Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain

Applied Soft Computing Journal 82 (2019) 105572

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain ∗

Ming Tang, Yilu Long, Huchang Liao , Zeshui Xu Business School, Sichuan University, Chengdu 610064, China

highlights • • • •

We We We We

define the inclusion measure for PLTSs and propose a family of formulas. introduce normalized axiomatic definitions of some information measures. explore the relationships among four information measures. design a clustering algorithm based on the inclusion measure.

article

info

Article history: Received 12 March 2019 Received in revised form 31 May 2019 Accepted 10 June 2019 Available online 18 June 2019 Keywords: Probabilistic linguistic term set Information measure Relationships Clustering algorithm Classifying cities

a b s t r a c t The probabilistic linguistic term set is a powerful tool to express and characterize people’s cognitive complex information and thus has obtained a great development in the last several years. To better use the probabilistic linguistic term sets in decision making, information measures such as the distance measure, similarity measure, entropy measure and correlation measure should be defined. However, as an important kind of information measure, the inclusion measure has not been defined by scholars. This study aims to propose the inclusion measure for probabilistic linguistic term sets. Formulas to calculate the inclusion degrees are put forward Then, we introduce the normalized axiomatic definitions of the distance, similarity and entropy measures of probabilistic linguistic term sets to construct a unified framework of information measures for probabilistic linguistic term sets. Based on these definitions, we present the relationships and transformation functions among the distance, similarity, entropy and inclusion measures. We believe that more formulas to calculate the distance, similarity, inclusion degree and entropy can be induced based on these transformation functions. Finally, we put forward an orthogonal clustering algorithm based on the inclusion measure and use it in classifying cities in the Economic Zone of Chengdu Plain, China. © 2019 Elsevier B.V. All rights reserved.

1. Introduction

{s0 = very poor, s1 = poor, s2 = slightly poor, s3 = medium, s4 = slightly high, s5 = high, s6 = very high}. Then, the quality

In real-world decision-making environment, people may tend to use qualitative linguistic approach to express their cognitive information. In 1975, Zadeh [1]’s work made the qualitative decision making a reality. People can use linguistic terms like ‘‘high’’ or ‘‘slightly bad’’ to give their evaluation information. Since then, a great deal of fruitful achievements have been achieved in the field of qualitative decision making. In 2012, Rodriguez, Martínez and Herrera [2] proposed the concept of hesitant fuzzy linguistic term set (HFLTS). The HFLTS can overcome the limitation of traditional qualitative linguistic approach which cannot describe the case that people hesitate among several linguistic terms. For instance, suppose that a linguistic term set (LTS) is given as S =

of a car can be represented by a hesitant fuzzy linguistic element (HFLE) [3] {hS = s5 , s6 }, meaning ‘‘between high and very high’’. In an HFLE, all possible linguistic terms have equal weight or importance degree. However, in many practical situations, people may have different preferences over different linguistic terms. For instance, in the former example, an expert thinks that the quality of the car is closer to ‘‘high’’ than ‘‘very high’’. That is to say, this expert vacillates between two linguistic labels s5 and s6 . Furthermore, he/she has a preference to choose s5 than s6 . It can be interpreted as this way: ‘‘60% sure that the quality is high and 20% sure the quality is very high’’. Then, such piece of preference information can be represented as {s5 (0.6), s6 (0.2)}. This kind of linguistic evaluation can also be explained as: given a decision group containing 10 experts, each expert still uses the above LTS to evaluate the quality of the car. There are 4 experts who think the quality is high, three experts who think the quality

∗ Corresponding author. E-mail addresses: [email protected] (M. Tang), [email protected] (Y.L. Long), [email protected] (H.C. Liao), [email protected] (Z.S. Xu). https://doi.org/10.1016/j.asoc.2019.105572 1568-4946/© 2019 Elsevier B.V. All rights reserved.

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M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

is very high and other experts cannot give their evaluations. To simulate these situations exactly, Ref. [4] introduced the concept of probabilistic linguistic term set (PLTS). Using the PLTS, experts can assign different linguistic terms with different weights using the probabilities. This improves the richness and flexibility of linguistic expression information. From this point of view, the PLTS is much more flexible than the HFLTS in expressing complex cognitive linguistic information. Therefore, it is necessary to research the theory of PLTSs. Information measures, including distance measure, similarity measure, entropy measure, correlation measure and inclusion measure, are the basis of many decision-making theories [5,6]. For PLTSs, some information measures have been developed. Pang, Wang and Xu [4] first gave the Euclidean distance between PLTSs. After that, different forms of distance measures have been introduced [7–11]. The similarity degree between PLTSs can be obtained by the relationship between the distance measure and similarity measure [6]. Wu et al. [9] proposed the similarity measure between PLTSs. Liu, Jiang, and Xu [12] proposed three kinds of entropy measures for PLTSs. Lin et al. [13] also defined a correlation measure for PLTSs and used it in clustering analysis. Wu and Liao [8] developed the Pearson correlation coefficient between PLTSs. However, there are few studies that focused on the relationship among these information measures of PLTSs, particularly on the systematic transformation functions. With these transformation functions, we can derive more formulas for the information measures of PLTSs. Existing studies have researched the distance measure, similarity measure, correlation measure and entropy measure of PLTSs. However, as a kind of important information measure in set theory, the inclusion measure of PLTSs has not been investigated by scholars. Without the inclusion measure, the integrated framework of information measures of PLTSs cannot be constructed. The inclusion measure depicts the degree to which a set is contained in another set [14]. Fuzzy set inclusion was introduced by Zadeh [14] in his pioneering work in 1965. He pointed out that the inclusion is a crisp property, that is, a set is included in another set definitely or not. After that, some studies investigated the inclusion measure of fuzzy sets and proposed some approaches to calculate it [15–17]. Fuzzy inclusion measure has been used in many fields such as mathematical morphology [18], fuzzy relational databases [19], approximate reasoning [20] and image processing [21]. Up to now, many extensions of fuzzy set have been developed and their corresponding inclusion measures have also been proposed, such as the inclusion measures of the interval-valued fuzzy sets [22], type-2 fuzzy sets [22], intuitionistic fuzzy sets [23], hesitant fuzzy sets [24], typical hesitant fuzzy sets [25] and HFLTSs [6]. This study first proposes the inclusion measure for PLTSs. In addition, even though an increasing number of papers have focused on the information measures of PLTSs, these information measures usually have different forms of definitions and lack the normalized axiomatic definitions under a unified framework. This study dedicates to filling this gap by putting forward the normalized axiomatizations of different information measures of PLTSs. Based on these axiomatic definitions, this study further investigates the relationships and transformation functions among the distance, similarity, entropy and inclusion measure for PLTSs in detail. To show the applicability of the proposed inclusion measures and the relationships between different measures of PLTSs, we develop an orthogonal clustering algorithm based on the inclusion measure of PLTSs. We highlight the innovations of our study as follows: (1) We define the inclusion measure for PLTSs and propose a family of formulas to calculate the inclusion degree.

(2) To better understand and lay the foundation of the followup theories, we introduce the normalized axiomatic definitions for the distance measure, similarity measure and entropy measure. (3) Based on these axiomatic definitions, we explore the relationships and transformation functions among the distance, similarity, inclusion and entropy measure systematically. (4) We design a clustering algorithm based on the inclusion measure proposed in this study, and use it in partitioning cities in the Economic Zone of Chengdu Plain, China. The organization of this study is as follows: Section 2 provides a brief review on the PLTS-related theories, including its definition, transformation function and comparison method. In Section 3, the axiomatic definition of inclusion measure and some formulas of the inclusion measure are proposed. Section 4 gives the normalized axiomatic definition of different information measures of PLTSs. Section 5 discusses the relationships among the similarity measure, distance measure, inclusion measure and entropy measure of PLTSs. In Section 6, we give a clustering algorithm based on the inclusion measure of PLTSs. Section 7 concludes this study. 2. Preliminaries The definition of the PLTS is derived from the HFLTS. Rodríguez et al. [20] first gave the definition of the HFLTS. Later, Liao et al. [3] introduced the mathematical form of an HFLTS as HS = {⟨x, hS (x)⟩|x ∈ X } where hS (x) is called the hesitant fuzzy linguistic element (HFLE), denoting the set of possible degrees of the linguistic valuable x to S. Pang, Wang and Xu [4] proposed the concept of the PLTS by adding a probability to each linguistic term, shown as follows: Definition 1 ([4]). Let xi ∈ X be fixed and S be an LTS. A PLTS on S is HS (p) = {⟨x, hS (p)(x)⟩|x ∈ X } with

{ hS (p)(x) =

sα (l) (p(l) )|sα (l) ∈ S , p(l) ≥ 0,

l = 1, 2, . . . , L,

L ∑

} p(l) ≤ 1

(1)

l=1

where sα (l) (p(l) ) is the lth linguistic term sα (l) associated with its probability p(l) , and L is the number of linguistic terms in hS (p). For convenience, hS (p) is called the probabilistic linguistic element (PLE). Pang, Wang and Xu [4] provided process for ∑L the(lnormalization ) PLTSs. Given a PLE hS (p) with p < 1, Pang, Wang and l=1 Xu [4] suggested that the ignorance of probability should be as˙ each linguistic }term in hS (p) averagely {signed (to ∑L such that hS (p) = sα (l) (p˙ l) )|l = 1, 2, . . . , L with p˙ (l) = p(l) / l=1 p(l) . In addition, in the process of decision-making, the numbers of linguistic terms in different PLEs are usually different. Thus, it is necessary to unify different PLTSs to make them { have the same number } of linguistic terms. Let h1S (p) = h2S (p) =

{

s2α (l) (p(l) )|l = 1, 2, . . . , L1

s1α (l) (p(l) )|l = 1, 2, . . . , L1

}

and

be two PLEs. If L1 > L2 , then

we add L1 − L2 linguistic terms to h2S (p). The added linguistic terms are supposed to be the smallest ones in h2S (p) and the probabilities of them are zero. Gou and Xu [26] defined a transformation function between the HFLE and the hesitant fuzzy element (HFE). Motivated by

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

Gou and Xu [26]’s idea, Bai et al. [27] put forward a transformation function for PLEs. Suppose that hS (p) is a PLE, then a transformation function and its inverse function are defined as: g: [−τ , τ ] → [0, 1], g(hS (p)) = {(α (l) /2τ + 1/2)(p(l) )} = hγ (p) g −1 : [0, 1] → [−τ , τ ], g −1 (γ (l) ) = {s(2γ (l) −1)τ (p

( ) γ (l )

= hS (p)

g: [−τ , τ ] → [0, 1], g(α (l) ) = {(α (l) /2τ + 1/2)(p(l) )} = hγ (p),

γ ∈ [0, 1] g −1 : [0, 1] → [−τ , τ ], g −1 (hγ (p)) = {((2γ − 1)τ )(p(γ ) )|γ ∈ [0, 1]} = α (l)

(l)

sα (l) (p )|sα (l) ∈ S , p

(l)

≥ 0;

(7)

3. Inclusion measures of PLTSs Zadeh [14] defined the inclusion measure of two fuzzy sets: A ⊆ B if mA (x) ≤ mB (x) for all x ∈ X , where mA (x) and mB (x) are the membership functions of A and B. This definition is a binary relation, indicating that a set is contained in another set completely or not. In the fuzzy set theory, it is reasonable to consider an indicator to measure the inclusion degree to which a set is the subset of another set. In the fuzzy set environment, three kinds of widely used axiomatizations were introduced by Kitainik [28], Sinha and Dougherty [16] and Young [15]. Bustince et al. [29] compared and verified these three axiomatizations. In this section, we define the axiomatic definition of inclusion measure for PLTSs using Young [15]’s axiomatization based on Ref. [29]. Before that, we first put forward the inclusion relation between PLEs based on the ideas in Refs. [14,18,21]. Definition 4. Let h1S (p) and h2S (p) be two PLEs on a nonempty set X . Then, the inclusion relation between h1S (p) and h2S (p) is defined as: (8)

where g(·) is a function defined in Definition 2.

