LNAI 7120 - Dominance-Based Soft Set Approach in ... - Springer Link

Dominance-Based Soft Set Approach in Decision-Making Analysis Awang Mohd Isa1, Ahmad Nazari Mohd Rose1, and Mustafa Mat Deris2 1

Faculty of Informatics Universiti Sultan Zainal Abidin, Terengganu, Malaysia 2 Faculty of Information Technology and Multimedia Universiti Tun Hussein Onn Malaysia, Johor, Malaysia {isa,anm}@unisza.edu.my, [email protected]

Abstract. Multi-criteria decision analysis, sometimes called multi-criteria decision making, is a discipline aimed at supporting decision makers faced with making numerous and sometimes conflicting evaluations. Multi-criteria decision analysis aims at highlighting these conflicts and providing a compromised solution in a transparent process. This paper introduces the application of soft-dominance relation based on soft set theory in the field of multi-criteria decision analysis. This relation is an extension of the soft set theory which deals with typical inconsistencies during the consideration of criteria and in preference-ordered decision classes. The paper also utilized softdominance relations based on soft set theory in obtaining the decision rules in dealing with problems in a multi-valued information system. Keywords: Multi-criteria decision analysis, Soft set theory, Multi-soft set, Soft-dominance relation, Multi-valued Information System.

1

Introduction

Many decision-making problems are characterized by the ranking of objects according to a set of criteria with pre-defined preference-ordered decision classes, such as credit approval [18], stock risk estimation [1], mobile phone alternatives estimation [10]. Models and algorithms were proposed for extracting and aggregating preference relations based on distinct criteria. The underlying objectives are to understand the decision process, to build decision models and to learn decisions rules from data. Rough set theory provides an effective tool for dealing with inconsistency and incomplete information [[19],[14]]. It has been widely applied in feature evaluation [17], attribute reduction and rule extraction [9]. Pawlak’s rough set model is constructed based on equivalence relations. These relations are viewed by many to be one of the main limitations when employing the model to complex decision tasks. However, there is an extension of the rough set [6] to deal with these limitations. In multiple criteria decision-making problems, there are preference structures between conditions and decisions. Greco et al. [6] introduced a dominance rough set J. Tang et al. (Eds.): ADMA 2011, Part I, LNAI 7120, pp. 299–310, 2011. © Springer-Verlag Berlin Heidelberg 2011

300

A.M. Isa, A.N.M. Rose, and M.M. Deris

model that is suitable for preference analysis. In [6], the decision-making problem with multiple attributes and multiple criteria were examined, where dominance relations were extracted from multiple criteria and similarity relations were constructed from numerical attributes and equivalence relations were constructed from nominal features. An extensive review of multi-criteria decision analysis based on dominance rough sets is given in [7]. Dominance rough sets have also been applied to ordinal attribute reduction and multi-criteria classification. While the theory rough set is well-known and often useful approach to describe uncertainty, it has inherent difficulties as pointed by Molodtsov [13]. Soft set theory proposed by Molodtsov [13] provides an effective tool for dealing with inconsistencies which is free from the difficulties affecting existing methods. As reported in [13], a wide range of applications of soft sets have been developed in many different fields. There has been a rapid growth of interest in soft set theory and its application especially in decision making in recent years. Maji et al. [12] discussed the application of soft set theory in a decision making. Based on fuzzy soft sets, Roy and Maji [15] presented a method of object recognition from an imprecise multiobserver data and applied it to decision making problems. Chaudhuri et al [2] define the concepts of Soft Relation and Fuzzy Soft Relation and then apply them to solve a number of Decision Making Problems. Feng et al [3] introduce an adjustable approach to fuzzy soft set and investigate the application of the weighted fuzzy soft set in decision making. Feng et al [4] also present the application of level soft sets in decision making based on interval-valued fuzzy soft sets. Jiang et al [11] present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets. Molotosv's proposal of soft set has been studied and applied by several authors in the cases of uncertainty in decision making. Currently, there are not much literature discussing the application of soft set in the area of multicriteria decision making dealing with uncertainty. Thus, our paper intends to study the feasibility of applying dominance relation based on soft set theory in situations shrouded by uncertainty during the process of multicriteria decision making. In the proposed scheme, the decision system is transformed into the equivalent multi-soft set where in each soft set, the predicates are ordered according to the preference order, and then the approximations will be obtained using dominance-based soft set approach (DSSA), that is an extension of the soft set theory. The rest of this paper is organized as follows. Section 2 describes the fundamental concept of information systems. In section 3, we present the concept of soft set theory for multi-valued information systems. The extension of soft set approach based on dominance principle is introduced in section 4. An illustrative example is given in section 5 followed by the conclusion of our work is described in section 6.

