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A dominance intuitionistic fuzzy-rough set approach and its applications
Applied Mathematical Modelling 37 (2013) 7128–7141
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A dominance intuitionistic fuzzy-rough set approach and its applications Bing Huang a,b,⇑, Yu-liang Zhuang a,b, Hua-xiong Li c, Da-kuan Wei d a
School of Information Sciences, Nanjing Audit University, Nanjing 211815, PR China Information Systems Auditing Experimental Center, Nanjing Audit University, Nanjing 211815, PR China c School of Engineering and Management, Nanjing University, Nanjing 210093, PR China d School of Information Engineering, Hunan Science and Technology University, Yongzhou 425100, PR China b
a r t i c l e
i n f o
Article history: Received 15 November 2011 Received in revised form 24 October 2012 Accepted 18 December 2012 Available online 27 December 2012 Keywords: Intuitionistic fuzzy set Rough set model Dominance relation Reduction Rule extraction
a b s t r a c t Although the rough set and intuitionistic fuzzy set both capture the same notion, imprecision, studies on the combination of these two theories are rare. Rule extraction is an important task in a type of decision systems where condition attributes are taken as intuitionistic fuzzy values and those of decision attribute are crisp ones. To address this issue, this paper makes a contribution of the following aspects. First, a ranking method is introduced to construct the neighborhood of every object that is determined by intuitionistic fuzzy values of condition attributes. Moreover, an original notion, dominance intuitionistic fuzzy decision tables (DIFDT), is proposed in this paper. Second, a lower/upper approximation set of an object and crisp classes that are confirmed by decision attributes is ascertained by comparing the relation between them. Third, making use of the discernibility matrix and discernibility function, a lower and upper approximation reduction and rule extraction algorithm is devised to acquire knowledge from existing dominance intuitionistic fuzzy decision tables. Finally, the presented model and algorithms are applied to audit risk judgment on information system security auditing risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It is known that rough set theory is introduced by Pawlak as an extension of the classical set theory [1,2]. The basic tools are relations which are the representatives of information systems or decision tables. In Pawlak’s rough set theory, the relation is an equivalence, whereas the relation is maybe a dominance, covering, similarity, tolerance, or other indiscernibility one within various generalized rough set models [3–7], in which the dominance rough set model and the fuzzy rough set theory are two types of the most important extended rough set models. To consider the ranking properties of criteria, Greco et al. [8–11] proposed a dominance-based rough set approach (DRSA) based on the substitution of the indiscernibility relation with a dominance relation. In the DRSA, condition attributes are the criteria used and classes are ranked by preference; therefore, the knowledge approximated is a collection of upward and downward unions of classes, and the dominance classes are sets of objects defined in a dominance relation. In recent years, a number of studies on DRSA have been conducted [12–17].
⇑ Corresponding author at: School of Information Sciences, Nanjing Audit University, Nanjing 211815, PR China. E-mail address: [email protected] (B. Huang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.12.009
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Fuzzy rough set is also a generalization of classic rough sets that are used to manage real-valued decision tables. Dubois and Prade [18] investigated the fuzzyfication of rough sets. The concepts of rough fuzzy set and fuzzy rough sets were proposed, in which crisp binary relations are replaced with fuzzy relations in the universe of discourse. When these concepts are used as bases, a proposed fuzzy rough set theory compensates for the deficiencies of the traditional rough set theory in several aspects [19–24]. Another important mathematical tool that can address imperfect and/or imprecise information is the intuitionistic fuzzy (IF) set initiated by Atanassov [25,26]. The IF set is naturally considered as an extension of Zadeh’s fuzzy sets, and is defined by a pair of membership functions: a fuzzy set provides only the a degree to which an element belongs to a universe, whereas an IF set yields gives both a membership degree and a non-membership degree. The membership and non-membership values generate an indeterminacy index that models the hesitancy in deciding the degree to which an object satisfies a particular property. Recently, IF set theory has been successfully applied in decision analysis and pattern recognition [27– 36]. Although rough sets and Atanassov’s intuitionistic fuzzy sets both capture particular facets of the same notion-imprecision, studies on the combination of intuitionistic fuzzy set theory and rough set theory are rare. In [37], Coker showed that a fuzzy rough set is in fact an intuitionistic L-fuzzy set, which appears to eliminate the likelihood of a new hybrid theory. Rough set approximations have recently been introduced into intuitionistic fuzzy sets [38–41,23,33]. On the basis of fuzzy rough sets in the context of Nanda and Majumdar [31], Chakrabarty et al. [40] and Jena and Ghosh [42] proposed the concept of intuitionistic fuzzy rough sets respectively, in which the lower and upper approximations are both intuitionistic fuzzy sets. Samanta and Mondal [43] also introduced this idea, which they call a rough intuitionistic fuzzy set. In this fuzzy set, in which the membership and non-membership functions are no longer fuzzy sets but fuzzy rough sets in the context of Nanda and Majumdar. Zhou and Wu [44,45] studied fuzzy rough approximation operators. Huang et al. [46,47] discussed dominance-based (interval-valued) intuitionistic fuzzy rough set models and their applications. Despite the aforementioned research efforts, hybrid models that combine an intuitionistic fuzzy set with a rough set are rarely developed. Knowledge reduction and rule extraction are both important tasks in classic and generalized rough set theory [48–58]. However, these issues have rarely been discussed in the context of intuitionistic fuzzy settings. To address this deficiency, the current paper focuses on the construction of dominance-based intuitionistic fuzzy-rough models and the simplification of decision rules in intuitionistic fuzzy information systems. First, we associate intuitionistic fuzzy-valued decision tables with a dominance relationto come up with a concept we call a dominance intuitionistic fuzzy-valued decision tables (DIFDT). Second, we establish a dominance-based rough set model, which is grounded primarily on the substitution of the indiscernibility relation in classic rough set theory with a dominance-based relation. Third, we propose two attribute reduction and rule extraction approaches for eliminating redundant information from the perspective of dominance intuitionistic fuzzy rules. Fourth, the proposed intuitionistic rough models and the approaches of attribute reduction and rule extraction are applied to information systems security audit risk judgement for CISA, candidate global supplier selection in a manufacturing company, and car classification. The rest of the paper is organized as follows: Section 2 briefly introduces the preliminary issues considered in the study, such as the notations of intuitionistic fuzzy-valued decision tables, some basic operations of intuitionistic fuzzy values and sets, and how to determine the dominance classes of an object with respect to its attribute values. In Section 3, we define the dominance-based intuitionistic fuzzy-rough model and present two definitions of the attribute reduction criterion. In Section 4, attribute reduction in the dominance-based intuitionistic fuzzy-rough model is investigated on the basis of discernibility matrices. Section 5 compares the presented dominance-based intuitionistic fuzzy-rough set model in DIFDT with other rough set models. In Section 6, three application examples are discussed to illustrate how to extract simpler dominancebased intuitionistic fuzzy rules are extracted using the attribute reduction approaches. Finally, Section 7 concludes the paper.
2. Preliminaries In this section, we introduce intuitionistic fuzzy values and basic operations, recall some of their properties, and propose some related notations of the intuitionistic fuzzy-valued decision table. Definition 2.1. Let hl; ci be an order pair, where 0 6 l; c 6 1 and 0 6 l þ c 6 1. Then we call hl; ci an intuitionistic fuzzy value. Definition 2.2 ([25,26]). Let ai ¼ hli ; ci ið1 6 i 6 2Þ be two intuitionistic fuzzy values, then (1) a1 ¼ a2 () l1 ¼ l2 ^ c1 ¼ c2 ; (2) a1 \ a2 ¼< minfl1 ; l2 g; maxfc1 ; c2 gi; (3) a1 [ a2 ¼< maxfl1 ; l2 g; minfc1 ; c2 gi. If a1 and a2 are two intuitionistic fuzzy values, then so are a1 \ a2 and a1 [ a2 . Given intersection and union of two intuitionistic fuzzy values, generalizing the arbitrary finite intuitionistic fuzzy values as follows is easily obtained.
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Definition 2.3. Let ai ¼ hli ; ci i (1 6 i 6 n) be n intuitionistic fuzzy values. Their intersection, denoted by S union, denoted by 16i6n ai are defined as follows:
\
16i6n
[ 16i6n
T
16i6n
ai , and their
ai ¼ hmin li ; max ci i; 16i6n
16i6n
ai ¼ hmax li ; min ci i: 16i6n
16i6n
Definition 2.4. An intuitionistic fuzzy-valued decision table (IFDT) is a quadruple S ¼ ðU; C [ D; V; f Þ, where U is a non-empty and finite set of objects called the universe, C is a non-empty and finite set of conditional attributes, D ¼ fdg denotes a singleton of decision attribute d, and C \ D ¼ /. V ¼ V 1 [ V 2 , where V 1 and V 2 are domains of condition and decision attributes, respectively. Information function f is a map from U A onto V, such that f ðx; cÞ 2 V c for all c 2 C; V c # V 1 , and f ðx; dÞ 2 V 2 for D ¼ fdg, where f ðx; cÞ is an intuitionistic fuzzy values denoted by f ðx; cÞ ¼ hlc ðxÞ; cc ðxÞi, and f ðx; dÞ is a crisp value. We call f ðx; cÞ the intuitionistic fuzzy value of object, x, under the condition attribute c and f ðx; dÞ the intuitionistic fuzzy value of x under the decision attribute d. In particular, f ðx; cÞ would degenerate into fuzzy value if lc ðxÞ ¼ 1 cc ðxÞ for every x 2 U. Under this consideration, we regard a fuzzy decision table as a special form of intuitionistic fuzzy-valued decision tables. In practical decision-making analysis, we always consider only a binary dominance relation between objects that are possibly dominant in terms of values of an attributes set in an intuitionistic fuzzy-valued decision table. In general, the preference for an increasing or decreasing rank is determined by a decision maker. If the domain of an attribute is arranged according to a decreasing or increasing order, then the attribute is a criterion. Without any loss of generality, we now consider only the increasing rank in dominance-based intuitionistic fuzzy-valued decision tables. Definition 2.5. We call an intuitionistic fuzzy-valued decision table a DIFDT if all the condition attributes and the decision attribute are criteria.
Example 2.1. A DIFDT is presented in Table 1, where, U ¼ fx1 ; x2 ; . . . ; x10 g, C ¼ fc1 ; c2 ; c3 ; c4 ; c5 g. In a DIFDT, key tasks are to compare and rank objects in terms of a condition attribute subset where every attribute takes intuitionistic fuzzy values. In the following, we introduce a kind of ranking method for two intuitionistic fuzzy values as follows:
Definition 2.6 [34]. Let a1 ¼ hl1 ; c1 i and a2 ¼ hl2 ; c2 i be intuitionistic fuzzy values, sða1 Þ ¼ l1 c1 and sða2 Þ ¼ l2 c2 the scores of a1 and a2 . Let hða1 Þ ¼ l1 þ c1 and hða2 Þ ¼ l2 þ c2 be the accuracy of a1 and a2 . We can rank a1 and a2 with the following criteria: (1) If sða1 Þ < sða2 Þ, then a1 a2 ; (2) If sða1 Þ ¼ sða2 Þ and (a) hða1 Þ ¼ hða2 Þ, then a1 ¼ a2 ; (b) hða1 Þ < hða2 Þ, then a1 a2 ; (c) hða1 Þ > hða2 Þ, then a1 a2 . For two intuitionistic fuzzy values, a1 ¼ h0:3; 0:5i and a2 ¼ h0:4; 0:6i, there are sða1 Þ ¼ sða2 Þ ¼ 0:2 and hða1 Þ ¼ 0:8hhða2 Þ ¼ 1:0; thus, a1 a2 . On the basis of Definition 2.6, we can design a method for ranking two objects whose attribute characters are described by means of intuitionistic fuzzy values.
Table 1 A DIFDT. U
c1
c2
c3
c4
c5
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:4; 0:5i h0:3; 0:5i h0:3; 0:5i h0:1; 0:8i h0:7; 0:3i h0:3; 0:6i h0:4; 0:5i h0:4; 0:6i h0:4; 0:6i h0:7; 0:3i
h0:3; 0:5i h0:4; 0:5i h0:4; 0:4i h0:1; 0:8i h0:4; 0:5i h0:4; 0:6i h0:4; 0:5i h0:4; 0:5i h0:7; 0:3i h0:7; 0:3i
h0:8; 0:2i h0:6; 0:1i h0:8; 0:1i h0:4; 0:5i h0:9; 0:1i h0:7; 0:2i h0:8; 0:2i h0:9; 0:1i h0:9; 0:1i h0:8; 0:2i
h0:4; 0:5i h0:4; 0:5i h0:4; 0:5i h0:1; 0:8i h0:5; 0:5i h0:5; 0:5i h0:4; 0:5i h0:7; 0:3i h0:4; 0:5i h0:9; 0:0i
h0:7; 0:1i h0:7; 0:3i h0:7; 0:3i h0:8; 0:2i h0:8; 0:1i h0:8; 0:2i h0:8; 0:2i h0:8; 0:2i h0:9; 0:0i h0:4; 0:5i
2 2 1 1 2 3 4 4 5 5
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Definition 2.7. Let DIFDT=ðU; C [ D; V; f Þ and B # C, for x; y 2 U, denoted by
xB y () f ðx; bÞ f ðy; bÞ () f ðx; bÞ f ðy; bÞ _ f ðx; bÞ ¼ f ðy; bÞ; 8b 2 B: Obviously, B is a binary relation in U, that is
B ¼ fðx; yÞ 2 U Ujf ðx; bÞ f ðy; bÞ; 8b 2 Bg: The binary relation defined in Definition 2.7 is called a dominance-based relation in DIFDT. Property 2.1. Let DIFDT¼ ðU; C [ D; V; f Þ and E # B # C, then (1) B is reflexive, transitive and non-symmetric; T (2) B ¼ fbg ; b2B (3) B # E . In terms of B # C, the dominance-based class induced by the dominance-based relation, B is the set of objects dominat ing x; i.e., ½x B ¼ fy 2 Ujðx; yÞ 2 B g, where ½xB describes the set of objects that may dominates x in terms of B # C in a DIFDT. This set is called the B-dominating set with respect to x 2 U. Meanwhile, the B-dominated set with respect to x 2 U can be denoted by ½x B ¼ fy 2 Ujðy; xÞ 2 B g. Example 2.2 (Continued from Example 2.1). Compute the dominating and dominated classes induced by C in Table 1. One can get all the dominating and dominated classes in Table 2 as follows.
