Semantic decision making using ontology-based soft sets

Mathematical and Computer Modelling 53 (2011) 1140–1149

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Semantic decision making using ontology-based soft sets Yuncheng Jiang a,b,∗ , Hai Liu a , Yong Tang a , Qimai Chen a a

School of Computer Science, South China Normal University, Guangzhou 510631, PR China

b

State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, PR China

article

info

Article history: Received 18 August 2010 Accepted 24 November 2010 Keywords: Soft sets Ontology-based soft sets Semantic decision making Query Description logics

abstract Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. Description Logics (DLs) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. To extend the expressive power of soft sets, ontology-based soft sets are presented by using the concepts of DLs to act as the parameters of soft sets. In this paper, we investigate soft set based decision making problems more deeply. Concretely, we first point out that the traditional approaches to (fuzzy) soft set based decision making are not fit to solve decision making problems involving user queries through some motivating examples. Furthermore, we present a novel approach to semantic decision making by using ontology-based soft sets and ontology (i.e., DL) reasoning. Lastly, the implementation method of semantic decision making is also discussed. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, there are many complicated problems in economics, engineering, environment, social science, medical science, etc., that involve data which are not always all crisp [1]. Classical methods are not always successful, because the uncertainties appearing in these domains may be of various types. While a wide range of theories such as probability theory, fuzzy set theory, intuitionistic fuzzy set theory, rough set theory, vague set theory, and interval mathematics are well known and often useful mathematical approaches to modeling uncertainty [2,3], each of these theories has its inherent difficulties as pointed out by Molodtsov [4]. The reason for these difficulties is, possibly, the inadequency of the parametrization tool of the theories. Consequently, Molodtsov initiated the soft set theory as a completely new approach for modeling vagueness and uncertainty which is free from the difficulties affecting existing methods [4]. This theory has proven useful in many different fields such as decision making [5,6,3,7–11], data analysis [12], forecasting [13] and simulation [14]. Up to the present, research on soft sets has been very active and many important results have been achieved in the theoretical aspect [3]. The concept and basic properties of soft set theory were presented in [1,4]. Concretely, Maji et al. [1] introduced several algebraic operations in the soft set theory and published a detailed theoretical study on soft sets. However, several assertions presented by Maji et al. [1] were not true in general [15]. Based on the analysis of several operations on soft sets introduced in [1], Ali et al. [15] presented some new algebraic operations for soft sets and proved that certain De Morgan’s laws hold in soft set theory with respect to these new definitions. Maji et al. [16], Majumdar and Samanta [17] extended (standard) soft sets to fuzzy soft sets. Maji et al. [18–20] extended (standard) soft sets to intuitionistic fuzzy soft sets. Yang et al. [21] presented the concept of the interval-valued fuzzy soft sets by combining the interval-valued

∗

Corresponding author at: School of Computer Science, South China Normal University, Guangzhou 510631, PR China. Tel.: +86 2085211352 306. E-mail addresses: [email protected], [email protected] (Y. Jiang).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.080

