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Irreducible bin packing and normality in routing open shop

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Abstract

The open shop is a classical scheduling problem known since 1976, which can be described as follows. A number of jobs have to be processed by a given set of machines, each machine should perform an operation on every job, and the processing times of all the operations are given. One has to construct a schedule to perform all the operations to minimize finish time also known as the makespan. The open shop problem is known to be NP-hard for three and more machines, while is polynomially solvable in the case of two machines. We consider the routing open shop problem, being a generalization of both the open shop problem and the metric traveling salesman problem. In this setting, jobs are located at nodes of a transportation network and have to be processed by mobile machines, initially located at a predefined depot. Machines have to process all the jobs and return to the depot to minimize makespan. A feasible schedule is referred to as normal if its makespan coincides with the standard lower bound. We introduce the Irreducible Bin Packing decision problem, use it to describe new sufficient conditions of normality for the two machine problem, and discuss the possibility to extend these results on the problem with three and more machines. To that end, we present two new computer-aided optima localization results.

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References

  1. Assmann, S.F., Johnson, D.S., Kleitman, D.J., Leung, J.Y.T.: On the dual of the one-dimensional bin packing problem. Journal of Algorithms 5(4), 502–525 (1984). https://doi.org/10.1016/0196-6774(84)90004-X

    Article  MathSciNet  Google Scholar 

  2. Averbakh, I., Berman, O.: Routing two-machine flowshop problems on networks with special structure. Transp. Sci. 30(4), 303–314 (1996). https://doi.org/10.1287/trsc.30.4.303

    Article  Google Scholar 

  3. Averbakh, I., Berman, O.: A simple heuristic for m-machine flow-shop and its applications in routing-scheduling problems. Oper. Res. 47(1), 165–170 (1999). https://doi.org/10.1287/opre.47.1.165

    Article  Google Scholar 

  4. Averbakh, I., Berman, O., Chernykh, I.: A 6/5-approximation algorithm for the two-machine routing open-shop problem on a two-node network. Eur. J. Oper. Res. 166(1), 3–24 (2005). https://doi.org/10.1016/j.ejor.2003.06.050

    Article  MathSciNet  Google Scholar 

  5. Averbakh, I., Berman, O., Chernykh, I.: The routing open-shop problem on a network: complexity and approximation. Eur. J. Oper. Res. 173(2), 531–539 (2006). https://doi.org/10.1016/j.ejor.2005.01.034

    Article  MathSciNet  Google Scholar 

  6. Bennell, J.A., Oliveira, J.: A tutorial in irregular shape packing problems. Journal of the Operational Research Society 60, S93–S105 (2009). https://doi.org/10.1057/jors.2008.169

    Article  Google Scholar 

  7. Bennell, J.A., Song, X.: A beam search implementation for the irregular shape packing problem. J. Heuristics 16(2), 167–188 (2010). https://doi.org/10.1007/s10732-008-9095-x

    Article  Google Scholar 

  8. Birgin, E.G., Martínez, J.M., Ronconi, D.P.: Optimizing the packing of cylinders into a rectangular container: a nonlinear approach. Eur. J. Oper. Res. 160, 19–33 (2005). https://doi.org/10.1016/j.ejor.2003.06.018

    Article  Google Scholar 

  9. Boyar, J., Epstein, L., Favrholdt, L.M., Kohrt, J.S., Larsen, K.S., Pedersen, M.M., Wøhlk, S.: The maximum resource bin packing problem. Theor. Comput. Sci. 362(1), 127–139 (2006). https://doi.org/10.1016/j.tcs.2006.06.001

    Article  MathSciNet  Google Scholar 

  10. Brucker, P., Knust, S., Edwin Cheng, T., Shakhlevich, N.: Complexity results for flow-shop and open-shop scheduling problems with transportation delays. Annals of Operations Research (129), 81–106. https://doi.org/10.1023/b:anor.0000030683.64615.c8(2004)