})

Definition 5. Let h1S (p), h2S (p) and h3S (p) be three PLEs. Then, In is

p(l) ≤ 1; α (l) ∈ [−τ , τ ]v

called an inclusion measure of PLEs if it satisfies the following conditions:

l=1

= {γ (l) |γ (l) = g(α (l) )} = hγ (p)

(In1) In(h1S (p), h2S (p)) = 1 if and only if h1S (p) ⊆ h2S (p); 1

({ G−1 : Θ → Φ , G−1 (hγ (p)) = G−1

l = 1, 2, . . . , L,

γ (l) ∈g(L)

h1S (p) ⊆ h2S (p) iff g(α 1(l) ) ≤ g(α 2(l) ), l = 1, 2, . . . , L

({

L ∑

) {(1 − γ (l) )(p(l) )} , l = 1, 2, . . . , L

∪

(3)

Then, the translation functions between hγ (p) and hS (p) are given as follows:

L ∑

(

(2)

Definition 2. Let S = {sα |α = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. Let hS (p) be a PLE with α (l) being the subscript of the linguistic term sα (l) in hS (p). Then, we have the following two equivalent transformation functions:

l = 1, 2, . . . , L,

The complement of a PLE is defined in Ref. [26]: 1

However, in these transformation functions, g is not a mapping from [−τ , τ ] to [0, 1] since hS (p) is a PLE instead of its subscript. Therefore, in this study, we rewrite the transformation functions below:

G: Φ → Θ , G(hS (p)) = G

σ (h1S (p)) = σ (h2S (p)), there is no difference between h1S (p) and h2S (p), and their relationship can be denoted as h1S (p) ∼ h2S (p). hS (p) = g −1

)|γ (l) ∈ [0, 1]}

3

(In2) In(h1S (p), hS (p)) = 0 if and only if G(h1S (p)) = {1}; (In3) If h1S (p) ⊆ h2S (p) ⊆ h3S (p), then In(h3S (p), h1S (p)) ≤ In(h2S (p),

sα (l) (p(l) )|sα (l) ∈ S , p(l) ≥ 0;

h1S (p)) and In(h3S (p), h1S (p)) ≤ In(h3S (p), h2S (p)).

}) p(l) ≤ 1

In analogous to the union and intersection of HFLEs [30], we define the union and intersection of PLEs:

l=1

h1S (p) ∪ h2S (p) = {sα (l) (p(l) )|g(α (l) ) ∈ (G(h1S (p)) ∪ G(h2S (p)))}

= {sα(l) |α (l) = g −1 (γ (l) )} = hS (p)

= {sα(l) (p(l) )|g(α (l) ) ≥ min(g(α 1(l) ), g(α 2(l) ))}

(4) To compare PLEs, Pang, Wang and Xu [4] proposed the score and the deviation degree of a PLE.

h1S (p) ∩ h2S (p) = {sα (l) (p(l) )|g(α (l) ) ∈ (G(h1S (p)) ∩ G(h2S (p)))}

Definition 3 ([4]). Let hS (p) = s(αl) (p(l) )|s(αl) ∈ S , l = 1, 2, . . . , L be a PLE, and α (l) be the subscript of s(αl) . Then, the score of hS (p)

Next, we introduce several formulas to calculate the inclusion degree between PLEs. Goguen [31] gave an inclusion degree between two fuzzy sets A and B as:

{

}

is defined as:

S(hS (p)) = sα , w here α =

L ∑

α (l) p(k) /

L ∑

p(l)

(5)

l=1

l=1

The deviation degree of hS (p) is defined as:

( σ (hS (p)) =

L ∑ l=1

)1/2 (p(l) (α (l) − α ))2

/

L ∑

p(l)

= {sα(l) (p(l) )|g(α (l) ) ≤ max(g(α 1(l) ), g(α 2(l) ))}

InGog (A, B) =

1∑ n

min(1, 1 − mA (x) + mB (x)), |X | = n

(9)

(10)

x

Similar to Eq. (10), we can define the inclusion degree between two PLEs h1S (p) and h2S (p): (6)

l=1

For two PLEs h1S (p) and h2S (p), S(h1S (p)) > S(h2S (p)) indicates that 1 hS (p) is superior to h2S (p), denoted as h1S (p) > h2S (p); S(h1S (p)) < S(h2S (p)) indicates that h1S (p) is inferior to h2S (p), denoted as h1S (p) < h2S (p). When S(h1S (p)) = S(h2S (p)), if σ (h1S (p)) > σ (h2S (p)), then

h1S (p) < h2S (p); if σ (h1S (p)) < σ (h2S (p)), then h1S (p) > h2S (p); if

In1 (h1S (p), h2S (p)) =

L 1∑

L

l=1

min(1, 1 − g(α 1(l) ) + g(α 2(l) )),

(11)

l = 1, 2, . . . , L Theorem 1. The measure defined in Eq. (11) is an inclusion measure for PLEs.

4

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

∑L

1(l) )+ Proof. (In1) If In1 (h1S (p), h2S (p)) = 1L l=1 min(1, 1 − g(α 1 2(l) 1(l) 2(l) g(α )) = 1, then g(α ) = g(α ), which implies hS (p) ⊆ h2S (p). Conversely, when h1S (p) ⊆ h2S (p), then g(α 1(l) ) ≤ g(α 2(l) ) for l = ∑L 1(l) 1, 2, . . . , L, which implies 1L ) + g(α 2(l) )) = l=1 min(1, 1 − g(α 1. ∑L 1 1(l) (In2) In1 (h1S (p), hS (p)) = 1L ) + 1 − l=1 min(1, 1 − g(α 1(l) 1(l) g(α )) = 0 ⇔ g(α ) = 1 ⇔ G(hS (p)) = {1} for l = 1, 2 · · · , L. (In3) If h1S (p) ⊆ h2S (p) ⊆ h3S (p), then g(α 1(l) ) ≤ g(α 2(l) ) ≤ ∑L g(α 3(l) ), thus g(α 3(l) ) − g(α 1(l) ) ≥ g(α 2(l) ) − g(α 1(l) ), 1L l=1 ∑ L 1(l) 2(l) min(1 , 1 − g( α ) + g( α )). min(1, 1−g(α 1(l) )+g(α 3(l) )) ≤ 1L l=1 Then, we have In(h3S (p), h1S (p)) ≤ In(h2S (p), h1S (p)). Similarly, it

holds for , proof of Theorem 1. In(h3S (p)

h1S (p))

≤

,

In(h3S (p)

h2S (p)).

This completes the

■

Inspired by the work of Zeng and Guo [32], we define another inclusion degree between two PLEs h1S (p) and h2S (p) as: In2 (h1S (p) g(α

2(l)

,

h2S (p))

=1−

L 1∑

L

|g(α

1(l)

) − min(g(α

1(l)

),

l=1

(12)

Theorem 2. The measure defined in Eq. (12) is an inclusion measure for PLEs. 1 L

∑L

1(l) ) − min(g(α 1(l) ), l=1 |g(α

≤ g(α 2(l) ) ⇔ h1S (p) ⊆ h2S (p). ∑L 1 1(l) ) − min(g(α 1(l) ), 1 − (In2) In2 (h1S (p), hS (p)) = 1 − 1L l=1 |g(α 1(l) g(α ))|= 0 ⇔ g(α 1 (l)) = 1 ⇔ G(hS (p)) = {1}. (In3) If h1S (p) ⊆ h2S (p) ⊆ h3S (p), then g(α 1(l) ) ≤ g(α 2(l) ) ≤ g(α 3(l) ). ∑L 1 1 3(l) ) − min(g(α 3(l) ), g(α 1(l) ))| ≥ Thus, l=1 |g(α L L ∑L 3 2(l) 2(l) 1(l) ) − min(g(α ), g(α ))|, which implies In(hS (p), l=1 |g(α

h1S (p)) ≤ In(h2S (p), h1S (p)). Similarly, it holds for In(h3S (p), h1S (p)) ≤ ■ In(h3S (p), h2S (p)). This completes the proof of Theorem 2.

Kaufmann [33] introduced an inclusion measure between two fuzzy sets A and B:

(13)

Similar to Eq. (13), we can define an inclusion measure for PLEs

L

1 L

l=1

∑L

l=1

g(α 2(l) )

max(g(α 1(l) ), g(α 2(l) ))

L

g(α 1(l) ) =

l=1

L 1∑

L

g(α 2(l) )

l=1

, other w ise

(15) Proof. The proof of Eq. (15) is similar to that of Eq. (12). The following theorem can be obtained easily: Theorem 4. Let In′ and In′′ be two inclusion measures for PLEs. Then, 1

2

(1) In′ (h1S (p), h2S (p)) · In′′ (hS (p), hS (p)) is an inclusion measure for PLEs;{ } 1 2 (2) max In′ (h1S (p), h2S (p)), In′′ (hS (p), hS (p)) is an inclusion measure { for PLEs;

1

}

2

sure for PLEs; 1 2 (4) λ1 In′ (h1S (p), h2S (p)) + λ2 In′′ (hS (p), hS (p)) is an inclusion measure { for PLEs, where λ1 , λ2 ∈ [0, 1], λ1 + λ2 = 1; } 1

2

(5) max In′ (h1S (p), h2S (p)) + In′′ (hS (p), hS (p)) − 1, 0 clusion { measure for PLEs;

1

2

(6) min In′ (h1S (p), h2S (p)) + In′′ (hS (p), hS (p)), 1

}

is an in-

is an inclusion

measure for PLEs.

Theorem 5. Let h1S (p), h2S (p) and h3S (p) be three PLEs. The following formulas hold: (1) In(h1S (p) ∩ h2S (p), h3S (p)) h3S (p))}; (2) In(h1S (p) ∪ h2S (p), h3S (p)) h3S (p))}; (3) In(h1S (p), h2S (p) ∩ h3S (p)) h3S (p))}; (4) In(h1S (p), h2S (p) ∪ h3S (p)) h3S (p))}.

= max{In(h1S (p), h3S (p)), In(h2S (p), = min{In(h1S (p), h3S (p)), In(h2S (p), = min{In(h1S (p), h2S (p)), In(h1S (p), = max{In(h1S (p), h2S (p)), In(h1S (p),

Proof. (1) For three PLEs, their containment relationships have six cases: (a) h1S (p) ⊆ h2S (p) ⊆ h3S (p) ⇒ In(h1S (p), h3S (p)) = 1, In(h2S (p), h3S (p)) = 1, h1S (p) ∩ h2S (p) = h1S (p)

⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = In(h1S (p), h3S (p)) = 1 ⇒ In(h1S (p) ∩ h2S (p), h3S (p))

as: In3 (h1S (p), h2S (p))

=

⎪ ⎪ ⎪ ⎪ ⎩ 1 ∑L

L 1∑

The theorem can be proved directly.

