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Design verification for bent sheet metal parts: A graphs approach
Pergamon
Int. Trans. Opl Res. Vol.4, No. 1, pp.67-73, 1997 © 1997 IFORS. Publishedby ElsevierScienceLid All rights reserved. Printed in Great Britain PII: S0969--6016(96)00030--5 0969-6016/97 $17.00 + 0.00
Design Verification for Bent Sheet Metal Parts: A Graphs Approach IR. JOOST R. D U F L O U Asian Institute of Technology, Thailand Design for manufacturing requires the flexible verification and optimization of design features with regard to the suitability to the planned production methods. The automatic identification and generation of a suitable process plan for large scale bent sheet metal parts would provide a tool that would allow efficient feedback in the design phase. In this paper, a number of problem-size reduction methods are proposed to bring automatic manufacturability verification a few steps closer to reality. The scale of the reformulated combinatorial problems is illustrated, indicating the suitability for solution by means of known search algorithms. © IFORS. Published by Elsevier Science Ltd.
Key words: Design, CAPP, bending, sheet metal, verification.
1. INTRODUCTION Optimizing a part or product design can be considered from different viewpoints, reflected in the objective function(s) selected for a detailed design analysis. Minimizing the part fabrication costs, minimizing the operation costs of the assembly, optimizing the reliability under dimensional or weight constraints, and minimizing maintenance expenses are some typical examples. A designer/manufacturer will normally attach a lot of importance to possible production cost reductions and, as part thereof, to the ease of manufacturing of a part design, in order to guarantee competitiveness in the market. As the total production costs are strongly correlated to the assembly costs, reduction of the number of components and parts is generally accepted as a cost saving strategy (Design for Assembly). This often leads to geometrically more complex, multifunctional parts which require more careful detailed design. In a Design for Manufacturing strategy, it is of crucial importance that the manufacturability of such parts can be evaluated, and preferably quantified, in an early phase of the design. For this purpose the required manufacturing expertise should be readily available to the designer. Consecutive design proposals can then be compared and an optimal set of design parameters, linked to a part that can be manufactured at minimal costs, identified. For the design of sheet metal parts, a quantified comparison will normally require the partial or even detailed generation or recuperation of process plans for the operations used in the evaluation. For a complete analysis, the following elements should be considered: the achievable efficiency in the nesting of the unfolded parts on the sheet material, the verification of the availability of the required tools, and the calculation of the expected processing time for punching, nibbling and/or the cutting of part blanks by means of a laser, plasma or waterjet cutter. However, prior to solving these 2D problems, the design should be checked for impossible or uneconomical bending operations. This requires a detailed analysis of the 3D geometry of the formed part design and a search for a bending sequence that allows production of the part on the available tools (see Toh et al., 1992). Generating process plans is, in general, a time consuming activity. However, for the 2D operations as mentioned above, semi-automatic process planning has been achieved in some dedicated, commercially available CAD/CAM packages. For 3D bending tasks, the scale of the average bending sequence search problem has, until now, prevented a systematic approach (de Vinet al., 1994; Vidlicka, 1993). The testing of the manufacturability of a single design proposal would by itself pose unrealistic memory and CPU-time requirements, effectively eliminating the possibility for an iterative design optimization. The author is investigating the possibilities of problem scale-reduction through an efficient graph representation combined with an analysis of the workpiece in a pre-processing step. The relevance of this approach is illustrated in this paper. Correspondence: Ir. Joost R. Duflou, Affiliated Faculty, Asian Institute of Technology, P.O. Box 2754, Bangkok, Thailand
67
I . J . R . Duflou--Bent Sheet Metal Parts
68
~
Ra._____mm Y-Axis
Operatork~N~Machine Side ~ Side
Punch X-Gauge
•
,,
Machine/~ ~
Table
..
......
I
L .....
o ,~....;~
-axi
4
.......
"
Fig. 1. Gauging alternatives for a single bend operation.
