Annotated Bibliography for below attached aricles

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017 1685

A New Aggregation Method-Based Error Analysis for Decision-Theoretic Rough Sets and Its

Application in Hesitant Fuzzy Information Systems Decui Liang , Zeshui Xu , Senior Member, IEEE, and Dun Liu

Abstract—Decision-theoretic rough sets (DTRSs) capture the de- cision semantics and can deduce three-way decisions with respect to the minimum expected risk. Considering the new evaluation format of hesitant fuzzy sets, we extend the range of applications of DTRSs to hesitant fuzzy information systems. In this case, the integrated approach by considering the interaction between information sys- tems and loss functions becomes one of challenges. Different from the results reported in most of the existing papers, we combine the hesitant fuzzy information system and loss functions together via error analysis. In the hesitant fuzzy information system, a new binary relation is first defined by utilizing the normalization of hesitant fuzzy elements and the distance function. Then, the cal- culations of the similarity class and the conditional probability are discussed. With the aid of the error analysis method, we effectively aggregate the loss functions presented in the similarity class and determine the expected losses in the format of the intervals. Based on the possibility degree, we further explore the decision rules by comparing the expected losses. With the above analysis, we design a decision-making procedure of three-way decisions in a hesitant fuzzy information system. Finally, we elaborate the application of three-way decisions in hesitant fuzzy information systems by an ex- ample of the security evaluation of peer-to-peer lending platforms and validate our methods.

Index Terms—Decision-theoretic rough sets (DTRSs), error analysis, hesitant fuzzy information systems, three-way decisions.

I. INTRODUCTION

A S AN extension model of rough sets [33], decision-theoretic rough sets (DTRSs) are an effective soft com- puting tool for dealing with uncertain decision-making prob- lems and play a significant role in rough sets [25], [46], [51],

Manuscript received March 11, 2016; revised July 10, 2016; accepted Septem- ber 7, 2016. Date of publication November 24, 2016; date of current version November 29, 2017. This work was supported in part by the National Science Foundation of China under Grant 71401026, Grant 71432003, Grant 71571148, Grant 61273209, and Grant 71571123, in part by the Fundamental Research Funds for the Central Universities of China under Grant ZYGX2014J100, in part by the Social Science Planning Project of the Sichuan Province under Grant SC15C009, and in part by the Sichuan Youth Science and Technology Innovation Team under Grant 2016TD0013.

D. Liang is with the School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]).

Z. Xu is with the Business School, Sichuan University, Chengdu 610065, China (e-mail: [email protected]).

D. Liu is with the School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2016.2632745

[53], [59]. In light of Bayesian decision procedure [4], [6], Yao [49], [50], [53], [55] proposed DTRSs, which bridge the rough set theory and the decision theory. DTRSs not only consider decision risk factors, but also provide a reasonable semantic in- terpretation in the aspect of determining the threshold values of probabilistic rough sets [51]. DTRSs finally deduce three-way decisions with respect to the minimum expected risk, namely, the acceptance, the noncommitment, and the rejection [28], [37]. Since DTRSs were proposed, they have attracted the attention of many researchers and have been used in many domains, such as information filtering [14], investment decision making [20], [27], text classification [15], cluster analysis [57], govern- ment decision making [29], risk decision making [16], [21], and web-based support systems [47], [48]. DTRSs have become an important research direction of the combination of rough sets and the decision analysis.

In the aspect of the research works of DTRSs, Azam and Yao [1] designed a game-theoretic method for the determination of the probabilistic thresholds. Hu [12] systematically discussed three-way decisions space and three-way decisions. Jia et al. [13] discussed the determination of the thresholds and the re- duction technology for DTRSs. Li et al. [19] further extended the results of Jia et al. [13] to the neighborhood system. Li and Zhou [16] considered the decision makers’ preferences and con- structed the corresponding three-way decision-making models with DTRSs. By combining the relative and absolute informa- tion, Li and Xu [17] proposed double quantification DTRSs. Considering the different decision scenarios, Liang et al. [20], [23]–[25], and Liang and Liu [21], [22] effectively estimated the loss functions in the format of triangular fuzzy numbers, intervals, intuitionistic fuzzy numbers, and hesitant fuzzy sets (HFSs), respectively. Liu et al. [30] provided a novel three-way decision model based on the incomplete information system. Qian et al. [36] developed a multigranulation DTRS model. Then, Li and Xu [18] proposed a multigranulation DTRS model in the ordered information systems. In order to adapt the prob- abilistic rough fuzzy set, Sun et al. [37] proposed a decision- theoretic rough fuzzy set. Zhou [62] provided a new formulation of multiclass DTRSs under the multiclass environment. Zhang and Miao [60] established a fundamental reduction framework for two-category DTRSs. Yao and Zhou [54] proposed a naive Bayesian DTRS model to estimate the conditional probability. These works extend the range of applications of DTRSs and enrich its research content.

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1686 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

With the aforementioned literature, DTRSs mainly rely on the loss functions and the conditional probability [18], [24], [30]. On one hand, the loss functions are evaluated by decision makers [27], [30], [49]–[51], [53]. On the other hand, the calculation of the conditional probability is derived from information systems [24], [30], [33], [54], [61]. The purpose of DTRSs is essentially to deduce three-way decisions from information systems [25], [46]. In the practical decision-making process, we are able to encounter one type of uncertain decision situations, where the evaluation result of the information system is not single, but sev- eral values simultaneously [26], [38], [42], [44]. HFSs proposed by Torra and Narukawa [38] exactly provide us a new evalua- tion format to solve these problems [22], [39], especially when we are hesitant to our decisions. The several values in HFSs are expressed by a set of possible values rather than a margin of error or some possibility distribution values, which are the memberships of an element to a set [44]. Nowadays, information systems under hesitant fuzzy environment are seldom discussed in the area of three-way decisions with DTRSs. Observed by this background, this paper focuses on discussing the application of DTRSs in the hesitant fuzzy information system. Under the hesitant fuzzy information system, we first define a new binary relation between two objects via the distance function. Then, the calculations of the similarity class and the conditional prob- ability are designed. When we deduce three-way decisions with DTRSs, the integrated approach by considering the interaction between information systems and loss functions becomes one of challenges. Most of the existing papers assume that each object in the information system has the same values of loss functions. In this case, the relationship between information systems and loss functions is constant. As that stated by Liu et al. [30], each object has a group of loss functions. Following the idea, we generalize the results reported in most of the existing papers and earnestly investigate the situation that each object presented in the hesitant fuzzy information system may have a group of loss functions. We analyze the aggregation of the loss functions presented in the similarity class. However, these existing meth- ods of [25], [46] are time-consuming and complex to obtain the expected losses. Moreover, because the information provided by the decision maker is important, it is necessary to cover the evaluation information as much as possible [7]. Fortunately, the error analysis technique can make the calculation process much easier and cover more information, which helps us obtain the decision results conveniently and precisely [7], [45], [56]. By utilizing the error analysis method, we ascertain the expected losses with the interval information granulation [34], [35]. It describes the possibility of expected losses, in which we adopt two expression strategies of the interval [31], [32]. With the aid of the possibility degree [3], [43], [45], we further explore the decision rules by comparing the expected losses. Finally, we design a decision-making procedure of three-way decisions with DTRSs in a hesitant fuzzy information system. This study elaborates the interaction between hesitant fuzzy information systems and loss functions.

The remainder of this paper is organized as follows: Section II provides the basic concepts of DTRSs and error analysis. Some applied questions of three-way decisions with DTRSs in hesi- tant fuzzy information systems are proposed in Section III. In

TABLE I LOSS FUNCTION REGARDING THE RISK OR COST OF ACTIONS

IN DIFFERENT STATES

C (P ) ¬C (N )

aP λP P λP N aB λBP λBN aN λN P λN N

Section IV, the decision-making procedure of three-way deci- sions in hesitant fuzzy information systems is designed, includ- ing the estimation of conditional probability and the aggregation of expected losses. Then, an example of the security evaluation of peer-to-peer (P2P) lending platforms is given to illustrate our approach in Section V. Section VI concludes the paper and outlines the future work.

