𝜆₀(1−𝜆₁)0.1, 𝜋⁻_𝐷(𝑥)=(1−𝜆₀)0.1.

Let 𝜇⁻=𝜇⁻(𝑥), 𝜇⁺=𝜇⁺(𝑥), 𝜈⁻=𝜈⁻(𝑥), 𝜈⁺=𝜈⁺(𝑥), 𝜋⁻=𝜋⁻(𝑥), 𝜋⁺=𝜋⁺(𝑥), and we obtain(1)Δ𝜋⁻(𝐴,𝐷)=𝜋⁻_𝐴−𝜋⁻_𝐷=0, Δ𝜋⁻(𝐵,𝐶)=𝜋⁻_𝐵−𝜋⁻_𝐶=−(1−𝜆₀)0.2→|Δ𝜋⁻(𝐴,𝐷)|≤|Δ𝜋⁻(𝐵,𝐶)|;(2)Δ𝜋⁺(𝐴,𝐷)=𝜋⁺_𝐴−𝜋⁺_𝐷=0<𝜋⁺_𝐵−𝜋⁺_𝐶=Δ𝜋⁺(𝐵,𝐶)=0.2→|Δ𝜋⁺(𝐴,𝐷)|<|Δ𝜋⁺(𝐵,𝐶)|; (3)Δ𝜇⁻(𝐴,𝐷)=𝜇⁻_𝐴−𝜇⁻_𝐷=0.1=𝜇⁻_𝐵−𝜇⁻_𝐶=Δ𝜇⁻(𝐵,𝐶)→|Δ𝜇⁻(𝐴,𝐷)|=|Δ𝜇⁻(𝐵,𝐶)|; (4)Δ𝜈⁻(𝐴,𝐷)=𝜈⁻_𝐴−𝜈⁻_𝐷=−0.1=−1×(𝜈⁻_𝐵−𝜈⁻_𝐶)=−Δ𝜈⁻(𝐵,𝐶)→|Δ𝜈⁻(𝐴,𝐷)|=|Δ𝜈⁻(𝐵,𝐶)|; (5)Δ𝜇⁻(𝐴,𝐷)−Δ𝜈⁻(𝐴,𝐷)=0.2>0=Δ𝜇⁻(𝐵,𝐶)−Δ𝜈⁻(𝐵,𝐶)→|Δ𝜇⁻(𝐴,𝐷)−Δ𝜈⁻(𝐴,𝐷)| >|Δ𝜇⁻(𝐵,𝐶)−Δ𝜈⁻(𝐵,𝐶)|;(6)Δ𝜇⁺(𝐴,𝐷)=𝜇⁺_𝐴−𝜇⁺_𝐷=0.1, Δ𝜇⁺(𝐵,𝐶)=𝜇⁺_𝐵−𝜇⁺_𝐶=0.1(1−2𝜆₀𝜆₁)→|Δ𝜇⁺(𝐴,𝐷)|≥|Δ𝜇⁺(𝐵,𝐶)|;(7)Δ𝜈⁺(𝐴,𝐷)=𝜈⁺_𝐴−𝜈⁺_𝐷=−0.1, Δ𝜈⁺(𝐵,𝐶)=𝜈⁺_𝐵−𝜈⁺_𝐶=0.1(1−2𝜆₀(1−𝜆₁))→|Δ𝜇⁺(𝐴,𝐷)|≥|Δ𝜇⁺(𝐵,𝐶)|;(8)Δ𝜇⁺(𝐴,𝐷)−Δ𝜈⁺(𝐴,𝐷)=0.2,Δ𝜇⁺(𝐵,𝐶)−Δ𝜈⁺(𝐵,𝐶)=0.2𝜆₀(1−2𝜆₁), →|Δ𝜇⁺(𝐴,𝐷)−Δ𝜈⁺(𝐴,𝐷)|≥|Δ𝜇⁺(𝐵,𝐶)−Δ𝜈⁺(𝐵,𝐶)|.

