Skip to main content

Advertisement

SpringerLink
Log in

Direct solution to constrained tropical optimization problems with application to project scheduling

  • Original Paper
  • Published:
Computational Management Science Aims and scope Submit manuscript

Abstract

We examine a new optimization problem formulated in the tropical mathematics setting as a further extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent semifield by using multiplicative conjugate transposition, subject to inequality constraints. As compared to the known problems, the new one has a more general objective function and additional constraints. We provide a complete solution in an explicit form to the problem by using an approach that introduces an auxiliary variable to represent the values of the objective function, and then reduces the initial problem to a parametrized vector inequality. The minimum of the objective function is evaluated by applying the existence conditions for the solution of this inequality. A complete solution to the problem is given by solving the parametrized inequality, provided the parameter is set to the minimum value. As a consequence, we obtain solutions to new special cases of the general problem. To illustrate the application of the results, we solve a real-world problem drawn from time-constrained project scheduling, and offer a representative numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Baccelli FL, Cohen G, Olsder GJ, Quadrat JP (1993) Synchronization and linearity. Wiley series in probability and statistics. Wiley, Chichester

    Google Scholar 

  • Butkovič P (2010) Max-linear systems. Springer monographs in mathematics. Springer, London. doi:10.1007/978-1-84996-299-5

  • Cuninghame-Green R (1979) Minimax algebra. Lecture notes in economics and mathematical systems, vol 166. Springer, Berlin. doi:10.1007/978-3-642-48708-8

  • Cuninghame-Green RA (1962) Describing industrial processes with interference and approximating their steady-state behaviour. Oper Res Q 13(1):95–100. doi:10.2307/3007584

    Article  Google Scholar 

  • Demeulemeester EL, Herroelen WS (2002) Project scheduling. International series in operations research and management science, vol 49. Springer, New York. doi:10.1007/b101924

  • Elsner L, van den Driessche P (2004) Max-algebra and pairwise comparison matrices. Linear Algebra Appl 385(1):47–62. doi:10.1016/S0024-3795(03)00476-2

    Article  Google Scholar 

  • Elsner L, van den Driessche P (2010) Max-algebra and pairwise comparison matrices, II. Linear Algebra Appl 432(4):927–935. doi:10.1016/j.laa.2009.10.005

  • Engel GM, Schneider H (1975) Diagonal similarity and equivalence for matrices over groups with 0. Czechoslovak Math J 25(3):389–403

    Google Scholar 

  • Golan JS (2003) Semirings and affine equations over them. Mathematics and its applications, vol 556. Kluwer Acad. Publ, Dordrecht. doi:10.1007/978-94-017-0383-3

  • Gondran M, Minoux M (2008) Graphs, dioids and semirings. Operations research/ computer science interfaces, vol 41. Springer, New York. doi:10.1007/978-0-387-75450-5

  • Gursoy BB, Mason O, Sergeev S (2013) The analytic hierarchy process, max algebra and multi-objective optimisation. Linear Algebra Appl 438(7):2911–2928. doi:10.1016/j.laa.2012.11.020

    Article  Google Scholar 

  • Heidergott B, Olsder GJ, van der Woude J (2006) Max-plus at work. Princeton series in applied mathematics. Princeton University Press, Princeton

  • Hoffman AJ (1963) On abstract dual linear programs. Naval Res Logist Q 10(1):369–373. doi:10.1002/nav.3800100131

    Article  Google Scholar 

  • Hudec O, Zimmermann K (1993) A service points location problem with min-max distance optimality criterion. Acta Univ Carolin Math Phys 34(1):105–112

    Google Scholar 

  • Kolokoltsov VN, Maslov VP (1997) Idempotent analysis and its applications. Mathematics and its applications, vol 401. Kluwer Acad. Publ, Dordrecht. doi:10.1007/978-94-015-8901-7

  • Krivulin N (2014) A constrained tropical optimization problem: complete solution and application example. In: Litvinov GL, Sergeev SN (eds) Tropical and idempotent mathematics and applications. Contemporary mathematics, vol 616. AMS, Providence, pp 163–177. doi:10.1090/conm/616/12308

  • Krivulin N (2015a) Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl 468:211–232. doi:10.1016/j.laa.2014.06.044

    Article  Google Scholar 

  • Krivulin N (2015b) A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization 64(5):1107–1129. doi:10.1080/02331934.2013.840624

    Article  Google Scholar 

  • Krivulin N (2015c) Tropical optimization problems in project scheduling. In: Hanzálek Z, Kendall G, McCollum B, Šůcha P (eds) MISTA 2015 proceedings, MISTA, pp 492–506

  • Krivulin N (2015d) Tropical optimization problems with application to project scheduling with minimum makespan. Ann Oper Res 1–18. doi:10.1007/s10479-015-1939-9

  • Krivulin NK (2011) An extremal property of the eigenvalue for irreducible matrices in idempotent algebra and an algebraic solution to a Rawls location problem. Vestnik St Petersburg Univ Math 44(4):272–281. doi:10.3103/S1063454111040078

    Article  Google Scholar 

  • Maclagan D, Sturmfels B (2015) Introduction to tropical geometry. Graduate studies in mathematics, vol 161. AMS, Providence

  • Neumann K, Schwindt C, Zimmermann J (2003) Project scheduling with time windows and scarce resources, 2nd edn. Springer, Berlin. doi:10.1007/978-3-540-24800-2

  • Pandit SNN (1961) A new matrix calculus. J SIAM 9(4):632–639. doi:10.1137/0109052

  • Romanovskiĭ IV (1964) Asymptotic behavior of dynamic programming processes with a continuous set of states. Soviet Math Dokl 5(6):1684–1687

    Google Scholar 

  • Superville L (1978) Various aspects of max-algebra. PhD thesis, The City University of New York, New York

  • T’kindt V, Billaut JC (2006) Multicriteria scheduling, 2nd edn. Springer, Berlin. doi:10.1007/b106275

  • Vorob’ev NN (1963) The extremal matrix algebra. Soviet Math Dokl 4(5):1220–1223

    Google Scholar 

  • Zimmermann K (1992) Optimization problems with unimodal functions in max-separable constraints. Optimization 24(1–2):31–41. doi:10.1080/02331939208843777

    Article  Google Scholar 

  • Zimmermann U (1981) Linear and combinatorial optimization in ordered algebraic structures. Annals of discrete mathematics, vol 10. Elsevier, Amsterdam

Download references

Acknowledgments

This work was supported in part by the Russian Foundation for Humanities (Grant No. 16-02-00059). The author thanks two referees for valuable comments and suggestions, which have been incorporated into the final version of the manuscript.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Saint Petersburg State University, Universitetsky Ave. 28, 198504, St. Petersburg, Russia

    Nikolai Krivulin

Authors
  1. Nikolai Krivulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolai Krivulin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Krivulin, N. Direct solution to constrained tropical optimization problems with application to project scheduling. Comput Manag Sci 14, 91–113 (2017). https://doi.org/10.1007/s10287-016-0259-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10287-016-0259-0

Keywords

Search

Navigation