⇔ min(g(α 1(l) ), g(α 2(l) )) = g(α 1(l) ) ⇔ g(α 1(l) )

⎧∑ |A ∩ B| ⎪ A (x), mB (x)) ⎨ x min(m ∑ , A ̸= ∅ = |A| InKos (A, B) = x mA (x) ⎪ ⎩ 1 ,A = ∅

, iff

(3) min In′ (h1S (p), h2S (p)), In′′ (hS (p), hS (p)) is an inclusion mea-

))|, l = 1, 2, . . . , L

Proof. (In1) In2 (h1S (p), h2S (p)) = 1 − g(α 2(l) ))|= 1

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨1

= max{In(h1S (p), h3S (p)), In(h2S (p), h3S (p))};

⎧ ∑L L 1 1(l) ⎪ ), g(α 2(l) )) 1 ∑ ⎪ l=1 min(g(α L ⎪ , g(α 1(l) ) ̸ = 0 ⎪ ∑ L ⎨ 1 L g(α 1(l) ) L

⎪ ⎪ ⎪ ⎪ ⎩1

l=1

l=1

,

L 1∑

L

(b) h1S (p) ⊆ h3S (p) ⊆ h2S (p) ⇒ In(h1S (p), h3S (p)) = 1, 0 ≤ (14)

g(α 1(l) ) = 0

l=1

Theorem 3. The measure defined in Eq. (14) is an inclusion measure for PLEs. Proof. The proof of Eq. (14) is similar to that of Eq. (12). Corollary 1. The following function is also an inclusion measure for PLEs: In4 (h1S (p), h2S (p))

In(h2S (p), h3S (p)) ≤ 1, h1S (p) ∩ h2S (p) = h1S (p)

⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = In(h1S (p), h3S (p)) ⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = max{In(h1S (p), h3S (p)), In(h2S (p), h3S (p))}; (c) h2S (p) h3S (p))

⊆ h1S (p) ⊆ h3S (p) ⇒ In(h1S (p), 3 2 1 2 2 = 1, In(hS (p), hS (p)) = 1, hS (p) ∩ hS (p) = hS (p)

⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = In(h2S (p), h3S (p)) = 1 ⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = max{In(h1S (p), h3S (p)), In(h2S (p), h3S (p))};

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

5

Table 1 Decision making matrix with PLEs. a1 a2 a3 a4

c1

c2

c3

c4

{s−2 (0.3), s−1 (0.7)} {s2 (0.4), s3 (0.6)} {s1 (0.8), s2 (0.2)} {s2 (0.5), s3 (0.5)}

{s−1 (0.2), s0 (0.8)} {s0 (0.5), s1 (0.5)} {s−2 (0.5), s−1 (0.5)} {s0 (0.3), s1 (0.7)}

{s2 (0.5), s3 (0.5)} {s1 (0.7), s2 (0.3)} {s0 (0.7), s1 (0.3)} {s1 (0.8), s2 (0.2)}

{s1 (0.4), s2 (0.6)} {s1 (0.9), s2 (0.1)} {s−1 (0.7), s0 (0.3)} {s1 (0.4), s2 (0.6)}

(d) h2S (p) ⊆ h3S (p) ⊆ h1S (p) ⇒ In(h2S (p), h3S (p)) = 1, 0 ≤ In(h1S (p), h3S (p)) ≤ 1, h1S (p) ∩ h2S (p) = h2S (p)

⇒ ⇒

In(h1S (p) In(h1S (p)

,

,

h2S (p) h3S (p)) In(h2S (p) h3S (p)) h2S (p) h3S (p)) 1 In(hS (p) h3S (p)) In(h2S (p) h3S (p))

∩

=1

,

∩

,

= max{

=

,

,

In(h1S (p), h3S (p)) ≤ 1 and

In(h2S (p), h3S (p)) ≤ In(h1S (p), h3S (p)), h1S (p) ∩ h2S (p) = h1S (p)

⇒ ⇒

∩

,

∩

,

,

h2S (p) h3S (p)) In(h1S (p) h3S (p)) h2S (p) h3S (p)) 1 In(hS (p) h3S (p)) In(h2S (p) h3S (p))

,

= max{

=

,

,

};

(f) h3S (p) ⊆ h2S (p) ⊆ h1S (p) ⇒ 0 ≤ In(h2S (p), h3S (p)) ≤ 1, 0 ≤ In(h1S (p), h3S (p)) ≤ 1 and

In(h1S (p), h3S (p)) ≤ In(h2S (p), h3S (p)), h1S (p) ∩ h2S (p) = h2S (p)

⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = In(h2S (p), h3S (p)) ⇒ In(h1S (p) ∩ h2S (p), h3S (p)) = max{In(h1S (p), h3S (p)), In(h2S (p), h3S (p))}. The proof for the rest formulas is similar. This completes the proof of Theorem 5. ■ We have investigated the inclusion measures for PLEs. The inclusion measure for PLTSs can be constructed by aggregating the inclusion degrees of PLEs. Below we propose several inclusion measures for PLTSs. Theorem 6. Let HSA (p) and HSB (p) be two PLTSs. Then, the following inclusion measures can be obtained:

∑n

(1) In1 (HSA (p), HSB (p)) = 1n (In(hA (pi )(xi ), hB (pi )(xi ))); ∑n i=1 (2) In2 (HSA (p), HSB (p)) = λ i=1 i (In(h ∑n A (pi )(xi ), hB (pi )(xi ))), where λi is a positive number with i=1 λi = 1; ∑ (3)

In3 (HSA (p)

,

HSB (p))

=

αi In(hA (pi )(xi ),hB (pi )(xi )) i=1,2,...,n∑ i=1,2,...,n

a1 a2 a3 a4

a1

a2

a3

a4

1 0.847 0.939 0.825

0.916 1 0.966 0.920

0.888 0.869 1 0.873

0.964 0.947 0.95 1

};

(e) h3S (p) ⊆ h1S (p) ⊆ h2S (p) ⇒ 0 ≤ In(h2S (p), h3S (p)) ≤ 1, 0 ≤

In(h1S (p) In(h1S (p)

Table 2 Results obtained by the first inclusion measure in Theorem 6.

αi

.

The theorem can be proved according to Definition 5. Example 1. Suppose that a financial company needs to move to a new location. There are four possible alternatives {a1 , a2 , a3 , a4 } to be invested and four criteria (c1 : running cost; c2 : market demand; c3 : government policy; c4 : labor condition) to be considered. The weight vector of these four criteria is ω = (0.3, 0.3, 0.2, 0.2)T . The think-tank experts use complex linguistic expressions to evaluate these four alternatives. The linguistic assessments can be transformed to PLEs and listed in Table 1. Here, we use In1 in Theorem 6 to compute the inclusion degrees and present the results in Table 2.

4. Normalized axiomatic definitions for distance, similarity and entropy measure Definitions of distance measure, similarity measure, entropy measure of PLTSs in different forms have been proposed. However, these definitions do not have normalized forms. In this section, we describe the normalized axiomatic definitions of these three kinds of information measures under a unified rule so as to discuss their relationships in the next section. 4.1. The normalized distance measure of PLEs The distance measure describes the deviation degree between two sets. A short distance denotes a low deviation degree between two sets. Next, we put forward the axiomatic concept of distance measure between PLTSs. Definition 6. Let h1S (p), h2S (p) and h3S (p) be three PLEs. Then, d is called a distance measure between PLEs if it satisfies the following conditions: (d1) 0 ≤ d(h1S (p), h2S (p)) ≤ 1; (d2) d(h1S (p), h2S (p)) = d(h2S (p), h1S (p));

(d3) d(h1S (p), h2S (p)) = 0 if and only if h1S (p) ∼ h2S (p); (d4) d(h1S (p), h2S (p)) = 1 if and only if G(h1S (p)) = {0} and G(h2S (p)) = {1} or G(h1S (p)) = {1} and G(h2S (p)) = {0}; (d5) If g(α 1(l) ) ≤ g(α 2(l) ) ≤ g(α 3(l) ), then d(h1S (p), h2S (p)) ≤ d(h1S (p), h3S (p)) and d(h2S (p), h3S (p)) ≤ d(h1S (p), h3S (p)). The existing distance measures between PLEs are based on this axiomatic definition. Pang, Wang and Xu [4] first gave the Euclidean distance between two PLEs:

  L ∑ 1 2 d(h (p), h (p)) = √ (p1(l) α 1(l) − p2(l) α 2(l) )2 /L S

S

(16)

l=1

Zhang et al. [7] introduced the Hamming distance between PLEs. However, Zhang et al. [7]’s measure has a limitation that the value is zero when p1(l) = p2(l) or α 1(l) = α 2(l) . Thus, Wu and Liao [8] proposed a new Euclidean distance measure to overcome this limitation. After that, Wu et al. [9] introduced an adjustment approach to operate PLEs. Based on the adjustment method, they proposed three kinds of distances. There are several other distance measures that have similar forms to these distances. 4.2. The normalized similarity measure of PLEs The similarity measure describes the closeness degree between two sets. What it aparts from the distance measure is that a high similarity degree denotes a low deviation degree between two sets.

6

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

Definition 7. Let h1S (p), h2S (p) and h3S (p) be three PLEs. Then, ρ is called a similarity measure between PLEs if it satisfies the conditions as follows: (s1) 0 ≤ ρ (h1S (p), h2S (p)) ≤ 1; (s2) ρ (h1S (p), h2S (p)) = ρ (h2S (p), h1S (p)); (s3) ρ

, ,

(h1S (p) (h1S (p)

h2S (p)) 1 if h2S (p)) 0 or G(h1S (p)) 1(l) 2(l)

(s4) ρ G(h2S (p)) = {1}

= =

and only if

h1S (p)

∼

h2S (p);

G(h1S (p))

if and only if = {0} and = {1} and G(h2S (p)) = {0}; (s5) If g(α ) ≤ g(α ) ≤ g(α 3(l) ), then ρ (h1S (p), h2S (p)) ≥ ρ (h1S (p), h3S (p)) and ρ (h2S (p), h3S (p)) ≥ ρ (h1S (p), h3S (p)). Wu et al. [9] also gave an axiomatic definition for the similarity measure of PLEs. The difference between their definition and Definition 7 is that Wu et al. [9]’s definition does not have the fifth condition. 4.3. The normalized entropy measure of PLEs The entropy measure of a fuzzy set was firstly developed by Zadeh [14] to describe the fuzziness degree of the set. Kaufmann [33] defined an entropy measure of fuzzy sets by calculating the metric distance between its membership function and that of its nearest crisp set. Then, Yager [34] gave the entropy measure of fuzzy sets in terms of a lack of distinction between the set and its complement set. Afterwards, many scholars have investigated the entropy measure for different extensions of fuzzy set [30,35,36]. Based on the expectation function, Liu, Jiang and Xu [12] introduced the definition of entropy measure for PLTSs and put forward three kinds of entropy measures: the fuzzy entropy, the hesitant entropy and the total entropy. Next, we give the normalized axiomatic definition of entropy measure for PLEs. Definition 8. Let h1S (p) and h2S (p) be two PLEs. Then, E is called an entropy measure of PLEs if it satisfies the following conditions: (e1) 0 ≤ E(h1S (p)) ≤ 1; (e2) E(h1S (p)) = 0 if and only if G(h1S (p)) = {0} or G(h1S (p)) = {1}; (e3) E(h1S (p)) = 1 if and only if g(α 1(l) ) + g(α 1(L1 −l+1) ) = 1;

(e4) E(h1S (p)) ≤ E(h2S (p)) if g(α 1(l) ) ≤ g(α 2(l) ) for g(α 2(l) ) + g(α 2(L−l+1) ) ≤ 1, or if g(α 1(l) ) ≥ g(α 2(l) ) for g(α 1(l) ) + g(α 1(L−l+1) ) ≥ 1; 1 (e5) E(h1S (p)) = E(hS (p)).

It is observed that Szmidt, Kacprzyk and Bujnowski [37] introduced a novel information measure called the knowledge measure for IFSs. Knowledge is related to the useful information considered in a particular context [38]. It measures the amount of knowledge contained in an IFS, and is regarded as the dual measure of the entropy measure.

Definition 9. Let HS1 (p) and HS2 (p) be two PLEs. Then, the correlation coefficient between these two PLTSs meets the following properties: (c1) C (HS1 (p), HS1 (p)) = 1; (c2) If HS1 (p) = HS2 (p), then C (HS1 (p), HS2 (p)) = 1;

(c3) C (HS1 (p), HS2 (p)) = C (HS2 (p), HS1 (p)).