Considering the necessary condition (P4), we draw a conclusion as follows. Δ𝜋⁺ does not meet the requirement of the ranking property, and if 𝜆₀≠1, Δ𝜋⁻ does not satisfy (P4) too. Δ𝑢⁻−Δ𝜈⁻ meets the necessary condition (P4), and Δ𝜇⁺, Δ𝑣⁺, and Δ𝜇⁺−Δ𝑣⁺ do so if 𝜆_𝑘∈(0,1) (𝑘=1,2). All the calculated results are shown in (32) and (33): |Δ𝜋⁻(𝐴,𝐷)|≤|Δ𝜋⁻(;|𝐵,𝐶)|Δ𝜋⁻(𝐴,𝐷)|=|Δ𝜋⁻(𝐵,𝐶)|⟺𝜆₀=1;Δ𝜋⁺(𝐴,𝐷)=Δ𝜋⁺(𝐵,𝐶);Δ𝜇⁻(𝐴,𝐷)=Δ𝜇⁻(;|𝐵,𝐶)Δ𝜈⁻(𝐴,𝐷)|=|Δ𝜈⁻(𝐵,𝐶)|;Δ𝜇⁻(𝐴,𝐷)−Δ𝜈⁻(>𝐴,𝐷)Δ𝜇⁻(𝐵,𝐶)−Δ𝜈⁻(𝐵,𝐶);Δ𝜇⁺(𝐴,𝐷)≥Δ𝜇⁺(𝐵,𝐶);Δ𝜇⁺(=𝐴,𝐷)Δ𝜇⁺(𝐵,𝐶)⟺𝜆₀=0,or𝜆₁=0,or𝜆₀=𝜆₁=1;Δ𝜈⁺(𝐴,𝐷)≥Δ𝜈⁺(𝐵,𝐶);Δ𝜈⁺(=𝐴,𝐷)Δ𝜈⁺(𝐵,𝐶)⟺𝜆₀=0,or𝜆₁=1,or𝜆₀=1,𝜆₁=0;Δ𝜇⁺(𝐴,𝐷)−Δ𝜈⁺(≥𝐴,𝐷)Δ𝜇⁺(𝐵,𝐶)−Δ𝜈⁺(𝐵,𝐶);Δ𝜇⁺(𝐴,𝐷)−Δ𝜈⁺(=𝐴,𝐷)Δ𝜇⁺(𝐵,𝐶)−Δ𝜈⁺(𝐵,𝐶)⟺𝜆₀=1,𝜆₁=0,or𝜆₀=1,𝜆₁=1. (32)(33)

In all, in order to meet the needs of necessary condition (P4), we conclude that the distance measure of DIVIFS should be composed of Δ𝜇⁻−Δ𝑣⁻, Δ𝜇⁺, Δ𝑣⁺, and Δ𝜇⁺−Δ𝑣⁺ and that Δ𝜇⁻ and Δ𝜈⁻ can be involved in the distance formula too. However, considering the contradiction between Δ𝜋 and (P4), Δ𝜋⁺ and Δ𝜋⁻ are not suitable for new distance measures.

In Xu’s papers [16, 17, 19, 20], many weighted similarity measures are proposed according to standardized distance measures based on the IFS theory and the IVIFS theory. And Xu presented them by defining the appropriate weights of membership function, non-membership function, and hesitancy function. According to Xu, we can also define the corresponding weighted distance measures on IVIFS as follows: 𝑑()=𝐴,𝐵_𝑛𝑖=1𝑤_𝑖𝐶₁Δ𝜇⁻𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝+𝐶₂Δ𝜇⁺𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝+𝐶₃Δ𝜈⁻𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝+𝐶₄Δ𝜈⁺𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝+𝐶₅Δ𝜇⁻𝐴𝑥_𝑖,𝐵𝑥_𝑖−𝐶₇Δ𝜈⁻𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝+𝐶₆Δ𝜇⁺𝐴𝑥_𝑖,𝐵𝑥_𝑖−𝐶₈Δ𝜈⁺𝐴𝑥_𝑖,𝐵𝑥_𝑖^𝑝1/𝑝;_𝑛𝑖=1𝑤_𝑖=1,𝑤_𝑖𝐶≥0;_𝑗≥0,𝑗=1,2,3,4,5,6,7,8. (34)