2. P R O B L E M S T A T E M E N T When preparing a process plan for a sheet metal bending operation, some attention needs to be paid to the capacity (maximum bending force) of the available press brake(s), as well as to the selection of appropriate tools (punch and die): based on bending angles and radii, maximum allowed forces, and geometrical considerations. As these required forces and the geometrical characteristics can easily be derived from the geometry database, a set consisting of a press brake and tools can be chosen automatically from a list of available tools, or a preliminary evaluation can be conducted on a proposed set. Once a set has been pre-selected, the principal problem remains to identify feasible bending sequences, according to geometric constraints, and to select an optimum solution based on ergonomical considerations. In principle any permutation of the series of bends (n) could be a possible solution for this combinatorial problem, resulting in the number of sequences to be evaluated to be N = n! During the positioning of the workpiece in preparation for a bending step, the sheet metal will need to be positioned against a programmable back gauge (X-axis). For parts with parallel edges or bends to both sides of the bend to be performed, more gauging solutions will be possible (Fig. I). In case two independently programmable back gauges (X1 and X2) are available on the selected press brake, gauging against non-parallel edges or bends could also be considered. However, in the normal case where only a single back gauge is provided, without an extra degree of freedom in the vertical direction (R-axis), the number of gauging options to be considered is normally limited to two (Fig. 1). An example of an exception is given in Fig. 2, where the four indicated gauging edges can be reduced to two part-positioning situations, (a) and (b). The necessity to evaluate both gauging directions for every bend to be performed increases the number of possible solutions to: N = 2n,2(n - 1),2(n - 2),... ,2(n - (n - 1)) = 2"n!
(1)
To allow tool selection, for example the most appropriate punch tool, from a list of t tools for every bend, would increase the number of possible solutions further to:
N = t"2"n!
(2)
A serious disadvantage of this approach, next to the significant increase in the size of the problem under consideration, is the loss of flexibility due to the requirement for an adjusted machine setup for every selected additional tool. As most CNC bending machines are not equipped with an automatic tool change option, the pre-selection of a suitable tool combination is both a more practical and a more strategic approach. The systematic search of all N = 2"n! permutations, without preliminary analysis of the workpiece characteristics, would require a detailed study of: 2n + 2n,2(n - 1) + ... + 2n,2(n - 1),... ,2(n - (n - 1)) =
~
21n!
(n - i)!
(3)
i=1
different bending operations, or in total 2"n.n! operations if memory and data storage facilities do not
International Transactions in Operational Research Vol. 4, No. I
-~
69
r b . ~ ',~j ° ~
Fig. 2. Reduction of multiple gauging solutions to two basic positioning situations.
T
Po.!t,o.e,/
_.~ |
~. M i n i m u m Retract | Distance
,_.., \ PosiUomng
Fig. 3. Example of a setup where timely back gauge retraction during the bending operation can avoid collision between the part and the gauge (X-axis).
allow to retrieve information related to already tested operations. A single bending verification implies the identification of a suitable gauging edge, interference checking during the positioning of the part, and checking for possible collision(s) between the folded part and the machine setup. An additional required check could be the verification of the necessity for gauge retraction during the actual bending operation (Fig. 3). While these tests already represent an intense calculation task for the simple case of a 2D profile (parts that can be completely geometrically defined by means of a 2D sectional view and a length specification), in the case of a 3D part design, these simulations require a very demanding number of geometrical interference checks. While this straightforward approach may be effective for smaller scale problems, it has proven to be inefficient for part designs with more than 10 bends, strictly due to memory and CPU time constraints. Selective search methods based on heuristics have been suggested as a way out, resulting in solutions for layouts with up to 18 bends, but with variable quality of the resulting process plans and processing times that do not allow an efficient practical application(Shpitalni and Saddan, 1994).
3. PROBLEM SIZE REDUCTION THROUGH REDUNDANCY ELIMINATION In the approach described above, a significant portion of redundancy can be identified. The transition from an intermediate state, in which a number of bends have already been completed, to a state with one additional bend formed, is independent of the sequence of the already formed bends. Indeed, the sequence followed to obtain a given intermediate state, is irrelevant to describe its momentary over all geometry. As a result, the permutations can be replaced by combinations and the identical, partially completed workpieces can be represented by a single node in a directed graph with a single origin and end node (Thulasiraman and Swany, 1992). The search problem is thus being
70
I.d.R.