2

Information System

An information system is a 4-tuple (quadruple), S = (U , A, V , f ) , where U is a non-

empty finite set of objects, A is a non-empty finite set of attributes, V = a∈ A V a , V a

Dominance-Based Soft Set Approach in Decision-Making Analysis

301

is the domain (value set) of attribute a, f : U × A → V is a total function such that

f (u , a ) ∈ V a , for every (u , a ) ∈ U × A , called information (knowledge) function. An information system is also called a knowledge representation systems or an attributevalued system that can be intuitively expressed in terms of an information table (as shown in Table 1). Table 1. An information system

a1

a2

ak

u1

f (u1 , a1 )

f (u1 , a 2 )

… …

u2

f (u 2 , a1 )

f (u 2 , a 2 )

…

f (u 2 , a k )

…

u3

f (u 3 , a1 )

f (u 3 , a 2 )

…

f (u 3 , a k )

…













U

uU

(

f u U , a1

)

(

f u U , a2

)

…

f (u1 , a k )

… …

(

f u U , ak

)

…

aA

( ) f (u , a ) f u1 , a A 2

(u

f

(

3

A

,

a

A



f uU ,a A

)

)

In many applications, there is an outcome of classification that is known. This knowledge, which is known as a posteriori knowledge is expressed by one (or more) distinguished attribute called decision attribute. This process is known as supervised learning. An information system of this kind is called a decision system. A decision system is an information system of the form D = (U , A  {d }, V , f ) , where d ∉ A is the decision attribute. The elements of A are called condition attributes. Condition attributes with value sets are ordered according to decreasing or increasing preference of a decision maker are called criteria.

3

Soft Set Theory

Throughout this section U refers to an initial universe, E is a set of parameters, P(U ) is the power set of U and A ⊆ E . Definition 1. (See [13].) A pair (F, A) is called a soft set over U, where F is a mapping given by

F : A → P(U ) . In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε ∈ A , F (ε ) may be considered as a set of ε -elements of the soft set (F, A) or as the set of ε -approximate elements of the soft set. Clearly, a soft set is not a (crisp) set. As for illustration, Molodtsov has considered several examples in [13]. The example shows that, soft set (F, A) can be viewed as a collection of approximations, where each approximation has two parts:-

302


(i) A predicate p ; and (ii) An approximate value-set

v (or simply to be called value-set v ).

We denote (F , A) = {p1 = v1 , p 2 = v 2 ,...., p n = v n } , where n is a number of predicates. Based on the definition of an information system and soft set, we then show that a soft set is a special type of information systems, i.e., a binary-valued information system. Proposition 2. (See [8].) If (F, A) is a soft set over the universe U, then (F, A) is a

binary-valued information system S = (U , A, V{0,1} , f ) .

Proof. Let (F, A) be a soft set over the universe U, we define a mapping F = {f1 , f 2 ,, f n } ,

where

1, x ∈ F (a i ) , for 1 ≤ i ≤ A . f i : U → Vi and f i ( x ) =  0, x ∉ F (a i )

Hence, if V =



ai ∈ A

V ai , where V ai = {0,1} , then a soft set (F , A) can be considered

as a binary-valued information system S = (U , A, V{0,1} , f ) .

From Proposition 2, it can be easily understood that a binary-valued information system can be represented using soft set theory. Thus, we can make a one-to-one correspondence between (F , E ) over U and S = (U , A, V{0,1} , f ) . Definition 3. (See [12].) The class of all value sets of a soft set (F , E ) is called value-

class of the soft set and is denoted by C( F , E ) . Proposition 4. (See [8].) If (F , A) = {(F , a i ) : 1 ≤ i ≤ A } is a multi-soft set over the

universe U, then (F , A) is a multi-valued information system S = (U , A, V , f ) . Proof.