  11. Chernykh, I.: Routing open shop with unrelated travel times. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) Discrete Optimization and Operations Research — 9th International Conference DOOR 2016, Vladivostok, Russia, September 19-23, 2016, Proceedings, Lecture Notes in Computer Science. 10.1007/978-3-319-44914-2_22, vol. 9869, pp 272–283 (2016)

  12. Chernykh, I.: Sufficient conditions of polynomial solvability of the two-machine preemptive routing open shop on a tree. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds.) Optimization and Applications. 9th International Conference, OPTIMA 2018, Petrovac, Montenegro, October 1–5, 2018, Revised Selected Papers, Communications in Computer and Information Science, vol. 974, pp. 97–110. Springer International Publishing. https://doi.org/10.1007/978-3-030-10934-9_7 (2019)

  13. Chernykh, I., Kononov, A.V., Sevastyanov, S.: Efficient approximation algorithms for the routing open shop problem. Computers & Operations Research 40 (3), 841–847 (2013). https://doi.org/10.1016/j.cor.2012.01.006

    Article  MathSciNet  Google Scholar 

  14. Chernykh, I., Krivonogiva, O.: Optima localization for the two-machine routing open shop on a tree (in Russian. Submitted to Diskretnyj Analiz i Issledovanie Operacij (2020)

  15. Chernykh, I., Krivonogova, O.: Efficient algorithms for the routing open shop with unrelated travel times on cacti. In: JaćimoviĆ, M., Khachay, M., Malkova, V., Posypkin, M. (eds.) Optimization and Applications. 10th International Conference, OPTIMA 2019, Petrovac, Montenegro, September 30 – October 4, 2019, Revised Selected Papers, Communications in Computer and Information Science, vol. 1145, pp. 1–15. Springer International Publishing. https://doi.org/10.1007/978-3-030-38603-0_1 (2020)

  16. Chernykh, I., Krivonogova, O.: On the optima localization for the three-machine routing open shop. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research, 19th International Conference, MOTOR 2020, Novosibirsk, Russia, July 6–10, 2020, Proceedings, Lecture Notes in Computer Science, vol. 12095, pp. 274–288. Springer International Publishing. 10.1007/978-3-030-49988-4_19 (2020)

  17. Chernykh, I., Kuzevanov, M.: Sufficient condition of polynomial solvability of two-machine routing open shop with preemption allowed (in Russian). Intellektual’nye sistemy 17(1-4), 552–556 (2013)

    Google Scholar 

  18. Chernykh, I., Lgotina, E.: The 2-machine routing open shop on a triangular transportation network. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) Discrete Optimization and Operations Research — 9th International Conference, DOOR 2016, Vladivostok, Russia, September 19-23, 2016, Proceedings, Lecture Notes in Computer Science, vol. 9869, pp. 284–297. https://doi.org/10.1007/978-3-319-44914-2_23 (2016)

  19. Chernykh, I., Lgotina, E.: How the difference in travel times affects the optima localization for the routing open shop. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research, MOTOR 2019, Ekaterinburg, Russia, July 8–12, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11548, pp. 187–201. https://doi.org/10.1007/978-3-030-22629-9_14 (2019)

  20. Chernykh, I., Lgotina, E.: Two-machine routing open shop on a tree: instance reduction and efficiently solvable subclass. Optimization Methods and Software. https://doi.org/10.1080/10556788.2020.1734802 (2020)

  21. Christensen, H.I., Khan, A., Pokutta, S., Tetali, P.: Multidimensional bin packing and other related problems: a survey. https://people.math.gatech.edu/~tetali/PUBLIS/CKPT.pdf (2016)

  22. Christofides, N.: Graph Theory: an Algorithmic Approach. Academic Press, New York (1975)

    MATH  Google Scholar 

  23. Coffman, E.G. Jr., Leung, J.Y., Ting, D.W.: Bin packing: maximizing the number of pieces packed. Acta Inf. 9(3), 263–271 (1978). https://doi.org/10.1007/BF00288885

    Article  MathSciNet  Google Scholar 

  24. Coffman, E.G. Jr., Csirik, J., Galambos, G., Martello, S., Vigo, D.: Bin Packing Approximation Algorithms: Survey and Classification, pp 455–531. Springer, New York (2013). https://doi.org/10.1007/978-1-4419-7997-1_35