Duflou--Bent Sheet Metal Parts
IE+26 B m IE+24
mj m
,/
IE+22 mJ
-
IE+20 --
~m--
Exhaustive Search
IE+I8 -_ IE+I6
~
Shortest Path Approach
J i jm
./
IE+I4 -
•J
IE+I2 IE+I0
•J
_
IE+08 -
7
-7
~ )
,
mj
1000000 10000 100
-- m
1 ~ U
~
m
I 3
~
~ I 5
,
~ ,
l
~
I 7
•
,
9
I 11
,
I 13
,
J 15
,
I 17
,
I 19
,
# o f b e n d s in part d e s i g n Fig. 4. C o m p a r i s o n of the m a x i m u m required n u m b e r of b e n d i n g e v a l u a t i o n s for an exhaustive search and a directed g r a p h a p p r o a c h based on c o m b i n a t o r i a l i n t e r m e d i a t e steps.
translated into a shortest path problem between two nodes in a directed graph (G). The maximum number of nodes (vertex set V) to be considered is: O(G)
=
C'. =
i=o
-
i
n -- i)?i?
(4)
and the theoretical number of possible links (edges E), taking double gauging options into consideration, is: .-1
.-1
n?
Z C~2(n - i) = 2 Z i?(n - 1-i)?"
i=0
i=O
(5)
The achieved improvement is clearly illustrated in Fig. 4. For a typical 3D problem with, for example, 14 bends to be sequenced, the number of bend operations to be evaluated would be reduced from 2.0 E + 16, for an exhaustive search based on permutations, to 229,376 (double gauging assumed possible for every bend); a scale reduction by a factor 8.7 E + 10.
4. P R O B L E M SIZE R E D U C T I O N T H R O U G H PRE-PROCESSING O F T H E PART GEOMETRY Although a very significant reduction of the problem size has been realized through the previous translation from a tree representation, based on permutations, to a directed graph with a unique origin to destination by shortest path search, the required residual processing can still be very demanding in terms of memory resources and processing time. For a large number of bends, this would still pose an obstacle to practical implementation of a shortest path search algorithm. However, a number of strategies can be simultaneously applied to further reduce the graph size. A common approach in a traditional exhaustive search strategy is to apply a backwards unfolding check that starts by identifying bends that can be performed as last in the process plan. If an array Seq(x) with n elements is defined to represent the bend sequence, identifying possible last bends would limit the possible values of Seq(n). For more complex part designs, it is not uncommon that only one bend can be identified as feasible for the last operation in the sequence. Parts that can not be produced at all by means of the pre-selected machine setup are normally also identified at this stage. In case a specific bend can be derived as necessarily the last operation, the graph could be redefined as illustrated for a simple case with five bends in Fig. 5.
International Transactions in Operational Research Fol. 4, No. 1
71
Fig. 5. Graph representation for a part containing five bends with operation 4 being identified as the only feasible last step in the sequence: the eliminated evaluation steps are indicated with bold lines. Single edges, representing both gauging options, are used for clarity's sake.
3/ / -"---~..~
5
//
Fig. 6. Example part with bend numbers indication.
The reduced problem contains a maximum number of edges to be evaluated:
Ci,_IC~_~_, + 1 = 2,
2, i
i!(n--2---i)! + 1 .
(6)
\i=O
Another generally applicable approach would be the identification of relative sequences based on geometrical analysis of subsets of bends. An example is given based on the part as shown in Fig. 6. When analyzing the detail formed by bends 1 and 2, and their adjoining flanges, it becomes clear that only one relative sequence for these bends is feasible for the given tool set, as illustrated by Fig. 7. If Seq(x) = 1 and Seq(y) = 2, then necessarily x > y, for x and y integers < n representing the position in the bend sequence for the respective bends. The identification of one relative sequence for two bends results in a reduction of the number of edges in the graph with: x
)
\i=0
)
(n - 2)!(n - 1 - i) ~(n _--~-S__-/)~ + 1
C'._2C._~_, + 1 = 2,
2,
\i=0
leaving a theoretical maximum number of required evaluations of:
(7)
I.J.R.
