Two-Stage Dynamic Programming in the Routing Problem with Decomposition

This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin...

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Veröffentlicht in:	Automation and remote control Jg. 84; H. 5; S. 543 - 563
Hauptverfasser:	Chentsov, A. G., Chentsov, P. A.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Moscow Pleiades Publishing 01.05.2023 Springer Nature B.V
Schlagworte:	Algorithms CAE) and Design Calculus of Variations and Optimal Control; Optimization Computer-Aided Engineering (CAD Constraints Control Cost function Cutting equipment Decomposition Dynamic programming Mathematical functions Mathematics Mathematics and Statistics Mechanical Engineering Mechatronics Numerical controls Operations Research Optimization Permutations Personal computers Robotics Set theory System Analysis Systems Theory Traveling salesman problem Workpieces dynamic programming route precedence conditions megalopolis
ISSN:	0005-1179, 1608-3032
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Abstract	This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin. In particular, this condition may arise in the tool control problem for thermal cutting machines with computer numerical control (CNC): if there are long parts among workpieces, the cutting process near a narrow material boundary should start with these workpieces since such parts are subject to thermal deformations, which may potentially cause rejects. The problem statement under consideration involves two zones for part processing. The aggregate routing process in the original problem includes a starting point, a route (a permutation of indices), and a particular track consistent with the route and the starting point. Each of the subproblems has specific precedence conditions, and the travel cost functions forming the additive criterion may depend on the list of pending tasks. A special two-stage procedure is introduced to apply dynamic programming as a solution method. The structure of the optimal solution is established and an algorithm based on this structure is developed. The algorithm is implemented on a personal computer and a computational experiment is carried out.
AbstractList	This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin. In particular, this condition may arise in the tool control problem for thermal cutting machines with computer numerical control (CNC): if there are long parts among workpieces, the cutting process near a narrow material boundary should start with these workpieces since such parts are subject to thermal deformations, which may potentially cause rejects. The problem statement under consideration involves two zones for part processing. The aggregate routing process in the original problem includes a starting point, a route (a permutation of indices), and a particular track consistent with the route and the starting point. Each of the subproblems has specific precedence conditions, and the travel cost functions forming the additive criterion may depend on the list of pending tasks. A special two-stage procedure is introduced to apply dynamic programming as a solution method. The structure of the optimal solution is established and an algorithm based on this structure is developed. The algorithm is implemented on a personal computer and a computational experiment is carried out.
Author	Chentsov, A. G. Chentsov, P. A.
Author_xml	– sequence: 1 givenname: A. G. surname: Chentsov fullname: Chentsov, A. G. email: chentsov@imm.uran.ru organization: Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Ural Federal University – sequence: 2 givenname: P. A. surname: Chentsov fullname: Chentsov, P. A. email: chentsov.p@mail.ru organization: Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Ural Federal University
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Cites_doi	10.1134/S0005117916110060 10.1287/opre.11.6.972 10.1080/00207540500070376 10.1134/S0005117914040122 10.1080/00207540600579615 10.1145/321105.321111
ContentType	Journal Article
Copyright	Pleiades Publishing, Ltd. 2023. ISSN 0005-1179, Automation and Remote Control, 2023, Vol. 84, No. 5, pp. 543–563. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Avtomatika i Telemekhanika, 2023, No. 5, pp. 133–164. Pleiades Publishing, Ltd. 2023.
Copyright_xml	– notice: Pleiades Publishing, Ltd. 2023. ISSN 0005-1179, Automation and Remote Control, 2023, Vol. 84, No. 5, pp. 543–563. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Avtomatika i Telemekhanika, 2023, No. 5, pp. 133–164. – notice: Pleiades Publishing, Ltd. 2023.