II. PRELIMINARIES

In this section, basic concepts of DTRSs [49], [50], [53] and error analysis [45], [56] are briefly reviewed as follows:

A. Decision-Theoretic Rough Sets (DTRSs)

Based on the Bayesian decision procedure, the DTRS model is composed of two states and three actions [53], [54]. The set of states is given by Ω = {C, ¬C} indicating that an object is in C and not in C, respectively. The set of actions is given by A = {aP , aB , aN }, where aP , aB , and aN represent three ac- tions when classifying object x, namely, deciding x ∈ POS(C), deciding x ∈ BND(C), and deciding x ∈ NEG(C), respec- tively. At this point, they correspond to the positive region, the boundary region and the negative region of rough sets [33]. The loss function regarding the risk or cost of actions in different states is given in Table I.

In Table I, λPP , λBP , and λNP denote the losses incurred for taking actions of aP , aB , and aN , respectively, when an object belongs to C. Similarly, λPN , λBN , and λNN denote the losses incurred for taking the same actions when the object belongs to ¬C. In this case, each loss function satisfies the condition λ•• ≥ 0(• = P, B, N ). Pr(C|[x]) is the conditional probability of an object x belonging to C given that the object is described by its equivalence class [x]. For an object x, the expected loss R(ai|[x]) associated with taking the individual action can be expressed as

R(aP |[x]) = λPP Pr(C|[x]) + λPN Pr(¬C|[x]), R(aB |[x]) = λBP Pr(C|[x]) + λBN Pr(¬C|[x]), R(aN |[x]) = λNP Pr(C|[x]) + λNN Pr(¬C|[x]). (1)

The Bayesian decision procedure suggests the following minimum-cost decision rules:

(P) If R(aP |[x]) ≤ R(aB |[x]) and R(aP |[x]) ≤ R(aN |[x]), decide x ∈ POS(C);

(B) If R(aB |[x]) ≤ R(aP |[x]) and R(aB |[x]) ≤ R(aN |[x]), decide x ∈ BND(C);

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LIANG et al.: NEW AGGREGATION METHOD-BASED ERROR ANALYSIS FOR DECISION-THEORETIC ROUGH SETS 1687

(N) If R(aN |[x]) ≤ R(aP |[x]) and R(aN |[x]) ≤ R(aB |[x]), decide x ∈ NEG(C)

where decision rules (P)–(N) are the three-way decisions pro- posed by Yao [53], [55]. Since Pr(C|[x]) + Pr(¬C|[x]) = 1, then we simplify the rules based only on the probability Pr(C|[x]) and 0 ≤ Pr(C|[x]) ≤ 1. By considering a reasonable kind of losses with the conditions

λPP ≤ λBP < λNP , λNN ≤ λBN < λPN . (2)

The decision rules (P)–(N) can be expressed concisely as fol- lows:

(P) If Pr(C|[x]) ≥ α and Pr(C|[x]) ≥ γ, decide x ∈ POS(C); (B) If Pr(C|[x]) ≤ α and Pr(C|[x]) ≥ β, decide x ∈ BND(C); (N) If Pr(C|[x]) ≤ β and Pr(C|[x]) ≤ γ, decide x ∈ NEG(C). The thresholds values α, β, γ are given in the form:

α = (λPN − λBN )

(λPN − λBN ) + (λBP − λPP ) ,

β = (λBN − λNN )

(λBN − λNN ) + (λNP − λBP ) ,

γ = (λPN − λNN )

(λPN − λNN ) + (λNP − λPP ) .

B. Error Analysis

In light of [7], [45], [56], let z = f (y1 , y2 , . . . , yn ) be a ran- dom function, where yi is a random variable (i = 1, 2, . . . , n). Assume that the random error of the variable yi is σ

2 y i

, then the random error of z is

σ2z = n∑

i= 1

( ∂f

∂yi

)2 σ2y i + 2

∑

1≤i< j≤n

∂f

∂yi

∂f

∂yj ρij σy i σy j (3)

where ρij is a correlation coefficient. In particular, when the random errors of the variables yi and yj are mutually indepen- dent, ρij is 0 (i, j = 1, 2, . . . , n). In this case, (3) is calculated as follows:

σ2z = n∑

i= 1

( ∂f

∂yi

)2 σ2y i . (4)

Considering the ranges of error Δyi replace the standard ran- dom error σy i , then (4) is transformed into the famous error propagation formula:

(Δz)2 = n∑

i= 1

( ∂f

∂yi

)2 (Δyi )

2 . (5)

III. QUESTION DESCRIPTIONS OF THREE-WAY DECISIONS WITH DTRSS IN HESITANT FUZZY INFORMATION SYSTEMS

In this section, we mainly discuss basic concepts of hesi- tant fuzzy information systems. Meantime, we also point out two pivotal questions during the application of three-way de- cisions with DTRSs. HFSs permit the membership degrees of

TABLE II HESITANT FUZZY INFORMATION SYSTEM

U a 1 a 2 a 3 a 4 d

x1 {0.3} {0.2} {0.08, 0.12} {0.15, 0.25} 1 x2 {0.9} {0.15, 0.25} {0.1} {0.15, 0.2, 0.25} 0 x3 {0.25, 0.35} {0.2} {0.1} {0.2} 1 x4 {0.75, 0.8, 0.85} {0.48, 0.52} {0.5, 0.7} {0.1} 0 x5 {0.8} {0.5} {0.5, 0.6, 0.7} {0.06, 0.14} 0 x6 {0.16, 0.24} {0.1} {0.5} {0.18, 0.22} 1 x7 {0.73, 0.87} {0.45, 0.5, 0.55} {0.55, 0.65} {0.1} 1 x8 {0.2} {0.1} {0.48, 0.52} {0.2} 0

an element to a set to be represented as several possible values between 0 and 1 [38], [39], [42]. The concept of HFSs is briefly reviewed as follows

Definition 1 ([38], [39]): Let S be a fixed set, an HFS on S is in terms of a function that when applied to S returns a subset of [0, 1]. To be easily understood, Xia and Xu [42] represented an HFS as the following mathematical symbol:

E = {< s, hE (s) > |s ∈ S} (6) where hE (s) is a set of values in [0, 1], denoting the possible membership degrees of the element s ∈ S to the set E. For the sake of simplicity, they called hE (s) as a hesitant fuzzy element (HFE). The number of the elements in hE (s) is #hE (s).

A hesitant fuzzy information system is a quadruple IS = (U, AT ∪ D, V, f ), where U is a nonempty finite set of ob- jects called the universe. AT is a nonempty and finite set of conditional attributes, D = {d} is a singleton of decision at- tribute and AT ∩ D = ∅. V = ∪a∈{AT∪D }Va and Va is a domain of attribute a. f : U × {AT ∪ D} → V is a function, such that f (x, a) ∈ Va for every a ∈ {AT}, f (x, d) ∈ Vd for d ∈ {D}, x ∈ U , where f (x, a) is an HFE and f (x, d) is precise. Sup- pose U/D = {C, ¬C}, Vd = {1, 0}. In what follows, a hesitant fuzzy information system is illustrated.

Example 1: During the evaluation of energy policy, Xu and Xia [44] summarized four attributes to be considered: a1 : eco- nomic, a2 : technological, a3 : environmental, a4 : socio-political. The decision attribute d is the intention of decision makers: 1: acceptance, 0: rejection. According to the set of attributes, a hes- itant fuzzy information system is constructed in Table II , where U = {x1 , x2 , . . . , x8}, AT = {a1 , a2 , a3 , a4}, and D = {d}.

From Table II, AT = {a1 , a2 , a3 , a4} is a set of conditional attributes and D = {d} is a decision attribute. The values of decision attribute correspond the classes C and ¬C, respectively. The domain of each conditional attribute is HFE. For instance, the value of the object x2 in the a2 is f (x2 , a2 ) = {0.15, 0.25}. The number of the elements in f (x2 , a2 ) is #f (x2 , a2 ) = 2.

In the framework of three-way decisions, the typical calcula- tion of conditional probability relies on the equivalence relation and the equivalence class [24], [30]. When we face the new hes- itant fuzzy information system, it poses one question: How to define the new binary relation and calculate the similarity class?

