American Institute of Mathematical Sciences

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May  2022, 18(3): 2033-2047. doi: 10.3934/jimo.2021055

A partitioning column approach for solving LED sorter manipulator path planning problems

Sheng-I Chen 1,2,, and  Yen-Che Tseng 1,2,

1. 

Department of Industrial Engineering and Management, National Yang Ming Chiao Tung Univeristy, Hsinchu, 30010, Taiwan
2. 

Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, 30010, Taiwan

* Corresponding author: Sheng-I Chen

Received  October 2020 Revised  December 2020 Published  May 2022 Early access  March 2021

Fund Project: The study is supported by MOST, Taiwan grant 109-2221-E-009-065-

This study considers the path planning problem of picking light-emitting diodes on a silicon wafer. The objective is to find the shortest walk for the sorter manipulator covering all nodes in a fully connected graph. We propose a partitioning column approach to reduce the original graph's size, where adjacent nodes at the same column are seen as a required edge, and the connection of vertices at different required edges is viewed as a non-required edge. The path planning problem turns to find the shortest closed walk to traverse required edges and is modeled as a rural postman problem with a solvable problem size. We formulate a mixed-integer program to obtain the exact solution for the transformed graph. We compare the proposed method with a TSP solver, Concorde. The result shows that our approach significantly reduces the problem size and obtains a near-optimal solution. For large problem instances, the proposed method can obtain a feasible solution in time, but not for the benchmarking solver.

Keywords: Light-emitting diode sorter, Manipulator path planning, Mixed-integer programming, Rural postman problems, Traveling salesman problem.
Mathematics Subject Classification: 05A18, 05C45, 90B06, 90C11.
Citation: Sheng-I Chen, Yen-Che Tseng. A partitioning column approach for solving LED sorter manipulator path planning problems. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2033-2047. doi: 10.3934/jimo.2021055
References:
[1]

D. Applegate, R. Bixby, V. Chvátal and W. Cook, Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems, Mathematical Programming, 97 (2003), 91-153.  doi: 10.1007/s10107-003-0440-4.

[2]

S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman and A. Y. Wu, An optimal algorithm for approximate nearest neighbor searching, Journal of the ACM, 45 (1998), 891-923.  doi: 10.1145/293347.293348.

[3]

N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Technical Report, 388, Graduate School of Industrial Administration, CMU, 1976.

[4]

N. Christofides, V. Campos, A. Corberan and E. Mota, An Algorithm for the Rural Postman Problem, , Imperical College, London, England, Tech. Report, 1981.

[5]

W. Cook, Concorde TSP solver, Available at: http://www.math.uwaterloo.ca/tsp/concorde/index.html. [Accessed on March 29, 2019], 2006.

[6]

A. Corberan and J. M. Sanchis, A polyhedral approach to therural postman problem, European Journal of Operational Research, 79 (1994), 95-114.  doi: 10.1016/0377-2217(94)90398-0.

[7]

G. Cornuéjols, J. Fonlupt and D. Naddef, The traveling salesman problem on a graph and some related integer polyhedra, Mathematical Programming, 33 (1985), 1-27.  doi: 10.1007/BF01582008.

[8]

H. Crowder and M. W. Padberg, Solving large-scale symmetric travelling salesman problems to optimality, Management Science, 26 (1980), 495-509.  doi: 10.1287/mnsc.26.5.495.

[9]

M. Drexl, On the generalized directed rural postman problem, Journal of the Operational Research Society, 65 (2014), 1143-1154.  doi: 10.1057/jors.2013.60.

[10]

M. Dror, Arc Routing: Theory, Solutions, and Applications, Kluwer, Boston, MA, 2000. doi: 10.1007/978-1-4615-4495-1.

[11]

H. A. Eiselt, M. Gendreau and G. Laporte, Arc routing problems, part Ⅱ: The rural postman problem, Operations Research, 43 (1995), 399-414.  doi: 10.1287/opre.43.3.399.

