Advances in Systems Science and Application (2015) Vol.15 No.3 202-219 A Novel Multi-attribute Decision Making Methodology and Application Yong Liu1 and Yi Lin 2 1 School of business, Jiangnan University, Wuxi, 214122, China; 2 Mathematics Department, Slippery Rock University of USA, Pennsylvania, USA. Abstract In the multi-attribute decision making problems, how to effectively extract the decision making rules and rank the schemes are much more important research contents. However, the acquisitions of the decision making rules often are ignored. In view of this, with respect to the problems that there exist a lot of preference information and fuzzy information in the real decision making information sys- tem, a novel decision making methodology based on dominance intuitionistic fuzzy rough set is constructed in the paper, and then it is applied to audit risk assessment and risk judgment. Based on the analysis of model and example, the result shows that the proposed model can well realize the extraction of the decision making rules and the ranking of the schemes, and effectively deal with the intuitionistic fuzzy information system with preference information. Keywords Preference information; Decision making rule; Dominance distance index 1 Introduction As a useful mathematical tool to deal with knowledge with inaccuracy, uncertain- ty and fuzziness, rough set theory was initially proposed by Pawlak [1]. It has been widely applied in many fields, such as knowledge discovery, data mining, decision analysis, and pattern recognition[2–4]. The classical rough set theory conducts data reasoning on the basis of equivalence relationship, while it is dif- ficult to satisfy the harsh conditions of equivalence relationship in the practical applications. At the same time, the binary relationship existing on the field of discourse is often a fuzzy relationship and a similarity relationship instead of an equivalence relationship. In view of this, based on the idea and method of the fuzzy set theory[5] put forward the fuzzy rough sets theory. Because this new theory can well describe the uncertainty of various types of knowledge and more objectively reflect the physical world, it has been rapidly becoming a research focus of rough set theory, leading to its rapid development. As a result of simultaneously considering the positive, negative and hesitancy degrees for an object to belong to a set, intuitionistic fuzzy sets possess stronger ability of information expression and well describe and portray delicate ambigu- ities of the nature of the objective world when compared with the traditional fuzzy sets[6, 7]. Therefore, intuitionistic fuzzy sets and rough sets are first pro- Advances in Systems Science and Application (2015) Vol.15 No.3 203 posed to hybrid, leading to the construction of the intuitionistic fuzzy rough set model [8]. Due to the important theoretical value and application implications, intuitionistic fuzzy rough set theory has soon become a hot academic research area. Currently, most of the related research on intuitionistic fuzzy rough set lies in the aspects of constructing different models and exploring their relevant properties. For the related researches on constructing different models and exploring their properties, the relationship between intuitionistic fuzzy set theory and rough set theory firstly is revealed, and then they employ intuitionistic fuzzy set to de- fine approximation operators in the intuitionistic fuzzy approximation space. By making use of the cut set of intuitionistic fuzzy sets, the upper and lower ap- proximation operators of intuitionistic fuzzy rough set and the axiomatic method of the approximation operators based on general binary intuitionistic fuzzy re- lationship are respectively constructed [8–10], and then it is well known that the upper and lower approximation sets of intuitionistic fuzzy rough sets are in- tuitionistic fuzzy sets by making the proof [11]. Based on intuitionistic fuzzy residual implication and intuitionistic fuzzy relationship, an intuitionistic fuzzy rough set model is established[12]. However, it is difficult to apply this model to deal with an information system with noise data. With the intuitionistic fuzzy triangle model T = min, intuitionistic fuzzy t-conorms S= max, and intuition- istic fuzzy inverse operator N, the approximation operators of the intuitionistic fuzzy rough sets is defined, and the