As the statement in [20], [30], for each object x of the hesitant fuzzy information system, it has a group of loss functions, i.e., λPP (x), λBP (x), λNP (x), λPN (x), λBN (x), and λNN (x), see Table III.

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1688 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

TABLE III COST TABLE (u IS A UNIT COST)

U λP P λBP λN P λP N λBN λN N

x1 u 3u 8u 40u 8.5u 0 x2 2u 9u 13u 14u 9u 0.05u x3 u 10u 15u 20u 9u 0.5u x4 u 10u 16u 30u 9.5u 1.5u x5 0.05u 10.2u 14u 18u 10u u x6 1.05u 10.2u 15u 20u 10u 1.5u x7 2u 14.2u 23u 21u 15u 2u x8 2.5u 15u 21u 23u 10u 1.5u

Compared with Table II, Table III adds the loss functions of Table I and measures the loss functions for each object. Taken the object x1 for example, its corresponding loss functions are λPP (x1 ) = u, λBP (x1 ) = 3u, λNP (x1 ) = 8u, λPN (x1 ) = 40u, λBN (x1 ) = 8.5u, and λNN (x1 ) = 0. Based on the result of [30], we define the three-way decisions-based hesitant fuzzy information system as follows

Definition 2: A three-way decisions-based hesitant fuzzy information system is TIS = (IS, CT), where IS = (U, AT ∪ D, V, f ) is a hesitant fuzzy information system, CT = (U, λPP

⋃ λBP

⋃ λNP

⋃ λPN

⋃ λBN

⋃ λNN ) is a cost table.

For the hesitant fuzzy information system, U is a nonempty finite set of objects called the universe. AT is a nonempty and finite set of conditional attributes, D = {d} is a singleton of decision attribute and AT ∩ D = ∅. V = ∪a∈{AT∪D }Va and Va is a domain of attribute a. f : U × {AT ∪ D} → V is a func- tion, such that f (x, a) ∈ Va for every a ∈ {AT}, f (x, d) ∈ Vd for d ∈ {D}, x ∈ U , where f (x, a) is an HFE and f (x, d) is precise. For the value of loss functions of the object x ∈ U , it is denoted as λ••(x)(• = P, B, N ).

From Definition 2, we construct a hybrid information table by combining the hesitant fuzzy information system and loss func- tion [30]. Considering the interaction between hesitant fuzzy information systems and loss functions, there also exists a ques- tion: How to deal with the aggregation of the loss functions pre- sented in the similarity class and determine the expected losses. In the background of the hesitant fuzzy information system, we utilize the error analysis to explore the application of three-way decisions by solving these above-mentioned questions.

IV. DECISION-MAKING PROCEDURE OF THREE-WAY DECISIONS IN HESITANT FUZZY INFORMATION SYSTEMS

As the statement of Section III, there are two questions: 1) How to define the new binary relation and calculate the similar- ity class? and 2) How to deal with the aggregation of the loss functions presented in the similarity class and determine the ex- pected losses? Especially when we deduce three-way decisions with DTRSs, the integrated approach by considering the inter- action between information systems and loss functions becomes one of challenges. In this section, we derive three-way decisions from hesitant information systems by exerting the error analy- sis method. Inspired by the results of [18], [30], we first need to define a new binary relation of objects presented in hesitant fuzzy information systems and generate the corresponding sim-

ilarity classes. For each object, we also compute its conditional probability. With the aid of error analysis, we obtain the aggre- gation of the loss functions presented in the similarity class and estimate the expected losses. The decision-making procedure based on hesitant information systems with three-way decisions with DTRSs is finally designed.

A. Estimation of the Conditional Probability

The fundamental analysis of classical three-way decisions with DTRSs is on the basis of equivalence relation [30], [53]. However, the equivalence relation is not suitable for hesitant fuzzy information systems. The main reason is that the evalua- tion results of conditional attributes are HFEs. In order to deal with numerical data, Hu et al. [9], [10] generalized rough sets with the distance-based neighborhood relations. Li et al. [19] also constructed the neighborhood-based DTRS model. For the distance measure of HFEs, Xu and Xia [44] discussed some typ- ical calculation methods in detail and indicated that the normal- ization of HFEs needs to be finished in advance. For example, we have two HFEs h1 = {0.1, 0.2, 0.3} and h2 = {0.4, 0.5}. In this case, #h1 = 3 and #h2 = 2. Obviously, #h1 = #h2 . To operate correctly in computation, we should extend the shorter one until both of them have the same length when we compare them. Hence, Xu and Xia [44] proposed that we could extend the shorter one by adding any value in it which mainly depends on the decision maker’s risk preference. To go along with this, two types of the normalization of [44] are summarized as follows:

Type I: As an optimist decision maker, he (she) may add the maximum value.

Type II: As a pessimist decision maker, he (she) may add the minimum value.

For two HFEs h1 = {0.1, 0.2, 0.3} and h2 = {0.4, 0.5}, we should extend h2 until it has the same length with h1 . The op- timist may extend h2 as h2 = {0.4, 0.5, 0.5} and the pessimist may extend it as h2 = {0.4, 0.4, 0.5}. Inspired by the distance measure of HFEs presented in [44] and the neighborhood re- lation of [9], we redefine a new binary (similar) relation in the hesitant fuzzy information system

Definition 3: Let IS = (U, AT ∪ D, V, f ) be a hesitant fuzzy information system, A ⊆ AT, for x, y ∈ U , the new binary re- lation is denoted by

NA = {(x, y) ∈ U × U|dA (x, y) ≤ δ} (7)

where δ is a constant and 0 ≤ δ. dA (x, y) is a distance function and the Euclidean distance is widely applied:

dA (x, y) = [

1 |A|

∑

a∈A

( 1 la

la∑

j = 1

(f σ (j ) (x, a) − f σ (j ) (y, a))2 )]1

2

(8) where |A| denotes the cardinality of set A and la = max{#f (x, a), #f (y, a)}. f σ (j ) (x, a) and f σ (j ) (y, a) are the jth largest values in f (x, a) and f (y, a). Note that f (x, a) and f (y, a) have finished the normalization step.

The new binary relation is a reflexive and symmetry relation, but may not be a transitive relation. In what follows, the concept of the similarity class is given.

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LIANG et al.: NEW AGGREGATION METHOD-BASED ERROR ANALYSIS FOR DECISION-THEORETIC ROUGH SETS 1689

TABLE IV DISTANCES BETWEEN x1 AND OTHER OBJECTS

dA (x, y) x1 x2 x3 x4 x5 x6 x7 x8

x1 0 0.3016 0.0367 0.3908 0.3866 0.2138 0.3905 0.2136

Definition 4: Let IS = (U, AT ∪ D, V, f ) be a hesitant fuzzy information system, given x ∈ U and A ⊆ AT, the similarity class of x with respect to A is defined as

δA (x) = {y|y ∈ U, dA (x, y) ≤ δ}. (9) From Definition 4, the similarity classes of the universe gen-

erated by the new binary relation forms a covering. For example, suppose that U/NA = {b1 , b2 , . . . , bn } is a covering of a uni- verse U , namely,

⋃n i= 1 bi = U and bi ∩ bj = ∅ for i = j. In

light of the results presented in [19], the conditional probability under the new binary relation NA can be redefined as follows.

Definition 5: Let IS = (U, AT ∪ D, V, f ) be a hesitant fuzzy information system, U/D = {C, ¬C}. Given x ∈ U and A ⊆ AT, the conditional probability of x belonging to C with respect to A is defined as

Pr(C|δA (x)) = |δA (x) ∩ C| |δA (x)|

(10)

where |δA (x)| denotes the cardinality of the set δA (x), Pr(¬C|δA (x)) = 1 − Pr(C|δA (x)).

Based on the similarity class of (9), Definition 5 provides a solution for estimating the conditional probability of the object in hesitant fuzzy information systems. As the statement in [51], (10) is determined by the rough membership function. In this case, the rough membership function is given by the conditional probability. For clarity, we use an example to explain it.