For other theorems and properties about the correlation measure of PLTSs, please refer to Ref. [13]. In Section 6, we will use Ref. [13]’s correlation measure to make a comparison. 5. Relationships among distance, similarity, entropy and inclusion measures of PLTSs In this section, we investigate the relationships and transformation functions among distance, similarity, entropy and inclusion measures of PLTSs (PLEs) based on their axiomatizations. A number of studies have paid attention to the relationships among the information measures of fuzzy sets and their extensions such as hesitant fuzzy sets (HFSs) [42], interval-valued fuzzy sets (IVFS) [43] and intuitionistic fuzzy sets (IFSs) [44]. For deeply understanding, the existing related works are summarized in Table 3. The second column is the research objects and the third column presents information measures discussed for the researched objects. However, as far as we know, there is no paper that systematically investigates the relationships among the information measures of PLTSs. Most related studies only focused on the transformation functions among two or three information measures. This study investigates mutual conversion relations of four kinds of information measures. 5.1. The relationship between distance and similarity measures Distance and similarity measures both describe the deviation degrees between two data objects. However, these two measures’ perspectives are just opposite. A shorter distance and a higher similarity denote a lower degree of deviation. Theorem 7. Suppose that d is a distance measure between two PLEs h1S (p) and h2S (p). Then, ρ (h1S (p), h2S (p)) = 1 − d(h1S (p), h2S (p)) is a similarity measure between them. The proof of Theorem 7 is obvious based on the axiomatic definitions of distance and similarity measures of PLEs. Based on Eq. (8) and Theorem 7, we can obtain that

  L ∑ 1 2 ρ (h (p), h (p)) = 1 − √ (p1(l) α 1(l) − p2(l) α 2(l) )2 /L S

S

(17)

l=1

4.4. The normalized correlation measure of PLTSs The correlation measure is also an important measure of PLEs. Correlation measure denotes the dependency of two data objects. It is often used to measure the relationships of data objects. The biggest difference between the correlation measure and other measures is that the range of correlation coefficient falls in [−1, 1]. In the field of fuzzy set, the correlation measure has also been widely researched. The correlation measure of many extensions of fuzzy set have been defined, such as intuitionistic fuzzy sets [39], hesitant fuzzy sets [40] and HFLTSs [3]. For PLTSs, Lin et al. [13] proposed the correlation coefficient of PLTSs based on the score values of PLTSs. Zhang, Xu, and Jia [41] put forward a hybrid correlation measure of PLTSs. In addition, Wu et al. [9] proposed a Pearson correlation coefficient based on the expectation function of PLTSs.

is a similarity measure between h1S (p) and h2S (p). Theorem 8. Given a real function f : [0, 1] → [0, 1]. If f is a strictly monotonically decreasing function and d is a distance measure between two PLEs h1S (p) and h2S (p), then,

ρ (h1S (p), h2S (p)) =

f (d(h1S (p), h2S (p))) − f (1) f (0) − f (1)

(18)

is a similarity measure between them. Proof. It is obvious for (s1) and (s2). (s3) ρ (h1S (p), h2S (p)) = 1 ⇔ f (d(h1S (p), h2S (p))) − f (1) = f (0) − f (1) ⇔ d(h1S (p), h2S (p)) = 0

⇔ h1S (p) ∼ h2S (p).

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

7

Table 3 A summarization of related studies on the relationships among information measures. Ref.

Generalization forms

Information measures

Young [15] Zhang and Zhang [43] Zeng and Li [45] Zeng and Guo [32] Zeng and Li [46] Zhang, Zhang and Mei [35] Mitchell [39] Zhang et al. [47] Das, Guha and Mesiar [48] Zhang et al. [38] Zhang et al. [49] Farhadinia [42] Zhang and Yang [25] Gou, Xu and Liao [36] Tang and Liao [6] Farhadinia [50] Wang and Qu [51]

FS FS FS IVFS IVFS IVFS IFS IFS IFS IVIFS IVIFS HFS, IVHFS THFS HFLTS HFLTS HFLTS VSS

Inclusion, entropy Inclusion, distance, similarity Inclusion, similarity, entropy Distance, similarity, inclusion, entropy Similarity, entropy Entropy, similarity Similarity, entropy Distance, entropy, inclusion Distance, entropy, knowledge Entropy, similarity, inclusion Entropy, similarity, inclusion Similarity, distance, entropy Inclusion, similarity, entropy Entropy, similarity Distance, similarity, entropy, inclusion Entropy, distance, similarity Entropy, similarity, distance

Note. FS: fuzzy sets; IVFS: interval-valued fuzzy sets; IFS: intuitionistic fuzzy sets; IVIFS: interval-valued intuitionistic fuzzy sets; HFS: hesitant fuzzy sets; THFS: typical hesitant fuzzy sets; HFLTS: hesitant fuzzy linguistic term sets; VSS: vague soft sets.

(s4) ρ (h1S (p), h2S (p)) = 0 ⇔ f (d(h1S (p), h2S (p))) − f (1) = 0 ⇔ d(h1S (p), h2S (p)) = 1 ⇔ G(h1S (p)) = {0} and G(h2S (p)) = {1} or G(h1S (p)) = {1} and G(h2S (p)) = {0}. (s5) Since g(α 1(l) ) ≤ g(α 2(l) ) ≤ g(α 3(l) ), then we have d(h1S (p),

, , ≤ , and ≤ Because f is a strictly monotonically decreasing function, it follows f (d(h1S (p), h2S (p))) ≥ f (d(h1S (p), h3S (p))), f (d(h2S (p), h3S (p))) ≥ d(h1S (p)

h2S (p))

,

h3S (p))

d(h2S (p)

f (d(h1S (p) h3S (p))). Therefore, h3S (p)) and (h2S (p) h3S (p)) proof.

ρ

,

h3S (p))

we have ρ

≥ ρ (h1S (p),

d(h1S (p)

,

h3S (p)).

≥ ρ

,

(h1S (p) h2S (p)) (h1S (p) 3 hS (p)). This completes the

■

One simple function of f can be given as f (x) = 1 − x. Theorem 7 belongs to this situation. If we choose the function f as f (x) = 1 − x2 , then the similarity measure is obtained as:

ρ2 (h1S (p), h2S (p)) = 1 − d2 (h1S (p), h2S (p))

(19)

We can also use the exponential operation in the function f . For example, if f is chosen as f (x) = 1 − xex−1 , then, the similarity measure is given as:

ρ3 (h1S (p), h2S (p)) = 1 − d(h1S (p), h2S (p))e1−d(hS (p),hS (p)) 1

2

(20)

5.2. The transformation from distance and similarity measure to entropy measure This section introduces a family of transformation functions from distance and similarity measure to entropy measure of PLEs. Theorem 9. Suppose that d and ρ are the distance and similarity measures of a PLE h1S (p), respectively. Then, E(hS (p)) = ρ (hS (p), hS (p)) = 1 − d(hS (p), hS (p)) is an entropy measure of it. Proof. (E1) 0 ≤ ρ (hS (p), hS (p)) ≤ 1 ⇔ 0 ≤ E(hS (p)) ≤ 1. (E2) E(hS (p)) = 0 ⇔ ρ (hS (p), hS (p)) = 0 ⇔ G(hS (p)) = {0} or G(hS (p)) = {1}. (E3) E(hS (p)) = 1 ⇔ ρ (hS (p), hS (p)) = 1 ⇔ hS (p) ∼ hS (p) ⇔ g(α 1(l) ) + g(α 1(L−l+1) ) = 1. (E4) If g(α 1(l) ) ≤ g(α 2(l) ) for g(α 2(l) ) + g(α 2(L−l+1) ) ≤ 1, we have g(α 1(l) ) ≤ g(α 2(l) ) ≤ 1 − g(α 2(L−l+1) ) ≤ 1 − g(α 1(L−l+1) ). Through 1 2 (s5) of Definition 7, we have ρ (h1S (p), hS (p)) ≤ ρ (h2S (p), hS (p)). 1 2 1(l) Thus, E(hS (p)) ≤ E(hS (p)). It also holds when g(α ) ≥ g(α 2(l) ) for g(α 1(l) ) + g(α 1(L−l+1) ) ≥ 1.

(E5) E(hS (p)) = ρ (hS (p), hS (p)) = ρ (hS (p), hS (p)) = E(hS (p)).

The proof regarding the distance measure is similar to the above process. Thus, we complete the proof of Theorem 9. ■ Corollary 2. Suppose that d and ρ are the distance and similarity ρ (hS (p),hS (p)) measures of a PLE hS (p), respectively. Then, E(hS (p)) =

=

1−d(hS (p),hS (p)) 1+d(hS (p),hS (p))

2−ρ (hS (p),hS (p))

is an entropy measurer of it.

The proof of Corollary 2 is similar to that of Theorem 9. We should note that in Theorem 9, f (x) = 1 − x is chosen as the relationship function. If we use other forms of relationship functions, different results will be obtained. Theorem 10. Suppose that d and ρ are the distance and similarity measures of a PLE hS (p), respectively. Then, E(hS (p)) = ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) = 1 − d(hS (p) ∩ hS (p), hS (p) ∪ hS (p)) is an entropy measure of it. Proof. (E1) 0 ≤ ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) ≤ 1 ⇔ 0 ≤ E(hS (p)) ≤ 1. (E2) E(hS (p)) = 0 ⇔ ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) = 0 ⇔ G(hS (p) ∩ hS (p)) = {0}, G(hS (p) ∪ hS (p)) = {1} ⇔ G(hS (p)) = {0} or G(hS (p)) = {1}. (E3) E(hS (p)) = 1 ⇔ ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) = 1

⇔ hS (p) ∩ hS (p) ∼ hS (p) ∪ hS (p) ⇔ hS (p) ∼ hS (p) ⇔ g(α 1(l) ) + g(α 1(L−l+1) ) = 1. (E4) If g(α 1(l) ) ≤ g(α 2(l) ) for g(α 2(l) ) + g(α 2(L−l+1) ) ≤ 1, we have g(α 1(l) ) ≤ g(α 2(l) ) ≤ 1 − g(α 2(L−l+1) ) ≤ 1 − g(α 1(L−l+1) ). Suppose 1 1 2 that h1S (p) ∩ hS (p) = hAS (p), h1S (p) ∪ hS (p) = hBS (p), h2S (p) ∩ hS (p) = 2

hCS (p), h2S (p) ∪ hS (p) = hDS (p). Then, we have g(α A(l) ) = g(α 1(l) ),

g(α ) = 1 − g(α 1(L−l+1) ), g(α C (l) ) = g(α 2(l) ), g(α D(l) ) = 1 − g(α 2(L−l+1) ), g(α A(L−l+1) ) = g(α 1(L−l+1) ), g(α C (L−l+1) ) = g(α 2(L−l+1) ), 1 − g(α B(L−l+1) ) = g(α 1(l) ), 1 − g(α D(L−l+1) ) = g(α 2(l) ). Therefore, g(α A(l) ) ≤ g(α C (l) ) ≤ g(α D(l) ) ≤ g(α B(l) ), g(α A(L−l+1) ) ≥ g(α C (L−l+1) ) ≥ g(α D(L−l+1) ) ≥ g(α B(L−l+1) ). Through (s5) of Def1 1 inition 7, we have ρ (h1S (p) ∩ hS (p), h1S (p) ∪ hS (p)) ≤ ρ (h2S (p) ∩ B(l)

2

2

hS (p), h2S (p) ∪ hS (p)), and e(h1S (p)) ≤ e(h2S (p)). It also holds when g(α 1(l) ) ≥ g(α 2(l) ) for g(α 1(l) ) + g(α 1(L−l+1) ) ≥ 1.