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fuzzy set and soft set models. Jiang et al. [22] combined the interval-valued intuitionistic fuzzy sets and soft sets, from which a new soft set model, i.e., interval-valued intuitionistic fuzzy soft set theory, was obtained. Aktas and Cagman [2] introduced the basic properties of soft sets, compared soft sets to the related concepts of fuzzy sets and rough sets, pointed out that every fuzzy set and every rough set may be considered a soft set, and gave a definition of soft groups. Aygunoglu and Aygun [23] introduced the concept of fuzzy soft groups. Jun [24] applied soft sets to the theory of BCK/BCI-algebras, and introduced the concept of soft BCK/BCI-algebras. Jun and Park [25] and Jun et al. [26,27] reported the applications of soft sets in ideal theory of BCK/BCI-algebras and d-algebras. Feng et al. [28] defined soft semirings and several related notions to establish a connection between soft sets and semirings. Recently, Xiao et al. [29] proposed the notion of exclusive disjunctive soft sets and studied some of its operations. Qin and Hong [30] deal with the algebraic structure of soft sets. In fact, soft sets, fuzzy sets, and rough sets are distinct but closely related soft computing models. As is shown in [2], both fuzzy sets and rough sets may be considered as soft sets. Thus, one may expect that soft set theory could provide a more general mathematical framework for dealing with uncertain data. Furthermore, Feng et al. [31] investigated the problem of combining soft sets with fuzzy sets and rough sets. In general, three different types of hybrid models are presented, which are called rough soft sets, soft rough sets, and soft rough fuzzy sets, respectively. At the same time, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making [3]. Maji et al. [10] first applied soft sets to solve the decision making problem with the help of a rough approach. Chen et al. [32] presented a new definition of soft set parametrization reduction so as to improve the soft set based decision making in [10]. They also pointed out the basic difference between parametrization reduction of soft sets and attributes reduction in rough sets. Furthermore, Kong et al. [33] introduced the definition of normal parameter reduction in soft sets and fuzzy soft sets. To cope with fuzzy soft set based decision making problems, Roy and Maji [11] presented a novel method of object recognition from an imprecise multiobserver data. The method involves construction of a Comparison Table from a fuzzy soft set in a parametric sense for decision making. Kong et al. [9] argued that the Roy–Maji method [11] was incorrect and they presented a revised algorithm. Recently, Feng et al. [3] gave deeper insights into decision making based on fuzzy soft sets. They discussed the validity of the Roy–Maji method [11] and showed its limitations. By means of level soft sets, Feng et al. presented an adjustable approach to fuzzy soft set based decision making. It is well-known that the parameters in soft sets are very simple. In other words, each parameter is only a word or a sentence, and expressive (or complex) parameters are not considered in soft sets [34]. In order to extend the expressive power of soft sets, Jiang et al. [34] used the concepts of Description Logics (DLs) [35–38] to act as the parameters of soft sets. That is, an extended soft set theory based on DLs was presented in [34]. From a semantic point of view, the parameters of soft sets have no semantics. In [34] Jiang et al. extended soft sets with DLs. The aim is to add semantics for the parameters of soft sets by using DLs to define the parameters. Hence, we can use a terminology (not a word or a sentence) to define the parameters of soft sets. On the other hand, the ontology languages such as OWL Lite, OWL DL and OWL 2 are equivalent to DLs [35,39,40,36], therefore, from the ontology point of view, the extended soft sets presented in [34] can also be called ontology-based soft sets. The purpose of the present paper is to investigate decision making problems using ontology-based soft sets. That is, we try to investigate the soft set based decision making problems more deeply. Concretely, we first point out that the traditional approaches to (fuzzy) soft set based decision making [3,9–11] are not fit to solve decision making problems involving user queries through some motivating examples. Furthermore, we present a novel approach to semantic decision making by using ontology-based soft sets and ontology (i.e., DL) reasoning. Since this new method can be successfully applied to some decision making problems that cannot be solved by using the methods in [3,9–11], therefore, this makes our new proposal more suitable for dealing with the real-life applications. The rest of this paper is organized as follows. The following section provides some motivating examples. In Section 3, we briefly review some background on ontologies, DLs and soft sets. In Section 4, we present an approach to semantic decision making for queries by using ontology-based soft sets and ontology (i.e., DL) reasoning. Section 5 discusses the implementation problem of semantic decision making. Finally, in Section 6, we draw conclusions and present some topics for future research. 2. Motivating examples In order to illustrate semantic decision making for queries using ontology-based soft sets, we provide two motivating examples. First, let us continue to consider the Purchase Houses example presented in [2,15,32,3,34,1,4]. Example 1. Suppose that there is one soft set ⟨F , A⟩ in a Real Estate Company as follows: ⟨F , A⟩ = {expensive houses = {h1 , h2 , h5 , h6 }, beautiful houses = {h1 , h2 , h3 , h6 }, large houses = {h2 , h3 , h4 , h5 }, convenient traffic houses = {h3 , h4 , h5 , h6 }, houses in green surroundings = {h1 , h3 , h4 , h6 }}. Table 1 gives the tabular representation of the soft set ⟨F , A⟩, where e1 , e2 , e3 , e4 , and e5 stand for ‘‘expensive’’, ‘‘beautiful’’, ‘‘large’’, ‘‘convenient traffic’’ and ‘‘in green surroundings’’, respectively. Now we consider that Mr. X is going to buy a house by means of a query. For example, Mr. X can input the following query: beautiful ∧ large ∧ convenient-traffic ∧ in-green-surroundings. Thus, the resultant soft set ⟨G, B⟩ can be obtained. The tabular representation of ⟨G, B⟩ with choice values is shown as in Table 2. From Table 2, it is clear that the maximum choice value is c3 = 4 and so the optimal decision is to select h3 .

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Y. Jiang et al. / Mathematical and Computer Modelling 53 (2011) 1140–1149 Table 1 Tabular representation of the soft set ⟨F , A⟩. U

e1

e2

e3

e4

e5

h1 h2 h3 h4 h5 h6

1 1 0 0 1 1

1 1 1 0 0 1

0 1 1 1 1 0

0 0 1 1 1 1

1 0 1 1 0 1

Table 2 Tabular representation of the resultant soft set ⟨G, B⟩ with choice values. U

e2

e3

e4

e5

Choice values

h1 h2 h3 h4 h5 h6

1 1 1 0 0 1

0 1 1 1 1 0

0 0 1 1 1 1

1 0 1 1 0 1

c1 c2 c3 c4 c5 c6

=2 =2 =4 =3 =2 =3

Table 3 Tabular representation of the resultant soft set ⟨G, B⟩ with choice values. U