It is easy to prove that (11) is equivalent to a special case of (15) when 𝑘=0.5. And (15) is also a special case of (34) when 𝐶₁=𝐶₂=𝐶₃=𝐶₄=𝐶₆=𝐶₈=0 and 𝐶₅=𝐶₇=1. To simplifiy the formula, let 𝐶₅=𝐶₇ and 𝐶₆=𝐶₈. Assume that 𝐴 and 𝐵 are two DIVIFS, and we have 𝑑()=𝐴,𝐵_𝑛𝑖=1𝑤_𝑖𝐶₁𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖^𝑝+𝐶₂𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖^𝑝+𝐶₃𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖+𝜆₀𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝+𝐶₄𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀1−𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝+𝐶₅𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖^𝑝+𝐶₆𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀2𝜆₁−1𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝1/𝑝,_𝑛𝑖=1𝑤_𝑖=1,𝑤_𝑖≥0;𝐶_𝑗≥0,𝑗=1,2,3,4,5,6. (35)

Comparing (15)–(21) with (34)-(35), we conclude that Xu’s methods are different from the DIVIFS method. Xu defined the measures by constructing the weights of membership function, non-membership function, and hesitancy function, respectively. However, we concentrate on the detachment of hesitancy function and present the DIVIFS method. If we use Xu’s methods solely, we can define the weights of certainty and uncertainty, but we cannot analyze the variation of the absent party. Moreover, in Xu’s research, and even in all the conventional research, there is no research on the ranking condition (P4). Therefore, the distance measures on DIVIFS above are an important supplement to Xu’s methods and the others’.

In order to simplify the calculation, we define the following distance measure: 𝑑()=𝐴,𝐵_𝑛𝑖=1𝑤_𝑖𝐶₁𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖^𝑝+𝐶₂𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖^𝑝+𝐶₃𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖+𝜆₀𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝+𝐶₄𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀1−𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝+𝐶₅𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖^𝑝+𝐶₆𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀2𝜆₁−1𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖^𝑝,_𝑛𝑖=1𝑤_𝑖=1,𝑤_𝑖𝐶≥0;_𝑗≥0,𝑗=1,2,3,4,5,6. (36)

If 𝑝=1, the distance measure is the Hamming distance; if 𝑝=2, it is the Euclidean distance. In this paper, the Euclidean distance on DIVIFS will be used in the simulation of pattern recognition: 𝑑()=𝐴,𝐵_𝑛𝑖=1𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖²+𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖²+𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖+𝜆₀𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖²+𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀1−𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖²,𝑑()=𝐴,𝐵_𝑛𝑖=1𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖²+𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀2𝜆₁−1𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖²,𝑑()=𝐴,𝐵_𝑛𝑖=1𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖²+𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖²+𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖+𝜆₀𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖²+𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀1−𝜆₁𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖²+𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖²+𝜇⁻_𝐴𝑥_𝑖−𝜇⁻_𝐵𝑥_𝑖−𝜈⁻_𝐴𝑥_𝑖−𝜈⁻_𝐵𝑥_𝑖+𝜆₀2𝜆₁−1𝜋⁺_𝐴𝑥_𝑖−𝜋⁺_𝐵𝑥_𝑖². (37)(38)(39)

5. Application to Medical Diagnosis Based on Pattern Recognition

To make a proper diagnosis 𝐷 for a patient with given values of symptoms 𝑆, a medical knowledge base is necessary that involves elements described in terms of IFS. We adopt the same data as those in [8, 9]: the set of disease diagnoses is 𝐷={Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}, and the set of symptoms is a universe of discourse 𝑆={temperature, headache, stomach pain, cough, chest pain}. The data are given in Table 1, where each symptom is described as follows: membership function 𝜇_𝐴(𝑥), non-membership function 𝜈_𝐴(𝑥), and hesitancy function 𝜋_𝐴(𝑥). For example, for malaria the temperature is high (𝜇_𝐴(𝑥)=0.7, 𝜈_𝐴(𝑥)=0, 𝜋_𝐴(𝑥)=0.3), while for chest problem the temperature is low (𝜇_𝐴(𝑥)=0.1, 𝜈_𝐴(𝑥)=0.8, 𝜋_𝐴(𝑥)=0.1), and so forth.

Table 1 

Symptoms characteristic of the diagnoses under consideration.