72
D u f l o u - - B e n t Sheet Metal Parts
® ®
first second
®
(~
first second
Fig. 7. S u b s e t a n a l y s i s of a typical t w o b e n d detail, resulting in a relative s e q u e n c e detection.
2.50E+07
Theoretic max. # of bend evaluations: eq. 5
2.00E+07
Relative sequence given: eq. 8
1.50E+07 ~A--
Last bend known: eq. 6
1.00E+07
5.00E+06
/ 0.00E+00
i
:
-.J 5
3
-.d 7
-
-
-J 9
®
d
=
I1
.I _ 13
-"
I 15
17
19
# o f b e n d s in p a r t d e s i g n
Fig. 8. P r o b l e m size r e d u c t i o n t h r o u g h p r e - p r o c e s s i n g of the p a r t g e o m e t r y : effects o f a last b e n d a n d a relative s e q u e n c e identification o n the n u m b e r of b e n d o p e r a t i o n s to be e v a l u a t e d .
.-1 \i=o i.(n
n!
"~2 (n - 2)!(n - l - i) - i)!
)
(8)
The extent to which the problem size is affected by any of the previous pre-processing outcomes is summarized in Fig. 8. The simultaneous application of several of the derived rules would reduce the problem scale accordingly.
5. CONCLUSION It has been demonstrated how the order of magnitude of a large-scale combinatorial search problem can be reduced in the specific case of a manufacturability evaluation for bent sheet metal parts. Through redundancy elimination and a preparatory analysis of the geometrical aspects of the part, in comparison with a straightforward search approach, the size of the problem can be significantly down-scaled in preparation for the actual evaluation of the manufacturability and the quality of the related process plan.
International Transactions in Operational Research Vol. 4, No. I
73
REFERENCES de Vin, L.J., de Vries, J., Streppel, A. H., Klaassen, E. J. W. & Kals, H. J. J. (1994). The generation of bending sequences in a CAPP system for sheet metal components. Journal of Materials Processing Technology 41, 331-339. See Toh, K. H., Nee, A.Y.C., Loh, H. T. & Leel, K. S. (1992). A PC-based CAPP systemfor press brake bending. Proceedings of the National Symposium on Manufacturing Technology Manutech. Shpitalni, M. & Saddan, D. (1994). Automatic determination of bending sequences in sheet metal products. Annals of the CIRP 43, 23-26. Thulasiraman, K. & Swamy, M. N. S. (1992). Graphs: Theory and Algorithms. John Wiley & Sons Inc., New York. Vidlicka, P. (1993) Computer-aided manufacture of sheet metal components. Sheet Metal Industries, pp. 23-25.
Int. Trans. Opl Res. Vol.4, No. 1, pp.67-73, 1997 © 1997 IFORS. Publishedby ElsevierScienceLid All rights reserved. Printed in Great Britain PII: S0969--6016(96)00030--5 0969-6016/97 $17.00 + 0.00
Design Verification for Bent Sheet Metal Parts: A Graphs Approach IR. JOOST R. D U F L O U Asian Institute of Technology, Thailand Design for manufacturing requires the flexible verification and optimization of design features with regard to the suitability to the planned production methods. The automatic identification and generation of a suitable process plan for large scale bent sheet metal parts would provide a tool that would allow efficient feedback in the design phase. In this paper, a number of problem-size reduction methods are proposed to bring automatic manufacturability verification a few steps closer to reality. The scale of the reformulated combinatorial problems is illustrated, indicating the suitability for solution by means of known search algorithms. © IFORS. Published by Elsevier Science Ltd.
Key words: Design, CAPP, bending, sheet metal, verification.