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References	Chentsov, A.G., On Routing Complexes of Jobs, Vestn. UdGU,Mat. Mekh. Komp’yut. Nauki, 2013, no. 1, pp. 58–82. Chentsov, A.G., Ekstremal’nye zadachi marshrutizatsii i raspredeleniya zadanii: voprosy teorii (Extremal Problems of Routing and Task Distribution: Theoretical Fundamentals), Moscow–Izhevsk: Regulyarnaya i Khaoticheskaya Dinamika, 2008. Melamed, I.I., Sergeev, S.I., and Sigal, I.Kh., The Traveling Salesman Problem, Autom. Remote Control, 1989, vol. 50, no. 9, pp. 1147–1173; no. 10, pp. 1303–1324; no. 11, pp. 1459–1479. BellmanR.Dynamic Programming Treatment of the Travelling Salesman ProblemJ. ACM19629616313570210.1145/321105.3211110106.14102 ChentsovA.G.ChentsovP.A.Routing under Constraints: Problem of Visit to MegalopolisesAutom. Remote Control20167719571974366447010.1134/S00051179161100601356.90153 KuratowskiK.MostowskiA.Set Theory1967AmsterdamNorth-Holland0165.01701 ChentsovA.G.Problem of Successive Megalopolis Traversal with the Precedence ConditionsAutom. Remote Control201475728744327589610.1134/S00051179140401221297.90169 Little, J.D., Murty, K.G., Sweeney, D.W., and Karel, C., An Algorithm for the Traveling Salesman Problem, Oper. Res., 1963, no. 11(6), pp. 972–989. Chentsov, A.G. and Chentsov, A.A., On Finding the Value of the Routing Problem with Constraints, Probl. Upravlen. Informat., 2016, no. 1, pp. 41–54. DieudonnéJ.Foundations of Modern Analysis1960New YorkAcademic0100.04201 PetuninA.A.ChentsovA.G.ChentsovP.A.Optimal Routing in Problems of Sequential Traversal of Megapolises in the Presence of ConstraintsChelyabinsk Physical and Mathematical Journal202272092331503.90071 Petunin, A.A., On Some Strategies for Constructing the Tool’s Route in the Development of Controlling Programs for Thermal Cutting Machines, Vestn. UGATU, Ser. Upravlen., Vychisl. Tekh. Informatika, 2009, vol. 13, no. 2(35), pp. 280–286. Chentsov, A.G., Chentsov, A.A., and Sesekin, A.N., Zadachi marshrutizatsii peremeshchenii s neadditivnym agregirovaniem zatrat (Movement Routing Problems with Non-additive Cost Aggregation), Moscow: Lenand, 2021. WargaJ.Optimal Control of Differential and Functional Equations1972New YorkAcademic0253.49001 CookW.J.In Pursuit of the Traveling Salesman. Mathematics at the Limits of Computation2012PrincetonPrinceton University Press1236.00007 ChentsovA.G.ChentsovP.A.Dynamic Programming in the Routing Problem: Decomposition Variant, Russian Universities Reports.Mathematics2022279512407579481 Petunin, A.A., Chentsov, A.G., and Chentsov, P.A., Optimal’naya marshrutizatsiya instrumenta mashin figurnoi listovoi rezki s chislovym programmnym upravleniem. Matematicheskie modeli i algoritmy (Optimal Tool Routing of Shaped Sheet Cutting Machines with Computer Numerical Control. Mathematical Models and Algorithms), Yekaterinburg: Ural Federal University, 2020. Frolovskii, V.D., Automating the Design of Controlling Programs for Heat Metal Cutting on Equipment with Digital Program Control, Inform. Tekhnol. Proektirovan. Proizvod., 2005, no. 4, pp. 63–66. ChentsovA.G.ChentsovP.A.An Extremal Two-Stage Routing Problem and Procedures Based on Dynamic Programming, Trudy Inst. Mat. i Mekh.Ural. Otd. Ross. Akad. Nauk202228215248 Gimadi, E.Kh. and Khachai, M.Yu., Ekstremal’nye zadachi na mnozhestvakh perestanovok (Extremal Problems on Permutation Sets), Yekaterinburg: UPI Training Center, 2016. HeldM.KarpR.A Dynamic Programming Approach to Sequencing ProblemsJ. SIAM1962101962101394930106.14103 CormenT.H.LeisersonC.E.RivestR.L.Introduction to Algorithms1990CambridgeMIT Press1158.68538 GutinG.PunnenA.The Traveling Salesman Problem and Its Variations2002BerlinSpringer0996.00026 Lee, M.-K. and Kwon, K.