3. Rough set approach of DIFDT In this section, we investigate set approximation and attribute reduction with respect to the dominance-based relation in DIFDT. Definition 3.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. The universe, U, is partitioned into m equivalence classes by the decision attribute, d. That is, U=fdg ¼ fU 1 ; U 2 ; . . . ; U m g, where U 1 U 2 . . . U m , and U i U j denotes that 8x 2 U i ; y 2 U j S S implies f ðx; dÞ < f ðy; dÞ. Denote U 16j6k U j ð1 6 k 6 mÞ; U k ¼ k6j6m U j ð1 6 k 6 mÞ, and let: k ¼
AppB ðU k Þ ¼ fxj½xB # U k gð1 6 k 6 mÞ; AppB ðU k Þ ¼ fxj½xB \ U k – /g ¼
[
½xB ð1 6 k 6 mÞ;
x2U k
BNDB ðU k Þ ¼ AppB ðU k Þ AppB ðU k Þ; AppB ðU k Þ ¼ fxj½xB # U k gð1 6 k 6 mÞ; AppB ðU k Þ ¼ fxj½xB \ U k – /g ¼
[
½xB ð1 6 k 6 mÞ;
x2U k
BNDB ðU k Þ ¼ AppB ðU k Þ AppB ðU k Þ;
Table 2 All the dominating and dominated classes induced by C and C in Table 1. U
½x C ðxÞ
½x C ðxÞ
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
fx1 g fx2 g fx2 ; x3 g fx4 g fx1 ; x2 ; x4 ; x5 ; x6 ; x7 g fx4 ; x6 g fx1 ; x2 ; x4 ; x7 g fx2 ; x4 ; x6 ; x8 g fx2 ; x3 ; x4 ; x9 g fx10 g
fx1 ; x5 ; x7 g fx2 ; x3 ; x5 ; x7 ; x8 ; x9 g fx3 ; x9 g fx4 ; x5 ; x6 ; x7 ; x8 ; x9 g fx5 g fx5 ; x6 ; x8 g fx5 ; x7 g fx8 g fx9 g fx10 g
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\
AppB ðxÞ ¼
U k ;
½x \U –/ B k
AppB ðxÞ ¼
\ ½x B
U k ;
# U k
BNDB ðxÞ ¼ AppB ðxÞ AppB ðxÞ; \
AppB ðxÞ ¼
U k ;
½x \U –/ B k
AppB ðxÞ ¼
\ ½x B
U k ;
# U k
BNDB ðxÞ ¼ AppB ðxÞ AppB ðxÞ: Then, AppB ðU k Þ; AppB ðU k Þ and BNDB ðU k Þ are the lower approximation, upper approximation and boundary of the dominated class, U k with respect to condition attribute subset B; AppB ðU k Þ; AppB ðU k Þ and BNDB ðU k Þ are the lower approximation, upper approximation and boundary of the dominating class, U k with respect to B; In terms of B, AppB ðxÞ; AppB ðxÞ and BNDB ðxÞ are the lower approximation, upper approximation and boundary of x; App B ðxÞ; AppB ðxÞ and BNDB ðxÞ are the lower approximation, upper approximation and boundary of x. We call Definition 3.1 a dominance-based intuitionistic fuzzy-rough set model.
Property 3.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ; B1 # B2 # C, and x 2 U; thus, (1) AppB1 ðU k Þ # AppB1 ðU k Þ; AppB1 ðU k Þ # AppB1 ðU k Þ,
App B1 ðxÞ # AppB1 ðxÞ; AppB1 ðxÞ # AppB1 ðxÞ;
(2) App B1 ðxÞ # AppB2 ðxÞ; AppB1 ðxÞ # AppB2 ðxÞ; (3) App B2 ðxÞ # AppB1 ðxÞ; AppB2 ðxÞ # AppB1 ðxÞ.
Property 3.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B1 # B2 # C; then, (1) AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t P kg,
(2)
AppB ðU kÞ
¼ fx 2
UjApp B ðxÞ
¼
U t ;t
6 kg,
AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t P kg; AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t 6 kg.
Proof. T (1) x 2 AppB ðU ½x #U ½x \U –/ U i ¼ U t ;t P k; x 2 AppB ðU k Þ () ½xB \ U k – / k Þ () k () ½xB \ U t 1 ¼ /;t 1 < k () AppB ðxÞ ¼ T B B i () AppB ðxÞ ¼ ½x \U –/ U i ¼ U t ;t P k; T B i (2) x 2 AppB ðU () ½x () App x 2 AppB ðU ½x \U –/ U i ¼ U t ; t 6 k; k Þ () ½xB # U k k Þ () B ðxÞ ¼ B \ U t 2 ¼ /; t 2 > k T B i ½x \ U – / ( ) App ðxÞ ¼ U ¼ U ; t 6 k. h B k t ½x \U –/ i B B
i
Consider an object y 2 ½x B , then T as App U ¼ U ½x \U –/ k k1 , B ðxÞ ¼ B
8b 2 B; f ðy; bÞ f ðx; bÞ and ½yB # ½xB . Because we can denote the lower approximation of x then the upper bound of the decision value of y should be f ðz1 ; dÞ; z1 2 U k1 , i.e., T k f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; We can also denote the upper approximation of x as App ½x # U U k ¼ U k2 , then the lower B ðxÞ ¼ B k bound of the decision value of y should be f ðz2 ; dÞ; z2 2 U k2 , i.e., f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 . We formulate these two conclusions above as follows:
8y 2 ½xB ) f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; AppB ðxÞ ¼ U k1 ; 8y 2 ½xB ) f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 ; AppB ðxÞ ¼ U k2 : Similarly, consider an object y 2 ½x B , then these facts can be obtained as follows:
8y 2 ½xB ) f ðy; dÞ P f ðz3 ; dÞ; z3 2 U k3 ; AppB ðxÞ ¼ U k3 ; 8y 2 ½xB ) f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; AppB ðxÞ ¼ U k4 : A special subset of the condition attribute set in DIFDT preserves the lower or upper bound rules of all the objects in the universe. To derive this subset, we provide the necessary definitions of attribute reduction in DIFDT as follows.
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Definition 3.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. (1) B is considered a () consistent lower approximation set of DIFDT if 8x 2 U, App B ðxÞ ¼ AppC ðxÞ (AppB ðxÞ ¼ AppC ðxÞ). If B is a () consistent lower approximation set of DIFDT and no proper subset of B is a () consistent lower approximation set of DIFDT, then B is considered a () lower approximation reduct of DIFDT. (2) B is considered a () consistent upper approximation set of DIFDT if 8x 2 U, App B ðxÞ ¼ AppC ðxÞ (AppB ðxÞ ¼ AppC ðxÞ). If B is a () consistent upper approximation set of DIFDT and no proper subset of B is a () consistent upper approximation set of DIFDT, then B is considered a () upper approximation reduct of DIFDT.
Consider a consistent lower approximation set B. If y 2 ½x i.e., 8b 2 B; f ðy; bÞ f ðx; bÞ, B, f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; App B ðxÞ ¼ AppC ðxÞ ¼ U k1 . In other words, B is a consistent lower approximation set, then
then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; AppC ðxÞ ¼ U k1 : Similarly, B is a consistent upper approximation set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 ; AppC ðxÞ ¼ U k2 : Hence, we can conclude that a consistent lower (upper) approximation set is a subset of condition attribute set that preserves the lower (upper) approximations of all objects. The lower (upper) bounds of all decision rules derived from the consistent lower (upper) approximation set are completely consistent with the ones derived from C; i.e., if two decision rules derived respectively from the reduced and the original systems are supported by the same object, their lower (upper) bounds must be identical. Analogously, a lower (upper) consistent set is a subset of condition attribute set that preserves all upper (lower) bounds of all decision rules. If B is a consistent lower approximation set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P f ðz3 ; dÞ; z3 2 U k3 ; AppC ðxÞ ¼ U k3 : If B is a upper approximation consistent set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; AppC ðxÞ ¼ U k4 : For B # C, if we simplify f ðz1 ; dÞ; z1 2 U k1 ; App B ðxÞ ¼ U k1 as jAppB ðxÞj ¼ f ðz1 ; dÞ (refereed as to lower approximation value of x in terms of B), f ðz2 ; dÞ; z2 2 U k2 ; App B ðxÞ ¼ U k2 as jAppB ðxÞj ¼ f ðz2 ; dÞ (refereed as to upper approximation value of x in terms of B), f ðz3 ; dÞ; z3 2 U k3 ; App B ðxÞ ¼ U k3 as jAppB ðxÞj ¼ f ðz3 ; dÞ (refereed as to lower approximation value of x in terms of B) and f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; App B ðxÞ ¼ U k4 as jAppB ðxÞj ¼ f ðz4 ; dÞ (refereed as to upper approximation value of x in terms of B), respectively, then in a straightway method, the () consistent lower (upper) approximation set, B, preserves the following () upper and lower bounds of all decision rules supported by x 2 U, respectively:
8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 jAppC ðxÞj: Example 3.1 (Continued from Example 2.2). Let U=fdg ¼ fU 1 ; U 2 ; U 3 ; U 4 ; U 5 g, where U 1 ¼ fx3 ; x4 g; U 2 ¼ fx1 ; x2 ; x5 g; U 3 ¼ fx6 g; U 4 ¼ fx7 ; x8 g; U 5 ¼ fx9 ; x10 g, Then, U 1 ¼ U 1 ; U 2 ¼ U 1 [ U 2 ; U 3 ¼ U 1 [ U 2 [ U 3 ; U 4 ¼ U 1 [ U 2 [ U 3 [ U 4 ; U 5 ¼ U, and U ¼ U; U ¼ U [ U [ U [ U ; U ¼ U [ U [ U ; U ¼ U [ U ; U ¼ U . One can get all the and low/upper approx5 5 5 5 2 3 4 3 4 4 5 1 2 3 4 imations with respect to C in Table 3 as follows. Table 3 and low/upper approximations of all objects with respect to C in Table 1. U
App C ðxÞ
App C ðxÞ
App C ðxÞ
App C ðxÞ
x1
U 2
U 2
U 2
U 4
x2
U 2
U 2
U 1
U 5
x3
U 2 U 1 U 4 U 3 U 4 U 4 U 5 U 5
U 1 U 1 U 1 U 1 U 1 U 1 U 1 U 5
U 1 U 1 U 2 U 2 U 2 U 4 U 5 U 5
U 5
x4 x5 x6 x7 x8 x9 x10
U 5 U 2 U 4 U 4 U 4 U 5 U 5
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4. Approximation reduction approach of DIFDT Susmaga et al. [48] introduced a discernibility matrix to dominance-based decision tables and addressed the computation of dominance-based reducts using the dominance information table in DRSA. In this section, for simplicity, we only investigate an attribute reduction approach for the and lower approximations in DIFDT. This approach can be easily generalized to the and upper approximations in DIFDT. Theorem 4.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. Then, (1) B is a consistent lower approximation set if and only if f ðy; dÞ > jApp C ðxÞj implies y R ½xB for x; y 2 U. (2) B is a consistent upper approximation set if and only if f ðy; dÞ < jApp ðxÞj implies y R ½x B for x; y 2 U. C
Proof. (1) ) If there are x; y 2 U, such that f ðy; dÞ > jApp C ðxÞj and y 2 ½xB , we obtain jAppB ðxÞj P f ðy; dÞ > jAppC ðxÞj; i.e., App ðxÞ – App ðxÞ. Therefore, B is not a consistent lower approximation set, which contradicts the assumption. B C ( For any two objects, x; y 2 U, when f ðy; dÞ > jApp C ðxÞj; y R ½xB ; i.e., 8y 2 ½xB ; f ðy; dÞ 6 jAppC ðxÞj, and then AppB ðxÞ AppC ðxÞ. According to (2) in Property 3.1, AppB ðxÞ # AppC ðxÞ. Therefore, AppB ðxÞ ¼ AppC ðxÞ for x; y 2 U. On the basis of (1) in Definition 3.2, we conclude that B is a consistent lower approximation set. (2) This proof is similar to that in (1). h
Theorem 4.1 provides an approach to verifying whether a subset of condition attributes is consistent. Unfortunately, we cannot directly use this theorem to determine all the or lower approximation reducts. The discernibility matrix is key to various reduction algorithms in rough set theory. Thus, we define additional practical approaches to determining all the and lower by constructing discernibility matrices. We provide the definition of discernibility matrices as follows. Definition 4.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and U ¼ fx1 ; x2 ; . . . ; xn g. We define the and lower approximation discernibility matrices of DIFDT as M ¼ ðd Þ and M ¼ ðd Þ , where ij nn ij nn
( dij
¼
and
fc 2 Cjf ðxj ; cÞ f ðxi ; cÞg f ðxj ; dÞ > jAppC ðxi Þj; C
( dij ¼
otherwise
fc 2 Cjf ðxj ; cÞ f ðxi ; cÞg f ðxj ; dÞ < jAppC ðxi Þj; C otherwise;
respectively. Theorem 4.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ; U ¼ fx1 ; x2 ; . . . ; xn g and B # C. Then, (1) B is a consistent lower approximation setif and only if B \ d ij – / for all dij ð1 6 i; j 6 nÞ. (2) B is a consistent lower approximation set if and only if B \ d – /for all d ij ij ð1 6 i; j 6 nÞ.