    Google Scholar 

  25. Epstein, L., Favrholdt, L.M., Kohrt, J.S.: Comparing online algorithms for bin packing problems. J. Sched. 15(1), 13–21 (2012). https://doi.org/10.1007/s10951-009-0129-5

    Article  MathSciNet  Google Scholar 

  26. Epstein, L., Imreh, C., Levin, A.: Bin covering with cardinality constraints. Discret. Appl. Math. 161, 1975–1987 (2013). https://doi.org/10.1016/j.dam.2013.03.020

    Article  MathSciNet  Google Scholar 

  27. Epstein, L., Levin, A.: On bin packing with conflicts. SIAM J. Optim. 19, 1270–1298 (2008). https://doi.org/10.1137/060666329

    Article  MathSciNet  Google Scholar 

  28. Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA (1979)

  29. Golovachev, M., Pyatkin, A.V.: Routing open shop with two nodes, unit processing times and equal number of jobs and machines. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research, MOTOR 2019, Ekaterinburg, Russia, July 8–12, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11548, pp. 264–276. https://doi.org/10.1007/978-3-030-22629-9_19 (2019)

  30. Gonzalez, T.F., Sahni, S.: Open shop scheduling to minimize finish time. J. ACM 23(4), 665–679 (1976). https://doi.org/10.1145/321978.321985

    Article  MathSciNet  Google Scholar 

  31. Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. Theor. Comput. Sci. 306, 543–551 (2003). https://doi.org/10.1016/s0304-3975(03)00363-3

    Article  MathSciNet  Google Scholar 

  32. Karp, R.M.: Reducibility Among Combinatorial Problems, pp 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9

    MATH  Google Scholar 

  33. Kleinau, U.: Two-machine shop scheduling problems with batch processing. Math. Comput. Model. 17(6), 55–66 (1993). https://doi.org/10.1016/0895-7177(93)90196-6

    Article  MathSciNet  Google Scholar 

  34. Kononov, A.: On the routing open shop problem with two machines on a two-vertex network. J. Appl. Ind. Math. 6(3), 318–331 (2012). https://doi.org/10.1134/s1990478912030064

    Article  MathSciNet  Google Scholar 

  35. Kononov, A.: \(O(\log m)\)-approximation for the routing open shop problem. RAIRO-Oper. Res. 49, 383–391 (2015). https://doi.org/10.1051/ro/2014051

    Article  MathSciNet  Google Scholar 

  36. Kononov, A., Sevastianov, S., Tchernykh, I.: When difference in machine loads leads to efficient scheduling in open shops. Ann. Oper. Res. 92, 211–239 (1999). https://doi.org/10.1023/a:1018986731638

    Article  MathSciNet  Google Scholar 

  37. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Chapter 9. sequencing and scheduling: algorithms and complexity. In: Logistics of Production and Inventory, Handbooks in Operations Research and Management Science, vol. 4, pp. 445–522. Elsevier. https://doi.org/10.1016/S0927-0507(05)80189-6(1993)

  38. Levin, M.S.: Bin packing problems (promising models and examples). Journal on Communications Technology and Electronics 63, 655–666 (2018). https://doi.org/10.1134/s1064226918060177

    Article  Google Scholar 

  39. Lodi, A., Martello, S., Monaci, M.: Two-dimensional packing problems: a survey. Eur. J. Oper. Res. 141(2), 241–252 (2002). https://doi.org/10.1016/S0377-2217(02)00123-6

    Article  MathSciNet  Google Scholar 

  40. Lushchakova, I., Soper, A., Strusevich, V.: Transporting jobs through a two-machine open shop. Nav. Res. Logist. 56, 1–18 (2009). https://doi.org/10.1002/nav.20323

    Article  MathSciNet  Google Scholar 

  41. Martello, S., Pisinger, D., Vigo, D.: The three-dimensional bin packing problem. Oper. Res. 48, 256–267 (2000). https://doi.org/10.1287/opre.48.2.256.12386