e1

e2

e3

e4

Choice values

h1 h2 h3 h4 h5 h6

1 1 0 0 1 1

1 1 1 0 0 1

0 1 1 1 1 0

0 0 1 1 1 1

c1 c2 c3 c4 c5 c6

=2 =3 =3 =2 =3 =3

However, if Mr. X inputs the following query: costly ∧ pretty ∧ big ∧ conv enient-communications, it is easy to know that the resultant soft set ⟨G, B⟩ is a null soft set. Obviously, the resultant soft set ⟨G, B⟩ (null soft set) is not correct from a semantics point of view since there exist some semantic relations between the parameters of ⟨F , A⟩ and the parameters of the query. For example, expensive ≡ costly, beautiful ≡ pretty, large ≡ big, and convenient traffic ≡ convenient communications. Thus, the correctly resultant soft set ⟨G, B⟩ should be as follows: ⟨G, B⟩ = {costly houses = {h1 , h2 , h5 , h6 }, pretty houses = {h1 , h2 , h3 , h6 }, big houses = {h2 , h3 , h4 , h5 }, convenient communications houses = {h3 , h4 , h5 , h6 }}. The tabular representation of ⟨G, B⟩ with choice values is shown as in Table 3, where e1 , e2 , e3 , and e4 stand for ‘‘costly’’, ‘‘pretty’’, ‘‘big’’ and ‘‘convenient communications’’, respectively. From Table 3, it is clear that the maximum choice value is c2 = c3 = c5 = c6 = 3, hence any one of hi (i = 2, 3, 5, 6) could be selected as the optimal alternative. Clearly, in the query ‘‘costly ∧ pretty ∧ big ∧ convenient-communications’’, for each parameter in this query, there exists an equivalent parameter in parameters of ⟨F , A⟩. In this situation, we can obtain a precise soft set for the query. Further, if Mr. X inputs the following query: high-price ∧ good-color ∧ large-area ∧ sev eral-roads ∧ multi-hills, it is also easy to know that the resultant soft set ⟨G, B⟩ is a null soft set. Obviously, the resultant soft set ⟨G, B⟩ (null soft set) is also not correct from a semantics point of view. However, not like the query ‘‘costly ∧ pretty ∧ big ∧ convenientcommunications’’, there does not exist an equivalent parameter in parameters of ⟨F , A⟩ for each parameter in this query ‘‘high-price ∧ good-color ∧ multi-rooms ∧ several-roads ∧ in-good-surroundings’’. That is to say, we cannot obtain a precise soft set for this query. Nevertheless, we can obtain an approximate soft set by using the following subsumption relations between the parameters of ⟨F , A⟩ and the parameters of the query: high price ⊑ expensive, good color ⊑ beautiful, multi rooms ⊑ large, several roads ⊑ convenient traffic, and in green surroundings ⊑ in good surroundings. Thus, we can obtain an approximately resultant soft set ⟨G, B⟩ for the query as follows: ⟨G, B⟩ = {high price houses = {h1 , h2 , h5 , h6 }, good color houses = {h1 , h2 , h3 , h6 }, multi rooms houses = {h2 , h3 , h4 , h5 }, several roads houses = {h3 , h4 , h5 , h6 }, in good surroundings houses = {h1 , h3 , h4 , h6 }}.

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Table 4 Tabular representation of the approximately resultant soft set ⟨G, B⟩ with choice values. U

ε1

ε2

ε3

ε4

ε5

Choice value (ci )

h1 h2 h3 h4 h5 h6

1 1 0 0 1 1

1 1 1 0 0 1

0 1 1 1 1 0

0 0 1 1 1 1

1 0 1 1 0 1

c1 c2 c3 c4 c5 c6

=3 =3 =4 =3 =3 =4

Table 5 Information system IS. U

a1

a2

a3

a4

p1 p2 p3 p4 p5 p6

DLs Data mining Ontology OWL LP ASP

Soft sets Databases Rule Ontology ASP Prolog

Ontology Machine learning DLs KR First Order Logic LP

Decision making Rule OWL DLs Prolog KR

Table 6 Tabular representation of the approximately resultant soft set ⟨F , A⟩ with choice values. U

a1

a2

a3

a4

Choice value (ci )