Proof. According to Definition 3.2 and Theorem 4.1, these two conclusions are straightforward. h Now, we use discernibility matrices to acquire all the and lower approximation reducts of DIFDT. Theorem 4.3. Let DIFDT¼ ðU; C [ fdg; V; f Þ. Transform the minimal disjunction form of the and lower approximation V W V W Wr¼s Vv ¼qr Wr¼t Vl¼pr discernibility formulas, D ¼ i;j ð d i;j ð dij Þ, to disjunction forms D ¼ r¼1 ð v ¼1 civ Þ and D ¼ r¼1 ð l¼1 cil Þ, ij Þ and D ¼ respectively. We set B r ¼ fciv : v ¼ 1; . . . ; qr gðr ¼ 1; . . . ; sÞ and Br ¼ fc il : l ¼ 1; . . . ; pr gðr ¼ 1; . . . ; tÞ. Then, Br ðr ¼ 1; . . . ; sÞ and B r ðr ¼ 1; . . . ; tÞ are the and lower approximation reducts, respectively.
Proof. This is a direct result of Theorem 4.2 and the definition of the minimal disjunction form. h Decision rules commonly act as knowledge that aid the decision making. Therefore, rule acquisition is more important than reducts under certain circumstances. We present the methods of deriving upper and lower bound rules based on discernibility matrices.
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Definition 4.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and U ¼ fx1 ; x2 ; . . . ; xn g. The lower and upper approximation discernibility matrices of DIFDT are M ¼ ðd M ¼ ðd Þ , respectively. We define the and lower discernibility functions of ij Þnn and V W ij nn V W xi as f ðxi Þ and f ðxi Þ, where, f ðxi Þ ¼ j ð dij Þ and f ðxi Þ ¼ j ð d Þ. ij Theorem 4.4. Let DIFDT¼ ðU; C [ fdg; V; f Þ. We transformthe minimal disjunction form of the and lower approximation V W V W W Vl¼ps discernibility formulas of xi , f ðxi Þ ¼ j ð d Þ and f ðxi Þ ¼ j ð d Þ, into disjunction forms f ðxi Þ ¼ r¼u r¼1 ð l¼1 cjl Þ and ij ij Wr¼w Vv ¼qs f ðxi Þ ¼ r¼1 ð v ¼1 cjv Þ, respectively. Let Br ðxi Þ ¼ fcjl : l ¼ 1; . . . ; ps g ðr ¼ 1; . . . ; uÞ and Br ðxi Þ ¼ fcjv : v ¼ 1; . . . ; qs g ðr ¼ 1; . . . ; wÞ. Then, B r ðxi Þ ðr ¼ 1; . . . ; uÞ and Br ðxi Þ ðr ¼ 1; . . . ; wÞ are the and lower approximation reducts of xi , respectively. According to Theorem 4.4, all the simplest upper and lower bound rules in terms of every object in DIFDT can be obtained using the and lower discernibility functions. 5. Comparison of the dominance dominance-based intuitionistic fuzzy-rough set model in DIFDT with the other rough set models A consistent or inconsistent comparison with related work would be important to emphasize the rationality of the presented rough set model. In this section, we will establish the relationships between the dominance-based intuitionistic fuzzy-rough set model in DIFDT with the other rough set models. (1) Comparison between the presented rough set model in DIFDT and Pawlak’s rough set model [3]. According to Definition 3.1 and Property 3.1, we can follow that the dominance-based intuitionistic fuzzy-rough set model in DIFDT is a extension of Pawlak’s rough set model. (2) Comparison between the presented dominance-based intuitionistic fuzzy-rough set model in DIFDT and DRSA [8–10]. To a certain extent, the proposed dominance-based intuitionistic fuzzy-rough set model approach (DIFRSA) is consistent with DRSA, in which the corresponding properties described by Property 3.2 hold. Their main differences lie in the methods used to compare objects and granule construction that conforms to the decision attribute. (3) Comparison between DIFRSA and S-DRSA [14]. The feature common to DIFRSA and stochastic dominance-based rough set approach (S-DRSA) [14] is the use of the dominance relation to construct a rough set model. Their core difference is that DIFRSA uses the scores and accuracy of intuitionistic fuzzy values to rank objects, whereas S-DRSA uses the maximum likelihood principle to determine the class to which an object belongs. That is, our method emphasizes the ranking approach for two objects, whereas SDRSA focuses primarily the relationship between an object and a class. (4) Comparison between DIFRSA and fuzzy rough set model in interval-valued fuzzy information systems [48]. This paper and the fuzzy rough set theory for the interval-valued fuzzy information systems [12] both research the rough set models and the corresponding reduction methods for some special circumstances. Sun et al. mainly considered the rough set model and its knowledge reduction of a type of information systems, where its condition and decision attributes both take interval-valued fuzzy numbers. While this paper addresses another information system, where the condition attributes are taken intuitionistic fuzzy values (not interval-valued fuzzy ones) and the decision attributes take crisp ones (not interval-valued fuzzy values). In addition, Sun et al. have not investigated the dominating or dominated relation between attribute values. In other words, Sun et al. have not taken into account the criterion under the interval-valued fuzzy environments. However, this paper mainly studies the rough set model and its reduction for the dominance-based intuitionistic fuzzy decision tables. (5) Comparison between DIFRSA and dominance-based rough set model in intuitionistic fuzzy information systems [46]. By introducing the dominance-based relation used in this paper to intuitionistic fuzzy information systems, where both condition and decision attribute values are taken as intuitionistic fuzzy ones, Huang et al. proposed a notion of dominance intuitionistic fuzzy information systems (DIFIS) and established a dominance-based rough set model in DIFIS. Thus, these two information systems, DIFDT and DIFIS are distinct even though the same dominance-based relation is introduced in these two information systems. (6) Comparison between DIFRSA and diminance-based rough set model in interval-valued intuitionistic fuzzy information systems [47]. In [47], Huang et al. also investigated dominance-based interval-valued intuitionistic fuzzy rough set model and rule extraction in interval-valued intuitionistic fuzzy information systems where both condition and decision attribute values are taken as interval-valued intuitionistic fuzzy ones. Therefore, the common and distinct characteristics are obvious between DIFRSA and dominance-based interval-valued intuitionistic fuzzy rough set model. 6. Application examples As a useful tool that manages imperfect data and information, as well as imprecise knowledge, the intuitionistic fuzzy approach has been successfully applied to perform multi-criteria decision-making, group decision, and grey relational analysis [59,39,36]. In a similar manner, the proposed DIFRSA can also be used in these domains. To demonstrate its potential, we present three applications: (i) information systems security audit risk judgement for certified information systems auditors
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(CISA), (ii) candidate global supplier selection in a manufacturing company, and (iii) cars classification. In these applications, we use our approach to extract the simplified lower and upper bound intuitionistic fuzzy rules. Example 6.1. Information systems security audit risk judgement for CISA. The result of audit designation is significantly influenced by the audit evidence collected when planning the information systems security audit and the amount of audit evidence that depends on the degree of audit risk judgement. Therefore, the more objective and accurate the audit risk judgement rules, the lower the audit costs and risk of the information systems security audit failure are [60]. These information systems security audit risk assessment rules can be drawn using the proposed approach on the basis of the theory of DIFDT proposed in Sections 3 and 4. The information systems security audit risk judgement data we use here comes from the practical research of the first author’s colleagues, as shown in Table 4. Table 4 shows a information systems security audit risk judgement decision table, which is a DIFDT. In this table, object set U ¼ fx1 ; x2 ; . . . ; x10 g includes 10 audited objects. The condition attribute set C ¼ fc1 ; c2 ; c3 ; c4 ; c5 g has five condition attributes, where c1 = ‘‘Better Systems Total Security,’’ c2 = ‘‘Better Systems Operation Security,’’ c3 = ‘‘Safer Data Center,’’ c4 = ‘‘Credible Hardware Device,’’ and c5 = ‘‘Credible Network Security’’. Every value which condition attribute is taken on has special actual meaning. For example, f ðx1 ; c1 Þ ¼ h0:2; 0:4i means that the membership degree of systems total security is 0.2, and non-membership degree of systems total security is 0.4. The decision attribute set, d=‘‘Risk Judgement Order of Information Systems Security Audit’’. The domain of d is f1; 2; 3g, where 1 means ‘‘Complete Examination,’’ 2 means ‘‘Major Examination,’’ and 3 means ‘‘No Examination’’. By using the definitions in the Sections 3 and 4, we can obtain all the and lower approximations presented in Table 5. The and lower approximation discernibility matrix are
0
C
BC B B BC B BC B B BC M ¼ B BC B B BC B BC B B @C
c4 c5
C
C
c1 c 2
C
C
c 1 c2 c 3 c 5
c1 c2 c3 c5
C
C
C
C
C
C
C
c1 c2 c3 c5
c1 c3 c4
C
C
c1 c 2
C
c1 c 2 c 3 c4
c 1 c 2 c3
c1 c2 c3 c5
c1 c3 c4
C
C
c1
C
c 1 c3 c 4
c 1 c2 c 3 c 4
c1 c3 c4 c5
C
C
C
C
C
C
C
C
c3 c4 c5
C
C
c1
C
C
C
c1 c2 c3 c5
C C
C C
C C
C C
C C
C C
C C
c1 c3 c5 c5
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
1
C
C C C c1 c2 c3 c4 C C c1 c3 c4 C C C C C C C C C C c3 c4 c5 C C c4 c5 C C C A C C
C
and
0
C
B c4 c5 B B B C B B C B B B c c2 1 M ¼ B B C B B B C B Bc c c c B 1 2 3 5 B @ c1 c2 c3 c5 C
C
C
C
C
C
C
C
C
C
C
c1 c3 c4
c1 c3 c4
C
c3 c 4 c 5
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c1 c2
c1
C
c1
C
C
C
1
C
C
C
C
C
C
C
C
C C
c1 c 2 c 3 c 4 c1 c2 c3
c1 c3 c4 c 1 c2 c 3 c 4
C C
C C
C C
C C
C C
c1 c2 c3 c5
c1 c 2 c 3 c 5
c 1 c3 c 4 c 5
C
c 1 c 2 c3 c 5
c1 c 3 c 5
c5
C
CC C C CC C CC C C CC C; CC C C CC C CC C C CA
C
c1 c 2 c 3 c 4
c1 c3 c4
C
C
c3 c 4 c 5
c4 c5
C
C
respectively. Table 4 An information systems security audit risk judgement decision table. U
c1
c2
c3
c4
c5
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:2; 0:4i h0:1; 0:7i h0:1; 0:8i h0:1; 0:9i h0:4; 0:6i h0:1; 0:6i h0:6; 0:4i h0:6; 0:2i h0:6; 0:2i h0:6; 0:4i
h0:1; 0:7i h0:1; 0:8i h0:1; 0:8i h0:6; 0:3i h0:2; 0:6i h0:2; 0:6i h0:6; 0:4i h0:6; 0:2i h0:6; 0:4i h0:6; 0:4i
h0:2; 0:6i h0:3; 0:6i h0:2; 0:8i h0:2; 0:7i h0:2; 0:8i h0:2; 0:8i h0:6; 0:4i h0:8; 0:2i h0:8; 0:2i h0:8; 0:2i
h0:6; 0:4i h0:5; 0:2i h0:5; 0:4i h0:2; 0:8i h0:2; 0:8i h0:2; 0:4i h0:7; 0:3i h0:4; 0:6i h0:1; 0:6i h0:8; 0:2i
h0:2; 0:8i h0:2; 0:7i h0:6; 0:4i h0:6; 0:4i h0:2; 0:8i h0:2; 0:8i h0:4; 0:6i h0:4; 0:5i h0:8; 0:2i h0:6; 0:4i
1 2 1 1 2 1 2 2 3 3
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U
½x C
App C ðxÞ
App C ðxÞ
U 1 U 2 U 1 U 1 U 2 U 1 U 2 U 2 U 3 U 3
U 1
x1
x1
x1 ; x7 ; x10
x2
x2
x2 ; x7 ; x10
x3
x3
x3 ; x10
x4
x4
x4 ; x9
x5
x5
x5 ; x7 ; x8 ; x9 ; x10
x6
x6
x6 ; x7 ; x8 ; x10
x7
x1 ; x2 ; x5 ; x6 ; x7
x7 ; x10
x8
x5 ; x6 ; x8
x8
x9
x4 ; x5 ; x9
x9
x10
x1 ; x2 ; x3 ; x5 ; x6 ; x7 ; x10
x10
U 2 U 1 U 1 U 2 U 1 U 2 U 2 U 3 U 3
According to Theorem 4.3, only a pair of and lower approximation reducts exists; i.e., fc1 ; c5 g. That is B ¼ B ¼ fc1 ; c5 g. On the basis of the obtained and lower discernibility matrix, we can derive the lower discernibility functions of some objects using Definition 4.2 and Theorem 4.4. For example, the lower discernibility functions of xi ð1 6 i 6 8Þ are
f ðx1 Þ ¼ ðc4 _ c5 Þ ^ ðc1 _ c2 Þ ^ ðc1 _ c2 _ c3 _ c5 Þ ¼ ðc1 ^ c4 Þ _ ðc1 ^ c5 Þ _ ðc2 ^ c4 Þ _ ðc2 ^ c5 Þ; f ðx2 Þ ¼ c1 _ c2 _ c3 _ c5 ; f ðx3 Þ ¼ ðc1 _ c2 Þ ^ ðc1 _ c3 _ c4 Þ ¼ c1 _ ðc2 ^ c3 Þ _ ðc2 ^ c4 Þ; f ðx4 Þ ¼ c1 ; f ðx5 Þ ¼ c1 _ c2 _ c3 _ c4 _ c5 ; f ðx6 Þ ¼ ðc1 ^ c3 Þ _ ðc1 ^ c4 Þ _ ðc1 ^ c5 Þ; f ðx7 Þ ¼ c3 _ c5 _ ðc1 ^ c4 Þ; f ðx8 Þ ¼ c5 : The simplified upper bound rules for information systems security audit risk judgement are drawn by the lower discernibility functions of objects as follows:
f ðx; c1 Þ h0:2; 0:4i ^ f ðx; c4 Þ h0:6; 0:4i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c1 Þ h0:2; 0:4i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:1; 0:7i ^ f ðx; c4 Þ h0:6; 0:4i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:1; 0:7i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c1 Þ h0:1; 0:7i _ f ðx; c2 Þ h0:1; 0:8i _ f ðx; c3 Þ h0:3; 0:6i_; f ðx; c5 Þ h0:2; 0:7i ) f ðx; dÞ 6 2ðsupported by x2 Þ; f ðx; c1 Þ h0:1; 0:8i _ ðf ðx; c2 Þ h0:1; 0:8i ^ f ðx; c3 Þ h0:2; 0:8iÞ_; ðf ðx; c2 Þ h0:1; 0:8i ^ f ðx; c4 Þ h0:5; 0:4iÞ ) f ðx; dÞ 6 1ðsupported by x3 Þ; f ðx; c1 Þ h0:1; 0:9i ) f ðx; dÞ 6 1ðsupported by x4 Þ; f ðx; c1 Þ h0:4; 0:6i _ f ðx; c2 Þ h0:2; 0:6i _ f ðx; c3 Þ h0:2; 0:8i_; f ðx; c4 Þ h0:2; 0:8i _ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 2ðsupported by x5 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c3 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x6 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c4 Þ h0:2; 0:4i ) f ðx; dÞ 6 1ðsupported by x6 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x6 Þ;
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f ðx; c3 Þ h0:6; 0:4i _ f ðx; c5 Þ h0:4; 0:6i ) f ðx; dÞ 6 2ðsupported by x7 Þ; f ðx; c1 Þ h0:6; 0:4i ^ f ðx; c4 Þ h0:7; 0:3i ) f ðx; dÞ 6 2ðsupported by x7 Þ; f ðx; c5 Þ h0:4; 0:5i ) f ðx; dÞ 6 2ðsupported by x8 Þ: The first rule can be interpreted as follows: If Better Systems Total Security is less than h0:2; 0:4i and Credible Hardware Device is less than h0:6; 0:4i, then Risk Judgement on Information Systems Security Audit is Complete Examination. Similarly, the lower approximation discernibility functions of x2 ; x5 ; x7 ; x8 ; x9 ; x10 and their corresponding simplified lower bound rules for information systems security audit risk judgement are obtained by using the lower approximation discernibility matrix as follows:
f ðx2 Þ ¼ c4 _ ðc1 ^ c5 Þ _ ðc3 ^ c5 Þ; f ðx5 Þ ¼ c1 ; f ðx7 Þ ¼ c1 _ c3 _ c4 ; f ðx8 Þ ¼ c1 _ c2 _ c3 ; f ðx9 Þ ¼ c5 ; f ðx10 Þ ¼ c4 _ ðc1 ^ c5 Þ _ ðc3 ^ c5 Þ; f ðx; c4 Þ h0:5; 0:2i _ ðf ðx; c1 Þ h0:1; 0:7i ^ f ðx; c5 Þ h0:2; 0:7iÞ_; ðf ðx; c3 Þ h0:3; 0:6i ^ f ðx; c5 Þ h0:2; 0:7iÞ ) f ðx; dÞ P 2ðsupported by x2 Þ; f ðx; c1 Þ h0:4; 0:6i ) f ðx; dÞ P 2ðsupported by x5 Þ; f ðx; c1 Þ h0:6; 0:4i _ f ðx; c3 Þ h0:6; 0:4i _ f ðx; c4 Þ h0:7; 0:3i; ) f ðx; dÞ P 2ðsupported by x7 Þ; f ðx; c1 Þ h0:6; 0:2i _ f ðx; c2 Þ h0:6; 0:2i _ f ðx; c3 Þ h0:8; 0:2i; ) f ðx; dÞ P 2ðsupported by x8 Þ; f ðx; c5 Þ h0:8; 0:2i ) f ðx; dÞ P 3ðsupported by x9 Þ; f ðx; c4 Þ h0:8; 0:2i _ ðf ðx; c1 Þ h0:6; 0:4i ^ f ðx; c5 Þ h0:6; 0:4iÞ_; ðf ðx; c3 Þ h0:6; 0:4i ^ f ðx; c5 Þ h0:6; 0:4iÞ ) f ðx; dÞ P 3ðsupported by x10 Þ; This rule, f ðx; c5 Þ h0:8; 0:2i ) f ðx; dÞ P 3 can be explicated as: If Credible Network Security is greater than h0:8; 0:2i, then Risk Judgement on Information Systems Security Audit is No Examination. Example 6.2. Candidate global supplier selection in a manufacturing company A multi-criteria decision making problem is concerned with a manufacturing company which wants to select the best global supplier according to the core competencies of suppliers. Four types of suppliers are found in a manufacturing company. These are denoted by U ¼ fx1 ; x2 ; x3 ; x4 g. The company can evaluate their core competencies from the following aspects (attributes): ‘‘Level of Technology Innovation’’ (c1 ), ‘‘Control Ability of Flow’’ (c2 ), ‘‘Ability of Management’’ (c3 ), and‘‘Level of Service’’ (c4 ). We assume that the manufacturing company is amenable only to providing its attribute values on each attribute as an intuitionistic value and those of the selection results (d) on the selectors as crisp ones. Here, the candidate global selection rules can generally be extracted using the proposed rule extraction approach in this paper. Inspired by Tan and Chen [61], we build a similar dataset (Table 6); partly taken from [61]). The domain of d is f1; 2; 3g, where 1 means ‘‘Rejection,’’ 2 means ‘‘Considerable Selection,’’ and 3 means ‘‘Priority Selection’’.
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Using the same procedure in Example 5.1, we derive the and lower approximation discernibility matrices
0
C
BC B M ¼ B @ c3 C
c3 c4
C
C
C
c3
C
c2 c3
C
C
1
C C C c1 c3 c4 A C C
and
0
C
Bc c B 3 4 M ¼ B @ C C
C
c3
C
c3
C
C
C
c1 c3 c4
C
1
c2 c3 C C C; C A C
respectively. There is only and lower approximation reduct fc3 g. All the simplified upper and lower bound rules for the selection of candidate global supplier are as follows:
f ðx; c3 Þ h0:7; 0:2i _ f ðx; c4 Þ h0:3; 0:1i ) f ðx; dÞ 6 2ðsupported by x1 Þ; f ðx; c3 Þ h0:5; 0:4i ) f ðx; dÞ 6 1ðsupported by x3 Þ; f ðx; c2 Þ h0:2; 0:5i _ f ðx; c3 Þ h0:4; 0:2i ) f ðx; dÞ 6 2ðsupported by x4 Þ; f ðx; c3 Þ h0:7; 0:2i ) f ðx; dÞ P 2ðsupported by x1 Þ; f ðx; c3 Þ h0:9; 0:1i ) f ðx; dÞ P 3ðsupported by x2 Þ; f ðx; c1 Þ h0:6; 0:2i _ f ðx; c3 Þ h0:4; 0:2i _ f ðx; c4 Þ h0:7; 0:1i; ) f ðx; dÞ P 2ðsupported by x4 Þ; This rule, f ðx; c3 Þ h0:7; 0:2i ) f ðx; dÞ P 2 can be interpreted as: If Ability of Management is greater than h0:7; 0:2i, then Candidate Global Supplier Selection is at least Considerable Selection. Example 6.3. Cars classification The car dataset contains the information of ten new cars to be classified in the Guangzhou car market in Guangdong, China. Let U ¼ fxi j1 6 i 6 10g be the cars, each of which is described by six attributes: fuel economy (c1 ), aerod degree (c2 ), price (c3 ), comfort (c4 ); design (c5 ), and safety (c6 ). The characteristics of the ten new cars under the six attributes are represented (Table 7; partly taken from [35]). Similar to Examples 6.1 and 6.2, some simplified upper and lower bound rules can be drawn as follows (Owing to space limitations, we only list partial rules):
f ðx; c2 Þ h0:2; 0:7i _ f ðx; c3 Þ h0:4; 0:5i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:3; 0:5i _ f ðx; c3 Þ h0:2; 0:6i ) f ðx; dÞ 6 1ðsupported by x6 Þ; ... f ðx; c1 Þ h0:8; 0:1i _ f ðx; c5 Þ h0:8; 0:2i _ ðf ðx; c2 Þ h0:7; 0:2i ^ f ðx; c6 Þ ; h0:2; 0:6iÞ _ ðf ðx; c3 Þ h0:7; 0:0i ^ f ðx; c4 Þ h0:4; 0:1i ^ f ðx; c6 Þ h0:2; 0:6iÞ; ) f ðx; dÞ P 4ðsupported by x5 Þ;
Table 6 Candidate global supplier selection for manufacturing company. U
c1
c2
c3
c4
d
x1 x2 x3 x4
h0:5; 0:3i h0:4; 0:3i h0:4; 0:1i h0:6; 0:2i
h0:5; 0:4i h0:3; 0:4i h0:5; 0:3i h0:2; 0:5i
h0:7; 0:2i h0:9; 0:1i h0:5; 0:4i h0:4; 0:2i
h0:3; 0:1i h0:5; 0:2i h0:6; 0:2i h0:7; 0:1i
2 3 1 2
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Table 7 Cars classification. U
c1
c2
c3
c4
c5
c6
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:3; 0:4i h0:4; 0:3i h0:4; 0:2i h0:3; 0:4i h0:8; 0:1i h0:4; 0:3i h0:6; 0:4i h0:9; 0:1i h0:4; 0:4i h0:9; 0:1i
h0:2; 0:7i h0:5; 0:1i h0:6; 0:1i h0:9; 0:0i h0:7; 0:2i h0:3; 0:5i h0:4; 0:2i h0:7; 0:2i h1:0; 0:0i h0:8; 0:0i
h0:4; 0:5i h0:6; 0:2i h0:8; 0:1i h0:8; 0:1i h0:7; 0:0i h0:2; 0:6i h0:7; 0:2i h0:7; 0:1i h0:9; 0:1i h0:6; 0:3i
h0:8; 0:1i h0:2; 0:7i h0:2; 0:6i h0:7; 0:1i h0:4; 0:1i h0:7; 0:1i h0:3; 0:6i h0:4; 0:5i h0:6; 0:2i h0:5; 0:2i
h0:4; 0:5i h0:3; 0:6i h0:3; 0:7i h0:1; 0:8i h0:8; 0:2i h0:5; 0:4i h0:3; 0:7i h0:4; 0:5i h0:2; 0:7i h0:8; 0:1i
h0:2; 0:7i h0:7; 0:2i h0:5; 0:2i h0:2; 0:8i h0:2; 0:6i h0:3; 0:6i h0:6; 0:1i h0:8; 0:0i h0:1; 0:8i h0:6; 0:4i
1 2 2 3 4 1 2 4 3 4
f ðx; c1 Þ h0:9; 0:1i _ f ðx; c6 Þ h0:8; 0:0i _ ðf ðx2 Þ h0:7; 0:2i ^ f ðx; c3 Þ ; h0:7; 0:1i ^ f ðx; c5 Þ h0:4; 0:5iÞ ) f ðx; dÞ P 4ðsupported by x5 Þ; f ðx; c1 Þ h0:9; 0:1i _ f ðx; c5 Þ h0:8; 0:1i _ ðf ðx; c2 Þ h0:8; 0:0i ^ f ðx; c6 Þ ; h0:6; 0:4iÞ _ ðf ðx; c3 Þ h0:6; 0:3i ^ f ðx; c4 Þ h0:5; 0:2i ^ f ðx; c6 Þ h0:6; 0:4iÞ; ) f ðx; dÞ P 4ðsupported by x10 Þ ... 7. Conclusion Rough set theory is a useful mathematical tool for dealing with uncertain information. However, when we consider ranking fuzzy-valued objects rather than classifying them, conventional rough set theory is unable to solve these problems. One of the extensions of the classic rough set approach is the dominance fuzzy-valued rough set approach. The intuitionistic fuzzy decision table is an important type of data one, and is a generalized form of fuzzy-valued information systems [62,63]. This paper focuses on the construction of a fuzzy-rough set model and rule extraction in DIFDT, which aids decision making under dominance-based intuitionistic fuzzy settings. The main contribution of this paper is the introduction of the ranking method for comparing two intuitionistic fuzzy values in DIFDT. First, we defined the notion of DIFDT and introduced a ranking method for all objects with the dominance-based relation between objects. Second, on the basis of the dominance-based relation, we established a fuzzy-rough set approach in DIFDT, which are grounded primarily based on the substitution of the indiscernibility relation with the dominance-based relation. Third, to extract the simplest dominance intuitionistic fuzzy lower and upper bound rules, we used the discernibility matrices to propose two attribute reduction approaches for eliminating redundant information. Finally, we applied these approaches to information systems security audit risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification. The application examples yielded valuable rules. The approaches simplified a DIFDT and enabled the direct identification of considerably simpler intuitionistic fuzzy lower and upper bound rules. Moreover, the derived rules can facilitate knowledge acquisition from DIFDT. Acknowledgements The authors thank the anonymous referees for their constructive suggestions. This paper was supported by the Natural Science Foundation of China (Grant Nos. 61170105 and 71201076), Qing Lan Project of Jiangsu Province, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (Audition Science and Technology). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A dominance intuitionistic fuzzy-rough set approach and its applications Bing Huang a,b,⇑, Yu-liang Zhuang a,b, Hua-xiong Li c, Da-kuan Wei d a
School of Information Sciences, Nanjing Audit University, Nanjing 211815, PR China Information Systems Auditing Experimental Center, Nanjing Audit University, Nanjing 211815, PR China c School of Engineering and Management, Nanjing University, Nanjing 210093, PR China d School of Information Engineering, Hunan Science and Technology University, Yongzhou 425100, PR China b
a r t i c l e
i n f o
Article history: Received 15 November 2011 Received in revised form 24 October 2012 Accepted 18 December 2012 Available online 27 December 2012 Keywords: Intuitionistic fuzzy set Rough set model Dominance relation Reduction Rule extraction
a b s t r a c t Although the rough set and intuitionistic fuzzy set both capture the same notion, imprecision, studies on the combination of these two theories are rare. Rule extraction is an important task in a type of decision systems where condition attributes are taken as intuitionistic fuzzy values and those of decision attribute are crisp ones. To address this issue, this paper makes a contribution of the following aspects. First, a ranking method is introduced to construct the neighborhood of every object that is determined by intuitionistic fuzzy values of condition attributes. Moreover, an original notion, dominance intuitionistic fuzzy decision tables (DIFDT), is proposed in this paper. Second, a lower/upper approximation set of an object and crisp classes that are confirmed by decision attributes is ascertained by comparing the relation between them. Third, making use of the discernibility matrix and discernibility function, a lower and upper approximation reduction and rule extraction algorithm is devised to acquire knowledge from existing dominance intuitionistic fuzzy decision tables. Finally, the presented model and algorithms are applied to audit risk judgment on information system security auditing risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It is known that rough set theory is introduced by Pawlak as an extension of the classical set theory [1,2]. The basic tools are relations which are the representatives of information systems or decision tables. In Pawlak’s rough set theory, the relation is an equivalence, whereas the relation is maybe a dominance, covering, similarity, tolerance, or other indiscernibility one within various generalized rough set models [3–7], in which the dominance rough set model and the fuzzy rough set theory are two types of the most important extended rough set models. To consider the ranking properties of criteria, Greco et al. [8–11] proposed a dominance-based rough set approach (DRSA) based on the substitution of the indiscernibility relation with a dominance relation. In the DRSA, condition attributes are the criteria used and classes are ranked by preference; therefore, the knowledge approximated is a collection of upward and downward unions of classes, and the dominance classes are sets of objects defined in a dominance relation. In recent years, a number of studies on DRSA have been conducted [12–17].