The set of patients is 𝑃={AI, Bob, Joe, Ted}. The symptoms are given in Table 2, where we need all three parameters (𝜇_𝐴(𝑥),𝜈_𝐴(𝑥),𝜋_𝐴(𝑥)) to describe each symptom (see [8, 9]) as before. We seek a diagnosis for each patient 𝑝_𝑗, 𝑗=1,2,3,4. In [8, 9], Szmidt and Kacprzyk [8, 9] proposed to solve the problem in the following way: (1) to calculate a distance from a set of symptoms 𝑠_𝑖(𝑖=1,2,3,4,5) (using the normalized Hamming distance, the Euclidean distance, and ratio-based measure of similarity) between each patient 𝑝_𝑗 (𝑗=1,2,3,4, Table 2) and each diagnosis 𝑑_𝑘, (𝑘=1,2,3,4,5, Table 1); (2) to determine that the shortest distance should be a proper diagnosis.

Table 2 

Symptoms characteristic of the patients under observation.

For (10), the normalized Hamming distance for all symptoms of patient 𝑗th from diagnosis 𝑘th is (40), and the normalized Euclidean distance for all symptoms of patient 𝑗th from diagnosis 𝑘th is (41). From (14), the ratio-based measure of similarity for all symptoms of patient 𝑗th from diagnosis 𝑘th is (42): 𝑑_𝐻𝑝_𝑗,𝑑_𝑘=∑⁵_𝑖=1𝜇⁻_𝑝𝑗𝑠_𝑖−𝜇⁻_𝑑𝑘𝑠_𝑖+𝜈⁻_𝑝𝑗𝑠_𝑖−𝜈⁻_𝑑𝑘𝑠_𝑖+𝜋⁺_𝑝𝑗𝑠_𝑖−𝜋⁺_𝑑𝑘𝑠_𝑖,𝑑10_𝐸𝑝_𝑗,𝑑_𝑘=110⁵_𝑖=1𝜇⁻_𝑝𝑗𝑠_𝑖−𝜇⁻_𝑑𝑘𝑠_𝑖²+𝜈⁻_𝑝𝑗𝑠_𝑖−𝜈⁻_𝑑𝑘𝑠_𝑖²+𝜋⁺_𝑝𝑗𝑠_𝑖−𝜋⁺_𝑑𝑘𝑠_𝑖²,𝑑_RB𝑝_𝑗,𝑑_𝑘=𝑑_𝐻𝑝_𝑗,𝑑_𝑘𝑑_𝐻𝑝_𝑗,𝑑^𝐶_𝑘. (40)(41)(42) Szmidt and Kacprzyk describe 𝑑^𝐶_𝑘 to be a set (𝜈_𝐴(𝑥),𝜇_𝐴(𝑥),𝜋_𝐴(𝑥)). For example, for malaria the temperature 𝑑_𝑘 is high (𝜇_𝐴(𝑥)=0.7, 𝜈_𝐴(𝑥)=0, 𝜋_𝐴(𝑥)=0.3), whereas 𝑑^𝐶_𝑘 is low (𝜇_𝐴(𝑥)=0, 𝜈_𝐴(𝑥)=0.7, 𝜋_𝐴(𝑥)=0.3), and so on.

From the obtained distance measures and similarity measures, Szmidt and Kacprzyk get the results as Table 3. Three types of distances given in Table 3 are described as follows: the first data is the Hamming distance as in (40), the second is the Euclidean distance as in (41), the third is the ratio-based measure as in (42). For example, 𝑑_𝐻(AI, viral fever), 𝑑_𝐸(AI, viral fever), and 𝑑_RB(AI, viral fever) are 0.28, 0.29, and 0.75 respectively. Similarly, we can get the other results. Obviously, Bob suffers from stomach problem and Joe from typhoid based on (40)–(42) above. AI suffers from Malaria and Ted from Viral fever calculated by the Hamming distance and the Euclidean distance, while AI suffers from Viral fever and Ted from Malaria calculated by the ratio-based measure.

Table 3 

Distance description between each patient and each disease based on IFS.

Let 𝑝=5 and 𝑤_𝑖=1/𝑛 for each 𝑖, and 𝑘 and 𝜆 are constants. Calculated by (15)–(21), Table 4 can be obtained.

Table 4 

Distance between each patient and each disease on IFS calculated by Xu’s methods.