(E5) E(hS (p)) = ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) = ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) = E(hS (p)). The proof regarding the distance measure is similar to the above process. Thus, we complete the proof of Theorem 10. ■

8

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

Corollary 3. Suppose that d and ρ are the distance and similarity measures of a PLE hS (p), respectively. Then, E(hS (p)) =

=

ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p))

Theorem 12. Suppose that ( E is an entropy ) measure for two PLEs h1S (p) and h2S (p). Then, E ξ (h1S (p), h2S (p)) is a similarity measure between them, where

ξ (h1S (p), h2S (p)) ( |g(α 1(1) ) − g(α 2(1) )| + 1 |g(α 1(2) ) − g(α 2(2) )| + 1 = , ,..., 2 2 ) |g(α 1(L) ) − g(α 2(L) )| + 1 .

2 − ρ (hS (p) ∩ hS (p), hS (p) ∪ hS (p)) 1 − d(hS (p) ∩ hS (p), hS (p) ∪ hS (p)) 1 + d(hS (p) ∩ hS (p), hS (p) ∪ hS (p))

is an entropy measure of it.

2

The proof of Corollary 3 is similar to that of Theorem 10. Theorem 11. Suppose that d and ρ are the distance and similarity measures of a PLE hS (p), respectively. Then, E(hS (p)) = d(hS (p)∪hS (p),S) = 1−ρ (hS (p)∪hS (p),S) is an entropy measure of it. d(hS (p)∩hS (p),S)

1−ρ (hS (p)∩hS (p),S)

Proof. (s1) and (s2) are obvious. (s3) E ξ (h1S (p), h2S (p)) = 1 ⇔

(

)

Proof. (E1) g(hS (p) ∪ hS (p)) ≥ g(hS (p) ∩ hS (p)) ⇒ d(hS (p) ∪ hS (p), S) ≤ d(hS (p) ∩ hS (p), S)

⇒0≤

d(hS (p) ∪ hS (p), S) d(hS (p) ∩ hS (p), S)

≤ 1 ⇒ 0 ≤ E(hS (p)) ≤ 1.

(E2) E(hS (p)) = 0 ⇔ d(hS (p) ∪ hS (p), S) = 0 ⇔ hS (p) ∪ hS (p) = S ⇔ G(hS (p)) = {1} or G(hS (p)) = {0}. (E3) E(hS (p)) = 1 ⇔ d(hS (p) ∪ hS (p), S) = d(hS (p) ∩ hS (p), S) ⇔ hS (p) ∪ hS (p) = hS (p) ∩ hS (p)

⇔ hS (p) ∼ hS (p) ⇔ g(α 1(l) ) + g(α 1(L−l+1) ) = 1. (E4) If g(α ) ≤ g(α ) for g(α ) + g(α ) ≤ 1, we have g(α 1(l) ) ≤ g(α 2(l) ) ≤ 1 − g(α 2(L−l+1) ) ≤ 1 − g(α 1(L−l+1) ). Suppose 1 1 2 that h1S (p) ∩ hS (p) = hAS (p), h1S (p) ∪ hS (p) = hBS (p), h2S (p) ∩ hS (p) = 1(l)

2(l)

2(l)

2(L−l+1)

2

hCS (p), h2S (p) ∪ hS (p) = hDS (p). Then, we have g(α A(l) ) = g(α 1(l) ),

g(α ) = 1 − g(α 1(L−l+1) ), g(α C (l) ) = g(α 2(l) ), g(α D(l) ) = 1 − g(α 2(L−l+1) ), g(α A(L−l+1) ) = g(α 1(L−l+1) ), g(α C (L−l+1) ) = g(α 2(L−l+1) ), 1 − g(α B(L−l+1) ) = g(α 1(l) ), 1 − g(α D(L−l+1) ) = g(α 2(l) ). Therefore, g(α A(l) ) ≤ g(α C (l) ) ≤ g(α D(l) ) ≤ g(α B(l) ), g(α A(L−l+1) ) ≥ g(α C (L−l+1) ) ≥ g(α D(L−l+1) ) ≥ g(α B(L−l+1) ). Through (d5) of Defi1 2 nition 6, we can obtain d(h1S (p) ∪ hS (p), S) ≥ d(h2S (p) ∪ hS (p), S) B(l)

1

2

and d(h1S (p) ∩ hS (p), S) ≤ d(h2S (p) ∩ hS (p), S). Therefore, e(h1S (p)) ≤

e(h2S (p)). It also holds when g(α 1(l) ) ≥ g(α 2(l) ) for g(α 1(l) ) + g(α 1(L−l+1) ) ≥ 1. (E5) E(hS (p)) =

d(hS (p)∪hS (p),S) d(hS (p)∩hS (p),S)

=

d(hS (p)∪hS (p),S) d(hS (p)∩hS (p),S)

= E(hS (p)).

The proof regarding the similarity measure is similar to the above process. Thus, we complete the proof of Theorem 11. ■

+

(2) E(hS (p)) = (3) E(hS (p)) =

d(hS (p) ∩ hS (p), ∅) d(hS (p) ∪ hS (p), ∅) d(hS (p) ∪ hS (p), S) d(hS (p) ∪ hS (p), ∅) d(hS (p) ∩ hS (p), ∅) d(hS (p) ∩ hS (p), S)

=

1 − ρ (hS (p) ∩ hS (p), ∅)

=

1 − ρ (hS (p) ∪ hS (p), S)

=

1 − ρ (hS (p) ∩ hS (p), ∅) 1 − ρ (hS (p) ∪ hS (p), ∅) 1 − ρ (hS (p) ∩ hS (p), ∅) 1 − ρ (hS (p) ∩ hS (p), S)

are entropy measures of it. The proof of Corollary 4 is similar to that of Theorem 11. 5.3. The transformation from entropy measure to similarity measure This section investigates the transformation functions from entropy measure to similarity measure of PLEs based on their axiomatic definitions.

2

|g(α 1(L−l+1) ) − g(α 2(L−l+1) )| + 1 2

=1

⇔ |g(α 1(l) ) − g(α 2(l) )| = 0, |g(α 1(L−l+1) ) − g(α 2(l) )| = 0 ⇔ h1S (p) ∼ h2S (p). ( ) (s4) E ξ (h1S (p), h2S (p)) = 0 ⇔ ξ (h1S (p), h2S (p)) = 0 or ξ (h1S (p), h2S (p)) = 0 ⇔ G(h1S (p)) = {0} and G(h2S (p)) = {1} or G(h1S (p)) = {1} and G(h2S (p)) = {0}. (s5) Suppose that there are three PLEs h1S (p), h2S (p) and h3S (p)

with g(α 1(l) ) ≤ g(α 2(l) ) ≤ g(α 3(l) ). Then, we have

|g(α 1(l) )−g(α 3(l) )|+1

|g(α 1(l) )−g(α 2(l) )|+1

2

and ξ (h1S (p), h3S (p)) ≥ ξ (h1S (p), h2S (p)). Based on the definition of ξ (h1S (p), h2S (p)), we have ξ (h1S (p), h2S (p))(l) ≥ ξ (h1S (p)(, h2S (p))(L−l+1) . )Through 8, we can ob( (e4) of Definition ) tain E ξ (h1S (p), h3S (p)) ≥ E ξ (h1S (p), h2S (p)) . It also holds when g(α 1(l) ) ≥ g(α 2(l) ) ≥ g(α 3(l) ). We complete the proof of Theorem 12. ■

≥

2

Corollary 5. Suppose that E is an entropy measure of two PLEs h1S (p) and h2S (p), then (1) E ξ (h1S (p), h2S (p))

)

(

is a similarity measure between them,

( ) where ξ ((h1S (p), h2S (p)) is) the complementary of ξ (h1S (p), h2S (p)). (2) E ζ (h1S (p), h2S (p)) is a similarity measure between them, where

ζ (h1S (p), h2S (p)) ( q q |g(α 1(1) ) − g(α 2(1) )| + 1 |g(α 1(2) ) − g(α 2(2) )| + 1 = , ,..., 2

Corollary 4. Suppose that d and ρ are the distance and similarity measures of a PLE hS (p), respectively. Then, (1) E(hS (p)) =

|g(α 1(l) ) − g(α 2(l) )| + 1

|g(α

1(L)

) − g(α 2

2

2(L)

q

)| + 1

) ,

q > 0. The proof of Corollary 5 is similar to that of Theorem 12. 5.4. The transformation from inclusion measure to distance and similarity measures In this section, we discuss the transformation functions from inclusion measure to similarity and distance measures of PLEs based on their axiomatic definitions. Theorem 13. Suppose that In is an entropy measure for two PLEs h1S (p) and h2S (p). Then, ρ (h1S (p), h2S (p)) = In(h1S (p), h2S (p)) · In(h2S (p), h1S (p)) is a similarity measure between them. Proof. (s1) It is obvious that ρ (h1S (p), h2S (p)) ∈ [0, 1].

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572 1

(s2) It is obvious that ρ (h1S (p), h2S (p)) = ρ (h2S (p), h1S (p)). (s3) ρ

⇔

,

(h1S (p)

In(h1S (p)

,

h2S (p))

h2S (p))

=

= 1,

In(h1S (p)

In(h1S (p)

,

,

h2S (p))

h2S (p))

·

,

In(h2S (p)

h1S (p))

h1S (p)

h2S (p)

⇔

∼

(In2) In(h1S (p), hS (p)) = ρ (h1S (p), h1S (p) ∩ h2S (p)) = 1 − d(h1S (p), ∩ h2S (p)) = 0

h1S (p)

=1

⇔ G(h1S (p)) = {1}.

.

(s4) ρ (h1S (p), h2S (p)) = In(h1S (p), h2S (p)) · In(h2S (p), h1S (p)) = 0

⇔ In(h1S (p), h2S (p)) = 0 or In(h2S (p), h1S (p)) = 0 ⇔ G(h1S (p)) = {0} and G(h2S (p)) = {1} or G(h1S (p)) = {1} and G(h2S (p)) = {0}. (s5) Since g(α 1(l) ) ≤ g(α 2(l) ) ≤ g(α 3(l) ), then In(h1S (p), h2S (p)) = 1 and In(h1S (p), h3S (p)) = 1. We can obtain In(h1S (p), h2S (p)) · In(h2S (p), h1S (p)) = In(h2S (p), h1S (p)) and In(h1S (p), h3S (p)) · In(h3S (p), h1S (p)) = In(h3S (p), h1S (p)). From (In3) of Definition 5, we have ρ (h1S (p), h2S (p)) ≥ ρ (h1S (p), h3S (p)). We complete the proof of Theorem 13.

■

Theorem 14. Suppose that In is an entropy measure of two PLEs h1S (p) and h2S (p). Then, ρ (h1S (p), h2S (p)) = min{In(h1S (p), h2S (p)), In(h2S (p), h1S (p))} is a similarity measure between them.

ρ

,

∈[ , ] , { , , , } = ∼ = ⇔ , , · , = , = = { } = { }

Proof. (s1) It is obvious that (h1S (p) h2S (p)) 0 1. (s2) It is obvious that (h1S (p) h2S (p)) (h2S (p) h1S (p)). (s3) (h1S (p) h2S (p)) min In(h1S (p) h2S (p)) In(h2S (p) h1S (p)) 2 1 h2S (p). 1 h1S (p) In(h1S (p) h2S (p)) 1 In(hS (p) hS (p)) (s4) (h1S (p) h2S (p)) In(h1S (p) h2S (p)) In(h2S (p) h1S (p)) 0 In(h1S (p) h2S (p)) 0 or In(h2S (p) h1S (p)) 0 1 and 1 or G(h1S (p)) 0 and G(h2S (p)) G(h1S (p)) 2 0 . G(hS (p)) (s5) Since g( 1(l) ) g( 2(l) ) g( 3(l) ), then In(h1S (p) h2S (p)) 1 and In(h1S (p) h3S (p)) 1. We can obtain min In(h1S (p) h2S (p)) In(h2S (p) h1S (p)) In(h2S (p) h1S (p)) and min In(h1S (p) h3S (p)) 3 3 In(hS (p) h1S (p)) In(hS (p) h1S (p)). From (In3) of Definition 5, we have (h1S (p) h2S (p)) (h1S (p) h3S (p)). We complete the proof of

ρ

ρ

,

,

⇔

ρ

⇔ ⇔

=

=

,

= = = { } ={ } α ≤ α , = , } = , }= , ρ , ≥ρ ,

Theorem 14.