    Article  MathSciNet  Google Scholar 

  42. Muritiba, A.E.F., Iori, M., Malaguti, E., Toth, P.: Algorithms for the bin packing problem with conflicts. INFORMS J. Comput. 22, 401–415 (2010). https://doi.org/10.1287/ijoc.1090.0355

    Article  MathSciNet  Google Scholar 

  43. Pyatkin, A., Chernykh, I.: The open shop problem with routing at a two-node network and allowed preemption. J. Appl. Ind. Math. 6(3), 346–354 (2012). https://doi.org/10.1134/s199047891203009x

    Article  MathSciNet  Google Scholar 

  44. Seiden, S.S., van Stee, R., Epstein, L.: New bounds for variable sized online bin packing. SIAM J. Comput. 32, 455–469 (2003). https://doi.org/10.1137/s0097539702412908

    Article  MathSciNet  Google Scholar 

  45. Sevastianov, S.V., Woeginger, G.J.: Makespan minimization in open shops: a polynomial time approximation scheme. Math. Program. 82(1-2, Ser. B), 191–198 (1998). https://doi.org/10.1007/BF01585871

    Article  MathSciNet  Google Scholar 

  46. Sevastyanov, S., Tchernykh, I.: Computer-aided way to prove theorems in scheduling. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds.) Algorithms — ESA’98, Lecture Notes in Computer Science, vol. 1461, pp. 502–513. https://doi.org/10.1007/3-540-68530-8_42 (1998)

  47. Strusevich, V.: A heuristic for the two-machine open-shop scheduling problem with transportation times. Discret. Appl. Math. 93(2), 287–304 (1999). https://doi.org/10.1016/S0166-218X(99)00115-8

    Article  MathSciNet  Google Scholar 

  48. van Bevern, R., Pyatkin, A.V.: Completing partial schedules for open shop with unit processing times and routing. In: Proceedings of the 11th International Computer Science Symposium in Russia (CSR’16), Lecture Notes in Computer Science, vol. 9691, pp. 73–87. https://doi.org/10.1007/978-3-319-34171-2_6 (2016)

  49. van Bevern, R., Pyatkin, A.V., Sevastyanov, S.V.: An algorithm with parameterized complexity of constructing the optimal schedule for the routing open shop problem with unit execution times. Siberian Electronic Mathematical Reports 16, 42–84 (2019). https://doi.org/10.33048/semi.2019.16.003

    MathSciNet  MATH  Google Scholar 

  50. Williamson, D.P., Hall, L.A., Hoogeveen, J.A., Hurkens, C.A.J., Lenstra, J.K., Sevast’janov, S.V., Shmoys, D.B.: Short shop schedules. Oper. Res. 45(2), 288–294 (1997). https://doi.org/10.1287/opre.45.2.288

    Article  MathSciNet  Google Scholar 

  51. Yu, V.F., Lin, S., Chou, S.: The museum visitor routing problem. Appl. Math. Comput. 216(3), 719–729 (2010). https://doi.org/10.1016/j.amc.2010.01.066

    MATH  Google Scholar 

  52. Yu, W., Zhang, G.: Improved approximation algorithms for routing shop scheduling. Lect. Notes Comput. Sci 7074, 30–39 (2011). https://doi.org/10.1007/978-3-642-25591-5_5

    Article  MathSciNet  Google Scholar 

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  1. Sobolev Institute of Mathematics, Koptyug ave. 4, Novosibirsk, 630090, Russia

    Ilya Chernykh & Artem Pyatkin

  2. Novosibirsk State University, Koptyug ave. 2, Novosibirsk, 630090, Russia

    Ilya Chernykh & Artem Pyatkin

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  1. Ilya Chernykh
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This research was supported by the Russian Foundation for Basic Research, project 20-01-00045, and by the program of fundamental scientific researches of the SB RAS, project 0314-2019-0014.

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Chernykh, I., Pyatkin, A. Irreducible bin packing and normality in routing open shop. Ann Math Artif Intell 89, 899–918 (2021). https://doi.org/10.1007/s10472-021-09759-x

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