Example 2: Continued with Example 1, we further use Ta- ble II to discuss the calculation of the conditional probabil- ity for the object x1 . In the hesitant fuzzy information sys- tem, U = {x1 , x2 , . . . , x8}, AT = {a1 , a2 , a3 , a4} and D = {d}. Let A = AT and U/D = {C, ¬C}. In this example, C = {x1 , x3 , x6 , x7}. We assume that the decision maker is an optimist and δ = 0.05. During the comparison, the normal- ization of HFEs is implemented by adding the maximum value. According to the Euclidean distance (8), the distances between x1 and other objects are computed in Table IV .

For Table IV, it shows the distance results between x1 and all objects presented in the hesitant fuzzy information system. We select the objects x1 and x2 to illustrate the calculation process of the distance. First, we compare the number of attribute values of these two objects in each at- tribute and confirm the normalization. Under the attribute a1 , f (x1 , a1 ) = {0.3} and f (x2 , a1 ) = {0.9}. The numbers of f (x1 , a1 ) and f (x2 , a1 ) are the same. In this situation, they do not need to carry out the normalization and la 1 = 1. Under the attribute a2 , f (x1 , a2 ) = {0.2} and f (x2 , a2 ) = {0.15, 0.25}. The numbers of f (x1 , a2 ) and f (x2 , a2 ) are differ- ent. With the help of the normalization, we extend f (x1 , a2 ) = {0.2} as f (x1 , a2 ) = {0.2, 0.2} and la 2 = 2. Analogously, the

normalization results of x1 and x2 under the at- tributes a3 and a4 are f (x1 , a3 ) = {0.08, 0.12}, f (x2 , a3 ) = {0.1, 0.1}, f (x1 , a4 ) = {0.15, 0.25, 0.25}, and f (x2 , a4 ) = {0.15, 0.2, 0.25}. Meantime, la 3 = 2 and la 4 = 3. According to (8), the distance between x1 and x2 is calculated as follows:

dA (x1 , x2 ) = [

1 |A|

∑

a∈A

( 1 la

la∑

j = 1

(f σ (j ) (x, a) − f σ (j ) (y, a))2 )]1

2

.

= 0.3016.

In a similar way, we can compute other distances, see Table IV. In light of (9) and δ = 0.05, we can determine δA (x1 ) = {x1 , x3}. On the basis of (10), we finally compute the conditional probability Pr(C|δA (x1 )) = |δA (x1 )∩C ||δA (x1 )| = 1.

B. Three-Way Decisions With DTRSs Using Error Analysis

According to the results reported in [23], [30] and Definition 2, each object may have a group of loss functions, namely, λ••(xi ), where xi denotes an object (i = 1, 2, . . . , m; • = P, B, N ). From another perspective, we also say that each loss function of Table I has multiple values. For example, the val- ues of λPN in Table III have λPN (x1 ) = 40u, λPN (x2 ) = 14u, · · · , λPN (x8 ) = 23u. In this case, λPP , λBP , λNP , λPN , λBN , and λNN of Table I are regarded as the random variables of the expected losses and they are mutually independent. Liang et al. [20] and Liu et al. [30] also proposed that each object has a group of loss functions. In this section, we consider the situa- tion that the loss functions of Table I may have multiple values for each similarity class. We mainly analyze the aggregation of loss functions by using error analysis, in which we adopt two classical expression strategies of the interval reported in [8], [11], [41], [45]. The error analysis, not only can save amount of time and simplify the calculation procedure for expected losses [7], [45], but also describes the possibility of expected losses by covering all the information in a simple and convenient way. In the view of the possibility degree of [3], [11], [21], [43], we further discuss the deduction of three-way decisions with DTRSs.

1) The Determinations of Expected Losses With the Aid of Error Analysis: Suppose that the probability Pr(C|δA (x)) of the hesitant fuzzy information system is constant and x ∈ {x1 , x2 , . . . , xm }. From the angle of error analysis, the expected losses of (1) are reinterpreted as follows:

R(a•|δA (x)) = f (λ•P , λ•N ), = λ•P Pr(C|δA (x)) + λ•N Pr(¬C|δA (x)) (11)

where • = P, B, N . Then, we obtain ∂f (λ•P , λ•N )

λ•P = Pr(C|δA (x)), (12)

∂f (λ•P , λ•N ) λ•N

= Pr(¬C|δA (x)). (13)

In light of the results reported in [22], [23], and (12) and (13), we obtain the variations of the expected losses with the loss functions of Table I.

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1690 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

Proposition 1: Let R(a•|δA (x)) = λ•P Pr(C|δA (x)) + λ•N Pr(¬C|δA (x)). When Pr(C|δA (x)) is constant, R(a•|δA (x)) is nonmonotonically decreasing with the increase of λ•P and λ•N (• = P, B, N ).

With the aid of error analysis, we can determine the value of expected losses with intervals in advance [34], [35]. At this point, the determination of reference point of the loss function is pivotal. Based on the results of [7], [45], the average value is normally used as a reference point. For each object of the hesitant fuzzy information system, the average value of the loss function λ•• of the similarity class of x is

λ••(x) = ∑

xi ∈δA (x) λ••(xi ). (14)

Meanwhile, the minimum and the maximum of loss function λ•• is

λ − ••(x) = min

xi ∈δA (x) λ••(xi ), (15)

λ + ••(x) = max

xi ∈δA (x) λ••(xi ) (16)

where • = P, B, N . Under the average value strategy, the cor- responding expected losses are calculated as follows:

R(a•|δA (x)) = λ•P (x)Pr(C|δA (x)) + λ•N (x)Pr(¬C|δA (x)). (17)

According to the results of (14), we deduce

R(a•|δA (x)) = ∑

xi ∈δA (x) λ•P (xi )Pr(C|δA (x))

+ ∑

xi ∈δA (x) λ•N (xi )Pr(¬C|δA (x)). (18)

Since the average of the loss function λ•• may not be the center of the interval [λ−••, λ

+ ••], then the corresponding error

values should be denoted as

Δλl••(x) = λ••(x) − λ−••(x), (19) Δλr••(x) = λ

+ ••(x) − λ••(x). (20)

By using the error propagation formula (5), we further measure the left error ΔR(a•|δA (x))l and the right error ΔR(a•|δA (x))r of the expected loss R(a•|δA (x)) as

(ΔR(a•|δA (x))l )2 = Pr(C|δA (x))2 (λ•P (x) − λ−•P (x))2

+ Pr(¬C|δA (x))2 (λ•N (x) − λ−•N (x))2 , (21) (ΔR(a•|δA (x))r )2 = Pr(C|δA (x))2 (λ+•P (x) − λ•P (x))2

+ Pr(¬C|δA (x))2 (λ+•N (x) − λ•N (x))2 . (22) On the basis of error analysis, we obtain an uncer-

tain expected loss for x: ˜R(a•|δA (x)) = [R(a•|δA (x))l , R(a•| δA (x))u ], namely, ˜R(a•|δA (x)) = [R(a•|δA (x)) − ΔR(a•| δA (x))l , R(a•|δA (x)) + ΔR(a•|δA (x))r ].

Theorem 1: Let R(a•|δA (x)), ΔR(a•|δA (x))l and ΔR(a•| δA (x))r be deduced from (18), (21), and (22), then we can derive the following relationships: 0 ≤

Fig. 1. Two classical interval expression strategies.

R(a•|δA (x)) −ΔR(a•|δA (x))l ≤R(a•|δA (x))≤R(a•|δA (x)) + ΔR(a•|δA (x))r (• = P, B, N ).

Proof: According to (17), (21), and (22), we obtain the following conditions: R(a•|δA (x)) ≥ 0, ΔR(a•| δA (x))l ≥ 0, ΔR(a•|δA (x))r ≥ 0. With the above re- sults, R(a•|δA (x)) − ΔR(a•|δA (x))l ≤ R(a•|δA (x)) ≤ R(a•|δA (x)) + ΔR(a•|δA (x))r holds. We prove the relation- ship between R(a•|δA (x)) and ΔR(a•|δA (x))l . Then, we have

(R(a•|δA (x)))2 − (ΔR(a•|δA (x))l )2

= 2λ•P (x)Pr(C|δA (x))λ•N (x)Pr(¬C|δA (x)) + Pr(C|δA (x))2 λ−•P (x)(2λ•P (x) − λ−•P (x)) + Pr(¬C|δA (x))2 λ−•N (x)(2λ•N (x) − λ−•N (x)).