1. INTRODUCTION Optimizing a part or product design can be considered from different viewpoints, reflected in the objective function(s) selected for a detailed design analysis. Minimizing the part fabrication costs, minimizing the operation costs of the assembly, optimizing the reliability under dimensional or weight constraints, and minimizing maintenance expenses are some typical examples. A designer/manufacturer will normally attach a lot of importance to possible production cost reductions and, as part thereof, to the ease of manufacturing of a part design, in order to guarantee competitiveness in the market. As the total production costs are strongly correlated to the assembly costs, reduction of the number of components and parts is generally accepted as a cost saving strategy (Design for Assembly). This often leads to geometrically more complex, multifunctional parts which require more careful detailed design. In a Design for Manufacturing strategy, it is of crucial importance that the manufacturability of such parts can be evaluated, and preferably quantified, in an early phase of the design. For this purpose the required manufacturing expertise should be readily available to the designer. Consecutive design proposals can then be compared and an optimal set of design parameters, linked to a part that can be manufactured at minimal costs, identified. For the design of sheet metal parts, a quantified comparison will normally require the partial or even detailed generation or recuperation of process plans for the operations used in the evaluation. For a complete analysis, the following elements should be considered: the achievable efficiency in the nesting of the unfolded parts on the sheet material, the verification of the availability of the required tools, and the calculation of the expected processing time for punching, nibbling and/or the cutting of part blanks by means of a laser, plasma or waterjet cutter. However, prior to solving these 2D problems, the design should be checked for impossible or uneconomical bending operations. This requires a detailed analysis of the 3D geometry of the formed part design and a search for a bending sequence that allows production of the part on the available tools (see Toh et al., 1992). Generating process plans is, in general, a time consuming activity. However, for the 2D operations as mentioned above, semi-automatic process planning has been achieved in some dedicated, commercially available CAD/CAM packages. For 3D bending tasks, the scale of the average bending sequence search problem has, until now, prevented a systematic approach (de Vinet al., 1994; Vidlicka, 1993). The testing of the manufacturability of a single design proposal would by itself pose unrealistic memory and CPU-time requirements, effectively eliminating the possibility for an iterative design optimization. The author is investigating the possibilities of problem scale-reduction through an efficient graph representation combined with an analysis of the workpiece in a pre-processing step. The relevance of this approach is illustrated in this paper. Correspondence: Ir. Joost R. Duflou, Affiliated Faculty, Asian Institute of Technology, P.O. Box 2754, Bangkok, Thailand
67
I . J . R . Duflou--Bent Sheet Metal Parts
68
~
Ra._____mm Y-Axis
Operatork~N~Machine Side ~ Side
Punch X-Gauge
•
,,
Machine/~ ~
Table
..
......
I
L .....
o ,~....;~
-axi
4
.......
"
Fig. 1. Gauging alternatives for a single bend operation.
2. P R O B L E M S T A T E M E N T When preparing a process plan for a sheet metal bending operation, some attention needs to be paid to the capacity (maximum bending force) of the available press brake(s), as well as to the selection of appropriate tools (punch and die): based on bending angles and radii, maximum allowed forces, and geometrical considerations. As these required forces and the geometrical characteristics can easily be derived from the geometry database, a set consisting of a press brake and tools can be chosen automatically from a list of available tools, or a preliminary evaluation can be conducted on a proposed set. Once a set has been pre-selected, the principal problem remains to identify feasible bending sequences, according to geometric constraints, and to select an optimum solution based on ergonomical considerations. In principle any permutation of the series of bends (n) could be a possible solution for this combinatorial problem, resulting in the number of sequences to be evaluated to be N = n! During the positioning of the workpiece in preparation for a bending step, the sheet metal will need to be positioned against a programmable back gauge (X-axis). For parts with parallel edges or bends to both sides of the bend to be performed, more gauging solutions will be possible (Fig. I). In case two independently programmable back gauges (X1 and X2) are available on the selected press brake, gauging against non-parallel edges or bends could also be considered. However, in the normal case where only a single back gauge is provided, without an extra degree of freedom in the vertical direction (R-axis), the number of gauging options to be considered is normally limited to two (Fig. 1). An example of an exception is given in Fig. 2, where the four indicated gauging edges can be reduced to two part-positioning situations, (a) and (b). The necessity to evaluate both gauging directions for every bend to be performed increases the number of possible solutions to: N = 2n,2(n - 1),2(n - 2),... ,2(n - (n - 1)) = 2"n!
(1)
To allow tool selection, for example the most appropriate punch tool, from a list of t tools for every bend, would increase the number of possible solutions further to:
N = t"2"n!
