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Thus, they should be performed together.In the classification procedure, first, all the bends arepided according to their bending directions (feed directionor perpendicular to it). Then in each direction, the bends areclassified. In other words, the bends which are parallel andperpendicular to the feed direction cannot be formed insimilar groups.3.3 Fuzzy set theoryThe handling rules and criteria needed for determining thebending sequence is discussed in this section. These rulesdeal with the selection of the next best bend for the bendingoperation. Each of these rules establishes relationshipsbetween pairs of bending operation groups. A highmembership grade indicated for a particular rule meansthat the bend group is a good selection for the nextoperation according to this rule. These relationshipsbetween the bending and criteria are represented as fuzzyrelations, and the membership grades of these fuzzy Fuzzy relations or sequencing rules describe the priority ofeach group. Thus, definition of these rules for a computer-aided system is important. According to previous studiesand experience of the authors [15], the following rules aresuggested:Rule 1: Distance rule This rule describes the influence ofthe shape of a bend on the sequencing strategy. The furthera bend is away from the mother plane, the higher its gradewill be and thus should be bent earlier. The fuzzy functionspresented in Fig. 6a are used to determine the grade ofmembership by this rule.Rule 2: Number of bends in a group The higher the numberof bends in a group, the more impact it will have on theoverall shape of the part and vice versa. The more impact agroup will have, the later it should be addressed in the Rule 3: Bending angle This rule is to determine the fuzzyrelationship value according to the angle of each plane.This is the angle between the mother plane and each rotatedplane. If this angle is greater than 90°, the bending processis pided into one or more processes. The fuzzy relation-ship value is unity in the case of a bend angle less than 90°and zero in other cases. These relationships according tobend angles are represented as fuzzy functions as shown inFig. 6c.Rule 4: Feeding direction [14] This rule is to determine thefuzzy relationship value of a fuzzy function according towhether or not the bend is in the feeding direction. Afterbending, an escape space is necessary in either the stripperplate of the upper die or the die plate of the lower die. Theescape space should be at a minimum considering the diestrength, the part to be fixed, the loss of die material, andthe manufacturing time.Bending processes requiring a large escape space shouldbe performed later to minimize the escape space. Because a bend perpendicular to the feeding direction requires asmaller escape space than a bend in the feeding direction,the former precedes the latter. The membership value forthe perpendicular feeding direction is unity, and zerootherwise.
The fuzzy membership function for this rule isshown in Fig. 6d.Note: To determine the grade of the membership foreach group, the maximum grade of its bends is chosen andlater used in the fuzzy matrix.3.3.2 Fuzzy matrixLet C={ci | i=1, 2,…….., n} 源:自*751~·论,文'网·www.751com.cn/ represent the set consisting ofall the remaining bend classes that are being considered forbending, where ci is one of the bend classes.Let R={rj | j=1, 2, 3, 4} represent the set of four criteriain 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 cfci ; rj 1Since related sets C and R are finite, a fuzzy relation f onC×R can be represented as a fuzzy matrix [M], the entriesof which are (Vij)c.The determination of the grade, which may varyanywhere between zero and unity, is based on thesequencing rules. The fuzzy matrix [M] is shown in Table 1.3.3.3 Determination of final value matrix (FVM) setFVM is a matrix consisting of fuzzy values for eachbend groups that are being considered. The fuzzy valuesare determined by implementing rules described inSection 3-3-1. These rules have been found to givesuitable results in bending operations. The higher value abend group has, the sooner it should be formed. TheFig. 8 Unfolded shape of the part and the feed direction relative importance of the rules is represented as a fuzzysetW[R],asshowninEq.(2).WR r1 * 1:2; r2 * 0:8; r3 * 0:6; r4 * 0:2 2Thus, the FVM set can be presented by Table 2 andexpressed as follows:FVM C V :W R 34 ExamplesTo evaluate the described method, four components areselected (Figs. 7, 11, 14, and 16) and will be presented asfollows:Example 1 Figure 7 presents a part which is used inelectrical components. To produce this part, five bending andseveral cutting operations are carried out. However, in thepresent paper, only bending operations will be investigated.According to the rules regarding the mother plane,described in Section 3-1, the central plane of the part isdetermined as the mother plane.The unfolded shape of the part is shown in Fig. 8.Amongst the bending operations, four are in perpendicularand one in feed direction. According to the classificationrules, the classes of this part are determined as follows:1. Bends b1and b5 are in one class (class one), accordingto rule 2.