From Table 4, we conclude that Bob suffers from stomach problem, Joe from typhoid, Ted from viral fever, and AI from viral fever according to most of Xu’s methods. But on the basis of 𝐷_𝐽0.1,0.6, AI can be diagnosed as a malaria sufferer. Using Xu’s methods of IVIFS from [20], we can get the same results as Table 4. The results above are also the same as those from [35] by Wei et al.

Let 𝜇⁻_𝐴(𝑥)=𝜇_𝐴(𝑥), 𝜈⁻_𝐴(𝑥)=𝜈_𝐴(𝑥), and 𝜋⁺_𝐴(𝑥)=𝜋_𝐴(𝑥). Using (37), we will get Table 5.

Table 5 

Distance description between each patient and each disease based on DIVIFS.

The distances given in Table 5 are described as follows: for example, from the first data of viral fever and malaria, we can describe the distance measure between AI and viral fever, and the distance measure between AI and malaria from (37), respectively. Similarly, we can get all the results (43), and obtain Table 5 from all (43): 𝑑𝐴𝐼,𝑉𝑖𝑟𝑎𝑙𝑓𝑒𝑣𝑒𝑟=1.06−0.24𝜆₀−0.14𝜆₀𝜆₁+0.31𝜆²₀−0.62𝜆²₀𝜆₁1−𝜆₁,𝑑(𝐴𝐼,𝑀𝑎𝑙𝑎𝑟𝑖𝑎)=1.08−0.2𝜆₀+0.2𝜆₀𝜆₁+0.1𝜆²₀−0.2𝜆²₀𝜆₁1−𝜆₁,𝑑𝐴𝐼,𝑇𝑦𝑝ℎ𝑜𝑖𝑑=1.6−0.18𝜆₀−0.04𝜆₀𝜆₁+0.2𝜆²₀−0.4𝜆²₀𝜆₁1−𝜆₁,𝑑𝐴𝐼,𝑆𝑡𝑜𝑚𝑎𝑐ℎ𝑝𝑟𝑜𝑏𝑙𝑒𝑚=5.28−0.42𝜆₀+0.56𝜆₀𝜆₁+0.14𝜆²₀−0.28𝜆²₀𝜆₁1−𝜆₁,𝑑𝐴𝐼,𝐶ℎ𝑒𝑠𝑡𝑝𝑟𝑜𝑏𝑙𝑒𝑚=6.44−0.36𝜆₀+0.44𝜆₀𝜆₁+0.14𝜆²₀−0.28𝜆²₀𝜆₁1−𝜆₁,𝑑𝐵𝑜𝑏,𝑉𝑖𝑟𝑎𝑙𝑓𝑒𝑣𝑒𝑟=3.38−0.86𝜆₀+1.34𝜆₀𝜆₁+0.19𝜆²₀−0.38𝜆²₀𝜆₁1−𝜆₁,𝑑(𝐵𝑜𝑏,𝑀𝑎𝑙𝑎𝑟𝑖𝑎)=6.12−0.62𝜆₀+1.12𝜆₀𝜆₁+0.06𝜆²₀−0.12𝜆²₀𝜆₁1−𝜆₁,𝑑𝐵𝑜𝑏,𝑇𝑦𝑝ℎ𝑜𝑖𝑑=2.04−0.52𝜆₀+0.84𝜆₀𝜆₁+0.1𝜆²₀−0.2𝜆²₀𝜆₁1−𝜆₁,𝑑𝐵𝑜𝑏,𝑆𝑡𝑜𝑚𝑎𝑐ℎ𝑝𝑟𝑜𝑏𝑙𝑒𝑚=0.28+0.02𝜆₀−0.16𝜆₀𝜆₁+0.06𝜆²₀−0.12𝜆²₀𝜆₁1−𝜆₁,𝑑𝐵𝑜𝑏,𝐶ℎ𝑒𝑠𝑡𝑝𝑟𝑜𝑏𝑙𝑒𝑚=3.96−0.46𝜆₀+0.64𝜆₀𝜆₁+0.14𝜆²₀−0.28𝜆²₀𝜆₁1−𝜆₁,𝑑𝐽𝑜𝑒,𝑉𝑖𝑟𝑎𝑙𝑓𝑒𝑣𝑒𝑟=1.7−0.34𝜆₀−0.18𝜆₀𝜆₁+0.43𝜆²₀−0.86𝜆²₀𝜆₁1−𝜆₁,𝑑(𝐽𝑜𝑒,𝑀𝑎𝑙𝑎𝑟𝑖𝑎)=3.12−0.64𝜆₀+0.6𝜆₀𝜆₁+0.34𝜆²₀−0.68𝜆²₀𝜆₁1−𝜆₁,𝑑𝐽𝑜𝑒,𝑇𝑦𝑝ℎ𝑜𝑖𝑑=1.12−0.36𝜆₀−0.24𝜆₀𝜆₁+0.48𝜆²₀−0.96𝜆²₀𝜆₁1−𝜆₁,𝑑𝐽𝑜𝑒,𝑆𝑡𝑜𝑚𝑎𝑐ℎ𝑝𝑟𝑜𝑏𝑙𝑒𝑚=4.