intuitionistic fuzzy rough set models based on the general intuitionistic fuzzy logic operators are developed[13–15]. By using the thought of intuitionistic fuzzy set and rough set, an improved intuitionistic fuzzy rough set model based on Hamming distance and establish the models such properties as interval, symmetry, complete similarity and complete dissimilarity are proposed[16], while the novel intuitionistic fuzzy rough set based on general intuitionistic fuzzy information systems is constructed in order to expand the model and its application[17]. the interval-valued intuitionistic fuzzy rough set based on the thought of implication is established, and then the related properties of the models are developed[18, 19]. By combining interval intuitionistic fuzzy set and rough set, the interval intuitionistic fuzzy rough set models based on interval intuitionistic fuzzy relationship are constructed[19–21]. By using interval-valued intuitionistic fuzzy compatibility relationship, the interval-valued intuitionistic fuzzy rough set model based on the concept of double universes and relevant properties is constructed, and then it is applied into the decision making[22, 23]. For the related literature on attribute reduction, a genetic algorithm is pro- posed to reduce attributes by making use of the characteristics of intuitionistic fuzzy information systems[24], while the kind of attribute reduction algorithm of intuitionistic fuzzy rough set based on mutual information by combining in- 204 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... formation entropy theory and intuitionistic fuzzy rough set is constructed and designed[25]. By using the intuitionistic fuzzy distance formula, an intuitionistic fuzzy rough model and an attribute reduction algorithm is constructed[26–28]. As discussed above, although there exist many related literatures on the in- tuitionistic fuzzy rough set model to construct models and discuss their theories from different levels, there are few intuitionistic fuzzy rough set models in the existing literatures those can be applied into the multi-attribute decision mak- ing and effectively deal with the real problems. In the multi-attribute group decision making problems, the dominated and dominating relationship for the attributes should be considered, and the subjective preferences of attribute val- ue from decision-makers are based on some methods to aggregate, so that the acquisition of the decision rules and the ranking of the schemes can be made. However, it is difficult for the group decision making methods based on rough set[29–38] to deal with multi-attribute decision making problems with noise data, preference information and fuzzy information. In view of this, with respect to the shortcoming of the size comparison of the intuitionistic fuzzy numbers, the concept of the intuitionistic fuzzy dominance distance index is defined, and then it is used to construct a novel intuitionistic fuzzy rough set model, finally an example illustrates the effectiveness and applicability of the proposed model. 2 Dominance Distance Index Intuitionistic fuzzy sets proposed by Atanassov[6, 7] are an expansion and further development of the traditional fuzzy sets. As a result of taking into account to the membership and non-membership information by adding a new attribute param- eter: non-membership function in intuitionistic fuzzy sets, it provides additional options for describing the properties of things and possesses a stronger capability of dealing with uncertainty information. That is, intuitionistic fuzzy sets can well describe and portray delicate ambiguities of the nature of the objective world. 2.1 Intuitionistic Fuzzy Dominance Distance Index Definition 1. (Intuitionistic fuzzy sets[6, 7]. Suppose thatX = {x1, x2, ..., xn} is a nonempty, finite set of objects with xi(i = 1, 2, ..., n) being the ith object. Then the set A = {< x, µA(x), υA(x) > |x ∈ X} of triplets is called an intuitionistic fuzzy set, where µA(x) and υA(x) are respectively known as the membership and non-membership for the object to belong to , that is, µA(x) : X → [0, 1], x ∈ X → µA(x) ∈ [0, 1] (1) υA(x) : X → [0, 1], x ∈ X → υA(x) ∈ [0, 1] (2) satisfying 0 ≤ µA(x) + υA(x) ≤ 1 , for any x ∈ X. And πA(x) = 1 − µA(x) − υA(x), x ∈ X stands for the degree of hesitation or uncertainty for the object to Advances in Systems Science and Application (2015) Vol.15 No.3 205 