Decision Making with Uncertainty Information Based on Lattice-Valued Fuzzy Concept Lattice

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1 Journal of Universal Computer Science vol. 16 no. 1 ( submitted: 1//09 accepted: 15/10/09 appeared: 1/1/10 J.UCS Decision aking with Uncertainty nformation Based on Lattice-Valued Fuzzy Concept Lattice Li Yang (ntelligent Control Development Center Southwest Jiaotong University Chengdu Sichuan P.R. China yangli667@sina.com Yang Xu (College of athematics Southwest Jiaotong University Chengdu Sichuan P.R. China xuyang@home.swjtu.edu.cn Abstract: For the processing of decision making with uncertainty information this paper establishes a decision model based on lattice-valued logic and researches the algorithm for extracting the maximum decision rules. Firstly we further research the lattice-valued fuzzy concept lattice by combining the lattice implication algebra and classical concept lattice; secondly we define the lattice-valued decision context as the equivalent form of decision information system and establish the single-target decision model and talk about some properties of the decision rules; finally we give the calculating methods of decision rules with different decision values and the algorithm for extracting the maximum decision rules. Keywords: Decision making Uncertainty Concept lattice Lattice-valued logic Decision rule Categories:..3 1 ntroduction n many cases decision making processes generally consist of some uncertainty problems [raj 08; Xu 08]. The increasing complexity of environment makes it too difficult or too ill-defined to be amenable for description in conventional quantitative expressions. t is more suitable to provide their decision information by means of various expressions rather than numerical ones such as linguistic values less better [Delgado 0; Herrera 00 05; Yang 08a] or symbolic values ++ + [Xu 07]. n order to possess a reasonable structure for these different expressions it has been the research hotspot to construct a decision model with a kind of information expression at present. Up to now several proposed decision models mainly include: (1 the decision model based on rough set [Zhang 05] which acquires the rules through decision matrixes and calculates the minimal decision rules by reduction theory; ( the decision model based on granular computing [Hobbs 85; Zadeh 97; Yao 03 04; Cheng 07] which decomposes complex information into many simpler sorts according to the characteristic and capability of decision information; (3 the decision model based on classical concept lattice [Liang 04; Zhang 06; Wang 07] which establishes the decision formal context according to the decision information and

2 160 Yang L. Xu Y.: Decision aking with Uncertainty nformation... acquires the rules by constructing the decision concept lattice which consists of the antecedent concept lattice and consequent concept lattice. But these models constructed in the certainty environment are inappropriate to deal with uncertainty information. Especially for the model (3 the necessary condition for utilizing it to extract the decision rules is to establish a surjection between the antecedent concept lattice and consequent concept lattice and the model (3 will not function properly if the decision formal context is not consistent i.e. there doesn t exist a surjection. However in most cases not all the decision concept lattices can find the relevant surjection and the model (3 can t deal with the uncertainty information. oreover for the decision making with a kind of uncertainty information --- the fuzzy information the general fuzzy concept lattice based on the fuzzy logic has no researches on it to this day. Classical concept lattice as a conceptual clustering method was proposed by Wille in 198 [Wille 8; anter 99] where an object-attribute view of data is developed. Due to the fact that the concept lattice can provides a theoretical framework for the design and discovery concept hierarchies from relational information system this paper still utilizes a kind of concept lattice to deal with the uncertainty problems in decision making. This kind of concept lattice --- the lattice-valued fuzzy concept lattice researched in this paper is established based on the combination of lattice implication algebra and classical concept lattice [Yang 08]. The introduction of the lattice-valued logic based on the lattice implication algebra into classical concept lattice is the important characteristic of the lattice-valued fuzzy concept lattice different from the general fuzzy concept lattice. For the non-numerical information the processing ability of the lattice-valued fuzzy concept lattice is better than that of other concept lattices. Especially for decision making with uncertainty information the lattice-valued fuzzy concept lattice can be looked as a useful mathematical tool. About the lattice-valued logic introduced in this paper Xu established it based on the lattice implications algebra [Xu 93; 99] in 1993 which is an alternative approach to treat fuzziness and incomparability and researched the corresponding reasoning theories and methods [Xu 94; 00; 01]. The lattice-valued fuzzy concept lattice is used as the rules base which is called the lattice-valued decision concept lattice and its characteristics different from the general rules bases are mainly as follows: (1 all the decision rules in the same lattice-valued decision concept lattice have the same decision value which is the essential difference from general rules bases. Such a rules base not only is beneficial to collect information and establish rules but also makes the extraction of potential rules simpler; ( the antecedent of decision rules in the lattice-valued fuzzy concept lattice consists of two types of sets which can depict the question more concretely; (3 the new decision rules can have their respective decision values according to the calculation of truth values by which the rules base can be enlarged into the one with different decision values. n particularly the single-target decision model established based on the lattice-valued fuzzy concept lattice is entirely different from the one based on classical concept lattice: on the one hand we need only to construct one lattice-valued fuzzy concept lattice and don t need to construct antecedent concept lattice and consequent concept lattice on the structure form which simplifies the decision model