76+0.38𝜆₀−1.36𝜆₀𝜆₁+0.3𝜆²₀−0.6𝜆²₀𝜆₁1−𝜆₁,𝑑𝐽𝑜𝑒,𝐶ℎ𝑒𝑠𝑡𝑝𝑟𝑜𝑏𝑙𝑒𝑚=6+0.28𝜆₀−1.24𝜆₀𝜆₁+0.34𝜆²₀−0.68𝜆²₀𝜆₁1−𝜆₁,𝑑𝑇𝑒𝑑,𝑉𝑖𝑟𝑎𝑙𝑓𝑒𝑣𝑒𝑟=0.92−0.12𝜆₀−0.08𝜆₀𝜆₁+0.16𝜆²₀−0.32𝜆²₀𝜆₁1−𝜆₁,𝑑(𝑇𝑒𝑑,𝑀𝑎𝑙𝑎𝑟𝑖𝑎)=1.46−0.4𝜆₀+0.54𝜆₀𝜆₁+0.13𝜆²₀−0.26𝜆²₀𝜆₁1−𝜆₁,𝑑𝑇𝑒𝑑,𝑇𝑦𝑝ℎ𝑜𝑖𝑑=2.06−0.42𝜆₀+0.46𝜆₀𝜆₁+0.19𝜆²₀−0.38𝜆²₀𝜆₁1−𝜆₁,𝑑𝑇𝑒𝑑,𝑆𝑡𝑜𝑚𝑎𝑐ℎ𝑝𝑟𝑜𝑏𝑙𝑒𝑚=3.42−0.16𝜆₀+0.02𝜆₀𝜆₁+0.15𝜆²₀−0.3𝜆²₀𝜆₁1−𝜆₁,𝑑𝑇𝑒𝑑,𝐶ℎ𝑒𝑠𝑡𝑝𝑟𝑜𝑏𝑙𝑒𝑚=4.54−0.44𝜆₀+0.5𝜆₀𝜆₁+0.19𝜆²₀−0.38𝜆²₀𝜆₁1−𝜆₁. (43) We aim to seek the minimum distance among 𝑑(𝐴𝐼,𝑑𝑖𝑠𝑒𝑎𝑠𝑒)s. Therefore, the minimum among all these 𝑑(𝐴𝐼,𝑑𝑖𝑠𝑒𝑎𝑠𝑒)s is the pattern we need. From Table 5, it is clear that if 𝜆₀=0, AI, Bob, Joe, and Ted are identified as having viral fever, stomach problem, typhoid, and viral fever, respectively; if 𝜆₀>0, Bob, Joe, and Ted are also identified as having stomach problem, typhoid, and viral fever, respectively, which is the same as Xu’s results in [20] and Wei’s results in [35], but AI is different from the above. When 𝑑(𝐴𝐼,𝑣𝑖𝑟𝑎𝑙𝑓𝑒𝑣𝑒𝑟)>𝑑(𝐴𝐼,𝑚𝑎𝑙𝑎𝑟𝑖𝑎), AI is identified as malaria; otherwise, AI is viral fever. All in all, we have the following results: if 0.02+0.04𝜆₀+0.34𝜆₀𝜆₁−0.21𝜆²₀+0.42𝜆²₀𝜆₁(1−𝜆₁)<0, we can infer that AI suffers from malaria, and on the contrary, AI from viral fever. From Table 5, we conclude that AI suffers from either malaria or viral fever, Bob from stomach problem, Joe from typhoid, and Ted suffers from viral fever. The pattern recognition results are somewhat different from those of the dynamic intuitionistic fuzzy sets (DIFS) method in [26–28], in which we only proposed a DIFS method derived from IFS. However, in this paper, we obtain a series of DIVIFS methods derived from IFS and IVIFS, which are the further study in deep from [26–28].