Let

S = (U , A, V , f )

be a multi-valued information system and

S = (U , a i , V ai , f ) , 1 ≤ i ≤ A be the A binary-valued information systems. From i

Proposition 2, we have  S 1 = (U , a 1 , V {0 ,1} , f ) ⇔ (F , a 1 )  2  S = (U , a 2 , V {0 ,1} , f ) ⇔ (F , a 2 ) S = (U , A , V , f ) =       S A = U , a A , V {0 ,1} , f ⇔ F , a A 

) (

(

=

((F , a

1

), (F , a ),  , (F , a 2

= {(F , a i ) : 1 ≤ i ≤ A }

)

A

))


It is proved that

(F , A) = {(F , a ) : 1 ≤ i ≤ A } i

303

is a multi-soft sets over universe U

representing a multi-valued information system S = (U , A, V , f ) . Since the definition of soft sets is based on the mapping of value sets to the set of objects, it can then only handle one kind of inconsistency of decision - the one related to the inclusion of objects into different value-set of attributes, i.e., predicates. While this is sufficient for classification of taxonomy type, the classical soft set approach fails in case of ordinal classification with monotonicity constraints [16], where the value-sets of attributes are preference ordered. In this case, decision examples may be inconsistent in the sense of violation of the dominance principle which requires that an object x dominating object y on all considered criteria (i.e., x having evaluations at least as good as y on all considered criteria).

4

Dominance Relation Based Soft Set Theory

In this section, we present the principle of dominance relation based on soft set theory or dominance-based soft set approach (DSSA), that can be used in decision-making analysis. As mention the previous section, multi-valued information system S = (U , A, V , f ) can be represented by a multi-soft set (F , A) = {(F , a i ) : 1 ≤ i ≤ A } , where A is a finite set of parameters representing the set of attributes in multi-valued information system. The set A is, in general, divided into set C of condition attributes and set D of decision attributes. Condition attributes with value sets ordered according to decreasing or increasing preference of a decision maker are called criteria. Since for each criterion c ∈ A is represented by criterion soft set C = ( F , c) in multi-soft set (F , A) , the value sets for criterion c is equivalence to the value sets of the soft set when the predicates are ordered according to decreasing or increasing preference. For soft set C = ( F , c) ,  c is an outranking relation on U with reference to soft set C ∈ ( F , A) such that x c y means “ x is at least as good as y with respect to soft set C ”. Definition 5. For criterion soft set C = {p i = vi , i = 1,2,..., n} , for all r , s ∈ {1,.., n} ,

such that r > s , predicate p r is dominance than predicate p s with respect to C , if the value of p r is preferred to p s , and we denote that by p r  c p s . Definition 6. For criterion soft set C = {p i = vi , i = 1,2,..., n} , the set of objects in

value-set v r are preferred to the set of objects in value-set v s with respect to C , if predicate p r is dominance than predicate p s , we denote it by v r  c v s iff p r  c p s ∀r , s ∈ {1,...n} .

Furthermore, let suppose that the set of decision attributes D is a singleton {d } and is represented by decision soft set D = ( F , d ) . The values of predicate in D make a

partition of universe U into a finite number of decision classes, Cl = {Cl t , t = 1,..., n} ,

304


such that each x ∈ U belongs to one and only one class Cl t ∈ Cl . It is supposed that

the classes are preference-ordered, i.e. for all r , s ∈ {1,.., n} , such that r > s , the

objects from Cl r are preferred to objects from Cl s . If  is a comprehensive weak x, y ∈ U , x  y means “ x is comprehensively at least as good as y ”, it is supposed: [x ∈ Clr , y ∈ Cls , r > s ] 

preference relation on U , i.e. if for all

[x y and not y x] . The above assumptions are typical for consideration of ordinal classification problems (also called multiple criteria sorting problems). The set to be approximated are called upward union and downward union of classes, respectively: Clt =  Cl s , Clt =  Cl s , t = 1,..., n . s ≥t

s ≤t

The statement x ∈ Cl t means “ x belongs to at least class Cl t ”, while x ∈ Cl t means “ x belongs to at most class Cl t ”. Let us remark that Cl 1 = Cl n = U , Cl n = Cl n and