3 Yang L. Xu Y.: Decision aking with Uncertainty nformation and dispenses with establishing the surjection on it; on the other hand we introduce the calculating method of decision values which provides more reasonable methods in decision making. Based on these analyses this paper puts forward the method of decision making with uncertainty information based on the lattice-valued fuzzy concept lattice. Section gives an overview of some basic theories of the concept lattice and the lattice implication algebra. n Section 3 we briefly research the related works of the lattice-valued fuzzy concept lattice. Successively we construct the single-target decision model in Section 4 where we firstly establish the lattice-valued single-target decision concept lattice and give the definition of decision rule; secondly we present the calculating method of decision rules with different decision values and talk about the decision rules properties; finally we discuss the algorithm of extracting the maximum decision rules from the lattice-valued decision context and establish the decision rules base. Concluding remarks and future researches are presented in Section 5. Preliminaries n this section we review briefly the classical concept lattice and the lattice implication algebra and they are the foundations of constructing the lattice-valued fuzzy concept lattice. Definition.1 [Birkhoff 67] A partial ordered set is a set in which a binary relation is defined which satisfies the following conditions: for any xyz (1 x x for any x (Reflexive; ( x y and y x implies x = y (Antisymmetry; (3 x y and y z implies x z (Transitivity. Definition. [Birkhoff 67] Let L be an arbitrary set and let there be given two binary operations on L denoted by and. Then the structure ( L is an algebraic structure with two binary operations. We call the structure ( L a lattice provided that it satisfies the following properties: (1 For any xyz L x ( y z = ( x y z and x ( y z = ( x y z ; ( For any xy L x y = y x and x y = y x; (3 For any x L x x = x and x x = x; (4 For any xy L x ( x y = x and x ( x y = x. Definition.3 [Xu 03] f a lattice has a smallest element denoted by O and a greatest element denoted by then it is called a bounded lattice. Definition.4 [Xu 93; 03] Let ( L O be a bounded lattice with an order-reversing involution ' and O the greatest and the smallest element of L respectively and : L L L a mapping. ( L O is called a lattice implication algebra if the following conditions hold for any xyz L: (1 x ( y z = y ( x z ; ( x x = ;

