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3.3 Fuzzy set theory
The handling rules and criteria needed for determining the bending sequence is discussed in this section. These rules deal with the selection of the next best bend for the bending operation. Each of these rules establishes relationships between pairs of bending operation groups. A high membership grade indicated for a particular rule means that the bend group is a good selection for the next operation according to this rule. These relationships between the bending and criteria are represented as fuzzy relations, and the membership grades of these fuzzy relations are determined through fuzzy membership function,as shown in Fig. 6.
3.3.1 Sequencing rules
Fuzzy relations or sequencing rules describe the priority of each group. Thus, definition of these rules for a computeraided system is important. According to previous studies and experience of the authors [15], the following rules are suggested:
Rule 1: Distance rule This rule describes the influence of the shape of a bend on the sequencing strategy. The further a bend is away from the mother plane, the higher its grade will be and thus should be bent earlier. The fuzzy functions presented in Fig. 6a are used to determine the grade of membership by this rule.
Rule 2: Number of bends in a group The higher the number of bends in a group, the more impact it will have on the overall shape of the part and vice versa. The more impact a group will have, the later it should be addressed in the operation. So the fuzzy relationship value of this rule can be represented as in Fig. 6b.
Rule 3: Bending angle This rule is to determine the fuzzy relationship value according to the angle of each plane. This is the angle between the mother plane and each rotated plane. If this angle is greater than 90°, the bending process is pided into one or more processes. The fuzzy relationship value is unity in the case of a bend angle less than 90° and zero in other cases. These relationships according to bend angles are represented as fuzzy functions as shown in Fig. 6c.
Rule 4: Feeding direction [14] This rule is to determine the fuzzy relationship value of a fuzzy function according to whether or not the bend is in the feeding direction. After bending, an escape space is necessary in either the stripper plate of the upper die or the die plate of the lower die. The escape space should be at a minimum considering the die strength, the part to be fixed, the loss of die material, and the manufacturing time.
Bending processes requiring a large escape space should be performed later to minimize the escape space. Because Fig. 6 Fuzzy membership function. a Distance rule. b Number of bends in a group. c Bending angle. d Feeding direction) a bend perpendicular to the feeding direction requires a smaller escape space than a bend in the feeding direction, the former precedes the latter. The membership value for the perpendicular feeding direction is unity, and zero otherwise. The fuzzy membership function for this rule is shown in Fig. 6d.
Note: To determine the grade of the membership for each group, the maximum grade of its bends is chosen and later used in the fuzzy matrix.
3.3.2 Fuzzy matrix
Let C={ci |i=1, 2,…….., n} represent the set consisting of all the remaining bend classes that are being considered for bending, where ci is one of the bend classes.
Let R={rj |j=1, 2, 3, 4} represent the set of four criteria in the handling rules, where rj represents one of the criteria.
A fuzzy relation is a mapping from C×R into [0, 1], such that (Vij)c, is expressed as follows:
(Vij)c =f( ci; rj)
Since related sets C and R are finite, a fuzzy relation f on C×R can be represented as a fuzzy matrix [M], the entries of which are (Vij)c.
The determination of the grade, which may vary anywhere between zero and unity, is based on the sequencing rules. The fuzzy matrix [M] is shown in Table 1.
3.3.3 Determination of final value matrix (FVM) set