⇑ Corresponding author at: School of Information Sciences, Nanjing Audit University, Nanjing 211815, PR China. E-mail address: [email protected] (B. Huang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.12.009
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Fuzzy rough set is also a generalization of classic rough sets that are used to manage real-valued decision tables. Dubois and Prade [18] investigated the fuzzyfication of rough sets. The concepts of rough fuzzy set and fuzzy rough sets were proposed, in which crisp binary relations are replaced with fuzzy relations in the universe of discourse. When these concepts are used as bases, a proposed fuzzy rough set theory compensates for the deficiencies of the traditional rough set theory in several aspects [19–24]. Another important mathematical tool that can address imperfect and/or imprecise information is the intuitionistic fuzzy (IF) set initiated by Atanassov [25,26]. The IF set is naturally considered as an extension of Zadeh’s fuzzy sets, and is defined by a pair of membership functions: a fuzzy set provides only the a degree to which an element belongs to a universe, whereas an IF set yields gives both a membership degree and a non-membership degree. The membership and non-membership values generate an indeterminacy index that models the hesitancy in deciding the degree to which an object satisfies a particular property. Recently, IF set theory has been successfully applied in decision analysis and pattern recognition [27– 36]. Although rough sets and Atanassov’s intuitionistic fuzzy sets both capture particular facets of the same notion-imprecision, studies on the combination of intuitionistic fuzzy set theory and rough set theory are rare. In [37], Coker showed that a fuzzy rough set is in fact an intuitionistic L-fuzzy set, which appears to eliminate the likelihood of a new hybrid theory. Rough set approximations have recently been introduced into intuitionistic fuzzy sets [38–41,23,33]. On the basis of fuzzy rough sets in the context of Nanda and Majumdar [31], Chakrabarty et al. [40] and Jena and Ghosh [42] proposed the concept of intuitionistic fuzzy rough sets respectively, in which the lower and upper approximations are both intuitionistic fuzzy sets. Samanta and Mondal [43] also introduced this idea, which they call a rough intuitionistic fuzzy set. In this fuzzy set, in which the membership and non-membership functions are no longer fuzzy sets but fuzzy rough sets in the context of Nanda and Majumdar. Zhou and Wu [44,45] studied fuzzy rough approximation operators. Huang et al. [46,47] discussed dominance-based (interval-valued) intuitionistic fuzzy rough set models and their applications. Despite the aforementioned research efforts, hybrid models that combine an intuitionistic fuzzy set with a rough set are rarely developed. Knowledge reduction and rule extraction are both important tasks in classic and generalized rough set theory [48–58]. However, these issues have rarely been discussed in the context of intuitionistic fuzzy settings. To address this deficiency, the current paper focuses on the construction of dominance-based intuitionistic fuzzy-rough models and the simplification of decision rules in intuitionistic fuzzy information systems. First, we associate intuitionistic fuzzy-valued decision tables with a dominance relationto come up with a concept we call a dominance intuitionistic fuzzy-valued decision tables (DIFDT). Second, we establish a dominance-based rough set model, which is grounded primarily on the substitution of the indiscernibility relation in classic rough set theory with a dominance-based relation. Third, we propose two attribute reduction and rule extraction approaches for eliminating redundant information from the perspective of dominance intuitionistic fuzzy rules. Fourth, the proposed intuitionistic rough models and the approaches of attribute reduction and rule extraction are applied to information systems security audit risk judgement for CISA, candidate global supplier selection in a manufacturing company, and car classification. The rest of the paper is organized as follows: Section 2 briefly introduces the preliminary issues considered in the study, such as the notations of intuitionistic fuzzy-valued decision tables, some basic operations of intuitionistic fuzzy values and sets, and how to determine the dominance classes of an object with respect to its attribute values. In Section 3, we define the dominance-based intuitionistic fuzzy-rough model and present two definitions of the attribute reduction criterion. In Section 4, attribute reduction in the dominance-based intuitionistic fuzzy-rough model is investigated on the basis of discernibility matrices. Section 5 compares the presented dominance-based intuitionistic fuzzy-rough set model in DIFDT with other rough set models. In Section 6, three application examples are discussed to illustrate how to extract simpler dominancebased intuitionistic fuzzy rules are extracted using the attribute reduction approaches. Finally, Section 7 concludes the paper.
2. Preliminaries In this section, we introduce intuitionistic fuzzy values and basic operations, recall some of their properties, and propose some related notations of the intuitionistic fuzzy-valued decision table. Definition 2.1. Let hl; ci be an order pair, where 0 6 l; c 6 1 and 0 6 l þ c 6 1. Then we call hl; ci an intuitionistic fuzzy value. Definition 2.2 ([25,26]). Let ai ¼ hli ; ci ið1 6 i 6 2Þ be two intuitionistic fuzzy values, then (1) a1 ¼ a2 () l1 ¼ l2 ^ c1 ¼ c2 ; (2) a1 \ a2 ¼< minfl1 ; l2 g; maxfc1 ; c2 gi; (3) a1 [ a2 ¼< maxfl1 ; l2 g; minfc1 ; c2 gi. If a1 and a2 are two intuitionistic fuzzy values, then so are a1 \ a2 and a1 [ a2 . Given intersection and union of two intuitionistic fuzzy values, generalizing the arbitrary finite intuitionistic fuzzy values as follows is easily obtained.
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Definition 2.3. Let ai ¼ hli ; ci i (1 6 i 6 n) be n intuitionistic fuzzy values. Their intersection, denoted by S union, denoted by 16i6n ai are defined as follows:
\
16i6n
[ 16i6n
T
16i6n
ai , and their
ai ¼ hmin li ; max ci i; 16i6n
16i6n
ai ¼ hmax li ; min ci i: 16i6n
16i6n
Definition 2.4. An intuitionistic fuzzy-valued decision table (IFDT) is a quadruple S ¼ ðU; C [ D; V; f Þ, where U is a non-empty and finite set of objects called the universe, C is a non-empty and finite set of conditional attributes, D ¼ fdg denotes a singleton of decision attribute d, and C \ D ¼ /. V ¼ V 1 [ V 2 , where V 1 and V 2 are domains of condition and decision attributes, respectively. Information function f is a map from U A onto V, such that f ðx; cÞ 2 V c for all c 2 C; V c # V 1 , and f ðx; dÞ 2 V 2 for D ¼ fdg, where f ðx; cÞ is an intuitionistic fuzzy values denoted by f ðx; cÞ ¼ hlc ðxÞ; cc ðxÞi, and f ðx; dÞ is a crisp value. We call f ðx; cÞ the intuitionistic fuzzy value of object, x, under the condition attribute c and f ðx; dÞ the intuitionistic fuzzy value of x under the decision attribute d. In particular, f ðx; cÞ would degenerate into fuzzy value if lc ðxÞ ¼ 1 cc ðxÞ for every x 2 U. Under this consideration, we regard a fuzzy decision table as a special form of intuitionistic fuzzy-valued decision tables. In practical decision-making analysis, we always consider only a binary dominance relation between objects that are possibly dominant in terms of values of an attributes set in an intuitionistic fuzzy-valued decision table. In general, the preference for an increasing or decreasing rank is determined by a decision maker. If the domain of an attribute is arranged according to a decreasing or increasing order, then the attribute is a criterion. Without any loss of generality, we now consider only the increasing rank in dominance-based intuitionistic fuzzy-valued decision tables. Definition 2.5. We call an intuitionistic fuzzy-valued decision table a DIFDT if all the condition attributes and the decision attribute are criteria.
Example 2.1. A DIFDT is presented in Table 1, where, U ¼ fx1 ; x2 ; . . . ; x10 g, C ¼ fc1 ; c2 ; c3 ; c4 ; c5 g. In a DIFDT, key tasks are to compare and rank objects in terms of a condition attribute subset where every attribute takes intuitionistic fuzzy values. In the following, we introduce a kind of ranking method for two intuitionistic fuzzy values as follows:
Definition 2.6 [34]. Let a1 ¼ hl1 ; c1 i and a2 ¼ hl2 ; c2 i be intuitionistic fuzzy values, sða1 Þ ¼ l1 c1 and sða2 Þ ¼ l2 c2 the scores of a1 and a2 . Let hða1 Þ ¼ l1 þ c1 and hða2 Þ ¼ l2 þ c2 be the accuracy of a1 and a2 . We can rank a1 and a2 with the following criteria: (1) If sða1 Þ < sða2 Þ, then a1 a2 ; (2) If sða1 Þ ¼ sða2 Þ and (a) hða1 Þ ¼ hða2 Þ, then a1 ¼ a2 ; (b) hða1 Þ < hða2 Þ, then a1 a2 ; (c) hða1 Þ > hða2 Þ, then a1 a2 . For two intuitionistic fuzzy values, a1 ¼ h0:3; 0:5i and a2 ¼ h0:4; 0:6i, there are sða1 Þ ¼ sða2 Þ ¼ 0:2 and hða1 Þ ¼ 0:8hhða2 Þ ¼ 1:0; thus, a1 a2 . On the basis of Definition 2.6, we can design a method for ranking two objects whose attribute characters are described by means of intuitionistic fuzzy values.
Table 1 A DIFDT. U
c1
c2
c3
c4
c5
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:4; 0:5i h0:3; 0:5i h0:3; 0:5i h0:1; 0:8i h0:7; 0:3i h0:3; 0:6i h0:4; 0:5i h0:4; 0:6i h0:4; 0:6i h0:7; 0:3i
h0:3; 0:5i h0:4; 0:5i h0:4; 0:4i h0:1; 0:8i h0:4; 0:5i h0:4; 0:6i h0:4; 0:5i h0:4; 0:5i h0:7; 0:3i h0:7; 0:3i
h0:8; 0:2i h0:6; 0:1i h0:8; 0:1i h0:4; 0:5i h0:9; 0:1i h0:7; 0:2i h0:8; 0:2i h0:9; 0:1i h0:9; 0:1i h0:8; 0:2i
h0:4; 0:5i h0:4; 0:5i h0:4; 0:5i h0:1; 0:8i h0:5; 0:5i h0:5; 0:5i h0:4; 0:5i h0:7; 0:3i h0:4; 0:5i h0:9; 0:0i
h0:7; 0:1i h0:7; 0:3i h0:7; 0:3i h0:8; 0:2i h0:8; 0:1i h0:8; 0:2i h0:8; 0:2i h0:8; 0:2i h0:9; 0:0i h0:4; 0:5i
2 2 1 1 2 3 4 4 5 5
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Definition 2.7. Let DIFDT=ðU; C [ D; V; f Þ and B # C, for x; y 2 U, denoted by
xB y () f ðx; bÞ f ðy; bÞ () f ðx; bÞ f ðy; bÞ _ f ðx; bÞ ¼ f ðy; bÞ; 8b 2 B: Obviously, B is a binary relation in U, that is
B ¼ fðx; yÞ 2 U Ujf ðx; bÞ f ðy; bÞ; 8b 2 Bg: The binary relation defined in Definition 2.7 is called a dominance-based relation in DIFDT. Property 2.1. Let DIFDT¼ ðU; C [ D; V; f Þ and E # B # C, then (1) B is reflexive, transitive and non-symmetric; T (2) B ¼ fbg ; b2B (3) B # E . In terms of B # C, the dominance-based class induced by the dominance-based relation, B is the set of objects dominat ing x; i.e., ½x B ¼ fy 2 Ujðx; yÞ 2 B g, where ½xB describes the set of objects that may dominates x in terms of B # C in a DIFDT. This set is called the B-dominating set with respect to x 2 U. Meanwhile, the B-dominated set with respect to x 2 U can be denoted by ½x B ¼ fy 2 Ujðy; xÞ 2 B g. Example 2.2 (Continued from Example 2.1). Compute the dominating and dominated classes induced by C in Table 1. One can get all the dominating and dominated classes in Table 2 as follows.