Cl 1 = Cl 1 . Furthermore, for t = 2,.., n, Cl t−1 = U − Cl t and Cl t = U − Cl t−1 . The key idea of the soft set approach is representation (approximation) of knowledge generated by decision soft set, using granules of knowledge generated by criterion soft sets. In DSSA, the knowledge to be represented is a collection of upward and downward unions of classes, and the granules of knowledge are sets of objects defined using a soft dominance relation. x dominates y with respect to P ⊆ C (shortly, x Psoft -dominates y), denoted by xD P y , if for every criterion soft set q ∈ P and x ∈ v i , y ∈ v j : v i  q v j . The relation Psoft -dominance is reflexive and transitive, i.e. it is a partial order. Given a set of soft sets P ⊆ C and x ∈ U , the “granules of knowledge” used for approximation in DSSA are: - a set of objects dominating x , called Psoft -dominating set, D P+ ( x) = {y ∈ U : yD P x} ,

-

a set of objects dominated by

x , called Psoft -dominated set,

D ( x) = {y ∈ U : xD P y} . From the above, it can be seen that the “granules of knowledge” have the form of upward (positive) and downward (negative) dominance cones in the evaluation space. Recall that the dominance principle requires that an object x dominating object y on all considered criterion soft sets or criteria (i.e., x having evaluation at least as good as y on all considered soft set) should also dominate y on decision soft set (i.e., x should be assigned to at least as good decision class as y ). This is the only principle widely agreed upon in the multiple criteria comparisons of objects. − P


305

Given P ⊆ C , the inclusion of an object x ∈ U to the upward union of classes Cl t (t = 2,..., n) is inconsistent with the dominance principle if one of the following conditions holds: -

x belongs to class Cl t or better, but it is Psoft -dominated by an object y belonging to a class worse than Cl t , i.e. x ∈ Cl t but D P+ ( x) ∩ Cl t−1 ≠ φ ,

-

x belongs to a worse class than Cl t but it Psoft -dominates an object y belonging to class Cl t or better, i.e. x ∉ Cl t but D P− ( x) ∩ Cl t−1 ≠ φ .

If, given a set of soft set P ⊆ C , the inclusion of x ∈ U to Cl t (t = 2,..., n) is inconsistent with the dominance principle, then x belongs to Cl t with some ambiguity. Thus,

x belongs to Clt without any ambiguity with respect to P ⊆ C , if

x ∈ Cl t and there is no inconsistency with the dominance principle. This means that all objects Psoft -dominating

x belong to Clt , i.e., DP+ ( x) ⊆ Cl t .

Furthermore, x possibly belongs to Cl t with respect to P ⊆ C if one of the following conditions holds: -

according to decision soft set D = ( F , d ) , object x belongs to Cl t ,

-

according to decision soft set D = ( F , d ) , object x does not belong to Cl t , but it is inconsistent in the sense of the dominance principle with an object y belonging to Clt .

In terms of ambiguity, x possibly belongs to Cl t with respect to P ⊆ C , if x possibly belongs to Cl t with or without ambiguity. Due to the reflexivity of the soft dominance relation D P , the above conditions can be summarized as follows: x possibly belongs to class Cl t or better, with respect to P ⊆ C , if among the objects Psoft -dominated by x there is an object y belonging to class Cl t or better, i.e., D P− ( x) ∩ Cl t ≠ φ . The Psoft -lower approximation of Cl t , denoted by P(Cl t ) , and the Psoft -upper approximation of Cl t , denoted by P(Cl t ) , are defined as follows (t=1,…,n): P(Cl t ) = {x ∈ U : DP+ ( x) ⊆ Cl t } ,

P(Cl t ) = {x ∈ U : D P− ( x) ∩ Cl t ≠ φ }.

Analogously, one can define the Psoft -lower approximation and the Psoft -upper approximation of Cl t as follows (t=1,…,n): P(Cl t ) = {x ∈ U : D P− ( x) ⊆ Cl t } ,

P(Cl t ) = {x ∈ U : DP+ ( x) ∩ Cl t ≠ φ } .