[12]

J. Edmond and E. L. Johnson, Matching, Euler tour and the chinese postman problem, Mathematical Programming, 5 (1973), 88-124.  doi: 10.1007/BF01580113.

[13]

E. Fernández, O. Meza, R. Garfinkel and M. Ortega, On the undirected rural postman problem: Tight bounds based on a new formulation, Operations Research, 51 (2003), 281-291.  doi: 10.1287/opre.51.2.281.12790.

[14]

J. Fonlupt and A. Nachef, Dynamic programming and the graphical traveling salesman problem, Journal of the ACM (JACM), 40 (1993), 1165-1187.  doi: 10.1145/174147.169803.

[15]

R. W. Folyd, Algorithm 97: Shortest path, Communications of the ACM, 5 (1962), 345. doi: 10.1145/367766.368168.

[16]

R. S. Garfinkel and I. R. Webb, On crossings, the crossing postman problem, and the rural postman problem, Networks: An International Journal, 34 (1999), 173-180.  doi: 10.1002/(SICI)1097-0037(199910)34:3<173::AID-NET1>3.0.CO;2-W.

[17]

R. Kala, A. Shukla and R. Tiwari, Robotic path planning using evolutionary momentum-based exploration, Journal of Experimental and Theoretical Artificial Intelligence, 23 (2011), 469-495.  doi: 10.1080/0952813X.2010.490963.

[18]

E. L. Lawler, J. K. Lenstra, A. R. Kan and D. B. Shmoys, The traveling salesman problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons, Ltd., Chichester, 1985.

[19]

S. Lin and B. W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Operations Research, 21 (1973), 498-516.  doi: 10.1287/opre.21.2.498.

[20]

C. E. Miller, A. W. Tucker and R. A. Zemlin, Integer programming formulation of traveling salesman problems, Journal of the ACM, 7 (1960), 326-329.  doi: 10.1145/321043.321046.

[21]

M. Padberg and G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Review, 33 (1991), 60-100.  doi: 10.1137/1033004.

[22]

D. J. Rosenkrantz, R. E. Stearns, P. M. Lewis and II, An analysis of several heuristics for the traveling salesman problem, SIAM Journal on Computing, 6 (1977), 563-581.  doi: 10.1137/0206041.

[23]

J. Sankaranarayanan, H. Samet and A. Varshney, A fast all nearest neighbor algorithm for applications involving large point-clouds, Computers and Graphics, 31 (2007), 157-174.  doi: 10.1016/j.cag.2006.11.011.

[24]

W. Sheng, N. Xi, M. Song and Y. Chen, Robot path planning for dimensional measurement in automotive manufacturing, Journal of Manufacturing Science and Engineering, 127 (2005), 420-428.  doi: 10.1115/1.1870013.

[25]

T. Wu, B. Li, L. Wang and Y. Huang, Study on path-optimization by grade sorting dies, In IEEE International Conference on Mechatronics and Automation, (2010), 876–880. doi: 10.1109/ICMA.2010.5589000.

show all references

References:
[1]

D. Applegate, R. Bixby, V. Chvátal and W. Cook, Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems, Mathematical Programming, 97 (2003), 91-153.  doi: 10.1007/s10107-003-0440-4.

[2]

S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman and A. Y. Wu, An optimal algorithm for approximate nearest neighbor searching, Journal of the ACM, 45 (1998), 891-923.  doi: 10.1145/293347.293348.

[3]

N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Technical Report, 388, Graduate School of Industrial Administration, CMU, 1976.

[4]

N. Christofides, V. Campos, A. Corberan and E. Mota, An Algorithm for the Rural Postman Problem, , Imperical College, London, England, Tech. Report, 1981.

[5]

W. Cook, Concorde TSP solver, Available at: http://www.math.uwaterloo.ca/tsp/concorde/index.html. [Accessed on March 29, 2019], 2006.

[6]

A. Corberan and J. M. Sanchis, A polyhedral approach to therural postman problem, European Journal of Operational Research, 79 (1994), 95-114.  doi: 10.1016/0377-2217(94)90398-0.