3. Rough set approach of DIFDT In this section, we investigate set approximation and attribute reduction with respect to the dominance-based relation in DIFDT. Definition 3.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. The universe, U, is partitioned into m equivalence classes by the decision attribute, d. That is, U=fdg ¼ fU 1 ; U 2 ; . . . ; U m g, where U 1 U 2 . . . U m , and U i U j denotes that 8x 2 U i ; y 2 U j S S implies f ðx; dÞ < f ðy; dÞ. Denote U 16j6k U j ð1 6 k 6 mÞ; U k ¼ k6j6m U j ð1 6 k 6 mÞ, and let: k ¼
AppB ðU k Þ ¼ fxj½xB # U k gð1 6 k 6 mÞ; AppB ðU k Þ ¼ fxj½xB \ U k – /g ¼
[
½xB ð1 6 k 6 mÞ;
x2U k
BNDB ðU k Þ ¼ AppB ðU k Þ AppB ðU k Þ; AppB ðU k Þ ¼ fxj½xB # U k gð1 6 k 6 mÞ; AppB ðU k Þ ¼ fxj½xB \ U k – /g ¼
[
½xB ð1 6 k 6 mÞ;
x2U k
BNDB ðU k Þ ¼ AppB ðU k Þ AppB ðU k Þ;
Table 2 All the dominating and dominated classes induced by C and C in Table 1. U
½x C ðxÞ
½x C ðxÞ
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
fx1 g fx2 g fx2 ; x3 g fx4 g fx1 ; x2 ; x4 ; x5 ; x6 ; x7 g fx4 ; x6 g fx1 ; x2 ; x4 ; x7 g fx2 ; x4 ; x6 ; x8 g fx2 ; x3 ; x4 ; x9 g fx10 g
fx1 ; x5 ; x7 g fx2 ; x3 ; x5 ; x7 ; x8 ; x9 g fx3 ; x9 g fx4 ; x5 ; x6 ; x7 ; x8 ; x9 g fx5 g fx5 ; x6 ; x8 g fx5 ; x7 g fx8 g fx9 g fx10 g
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\
AppB ðxÞ ¼
U k ;
½x \U –/ B k
AppB ðxÞ ¼
\ ½x B
U k ;
# U k
BNDB ðxÞ ¼ AppB ðxÞ AppB ðxÞ; \
AppB ðxÞ ¼
U k ;
½x \U –/ B k
AppB ðxÞ ¼
\ ½x B
U k ;
# U k
BNDB ðxÞ ¼ AppB ðxÞ AppB ðxÞ: Then, AppB ðU k Þ; AppB ðU k Þ and BNDB ðU k Þ are the lower approximation, upper approximation and boundary of the dominated class, U k with respect to condition attribute subset B; AppB ðU k Þ; AppB ðU k Þ and BNDB ðU k Þ are the lower approximation, upper approximation and boundary of the dominating class, U k with respect to B; In terms of B, AppB ðxÞ; AppB ðxÞ and BNDB ðxÞ are the lower approximation, upper approximation and boundary of x; App B ðxÞ; AppB ðxÞ and BNDB ðxÞ are the lower approximation, upper approximation and boundary of x. We call Definition 3.1 a dominance-based intuitionistic fuzzy-rough set model.
Property 3.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ; B1 # B2 # C, and x 2 U; thus, (1) AppB1 ðU k Þ # AppB1 ðU k Þ; AppB1 ðU k Þ # AppB1 ðU k Þ,
App B1 ðxÞ # AppB1 ðxÞ; AppB1 ðxÞ # AppB1 ðxÞ;
(2) App B1 ðxÞ # AppB2 ðxÞ; AppB1 ðxÞ # AppB2 ðxÞ; (3) App B2 ðxÞ # AppB1 ðxÞ; AppB2 ðxÞ # AppB1 ðxÞ.
Property 3.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B1 # B2 # C; then, (1) AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t P kg,
(2)
AppB ðU kÞ
¼ fx 2
UjApp B ðxÞ
¼
U t ;t
6 kg,
AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t P kg; AppB ðU k Þ ¼ fx 2 UjAppB ðxÞ ¼ U t ; t 6 kg.
Proof. T (1) x 2 AppB ðU ½x #U ½x \U –/ U i ¼ U t ;t P k; x 2 AppB ðU k Þ () ½xB \ U k – / k Þ () k () ½xB \ U t 1 ¼ /;t 1 < k () AppB ðxÞ ¼ T B B i () AppB ðxÞ ¼ ½x \U –/ U i ¼ U t ;t P k; T B i (2) x 2 AppB ðU () ½x () App x 2 AppB ðU ½x \U –/ U i ¼ U t ; t 6 k; k Þ () ½xB # U k k Þ () B ðxÞ ¼ B \ U t 2 ¼ /; t 2 > k T B i ½x \ U – / ( ) App ðxÞ ¼ U ¼ U ; t 6 k. h B k t ½x \U –/ i B B
i
Consider an object y 2 ½x B , then T as App U ¼ U ½x \U –/ k k1 , B ðxÞ ¼ B
8b 2 B; f ðy; bÞ f ðx; bÞ and ½yB # ½xB . Because we can denote the lower approximation of x then the upper bound of the decision value of y should be f ðz1 ; dÞ; z1 2 U k1 , i.e., T k f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; We can also denote the upper approximation of x as App ½x # U U k ¼ U k2 , then the lower B ðxÞ ¼ B k bound of the decision value of y should be f ðz2 ; dÞ; z2 2 U k2 , i.e., f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 . We formulate these two conclusions above as follows:
8y 2 ½xB ) f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; AppB ðxÞ ¼ U k1 ; 8y 2 ½xB ) f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 ; AppB ðxÞ ¼ U k2 : Similarly, consider an object y 2 ½x B , then these facts can be obtained as follows:
8y 2 ½xB ) f ðy; dÞ P f ðz3 ; dÞ; z3 2 U k3 ; AppB ðxÞ ¼ U k3 ; 8y 2 ½xB ) f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; AppB ðxÞ ¼ U k4 : A special subset of the condition attribute set in DIFDT preserves the lower or upper bound rules of all the objects in the universe. To derive this subset, we provide the necessary definitions of attribute reduction in DIFDT as follows.
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Definition 3.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. (1) B is considered a () consistent lower approximation set of DIFDT if 8x 2 U, App B ðxÞ ¼ AppC ðxÞ (AppB ðxÞ ¼ AppC ðxÞ). If B is a () consistent lower approximation set of DIFDT and no proper subset of B is a () consistent lower approximation set of DIFDT, then B is considered a () lower approximation reduct of DIFDT. (2) B is considered a () consistent upper approximation set of DIFDT if 8x 2 U, App B ðxÞ ¼ AppC ðxÞ (AppB ðxÞ ¼ AppC ðxÞ). If B is a () consistent upper approximation set of DIFDT and no proper subset of B is a () consistent upper approximation set of DIFDT, then B is considered a () upper approximation reduct of DIFDT.
Consider a consistent lower approximation set B. If y 2 ½x i.e., 8b 2 B; f ðy; bÞ f ðx; bÞ, B, f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; App B ðxÞ ¼ AppC ðxÞ ¼ U k1 . In other words, B is a consistent lower approximation set, then
then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 f ðz1 ; dÞ; z1 2 U k1 ; AppC ðxÞ ¼ U k1 : Similarly, B is a consistent upper approximation set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P f ðz2 ; dÞ; z2 2 U k2 ; AppC ðxÞ ¼ U k2 : Hence, we can conclude that a consistent lower (upper) approximation set is a subset of condition attribute set that preserves the lower (upper) approximations of all objects. The lower (upper) bounds of all decision rules derived from the consistent lower (upper) approximation set are completely consistent with the ones derived from C; i.e., if two decision rules derived respectively from the reduced and the original systems are supported by the same object, their lower (upper) bounds must be identical. Analogously, a lower (upper) consistent set is a subset of condition attribute set that preserves all upper (lower) bounds of all decision rules. If B is a consistent lower approximation set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P f ðz3 ; dÞ; z3 2 U k3 ; AppC ðxÞ ¼ U k3 : If B is a upper approximation consistent set, then
8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; AppC ðxÞ ¼ U k4 : For B # C, if we simplify f ðz1 ; dÞ; z1 2 U k1 ; App B ðxÞ ¼ U k1 as jAppB ðxÞj ¼ f ðz1 ; dÞ (refereed as to lower approximation value of x in terms of B), f ðz2 ; dÞ; z2 2 U k2 ; App B ðxÞ ¼ U k2 as jAppB ðxÞj ¼ f ðz2 ; dÞ (refereed as to upper approximation value of x in terms of B), f ðz3 ; dÞ; z3 2 U k3 ; App B ðxÞ ¼ U k3 as jAppB ðxÞj ¼ f ðz3 ; dÞ (refereed as to lower approximation value of x in terms of B) and f ðy; dÞ 6 f ðz4 ; dÞ; z4 2 U k4 ; App B ðxÞ ¼ U k4 as jAppB ðxÞj ¼ f ðz4 ; dÞ (refereed as to upper approximation value of x in terms of B), respectively, then in a straightway method, the () consistent lower (upper) approximation set, B, preserves the following () upper and lower bounds of all decision rules supported by x 2 U, respectively:
8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ P jAppC ðxÞj; 8y 2 U; 8b 2 B; f ðy; bÞ f ðx; bÞ ) f ðy; dÞ 6 jAppC ðxÞj: Example 3.1 (Continued from Example 2.2). Let U=fdg ¼ fU 1 ; U 2 ; U 3 ; U 4 ; U 5 g, where U 1 ¼ fx3 ; x4 g; U 2 ¼ fx1 ; x2 ; x5 g; U 3 ¼ fx6 g; U 4 ¼ fx7 ; x8 g; U 5 ¼ fx9 ; x10 g, Then, U 1 ¼ U 1 ; U 2 ¼ U 1 [ U 2 ; U 3 ¼ U 1 [ U 2 [ U 3 ; U 4 ¼ U 1 [ U 2 [ U 3 [ U 4 ; U 5 ¼ U, and U ¼ U; U ¼ U [ U [ U [ U ; U ¼ U [ U [ U ; U ¼ U [ U ; U ¼ U . One can get all the and low/upper approx5 5 5 5 2 3 4 3 4 4 5 1 2 3 4 imations with respect to C in Table 3 as follows. Table 3 and low/upper approximations of all objects with respect to C in Table 1. U
App C ðxÞ
App C ðxÞ
App C ðxÞ
App C ðxÞ
x1
U 2
U 2
U 2
U 4
x2
U 2
U 2
U 1
U 5
x3
U 2 U 1 U 4 U 3 U 4 U 4 U 5 U 5
U 1 U 1 U 1 U 1 U 1 U 1 U 1 U 5
U 1 U 1 U 2 U 2 U 2 U 4 U 5 U 5
U 5
x4 x5 x6 x7 x8 x9 x10
U 5 U 2 U 4 U 4 U 4 U 5 U 5
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4. Approximation reduction approach of DIFDT Susmaga et al. [48] introduced a discernibility matrix to dominance-based decision tables and addressed the computation of dominance-based reducts using the dominance information table in DRSA. In this section, for simplicity, we only investigate an attribute reduction approach for the and lower approximations in DIFDT. This approach can be easily generalized to the and upper approximations in DIFDT. Theorem 4.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and B # C. Then, (1) B is a consistent lower approximation set if and only if f ðy; dÞ > jApp C ðxÞj implies y R ½xB for x; y 2 U. (2) B is a consistent upper approximation set if and only if f ðy; dÞ < jApp ðxÞj implies y R ½x B for x; y 2 U. C
Proof. (1) ) If there are x; y 2 U, such that f ðy; dÞ > jApp C ðxÞj and y 2 ½xB , we obtain jAppB ðxÞj P f ðy; dÞ > jAppC ðxÞj; i.e., App ðxÞ – App ðxÞ. Therefore, B is not a consistent lower approximation set, which contradicts the assumption. B C ( For any two objects, x; y 2 U, when f ðy; dÞ > jApp C ðxÞj; y R ½xB ; i.e., 8y 2 ½xB ; f ðy; dÞ 6 jAppC ðxÞj, and then AppB ðxÞ AppC ðxÞ. According to (2) in Property 3.1, AppB ðxÞ # AppC ðxÞ. Therefore, AppB ðxÞ ¼ AppC ðxÞ for x; y 2 U. On the basis of (1) in Definition 3.2, we conclude that B is a consistent lower approximation set. (2) This proof is similar to that in (1). h
Theorem 4.1 provides an approach to verifying whether a subset of condition attributes is consistent. Unfortunately, we cannot directly use this theorem to determine all the or lower approximation reducts. The discernibility matrix is key to various reduction algorithms in rough set theory. Thus, we define additional practical approaches to determining all the and lower by constructing discernibility matrices. We provide the definition of discernibility matrices as follows. Definition 4.1. Let DIFDT¼ ðU; C [ fdg; V; f Þ and U ¼ fx1 ; x2 ; . . . ; xn g. We define the and lower approximation discernibility matrices of DIFDT as M ¼ ðd Þ and M ¼ ðd Þ , where ij nn ij nn
( dij
¼
and
fc 2 Cjf ðxj ; cÞ f ðxi ; cÞg f ðxj ; dÞ > jAppC ðxi Þj; C
( dij ¼
otherwise
fc 2 Cjf ðxj ; cÞ f ðxi ; cÞg f ðxj ; dÞ < jAppC ðxi Þj; C otherwise;
respectively. Theorem 4.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ; U ¼ fx1 ; x2 ; . . . ; xn g and B # C. Then, (1) B is a consistent lower approximation setif and only if B \ d ij – / for all dij ð1 6 i; j 6 nÞ. (2) B is a consistent lower approximation set if and only if B \ d – /for all d ij ij ð1 6 i; j 6 nÞ.