4 16 Yang L. Xu Y.: Decision aking with Uncertainty nformation... (3 x y = y x ; (4 x y = y x = implies x = y; (5 ( x y y = ( y x x ; (6 ( x y z = ( x z ( y z ; (7 ( x y z = ( x z ( y z. Definition.5 [Xu 03] A Wajsberg algebra is an algebra ( L 1 of type (10 such that (1 1 x = x; ( ( x y (( y z ( x z = 1; (3 ( x y y = ( y x x ; (4 ( x y ( y x = 1. There is a one-to-one corresponding between Wajsberg algebras and V-algebras [Höhle 95; 95a]. Theorem.1 Let ( L O be a lattice implication algebra define ' = = 1 then ( L 1 is a Wajsberg algebra. Proof. ( x x = ( x x = x ( x = x = x x = 1 x; ( x y (( y z ( x z = ( x y ( x (( y z z = ( x y ( x (( z y y = ( x y (( z y ( x y = ( z y (( x y ( x y = ( z y = 1 ; x y = y x ( y x ( x y = ( x y ( y x = ( x y ( y x = 1. Example.1 [Xu 03] (Boolean Algebra Let ( L ' be a Boolean lattice for any xy L define x y = x y then ( L ' 01 is a lattice implication algebra. Example. [Xu 03] (Lukasiewicz implication algebra on [01] f the operations on [01] are defined respectively as follows: x y = max x y { } x y = min { x y} x = 1 x

5 Yang L. Xu Y.: Decision aking with Uncertainty nformation then ([01] 01 { } x y = min 11 x+ y is a lattice implication algebra. Definition.6 [anter 99] A formal context in classical concept lattice is defined as a set structure ( consisting of the finite sets and and a binary relation. The elements of and are called objects and attributes respectively and the relationship gm is read: the object g has the attribute m. For a set of objects A and a set of attributes B A* is defined as the set of features shared by all the objects in A B* is defined as the set of objects that posses all the features in B that is A = { m gm g A} B { g gm m B} =. Definition.7 [anter 99] A formal concept of the context ( is defined as a pair ( A B with A B and A* = B B* = A. The set A is called the extent and B the intent of the concept ( A B. 3 Lattice-Valued Fuzzy Concept Lattice n order to provide a mathematical tool for incomparability fuzzy information processing we further research the lattice-valued fuzzy concept lattice which is the combination of classical concept lattice and lattice implication algebra and totally different from the general fuzzy concept lattice. ts ideological core is constructing the lattice-valued fuzzy relation between objects and attributes. n this section we study the definitions and properties of the lattice-valued fuzzy concept lattice and give an example to illustrate it. Definition 3.1 [Yang 08] A 4-tuple K = ( L is called lattice-valued fuzzy context where = { g1 g g p } is the non-empty finite objects set = { } is the non-empty finite attributes set ( L O is a m1 m m q lattice implication algebra is a fuzzy relation between and i.e. : L. Let be a non-empty finite objects set and ( L O be a lattice implication algebra. Denote the set of all the fuzzy subsets on as L for any A 1 A L and g A 1 A A 1( g A ( g then ( L is a partial ordered set. Let be a non-empty finite attributes set and ( L O be a lattice implication algebra. Denote the set of all the fuzzy subsets on as L for any B 1 B L and m B 1 B B 1( m B ( m then ( L is a partial ordered set. Theorem 3.1 [Yang 08] Let K = ( L be a lattice-valued fuzzy context and L a lattice implication algebra define mappings f h between L and L

6 164 Yang L. Xu Y.: Decision aking with Uncertainty nformation... ( ( f : L L f A ( m = A( g ( g m g h: L L h( B ( g = ( B ( m ( g m m then for any A L B L ( f h is a alois connection. Definition 3. [Yang 08] A lattice-valued fuzzy concept of K = ( L is defined as a pair ( AB with A L B L and f ( A = B hb ( = A. For any lattice-valued fuzzy concepts ( A 1 B 1 and ( A B define ( A B ( A B A A ( or B B and denote the set { } C( K = ( AB f( A = BhB ( = A be the lattice-valued fuzzy concept lattice. Theorem 3. [Yang 08] Let K = ( L be a lattice-valued fuzzy context and ( f h the alois connection on it for any A 1 A A L B 1 B B L there are following properties: (1 A 1 A f( A f( A 1 B 1 B h( B h( B 1 ; ( A hf( A B fh ( B ; (3 f ( A = fhf( A hb ( = hfhb ( ; (4 f ( A 1 A = f( A 1 f( A hb ( 1 B = hb ( 1 hb (. Example 3.1 Let us consider a lattice-valued fuzzy context K = ( L depicted in Table 1 where = { g1 g} = { m1 m m3 m4} ( L O is a lattice implication algebra and its Hasse diagram is shown as Figure 1 and implication operator as Table : m 1 m m 3 m 4 g a b c 1 g d c O b Table 1: Lattice-valued fuzzy context ( L a b d c O Figure 1: Hasse diagram of L = { Oabcd } 6