-B., Cutting Path Optimization in CNC Cutting Processes Using a Two-Step Genetic Algorithm, Int. J. Product. Res., 2006, no. 44, pp. 5307–5326. WangG.G.XieS.Q.Optimal Process Planning for a Combined Punch-and-Laser Cutting Machine Using Ant Colony OptimizationInt. J. Product. Res.2005432195221610.1080/00207540500070376 Lawler, E.L., Efficient Implementation of Dynamic Programming Algorithms for Sequencing Problems, Report BW106, Amsterdam: Mathematisch Centrum, 1979, pp. 1–16. G.G. Wang (2407_CR17) 2005; 43 2407_CR25 A.G. Chentsov (2407_CR11) 2022; 27 W.J. Cook (2407_CR2) 2012 A.A. Petunin (2407_CR26) 2022; 7 R. Bellman (2407_CR6) 1962; 9 2407_CR23 K. Kuratowski (2407_CR19) 1967 2407_CR24 A.G. Chentsov (2407_CR13) 2014; 75 M. Held (2407_CR7) 1962; 10 A.G. Chentsov (2407_CR14) 2016; 77 2407_CR8 2407_CR5 2407_CR9 A.G. Chentsov (2407_CR12) 2022; 28 2407_CR16 2407_CR15 J. Dieudonné (2407_CR20) 1960 2407_CR3 2407_CR4 G. Gutin (2407_CR1) 2002 2407_CR18 T.H. Cormen (2407_CR21) 1990 J. Warga (2407_CR22) 1972 2407_CR10
References_xml	– reference: ChentsovA.G.ChentsovP.A.Dynamic Programming in the Routing Problem: Decomposition Variant, Russian Universities Reports.Mathematics2022279512407579481 – reference: KuratowskiK.MostowskiA.Set Theory1967AmsterdamNorth-Holland0165.01701 – reference: BellmanR.Dynamic Programming Treatment of the Travelling Salesman ProblemJ. ACM19629616313570210.1145/321105.3211110106.14102 – reference: Chentsov, A.G., Ekstremal’nye zadachi marshrutizatsii i raspredeleniya zadanii: voprosy teorii (Extremal Problems of Routing and Task Distribution: Theoretical Fundamentals), Moscow–Izhevsk: Regulyarnaya i Khaoticheskaya Dinamika, 2008. – reference: CookW.J.In Pursuit of the Traveling Salesman. Mathematics at the Limits of Computation2012PrincetonPrinceton University Press1236.00007 – reference: ChentsovA.G.Problem of Successive Megalopolis Traversal with the Precedence ConditionsAutom. Remote Control201475728744327589610.1134/S00051179140401221297.90169 – reference: GutinG.PunnenA.The Traveling Salesman Problem and Its Variations2002BerlinSpringer0996.00026 – reference: WargaJ.Optimal Control of Differential and Functional Equations1972New YorkAcademic0253.49001 – reference: Chentsov, A.G., Chentsov, A.A., and Sesekin, A.N., Zadachi marshrutizatsii peremeshchenii s neadditivnym agregirovaniem zatrat (Movement Routing Problems with Non-additive Cost Aggregation), Moscow: Lenand, 2021. – reference: Gimadi, E.Kh. and Khachai, M.Yu., Ekstremal’nye zadachi na mnozhestvakh perestanovok (Extremal Problems on Permutation Sets), Yekaterinburg: UPI Training Center, 2016. – reference: HeldM.KarpR.A Dynamic Programming Approach to Sequencing ProblemsJ. SIAM1962101962101394930106.14103 – reference: Little, J.D., Murty, K.G., Sweeney, D.W., and Karel, C., An Algorithm for the Traveling Salesman Problem, Oper. Res., 1963, no. 11(6), pp. 972–989. – reference: Petunin, A.A., On Some Strategies for Constructing the Tool’s Route in the Development of Controlling Programs for Thermal Cutting Machines, Vestn. UGATU, Ser. Upravlen., Vychisl. Tekh. Informatika, 2009, vol. 13, no. 2(35), pp. 280–286. – reference: WangG.G.XieS.Q.Optimal Process Planning for a Combined Punch-and-Laser Cutting Machine Using Ant Colony OptimizationInt. J. Product. Res.2005432195221610.1080/00207540500070376 – reference: PetuninA.A.ChentsovA.G.ChentsovP.A.Optimal Routing in Problems of Sequential Traversal of Megapolises in the Presence of ConstraintsChelyabinsk Physical and Mathematical Journal202272092331503.90071 – reference: CormenT.H.LeisersonC.E.RivestR.L.Introduction to Algorithms1990CambridgeMIT Press1158.68538 – reference: DieudonnéJ.Foundations of Modern Analysis1960New YorkAcademic0100.04201 – reference: Lee, M.-K. and Kwon, K.