(2)
A serious disadvantage of this approach, next to the significant increase in the size of the problem under consideration, is the loss of flexibility due to the requirement for an adjusted machine setup for every selected additional tool. As most CNC bending machines are not equipped with an automatic tool change option, the pre-selection of a suitable tool combination is both a more practical and a more strategic approach. The systematic search of all N = 2"n! permutations, without preliminary analysis of the workpiece characteristics, would require a detailed study of: 2n + 2n,2(n - 1) + ... + 2n,2(n - 1),... ,2(n - (n - 1)) =
~
21n!
(n - i)!
(3)
i=1
different bending operations, or in total 2"n.n! operations if memory and data storage facilities do not
International Transactions in Operational Research Vol. 4, No. I
-~
69
r b . ~ ',~j ° ~
Fig. 2. Reduction of multiple gauging solutions to two basic positioning situations.
T
Po.!t,o.e,/
_.~ |
~. M i n i m u m Retract | Distance
,_.., \ PosiUomng
Fig. 3. Example of a setup where timely back gauge retraction during the bending operation can avoid collision between the part and the gauge (X-axis).
allow to retrieve information related to already tested operations. A single bending verification implies the identification of a suitable gauging edge, interference checking during the positioning of the part, and checking for possible collision(s) between the folded part and the machine setup. An additional required check could be the verification of the necessity for gauge retraction during the actual bending operation (Fig. 3). While these tests already represent an intense calculation task for the simple case of a 2D profile (parts that can be completely geometrically defined by means of a 2D sectional view and a length specification), in the case of a 3D part design, these simulations require a very demanding number of geometrical interference checks. While this straightforward approach may be effective for smaller scale problems, it has proven to be inefficient for part designs with more than 10 bends, strictly due to memory and CPU time constraints. Selective search methods based on heuristics have been suggested as a way out, resulting in solutions for layouts with up to 18 bends, but with variable quality of the resulting process plans and processing times that do not allow an efficient practical application(Shpitalni and Saddan, 1994).
3. PROBLEM SIZE REDUCTION THROUGH REDUNDANCY ELIMINATION In the approach described above, a significant portion of redundancy can be identified. The transition from an intermediate state, in which a number of bends have already been completed, to a state with one additional bend formed, is independent of the sequence of the already formed bends. Indeed, the sequence followed to obtain a given intermediate state, is irrelevant to describe its momentary over all geometry. As a result, the permutations can be replaced by combinations and the identical, partially completed workpieces can be represented by a single node in a directed graph with a single origin and end node (Thulasiraman and Swany, 1992). The search problem is thus being
70
I.d.R.
Duflou--Bent Sheet Metal Parts
IE+26 B m IE+24
mj m
,/
IE+22 mJ
-
IE+20 --
~m--
Exhaustive Search
IE+I8 -_ IE+I6
~
Shortest Path Approach
J i jm
./
IE+I4 -
•J
IE+I2 IE+I0
•J
_
IE+08 -
7
-7
~ )
,
mj
1000000 10000 100
-- m
1 ~ U
~
m
I 3
~
~ I 5
,
~ ,
l
~
I 7
•
,
9
I 11
,
I 13
,
J 15
,
I 17
,
I 19
,
# o f b e n d s in part d e s i g n Fig. 4. C o m p a r i s o n of the m a x i m u m required n u m b e r of b e n d i n g e v a l u a t i o n s for an exhaustive search and a directed g r a p h a p p r o a c h based on c o m b i n a t o r i a l i n t e r m e d i a t e steps.
translated into a shortest path problem between two nodes in a directed graph (G). The maximum number of nodes (vertex set V) to be considered is: O(G)
=
C'. =
i=o
-
i
n -- i)?i?
(4)
and the theoretical number of possible links (edges E), taking double gauging options into consideration, is: .-1
.-1
n?
Z C~2(n - i) = 2 Z i?(n - 1-i)?"
i=0
i=O
(5)
The achieved improvement is clearly illustrated in Fig. 4. For a typical 3D problem with, for example, 14 bends to be sequenced, the number of bend operations to be evaluated would be reduced from 2.0 E + 16, for an exhaustive search based on permutations, to 229,376 (double gauging assumed possible for every bend); a scale reduction by a factor 8.7 E + 10.