Based on (14) and (15), we obtain 2λ•P (x) − λ−•P (x) ≥ 0 and 2λ•N (x) − λ−•N (x) ≥ 0. In this case, (R(a•|δA (x)))2 − (ΔR(a•|δA (x))l )2 ≥ 0, i.e., R(a•|δA (x)) − ΔR(a•|δA (x))l ≥ 0. Therefore, the statement of Theorem 1 holds.

Theorem 1 roughly depicts the range of uncertain expected

losses ˜R(a•|δA (x))(• = P, B, N ). As the results reported in [7], [45], the error analysis can save an amount of time and sim- plify the calculation procedure of the expected losses for each object. Following the interval information granulation of [7], [34], [45], the final result of each expected loss generates an in-

terval number ˜R(a•|δA (x)) = [R(a•|δA (x))l , R(a•|δA (x))u ]. With respect to the interval number, it is generally indicated by

the lower bound and the upper bound. In most of the literature, it is considered that the values of the interval number are uniformly distributed [7], [45]. Each value of the interval can be considered of equal opportunity, e.g., the interval R̃ = [Rl , Ru ] is described as (a) of Fig. 1. Bu and Zhang [2] summarized that the interval with two parameters may produce large errors. To do this, they proposed a new representation of interval number with three parameters, e.g., (b) of Fig. 1.

From Fig. 1, it shows two classical distributions of the interval R̃. R denotes the most likely value of the interval R̃ = [Rl , Ru ], which can be measured by the average. The interval number with three parameters not only maintains the ranges of parameters, but also highlights the most likely value [8], [11], [41]. Based on the new representation, Hu et al. [8], [11] designed some new methods in the multiple attribute decision making. Wang

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LIANG et al.: NEW AGGREGATION METHOD-BASED ERROR ANALYSIS FOR DECISION-THEORETIC ROUGH SETS 1691

TABLE V RESULTS OF EXPECTED LOSSES WITH DIFFERENT EXPRESSION STRATEGIES

The expected loss The interval with two parameters The interval with three parameters

˜R(aP |δA (x)) [R(aP |δA (x))l , R(aP |δA (x))u ] [R(aP |δA (x))l , R(aP |δA (x)), R(aP |δA (x))u ] ˜R(aB |δA (x)) [R(aB |δA (x))l , R(aB |δA (x))u ] [R(aB |δA (x))l , R(aB |δA (x)), R(aB |δA (x))u ] ˜R(aN |δA (x)) [R(aN |δA (x))l , R(aN |δA (x))u ] [R(aN |δA (x))l , R(aN |δA (x)), R(aN |δA (x))u ]

and Liu [41] applied the representation into the DEA model. By using the three parameters, the interval R̃ can be denoted as R̃ = [Rl , R, Ru ]. For example, Hu et al. [11] also defined the corresponding membership function of R̃, which is shown as follows

μ R̃ (x) = [Rl , R, Ru ] =

⎧ ⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

x − Rl R − Rl , R

l ≤ x ≤ R; x − Ru R − Ru , R ≤ x ≤ R

u ;

0, otherwise.

(23)

Considering the different expression strategies of the interval,

we finally determine the expected losses ˜R(a•|δA (x)). Their results are shown in Table V.

2) Three-Way Decisions Deduced by Comparing Expected Losses: In light of the Bayesian decision procedure, it suggests the following minimum-cost decision rules:

(P) If ˜R(aP |δA (x)) � ˜R(aB |δA (x)) and ˜R(aP |δA (x)) � ˜R(aN |δA (x)), decide x ∈ POS(C);

(B) If ˜R(aB |δA (x)) � ˜R(aP |δA (x)) and ˜R(aB |δA (x)) � ˜R(aN |δA (x)), decide x ∈ BND(C);

(N) If ˜R(aN |δA (x)) � ˜R(aP |δA (x)) and ˜R(aN |δA (x)) � ˜R(aB |δA (x)), decide x ∈ NEG(C)

where the decision rules (P)–(N) are the three-way decisions [49]–[51], [53]. The three-way decisions comprise the positive rules (x ∈ POS(C)), the boundary rules (x ∈ BND(C)), and the negative rules (x ∈ NEG(C)). In viewpoint of the general semantics, the positive rules make decisions of acceptance. The negative rules make decisions of rejection, and the boundary rules make decisions of noncommitment. In this case, the ex- pected losses are intervals. The final decision result mainly de- pends on pairwise comparisons of the expected losses, which is essentially the possibility degree of [3], [11], [21], [43]. In view of the different expression strategies of intervals of Table V, we discuss the comparisons among expected losses, respectively.

a) Strategy 1: The interval with two parameters. Under the two parameters strategy, the expected loss ˜R(a•|δA (x)) is ˜R(a•|δA (x)) = [R(a•|δA (x))l , R(a•|δA (x))u ],

where R(a•|δA (x))l = R(a•|δA (x)) − ΔR(a•|δA (x))l and R (a•|δA (x))u = R(a•|δA (x)) + ΔR(a•|δA (x))r (•= P, B, N ). For the result of Xu and Da [43], given two intervals

R̃1 = [Rl1 , R r 1 ], R̃2 = [R

l 2 , R

r 2 ], then the possibility degree of

R̃1 ≥ R̃2 is defined as:

p(R̃1 ≥ R̃2 )1 = max {

1 − max (

Rr2 − Rl1 d(R̃1 ) + d(R̃2 )

, 0

) , 0

} .

(24) where d(R̃1 ) = Rr1 − Rl1 and d(R̃2 ) = Rr2 − Rl2 . To rank the expected losses ˜R(a•|δA (x)), we can compare each pair of the expected losses by using the possible degree formula.

b) Strategy 2: The interval with three parameters. Under the three parameters strategy, the expected loss is ˜R(a•|δA (x)) = [R(a•|δA (x))l , R(a•|δA (x)), R(a•|δA (x))u ],

where R(a•|δA (x))l = R(a•|δA (x)) − ΔR(a•|δA (x))l and R (a•|δA (x))u = R(a•|δA (x)) + ΔR(a•|δA (x))r (• = P, B, N ). For the result of Chen and Lee [3], given two intervals R̃1 = [Rl1 , R1 , R

r 1 ], R̃2 = [R

l 2 , R2 , R

r 2 ], then the possibility

degree of R̃1 ≥ R̃2 is defined as:

p(R̃1 ≥ R̃2 )2 = max {

1 − max (

f1 f2

, 0 )

, 0 }

. (25)

where

f1 = max(R l 2 − Rl1 , 0) + 2max(R2 − R1 , 0)

+ max(Rr2 − Rr1 , 0) + Rr2 − Rl1 , f2 = |Rl2 − Rl1| + 2|R2 − R1| + |Rr2 − Rr1|

+ Rr1 − Rl1 + Rr2 − Rl2 .

To rank the expected losses ˜R(a•|δA (x)), we also can compare each pair of the expected losses by using the possible degree formula (25).

To rank intervals, Xu and Da [43] constructed a possibility degree matrix and sum all elements in each line of the matrix as principle to rank intervals. According to (24) and (25), we further discuss the derivation of the decision rules. On the basis of the results reported in [21], the possibility degree matrix P∇

is constructed as shown in Table VI (∇ = 1, 2). According to the properties of the possibility degree [3], [43],

we have: p∇ij ≥ 0, p∇ij + p∇j i = 1, p∇ii = 0.5 (i, j = 1, 2, 3; ∇ = 1, 2). Following the idea of [21], the possibility degree matrix P∇ is further simplified as

P∇ =

⎛

⎜⎜ ⎝

0.5 p∇12 p ∇ 13

1 − p∇12 0.5 p∇23 1 − p∇13 1 − p∇23 0.5

⎞

⎟⎟ ⎠ .