-B., Cutting Path Optimization in CNC Cutting Processes Using a Two-Step Genetic Algorithm, Int. J. Product. Res., 2006, no. 44, pp. 5307–5326. – reference: ChentsovA.G.ChentsovP.A.Routing under Constraints: Problem of Visit to MegalopolisesAutom. Remote Control20167719571974366447010.1134/S00051179161100601356.90153 – reference: Frolovskii, V.D., Automating the Design of Controlling Programs for Heat Metal Cutting on Equipment with Digital Program Control, Inform. Tekhnol. Proektirovan. Proizvod., 2005, no. 4, pp. 63–66. – reference: ChentsovA.G.ChentsovP.A.An Extremal Two-Stage Routing Problem and Procedures Based on Dynamic Programming, Trudy Inst. Mat. i Mekh.Ural. Otd. Ross. Akad. Nauk202228215248 – reference: Petunin, A.A., Chentsov, A.G., and Chentsov, P.A., Optimal’naya marshrutizatsiya instrumenta mashin figurnoi listovoi rezki s chislovym programmnym upravleniem. Matematicheskie modeli i algoritmy (Optimal Tool Routing of Shaped Sheet Cutting Machines with Computer Numerical Control. Mathematical Models and Algorithms), Yekaterinburg: Ural Federal University, 2020. – reference: Chentsov, A.G., On Routing Complexes of Jobs, Vestn. UdGU,Mat. Mekh. Komp’yut. Nauki, 2013, no. 1, pp. 58–82. – reference: Chentsov, A.G. and Chentsov, A.A., On Finding the Value of the Routing Problem with Constraints, Probl. Upravlen. Informat., 2016, no. 1, pp. 41–54. – reference: Melamed, I.I., Sergeev, S.I., and Sigal, I.Kh., The Traveling Salesman Problem, Autom. Remote Control, 1989, vol. 50, no. 9, pp. 1147–1173; no. 10, pp. 1303–1324; no. 11, pp. 1459–1479. – reference: Lawler, E.L., Efficient Implementation of Dynamic Programming Algorithms for Sequencing Problems, Report BW106, Amsterdam: Mathematisch Centrum, 1979, pp. 1–16. – ident: 2407_CR9 – volume: 77 start-page: 1957 year: 2016 ident: 2407_CR14 publication-title: Autom. Remote Control doi: 10.1134/S0005117916110060 – volume-title: Foundations of Modern Analysis year: 1960 ident: 2407_CR20 – ident: 2407_CR5 doi: 10.1287/opre.11.6.972 – ident: 2407_CR16 – volume-title: The Traveling Salesman Problem and Its Variations year: 2002 ident: 2407_CR1 – ident: 2407_CR3 – volume: 43 start-page: 2195 year: 2005 ident: 2407_CR17 publication-title: Int. J. Product. Res. doi: 10.1080/00207540500070376 – ident: 2407_CR24 – volume: 75 start-page: 728 year: 2014 ident: 2407_CR13 publication-title: Autom. Remote Control doi: 10.1134/S0005117914040122 – ident: 2407_CR10 – volume-title: Optimal Control of Differential and Functional Equations year: 1972 ident: 2407_CR22 – ident: 2407_CR8 – ident: 2407_CR15 – ident: 2407_CR18 doi: 10.1080/00207540600579615 – ident: 2407_CR4 – volume: 10 start-page: 196 year: 1962 ident: 2407_CR7 publication-title: J. SIAM – volume: 28 start-page: 215 year: 2022 ident: 2407_CR12 publication-title: Ural. Otd. Ross. Akad. Nauk – volume-title: In Pursuit of the Traveling Salesman. Mathematics at the Limits of Computation year: 2012 ident: 2407_CR2 – ident: 2407_CR25 – volume-title: Introduction to Algorithms year: 1990 ident: 2407_CR21 – ident: 2407_CR23 – volume-title: Set Theory year: 1967 ident: 2407_CR19 – volume: 7 start-page: 209 year: 2022 ident: 2407_CR26 publication-title: Chelyabinsk Physical and Mathematical Journal – volume: 27 start-page: 95 year: 2022 ident: 2407_CR11 publication-title: Mathematics – volume: 9 start-page: 61 year: 1962 ident: 2407_CR6 publication-title: J. ACM doi: 10.1145/321105.321111
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Snippet	This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary...
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SubjectTerms	Algorithms CAE) and Design Calculus of Variations and Optimal Control; Optimization Computer-Aided Engineering (CAD Constraints Control Cost function Cutting equipment Decomposition Dynamic programming Mathematical functions Mathematics Mathematics and Statistics Mechanical Engineering Mechatronics Numerical controls Operations Research Optimization Permutations Personal computers Robotics Set theory System Analysis Systems Theory Traveling salesman problem Workpieces
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Title	Two-Stage Dynamic Programming in the Routing Problem with Decomposition
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