p1 p2 p3 p4 p5 p6

1 0 1 1 1 1

0 0 1 1 1 0

1 0 1 1 1 1

0 1 1 1 0 1

c1 c2 c3 c4 c5 c6

=2 =1 =4 =4 =3 =3

The tabular representation of ⟨G, B⟩ with choice values is shown as in Table 4, where ε1 , ε2 , ε3 , ε4 , and ε5 stand for ‘‘high price’’, ‘‘good color’’, ‘‘multi rooms’’, ‘‘several roads’’ and ‘‘in good surroundings’’, respectively. From Table 4, it is clear that the maximum choice value is c3 = c6 = 4, hence either h3 or h6 could be selected as the optimal alternative. The next example illustrates the decision making in information systems. Example 2. Suppose that there six papers U = {p1 , p2 , p3 , p4 , p5 , p6 } in an information system IS and the set of attributes is given by A = {a1 , a2 , a3 , a4 }, where ai (1 ≤ i ≤ 4) stand for ‘‘keyword1’’, ‘‘keyword2’’, ‘‘keyword3’’, and ‘‘keyword4’’, respectively. The information system IS is shown as in Table 5. Assume that Mr. X wants to obtain a closely related research paper in the area of ‘‘Semantic Web’’ using the following query: ‘‘keyword1’’ = Semantic Web, ‘‘keyword2’’ = Description Logics, ‘‘keyword3’’ = Web Ontology Language, and ‘‘keyword4’’ = Reasoning rule. It is easy to know that the information system IS will be translated into a null soft set for the query. In fact, there exist some semantic relations between the information system IS and the query. For instance, we have the following semantic relations: DLs ⊑ semantic web, Ontology ⊑ semantic web, Rule ⊑ semantic web, OWL ⊑ semantic web, KR ⊑ semantic web, LP ⊑ semantic web, ASP ⊑ semantic web, First Order Logic ⊑ semantic web, DLs ≡ Description Logics, OWL ≡ Web Ontology Language, and Reasoning rule ⊑ Rule. Thus, we can obtain an approximately resultant soft set ⟨F , A⟩ for the query. The tabular representation of ⟨F , A⟩ with choice values is shown as in Table 6. From Table 6, it is clear that the maximum choice value is c3 = c4 = 4, hence either p3 or p4 could be selected as the optimal alternative. To cope with the situations mentioned in Examples 1 and 2, we have to provide a semantic extension of the proposal presented in [3,9–11]. The purpose of the present paper is to provide a novel approach to semantic decision making by using ontology-based soft sets and ontology (i.e., DL) reasoning.

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3. Preliminaries For completeness of presentation and convenience of subsequent discussions, in the current section we will briefly recall some basic notions of ontologies, DLs and soft sets. See especially [2,15,35,34,1,4] for further details and properties. 3.1. Ontologies and description logics Ontologies, defined as ‘‘formal, explicit specifications of a shared conceptualization’’ [41], encode machine-interpretable descriptions of the concepts and the relations in a domain using abstractions as class, role or instance, which are qualified using logical axioms. Nowadays, properties and semantics of ontology constructs mainly are determined by Description Logics (DLs) [35], a family of logics for representing structured knowledge which have proved to be very useful as ontology languages [42,38]. Formally, an ontology is a triple O = ⟨RB , T B , AB ⟩, where RB (the Role Box or RBox) and T B (the Terminological Box or TBox) comprise the intensional knowledge, i.e., general knowledge about the world to be described (statements about roles and concepts, respectively), and AB (the Assertional Box or ABox) the extensional knowledge, i.e., particular knowledge about a specific instantiation of this world (statements about individuals in terms of concepts and roles) [42]. In the following, we introduce the DL ALC [43], which is a significant representative of DLs. It should be noted that the ontology-based soft sets (i.e., extended soft sets with DLs) [34] is not restricted to ALC . It applies to arbitrary (decidable) DLs, provided that the DL allows for negation. We assume three alphabets of symbols, called atomic concepts (denoted by A), atomic roles (denoted by R) and individuals (denoted by a and b). A concept (denoted by C and D) of the language ALC is built out of atomic concepts according to the following syntax rules: C , D → ⊤| (top concept) ⊥| (bottom concept) A| (atomic concept) ¬C | (concept negation) C ⊓ D| (concept conjunction) C ⊔ D| (concept disjunction) ∃R · C | (existential quantification) ∀R · C | (universal quantification). From a semantic point of view, concepts are interpreted as subsets of an abstract domain, while roles are interpreted as binary relations over such a domain. More precisely, an interpretation I = (∆I , •I ) consists of a domain of interpretation ∆I , and an interpretation function •I mapping every atomic concept A to a subset of ∆I and every atomic role R to a subset of ∆I × ∆I . The interpretation function •I is extended to complex concepts of ALC (note that in ALC roles are always atomic) as follows:

• • • • • • • • •

AI ⊆ ∆I ; RI ⊆ ∆I × ∆I ; ⊤I = ∆I ; ⊥I = φ ; (¬C )I = ∆I \ C I ; (C ⊓ D)I = C I ∩ DI ; (C ⊔ D)I = C I ∪ DI ; (∃R · C )I = {x ∈ ∆I |∃y ∈ ∆I , ⟨x, y⟩ ∈ RI ∧ y ∈ C I }; (∀R · C )I = {x ∈ ∆I |∀y ∈ ∆I , ⟨x, y⟩ ∈ RI → y ∈ C I }.