Proof. According to Definition 3.2 and Theorem 4.1, these two conclusions are straightforward. h Now, we use discernibility matrices to acquire all the and lower approximation reducts of DIFDT. Theorem 4.3. Let DIFDT¼ ðU; C [ fdg; V; f Þ. Transform the minimal disjunction form of the and lower approximation V W V W Wr¼s Vv ¼qr Wr¼t Vl¼pr discernibility formulas, D ¼ i;j ð d i;j ð dij Þ, to disjunction forms D ¼ r¼1 ð v ¼1 civ Þ and D ¼ r¼1 ð l¼1 cil Þ, ij Þ and D ¼ respectively. We set B r ¼ fciv : v ¼ 1; . . . ; qr gðr ¼ 1; . . . ; sÞ and Br ¼ fc il : l ¼ 1; . . . ; pr gðr ¼ 1; . . . ; tÞ. Then, Br ðr ¼ 1; . . . ; sÞ and B r ðr ¼ 1; . . . ; tÞ are the and lower approximation reducts, respectively.
Proof. This is a direct result of Theorem 4.2 and the definition of the minimal disjunction form. h Decision rules commonly act as knowledge that aid the decision making. Therefore, rule acquisition is more important than reducts under certain circumstances. We present the methods of deriving upper and lower bound rules based on discernibility matrices.
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Definition 4.2. Let DIFDT¼ ðU; C [ fdg; V; f Þ and U ¼ fx1 ; x2 ; . . . ; xn g. The lower and upper approximation discernibility matrices of DIFDT are M ¼ ðd M ¼ ðd Þ , respectively. We define the and lower discernibility functions of ij Þnn and V W ij nn V W xi as f ðxi Þ and f ðxi Þ, where, f ðxi Þ ¼ j ð dij Þ and f ðxi Þ ¼ j ð d Þ. ij Theorem 4.4. Let DIFDT¼ ðU; C [ fdg; V; f Þ. We transformthe minimal disjunction form of the and lower approximation V W V W W Vl¼ps discernibility formulas of xi , f ðxi Þ ¼ j ð d Þ and f ðxi Þ ¼ j ð d Þ, into disjunction forms f ðxi Þ ¼ r¼u r¼1 ð l¼1 cjl Þ and ij ij Wr¼w Vv ¼qs f ðxi Þ ¼ r¼1 ð v ¼1 cjv Þ, respectively. Let Br ðxi Þ ¼ fcjl : l ¼ 1; . . . ; ps g ðr ¼ 1; . . . ; uÞ and Br ðxi Þ ¼ fcjv : v ¼ 1; . . . ; qs g ðr ¼ 1; . . . ; wÞ. Then, B r ðxi Þ ðr ¼ 1; . . . ; uÞ and Br ðxi Þ ðr ¼ 1; . . . ; wÞ are the and lower approximation reducts of xi , respectively. According to Theorem 4.4, all the simplest upper and lower bound rules in terms of every object in DIFDT can be obtained using the and lower discernibility functions. 5. Comparison of the dominance dominance-based intuitionistic fuzzy-rough set model in DIFDT with the other rough set models A consistent or inconsistent comparison with related work would be important to emphasize the rationality of the presented rough set model. In this section, we will establish the relationships between the dominance-based intuitionistic fuzzy-rough set model in DIFDT with the other rough set models. (1) Comparison between the presented rough set model in DIFDT and Pawlak’s rough set model [3]. According to Definition 3.1 and Property 3.1, we can follow that the dominance-based intuitionistic fuzzy-rough set model in DIFDT is a extension of Pawlak’s rough set model. (2) Comparison between the presented dominance-based intuitionistic fuzzy-rough set model in DIFDT and DRSA [8–10]. To a certain extent, the proposed dominance-based intuitionistic fuzzy-rough set model approach (DIFRSA) is consistent with DRSA, in which the corresponding properties described by Property 3.2 hold. Their main differences lie in the methods used to compare objects and granule construction that conforms to the decision attribute. (3) Comparison between DIFRSA and S-DRSA [14]. The feature common to DIFRSA and stochastic dominance-based rough set approach (S-DRSA) [14] is the use of the dominance relation to construct a rough set model. Their core difference is that DIFRSA uses the scores and accuracy of intuitionistic fuzzy values to rank objects, whereas S-DRSA uses the maximum likelihood principle to determine the class to which an object belongs. That is, our method emphasizes the ranking approach for two objects, whereas SDRSA focuses primarily the relationship between an object and a class. (4) Comparison between DIFRSA and fuzzy rough set model in interval-valued fuzzy information systems [48]. This paper and the fuzzy rough set theory for the interval-valued fuzzy information systems [12] both research the rough set models and the corresponding reduction methods for some special circumstances. Sun et al. mainly considered the rough set model and its knowledge reduction of a type of information systems, where its condition and decision attributes both take interval-valued fuzzy numbers. While this paper addresses another information system, where the condition attributes are taken intuitionistic fuzzy values (not interval-valued fuzzy ones) and the decision attributes take crisp ones (not interval-valued fuzzy values). In addition, Sun et al. have not investigated the dominating or dominated relation between attribute values. In other words, Sun et al. have not taken into account the criterion under the interval-valued fuzzy environments. However, this paper mainly studies the rough set model and its reduction for the dominance-based intuitionistic fuzzy decision tables. (5) Comparison between DIFRSA and dominance-based rough set model in intuitionistic fuzzy information systems [46]. By introducing the dominance-based relation used in this paper to intuitionistic fuzzy information systems, where both condition and decision attribute values are taken as intuitionistic fuzzy ones, Huang et al. proposed a notion of dominance intuitionistic fuzzy information systems (DIFIS) and established a dominance-based rough set model in DIFIS. Thus, these two information systems, DIFDT and DIFIS are distinct even though the same dominance-based relation is introduced in these two information systems. (6) Comparison between DIFRSA and diminance-based rough set model in interval-valued intuitionistic fuzzy information systems [47]. In [47], Huang et al. also investigated dominance-based interval-valued intuitionistic fuzzy rough set model and rule extraction in interval-valued intuitionistic fuzzy information systems where both condition and decision attribute values are taken as interval-valued intuitionistic fuzzy ones. Therefore, the common and distinct characteristics are obvious between DIFRSA and dominance-based interval-valued intuitionistic fuzzy rough set model. 6. Application examples As a useful tool that manages imperfect data and information, as well as imprecise knowledge, the intuitionistic fuzzy approach has been successfully applied to perform multi-criteria decision-making, group decision, and grey relational analysis [59,39,36]. In a similar manner, the proposed DIFRSA can also be used in these domains. To demonstrate its potential, we present three applications: (i) information systems security audit risk judgement for certified information systems auditors
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(CISA), (ii) candidate global supplier selection in a manufacturing company, and (iii) cars classification. In these applications, we use our approach to extract the simplified lower and upper bound intuitionistic fuzzy rules. Example 6.1. Information systems security audit risk judgement for CISA. The result of audit designation is significantly influenced by the audit evidence collected when planning the information systems security audit and the amount of audit evidence that depends on the degree of audit risk judgement. Therefore, the more objective and accurate the audit risk judgement rules, the lower the audit costs and risk of the information systems security audit failure are [60]. These information systems security audit risk assessment rules can be drawn using the proposed approach on the basis of the theory of DIFDT proposed in Sections 3 and 4. The information systems security audit risk judgement data we use here comes from the practical research of the first author’s colleagues, as shown in Table 4. Table 4 shows a information systems security audit risk judgement decision table, which is a DIFDT. In this table, object set U ¼ fx1 ; x2 ; . . . ; x10 g includes 10 audited objects. The condition attribute set C ¼ fc1 ; c2 ; c3 ; c4 ; c5 g has five condition attributes, where c1 = ‘‘Better Systems Total Security,’’ c2 = ‘‘Better Systems Operation Security,’’ c3 = ‘‘Safer Data Center,’’ c4 = ‘‘Credible Hardware Device,’’ and c5 = ‘‘Credible Network Security’’. Every value which condition attribute is taken on has special actual meaning. For example, f ðx1 ; c1 Þ ¼ h0:2; 0:4i means that the membership degree of systems total security is 0.2, and non-membership degree of systems total security is 0.4. The decision attribute set, d=‘‘Risk Judgement Order of Information Systems Security Audit’’. The domain of d is f1; 2; 3g, where 1 means ‘‘Complete Examination,’’ 2 means ‘‘Major Examination,’’ and 3 means ‘‘No Examination’’. By using the definitions in the Sections 3 and 4, we can obtain all the and lower approximations presented in Table 5. The and lower approximation discernibility matrix are
0
C
BC B B BC B BC B B BC M ¼ B BC B B BC B BC B B @C
c4 c5
C
C
c1 c 2
C
C
c 1 c2 c 3 c 5
c1 c2 c3 c5
C
C
C
C
C
C
C
c1 c2 c3 c5
c1 c3 c4
C
C
c1 c 2
C
c1 c 2 c 3 c4
c 1 c 2 c3
c1 c2 c3 c5
c1 c3 c4
C
C
c1
C
c 1 c3 c 4
c 1 c2 c 3 c 4
c1 c3 c4 c5
C
C
C
C
C
C
C
C
c3 c4 c5
C
C
c1
C
C
C
c1 c2 c3 c5
C C
C C
C C
C C
C C
C C
C C
c1 c3 c5 c5
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
1
C
C C C c1 c2 c3 c4 C C c1 c3 c4 C C C C C C C C C C c3 c4 c5 C C c4 c5 C C C A C C
C
and
0
C
B c4 c5 B B B C B B C B B B c c2 1 M ¼ B B C B B B C B Bc c c c B 1 2 3 5 B @ c1 c2 c3 c5 C
C
C
C
C
C
C
C
C
C
C
c1 c3 c4
c1 c3 c4
C
c3 c 4 c 5
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c1 c2
c1
C
c1
C
C
C
1
C
C
C
C
C
C
C
C
C C
c1 c 2 c 3 c 4 c1 c2 c3
c1 c3 c4 c 1 c2 c 3 c 4
C C
C C
C C
C C
C C
c1 c2 c3 c5
c1 c 2 c 3 c 5
c 1 c3 c 4 c 5
C
c 1 c 2 c3 c 5
c1 c 3 c 5
c5
C
CC C C CC C CC C C CC C; CC C C CC C CC C C CA
C
c1 c 2 c 3 c 4
c1 c3 c4
C
C
c3 c 4 c 5
c4 c5
C
C
respectively. Table 4 An information systems security audit risk judgement decision table. U
c1
c2
c3
c4
c5
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:2; 0:4i h0:1; 0:7i h0:1; 0:8i h0:1; 0:9i h0:4; 0:6i h0:1; 0:6i h0:6; 0:4i h0:6; 0:2i h0:6; 0:2i h0:6; 0:4i
h0:1; 0:7i h0:1; 0:8i h0:1; 0:8i h0:6; 0:3i h0:2; 0:6i h0:2; 0:6i h0:6; 0:4i h0:6; 0:2i h0:6; 0:4i h0:6; 0:4i
h0:2; 0:6i h0:3; 0:6i h0:2; 0:8i h0:2; 0:7i h0:2; 0:8i h0:2; 0:8i h0:6; 0:4i h0:8; 0:2i h0:8; 0:2i h0:8; 0:2i
h0:6; 0:4i h0:5; 0:2i h0:5; 0:4i h0:2; 0:8i h0:2; 0:8i h0:2; 0:4i h0:7; 0:3i h0:4; 0:6i h0:1; 0:6i h0:8; 0:2i
h0:2; 0:8i h0:2; 0:7i h0:6; 0:4i h0:6; 0:4i h0:2; 0:8i h0:2; 0:8i h0:4; 0:6i h0:4; 0:5i h0:8; 0:2i h0:6; 0:4i
1 2 1 1 2 1 2 2 3 3
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U
½x C
App C ðxÞ
App C ðxÞ
U 1 U 2 U 1 U 1 U 2 U 1 U 2 U 2 U 3 U 3
U 1
x1
x1
x1 ; x7 ; x10
x2
x2
x2 ; x7 ; x10
x3
x3
x3 ; x10
x4
x4
x4 ; x9
x5
x5
x5 ; x7 ; x8 ; x9 ; x10
x6
x6
x6 ; x7 ; x8 ; x10
x7