belong to . So the intuitionistic fuzzy number is denoted as α =< µA(x), υA(x) >. Definition 2. (Intuitionistic fuzzy sets[6, 7]. For any intuitionistic fuzzy number α =< µ, ν >, the score function S(α) of this number is defined as follows: S(α) = µ− ν, S(α) ∈ [−1, 1] (3) The larger S(α) is, the greater the intuitionistic fuzzy number α =< µ, ν > is. For example, assume that both intuitionistic fuzzy numbers are respectively α1 =< 0.8, 0.1 > and α1 =< 0.9, 0.1 >, because S(α1) = 0.7 and S(α2) = 0.8, then S(α1) < S(α2). So we can regard the intuitionistic fuzzy number α2 as being greater than α2. Definition 3. (Intuitionistic fuzzy sets[6, 7]. For any intuitionistic fuzzy number α =< µ, ν >, the accuracy function H(α) of this number is defined as follows: H(α) = µ+ ν 2 (4) The larger H(α) is, the greater the intuitionistic fuzzy number α =< µ, ν > is. For example, for intuitionistic fuzzy numbers α3 =< 0.7, 0.2 > and α4 =< 0.4, 0.2 >, according to Definition 4, the accuracy functions of these intuitionistic fuzzy numbers are respectively H(α3) = 0.45 and H(α3) = 0.3. Therefore,α3 > α4. According to the definition 2 and 3, based on the score function S(α) and precision function H(α), the intuitionistic fuzzy numbers are compared. For any both intuitionistic fuzzy numbers αi =< µi, υi > and αk =< µk, υk >, if S(αi) ≥ S(αk), then αi ≥ αk; if S(αi) = S(αk) and H(αi) ≥ H(αk), then αi ≥ αk. For example, for intuitionistic fuzzy numbers α5 =< 0.8, 0.1 > and α6 =< 0.6, 0.3 >, due to S(α5) = 0.7, S(α6) = 0.3, therefore α5 > α6; while for the intuitionistic fuzzy numbers α7 =< 0.7, 0.3 > and α8 =< 0.5, 0.1 >, due to S(α7) = 0.4, S(α8) = 0.4, H(α7) = 0.5, H(α8) = 0.3, therefore α7 > α8. However, there exist the shortcomings that how much their uncertainty allotted to the membership and non-membership for two intuitionistic fuzzy numbers, so that the size of the intuitionistic fuzzy numbers cannot be exactly determined, for example, the intuitionistic fuzzy numbers α7, α8. In order to determine the size relationship of the intuitionistic fuzzy numbers, a novel method should be proposed. Definition 4. For any given two intuitionistic fuzzy numbers αi =< µi, υi > and αk =< µk, υk >, then the dominance distance index IFDD(xi, xk) of the intuitionistic fuzzy numbers αi and αk can be defined as follows: IFDD(xi, xk) =  1 µi ≥ µk, υi ≤ υk 0 µi < µk, υi > υk 1 2 + 1 4 µi−υi−µk+υk µ(Ω) other (5) 206 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... Where, µ(Ω) stands for the background measure, and it is generally taken as µ(Ω) = max {1− υ1, 1− υ2, ..., 1− υn}−min {µ1, µ2, ..., µn}, while IFDD(xi, xk) expresses the degree of intuitionistic fuzzy numbers vi better than the intuition- istic fuzzy number vk. Based on the definition 4, intuitionistic fuzzy numbers can be compared. For example, suppose that the background measure µ(Ω) is 1, for the intuitionistic fuzzy numbers α9 =< 0.8, 0.1 > and α10 =< 0.5, 0.3 >, such that IFDD(xi, xk) = 0.55, therefore, α11is greater than α12 by the possibility of 55%. 2.2 Dominance Relationship Based on Dominance Distance Index Definition 5. Assume that an ordered 4-tuple S = (U,A, V, f) is known as an preference information system, if U = {U1, U2, ..., Un} is a finite and nonemp- ty set, known as the universe; A = C ∪ D is a finite, nonempty attribute set, while C = {a1, a2, ..., am} and D are respectively the condition attribute set and the decision attribute set; V = ∪Va stands for the value domain of the in- formation system S, where Va is the value of U with respect to the attribute a ∈ A; f : U ×A → V is an information function. If Va is an intuitionistic fuzzy number αa =< µa, υa >, then the information system S is called as the intuition- istic fuzzy preference information system, and denoted as IFS. Definition 6. Suppose that there exist an intuitionistic fuzzy preference infor- mation system IFS = (U,A, V, f), for ∀a ∈ P ⊆ A, xi, xk ∈ U, f(xi, a) = αia =< µia, υia >∈ V , f(xk, a) = αka =< µka, υka >∈ V , λ ∈ (0.5, 1], if IFDD(xi, xk) ≥ λ, then the dominance relationship between the objects xi and xk with respect to the α attribute or the attribute set P can be called as a dom- inance relationship based on the dominance distance index with threshold value λ, written as (xi, xk) ∈ R≥λ a , (xi, xk) ∈ R≥λ P . Property 1.Suppose that there exist an intuitionistic fuzzy preference infor- mation system IFS = (U,A, V, f), ∀a ∈ P ⊆ A, ∀xi, xk ∈ U, f(xi, a) = αia =< µia, υia >∈ V, f(xk, a) = αka =< µka, υka >∈ V , it holds true: 0 ≤ IFDDa(xi, xk) ≤ 1 (6) 0 ≤ IFDDP (xi, xk) ≤ 1 (7) Proof. According to the definition 4, (1)if µia ≥ µka, υia ≤ υka, then IFDDa(xi, xk) = 1; (2)if µia < µka, υia > υka, then IFDDa(xi, xk) = 0; (3)if µia ≥ µka, υia ≥ υka or µia ≤ µka, υia ≤ υka, because of the background value measure µ(Ω) = max {1− υ1, 1− υ2, ..., 1− υn}−min {µ1, µ2, ..., µn}, then 2(min i (µia)−max i (1−υia)) ≤ µia−υia−µka+υka ≤ 2(max i (1−υia)−min i (µia)), so that −2 ≤ µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 2, and then there exists Advances in Systems Science and Application (2015) Vol.15 No.3 207 0 ≤ 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 1, therefore, 0 ≤ IFDDa(xi, xk) ≤ 1. In conclusion, 0 ≤ IFDDa(xi, xk) ≤ 1 can be obtained. Let the weight vector of attributes is w = (w1, w2, ..., wp), and it satisfies wt > 0 and p∑ t=1 wt = 1, and then there exists IFDDP (xi, xk) = p∑ t=1 wtIFDDat(xi, xk), therefore, 0 ≤ IFDDP (xi, xk) ≤ 1. Property 2. For ∀a ∈ P ⊆ A, xi, xk ∈ U , f(xi, a) = αia =< µia, υia >∈ V and f(xk, a) = αka =< µka, υka >∈ V , λ ∈ (0.5, 1], if (xk, xs) ∈ R≥λ a , the follow- ing holds true: IFDDa(xi, xk) ≤ IFDDa(xi, xs). Proof. It suffices to show IFDDa(xi, xk)− IFDDa(xi, xs) ≤ 0. For IFDDa(xi, xk), there exist three cases:IFDDa(xi, xk) = 1. IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) and IFDDa(xi, xk) = 0. For IFDDa(xi, xs), there also are three cases. For (xk, xs) ∈ R≥ a , according to the dominance relationship R≥λ a , that is, IFDDa(xk, xs) ≥ λ By analyzing the situation, there exit the following 6 cases. (1) When the object xi is definitely better than the objects xk and xs, that is αi > αk and αi > αs, thus IFDDa(xi, xs) = 1, IFDDa(xi, xk) = 1, therefore, IFDDa(xi, xs)− IFDDa(xi, xk) = 0. (2)When the objectxi is definitely better than the object xs, and not nec- essarily better than the object xk, IFDDa(xi, xs) = 1 and IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) can be acquired. Therefore, the following holds true: IFDDa(xi, xs) − IFDDa(xi, xk) = 1 2 − 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) . According to Def- inition 4, there exits −2 ≤ µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 2, That is, IFDDa(xi, xk) − IFDDa(xi, xs) ≤ 0. (3) When the object xi is definitely better than the objects xs and must be inferior to the object xk, IFDDa(xi, xs) = 1 and IFDDa(xi, xk) = 0 can be obtained. Thus, there exists IFDDa(xi, xs) − IFDDa(xi, xk) = 1 that is, IFDDa(xi, xk)− IFDDa(xi, xs) ≤ 0. (4) When the object xi may be superior to the object xs and may be superior to the object xk, IFDDa(xi, xs) = 1 2+ 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) and IFDDa(xi, xk) = 1 2+ 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) can be acquired. Thus, IFDDa(xi, xs)−IFDDa(xi, xk) = 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) Due to IFDDa(xk, xs) ≥ 1 2 , thus µka − υka − µsa + υsa ≥ 0, therefore, it then follows that IFDDa(xi, xs)− IFDDa(xi, xk) ≥ 0. When the object xi may be superior to the object xs and must be inferior to the object xk, IFDDa(xi, xs) = 1 2+ 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) and IFDDa(xi, xk) = 0 can be obtained. Thus IFDDa(xi, xs)− IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) . 208 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... From the fact that −2 ≤ µia−υia−µsa+υsa max i (1−υia)−min i (µia) ≤ 2, it follows that IFDDa(xi, xs)− IFDDa(xi, xk) ≥ 0. When the object xi must be inferior to the objects xs and xk, there ex- ist IFDDa(xi, xs) = 0 and IFDDa(xi, xk) = 0, and then IFDDa(xi, xs) − IFDDa(xi, xk) = 0 can be concluded. In conclusion, IFDDa(xi, xs)−IFDDa(xi, xk) ≥ 0 therefore, IFDDa(xi, xk) ≤ IFDDa(xi, xs). QED Property 3. The dominance relationship based on dominance distance index satisfies the condition of transitivity. Proof. It suffices to show that for ∀a ∈ P ⊆ A, xi, xk ∈ U, if IFDDa(xi, xk) ≥ 1 2 and IFDDa(xk, xs) ≥ 1 2 , then IFDDa(xi, xs) ≥ 1 2 . When IFDDa(xi, xk) ≥ 1 2 , there exists µia − υia − µka + υka ≥ 0 . Accord- ingly there are two cases which are respectively certainly better and perhaps superior for the object xi than the object xk. That is, if µi ≥ µk, υi ≤ υk, then IFDDa(xi, xk) = 1. If IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) and µia − υia − µka + υka ≥ 0 then IFDDa(xi, xk) ≥ 1 2 . when IFDDa(xk, xs) ≥ 