,

=ρ

α

≤

,

{ {

,

, ,

= , ,

,

■

Through the relationship between the distance and similarity measures, we can also obtain the transformation function from the inclusion measure to the distance measure. Corollary 6. Suppose that In is an inclusion measure of two PLEs h1S (p) and h2S (p). Then, the (1) d(h1S (p), h2S (p)) = 1 − In(h1S (p), h2S (p)) · In(h2S (p), h1S (p)) (2) d(h1S (p), h2S (p)) = 1 − min{In(h1S (p), h2S (p)), In(h2S (p), h1S (p))} are distance measures between them. The proof of Corollary 6 is similar to those of Theorems 13 and 14.

(In3) Since h1S (p) ⊆ h2S (p) ⊆ h3S (p), then ρ (h2S (p), h2S (p) ∩ h1S (p)) = ρ (h2S (p) ∩ h1S (p)) and ρ (h3S (p), h3S (p) ∩ h1S (p)) = ρ (h3S (p) ∩

h1S (p)). From (s5) of Definition 7, we have ρ (h3S (p), h1S (p)) ≤ ρ (h2S (p), h1S (p)). Thus, In(h3S (p), h1S (p)) ≤ In(h2S (p), h1S (p)). It also holds for In(h3S (p), h1S (p)) ≤ In(h3S (p), h2S (p)). We complete the proof of Theorem 15. ■ Corollary 7. Suppose that d and ρ are the distance and similarity measures between two PLEs h1S (p)and h2S (p), respectively. Then In(h1S (p), h2S (p)) = ρ (h2S (p), h1S (p) ∪ h2S (p)) = 1 − d(h2S (p), h1S (p) ∪ h2S (p)) is an inclusion measure between them. The proof for Corollary 7 is similar to that of Theorem 15. 5.6. The relationship between the inclusion measure and the entropy measure In this section, we investigates the transformation rules from the inclusion measure to the entropy measure of PLEs. Theorem 16. Suppose that In is an inclusion measure of a PLE hS (p). Then, E(hS (p)) = In(hS (p) ∪ hS (p), hS (p) ∩ hS (p)) is an entropy measure of it. Proof. (e1) It is obvious. (e2) E(hS (p)) = In(hS (p) ∪ hS (p), hS (p) ∩ hS (p)) = 0 ⇔ G(hS (p) ∪ hS (p)) = {1} and G(hS (p) ∩ hS (p)) = {0} ⇔ G(hS (p)) = {1} or G(hS (p)) = {0}. (e3) E(hS (p)) = In(hS (p) ∪ hS (p), hS (p) ∩ hS (p)) = 1

⇔ hS (p) ∪ hS (p) ⊆ hS (p) ∩ hS (p)hS (p) ∪ hS (p) ∼ hS (p) ∩ hS (p) ⇔ hS (p) ∼ hS (p) ⇔ g(α 1(l) ) + g(α 1(L1 −l+1) ) = 1. (e4) If g(α 1(l) ) ≤ g(α 2(l) ) for g(α 2(l) ) + g(α 2(L−l+1) ) ≤ 1, we have g(α 1(l) ) ≤ g(α 2(l) ) ≤ 1 − g(α 2(L−l+1) ) ≤ 1 − g(α 1(L−l+1) ). Suppose 1 1 2 that h1S (p) ∩ hS (p) = hAS (p), h1S (p) ∪ hS (p) = hBS (p), h2S (p) ∩ hS (p) = 2

hCS (p), h2S (p) ∪ hS (p) = hDS (p). Then, we have g(α A(l) ) = g(α 1(l) ),

g(α B(l) ) = 1 − g(α 1(L−l+1) ), g(α C (l) ) = g(α 2(l) ), g(α D(l) ) = 1 − g(α 2(L−l+1) ), g(α A(L−l+1) ) = g(α 1(L−l+1) ), g(α C (L−l+1) ) = g(α 2(L−l+1) ), 1 − g(α B(L−l+1) ) = g(α 1(l) ), 1 − g(α D(L−l+1) ) = g(α 2(l) ). Therefore, g(α A(l) ) ≤ g(α C (l) ) ≤ g(α D(l) ) ≤ g(α B(l) ), g(α A(L−l+1) ) ≥ g(α C (L−l+1) ) ≥ g(α D(L−l+1) ) ≥ g(α B(L−l+1) ). Through (In3) of 1 1 Definition 5, we have In(h1S (p) ∩ hS (p), h1S (p) ∪ hS (p)) ≤ In(h2S (p) ∩ 2

5.5. The transformation from distance and similarity measures to inclusion measure This section discusses the transformation functions from the distance and similarity measures to the inclusion measure of PLEs. Theorem 15. Suppose that d and ρ are the distance and similarity measures between two PLEs h1S (p)and h2S (p), respectively. Then, In(h1S (p), h2S (p)) = ρ (h1S (p), h1S (p) ∩ h2S (p)) = 1 − d(h1S (p), h1S (p) ∩ h2S (p)) is an inclusion measure between them.

,

Proof. (In1) In(h1S (p) d(h1S (p) h1S (p) h2S (p))

,

∩

h2S (p))

= ρ

(h1S (p)

,

h1S (p)

=1

⇔ h1S (p) ∼ h1S (p) ∩ h2S (p) ⇔ h1S (p) ⊆ h2S (p).

∩

h2S (p))

9

= 1−

2

hS (p), h2S (p) ∪ hS (p)), and e(h1S (p)) ≤ e(h2S (p)). (e5) It is obvious that E(hS (p)) = E(hS (p)). We complete the proof of Theorem 16. ■ In this section, we discussed the relationships among distance, entropy, similarity and inclusion measures of PLEs based on their normalized axiomatic definitions. A summarization of these relationships is presented in Fig. 1. The direction of the arrow indicates that one measure can be converted to another. The theorems and corollaries accompanied with the arrow denote the transformation rules between two measures. 6. An orthogonal clustering algorithm based on the inclusion measure of PLEs and its application Based on the inclusion measure of PLEs introduced in this study, we design a clustering algorithm to show the applicability

10

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

Inspired by the work of Ref. [55], based on Definition 10, we define the mutual inclusion matrix. Definition 11. Let HSi (p)(i = 1, 2, . . . , m) be m PLTSs. M = (muInij )m×m is a mutual inclusion matrix, where muInij is the j mutual inclusion degree of HSi (p) and HS (p), satisfying: (1) muInij ∈ [0, 1]; j (2) muInij = 1 if and only if HSi (p) ∼ HS (p); (3) muInij = muInji for all i, j = 1, 2, . . . , m. Definition 12. Let M = (muInij )m×m be a mutual inclusion matrix. If M 2 = M ◦ M = (muInij )m×m , then, M 2 is called a composition Fig. 1. Relationships among four kinds of information measures of PLEs.

of the information measures of PLEs. Furthermore, an illustrative example about partitioning cities in the Economic Zone of Chengdu Plain, China, is adopted to verify the algorithm. 6.1. Designing an orthogonal clustering algorithm based on the inclusion measure of PLEs Clustering analysis is a kind of unsupervised algorithm. Its aim is to partition unknown labeled data objects into different subgroups according to the similarity degrees of them [52]. The data objects in the same sub-group have high degrees of similarity, that is, they are homogeneous; while the data objects in different sub-groups should be as heterogeneous as possible [53]. There are a large number of clustering algorithms. In 1965, Zadeh [14] introduced the fuzzy sets. Since then, various forms of extensions to depict and handle uncertain and imprecise information have been developed. In the meantime, clustering algorithms based on these extensions also have a great development, such as intuitionistic fuzzy clustering [54–56], hesitant fuzzy clustering [40, 57] and type-2 fuzzy clustering [58–60]. The distance measures play a critical role among these clustering algorithms [54,57– 60]. There are some studies that used similarity measures [56] or correlation measures [40,55] to design the clustering algorithm. However, few papers adopted the inclusion measures in clustering algorithms. As mentioned above, the inclusion measure depicts the degree to which a set is contained in another set. If the clusters are determined, the inclusion degree to which a data object contains in different clusters should be as small as possible. However, unlike the distance, similarity and correlation measures, the inclusion measure is asymmetrical. To better use the inclusion measure in clustering analysis, this study proposes the concept of mutual inclusion degree of PLEs. Then, we use the mutual inclusion degree in the orthogonal clustering algorithm. Definition 10. Let h1S (p) and h2S (p) be two PLEs on a nonempty set X . The mutual inclusion degree of h1S (p) and h2S (p) is defined as: muIn(h1S (p), h2S (p)) = (In(h1S (p), h2S (p)), In(h2S (p), h1S (p)))1/2

(21)

Through the definition of mutual inclusion degree, we can find some properties of it: (1) muIn ∈ [0, 1]. muIn = 0 when G(h1S (p)) = {1}, G(h2S (p)) = {0} or G(h1S (p)) = {0}, G(h2S (p)) = {1}; muIn = 1 if and only if h1S (p) contains h2S (p) completely and vice verse, that is, h1S (p) ∼ h2S (p). (2) muIn(h1S (p), h2S (p)) = muIn(h2S (p), h1S (p)). (3) muIn(h1S (p), h1S (p)) = 1.

matrix of M, where muInij = maxk {min{muInik , muInkj }}, i, j = 1, 2, . . . , m. Based on Definition 12, we can obtain the following theorem: Theorem 17. Let M = (muInij )m×m be a mutual inclusion matrix. Then, the composition matrix M 2 is also a mutual inclusion matrix. Proof. (1) Because M = (muInij )m×m is a mutual inclusion matrix, we have muInij ∈ [0, 1]. Thus, muInij = maxk {min{muInik , muInkj }} ∈ [0, 1], for all i, j = 1, 2, . . . , m. j (2) Since muInij = 1 if and only if HSi (p) ∼ HS (p), then muInij = maxk {min{muInik , muInkj }} = 1 if and only if HSi (p) ∼ j HS (p) ∼ HSk (p). (3) Since muInij = muInji , then for all i, j = 1, 2, . . . , m, we have { } muInij = maxk min{muInik , muInkj } = maxk {min{muInki , { } } muInjk } = maxk min{muInjk , muInki } = muInji . This completes the proof. ■ Based on Theorem 17, the following theorem can be derived. Theorem 18. Let M = (muInij )m×m be a mutual inclusion matrix. k+1 Then, for any nonnegative integer k, the composition matrix M 2 k + 1 k + 1 k + 1 is also a mutual inclusion deduced from M 2 = M2 ◦ M2 matrix. Definition 13. Let M = (muInij )m×m be a mutual inclusion matrix. If M 2 ⊆ M, i.e., maxk {min{muInik , muInkj } ≤ muInij for all i, j = 1, 2, . . . , m, then, M is called an equivalent mutual inclusion matrix. Theorem 19. Let M = (muInij )m×m be a mutual inclusion matrix. k (k+1) There is a positive integer k such that M 2 = M 2 after finite times of compositions. Next, we give the concept of λ-cutting matrix corresponding to the equivalent mutual inclusion matrix. Definition 14. Let M = (muInij )m×m be a mutual inclusion matrix. The cut relation Mλ = (λ muInij )m×m of M is called the λ-cutting matrix, which can be represented as:

⎡

⎤ · · · λ muIn1j · · · λ muIn1m .. .. .. ⎢ ⎥ .. . . .. ⎢ ⎥ . . . ⎢ ⎥ Mλ= ⎢ λ muIni1 · · · λ muInij · · · λ muInim ⎥ ⎢ ⎥ .. .. .. . .. ⎣ ⎦ .. . . . . · · · λ muInmj · · · λ muInmm λ muInm1 { 0 if muInij < λ, where λ muInij = i, j = 1, 2, . . . , m with 1 if muInij ≥ λ, λ ∈ [0, 1] being the confidence level. The confidence level λ muIn11

is selected to classify samples. After obtaining the equivalence

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

11

relation, we can intercept it at an appropriate level of λ and get the ordinary equivalence relation (λ-cutting matrix). Different λ determines different ordinary equivalence relations, and thus different classifications are determined. If M is an equivalence relation on X , then, its cut relation is a classical equivalence relation. They can make a partition on X . We can obtain a family of partition when λ drops from 1 to 0. As λ declines, Mλ ’s classification becomes more and more coarse. We can choose a certain level of division according to actual needs. After this, we can construct the corresponding λ-cutting matrix. From the above analysis, a clustering algorithm based on the mutual inclusion degrees of PLTSs is proposed as follows: Algorithm 1 (An Orthogonal Clustering Algorithm Based on the Inclusion Measure of PLEs). Step 1. Use Eq. (21) to compute the mutual inclusion degrees of PLTSs and construct the mutual inclusion matrix M = (muInij )m×m , i, j = 1, 2, . . . , m. Step 2. Construct the λ-cutting matrix Mλ = (λ muInij )m×m of M by using Definition 14 if the mutual inclusion matrix is an equivalent mutual inclusion matrix; otherwise, derive the equivalent mutual inclusion matrix M and construct the λ-cutting matrix M λ = (λ muInij )m×m . Step 3. If all elements of the ith line in the λ-cutting matrix Mλ are the same as the corresponding elements of the jth line, then j the PLTSs HSi (p) and HS (p) belong to the same type. Based on this principle, all PLTSs can be classified. The Pseudo code of this clustering algorithm is provided below.

Fig. 2. Flowchart of the proposed algorithm.

the fuzzy similarity matrix. Fu [63] put forward an algorithm to calculate a fuzzy similarity matrix’s transitive closure. Their algorithm had a complexity of O( 61 m3 ). Simplifying the matrix computation and using our method to related topics are worth studying in the future. Fig. 2 presents the flowchart that uses the proposed clustering algorithm to classify data objects. 6.2. An illustrative example: classifying cities in the Economic Zone of Chengdu Plain, China

The time complexity of this clustering algorithm is O(m3 log m) [61], where n is the number of data objects. There are other studies that focused on accelerating the computation based on the fuzzy equivalence relation. For instance, Larsen and Yager [62] introduced an algorithm to construct a binary tree representation of the fuzzy equivalence relation based on a fuzzy similarity matrix. Their total worst-case time to create fuzzy equivalence matrix is O(nm2 ), where n is the number of non-zero values in

The Urban Economic Circle is a complex dynamic system with strong regionalism and structural characteristic. Classically, the Urban Economic Circle takes one or several economically developed central cities as its core, and contains a number of surrounding cities or towns with economic internal links. Central cities’ economic attraction and economic radiation capacity can reach and promote the largest regional scope of economic development in the corresponding regions. As an important economic map of southwest China, the Economic Zone of Chengdu Plain has its unique circle structure and system function zoning. The Economic Zone of Chengdu Plain assumes unbalanced and structured differential development influenced by differences in development history and economic basis. This study presents a clustering analysis for cities in the Economic Zone of Chengdu Plain so as to provide macro-economic regulation for healthy development. There are eight important cities in the Economic Zone of Chengdu Plain, including a1 (Chengdu), a2 (Deyang), a3 (Mianyang), a4 (Leshan), a5 (Meishan), a6 (Yaan), a7 (Ziyang) and a8 (Neijiang). Fig. 3 displays the geographical distribution of the Economic Zone of Chengdu Plain, which is shown in green color. The green part, together with the yellow part around it, represents Sichuan Province. To comprehensively evaluate the

12

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

Fig. 3. Geographical distribution of the Economic Zone of Chengdu Plain. Table 4 Decision making matrix with PLEs. a1 a2 a3 a4 a5 a6 a7 a8

c1

c2

c3

c4

{s−1 (0.4), s0 (0.6)} {s2 (0.4)} {s0 (0.6), s1 (0.4)} {s0 (0.6), s1 (0.4)} {s1 (0.4), s2 (0.4)} {s1 (0.5), s2 (0.5)} {s1 (0.5)} {s−1 (0.6), s0 (0.4)}

{s3 (0.7), s4 (0.3)} {s−1 (0.2), s0 (0.4), s1 (0.2)} {s1 (0.5), s2 (0.5)} {s0 (0.3), s1 (0.7)} {s−2 (0.5), s−1 (0.3)} {s−3 (0.2), s−2 (0.8)} {s−2 (0.6), s−1 (0.4)} {s−1 (1)}

{s2 (0.4), s3 (0.6)} {s1 (0.7), s2 (0.3)} {s0 (0.1), s1 (0.9)} {s1 (0.8)} {s0 (0.7)} {s−3 (0.2), s−2 (0.8)} {s−1 (0.6), s1 (0.2)} {s−1 (0.8), s0 (0.2)}

{s3 (0.4), s4 (0.6)} {s1 (0.5), s2 (0.5)} {s1 (0.7), s2 (0.3)} {s1 (0.4), s2 (0.6)} {s−2 (0.2), s−1 (0.6), s0 (0.2)} {s0 (0.7)} {s0 (0.4), s1 (0.6)} {s−3 (0.3), s−2 (0.7)}

Table 5 Clustering results of the illustrative example by our method.

Table 6 Clustering results of the illustrative example by the method in Ref. [13].

Class

Confidence level

Clustering result

Class

Confidence level

Clustering result

8 7 6 5 4 3 2 1

0.9460 < λ ≤ 1 0.9352 < λ ≤ 0.9460 0.9336 < λ ≤ 0.9352 0.9259 < λ ≤ 0.9336 0.9083 < λ ≤ 0.9259 0.9077 < λ ≤ 0.9083 0.8998 < λ ≤ 0.9077 0 ≤ λ ≤ 0.8998

{a1 }, {a2 }, {a3 }, {a4 }, {a5 }, {a6 }, {a7 }, {a8 } {a1 }, {a2 , a7 }, {a3 }, {a4 }, {a5 }, {a6 }, {a8 } {a1 }, {a2 , a5 , a7 }, {a3 }, {a4 }, {a6 }, {a8 } {a1 }, {a2 , a5 , a7 , a8 }, {a3 }, {a4 }, {a6 } {a1 }, {a2 , a4 , a5 , a7 , a8 }, {a3 }, {a6 } {a1 }, {a2 , a4 , a5 , a6 , a7 , a8 }, {a3 } {a1 }, {a2 , a3 , a4 , a5 , a6 , a7 , a8 } {a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 }

8 7 6 5 4 3 2 1

0.9495 < λ ≤ 1 0.9494 < λ ≤ 0.9495 0.9468 < λ ≤ 0.9494 0.8742 < λ ≤ 0.9468 0.8112 < λ ≤ 0.8742 0.7306 < λ ≤ 0.8112 0.6583 < λ ≤ 0.7306 0 ≤ λ ≤ 0.6583

{a1 }, {a2 }, {a3 }, {a4 }, {a5 }, {a6 }, {a7 }, {a8 } {a1 , a5 }, {a2 }, {a3 }, {a4 }, {a6 }, {a7 }, {a8 } {a1 , a3 , a5 }, {a2 }, {a4 }, {a6 }, {a7 }, {a8 } {a1 , a3 , a5 }, {a2 }, {a4 }, {a6 , a7 }, {a8 } {a1 , a3 , a5 }, {a4 }, {a2 , a6 , a7 }, {a8 } {a1 , a3 , a4 , a5 }, {a2 , a6 , a7 }, {a8 } {a1 , a2 , a3 , a4 , a5 , a6 , a7 }, {a8 } {a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 }

economic performance of these eight cities, four criteria are considered, such as: x1 (population density), x2 (per capita GDP), x3 (proportion of tertiary industry), and x4 (urbanization rate). The weight vector of these four criteria is ω = (0.1, 0.4, 0.2, 0.3)T . Table 4 illustrates the evaluate results expressed by linguistic expressions and then transformed to PLEs. Next, the illustrative example is solved by Algorithm 1 and the steps are presented as follows: Step 1. The mutual inclusion degrees between pairwise PLEs are calculated by Eq. (21), and the mutual inclusion matrix is constructed as follows (see Box I): Step 2. Construct the equivalent mutual inclusion matrix as (see Box II): Since M 16 = M 8 , then, M 8 is the equivalent mutual inclusion matrix. Step 3. Construct the λ-cutting matrices regarding different confidence level λ. Based on the results, all possible partitions are presented in Table 5.

Based on the correlation coefficient, Lin et al. [13] designed a clustering algorithm. Their algorithm is also an orthogonal clustering method and has similar calculation process. Next, we use Lin et al. [13]’s method to solve the example. Step 1. The correlation matrix based the weighted correlation coefficients in Ref. [13] is constructed as (see Box III): Step 2. Construct the equivalent correlation matrix in the following way (see Box IV): Since C 16 = C 8 , then C 8 is the equivalent correlation matrix. Step 3. Construct the λ-cutting matrices regarding different confidence level λ. Based on the results, all possible partitions are presented in Table 6. We should note that the correlation coefficient proposed by Lin et al. [13] does not have the normalization process. The correlation coefficient is a statistical index used to reflect the degree of close correlation between variables. It is calculated by the product moment method, which is based on the difference between two variables and their respective average values. It describes the degree of linear fit between random variables. The

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

⎧ 1.0000 ⎪ ⎪ ⎪0.8887 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎨ 0.8718 M= 0.8533 ⎪ ⎪ ⎪ ⎪ 0.8300 ⎪ ⎪ ⎪ ⎪ ⎩0.8550 0.8554

0.8887 1.0000 0.8944 0.9249 0.9187 0.8991 0.9460 0.8895

0.8998 0.8944 1.0000 0.8897 0.8749 0.9077 0.8792 0.8802

0.8718 0.9249 0.8897 1.0000 0.9259 0.8656 0.9214 0.8698

0.8533 0.9187 0.8749 0.9259 1.0000 0.9083 0.9352 0.9336

0.8300 0.8991 0.9077 0.8656 0.9083 1.0000 0.8994 0.8718

0.8550 0.9460 0.8792 0.9214 0.9352 0.8994 1.0000 0.9200

0.8554⎪ ⎪ 0.8895⎪ ⎪ ⎪ ⎪ 0.8802⎪ ⎪ ⎬ 0.8698 0.9336⎪ ⎪ ⎪ 0.8718⎪ ⎪ ⎪ 0.9200⎪ ⎪ ⎭ 1.0000

⎫

Box I.