An ALC TBox T B consists of a finite set of general concept inclusion (GCI) axioms of the form C ⊑ D, which means that concept C is more specific than D, i.e., D subsumes C . A concept definition C ≡ D (C and D are equivalent) is an abbreviation of the pair of axioms C ⊑ D and D ⊑ C . In T B the interpretations of concepts can be restricted to the models of T B . Based on this model–theoretic semantics, concepts can be checked for unsatisfiability: whether they are necessarily interpreted as the empty set. Another useful semantic implication is subsumption of two concepts C and D (a subset relation C I and DI w.r.t. all models I of T B ) denoted by T B |H C ⊑ D. An ALC ABox AB consists of a finite set of assertions about individuals. An assertion is either a concept assertion C (a) (meaning that a is an instance of C ) or a role assertion R(a, b) (meaning (a, b) is an instance of R). The semantics is a straightforward extension of the previous definition: an interpretation I is a model for an assertion C (a) and R(a, b) if and only if aI ∈ C I and (aI , bI ) ∈ RI . In ALC there is no RBox RB , since no axioms involving roles are allowed. In more expressive DLs such as SROIQ [39,40], RB consists of a finite set of role axioms stating restrictions as subsumption, transitivity, cardinality, etc. [42]. A DL ontology not only stores axioms and assertions, but also offers some reasoning services, such as KB satisfiability (or consistency), concept satisfiability, subsumption or instance checking. Regarding other expressive DLs, the interested reader is referred to the handbook [35].

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3.2. Soft sets We assume that U refers to an initial universe, E is a set of parameters, P (U ) is the power set of U, and A ⊆ E. Formally, Molodtsov [4] defined the soft set in the following way: A pair ⟨F , A⟩ is called a soft set over U, where F is a mapping given by F : A → P (U ). Usually, parameters are attributes, characteristics, or properties of objects in U [15]. Concretely, the set of parameters E consists of a word or sentence. In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε ∈ E , F (ε) may be considered as the set of ε -elements of the soft set ⟨F , A⟩, or as the set of ε -approximate elements of the soft set [2,4]. Clearly, a soft set is not a set. For two soft sets ⟨F , A⟩ and ⟨G, B⟩ over U , ⟨F , A⟩ is called a soft subset of ⟨G, B⟩ [1] if (1) A ⊆ B and (2) ∀ ε ∈ A, F (ε) and G(ε) are identical approximations. This relationship is denoted by ⟨F , A⟩ b ⟨G, B⟩. Similarly, ⟨F , A⟩ is called a soft superset of ⟨G, B⟩ if ⟨G, B⟩ is a soft subset of ⟨F , A⟩. This relationship is denoted by ⟨F , A⟩ c ⟨G, B⟩. Two soft sets ⟨F , A⟩ and ⟨G, B⟩ over U are called soft equal if ⟨F , A⟩ is a soft subset of ⟨G, B⟩ and ⟨G, B⟩ is a soft subset of ⟨F , A⟩ [1]. In the following, we briefly introduce the notion of ontology-based soft sets. We assume that DL is an arbitrary (decidable) description logic such as ALC [43] or SROIQ [39,40], provided that the DL allows for negation. Let DL be an arbitrary DL, Σ be a set of DL-concepts, I = (∆I , •I ) be a model of the DL-knowledge base (i.e., DLontology). Let P (U ) denote the power of U , U ⊆ ∆I , and M ⊂ Σ . A pair ⟨I , M ⟩ is called an ontology-based soft set over ∆I . Clearly, I |M is a mapping given by I |M : M → P (∆I ), i.e., ∀ C ∈ M , C I ⊆ ∆I . In other words, an ontology-based soft set over ∆I is a parameterized family of subsets of the domain of interpretation ∆I . Every set C I , C ∈ Σ , from this family may be considered as the set of C -elements (or C -individuals) of the ontology-based soft set ⟨I , M ⟩, or as the set of C -approximate elements (or C -approximate individuals) of the ontology-based soft set ⟨I , M ⟩. Regarding the mathematical properties of (ontology-based) soft set theoretic operations, the interested reader is referred to [15,34,1] for more details. 4. Semantic decision making for queries In this section, we present an approach to semantic decision making for queries by using ontology-based soft sets and ontology (i.e., DL) reasoning. We first define some ontologies for semantic decision making. We then provide an approach to semantic decision making. 4.1. Ontologies for semantic decision making In order to achieve semantic decision making, we need two kinds of ontologies: soft set ontology and domain ontology. In the rest of this paper, we assume that DL is an arbitrary (decidable) description logic such as ALC [43] or SROIQ [39,40], provided that the DL allows for negation. Soft Set Ontology (SSO): Informally, to obtain an ontology-based soft set, we have to construct an ontology firstly. This ontology is called an SSO. In other words, given an SSO, we can get an ontology-based soft set. Formally, given an SSO (or DL-knowledge base) SSO = ⟨RB S , T B S , AB S ⟩, where RB S , T B S , and AB S are the RBox, TBox, and ABox of SSO respectively, we can obtain an interpretation I for SSO according to the semantics of DLs [35]. If I |H RB S , I |H T B S and I |H AB S , then we have that I is a model of SSO . Assume that the set of decision parameters M = {C1 , C2 , . . . , Cn }. Hence, for each parameter Ci ∈ M , i ∈ {1, 2, . . . , n}, we have CiI ⊆ ∆I . Thus, we obtain an ontologybased soft set as follows:

⟨I , M ⟩ = {(C1 , C1I ), (C2 , C2I ), . . . , (Cn , CnI )}. Example 3 (Example 1 of [34] Cont’d). Let us consider an ontology-based soft set ⟨I , M ⟩ which describes the ‘‘attractiveness of houses’’ that Mr. X is considering for purchase. Suppose that there six houses in the domain ∆I = {h1 , h2 , h3 , h4 , h5 , h6 } under consideration, and that Σ = {C1 , C2 , C3 , C4 , C5 } is a set of decision parameters (i.e., DL-concepts). The Ci (i = 1, 2, 3, 4, 5) stand for the parameters (i.e., DL-concepts) ‘‘expensive’’, ‘‘beautiful’’, ‘‘large’’, ‘‘convenient traffic’’, and ‘‘in green surroundings’’, respectively. Let SSO = ⟨RB S , T B S , AB S ⟩ be a soft set ontology, where RB S = φ, T B S and AB S are defined in Example 1 of [34]. Thus, we can obtain an ontology-based soft set ⟨I , M ⟩. The tabular representation of ⟨I , M ⟩ is shown as in Table 7.

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Y. Jiang et al. / Mathematical and Computer Modelling 53 (2011) 1140–1149 Table 7 Tabular representation of the ontologybased soft set ⟨I , M ⟩. U

C1

C2

C3

C4

C5

h1 h2 h3 h4 h5 h6

1 0 1 0 0 0

1 1 1 0 0 1

1 1 1 1 0 1

1 1 1 1 0 1

1 1 1 0 1 0

Domain Ontology (DO): Informally, domain ontology is used to express semantic relations between the parameters of soft set and the parameters of query. Since we cannot know in advance the parameters of the query, so we have to build a comprehensive domain ontology in order to realize the semantic decision making. Formally, a domain ontology (or DL-knowledge base) DO = ⟨RB D , T B D ⟩, where RB D and T B D are the RBox and TBox of DO respectively. It should be noted that the ABox of DO is empty, that is, AB D = φ . This is because there are no individuals in the query, in other words, we do not need to consider individuals in DO . Let us consider the domain ontology of the Purchase Houses example. Example 4. The domain ontology of Purchase Houses example DO = ⟨RB D , T B D ⟩ is defined as follows:

RB D = φ. T B D = {expensiv e ≡ costly, expensiv e ≡ big-ticket , expensiv e ≡ ¬cheap, inexpensiv e ≡ cheap, beautiful ≡ pretty, beautiful ≡ ¬ugly, beautiful ≡ brilliant , large ≡ big , large ≡ great , large ≡ ¬small, large ≡ bulky, large ≡ ¬diminutiv e, conv enient traffic ≡ conv enient communications, high price ⊑ expensiv e, low price ⊑ cheap, good color ⊑ beautiful, bad color ⊑ ugly, multi rooms ⊑ large, one room ⊑ small, sev eral roads ⊑ conv enient traffic , sev eral buses ⊑ conv enient traffic , in green surroundings ⊑ in good surroundings}.