[7]

G. Cornuéjols, J. Fonlupt and D. Naddef, The traveling salesman problem on a graph and some related integer polyhedra, Mathematical Programming, 33 (1985), 1-27.  doi: 10.1007/BF01582008.

[8]

H. Crowder and M. W. Padberg, Solving large-scale symmetric travelling salesman problems to optimality, Management Science, 26 (1980), 495-509.  doi: 10.1287/mnsc.26.5.495.

[9]

M. Drexl, On the generalized directed rural postman problem, Journal of the Operational Research Society, 65 (2014), 1143-1154.  doi: 10.1057/jors.2013.60.

[10]

M. Dror, Arc Routing: Theory, Solutions, and Applications, Kluwer, Boston, MA, 2000. doi: 10.1007/978-1-4615-4495-1.

[11]

H. A. Eiselt, M. Gendreau and G. Laporte, Arc routing problems, part Ⅱ: The rural postman problem, Operations Research, 43 (1995), 399-414.  doi: 10.1287/opre.43.3.399.

[12]

J. Edmond and E. L. Johnson, Matching, Euler tour and the chinese postman problem, Mathematical Programming, 5 (1973), 88-124.  doi: 10.1007/BF01580113.

[13]

E. Fernández, O. Meza, R. Garfinkel and M. Ortega, On the undirected rural postman problem: Tight bounds based on a new formulation, Operations Research, 51 (2003), 281-291.  doi: 10.1287/opre.51.2.281.12790.

[14]

J. Fonlupt and A. Nachef, Dynamic programming and the graphical traveling salesman problem, Journal of the ACM (JACM), 40 (1993), 1165-1187.  doi: 10.1145/174147.169803.

[15]

R. W. Folyd, Algorithm 97: Shortest path, Communications of the ACM, 5 (1962), 345. doi: 10.1145/367766.368168.

[16]

R. S. Garfinkel and I. R. Webb, On crossings, the crossing postman problem, and the rural postman problem, Networks: An International Journal, 34 (1999), 173-180.  doi: 10.1002/(SICI)1097-0037(199910)34:3<173::AID-NET1>3.0.CO;2-W.

[17]

R. Kala, A. Shukla and R. Tiwari, Robotic path planning using evolutionary momentum-based exploration, Journal of Experimental and Theoretical Artificial Intelligence, 23 (2011), 469-495.  doi: 10.1080/0952813X.2010.490963.

[18]

E. L. Lawler, J. K. Lenstra, A. R. Kan and D. B. Shmoys, The traveling salesman problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons, Ltd., Chichester, 1985.

[19]

S. Lin and B. W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Operations Research, 21 (1973), 498-516.  doi: 10.1287/opre.21.2.498.

[20]

C. E. Miller, A. W. Tucker and R. A. Zemlin, Integer programming formulation of traveling salesman problems, Journal of the ACM, 7 (1960), 326-329.  doi: 10.1145/321043.321046.

[21]

M. Padberg and G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Review, 33 (1991), 60-100.  doi: 10.1137/1033004.

[22]

D. J. Rosenkrantz, R. E. Stearns, P. M. Lewis and II, An analysis of several heuristics for the traveling salesman problem, SIAM Journal on Computing, 6 (1977), 563-581.  doi: 10.1137/0206041.

[23]

J. Sankaranarayanan, H. Samet and A. Varshney, A fast all nearest neighbor algorithm for applications involving large point-clouds, Computers and Graphics, 31 (2007), 157-174.  doi: 10.1016/j.cag.2006.11.011.

[24]

W. Sheng, N. Xi, M. Song and Y. Chen, Robot path planning for dimensional measurement in automotive manufacturing, Journal of Manufacturing Science and Engineering, 127 (2005), 420-428.  doi: 10.1115/1.1870013.

[25]

T. Wu, B. Li, L. Wang and Y. Huang, Study on path-optimization by grade sorting dies, In IEEE International Conference on Mechatronics and Automation, (2010), 876–880. doi: 10.1109/ICMA.2010.5589000.