4. P R O B L E M SIZE R E D U C T I O N T H R O U G H PRE-PROCESSING O F T H E PART GEOMETRY Although a very significant reduction of the problem size has been realized through the previous translation from a tree representation, based on permutations, to a directed graph with a unique origin to destination by shortest path search, the required residual processing can still be very demanding in terms of memory resources and processing time. For a large number of bends, this would still pose an obstacle to practical implementation of a shortest path search algorithm. However, a number of strategies can be simultaneously applied to further reduce the graph size. A common approach in a traditional exhaustive search strategy is to apply a backwards unfolding check that starts by identifying bends that can be performed as last in the process plan. If an array Seq(x) with n elements is defined to represent the bend sequence, identifying possible last bends would limit the possible values of Seq(n). For more complex part designs, it is not uncommon that only one bend can be identified as feasible for the last operation in the sequence. Parts that can not be produced at all by means of the pre-selected machine setup are normally also identified at this stage. In case a specific bend can be derived as necessarily the last operation, the graph could be redefined as illustrated for a simple case with five bends in Fig. 5.
International Transactions in Operational Research Fol. 4, No. 1
71
Fig. 5. Graph representation for a part containing five bends with operation 4 being identified as the only feasible last step in the sequence: the eliminated evaluation steps are indicated with bold lines. Single edges, representing both gauging options, are used for clarity's sake.
3/ / -"---~..~
5
//
Fig. 6. Example part with bend numbers indication.
The reduced problem contains a maximum number of edges to be evaluated:
Ci,_IC~_~_, + 1 = 2,
2, i
i!(n--2---i)! + 1 .
(6)
\i=O
Another generally applicable approach would be the identification of relative sequences based on geometrical analysis of subsets of bends. An example is given based on the part as shown in Fig. 6. When analyzing the detail formed by bends 1 and 2, and their adjoining flanges, it becomes clear that only one relative sequence for these bends is feasible for the given tool set, as illustrated by Fig. 7. If Seq(x) = 1 and Seq(y) = 2, then necessarily x > y, for x and y integers < n representing the position in the bend sequence for the respective bends. The identification of one relative sequence for two bends results in a reduction of the number of edges in the graph with: x
)
\i=0
)
(n - 2)!(n - 1 - i) ~(n _--~-S__-/)~ + 1
C'._2C._~_, + 1 = 2,
2,
\i=0
leaving a theoretical maximum number of required evaluations of:
(7)
I.J.R.
72
D u f l o u - - B e n t Sheet Metal Parts
® ®
first second
®
(~
first second
Fig. 7. S u b s e t a n a l y s i s of a typical t w o b e n d detail, resulting in a relative s e q u e n c e detection.
2.50E+07
Theoretic max. # of bend evaluations: eq. 5
2.00E+07
Relative sequence given: eq. 8
1.50E+07 ~A--
Last bend known: eq. 6
1.00E+07
5.00E+06
/ 0.00E+00
i
:
-.J 5
3
-.d 7
-
-
-J 9
®
d
=
I1
.I _ 13
-"
I 15
17
19
# o f b e n d s in p a r t d e s i g n
Fig. 8. P r o b l e m size r e d u c t i o n t h r o u g h p r e - p r o c e s s i n g of the p a r t g e o m e t r y : effects o f a last b e n d a n d a relative s e q u e n c e identification o n the n u m b e r of b e n d o p e r a t i o n s to be e v a l u a t e d .
.-1 \i=o i.(n
n!
"~2 (n - 2)!(n - l - i) - i)!
)
(8)
The extent to which the problem size is affected by any of the previous pre-processing outcomes is summarized in Fig. 8. The simultaneous application of several of the derived rules would reduce the problem scale accordingly.
5. CONCLUSION It has been demonstrated how the order of magnitude of a large-scale combinatorial search problem can be reduced in the specific case of a manufacturability evaluation for bent sheet metal parts. Through redundancy elimination and a preparatory analysis of the geometrical aspects of the part, in comparison with a straightforward search approach, the size of the problem can be significantly down-scaled in preparation for the actual evaluation of the manufacturability and the quality of the related process plan.
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