There is one remark here. In Example 4, we only give part of the domain ontology of Purchase Houses example. Regarding construction of a comprehensive domain ontology, we have to consult experts in this field and use ontology editors such as protégé [48]. 4.2. Semantic decision making Like most decision making problems, (ontology-based) soft sets based semantic decision making involves the evaluation of all the objects which are decision alternatives through evaluating decision parameters. However, in practical applications, users often do not know the exact decision parameters of the soft set. Then, users tend to make decisions through queries. In what follows, we present an approach to semantic decision making for queries by using DL reasoners such as Pellet [44] and HermiT [45]. Informally, a query is a set of query parameters, where each query parameter is a concept of DL-ontology (i.e., DLconcept). Formally, a query Q = {Q1 , Q2 , . . . , Qk }, where for each query parameter Qi ∈ Q , i ∈ {1, 2, . . . , k}, Qi is a DLconcept. Now we give the semantic decision making algorithm for a query. Algorithm 1. Semantic decision making algorithm for a query. Input: An ontology-based soft set ⟨I , M ⟩ = {(C1 , C1I ), (C2 , C2I ), . . . , (Cn , CnI )}, a soft set ontology SSO , a domain ontology DO , and a query Q = {Q1 , Q2 , . . . , Qk } (k ≤ n). Output: The optimal alternative. Step: 1. Obtain an ontology-based soft set ⟨J , N ⟩ for the query Q in the following way: for each query parameter Qi ∈ Q (1 ≤ i ≤ k). (1) If there exists a decision parameter Cj of ⟨I , M ⟩ (1 ≤ j ≤ n) such that {SSO , DO } |H Qi ≡ Cj , then we obtain an element (Qi , CjI ) of ⟨J , N ⟩. Goto (4). (2) If there exists a decision parameter Cj of ⟨I , M ⟩ (1 ≤ j ≤ n) such that {SSO , DO } |H Qi ⊒ Cj , then we obtain an element (Qi , CjI ) of ⟨J , N ⟩. Goto (4). (3) If there exists a decision parameter Cj of ⟨I , M ⟩ (1 ≤ j ≤ n) such that {SSO , DO } |H Qi ⊑ Cj , then we obtain an element (Qi , CjI ) of ⟨J , N ⟩. Goto (4). (4) If any query parameter Qi ∈ Q (1 ≤ i ≤ k) has Qi -elements (or Qi -individuals), then Goto (5); else, that is, there exists one query parameter Qi ∈ Q (1 ≤ i ≤ k) such that Qi has not Qi -elements (or Qi -individuals), then output ‘‘Cannot make decisions and please input different decision making parameters’’ and Goto 5.

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Table 8 Tabular representation of the approximate soft set ⟨J , N ⟩ with choice values. U

C1

C2

C3

C4

Choice values

h1 h2 h3 h4 h5 h6

0 0 1 1 0 0

1 1 1 0 0 1

0 1 1 1 1 0

1 0 1 1 0 1

c1 c2 c3 c4 c5 c6

=2 =2 =4 =3 =1 =2

(5) Obtain the ontology-based soft set ⟨J , N ⟩ = {(Q1 , C1I ), (Q2 , C2I ), . . . , (Qk , CkI )}. 2. 3. 4. 5.

Present the ontology-based soft set ⟨J , N ⟩ in tabular form and compute the choice value ci of oi , ∀i. The optimal decision is to select ol if cl = maxi ci and output ol . If l has more than one value then any one of ol may be chosen. Algorithm end.

There are three remarks here. Firstly, it should be noted that in the steps (1), (2) and (3) of the above algorithm, we should obtain a unique concept Cj , since the decision parameters of ⟨I , M ⟩ should be different from a semantics point of view. That is, for any decision parameters Ci and Cj of ⟨I , M ⟩, we have that {SSO , DO } |H Ci ̸= Cj . Secondly, in the steps (1), (2) and (3) of the above algorithm, to decide whether {SSO , DO } |H Qi ≡ Cj , {SSO , DO } |H Qi ⊒ Cj , or {SSO , DO } |H Qi ⊑ Cj , we must rely on DL reasoners such as Pellet [44] and HermiT [45]. Thirdly, it is easy to extend the above algorithm to weighted (ontology-based) soft set based decision making. Concretely, in step 2 of the above algorithm, we need to compute the weighted choice value. Now we will illustrate the whole algorithm with an example. Example 5. Let SSO = ⟨RB S , T B S ⟩ be a soft set ontology which is defined in Example 3, ⟨I , M ⟩ be an ontology-based soft set defined in Example 1 (i.e., Table 1), and DO = ⟨RB D , T B D ⟩ be a domain ontology of Purchase Houses example defined in Example 4. Assume that Mr. X inputs the following query: costly ∧ pretty ∧ big ∧ conv enient-communications. Since {SSO , DO } |H expensive ≡ costly, {SSO , DO } |H beautiful ≡ pretty, {SSO , DO } |H large ≡ big, and {SSO , DO } |H convenient traffic ≡ convenient communications, then we obtain the following soft set: ⟨J , N ⟩ = {costly houses = {h1 , h2 , h5 , h6 }, pretty houses = {h1 , h2 , h3 , h6 }, big houses = {h2 , h3 , h4 , h5 }, convenient communications houses = {h3 , h4 , h5 , h6 }}. The tabular representation of ⟨J , N ⟩ with choice values is shown as in Table 3. Now we assume that Mr. X inputs the following query: cheap ∧ pretty ∧ multirooms ∧ in good surroundings. Since {SSO , DO } |H cheap ≡ ¬ expensive, then we have cheapI = (¬expensive)I = ∆I − expensiveI = {h3 , h4 }. Since {SSO , DO } |H beautiful ≡ pretty, then we have prettyI = {h1 , h2 , h3 , h6 }. Since {SSO , DO } |H multi rooms ⊑ large, then we have multi roomsI = {h2 , h3 , h4 , h5 }. Since {SSO , DO } |H in green surroundings ⊑ in good surroundings, then we have in good surroundingsI = {h1 , h3 , h4 , h6 }. Thus, we obtain the following approximate soft set: ⟨J , N ⟩ = {cheap houses = {h3 , h4 }, pretty houses = {h1 , h2 , h3 , h6 }, multi rooms houses = {h2 , h3 , h4 , h5 }, in good surroundings houses = {h1 , h3 , h4 , h6 }}. The tabular representation of ⟨J , N ⟩ with choice values is shown as in Table 8, where C1 , C2 , C3 , and C4 stand for ‘‘cheap’’, ‘‘pretty’’, ‘‘multi rooms’’ and ‘‘in good surroundings’’, respectively. From Table 8, it is clear that the maximum choice value is c3 = 4, hence h3 could be selected as the optimal alternative. 5. Implementation In this section we will discuss the implementation problem of semantic decision making. From the algorithm of semantic decision making (see Algorithm 1 in Section 4.2), we know that the implementation of semantic decision making needs to use reasoning of DLs. Concretely, we need to obtain the initial soft set ⟨I , M ⟩ from the soft set ontology SSO by using DL reasoners such as Pellet [44] and HermiT [45]. And then we need to obtain the soft set ⟨J , N ⟩ from the soft set ⟨I , M ⟩, the domain ontology DO , and the query Q by using DL reasoners. Fig. 1 illustrates the architecture of the implementation of semantic decision making. Computational complexity of the semantic decision making using ontology-based soft sets and DL reasoning is conditioned by complexity of reasoning in soft set ontology and domain ontology, since computing the choice value in the