1 2 similarly there are also two cases which are respec- tively and certainly better and perhaps superior for the object xk than the object xs. Thus there exist the following 4 cases where the object xi is better than the object xs. (1)If IFDDa(xi, xk) = 1 and IFDDa(xk, xs) = 1, then µi ≥ µk, υi ≤ υk and µk ≥ µs, υk ≤ υs, and then µi ≥ µs, υi ≤ υs. Therefore IFDDa(xi, xs) = 1. (2)If IFDDa(xi, xk) = 1, IFDDa(xk, xs) = 1 2 + 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µka−υka−µsa+υsa ≥ 0, then µia−υia−µsa+υsa ≥ 0. According to Definition 4, IFDDa(xk, xs) ≥ 1 2 can be obtained. (3)IFDDa(xk, xs) = 1 2+ 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µia−υia−µka+υka ≥ 0 and IFDDa(xk, xs) = 1 2 + 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µka − υka − µsa + υsa ≥ 0, such that µia − υia − µsa + υsa ≥ 0 And according to the definition of the dominance distance index, IFDDa(xk, xs) ≥ 1 2 can be obtained. (4) If IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) , µia−υia−µka+υka ≥ 0 and IFDDa(xk, xs) = 1, then µia − υia − µsa + υsa ≥ 0. According to the definition of the dominance distance index, therefore, IFDDa(xk, xs) ≥ 1 2 . In conclusion, IFDDa(xk, xs) ≥ 1 2 can be always obtained. QED. Property 4. For ∀a ∈ P ⊆ A, xi, xk ∈ U , αia =< µia, υia >∈ V , αka =< µka, υka >∈ V ,then the following hold true: IFDDa(xi, xk)+IFDDa(xk, xi) = 1. Proof. According to equation (5), (1)If µia ≥ µka, υia ≤ υka, then IFDDa(xi, xk) = 1 and IFDDa(xk, xi) = 0, therefore IFDDa(xi, xk) + IFDDa(xk, xi) = 1 can be obtained. Advances in Systems Science and Application (2015) Vol.15 No.3 209 (2) When µia < µka, υia > υka, it follows that IFDDa(xi, xk) = 0 and GDDa(xk, xi) = 1. Thus, we have GDDa(xi, xk) +GDDa(xk, xi) = 1. (3) When the other conditions, IFDDa(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka µa(Ω) and IFDDa(xk, xi) = 1 2 + 1 4 µka−υka−µia+υia µa(Ω) can be acquired. Thus, IFDDa(xi, xk) + IFDDa(xk, xi) = 1 can be obtained. QED. 3 Dominance Intuitionistic Fuzzy Variable Precision Rough Set Model In order to deal with preference attributes, with respect to the information system with preference value[39? –41], by introducing dominance relationship into the rough set mode and taking advantage of it to substitute for the indistinguishable relationship constructed the dominance rough set model, and then acquired the decision making rules. Based on the thought of dominance rough set model, the dominance relationship based on intuitionistic fuzzy dominance distance index can be used to substitute the equivalence relationship of rough set, and then the dominance intuitionistic fuzzy rough set model is constructed. 3.1 The Construction of Model Definition 7. Suppose that there exist an intuitionistic fuzzy preference in- formation system IFS = (U,A, V, f), for P ⊆ A, Cl≥t ⊆ D, a threshold value λ ∈ (0.5, 1] and β ∈ (0.5, 1], if D+ P (x) = {y ∈ U |IFDDa(x) ≥ λ, ∀a ∈ P } and D− P (x) = {y ∈ U |IFDDa(x) < λ, ∀a ∈ P } respectively stand for the λ−P dom- inating set and dominated set with respect to x, and then the lower approxima- tion and upper approximation of the decision making class Cl≥t are respectively defined as follows: aprλ P (Cl≥t ) = ∪{x ∈ U : D+ P (x) ⊆ Cl≥t } (8) aprλP (Cl≥t ) = ∪{x ∈ U : D− P (x) ∩ Cl≥t ̸= ∅} (9) And thus, [ aprλ P (Cl≥t ), apr λ P (Cl≥t ) ] is called as the dominance intuitionistic fuzzy rough set. The λ−P -lower approximation aprλ P (Cl≥t ) of (Cl≥t ) can be called as the posi- tive domain of the dominance intuitionistic fuzzy rough set and interpreted as the set of the union of all condition classes with confidence threshold value , where the classified objects definitely belong to the upward union (Cl≥t ). Accordingly, the λ − P -upper approximation aprλ P (Cl≥t ) of (Cl≥t ) can be interpreted as the union of all classes with confidence threshold value λ, where the classified objects possibly belong to the upward union (Cl≥t ). Accordingly, based on the lower approximation and upper approximation of the intuitionistic fuzzy rough set, the λ boundary domain and classification quality of the set (Cl≥t ) can be defined as follows: bndλP (Cl≥t ) = aprλP (Cl≥t )− aprλ P (Cl≥t ) (10) 210 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... αλ P (Cl≥t ) = |aprλ P (Cl≥t )| |aprλP (Cl≥t )| (11) Analogously, the λ -lower and λ -upper approximations of can be respectively defined as follows: aprλ P (Cl≤t ) = ∪{x ∈ U : D− P (x) ⊆ Cl≤t } (12) aprλP (Cl≤t ) = ∪{x ∈ U : D+ P (x) ∩ Cl≤t ̸= ∅} (13) And thus, [ aprλ P (Cl≤t ), apr λ P (Cl≤t ) ] stands for the dominance intuitionistic fuzzy rough set. Accordingly, based on the lower approximation and upper approximation of the intuitionistic fuzzy rough set, the λ boundary domain and classification quality of the set (Cl≥t ) can be defined as follows: bndλP (Cl≤t ) = aprλP (Cl≤t )− aprλ P (Cl≤t ) (14) αλ P (Cl≤t ) = |aprλ P (Cl≤t )| |aprλP (Cl≤t )| (15) 3.2 Attribute Reduction and Preference Rules Definition 8. Suppose that that there exist an intuitionistic fuzzy preference information system IFS = (U,A, V, f), for P ⊆ A and the given threshold value λ ∈ (0.5, 1], the classification quality of Cl can be defined as follows: γλP (Cl) = |U − ((∪bnd(Cl≥t )) ∪ (∪bnd(Cl≤t )))| |U | (16) The classification quality γλP (Cl) of Cl stands for the ratio of the relation be- tween all the correctly classified objects and all the objects with respect to the attribute set P in the information system. For every minimal subset P ⊆ C, the attribute set P satisfying γλP (Cl) = γλC(Cl) s called as the reduction of C with respect to Cl and denoted byREDCl(P ). The preferential decision rule is one kind of dependence form between condition preference attribute and decision preference attribute. Based on dominance rela- tionship with the dominance distance index, the rough approximation is acquired, and then the preferential decision rule can be induced and shown as follows: For the given threshold value λ, D≥ -decision rules can take on the following form: If f(x, q1) ≥ rq1 ∧ f(x, q2) ≥ rq2... ∧ f(x, qp) ≥ rqpthenx ∈ Cl≥t ; For the given threshold value λ, D≤ -decision rules can take on the following Advances in Systems Science and Application (2015) Vol.15 No.3 211 form: If f(x, q1) ≤ rq1 ∧ f(x, q2) ≤ rq2... ∧ f(x, qp) ≤ rqpthenx ∈ Cl≥t ; where {q1, q2, ...qp} ⊆ C, iff(x, q1) ≤ rq1 ∧ f(x, q2) ≤ rq2... ∧ f(x, qp) ≤ rqpthenx ∈ Cl≥t , t ∈ {1, 2, ..., l}. 3.3 Comprehensive Dominance Degree Definition 9. Suppose that the intuitionistic fuzzy preference information sys- tem IFS = (U,A, V, f), for ∀aj ∈ P ⊆ A, xi, xk ∈ U , f(xi, aj) = αij =< µij , υij >∈ V , f(xk, aj) = αkj =< µkj , υkj >∈ V the following is called as the dominance degree with equal weight with respect to the attribute set P for the object xi over the object xk based on the dominance distance index: IFDDP (xi, xk) = 1 |P | ∑ ∀a∈P IFDDa(xi, xk) (17) where |P | is the cardinality of the attribute set P . In the situation of real-life decision making, due to the fact that different deci- sion makers may very well weigh the attributes differently, the results, obtained on the basis of this uniform treatment of equal weights, can be obviously expect- ed to be inconsistent with the reality. Therefore, there is a need to study the situation that the attributes are given different weights. Definition 10. Suppose that the intuitionistic fuzzy preference information sys- tem IFS = (U,A, V, f), for ∀aj ∈ P ⊆ A, xi, xk ∈ U , f(xi, aj) = αij =< µij , υij >∈ V , f(xk, aj) = αkj =< µkj , υkj >∈ V , let the weight vector of the attribute be w = (w1, ..., wt, ..., w|P |), satisfying wt > 0 and |P |∑ t=1 wt = 1. Then IFDDP (xi, xk) = |P |∑ t=1 wtIFDDa(xi, xk) (18) is referred to as the different weight dominance degree with respect to the at- tribute set P for the object xi over the object xk based on the dominance distance index. Definition 11. Suppose that the intuitionistic fuzzy preference information sys- tem is IFS = (U,A, V, f), for ∀xi ∈ U , P ⊆ C, the comprehensive dominance degree of the object xi in all the objects based on the dominance distance index is defined as follows: IFDDP (xi) = 1 |U | − 1 ∑ i ̸=k IFDDP (xi, xk) = 1 |U | − 1 ∑ i̸=k |P |∑ t=1 wtIFDDa(xi, xk) (19) 212 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... 3.4 The Decision Steps Based on Dominance Intuitionistic Fuzzy Rough Set To sum up, the decision steps are following: Step1: by adjusting the given threshold value , based on the condition at- tribute set, the different classifications can be determined. Step2: based on the decision attribute set, all classifications can be deter- mined. Step3: according to all classifications based on the condition attribute set and the decision attribute set, the lower approximation and the upper approximation of the decision classifications can be acquired, and then the attribute reduction can be obtained, so that t the decision rules can be extracted. Step4: by the classification results based on the condition attribute set or the decision attribute set, based on the expressions (18) and (19), the comprehensive dominance degrees of the objects are calculated, and then the ranking of the objects can be determine. 