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1692 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

TABLE VI POSSIBILITY DEGREE MATRIX BETWEEN THE EXPECTED LOSSES

P ˜R(aP |δA (x)) ˜R(aB |δA (x)) ˜R(aN |δA (x))

˜R(aP |δA (x)) p∇1 1 = p( ˜R(aP |δA (x)) ≥ ˜R(aP |δA (x)))∇ p∇1 2 = p( ˜R(aP |δA (x)) ≥ ˜R(aB |δA (x)))∇ p∇1 3 = p( ˜R(aP |δA (x)) ≥ ˜R(aN |δA (x)))∇ ˜R(aB |δA (x)) p∇2 1 = p( ˜R(aB |δA (x)) ≥ ˜R(aP |δA (x)))∇ p∇2 2 = p( ˜R(aB |δA (x)) ≥ ˜R(aB |δA (x)))∇ p∇2 3 = p( ˜R(aB |δA (x)) ≥ ˜R(aN |δA (x)))∇ ˜R(aN |δA (x)) p∇3 1 = p( ˜R(aN |δA (x))) ≥ ˜R(aP |δA (x))∇ p∇3 2 = p( ˜R(aN |δA (x))) ≥ ˜R(aB |δA (x))∇ p∇3 3 = p( ˜R(aN |δA (x))) ≥ ˜R(aN |δA (x))∇

Note: In the table, ∇ = 1, 2 denotes the different strategies.

In this case, the matrix needs only to calculate p∇12 , p ∇ 13 , and p

∇ 23 .

In light of the results of [43], [45], all elements in each line of the matrix P∇ are summarized:

p∇i = 3∑

j = 1

p∇ij (26)

where i = 1, 2, 3. More specifically, all elements in each line of the matrix P∇ are calculated as follows:

p∇1 = 0.5 + p ∇ 12 + p

∇ 13 ; (27)

p∇2 = 1.5 + p ∇ 23 − p∇12 ; (28)

p∇3 = 2.5 − p∇13 − p∇23 (29) where p∇1 corresponds to the total possibility degree of

˜R(aP |δA (x)), p∇2 is the total possibility degree of ˜R(aB |δA (x)), and p∇3 is the total possibility degree of ˜R(aN |δA (x)). Based on the decision rules (P)–(N), we then rank the expected losses in accordance with the value of p∇i . Hence, the decision rules (P)–(N) can be restated as

(P1) If p∇1 ≤ p∇2 and p∇1 ≤ p∇3 , decide x ∈ POS(C); (B1) If p∇2 ≤ p∇1 and p∇2 ≤ p∇3 , decide x ∈ BND(C); (N1) If p∇3 ≤ p∇1 and p∇3 ≤ p∇1 , decide x ∈ NEG(C)

where the decision rules (P1)–(N1) are three-way decisions.

C. Decision-Making Procedure

With the help of the above-mentioned results, we design a decision-making procedure of three-way decisions in hesitant fuzzy information systems. It is described in Fig. 2.

For Fig. 2, the key steps are elaborated as follows: Step 1: Based on the practical context, we collect the

evaluations and obtain a three-way decisions based hesitant fuzzy information system TIS = (IS, CT), where the hesitant fuzzy information system is indicated as IS = (U, AT ∪ D, V, f ) and the cost table is CT = (U, λPP

⋃ λBP

⋃ λNP⋃

λPN ⋃

λBN ⋃

λNN ). Step 2: Let A = AT and we determine the meaning of C.

According to (8) and the normalization method of HFEs, we compute the distance dA (x, y) between any two objects in the information system and construct the corresponding distance matrix d∇A (∇ = 1, 2).

Step 3: Given the value of δ, for each object x ∈ U , we ascertain its corresponding similarity class δA (x) based on (9) and d∇A (∇ = 1, 2).

Fig. 2. Decision procedure of the application of three-way decisions in a hesitant fuzzy information system.

Step 4: On one hand, the conditional probability of x be- longing to C is computed by using (10), which is denoted as Pr(C|δA (x)). On the other hand, a group of loss functions of the object x are obtained, i.e., λ••(xi ) and xi ∈ δA (x)(• = P, B, N ).

Step 5: Considering the application background, the expres- sion strategy of the interval is selected. On the basis of (18), (21), and (22), we determine the expected losses for each object with the aid of error analysis.

Step 6: In light of the possibility degree formula (24) or (25), the corresponding possibility degree matrix P∇ for each object is built. According to (26)–(28), we further confirm the judgement criteria (P1)–(N1)(∇ = 1, 2).

In view of the above steps, the normalization method of HFEs and the comparison methods of the intervals play an important role in our decision analysis. For this, there are four cases for supporting the decision making, which is summarized in Ta- ble VII.

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LIANG et al.: NEW AGGREGATION METHOD-BASED ERROR ANALYSIS FOR DECISION-THEORETIC ROUGH SETS 1693

TABLE VII METHODS BASED ON THE COMBINATIONS OF BETWEEN THE NORMALIZATION METHODS OF HFES AND THE EXPRESSION

STRATEGIES OF THE EXPECTED LOSSES

Analytical methods The expression strategy of the interval

Strategy 1 Strategy 2

The normalization Type I Method 1 Method 2 method of HFE Type II Method 3 Method 4

From Table VII, Methods 1–4 mainly depend on the com- binations of between the normalization methods of HFEs and the expression strategies of the expected losses. For example, Method 1 is obtained by combining Type I and Strategy 1. In the application, we successively analyze these methods and com- pare them.

V. ILLUSTRATIVE EXAMPLE

In recent years, Internet finance opens up a revolutionary means of financing other than resorting to the capital markets or conventional banks. It heralds an era in which all market participants can borrow and lend directly on the Internet with few information barriers. In China, it has attracted great atten- tion by some companies and persons. One of popular modes in the Internet finance product is online P2P lending [58], such as CreditEase, Lufax, Tuandai, China Rapid Finance, and Di- anRong. This loan market could offer certain benefits to both borrowers and lenders [5]. The lenders often earn higher returns compared to savings and investment products offered by banks, while the borrowers can borrow money at lower interest rates [40]. Until the end of December 2015, the P2P platform has reached 2595. At the same time, we also often hear the collapse of the platform one by one. The lender’s investment in the loan is not normally protected by any government guarantee [40]. Aiming at this risk problem, we will illustrate the application of three-way decisions in hesitant fuzzy information systems by an example of the security evaluation of P2P lending platforms in this section.

Based on the P2P lending, we collect the data and construct a model of three-way decisions based on the hesitant fuzzy information system TIS = (IS, CT). The information system is shown in Table VIII.

For Table VIII, there are 12 P2P lending platforms xi (i = 1, 2, . . . , 12), and four factors to be considered:

1) a1 : credit risks; 2) a2 : ethical risk; 3) a3 : policy risk; 4) a4 : system risk. The decision attribute d denotes the security of P2P lend-

ing platform: 1: trustworthiness, 0: untrustworthiness, which corresponds to the states Ω = {C, ¬C}. In light of the hes- itant fuzzy information system IS = (U, AT ∪ D, V, f ), U = {x1 , x2 , . . . , x12}, AT = A = {a1 , a2 , a3 , a4} and the evalua- tions of the conditional attributes are HFEs. In the background of three-way decisions, the set of the states Ω = {C, ¬C} in- dicates that the P2P lending platform is reliable or unreliable,

respectively. In this case, C = {x1 , x3 , x6 , x9 , x10 , x12}. The set of actions for the P2P lending platform is given by A = {aP , aB , aN }, where aP , aB , and aN represent the acceptance decision, the deferment decision, and the rejection decision. For the cost table CT, we consider six types of loss functions, i.e., λPP , λBP , λNP , λPN , λBN , and λNN . On the basis of the re- sults presented in Table I, λPP , λBP , and λNP denote the losses incurred for taking actions of the acceptance, the deferment decision and the rejection decision respectively, when the P2P lending platform belongs to a reliable platform. Similarly, λPN , λBN and λNN denote the losses incurred for taking actions of the acceptance, the deferment decision and the rejection deci- sion when the platform is unreliable. The conditional probability Pr(C|δA (xi )) is the probability of a platform xi being a reliable one.

In light of the decision procedure of Section IV, we utilize the methods of Table VII to elaborate the application of three- way decisions and compare the results of different methods, i.e., Methods 1–4. For the sake of brevity, we mainly introduce the procedures of Methods 1 and 4.