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Fig. 1. Architecture of the implementation of semantic decision making.

ontology-based soft set ⟨J , N ⟩ can be carried out in polynomial time. In fact, computational complexity of semantic decision making is determined by the more expressive ontology between soft set ontology and domain ontology. For example, if the soft set ontology adopts SROIQ(D)-ontology and the domain ontology adopts SH OIQ-ontology, then computational complexity of semantic decision making is determined by SROIQ(D)-ontology reasoning. Regarding the reasoning complexity of ontologies (DL knowledge bases), the interested reader is referred to [35] for more details. 6. Conclusion In this paper, we investigate soft set based decision making problems more deeply. Concretely, we first point out that the traditional approaches to (fuzzy) soft set based decision making are not fit to solve decision making problems involving user queries through some motivating examples. And then we present a novel approach to semantic decision making by using ontology-based soft sets and DL reasoning. Furthermore, the implementation method of semantic decision making is also discussed. Current research effort is to implement the semantic decision making algorithm and to perform an empirical evaluation in real scenarios. To extend this work, one may possibly use ontology based soft sets to address group decision making problems [46,47]. An interesting topic of future research is to investigate semantic decision making using ontology-based (intuitionistic) fuzzy soft sets and (intuitionistic) fuzzy DL reasoning. It is also desirable to further apply ontologies (or DLs) to other practical applications based on (fuzzy) soft sets. Acknowledgements The works described in this paper are supported by the National Natural Science Foundation of China under Grant Nos. 60663001, 60673135, 60970044 and 60736020; the Natural Science Foundation of Guangdong Province of China under Grant No. 10151063101000031; the Foundation of the State Key Laboratory of Computer Science of Chinese Academy of Sciences under Grant No. SYSKF0904; the Natural Science Foundation of Guangxi Province of China under Grant Nos. 0991100; and the Science Research and Technology Development Project of Guangdong Province of China under Grant Nos. 2009B090300326 and 2009B010800036. References [1] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Computers & Mathematics with Applications 45 (4–5) (2003) 555–562. [2] H. Aktas, N. Cagman, Soft sets and soft groups, Information Sciences 177 (13) (2007) 2726–2735. [3] F. Feng, Y.B. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234 (1) (2010) 10–20. [4] D. Molodtsov, Soft set theory—first results, Computers & Mathematics with Applications 37 (4–5) (1999) 19–31. [5] N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Computers & Mathematics with Applications 59 (10) (2010) 3308–3314. [6] N. Cagman, S. Enginoglu, Soft set theory and uni-int decision making, European Journal of Operational Research 207 (2) (2010) 848–855. [7] F. Feng, Y. Li, V. Leoreanu-Fotea, Application of level soft sets in decision making based on interval-valued fuzzy soft sets, Computers & Mathematics with Applications 60 (6) (2010) 1756–1767. [8] Y. Jiang, Y. Tang, Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling 35 (2) (2011) 824–836. [9] Z. Kong, L. Gao, L. Wang, Comment on a fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 223 (2) (2009) 540–542. [10] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers & Mathematics with Applications 44 (8–9) (2002) 1077–1083. [11] A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2) (2007) 412–418. [12] Y. Zou, Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems 21 (8) (2008) 941–945. [13] Z. Xiao, K. Gong, Y. Zou, A combined forecasting approach based on fuzzy soft sets, Journal of Computational and Applied Mathematics 228 (1) (2009) 326–333.

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