M2 =

M4 =

M8 =

M 16 =

⎧ 1.0000 ⎪ ⎪ ⎪ 0.8944 ⎪ ⎪ ⎪ ⎪0.8998 ⎪ ⎨ 0.8897 0.8887 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎪ ⎪ ⎪ ⎩0.8887 0.8887 ⎧ 1.0000 ⎪ ⎪ ⎪0.8991 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎨ 0.8998 0.8998 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎪ ⎪ ⎪ ⎩0.8998 0.8998 ⎧ 1.0000 ⎪ ⎪ ⎪0.8998 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎨ 0.8998 0.8998 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎪ ⎪ ⎪ ⎩0.8998 0.8998 ⎧ 1.0000 ⎪ ⎪ ⎪0.8998 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎨ 0.8998 0.8998 ⎪ ⎪ ⎪ ⎪ 0.8998 ⎪ ⎪ ⎪ ⎪ ⎩0.8998 0.8998

0.8944 1.0000 0.8991 0.9249 0.9352 0.9083 0.9460 0.9220

0.8998 0.8991 1.0000 0.8944 0.9077 0.9077 0.8994 0.8895

0.8897 0.9249 0.8944 1.0000 0.9259 0.9083 0.9259 0.9259

0.8887 0.9352 0.9077 0.9259 1.0000 0.9083 0.9352 0.9336

0.8998 0.9083 0.9077 0.9083 0.9083 1.0000 0.9083 0.9083

0.8887 0.9460 0.8994 0.9259 0.9352 0.9083 1.0000 0.9336

0.8887⎪ ⎪ 0.9220⎪ ⎪ ⎪ ⎪ 0.8895⎪ ⎪ ⎬ 0.9259 0.9336⎪ ⎪ ⎪ 0.9083⎪ ⎪ ⎪ 0.9336⎪ ⎪ ⎭ 1.0000

0.8991 1.0000 0.9077 0.9259 0.9352 0.9083 0.9460 0.9336

0.8998 0.9077 1.0000 0.9077 0.9077 0.9077 0.9077 0.9077

0.8998 0.9259 0.9077 1.0000 0.9259 0.9083 0.9259 0.9259

0.8998 0.9352 0.9077 0.9259 1.0000 0.9083 0.9352 0.9336

0.8998 0.9083 0.9077 0.9083 0.9083 1.0000 0.9083 0.9083

0.8998 0.9460 0.9077 0.9259 0.9352 0.9083 1.0000 0.9336

0.8998⎪ ⎪ 0.9336⎪ ⎪ ⎪ ⎪ 0.9077⎪ ⎪ ⎬ 0.9259 0.9336⎪ ⎪ ⎪ 0.9083⎪ ⎪ ⎪ 0.9336⎪ ⎪ ⎭ 1.0000

0.8998 1.0000 0.9077 0.9259 0.9352 0.9083 0.9460 0.9336

0.8998 0.9077 1.0000 0.9077 0.9077 0.9077 0.9077 0.9077

0.8998 0.9259 0.9077 1.0000 0.9259 0.9083 0.9259 0.9259

0.8998 0.9352 0.9077 0.9259 1.0000 0.9083 0.9352 0.9336

0.8998 0.9083 0.9077 0.9083 0.9083 1.0000 0.9083 0.9083

0.8998 0.9460 0.9077 0.9259 0.9352 0.9083 1.0000 0.9336

0.8998⎪ ⎪ 0.9336⎪ ⎪ ⎪ ⎪ 0.9077⎪ ⎪ ⎬ 0.9259 0.9336⎪ ⎪ ⎪ 0.9083⎪ ⎪ ⎪ 0.9336⎪ ⎪ ⎭ 1.0000

0.8998 1.0000 0.9077 0.9259 0.9352 0.9083 0.9460 0.9336

0.8998 0.9077 1.0000 0.9077 0.9077 0.9077 0.9077 0.9077

0.8998 0.9259 0.9077 1.0000 0.9259 0.9083 0.9259 0.9259

0.8998 0.9352 0.9077 0.9259 1.0000 0.9083 0.9352 0.9336

0.8998 0.9083 0.9077 0.9083 0.9083 1.0000 0.9083 0.9083

0.8998 0.9460 0.9077 0.9259 0.9352 0.9083 1.0000 0.9336

0.8998⎪ ⎪ 0.9336⎪ ⎪ ⎪ ⎪ 0.9077⎪ ⎪ ⎬ 0.9259 0.9336⎪ ⎪ ⎪ 0.9083⎪ ⎪ ⎪ 0.9336⎪ ⎪ ⎭ 1.0000

⎫

⎫

⎫

⎫

Box II.

⎧ 1.0000 ⎪ ⎪ ⎪0.1243 ⎪ ⎪ ⎪ ⎪ 0.9494 ⎪ ⎨ 0.8112 C = 0.9495 ⎪ ⎪ ⎪ ⎪ 0.5925 ⎪ ⎪ ⎪ ⎪ ⎩−0.5575 −0.8033

0.1243 1.0000 −0.3748 0.4756 0.3923 0.7022 0.8742 −0.4228

0.9494 −0.3748 1.0000 0.6300 −0.9992 −0.7053 −0.7061 −0.6774

0.8112 0.4756 0.6300 1.0000 −0.6183 −0.1411 0.0296 −0.9742

0.9495 0.3923 −0.9992 −0.6183 1.0000 0.7306 0.7279 0.6583

0.5925 0.7022 −0.7053 −0.1411 0.7306 1.0000 0.9468 0.0777

Box III.

−0.5575 0.8742 −0.7061 0.0296 0.7279 0.9468 1.0000 −0.0566

⎫ −0.8033⎪ ⎪ −0.4228⎪ ⎪ ⎪ ⎪ −0.6774⎪ ⎪ ⎬ −0.9742 0.6583 ⎪ ⎪ ⎪ 0.0777 ⎪ ⎪ ⎪ −0.0566⎪ ⎪ ⎭ 1.0000

13

14

M. Tang, Y.L. Long, H.C. Liao et al. / Applied Soft Computing Journal 82 (2019) 105572

C2 =

C4 =

C8 =

C 16 =

⎧ 1.0000 ⎪ ⎪ ⎪0.5925 ⎪ ⎪ ⎪ ⎪ 0.9494 ⎪ ⎨ 0.8112 0.9495 ⎪ ⎪ ⎪ ⎪ 0.7306 ⎪ ⎪ ⎪ ⎪ ⎩0.7279 0.6583 ⎧ 1.0000 ⎪ ⎪ ⎪0.7306 ⎪ ⎪ ⎪ ⎪0.9494 ⎪ ⎨ 0.8112 0.9495 ⎪ ⎪ ⎪ ⎪ 0.7306 ⎪ ⎪ ⎪ ⎪ ⎩0.7306 0.6583 ⎧ 1 .0000 ⎪ ⎪ ⎪0.7306 ⎪ ⎪ ⎪ ⎪ 0.9494 ⎪ ⎨ 0.8112 0.9495 ⎪ ⎪ ⎪ ⎪ 0.7306 ⎪ ⎪ ⎪ ⎪ ⎩0.7306 0.6583 ⎧ 1.0000 ⎪ ⎪ ⎪0.7306 ⎪ ⎪ ⎪ ⎪ 0.9494 ⎪ ⎨ 0.8112 0.9495 ⎪ ⎪ ⎪ ⎪ 0.7306 ⎪ ⎪ ⎪ ⎪ ⎩0.7306 0.6583

0.5925 1.0000 0.4756 0.4756 0.7279 0.8742 0.8742 0.3923

0.9494 0.4756 1.0000 0.8112 0.9494 0.5925 0.0296 −0.3748

0.8112 0.4756 0.8112 1.0000 0.8112 0.5925 0.4756 −0.0566

0.9495 0.7279 0.9494 0.8112 1.0000 0.7306 0.7279 0.6583

0.7306 0.8742 0.5925 0.5925 0.7306 1.0000 0.9468 0.6583

0.7279 0.8742 0.0296 0.4756 0.7279 0.9468 1.0000 0.6583

0.6583 ⎪ ⎪ 0.3923 ⎪ ⎪ ⎪ ⎪ −0.3748⎪ ⎪ ⎬ −0.0566 0.6583 ⎪ ⎪ ⎪ 0.6583 ⎪ ⎪ ⎪ 0.6583 ⎪ ⎪ ⎭ 1.0000

0.7306 1.0000 0.7279 0.7279 0.7306 0.8742 0.8742 0.6583

0.9494 0.7279 1.0000 0.8112 0.9494 0.7306 0.7279 0.6583

0.8112 0.7279 0.8112 1.0000 0.8112 0.7306 0.7279 0.6583

0.9495 0.7306 0.9494 0.8112 1.0000 0.7306 0.7306 0.6583

0.7306 0.8742 0.7306 0.7306 0.7306 1.0000 0.9468 0.6583

0.7306 0.8742 0.7279 0.7279 0.7306 0.9468 1.0000 0.6583

0.6583⎪ ⎪ 0.6583⎪ ⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎬ 0.6583 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎭ 1.0000

0.7306 1.0000 0.7279 0.7279 0.7306 0.8742 0.8742 0.6583

0.9494 0.7306 1.0000 0.8112 0.9494 0.7306 0.7279 0.6583

0.8112 0.7306 0.8112 1.0000 0.8112 0.7306 0.7279 0.6583

0.9495 0.7306 0.9494 0.8112 1.0000 0.7306 0.7306 0.6583

0.7306 0.8742 0.7306 0.7306 0.7306 1.0000 0.9468 0.6583

0.7306 0.8742 0.7306 0.7306 0.7306 0.9468 1.0000 0.6583

0.6583⎪ ⎪ 0.6583⎪ ⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎬ 0.6583 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎭ 1.0000

0.7306 1.0000 0.7306 0.7306 0.7306 0.8742 0.8742 0.6583

0.9494 0.7306 1.0000 0.8112 0.9494 0.7306 0.7306 0.6583

0.8112 0.7306 0.8112 1.0000 0.8112 0.7306 0.7306 0.6583

0.9495 0.7306 0.9494 0.8112 1.0000 0.7306 0.7306 0.6583

0.7306 0.8742 0.7306 0.7306 0.7306 1.0000 0.9468 0.6583

0.7306 0.8742 0.7306 0.7306 0.7306 0.9468 1.0000 0.6583

0.6583⎪ ⎪ 0.6583⎪ ⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎬ 0.6583 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎪ 0.6583⎪ ⎪ ⎭ 1.0000

⎫

⎫

⎫

⎫

Box IV.

mutual inclusion degree proposed in our study describes the coincidence degree of two sets. Therefore, there are some differences in the clustering results calculated by these two algorithms. 7. Conclusions Many researches about information measures including the distance measure, similarity measure, entropy measure and correlation measure have been developed for PLTSs in the last several years. However, as an important information measure, the inclusion measure of PLTSs has not been proposed yet. This study introduced the concept of the inclusion measure for PLTSs. Based on the inclusion measure, we put forward some formulas to calculate it. We can select an appropriate formula according to specific situations. To better understand and lay the foundation for the following theories, we provided the normalized axiomatic definitions for the distance, similarity and entropy measures of PLEs. These normalized axiomatic definitions constructed the framework of information measures for PLTSs. Next, a family of transformation functions among the distance, similarity, inclusion and entropy measures were investigated in detail. We believed that these transformation functions will induce more formulas to calculate the distance, similarity, inclusion and entropy measures of PLTSs. Furthermore, to better use the inclusion measure proposed by this work in applications, we introduced a clustering algorithm and adopted it in classifying cities in the Economic Zone of Chengdu Plain, China. These conclusions can

be used in many other attractive fields such as image processing, fuzzy reasoning and pattern recognition. This study also has some limitations. For instance, the clustering algorithm is not verified by using some manual datasets. In the near future, we will research this topic. Furthermore, we will investigate the knowledge measures of PLTSs and HFLTSs. The relationships between the knowledge measure and other information measures also need to be discussed. Implementing the proposed inclusion measures and the clustering algorithm to practical applications are also good research topics. Acknowledgments The work was supported by the National Natural Science Foundation of China (71571156), the 2019 Sichuan Planning Project of Social Science, China (No. SC18A007), and the 2018 Key Project of the Key Research Institute of Humanities and Social Sciences in Sichuan Province, China (No. Xq18A01). Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105572.

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