A note on resolving the inconsistency of one-sided max-plus linear equations

Pingke Li

Kybernetika (2019)

  • Volume: 55, Issue: 3, page 531-539
  • ISSN: 0023-5954

Abstract

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When a system of one-sided max-plus linear equations is inconsistent, its right-hand side vector may be slightly modified to reach a consistent one. It is handled in this note by minimizing the sum of absolute deviations in the right-hand side vector. It turns out that this problem may be reformulated as a mixed integer linear programming problem. Although solving such a problem requires much computational effort, it may propose a solution that just modifies few elements of the right-hand side vector, which is a desired property in some practical situations.

How to cite

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Li, Pingke. "A note on resolving the inconsistency of one-sided max-plus linear equations." Kybernetika 55.3 (2019): 531-539. <http://eudml.org/doc/294573>.

@article{Li2019,
abstract = {When a system of one-sided max-plus linear equations is inconsistent, its right-hand side vector may be slightly modified to reach a consistent one. It is handled in this note by minimizing the sum of absolute deviations in the right-hand side vector. It turns out that this problem may be reformulated as a mixed integer linear programming problem. Although solving such a problem requires much computational effort, it may propose a solution that just modifies few elements of the right-hand side vector, which is a desired property in some practical situations.},
author = {Li, Pingke},
journal = {Kybernetika},
keywords = {max-plus algebra; max-plus linear systems; mixed integer programming},
language = {eng},
number = {3},
pages = {531-539},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on resolving the inconsistency of one-sided max-plus linear equations},
url = {http://eudml.org/doc/294573},
volume = {55},
year = {2019},
}

TY - JOUR
AU - Li, Pingke
TI - A note on resolving the inconsistency of one-sided max-plus linear equations
JO - Kybernetika
PY - 2019
PB - Institute of Information Theory and Automation AS CR
VL - 55
IS - 3
SP - 531
EP - 539
AB - When a system of one-sided max-plus linear equations is inconsistent, its right-hand side vector may be slightly modified to reach a consistent one. It is handled in this note by minimizing the sum of absolute deviations in the right-hand side vector. It turns out that this problem may be reformulated as a mixed integer linear programming problem. Although solving such a problem requires much computational effort, it may propose a solution that just modifies few elements of the right-hand side vector, which is a desired property in some practical situations.
LA - eng
KW - max-plus algebra; max-plus linear systems; mixed integer programming
UR - http://eudml.org/doc/294573
ER -

References

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  1. Allamigeon, X., Benchimol, P., Gaubert, S., 10.1137/130936464, SIAM J Discrete Math. 29 (2015), 751-795. MR3336300DOI10.1137/130936464
  2. Aminu, A., Butkovič, P., 10.1093/imaman/dpq020, IMA J. Management Math. 23 (2012), 41-66. MR2874172DOI10.1093/imaman/dpq020
  3. Baccelli, F., Cohen, G., Olsder, G. J., Quadrat, J.-P., Synchronization and Linearity: An Algebra for Discrete Event Systems., Wiley, Chichester 1992. MR1204266
  4. Butkovič, P., 10.1007/978-1-84996-299-5, Springer, Berlin 2010. Zbl1202.15032MR2681232DOI10.1007/978-1-84996-299-5
  5. Butkovič, P., Aminu, A., 10.1093/imaman/dpn029, IMA J. Management Math. 20 (2009), 233-249. MR2511497DOI10.1093/imaman/dpn029
  6. Cechlárová, K., 10.1016/s0024-3795(00)00060-4, Linear Algebra Appl. 310 (2000), 123-128. MR1753171DOI10.1016/s0024-3795(00)00060-4
  7. Cechlárová, K., Cuninghame-Green, R. A., Soluble approximation of linear systems in max-plus algebra., Kybernetika 39 (2003), 137-141. MR1996552
  8. Cechlárová, K., Diko, P., 10.1016/s0024-3795(98)10248-3, Linear Algebra Appl. 290 (1999), 267-273. MR1672997DOI10.1016/s0024-3795(98)10248-3
  9. Cimler, R., Gavalec, M., Zimmermann, K., 10.1016/j.fss.2017.05.004, Fuzzy Sets and Systems 341 (2018), 113-122. MR3787606DOI10.1016/j.fss.2017.05.004
  10. Cuninghame-Green, R. A., Cechlárová, K., 10.1016/0165-0114(94)00252-3, Fuzzy Sets and Systems 71 (1995), 227-239. MR1329610DOI10.1016/0165-0114(94)00252-3
  11. Gondran, M., Minoux, M., Graphs, Dioids and Semirings: New Models and Algorithms., Springer, New York 2008. Zbl1201.16038MR2389137
  12. Heidergott, B., Olsder, G. J., Woude, J. van der, 10.1515/9781400865239, Princeton University Press, Princeton 2005. MR2188299DOI10.1515/9781400865239
  13. Krivulin, N., 10.1080/02331934.2013.840624, Optimization 64 (2015), 1107-1129. MR3316792DOI10.1080/02331934.2013.840624
  14. Li, P., Fang, S.-C., Chebyshev approximation of inconsistent fuzzy relational equations with Max- T composition., In: Fuzzy Optimization (W. A. Lodwick and J. Kacprzyk, eds.), Springer, Berlin 2010, pp. 109-124. MR2722985
  15. Tharwat, A., Zimmermann, K., Some optimization problems on solubility sets of separable Max-Min equations and inequalities., Acta Univ. Carolinae. Math. Phys. 38 (1997), 45-57. MR1614039
  16. Zimmermann, K., Optimization problems under max-min separable equation and inequality constraints., In: Decision Making and Optimization: Special Matrices and Their Applications in Economics and Management (M. Gavalec, J. Ramík, and K. Zimmermann, eds.), Springer, Cham 2015, pp. 119-161. 

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