Figure 1.  The problem instances of LED distribution
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Figure 2.  The optimal path for the transformed graph
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Figure 3.  The example for partitioning columns
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Figure 4.  The worst case of discarded diodes
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Figure 5.  The coverage percentages of diodes in transformed graphs using different maximum mismatch ratios
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Table 1.  Notations
Mu The $ y-position $ of the uppermost diode
Ml The $ y-position $ of the lowermost diode
r The maximum mismatch ratio
nx The number of LEDs at the $ \textit{x-th} $ column
$ \theta $ The threshold of empty positions in the column $ (\theta=\lceil\textit{nx}r\rceil) $
$ \bar{b} $ The lower bound of the upper component (initialized as Mu)
$ \underline{b} $ The upper bound of the lower component (initialized as Ml)
$ \textbf{X} $ The collection of diode's $ \textit{y-position} $
$ \textbf{Y} $ The set of grades of diodes
$ \textbf{g} $ A function mapping $ \textbf{X} $ into $ \textbf{Y} $
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Mu The $ y-position $ of the uppermost diode
Ml The $ y-position $ of the lowermost diode
r The maximum mismatch ratio
nx The number of LEDs at the $ \textit{x-th} $ column
$ \theta $ The threshold of empty positions in the column $ (\theta=\lceil\textit{nx}r\rceil) $
$ \bar{b} $ The lower bound of the upper component (initialized as Mu)
$ \underline{b} $ The upper bound of the lower component (initialized as Ml)
$ \textbf{X} $ The collection of diode's $ \textit{y-position} $
$ \textbf{Y} $ The set of grades of diodes
$ \textbf{g} $ A function mapping $ \textbf{X} $ into $ \textbf{Y} $
Table 2.  The procedure to find component boundaries
1.    Set $ \bar{b} = i_{1} = M_{u}, \underline{b} = i_{2} = M_{1}, i = 0, \theta = \lceil n_{x} r \rceil $, and $ d = True; $
2.    If($ M_{u}-M_{l}+1 $)$ -n_{x} >\theta $
3.       While $ i <\vert n_{x} \vert $ and $ \theta > 0 $
4.          If $ d = True $ then
5.             If $ g(i_{1}) = g(i_{1}-1) $ then
6.                $ \bar{b}:=i_{1}-1; $
7.             Else
8.                $ d = False; $
9.                $ \theta$ – –;
10.             End if
11.             $ i{1} $ – –;
12.          Else
13.             If $ g(i_{2})=g(i_{2}+1) $ then
14.                $ \underline{b}:=i_{2}+1; $
15.             Else
16.                $ d = True; $
17.                $ \theta $ – –;
18.             End if
19.             $ i_{2}++; $
20.          End if
21.          $ i++; $
22.       End while
23.    End if
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1.    Set $ \bar{b} = i_{1} = M_{u}, \underline{b} = i_{2} = M_{1}, i = 0, \theta = \lceil n_{x} r \rceil $, and $ d = True; $
2.    If($ M_{u}-M_{l}+1 $)$ -n_{x} >\theta $
3.       While $ i <\vert n_{x} \vert $ and $ \theta > 0 $
4.          If $ d = True $ then
5.             If $ g(i_{1}) = g(i_{1}-1) $ then