4 Case Analysis Information system security design is that by testing the legal compliance, the confidentiality, the usability, and the reliability of the audiles, the security risk of information system is assessed, and then some audit opinions and suggestions of information system security can be given. While the risk guidance informa- tion system security audit based on the risk identification and evaluation is that through the comprehensive analysis of the audiles operation environment and in- formation system running environment, the factors affected the system security are extracted, and then according to the risk assessment, the implementation audit scope and key target are determined, so that the substantive tests are im- plemented. Therefore, the evaluation and judgment of audit risk possess are much more important in the information system security audit process. Currently, few scholars take advantage of rough set to make the audit risk judgment of the in- formation system security, and the evaluation is mainly based on the knowledge and experience of audit personnel in the practice process, however, it lacks the scientific and rationality, which has effect on the implementation of information system security audit, and ultimately affects the audit results. In recent years, the audit risk assessment and risk judgment expert system is established to ef- fectively decrease audit judgment deviation and reduce the audit risk. however, due to the complexity and uncertainty of the objective world, as well as the lim- itation of human ability to understand, there always exist a variety of preference information and fuzzy information in the audit risk assessment and risk judgment expert system, while it is difficult for the existing representation and processing methods of the information system security audit based on the expert experience to acquire exactly knowledge. In view of this, based on the proposed model in Advances in Systems Science and Application (2015) Vol.15 No.3 213 this paper, the attribute reduction is used to extract the decision rule of the audit risk assessment of the information system security and acquire the key factors and bottleneck factors of affecting the audit risk assessment of the information system security, and then the ranking of the different information system security can be determined, so that the proposed model can provide for audit personals a much reasonable and effective audit risk assessment method and tool. According to the actual audit cases, the related data can be collected and shown in the table 1. In the table 1, there exist 10 audited objects denoted as U = {x1, x2, ..., x10}, and the five attributes C = {a1, a2, a3, a4, a5} which re- spectively good system environment, good system control, reliable financial data, reliable audit software and standard operation. For each condition attribute, based on the audit results and their own professional quality, its values can be obtained by the comprehensive judgment from the information system audit ex- perts, for example, f(x2, a1) =< 0.5, 0.4 > stands for the fact that 50% experts think that the system environment of the audited object x2 is good, while 40% experts think that it is bad and 10% experts hesitate. The decision attribute set is denoted as D = {d}, and it indicates whether the audit risk of information system security is acceptable. For example, f(x2, d) = 1 expresses that experts regard that the audit risk of the audited object x2 is acceptable. Table 1 the audit risk assessment intuitionistic fuzzy information system of the information system security U a1 a2 a3 a4 a5 d U1 < 0.2, 0.6 > < 0.1, 0.7 > < 0.4, 0.4 > < 0.5, 0.4 > < 0.4, 0.4 > 0 U2 < 0.5, 0.4 > < 0.3, 0.6 > < 0.3, 0.6 > < 0.5, 0.2 > < 0.5, 0.4 > 0 U3 < 0.2, 0.6 > < 0.1, 0.8 > < 0.2, 0.7 > < 0.3, 0.6 > < 0.3, 0.6 > 0 U4 < 0.7, 0.1 > < 0.6, 0.4 > < 0.7, 0.2 > < 0.8, 0.2 > < 0.7, 0.2 > 1 U5 < 0.3, 0.6 > < 0.2, 0.7 > < 0.2, 0.7 > < 0.2, 0.6 > < 0.3, 0.7 > 0 U6 < 0.6, 0.3 > < 0.6, 0.4 > < 0.6, 0.3 > < 0.7, 0.2 > < 0.5, 0.4 > 1 U7 < 0.2, 0.6 > < 0.2, 0.6 > < 0.5, 0.4 > < 0.5, 