306


The Psoft -lower and Psoft -upper approximation defined above, satisfy the following

properties for each t ∈ {1,..., n} and for any P ⊆ C :

P(Cl t ) ⊆ Cl t ⊆ P(Cl t ) , P(Cl t ) ⊆ Cl t ⊆ P (Cl t ) . The Psoft -lower and Psoft -upper approximations of Cl t and Cl t have an important complementary property, according to which, P(Cl t ) = U − P(Cl t−1 ) and P(Cl t ) = U − P(Cl t−1 ), t = 2,..., n , P(Cl t ) = U − P(Cl t+1 ) and P(Cl t ) = U − P(Cl t+1 ), t = 1,..., n − 1 . The

Psoft -boundaries of Cl t and Cl t , denoted by

Bn P (Cl t ) and

Bn P (Cl t )

respectively, and defined as follows (t=1,...,n): Bn P (Cl t ) = P(Cl t ) − P(Cl t ) , and BnP (Clt ) = P(Clt ) − P(Clt ) . Due to complementary property, Bn P (Cl t ) = Bn P (Cl t−1 ) , for t=2,…,n. For any criterion soft set P ⊆ C , we define the accuracy of approximation of  Cl t and Cl t for all t ∈ T respectively as

α P (Cl t ) =

P(Cl t ) P(Cl t )

, α P (Cl t ) =

P(Cl t ) P(Cl t )

.

The quality of approximation of the ordinal classification Cl by a set of soft set P is defined as the ration of the number of objects Psoft -consistent with the dominance principle and the number of all objects in U . Since the Psoft -consistent objects are those which do not belong to any Psoft -boundary Bn P (Cl t ) , t=2,…,n, or Bn P (Cl t ) , t=1,…,n-1, the quality of approximation of the ordinal classification Cl by a set of soft set P , can be written as

γ P (Cl ) =

    U −   Bn P (Cl t )    Bn P (Cl t )     t∈T    t∈T





U

.

γ P (Cl ) can be seen as a degree of consistency of the objects from U , where P is the set of criterion soft set and Cl is the considered ordinal classification. Every minimal subset P ⊆ C such that γ P (Cl ) = γ C (Cl ) is called a reduct of Cl and is denoted by REDCl . Moreover, for a given set of U one may have more than one reduct. The intersection of all reducts is known as the core, denoted by CORE Cl . The dominance-based soft approximations of upward and downward unions of classes can serve to induce “if . . ., then . . .” decision rules. It is therefore more meaningful to consider the following three types of decision:


307

1. D - decision rules that have the following form: if f ( x, q1) ≥ rq1 and f ( x, q 2) ≥ rq 2 and ... f ( x, qp) ≥ rqp then x ∈ Cl t , where P = {q1, q 2,..., qp} ⊆ C , (rq1 , rq 2 ,..., rqp )∈ V q1 × V q 2 × ... × V qp and t ∈ T . These rules

are supported only by objects from the Psoft -lower approximation of the upward unions of classes Cl t . 2. D - decision rules that have the following form: if f ( x, q1) ≤ rq1 and f ( x, q 2) ≤ rq 2 and ... f ( x, qp) ≤ rqp then x ∈ Cl t , where P = {q1, q 2,..., qp} ⊆ C , (rq1 , rq 2 ,..., rqp )∈ V q1 × V q 2 × ... × V qp and t ∈ T . These rules

are supported only by objects from the Psoft -lower approximation of the downward unions of classes Cl t . 3. D  - decision rules that have the following form: if f ( x, q1) ≥ rq1 and f ( x, q 2) ≥ rq 2 and ... f ( x, qk ) ≥ rqk and f ( x, qk + 1) ≤ rqk +1 and

...

f ( x, qp) ≤ rqp

then

x ∈ Cl t ∪ Cl t +1 ∪ ... ∪ Cl s ,

where

P = {q1, q 2,..., qp} ⊆ C , (rq1 , rq 2 ,..., rqp )∈ V q1 × V q 2 × ... × V qp and s, t ∈ T such that

t < s . These rules are supported only by objects from the Psoft -boundaries of the unions of classes Cl t and Cl t .

