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step2. If 1 SM SM then go to step 3, else replace SM with SM1 and go to step1. step3. Calculate the Ȝ-matrix of SM, where (0,1] O step4. Determine the clusters based on the rule if SMpq=1, then case p and case q belong to the same cluster. For the similarity matrix SM mentioned above, let Ȝ=0.87 and the Ȝ-matrix is as follows: 1 0.56 0.57 0.57 0.82 0.54 0.55 0.550.56 1 0.56 0.56 0.56 0.54 0.55 0.550.57 0.56 1 1 0.57 0.54 0.55 0.550.57 0.56 1 1 0.57 0.54 0.55 0.550.82 0.56 0.57 0.57 1 0.54 0.55 0.550.54 0.54 0.54 0.54 0.54 1 0.50.55 0.55 0.55 0.550.55 0.55 0.55 0.55!#%#" 40.540.45 0.54 1 10.55 0.54 1 1§•¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸©¹ Based on the Ȝ-matrix, cases in table 1 is partitioned to 22 clusters as follows: ^ ` ^ ` 1 2 3 22 1 2 3 4 5 52 53 , , ,..., { },{ },{ , , },...,{ , } Eccc c uuuuu uu 4. Mining rules by fuzzy-rough approach In practical environment, a design rule looks like if a U-type bending has a small relative bend radius, then the method of exerting backpressure can be used to compensate the springback. Fuzzy and linguistic terms are usually used in these rules to express design experiences. To obtain similar rules, the numerical attributes should be fuzzified into linguistic terms. Each attribute of bend feature is assigned some linguistic terms and the number of linguistic terms for an attribute is dependent. For example, the attribute of bend radius has five terms: very big, big, normal, small, very Small. Typical triangular membership functions are used to calculate the membership degree of each attribute value and the term with maximum membership degree will be chosen. More details of fuzzification are available from [8]. The result of fuzzification is shown in table 1. Table 1 is called a decision table. Every row in table 1 determines a sequence, called a decision rule, and can be denoted by 1( ), 2( ), . . .
11( ) iaxa x a x c o (x), where x is a row in the table. In this table, the related die design for a stamping part is replaced with the cluster to which the case belongs. RST(Rough Set Theory), introduced by Pawlak[9] in the early 1980s, provide a promising framework for automatically transforming data into knowledge. According to RST, an information system is a pair S=<U, A>, where U is a non-empty and finite set, called the universe, and A is a non-empty, finite set of attributes. Take Table 1 as example, the universe is the case base with 53 cases and the A is a set of attributes including the 11 attributes of a bend feature and the cluster number . An indiscernibility relation[10] is defined for any subset P of A as: ( ) {( , ) : every a P, a(x)=a(y)} IND P x y U U for u . A major feature of the Rough Set theory is to find a minimal set of data from a data table that preserving its basic properties. In this sense, the reduction technique is important to remove superfluous data. A reduct is a minimal subset of attributes that enables the same classification of elements of the universe as the whole set of attributes. The reduct can be defined as follows: 1) P is an independent set of features if there does not exist a subset P of P such that IND(P)=IND(P). 2) A Set RP is a reduct of P if it is independent and IND(R) =IND (P) There are many ways to compute a reduct of a rough set. In this paper, the Johnsons algorithm is selected and Rosetta, a rough set toolkit[11], is used to obtain the reduct for the attributes of a bend feature. A attributes set usually has one or more reducts. In this paper the reduct with minimum number of attributes is chosen as defined: Red*(A)=Min (Card (Red (A))). Among the resulting reducts from Rosetta, the reduct Red*(A)= (BT, ASBM, T, LDA, ADA) is chosen to construct rules. So the other attributes will be dropped from the decision table. Based on the reduct Red*, totally 33 rules were extracted from the decision table, they are: Rule 1: BT (Rectangle type) and ASBM (high) and T (thin) and LDA (normal) and ADA (normal) Шc1 Rule 2: BT (U type) and ASBM (low) and T (thin) and LDA (normal) and ADA (normal) Шc2 Rule 3: BT (circle type) and ASBM (low) and T (thin) and LDA (normal) and ADA (normal) Шc3 . Rule 33: BT (U type) and ASBM (high) and T (thin) and LDA (very high) and ADA (high) Шc6 Each rule can be used as a guide for stamping die design. Take the Rule2 for example, suppose a designer is given a stamping part: the bend type of the part is U type, the stamping material is thin and resistant to springback, and the requirement for linear dimension accuracy and angle dimension accuracy is normal. Then the cluster 1 is the decision and the die design d1 will be picked up from the case base for the designer to reuse the specific experiences in the drawings. If a cluster has more than one case, all of these cases will be brought to designers. The extracted rules can be integrated into conventional RBR system and die designers can be supported directly by the die designs for similar stamping parts. Furthermore, since a cluster of successful designs meeting the requirement is collected, an experienced designer can induce common rules from these designs and consequently the knowledge of design will be mined in such a dynamic way. In the case retrieval stage of a CBR system, this approach can be used to quickly find a union of clusters, each of which contains the die designs for a feature of the target stamping part.