0.4 > < 0.2, 0.6 > 1 U8 < 0.1, 0.6 > < 0.2, 0.6 > < 0.4, 0.5 > < 0.3, 0.6 > < 0.2, 0.7 > 0 U9 < 0.7, 0.2 > < 0.6, 0.4 > < 0.8, 0.1 > < 0.6, 0.3 > < 0.8, 0.2 > 1 U10 < 0.6, 0.2 > < 0.6, 0.2 > < 0.8, 0.2 > < 0.4, 0.5 > < 0.4, 0.5 > 1 Step1: according to the condition attribute set C, when λ = 0.55, based on the dominance distance index, the universe can be divided into the following classifications: U/C = {X1, X2, X3, X4} where X1 = {x1, x3, x5} , X2 = {x7, x8} , X3 = {x2, x6, x9, x10} , X4 = {x4} and 214 S. Baizakov, N.Baizakov and Jeffrey Forrest: A Novel Multi-attribute Decision... X1 < X2 < X3 < X4. Step2:according to the decision attribute set D, the universe can be divided into the following classifications: U/D = {Cl1, Cl2}. where Cl1 = {x1, x2, x3, x5, x7, x8}, U/D = {Cl1, Cl2}. Step3: according to the classifications based on the condition attribute set and the decision attribute set, the lower approximation aprλ P (Cl≥1 ) of (Cl≥1 ) and the upper approximation aprλ P (Cl≥1 ).the lower approximation aprλ P (Cl≤2 ) of (Cl≤2 ) and the upper approximation aprλ P (Cl≤2 ) respectively are: aprλ P (Cl≤1 ) = {x1, x3, x5, x7, x8} , aprλP (Cl≤1 ) = {x1, x2, x3, x5, x6, x7, x8, x9, x10} aprλ P (Cl≥2 ) = {x4} , aprλP (Cl≥2 ) = {x2, x4, x6, x9, x10} γλ,βP (Cl) = | {x1, x3, x5, x7, x8} ∪ {x4} | | {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10} | = 6 10 = 0.6 When λ = 0.55, the reduction {a1, a2} based on the genetic algorithm can be acquired, therefore, it is well known that the factors of good system environment and good system control should be considered in the audit risk assessment of in- formation system. According to the reduction {a1, a2}, the probabilistic decision rules can be generated and shown in the table 2. Table 2 The probabilistic decision rules based on the reduction {a1, a2} with λ = 0.55. rules support number confidence a1 6< 0.3, 0.6 > and a2 6< 0.2, 0.7 > 100%−−−→ d = 0 2 100% a1 6< 0.2, 0.6 > and a2 6< 0.2, 0.6 > 100%−−−→ d = 0 3 100% a1 >< 0.6, 0.3 > and a2 >< 0.6, 0.4 > 100%−−−→ d = 1 4 100% Step4: Because there are four classifications based on the condition attribute set, and it satisfies X1 < X2 < X3 < X4, that is, {x1, x3, x5} < {x7, x8} < {x2, x6, x9, x10} < x4. For the object x4, due to the fact that there only exist- s the object x4 in the X4, so its comprehensive dominance degree no longer needs to calculate; for the classes {x1, x3, x5} , {x7, x8} and {x2, x6, x9, x10}, their comprehensive dominance degrees should be computed. According to the attribute dependency based on the proposed the model, the attribute depen- dency degree of each attribute can be acquired and then they are standard- ized, thus the weight vector is obtained as follows w = (w1, w2, w3, w4, w5) = (0.2125, 0.2024, 0.1964, 0.1998, 0.1889). For {x1, x3, x5}, based on the formula (18) and (19), IFDDA(x1) = 0.5204, Advances in Systems Science and Application (2015) Vol.15 No.3 215 IFDDA(x3) = 0.1745 and IFDDA(x5) = 0.3035 can be calculated, therefore, x1 > x5 > x3. For {x7, x8}, because the both objects are in the classifica- tion, it is necessary to only calculate the dominance degree with weights, and then it will exist IFDDA(x7) = IFDDA(x7, x8) = 0.8167 and IFDDA(x8) = IFDDA(x8, x7) = 0.1833, therefore, x7 > x8. For {x2, x6, x9, x10}, based on the formula (18) and (19), IFDDA(x2) = 0.1469, IFDDA(x6) = 0.4039, IFDDA(x9) = 0.5743 and IFDDA(x10) = 0.3563 can be calculated, therefore, x2 < x10 < x6 < x9. To sum up, the ranking of the audited objects is x3 < x5 < x1 < x7 < x8 < x2 < x10 < x6 < x9 < x4. Based on the above calculation and analysis, the proposed model possesses cer- tain fault-tolerant ability by adjusting parameter λ, and it can well do with the intuitionistic fuzzy information system with preference information and realize the extraction of group decision rules and the ranking of schemes, so that it can well deal with the real multi-attribute decision making problems with preference information and fuzzy information. 5 Conclusion In order to achieve the law of mining the real decision information system and extract decision rules, with respect to the intuitionistic fuzzy preference informa- tion system, we construct the dominance intuitionistic fuzzy rough set based on dominance distance index, and then exploit it to extract decision-making rules and determine the ranking of decision schemes. 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