The experimental results above show that there is a slight difference between the recognition results of DIVIFS and those of IFS. Conventional IFS method is simple, but its recognition results are fixed on the basis of the conventional distance formulas. Therefore, it is difficult to reveal the potential law from all available information when the IFS method is used, whereas the results of the DIVIFS method are flexible, which can be adjusted to sample data with the variables. Furthermore, if the fixed pattern recognition results are different from those of the sample data, the IFS method will fail. However, when using the DIVIFS method, we can meet the requirements of the sample data by adjusting variables to appropriate values. All the results above show that the DIVIFS method is more comprehensive and flexible than the IFS method.

6. Simulation to Medical Diagnosis

In the following, we will compare the DIVIFS method with the conventional method in medical diagnosis. We use a sample of 77 women of embryos transferred, which is from Laboratory of Reproductive Medicine, The First Affiliated Hospital, Southern Medical University. And the set of disease diagnoses is 𝐷={Not successful pregnancy and no symptoms of OHSS, Successful pregnancy and no symptoms of OHSS, Not successful pregnancy and symptoms of OHSS, Successful pregnancy and symptoms of OHSS}, and the set of symptoms is a universe of discourse 𝑆={Basic Hormone FSH, Basic Hormone P, Basic Hormone E2, Basic Hormone T, BMI, Day of HCG E2, Day of HCG endometrial thickness}, where each symptom is described as follows: membership function 𝜇_𝐴(𝑥), non-membership function 𝜈_𝐴(𝑥), and hesitancy function 𝜋_𝐴(𝑥).

Applying the conventional distance measures on IFS and DIVIFS to medical diagnosis, we have the results in Table 6. From Table 6, we conclude that the most accurate results of medical diagnosis are from Xu’s equation (15) when 𝑘=0.5, which is also only a special case of (35) when 𝐶₁=𝐶₂=𝐶₃=𝐶₄=𝐶₆=𝐶₈=0 and 𝐶₅=𝐶₇=1. And in 77 individuals, 74% of all the women can be classified correctly. However, using (38), the recognition rate will be 83%. Some results of medical diagnosis have been shown in Table 7.

Table 6 

The optimal results of medical diagnosis based on the distances of IFS and DIVIFS.

Table 7 

Comparison between classification of DIVIFS and existing classification calculated by similarity (𝑆=1−𝐷, where 𝐷 denotes the standardized distance).

7. Conclusion

We propose a method for the evaluation of a degree of agreement in a group of individuals by calculating distances between DIVIFS. By analyzing the variation on the degree of hesitancy, we introduce a novel DIVIFS method derived from IVIFS, which is an effective method for us to construct dynamic IVIFS model from IFS and IVIFS. And then we propose a novel ranking condition for the distance measures on IFS, IVIFS, and DIVIFS. According to the new condition, we show the construction of distance measures on DIVIFS in theory. Finally, we apply some distances on DIVIFS to medical diagnosis based on pattern recognition. The experimental results show that the DIVIFS method and its distance measures are more comprehensive and more effective than the conventional distance measures on IFS and IVIFS.

Acknowledgments

This paper is funded by the National Natural Science Foundation of China (Grant no. 71271061, Grant no. 70801020), the “Twelfth Five-Years” Philosophy and Social Sciences Planning Project of Guangdong Province (Grant no. GD12XGL14), the “Twelfth Five-Year” Education Planning Project of Guangdong Province (Grant no. 2012JK129), Major Education Foundation of Guangdong University of Foreign Studies (Grant no. GYJYZDA12011), Business Intelligence Key Team of Guangdong University of Foreign Studies (Grant no. TD1202), Science and Technology Planning Project of Guangdong Province (Grant no. 2010B010600034), and High-Level Scientific Research Project of the Department of Education of Guangdong Province (Grant no. 396-GK100017).