7 Yang L. Xu Y.: Decision aking with Uncertainty nformation a b c d O a b c d O a a c c d b c c c c a b a b d a a O Table : mplication Operator of L = { Oabcd } 6 For Table 1 we can calculate its lattice-valued fuzzy concepts according to the Definition 3. and Theorem 3.1 as follows: C doob C a ddob C a doda C = c abob C C C C C C = ( = ( 3 = ( 4 ( = ( b dcob C6 = ( aa cdda C7 = ( b doc C8 = ( d abda = ( ac aaob C10 = ( cc abbb C11 = ( ba ccda C1 = ( ab cdc = ( ad ada C14 = ( bc aob C15 = ( O abc C16 ( cd abaa = ( dc aabb C18 = ( bb ccc C19 = ( ao ac C0 = ( bd da = ( co ab C = ( dd aaa C3 = ( Oc abb C4 ( bo c = ( do a C = ( Od aa C = ( OO = = n the above lattice-valued fuzzy concepts we illustrate the calculation method C = a ddob as follows: by the fuzzy concept ( When A = { a } we will firstly calculate f( A = { f( A ( m1 f( A ( m f( A ( m3 f( A ( m4} by Theorem 3.1 in which f ( A ( m = ( A ( g ( g m = ( a a ( d = d = d 1 1 g f ( A ( m = ( A ( g ( g m = ( a b ( c = a c = d g f( A ( m = ( A ( g ( g m = ( a c ( O = c O = O 3 3 g f ( A ( m = ( A ( g ( g m = ( a ( b = b = b 4 4 g i.e. f ( A = { d d O b} denoted by B = { ddob } ; We will secondly calculate hb ( = { hb ( ( g1 hb ( ( g} by Theorem 3.1 in which hb ( ( g = ( Bm ( ( g m = ( d a ( d b ( O c ( b = a 1 1 m

8 166 Yang L. Xu Y.: Decision aking with Uncertainty nformation... hb ( ( g = ( Bm ( ( g m = ( d d ( d c ( O O ( b b = m i.e. hb ( = { a } = A so C = ( a ddob is a lattice-valued fuzzy concept. All the above lattice-valued fuzzy concepts can be constructed into a concept lattice by Definition 3. and its Hasse diagram as Figure : C 1 C C 3 C 4 C 5 C 11 C6 C 1 C15 C 7 C 13 C 8 C 9 C 14 C 16 C 10 C 17 C 18 C 19 C 0 C 1 C C 3 C 4 C 5 C 6 C 7 Figure : Hasse diagram of C ( K 4 Decision aking Based on Lattice-Valued Fuzzy Concept Lattice 4.1 Decision aking with Uncertainty nformation n this section we mainly define the lattice-valued decision context according to the concrete decision information system and establish the lattice-valued decision concept lattice as the single-target decision model. Definition 4.1 A 4-tuple K = ( { d} L is called a lattice-valued single-target decision context where is the non-empty finite objects set is the non-empty finite attributes set and d is a decision attribute ( L O is a lattice implication algebra is a relation between and { d} i.e. : { d} L. Definition 4. Let K = ( { d} L decision context a 4-tuple K α = ( α { d} L α is called a lattice-valued single-target = g g d = L decision context with decision value α where α { α αα }