6.                $ \bar{b}:=i_{1}-1; $
7.             Else
8.                $ d = False; $
9.                $ \theta$ – –;
10.             End if
11.             $ i{1} $ – –;
12.          Else
13.             If $ g(i_{2})=g(i_{2}+1) $ then
14.                $ \underline{b}:=i_{2}+1; $
15.             Else
16.                $ d = True; $
17.                $ \theta $ – –;
18.             End if
19.             $ i_{2}++; $
20.          End if
21.          $ i++; $
22.       End while
23.    End if
Table 3.  The size of the original and transformed problems (maximum mismatch ratio = 0.3)
Instance Original problem size Transformed problem size Reduction
$ \vert \boldsymbol{V} \vert $ $ \vert \boldsymbol{E} \vert $ $ \vert \boldsymbol{V_{r}} \vert $ $ \vert \boldsymbol{E_{r}} \vert $ $ 1-\vert \boldsymbol{V_{r}} \vert / \vert \textbf{V} \vert $ $ 1-\vert \boldsymbol{E_{r}} \vert / \vert \textbf{E} \vert $
Grade1 48 2,256 30 870 38% 61%
Grade2 74 5,402 108 11,556 -46% -114%
Grade3 118 13,806 80 6,320 32% 54%
Grade4 396 156,420 126 15,750 68% 90%
Grade5 1,849 3,416,952 226 50,850 88% 99%
Grade6 3,921 15,370,320 164 26,732 96% 100%
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Instance Original problem size Transformed problem size Reduction
$ \vert \boldsymbol{V} \vert $ $ \vert \boldsymbol{E} \vert $ $ \vert \boldsymbol{V_{r}} \vert $ $ \vert \boldsymbol{E_{r}} \vert $ $ 1-\vert \boldsymbol{V_{r}} \vert / \vert \textbf{V} \vert $ $ 1-\vert \boldsymbol{E_{r}} \vert / \vert \textbf{E} \vert $
Grade1 48 2,256 30 870 38% 61%
Grade2 74 5,402 108 11,556 -46% -114%
Grade3 118 13,806 80 6,320 32% 54%
Grade4 396 156,420 126 15,750 68% 90%
Grade5 1,849 3,416,952 226 50,850 88% 99%
Grade6 3,921 15,370,320 164 26,732 96% 100%
Table 4.  Vertex reduction percentages in transformed graphs using different mismatch ratios
Instance $ r=0.01 $ $ r=0.05 $ $ r=0.1 $ $ r=0.15 $ $ r=0.2 $ $ r=0.25 $ $ r=0.3 $ $ r=0.33 $ $ r=0.35 $ $ r=0.5 $ $ r=0.65 $
Grade1 33% 33% 33% 38% 38% 38% 38% 38% 38% 38% 38%
Grade2 -46% -46% -46% -46% -46% -46% -46% -46% -46% -46% -46%
Grade3 32% 32% 32% 32% 32% 32% 32% 32% 32% 32% 32%
Grade4 67% 67% 67% 67% 67% 68% 68% 68% 68% 68% 69%
Grade5 87% 87% 87% 87% 88% 88% 88% 88% 88% 88% 88%
Grade6 95% 96% 96% 96% 96% 96% 96% 96% 96% 96% 96%
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Instance $ r=0.01 $ $ r=0.05 $ $ r=0.1 $ $ r=0.15 $ $ r=0.2 $ $ r=0.25 $ $ r=0.3 $ $ r=0.33 $ $ r=0.35 $ $ r=0.5 $ $ r=0.65 $
Grade1 33% 33% 33% 38% 38% 38% 38% 38% 38% 38% 38%
Grade2 -46% -46% -46% -46% -46% -46% -46% -46% -46% -46% -46%
Grade3 32% 32% 32% 32% 32% 32% 32% 32% 32% 32% 32%
Grade4 67% 67% 67% 67% 67% 68% 68% 68% 68% 68% 69%
Grade5 87% 87% 87% 87% 88% 88% 88% 88% 88% 88% 88%
Grade6 95% 96% 96% 96% 96% 96% 96% 96% 96% 96% 96%
Table 5.  Edge reduction percentages in transformed graphs using different maximum mismatch ratios
Instance $ r=0.01 $ $ r=0.05 $ $ r=0.1 $ $ r=0.15 $ $ r=0.2 $ $ r=0.25 $ $ r=0.3 $ $ r=0.33 $ $ r=0.35 $ $ r=0.5 $ $ r=0.65 $
Grade1 55% 55% 55% 61% 61% 61% 61% 61% 61% 61% 61%