x1 ; x2 ; x5 ; x6 ; x7
x7 ; x10
x8
x5 ; x6 ; x8
x8
x9
x4 ; x5 ; x9
x9
x10
x1 ; x2 ; x3 ; x5 ; x6 ; x7 ; x10
x10
U 2 U 1 U 1 U 2 U 1 U 2 U 2 U 3 U 3
According to Theorem 4.3, only a pair of and lower approximation reducts exists; i.e., fc1 ; c5 g. That is B ¼ B ¼ fc1 ; c5 g. On the basis of the obtained and lower discernibility matrix, we can derive the lower discernibility functions of some objects using Definition 4.2 and Theorem 4.4. For example, the lower discernibility functions of xi ð1 6 i 6 8Þ are
f ðx1 Þ ¼ ðc4 _ c5 Þ ^ ðc1 _ c2 Þ ^ ðc1 _ c2 _ c3 _ c5 Þ ¼ ðc1 ^ c4 Þ _ ðc1 ^ c5 Þ _ ðc2 ^ c4 Þ _ ðc2 ^ c5 Þ; f ðx2 Þ ¼ c1 _ c2 _ c3 _ c5 ; f ðx3 Þ ¼ ðc1 _ c2 Þ ^ ðc1 _ c3 _ c4 Þ ¼ c1 _ ðc2 ^ c3 Þ _ ðc2 ^ c4 Þ; f ðx4 Þ ¼ c1 ; f ðx5 Þ ¼ c1 _ c2 _ c3 _ c4 _ c5 ; f ðx6 Þ ¼ ðc1 ^ c3 Þ _ ðc1 ^ c4 Þ _ ðc1 ^ c5 Þ; f ðx7 Þ ¼ c3 _ c5 _ ðc1 ^ c4 Þ; f ðx8 Þ ¼ c5 : The simplified upper bound rules for information systems security audit risk judgement are drawn by the lower discernibility functions of objects as follows:
f ðx; c1 Þ h0:2; 0:4i ^ f ðx; c4 Þ h0:6; 0:4i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c1 Þ h0:2; 0:4i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:1; 0:7i ^ f ðx; c4 Þ h0:6; 0:4i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:1; 0:7i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c1 Þ h0:1; 0:7i _ f ðx; c2 Þ h0:1; 0:8i _ f ðx; c3 Þ h0:3; 0:6i_; f ðx; c5 Þ h0:2; 0:7i ) f ðx; dÞ 6 2ðsupported by x2 Þ; f ðx; c1 Þ h0:1; 0:8i _ ðf ðx; c2 Þ h0:1; 0:8i ^ f ðx; c3 Þ h0:2; 0:8iÞ_; ðf ðx; c2 Þ h0:1; 0:8i ^ f ðx; c4 Þ h0:5; 0:4iÞ ) f ðx; dÞ 6 1ðsupported by x3 Þ; f ðx; c1 Þ h0:1; 0:9i ) f ðx; dÞ 6 1ðsupported by x4 Þ; f ðx; c1 Þ h0:4; 0:6i _ f ðx; c2 Þ h0:2; 0:6i _ f ðx; c3 Þ h0:2; 0:8i_; f ðx; c4 Þ h0:2; 0:8i _ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 2ðsupported by x5 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c3 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x6 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c4 Þ h0:2; 0:4i ) f ðx; dÞ 6 1ðsupported by x6 Þ; f ðx; c1 Þ h0:1; 0:6i ^ f ðx; c5 Þ h0:2; 0:8i ) f ðx; dÞ 6 1ðsupported by x6 Þ;
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f ðx; c3 Þ h0:6; 0:4i _ f ðx; c5 Þ h0:4; 0:6i ) f ðx; dÞ 6 2ðsupported by x7 Þ; f ðx; c1 Þ h0:6; 0:4i ^ f ðx; c4 Þ h0:7; 0:3i ) f ðx; dÞ 6 2ðsupported by x7 Þ; f ðx; c5 Þ h0:4; 0:5i ) f ðx; dÞ 6 2ðsupported by x8 Þ: The first rule can be interpreted as follows: If Better Systems Total Security is less than h0:2; 0:4i and Credible Hardware Device is less than h0:6; 0:4i, then Risk Judgement on Information Systems Security Audit is Complete Examination. Similarly, the lower approximation discernibility functions of x2 ; x5 ; x7 ; x8 ; x9 ; x10 and their corresponding simplified lower bound rules for information systems security audit risk judgement are obtained by using the lower approximation discernibility matrix as follows:
f ðx2 Þ ¼ c4 _ ðc1 ^ c5 Þ _ ðc3 ^ c5 Þ; f ðx5 Þ ¼ c1 ; f ðx7 Þ ¼ c1 _ c3 _ c4 ; f ðx8 Þ ¼ c1 _ c2 _ c3 ; f ðx9 Þ ¼ c5 ; f ðx10 Þ ¼ c4 _ ðc1 ^ c5 Þ _ ðc3 ^ c5 Þ; f ðx; c4 Þ h0:5; 0:2i _ ðf ðx; c1 Þ h0:1; 0:7i ^ f ðx; c5 Þ h0:2; 0:7iÞ_; ðf ðx; c3 Þ h0:3; 0:6i ^ f ðx; c5 Þ h0:2; 0:7iÞ ) f ðx; dÞ P 2ðsupported by x2 Þ; f ðx; c1 Þ h0:4; 0:6i ) f ðx; dÞ P 2ðsupported by x5 Þ; f ðx; c1 Þ h0:6; 0:4i _ f ðx; c3 Þ h0:6; 0:4i _ f ðx; c4 Þ h0:7; 0:3i; ) f ðx; dÞ P 2ðsupported by x7 Þ; f ðx; c1 Þ h0:6; 0:2i _ f ðx; c2 Þ h0:6; 0:2i _ f ðx; c3 Þ h0:8; 0:2i; ) f ðx; dÞ P 2ðsupported by x8 Þ; f ðx; c5 Þ h0:8; 0:2i ) f ðx; dÞ P 3ðsupported by x9 Þ; f ðx; c4 Þ h0:8; 0:2i _ ðf ðx; c1 Þ h0:6; 0:4i ^ f ðx; c5 Þ h0:6; 0:4iÞ_; ðf ðx; c3 Þ h0:6; 0:4i ^ f ðx; c5 Þ h0:6; 0:4iÞ ) f ðx; dÞ P 3ðsupported by x10 Þ; This rule, f ðx; c5 Þ h0:8; 0:2i ) f ðx; dÞ P 3 can be explicated as: If Credible Network Security is greater than h0:8; 0:2i, then Risk Judgement on Information Systems Security Audit is No Examination. Example 6.2. Candidate global supplier selection in a manufacturing company A multi-criteria decision making problem is concerned with a manufacturing company which wants to select the best global supplier according to the core competencies of suppliers. Four types of suppliers are found in a manufacturing company. These are denoted by U ¼ fx1 ; x2 ; x3 ; x4 g. The company can evaluate their core competencies from the following aspects (attributes): ‘‘Level of Technology Innovation’’ (c1 ), ‘‘Control Ability of Flow’’ (c2 ), ‘‘Ability of Management’’ (c3 ), and‘‘Level of Service’’ (c4 ). We assume that the manufacturing company is amenable only to providing its attribute values on each attribute as an intuitionistic value and those of the selection results (d) on the selectors as crisp ones. Here, the candidate global selection rules can generally be extracted using the proposed rule extraction approach in this paper. Inspired by Tan and Chen [61], we build a similar dataset (Table 6); partly taken from [61]). The domain of d is f1; 2; 3g, where 1 means ‘‘Rejection,’’ 2 means ‘‘Considerable Selection,’’ and 3 means ‘‘Priority Selection’’.
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Using the same procedure in Example 5.1, we derive the and lower approximation discernibility matrices
0
C
BC B M ¼ B @ c3 C
c3 c4
C
C
C
c3
C
c2 c3
C
C
1
C C C c1 c3 c4 A C C
and
0
C
Bc c B 3 4 M ¼ B @ C C
C
c3
C
c3
C
C
C
c1 c3 c4
C
1
c2 c3 C C C; C A C
respectively. There is only and lower approximation reduct fc3 g. All the simplified upper and lower bound rules for the selection of candidate global supplier are as follows:
f ðx; c3 Þ h0:7; 0:2i _ f ðx; c4 Þ h0:3; 0:1i ) f ðx; dÞ 6 2ðsupported by x1 Þ; f ðx; c3 Þ h0:5; 0:4i ) f ðx; dÞ 6 1ðsupported by x3 Þ; f ðx; c2 Þ h0:2; 0:5i _ f ðx; c3 Þ h0:4; 0:2i ) f ðx; dÞ 6 2ðsupported by x4 Þ; f ðx; c3 Þ h0:7; 0:2i ) f ðx; dÞ P 2ðsupported by x1 Þ; f ðx; c3 Þ h0:9; 0:1i ) f ðx; dÞ P 3ðsupported by x2 Þ; f ðx; c1 Þ h0:6; 0:2i _ f ðx; c3 Þ h0:4; 0:2i _ f ðx; c4 Þ h0:7; 0:1i; ) f ðx; dÞ P 2ðsupported by x4 Þ; This rule, f ðx; c3 Þ h0:7; 0:2i ) f ðx; dÞ P 2 can be interpreted as: If Ability of Management is greater than h0:7; 0:2i, then Candidate Global Supplier Selection is at least Considerable Selection. Example 6.3. Cars classification The car dataset contains the information of ten new cars to be classified in the Guangzhou car market in Guangdong, China. Let U ¼ fxi j1 6 i 6 10g be the cars, each of which is described by six attributes: fuel economy (c1 ), aerod degree (c2 ), price (c3 ), comfort (c4 ); design (c5 ), and safety (c6 ). The characteristics of the ten new cars under the six attributes are represented (Table 7; partly taken from [35]). Similar to Examples 6.1 and 6.2, some simplified upper and lower bound rules can be drawn as follows (Owing to space limitations, we only list partial rules):
f ðx; c2 Þ h0:2; 0:7i _ f ðx; c3 Þ h0:4; 0:5i ) f ðx; dÞ 6 1ðsupported by x1 Þ; f ðx; c2 Þ h0:3; 0:5i _ f ðx; c3 Þ h0:2; 0:6i ) f ðx; dÞ 6 1ðsupported by x6 Þ; ... f ðx; c1 Þ h0:8; 0:1i _ f ðx; c5 Þ h0:8; 0:2i _ ðf ðx; c2 Þ h0:7; 0:2i ^ f ðx; c6 Þ ; h0:2; 0:6iÞ _ ðf ðx; c3 Þ h0:7; 0:0i ^ f ðx; c4 Þ h0:4; 0:1i ^ f ðx; c6 Þ h0:2; 0:6iÞ; ) f ðx; dÞ P 4ðsupported by x5 Þ;
Table 6 Candidate global supplier selection for manufacturing company. U
c1
c2
c3
c4
d
x1 x2 x3 x4
h0:5; 0:3i h0:4; 0:3i h0:4; 0:1i h0:6; 0:2i
h0:5; 0:4i h0:3; 0:4i h0:5; 0:3i h0:2; 0:5i
h0:7; 0:2i h0:9; 0:1i h0:5; 0:4i h0:4; 0:2i
h0:3; 0:1i h0:5; 0:2i h0:6; 0:2i h0:7; 0:1i
2 3 1 2
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Table 7 Cars classification. U
c1
c2
c3
c4
c5
c6
d
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
h0:3; 0:4i h0:4; 0:3i h0:4; 0:2i h0:3; 0:4i h0:8; 0:1i h0:4; 0:3i h0:6; 0:4i h0:9; 0:1i h0:4; 0:4i h0:9; 0:1i
h0:2; 0:7i h0:5; 0:1i h0:6; 0:1i h0:9; 0:0i h0:7; 0:2i h0:3; 0:5i h0:4; 0:2i h0:7; 0:2i h1:0; 0:0i h0:8; 0:0i
h0:4; 0:5i h0:6; 0:2i h0:8; 0:1i h0:8; 0:1i h0:7; 0:0i h0:2; 0:6i h0:7; 0:2i h0:7; 0:1i h0:9; 0:1i h0:6; 0:3i
h0:8; 0:1i h0:2; 0:7i h0:2; 0:6i h0:7; 0:1i h0:4; 0:1i h0:7; 0:1i h0:3; 0:6i h0:4; 0:5i h0:6; 0:2i h0:5; 0:2i
h0:4; 0:5i h0:3; 0:6i h0:3; 0:7i h0:1; 0:8i h0:8; 0:2i h0:5; 0:4i h0:3; 0:7i h0:4; 0:5i h0:2; 0:7i h0:8; 0:1i
h0:2; 0:7i h0:7; 0:2i h0:5; 0:2i h0:2; 0:8i h0:2; 0:6i h0:3; 0:6i h0:6; 0:1i h0:8; 0:0i h0:1; 0:8i h0:6; 0:4i
1 2 2 3 4 1 2 4 3 4
f ðx; c1 Þ h0:9; 0:1i _ f ðx; c6 Þ h0:8; 0:0i _ ðf ðx2 Þ h0:7; 0:2i ^ f ðx; c3 Þ ; h0:7; 0:1i ^ f ðx; c5 Þ h0:4; 0:5iÞ ) f ðx; dÞ P 4ðsupported by x5 Þ; f ðx; c1 Þ h0:9; 0:1i _ f ðx; c5 Þ h0:8; 0:1i _ ðf ðx; c2 Þ h0:8; 0:0i ^ f ðx; c6 Þ ; h0:6; 0:4iÞ _ ðf ðx; c3 Þ h0:6; 0:3i ^ f ðx; c4 Þ h0:5; 0:2i ^ f ðx; c6 Þ h0:6; 0:4iÞ; ) f ðx; dÞ P 4ðsupported by x10 Þ ... 7. Conclusion Rough set theory is a useful mathematical tool for dealing with uncertain information. However, when we consider ranking fuzzy-valued objects rather than classifying them, conventional rough set theory is unable to solve these problems. One of the extensions of the classic rough set approach is the dominance fuzzy-valued rough set approach. The intuitionistic fuzzy decision table is an important type of data one, and is a generalized form of fuzzy-valued information systems [62,63]. This paper focuses on the construction of a fuzzy-rough set model and rule extraction in DIFDT, which aids decision making under dominance-based intuitionistic fuzzy settings. The main contribution of this paper is the introduction of the ranking method for comparing two intuitionistic fuzzy values in DIFDT. First, we defined the notion of DIFDT and introduced a ranking method for all objects with the dominance-based relation between objects. Second, on the basis of the dominance-based relation, we established a fuzzy-rough set approach in DIFDT, which are grounded primarily based on the substitution of the indiscernibility relation with the dominance-based relation. Third, to extract the simplest dominance intuitionistic fuzzy lower and upper bound rules, we used the discernibility matrices to propose two attribute reduction approaches for eliminating redundant information. Finally, we applied these approaches to information systems security audit risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification. The application examples yielded valuable rules. The approaches simplified a DIFDT and enabled the direct identification of considerably simpler intuitionistic fuzzy lower and upper bound rules. Moreover, the derived rules can facilitate knowledge acquisition from DIFDT. Acknowledgements The authors thank the anonymous referees for their constructive suggestions. This paper was supported by the Natural Science Foundation of China (Grant Nos. 61170105 and 71201076), Qing Lan Project of Jiangsu Province, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (Audition Science and Technology). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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