9 Yang L. Xu Y.: Decision aking with Uncertainty nformation α : α L accordingly C ( K α is called a lattice-valued single-target decision concept lattice with decision value α. Definition 4.3 Let K α = ( α { d} L α decision context with decision value α for any A L α and B L define the alois connection ( fα hα between L α and L as follows: α fα : L L fα ( A ( m = ( A( g α( g m g α. α hα : L L hα ( B ( g = ( B( m α( g m m Let decision context with decision value α denote: C ( K { A ( AB ( K } α α = C α C ( K α = { B ( AB C( K α }. n the following // is expressed as the incomparability. Definition 4.4 Let K α = ( α { d} L α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision value α and ( fα hα the ( AB d( AB = α i.e. if alois connection the decision rule is defined as { } A and B then the decision value of ( AB is α if A and B satisfy at least one of the conditions as follows: (1 B = fα ( A or B // fα ( A for any A C ( K α α B L ; ( A = hα ( B or A // hα ( B for any B ( K C α A L α. For the decision rules mined from the lattice-valued single-target decision context it is necessary for us to find out the redundant rules and the matrix rules existed in this model. Definition 4.5 Let K α = ( α { d} L α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision value α and ( fα hα the ( AB d( AB = α is called as the alois connection the decision rule { } redundant rule if A and B satisfy at least one of the conditions as follows: (1 B // fα ( A for any A C ( K α α B L ; ( A // h ( B for any B C ( A L α. α K α Definition 4.6 Let { } K α = ( α d L α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision valueα for any A L α and B L ( AB d( AB = α is called as the maximum rule if the decision rule { }

10 168 Yang L. Xu Y.: Decision aking with Uncertainty nformation... ( AB C ( K α. n this decision model we can get not only the decision rules with decision value α but also the decision rules with decision values different from α according to the following two definitions. Let ( L O be a lattice implication algebra we denote: > α L = { β β > α β L} < α L = { β β < α β L} Definition 4.7 Let K = ( { d} L α α α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision value α and ( fα hα the alois connection for any A L α and B L the decision rule is defined as ( AB { d( AB > L α } i.e. if A and B then the decision value of ( AB is more than α if A and B satisfy at least one of the conditions as follows: (1 A C ( K α α and B fα ( A ; ( B C ( and A h ( B. K α Definition 4.8 Let { } α K α = ( α d L α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision value α and ( fα hα the alois connection for any A L α and B L the decision rule is defined as ( AB d( AB < L α i.e. if A and B then the decision value of ( AB { } is less than α if A and B satisfy at least one of the conditions as follows: (1 A C ( K α α and B fα ( A ; ( B C ( and A h ( B. K α α For the decision rules with decision values different from α their concrete decision values can be calculated by the following definitions and theorems. Definition 4.9 Let ( L be a lattice-valued fuzzy context A 1 A L α and A A define the degree to which 1 A 1 is more than A as: 1 L and B B 1 B B ( S( A A = A ( g A ( g ; 1 1 g define the degree to which B 1 is more than B as: SB ( B = B ( m B ( m. ( 1 1 m Theorem 4.1 Let ( L be a lattice-valued fuzzy context then (1 S( A A 3 S( A 1 A 3 if A 3 A A for any 1 A 1 A A 3 L ;