Grade2 -114% -114% -114% -114% -114% -114% -114% -114% -114% -114% -114%
Grade3 54% 54% 54% 54% 54% 54% 54% 54% 54% 54% 54%
Grade4 89% 89% 89% 89% 89% 90% 90% 90% 90% 90% 90%
Grade5 98% 98% 98% 98% 98% 98% 98% 99% 99% 99% 99%
Grade6 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%
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Instance $ r=0.01 $ $ r=0.05 $ $ r=0.1 $ $ r=0.15 $ $ r=0.2 $ $ r=0.25 $ $ r=0.3 $ $ r=0.33 $ $ r=0.35 $ $ r=0.5 $ $ r=0.65 $
Grade1 55% 55% 55% 61% 61% 61% 61% 61% 61% 61% 61%
Grade2 -114% -114% -114% -114% -114% -114% -114% -114% -114% -114% -114%
Grade3 54% 54% 54% 54% 54% 54% 54% 54% 54% 54% 54%
Grade4 89% 89% 89% 89% 89% 90% 90% 90% 90% 90% 90%
Grade5 98% 98% 98% 98% 98% 98% 98% 99% 99% 99% 99%
Grade6 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%
Table 6.  Computational results of solving the original and transformed problems
Instance Partitioning column MTZ formulation
Run time(s) Total cost MIPGap Run time(s) Total cost MIPGap
Grade1 <1 227 0% 7,200 229 42%
Grade2 14 386 0% 184 386 0%
Grade3 8 393 0% 7,200 444 61%
Grade4 78 660 0% $ - $ $ - $ $ - $
Grade5 260 2,078 0% $ - $ $ - $ $ - $
Grade6 6,356 4,205 0% $ - $ $ - $ $ - $
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Instance Partitioning column MTZ formulation
Run time(s) Total cost MIPGap Run time(s) Total cost MIPGap
Grade1 <1 227 0% 7,200 229 42%
Grade2 14 386 0% 184 386 0%
Grade3 8 393 0% 7,200 444 61%
Grade4 78 660 0% $ - $ $ - $ $ - $
Grade5 260 2,078 0% $ - $ $ - $ $ - $
Grade6 6,356 4,205 0% $ - $ $ - $ $ - $
Table 7.  Comparing the performance of Concorde in solving problem instances with integral and fractional edge distances
Instance Concord
(with rounding to the nearest integer)		 Concorde
(with four decimal place)
Run time(s) Total cost Rounding Error % Run time(s) Total cost
Grade1 <1 227 0% 1 227
Grade2 <1 386 0% 1 386
Grade3 1 393 0% 1 394
Grade4 5 582 4% 18 605
Grade5 $ - $ $ - $ $ - $ $ - $ $ - $
Grade6 42 3,926 $ - $ $ - $ $ - $
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Instance Concord
(with rounding to the nearest integer)		 Concorde
(with four decimal place)
Run time(s) Total cost Rounding Error % Run time(s) Total cost
Grade1 <1 227 0% 1 227
Grade2 <1 386 0% 1 386
Grade3 1 393 0% 1 394
Grade4 5 582 4% 18 605
Grade5 $ - $ $ - $ $ - $ $ - $ $ - $
Grade6 42 3,926 $ - $ $ - $ $ - $
Table 8.  The benchmark of Concorde and the proposed algorithm using Manhattan and Euclidean distances
Instance Euclidean distance Manhattan distance
Concorde Proposed algorithm Concorde Proposed algorithm
Grade1 227 227 258 258
Grade2 386 386 462 462
Grade3 394 393 430 428
Grade4 605 660 692 742
Grade5 $ - $ 2,078 $ - $ 2,154
Grade6 $ - $ 4,205 $ - $ 4,286
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Instance Euclidean distance Manhattan distance
Concorde Proposed algorithm Concorde Proposed algorithm
Grade1 227 227 258 258
Grade2 386 386 462 462
Grade3 394 393 430 428
Grade4 605 660 692 742
Grade5 $ - $ 2,078 $ - $ 2,154
Grade6 $ - $ 4,205 $ - $ 4,286
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