5

Experiments

On the basis of data proposed by Grabisch [5], for an evaluation in a high school, an example is taken to illustrate the application of the method. The director of the school wants to assign students to two classes: bad and good. To fix the classification rules the director is asked to present some examples. The examples concern with six students described by means of four attributes (see Table 2 below):

• •

A1 : level in Mathematics,

• •

A3 : level in Literature,

A2 : level in Physics, A4 : global evaluation (decision class). Table 2. Multi-valued information table with examples of classification

Student

A1 (Mathematics)

1 2 3 4 5 6

good medium medium bad medium good

A2 (Physics) good bad bad bad good bad

A3 (Literature)

A4 (Evaluation)

bad bad bad bad good good

good bad good bad bad good

308


The components of the multi-valued information table S are: U = {1,2,3,4,5,6}

A = {A1 , A2 , A3 , A4 }

V1 = {bad , medium, good }

V 2 = {bad , good } V3 = {bad , good }

V 4 = {bad , good } the information function f ( x, q) , taking values f (1, A1 ) =good, f (1, A2 ) = good, and so on. Within this approach we approximate the class Cl t of “(at most) bad” students and the class Cl t of “(at least) good” students. Since information table as in table 1 above consist of only two decision classes, we have Cl1 = Cl1 , and Cl 2≥ = Cl 2 .

Moreover, C = {A1 , A2 , A3 } and D = {A4 } . In this case, however, A1 , A2 and A3 are criteria and the classes are preference-ordered. Furthermore, the multi-soft set equivalent to the multi-valued information table as in table 1 is given below:

( F , a1 ) = {bad = {4}, medium = {2,3,5}, good = {1,6}}  ( F , a 2 ) = {bad = {2,3,4,6}, good = {1,5}} ( F , A) =  . ( F , a 3 ) = {bad = {1,2,3,4}, good = {5,6}} ( F , d ) = {bad = {2,4,5}, good = {1,3,6}} And, with respect to each criterion soft set:

• • •

( F , a 3 ) : p1 = bad, and p2 =good; and

•

( F , d ) : p1 = bad, and p2 =good.

( F , a1 ) : p1 = bad, p2 = medium, and p3 =good; ( F , a 2 ) : p1 = bad, and p2 =good;

Our experiment obtained the following results. The Psoft -lower approximations, Psoft upper approximations and the Psoft -boundaries of classes Cl1 and Cl 2 are equal to, respectively: P(Cl1 ) = {4} , P(Cl1 ) = {2,3,4,5} , Bn P (Cl1 ) = {2,3,5} ,

P(Cl 2 ) = {1,6} , P(Cl 2 ) = {1,2,3,5,6} , Bn P (Cl 2 ) = {2,3,5} . Therefore, the accuracy of the approximation is 0.25 for Cl1 and 0.4 for Cl 2 , while the quality of approximation is equal to 0.5. There are only one reduct which is also the core, i.e. Re d Cl = CoreCl = {A1 } . The minimal set of decision rules that are derived from the experiment, are shown below (within the parenthesis are the objects that support the corresponding rule):


309

1. if f ( x, A1 ) ≥ good, then x ∈ Cl 2 (1,6) 2. if f ( x, A1 ) ≤ bad, then x ∈ Cl1 (4) 3. if f ( x, A1 ) ≥ medium and f ( x, A1 ) ≤ medium (i.e. f ( x, A1 ) is medium), then x ∈ Cl1 ∪ Cl 2 (2,3,5). Let us notice from table 1 that student 5 dominates student 3, i.e. student 5 is at least as good as student 3 with respect to all the three criteria, however, student 5 has a global evaluation worse than student 3. Therefore, this can be seen as an inconsistency revealed by the approximation based on soft-dominance which cannot be captured by the approximation based on the mapping of parameters to the set of objects under consideration. Moreover, let us remark that the decision rules induced from approximation obtained from soft-dominance relations give a more synthetic representation of knowledge contained in the decision system.

6

Conclusion

In this paper we have presented the applicability of soft set theory in multi-criteria decision analysis (MCDA). Dominance-based soft set approach is an extension of soft set theory for MCDA which permits the dealing of typical inconsistencies during the consideration of criteria or preference-ordered decision classes. Based on the approximations obtained through the soft-dominance relation, it is possible to induce a generalized description of the preferential information contained in the decision system, in terms of decision rules. The decision rules are expressions of the form if (condition) then (consequent), represented in a form of dependency between condition and decision criteria. When the proposed approach is applied to the multi-valued decision system through simulation, the decisions rules obtained are equivalent to the one that obtained by the previous technique [7] using rough-set based approach.

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