11 Yang L. Xu Y.: Decision aking with Uncertainty nformation if B 3 B B for any 1 B 1 B B 3 L. Proof. (1 A 1 A A 3 L and A A A for any g we can get 3 1 A ( g A ( g A ( g A ( g ( SB ( B3 SB ( 1 B = ( 1 3 ( A ( g A 3( g S( A A A ( g A ( g g g = S( A A. 3 ( can be proved similarly. Definition 4.10 Let K α = ( α { d} L α decision context with decision value α and ( fα hα the alois connection for any decision rule ( AB { d( AB > L α } satisfying A C ( K α α and B fα ( A then d ( A B can be calculated as: d( A B = α S( B fα ( A. Definition 4.11 Let K α = ( α { d} L α decision context with decision value α and ( fα hα the alois connection for any decision rule ( AB { d( AB > L α } satisfying B C ( K α and A h ( B then d( A B can be calculated as: α ( α d( A B = α S A h ( B. Similarly for the decision rules with the decision values less than α the following Definition 4.1 and Definition 4.13 also show that the size degree of the two L-fuzzy attribute subsets and the size degree of the two L-fuzzy object subsets can be looked as the decrease degree of α. Definition 4.1 Let K α = ( α { d} L α decision context with decision value α and ( fα hα the alois connection for ( AB d( AB < L α satisfying A C ( K and any decision rule { } B fα ( A then d ( A B can be calculated as: d( A B = α S ( fα ( A B. Definition 4.13 Let K α = ( α { d} L α decision context with decision value α and ( fα hα the alois connection for any decision rule ( AB { d( AB < L α } satisfying B C ( K α and A h ( B then d( A B can be calculated as: α ( α d( A B = α S h ( B A. n the following we will talk about the properties of the formulas such as coherent isotone [Zhang 05]. Theorem 4. The formulas defined by Definition 4.10 and 4.11 are coherent. α α

12 170 Yang L. Xu Y.: Decision aking with Uncertainty nformation... Proof. We need only to prove that A 1 A L α B 1 B L d( A B = α when A = A 1 and B = B it is obviously that SA ( 1 A 1 = SB ( B 1 = O so d( A B = α. Theorem 4.3 The formulas defined by Definition 4.1 and 4.13 are coherent. Proof. The proof is similar to the one of Theorem 4.. Theorem 4.4 The formulas defined by Definition 4.10 and 4.11 are isotone. Proof. We firstly prove that the formula defined by Definition 4.10 is isotone i.e. AA 1 A L α B B 1 B L ( AB d( AB = α and B = B 1 = B A A A it follows that 1 { } d( A B = α S( A A = α SA ( A α SA ( 1 A = α SA ( A = d( A 1 B 1 ; The isotone of the fomular defined by Definition 4.11 can be proved similarly. So the formulas defined by Definition 4.10 and 4.11 are isotone. Theorem 4.5 The formulas defined by Definition 4.1 and 4.13 are isotone. Proof. We firstly prove that the formula defined by Definition 4.1 is isotone i.e. AA 1 A L α B B 1 B L ( AB d ( AB = α and B = B 1 = B A A 1 A it follows that d( A B = α S ( A A 1 ( α SA ( A = ( α SA ( A 1 = α π ( A A = d( A 1 B 1 ; The isotone of the formula defined by Definition 4.13 can be proved similarly. So the formulas defined by Definition 4.1 and 4.13 are isotone. 4. Properties of decision rules The properties of decision rules will play a certain extent role on mining all the decision rules so we talk about them in the following. Theorem 4.6 Let K α = ( α { d} L α decision context with decision value α. f AA 1 A L α B L s.t. ( AB d( AB = α is a maximum decision rule and d( A 1 B > L α { } d( A B L > α then 1

13 Yang L. Xu Y.: Decision aking with Uncertainty nformation d( A A B d( A B d( A B ; ( d( A 1 A B d( A 1 B d( A B. Proof. (1 AA 1 A L α B L d( A 1 A B = α S( A 1 A A = α S( A A A (1 1 = 1 g 1 ( ( A 1 A ( g A ( g = α ( 1 ( ( ( ( ( = α A g Ag A g Ag g g ( A 1 ( g Ag ( α ( A ( g Ag ( = α g g ( α S ( A 1 A ( α S ( A A = = d( A 1 B d( A B ; ( d( A 1 A B = α S( A 1 A A = α S( A A A g α 1 ( ( A 1 A ( g A ( g = α g α (( A 1 ( g A ( g ( A ( g A ( g = α ( 1 ( ( ( ( ( α A g A g A g A g g α g α ( 1 ( ( ( ( ( = α A g A g α A g A g g α g α = α S ( A A α S ( A A ( 1 ( 1 ( α S ( A 1 A ( α S ( A A = = d( A 1 B d( A B. K = ( d L Theorem 4.7 Let { } α α α decision context with decision value α. f AA 1 A L α ( AB { d( AB = α} B L is a maximum decision rule and d( A 1 B < d( A B L < α then (1 d( A 1 A B d( A 1 B d( A B ; d( A A B = d( A B d( A B. ( 1 1 s.t. L α