1) Method 1: Type I and Strategy 1 For Method 1, it shows that the normalization method of HFEs

is implemented by adding the maximum value. According to (8), we can compute the distance dA (x, y) between any two objects of Table VIII and obtain the corresponding distance matrix d1A . Then, we further determine the similarity class of each object based on (9) when the value of δ is constant. The amount of the objects presented in the similarity class under the different values of δ is summarized in Fig. 3.

From Fig. 3, we discuss three kinds of δ, namely, δ = 0.02, δ = 0.1, and δ = 0.2. With the increase of δ, the amount of the objects presented in the similarity class are inclined to amplify. For each object x ∈ U , given the value of δ is 0.1, we confirm its corresponding similarity class δA (x) and Pr(C|δA (x)) based on (9) and (10), respectively. The results are shown in Table IX.

On the basis of (18), (21), and (22), we determine the expected losses for each object with the aid of error analysis. In this situation, we use Strategy 1 to express the expected losses, i.e., the interval with two parameters, see Table X.

For each object, we compute the total possibility degrees of the expected losses based on (24) and (26)–(28). The results of p11 , p

1 2 , and p

1 3 of each object are shown in Fig. 4.

On the basis of the judgement criteria (P1)–(N1) and the results presented in Fig. 4, we can obtain the following results: x1 , x2 , x3 , x10 ∈ POS(C) and x4 , x5 , x6 , x7 , x8 , x9 , x11 , x12 ∈ NEG(C). They imply that the P2P lending plat- forms x1 , x2 , x3 , x10 can be accepted, while x4 , x5 , x6 , x7 , x8 , x9 , x11 , x12 should be rejected.

2) Method 4: Type II and Strategy 2 For Method 4, it shows that the normalization method of HFEs

is implemented by adding the minimum value. According to (8), we also can compute the distance matrix d2A . Suppose that the value of δ is 0.1, we confirm the corresponding similarity class δA (x) and Pr(C|δA (x)) based on (9) and (10). The results are the same with those of Method 1, which are summarized in Table IX. With the aid of error analysis method, we use Strategy 2 to denote the expected losses, i.e., the interval with three parameters. The results are shown in Table XI.

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1694 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

TABLE VIII THREE-WAY DECISIONS-BASED HESITANT FUZZY INFORMATION SYSTEM (u IS A UNIT COST)

U a1 a2 a3 a4 d λP P λBP λN P λP N λBN λN N

x1 {0.7} {0.8} {0.88, 0.92} {0.75, 0.85} 1 u 3u 8u 15u 8.5u 0.8u x2 {0.7} {0.75, 0.85} {0.85} {0.8, 0.82} 0 0.5u 4u 7u 12.5u 10u u x3 {0.65, 0.75} {0.74} {0.9} {0.8} 1 2u 3u 6u 13u 12u 1.5u x4 {0.35, 0.4, 0.45} {0.48, 0.52} {0.3, 0.5} {0.6} 0 2.5u 5u 21u 23u 18u 1.5u x5 {0.4} {0.5} {0.3, 0.4, 0.5} {0.56, 0.64} 0 3u 7u 24u 26u 21u 2.5u x6 {0.76, 0.84} {0.9} {0.7} {0.86, 0.94} 1 1.05u 6u 15u 14u 10u 2u x7 {0.33, 0.47} {0.45, 0.5, 0.55} {0.35, 0.45} {0.6} 0 2u 6.8u 22u 24u 23.4u 2u x8 {0.8} {0.9} {0.68, 0.72} {0.9} 0 2.5u 8.2u 10u 12u 8u 2.5u x9 {0.4} {0.5} {0.25, 0.55} {0.6} 1 2.8u 8.2u 21u 26.6u 24u 3u x1 0 {0.68, 0.72} {0.78} {0.86, 0.88} {0.8} 1 3u 6u 9u 11u 9u 2.5u x1 1 {0.85, 0.88} {0.9} {0.66, 0.7} {0.82, 0.88, 0.9} 0 1.8u 7u 12u 13.8u 11u 1.8u x1 2 {0.8} {0.88, 0.92} {0.68, 0.72} {0.88, 0.92} 1 2u 9.2u 13u 14u 12u 2u

Fig. 3. Amount of the objects presented in the similarity class under the different values of δ.

TABLE IX SIMILARITY CLASS AND ITS CORRESPONDING CONDITIONAL PROBABILITY

OF EACH OBJECT TO C WITH δ = 0.1

U δA (x) Pr(C|δA (x))

x1 {x1 , x2 , x3 , x1 0 } 3/4 x2 {x1 , x2 , x3 , x1 0 } 3/4 x3 {x1 , x2 , x3 , x1 0 } 3/4 x4 {x4 , x5 , x7 , x9 } 1/4 x5 {x4 , x5 , x7 , x9 } 1/4 x6 {x6 , x8 , x1 1 , x1 2 } 1/2 x7 {x4 , x5 , x7 , x9 } 1/4 x8 {x6 , x8 , x1 1 , x1 2 } 1/2 x9 {x4 , x5 , x7 , x9 } 1/4 x1 0 {x1 , x2 , x3 , x1 0 } 3/4 x1 1 {x6 , x8 , x1 1 , x1 2 } 1/2 x1 2 {x6 , x8 , x1 1 , x1 2 } 1/2

For each object, we compute the total possibility degrees of the expected losses based on (25) and (26)–(28). The results of p21 , p

2 2 , and p

2 3 of each object are shown in Fig. 5.

TABLE X RESULTS OF EXPECTED LOSSES BASED ON STRATEGY 1

U ˜R(aP |δA (x)) ˜R(aB |δA (x)) ˜R(aN |δA (x))

x1 [3.4723u, 5.5975u] [4.6437u, 7.0600u] [4.8508u, 7.1427u] x2 [3.4723u, 5.5975u] [4.6437u, 7.0600u] [4.8508u, 7.1427u] x3 [3.4723u, 5.5975u] [4.6437u, 7.0600u] [4.8508u, 7.1427u] x4 [17.8865u, 20.5982u] [15.1523u, 19.7236u] [6.5719u, 7.9401u] x5 [17.8865u, 20.5982u] [15.1523u, 19.7236u] [6.5719u, 7.9401u] x6 [6.8187u, 8.0743u] [7.5446u, 10.1106u] [6.0300u, 8.5554u] x7 [17.8865u, 20.5982u] [15.1523u, 19.7236u] [6.5719u, 7.9401u] x8 [6.8187u, 8.0743u] [7.5446u, 10.1106u] [6.0300u, 8.5554u] x9 [17.8865u, 20.5982u] [15.1523u, 19.7236u] [6.5719u, 7.9401u] x1 0 [3.4723u, 5.5975u] [4.6437u, 7.0600u] [4.8508u, 7.1427u] x1 1 [6.8187u, 8.0743u] [7.5446u, 10.1106u] [6.0300u, 8.5554u] x1 2 [6.8187u, 8.0743u] [7.5446u, 10.1106u] [6.0300u, 8.5554u]

From Fig. 5, it has the same trend of the results presented in Fig. 4 and shows the relationship of different possible degrees. It also implies that the P2P lending platforms x1 , x2 , x3 , x10 can be accepted. At the same time, x4 , x5 , x6 , x7 , x8 , x9 , x11 , x12

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LIANG et al.: NEW AGGREGATION METHOD-BASED ERROR ANALYSIS FOR DECISION-THEORETIC ROUGH SETS 1695

Fig. 4. Total possibility degrees of the expected losses for each object based on (24).