14 17 Yang L. Xu Y.: Decision aking with Uncertainty nformation... Proof. (1 d( A 1 A B = α S ( A A A 1 ( α S ( A A 1 A = ( ( ( 1 ( = α Ag A A g g α ( ( 1 ( ( ( ( Ag A g Ag A g α g α g α ( α S ( A A 1 ( α S ( A A ( α S ( A A 1 ( α S ( A A = = = d( A 1 B d( A B. ( d( A 1 A B = α S ( A A A 1 ( α S ( A A 1 A = ( ( ( 1 ( = α Ag A A g g α ( ( 1 ( ( ( ( Ag A g Ag A g = α g α g α ( α S ( A A 1 ( α S ( A A ( α S ( A A 1 ( α S ( A A = = = d( A 1 B d( A B. The following theorems can be immediately obtained from Theorem 4.6 and Theorem 4.7. Theorem 4.8 Let K α = ( α { d} L α decision context with decision value α. f B B 1 B L for A L α ( AB d( AB = α is a maximum decision rule and d( A B > 1 L α { } > then ( ( ( ; ( ( (. d( A B L α (1 d A B 1 B = d A B 1 d A B ( d A B 1 B d A B 1 d A B K α = ( α d L α decision context with decision value α. f B B 1 B L for A L α Theorem 4.9 Let { }

15 ( AB { d( AB = α} is a maximum decision rule and d( A B < 1 < then ( ( ( ; ( ( (. d( A B L α (1 d A B 1 B d A B 1 d A B ( d A B 1 B = d A B 1 d A B K α = ( α d L α decision context with decision value α and C ( K α the lattice-valued single-target decision concept lattice with decision value α if A 1 A L α B 1 B L ( A { } 1 B 1 d( A 1 B 1 = α and ( A { } B d( A B = α are the maximum decision rules then (1 d( A 1 A B 1 B L > α ; ( d( A 1 A B 1 B L < α. Proof. (1 By Definition 4.6 ( A 1 B 1 L( K α and ( A B L( K α hold for any A 1 A L α B 1 B L then B 1 = f( A 1 B = f( A Theorem 4.10 Let { } ( A 1 A f( A 1 A C( K α i.e. ( 1 ( 1 ( 1 ( 1 { } A A f A A d A A f A A = α is a maximum decision rule. By the properties of lattice-valued fuzzy concept lattice d( A A B B = d A A f( A f( A d( A A B B L α So 1 1 ( ( 1 ( 1 ( 1 ( >. = d A A f A d A A f A α. ( Similarly ( A 1 A f( A 1 A C( K α 1 L ( ( 1 ( 1 1 ( 1 L α holds for any A 1 A L α { } B B A A f A A d A A f A A = α is a maximum decision rule then d( A A B B = d A A f( A f( A d( A A B B L < α. So 1 1 ( ( 1 ( 1 ( 1 ( = d A A f A d A A f A α. Theorem 4.11 Let { } K α = ( α d L α decision context with decision value α. f AA 1 A L α B L s.t. ( AB d( AB = α is a maximum decision rule d( A 1 B L and { } > then 1 d( A B L α Yang L. Xu Y.: Decision aking with Uncertainty nformation... d( A A B L > α. 173

16 174 Yang L. Xu Y.: Decision aking with Uncertainty nformation... Proof. Suppose that ( fα gα is the alois connection on K α. A = h ( B α A holds for any AA 1 A L α and B L by Definition 4.7 then A A 1 A holds by A ( A 1 A = A 1 ( A A = so d( A 1 A B > d( A B i.e. d( A 1 A B L > α. K = ( d L Theorem 4.1 Let { } α α α decision context with decision value α. f B B 1 B L ( AB { d( AB = α} A L α s.t. is a maximum decision rule d( A B 1 L and d( A B L > α then d( A B 1 B L > α. Proof. Suppose that ( fα gα is the alois connection on K α. B = f ( A α B holds for any B B 1 B L A L α by Definition 4.7 then B B 1 B holds by B ( B 1 B = B 1 ( B B = so d( A B 1 B > d( A B i.e. d( A B B L > α Extracting Algorithm of aximum Decision Rules Let { } K = ( d L decision context α α α with decision value α where α = { g1 g g r } = { m1 m m s } L = ( Ln O is a lattice implication algebra α : α L and ( fα hα is the alois connection. The extracting algorithm of maximum decision rules as follows: K = ( d L nput: the lattice-valued single-target decision context { } Output: the maximum decision rules α α α Begin while ( K α Φ do Calculate the fuzzy subsets A k = ( A k( g1 A k( g A k( gr ( OO O A ( 1 k n k r For k 1 to n do for j 1 to s do for i 1 to r do f ( g m : = A ( g ( g m α i j k i α i j B ( m : = B ( m f ( g m endfor; endfor; for i 1 to r do j j α i j r of L α

17 Yang L. Xu Y.: Decision aking with Uncertainty nformation for j 1 to s do ( h B ( m : = B ( m ( g m α j j α i j ( Cg ( : = Cg ( h Bm ( i i α j endfor; if Cg ( i = A k( gi then endif; endfor; endfor; end; This above algorithm for extracting the maximum decision rules depends on the lattice-valued single-target decision concept lattice and the decision rules basis are different with the changeable decision values of single-target decision concept lattices which provides us a direct and convenient mathematical tool. 5 Conclusions For dealing with uncertainty problems in decision making this paper proposed a new decision model and researched the methods of mining decision rules. Based on the lattice-valued logic we utilized the lattice implication algebra to depict the uncertainty information and constructed the lattice-valued single-target decision concept lattice as the decision model. Concretely we gave the definition of decision rule and the calculating method of decision rules with different decision values. We discussed the properties of decision rules and proposed the algorithm for mining the maximum decision rules which provides a feasible method for decision making. n the future work we will be devoted to software implementation of this decision method and apply it into practice. Obviously other decision models and decision methods for dealing with uncertainty information will be also our next researches. Acknowledgements The work was partially supported by the Natural Science Foundation of PR China (rant no and the specialized Research Fund for the Doctoral Program of Higher Education of China (rant no References [Birkhoff 67] Birkhoff.: Lattice Theory American athematical Society Colloquium Publications 1967 XXV Vol.5 American athematical Society Providence R. [Cheng 07] Cheng X. Liang J.Y. Qian Y.H.: Dynamic ining Decision Rules Based on Partial ranulation Computer Applications (3: [Delgado 0] Delgado. Herrera F. and Herrera-Viedma E.: A communication model based on the -tuple fuzzy linguistic representation for a distributed intelligent agent system on internet Soft Computing 00 6:

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19 Yang L. Xu Y.: Decision aking with Uncertainty nformation Lattice for Decision-making The 8th nternational FLNS Conference on Computational ntelligence in Decision and Control (adrid Spain 008 9: 1-4. [Yao 03] Yao Y.Y. Liau C.J. Zhong N.: ranular Computing Based on Rough Sets Quotient Space Theory and Belief Functions LNA : [Yao 04] Yao Y.Y.: A Partition odel of ranular Computing LNCS :3-53. [Zadeh 97] Zadeh L.A.: Towards a Theory of Fuzzy nformation ranulation and its Centrality in Human Reasoning and Fuzzy Logic Fuzzy Sets and Systems : [Zhang 05] Zhang W.X. Qiu.F.: Uncertain Decision aking Based on Rough Sets Tsinghua University Press Beijing: 005. [Zhang 06] Zhang W.X. Yao Y.Y. Liang Y.: Rough Set and Concept Lattice Xi an Jiaotong University Press 006.