TABLE XI RESULTS OF EXPECTED LOSSES BASED ON STRATEGY 2

U ˜R(aP |δA (x)) ˜R(aB |δA (x)) ˜R(aN |δA (x))

x1 [3.4723u, 4.4375u, 5.5975u] [4.6437u, 5.4688u, 7.0600u] [4.8508u, 5.9875u, 7.1427u] x2 [3.4723u, 4.4375u, 5.5975u] [4.6437u, 5.4688u, 7.0600u] [4.8508u, 5.9875u, 7.1427u] x3 [3.4723u, 4.4375u, 5.5975u] [4.6437u, 5.4688u, 7.0600u] [4.8508u, 5.9875u, 7.1427u] x4 [17.8865u, 19.3188u, 20.5982u] [15.1523u, 17.8875u, 19.7236u] [6.5719u, 7.1875u, 7.9401u] x5 [17.8865u, 19.3188u, 20.5982u] [15.1523u, 17.8875u, 19.7236u] [6.5719u, 7.1875u, 7.9401u] x6 [6.8187u, 7.6438u, 8.0743u] [7.5446u, 8.9250u, 10.1106u] [6.0300u, 7.2875u, 8.5554u] x7 [17.8865u, 19.3188u, 20.5982u] [15.1523u, 17.8875u, 19.7236u] [6.5719u, 7.1875u, 7.9401u] x8 [6.8187u, 7.6438u, 8.0743u] [7.5446u, 8.9250u, 10.1106u] [6.0300u, 7.2875u, 8.5554u] x9 [17.8865u, 19.3188u, 20.5982u] [15.1523u, 17.8875u, 19.7236u] [6.5719u, 7.1875u, 7.9401u] x1 0 [3.4723u, 4.4375u, 5.5975u] [4.6437u, 5.4688u, 7.0600u] [4.8508u, 5.9875u, 7.1427u] x1 1 [6.8187u, 7.6438u, 8.0743u] [7.5446u, 8.9250u, 10.1106u] [6.0300u, 7.2875u, 8.5554u] x1 2 [6.8187u, 7.6438u, 8.0743u] [7.5446u, 8.9250u, 10.1106u] [6.0300u, 7.2875u, 8.5554u]

Fig. 5. Total possibility degrees of the expected losses for each object based on (25).

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1696 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 25, NO. 6, DECEMBER 2017

TABLE XII DECISION RESULTS OF P2P LENDING PLATFORMS

BY USING DIFFERENT METHODS

Methods POS(C ) NEG(C )

Method 1 {x1 , x2 , x3 , x1 0 } {x4 , x5 , x6 , x7 , x8 , x9 , x1 1 , x1 2 } Method 2 {x1 , x2 , x3 , x1 0 } {x4 , x5 , x6 , x7 , x8 , x9 , x1 1 , x1 2 } Method 3 {x1 , x2 , x3 , x1 0 } {x4 , x5 , x6 , x7 , x8 , x9 , x1 1 , x1 2 } Method 4 {x1 , x2 , x3 , x1 0 } {x4 , x5 , x6 , x7 , x8 , x9 , x1 1 , x1 2 }

should be rejected. When the value of δ is 0.1, Types I and II have the same results. In light of the possibility degree formula (25), Strategy 2 of Method 4 wields more information of the interval with respect to Strategy 1, e.g., the most likely value. Through the comparison between Figs. 4 and 5, we find that Method 4 can be better to distinguish the total possibility degrees of the expected losses with respect to Method 1.

In light of the aforementioned results, we deduce that the decision results analyzed by Methods 2 and 3 are the same as those of Methods 4 and 1, when the value of δ is 0.1. Hence, we can summarize the decision results by using different methods of Table VII. Their results are shown in Table XII.

From Table XII, the decision results of each object are con- sistent in Methods 1–4. Generally speaking, with the help of three-way decisions with DTRSs, it suggests that the platforms x1 , x2 , x3 , x10 are reliable and can be invested. For the platforms x4 , x5 , x6 , x7 , x8 , x9 , x11 , x12 , we think that they are unreliable and should be rejected.

VI. CONCLUSION

In this paper, we investigate the decision making of DTRSs in the hesitant fuzzy information system. We mainly study two di- rections, one is the hesitant fuzzy information system, the other is the loss functions of DTRSs. More specifically, we explore the calculations of the similarity class and the conditional probabil- ity based on a new binary relation. With the aid of error analysis method, the integrated approach by considering the interaction between information systems and loss functions is further dis- cussed in detail. On the basis of the hesitant fuzzy information system, we finally design a decision-making procedure of three- way decisions and summarize four methods, i.e., Methods 1–4. Our study solves the applications of DTRSs to hesitant fuzzy information systems and enriches its research content. It can be useful in dealing with many operational research problems in the context of risk and uncertainty [22], such as the project investments and the product inventory management. Future re- search work will focus on discussing the determination of the related parameters of our decision procedure with the help of machine learning algorithms.

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Decui Liang received the B.S. degree in information management and information system, and the Ph.D. degree in management science and engineering from Southwest Jiaotong University, Chengdu, China, in 2008 and 2013, respectively.

In 2012, he was a visiting Ph.D. student in the Department of Electrical and Computer Engineer- ing, University of Alberta, Edmonton, AB, Canada. He is currently an Associate Professor in the School of Management and Economics, University of Elec- tronic Science and Technology of China, Chengdu.

His research interests include three-way decisions, rough set theory, granular computing, and multiple-attribute decision making with uncertainty. In these areas, he has published more than 20 papers in leading international journals or conference proceedings, such as the IEEE TRANSACTIONS ON FUZZY SYSTEMS, Information Sciences, the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CY- BERNETICS: SYSTEMS, Knowledge-Based Systems, and the International Journal of Approximate Reasoning in the above areas.

Zeshui Xu (M’08–SM’09) received the Ph.D. degree in management science and engineering from South- east University, Nanjing, China, in 2003.

From April 2003 to May 2005, he was a Post- doctoral Researcher in the School of Economics and Management, Southeast University. From October 2005 to December 2007, he was a Postdoctoral Re- searcher in the School of Economics and Manage- ment, Tsinghua University, Beijing, China. He is a Distinguished Young Scholar of the National Natural Science Foundation of China and the Chang Jiang

Scholars of the Ministry of Education of China. He is currently a Professor in the Business School, Sichuan University, Chengdu, China. He has been selected as a 2014 Thomson Reuters Highly Cited Researcher (in the fields of Computer Science and Engineering, respectively) and also included in The World’s Most Influential Scientific Minds 2014. His h-index is 86, and has authored the fol- lowing books: Uncertain Multi-Attribute Decision Making: Methods and Appli- cations (Springer, 2015), Intuitionistic Fuzzy Information Aggregation: Theory and Applications (Science Press and Springer, 2012), Linguistic Decision Mak- ing: Theory and Methods (Science Press and Springer, 2012), Intuitionistic Fuzzy Preference Modeling and Interactive Decision Making (Springer, 2013), Intuitionistic Fuzzy Aggregation and Clustering (Springer, 2013), and Hesitant Fuzzy Sets Theory (Springer, 2014). He has contributed more than 400 journal articles to professional journals. His current research interests include infor- mation fusion, group decision making, computing with words, and aggregation operators.

Dr. Xu is the Advisory Member of the journal: Granular Computing, the Associate Editor of Fuzzy Optimization and Decision Making and the Jour- nal of Intelligence Systems, the Section Editor of the Asian Journal of Social and Economic Sciences, and the Chief Editor of the Scholars Journal of Eco- nomics, Business and Management. He is also a Member of the Editorial Boards of Information Fusion, Information: An International Journal, International Journal of Applied Management Science, International Journal of Data Anal- ysis Techniques and Strategies, Journal of Applied and Computational Math- ematics, International Journal of Research in Industrial Engineering, System Engineering-Theory and Practice, Fuzzy Systems and Mathematics, Journal of Systems Engineering, and Chinese Journal of Management Science.

Dun Liu received the B.S. degree in information and computing science, and the Ph.D. degree in manage- ment science and engineering from Southwest Jiao- tong University, Chengdu, China, in 2005 and 2011, respectively.

He was a visiting Ph.D. student at the Univer- sity of Regina, Regina, SK, Canada, in 2009. He is currently an Associate Professor in the School of Eco- nomics and Management, Southwest Jiaotong Uni- versity, and a Postdoctoral Researcher in the School of Economics and Management, Tsinghua Univer-

sity, Beijing, China. His research interests include data mining and knowledge discovery, rough sets, granular computing, decision support systems and man- agement information systems. Since 2005, he has published more than 80 re- search papers in refereed journals and conferences. He has served as ISKE2009, RSKT2011-RSKT2014, IJCRS 2015 Session Chair. He is a member of ACM, a senior member of IRSS, and a senior member of China Computer Federation.

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