Skip to main content
SpringerLink
Log in

Outcome Range Problem in Interval Linear Programming: An Exact Approach

  • Conference paper
  • First Online:
Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12482))

  • 572 Accesses

Abstract

Interval programming provides a mathematical model for uncertain optimization problems, in which the input data can be perturbed independently within the given lower and upper bounds. This paper discusses the recently proposed outcome range problem in the context of interval linear programming. The motivation for the outcome range problem is to assess further impacts and consequences of optimal decision making, modeled in the program by an additional linear outcome function. Specifically, the goal is to compute a lower and an upper bound on the value of the given outcome function over the optimal solution set of the interval program. In this paper, we focus mainly on programs with interval coefficients in the objective function and the right-hand-side vector. For this special class of interval programs, we design an algorithm for computing the outcome range exactly, based on complementary slackness and guided basis enumeration. Finally, we perform a series of computational experiments to evaluate the performance of the proposed method.

E. Garajová and M. Rada were supported by the Czech Science Foundation under Grant P403-20-17529S. M. Hladík was supported by the Czech Science Foundation under Grant P403-18-04735S. E. Garajová and M. Hladík were also supported by the Charles University project GA UK No. 180420.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Allahdadi, M., Mishmast Nehi, H.: The optimal solution set of the interval linear programming problems. Optim. Lett. 7(8), 1893–1911 (2012). https://doi.org/10.1007/s11590-012-0530-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Chaiyakan, S., Thipwiwatpotjana, P.: Mean Absolute deviation portfolio frontiers with interval-valued returns. In: Seki, H., Nguyen, C.H., Huynh, V.-N., Inuiguchi, M. (eds.) IUKM 2019. LNCS (LNAI), vol. 11471, pp. 222–234. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-14815-7_19

    Chapter  Google Scholar 

  3. Corsaro, S., Marino, M.: Interval linear systems: the state of the art. Comput. Stat. 21(2), 365–384 (2006). https://doi.org/10.1007/s00180-006-0268-5

    Article  MathSciNet  Google Scholar 

  4. D’Ambrosio, C., Gentili, M., Cerulli, R.: The optimal value range problem for the Interval (immune) transportation problem. Omega 95, 102059 (2020). https://doi.org/10.1016/j.omega.2019.04.002

    Article  Google Scholar 

  5. Garajová, E., Hladík, M.: On the optimal solution set in interval linear programming. Comput. Optim. Appl. 72(1), 269–292 (2018). https://doi.org/10.1007/s10589-018-0029-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Garajová, E., Hladík, M., Rada, M.: Interval linear programming under transformations: optimal solutions and optimal value range. CEJOR 27(3), 601–614 (2018). https://doi.org/10.1007/s10100-018-0580-5

    Article  MathSciNet  MATH  Google Scholar 

  7. Gurobi Optimization, LLC: Gurobi optimizer reference manual (2020). http://www.gurobi.com

  8. Hladík, M.: Optimal value range in interval linear programming. Fuzzy Optim. Decis. Making 8(3), 283–294 (2009). https://doi.org/10.1007/s10700-009-9060-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Hladík, M.: An interval linear programming contractor. In: Ramík, J., Stavárek, D. (eds.) Proceedings 30th International Conference on Mathematical Methods in Economics 2012, Karviná, Czech Republic, pp. 284–289 (Part I), Silesian University in Opava, School of Business Administration in Karviná, September 2012

    Google Scholar 

  10. Hladík, M.: The worst case finite optimal value in interval linear programming. Croatian Oper. Res. Rev. 9(2), 245–254 (2018). https://doi.org/10.17535/crorr.2018.0019

  11. Inuiguchi, M.: Enumeration of all possibly optimal vertices with possible optimality degrees in linear programming problems with a possibilistic objective function. Fuzzy Optim. Decis. Making 3(4), 311–326 (2004). https://doi.org/10.1007/s10700-004-4201-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Jansson, C., Rump, S.M.: Rigorous solution of linear programming problems with uncertain data. ZOR - Methods Models Oper. Res. 35(2), 87–111 (1991). https://doi.org/10.1007/BF02331571

    Article  MathSciNet  MATH  Google Scholar 

  13. Jansson, C.: A self-validating method for solving linear programming problems with interval input data. In: Kulisch, U., Stetter, H.J. (eds.) Scientific Computation with Automatic Result Verification. Computing Supplementum, pp. 33–45. Springer, Vienna (1988). https://doi.org/10.1007/978-3-7091-6957-5_4

  14. Lu, H.W., Cao, M.F., Wang, Y., Fan, X., He, L.: Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Appl. Math. Model. 38(3), 1092–1100 (2014). https://doi.org/10.1016/j.apm.2013.07.030

    Article  MathSciNet  MATH  Google Scholar 

  15. MathWorks: MATLAB fmincon. https://www.mathworks.com/help/optim/ug/fmincon.html

  16. Mishmast Nehi, H., Ashayerinasab, H.A., Allahdadi, M.: Solving methods for interval linear programming problem: a review and an improved method. Oper. Res. Int. J. 20(3), 1205–1229 (2018). https://doi.org/10.1007/s12351-018-0383-4

    Article  Google Scholar 

  17. Mohammadi, M., Gentili, M.: Bounds on the worst optimal value in interval linear programming. Soft. Comput. 23(21), 11055–11061 (2018). https://doi.org/10.1007/s00500-018-3658-z

    Article  MATH  Google Scholar 

  18. Mohammadi, M., Gentili, M.: The outcome range problem. arXiv:1910.05913 [math], April 2020. http://arxiv.org/abs/1910.05913

  19. Mráz, F.: Calculating the exact bounds of optimal values in LP with interval coefficients. Ann. Oper. Res. 81, 51–62 (1998). https://doi.org/10.1023/A:1018985914065

    Article  MathSciNet  MATH  Google Scholar 

  20. Neumaier, A.: Interval Methods for Systems of Equations. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1991). https://doi.org/10.1017/CBO9780511526473

  21. Rohn, J.: Interval linear programming. In: Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K. (eds.) Linear Optimization Problems with Inexact Data, Boston, MA, US, pp. 79–100. Springer (2006). https://doi.org/10.1007/0-387-32698-7_3

Download references

Acknowledgements

The authors would like to thank M. Mohammadi and M. Gentili for providing the test instances and results of the fmincon method.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, Malostranské nám. 25, Prague, Czech Republic

    Elif Garajová & Milan Hladík

  2. Department of Econometrics, University of Economics, nám. W. Churchilla 4, Prague, Czech Republic

    Elif Garajová, Miroslav Rada & Milan Hladík

  3. Department of Financial Accounting and Auditing, University of Economics, nám. W. Churchilla 4, Prague, Czech Republic

    Miroslav Rada

Authors
  1. Elif Garajová
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miroslav Rada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milan HladĂ­k
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elif Garajová .

Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan

    Van-Nam Huynh

  2. University of Hyogo, Kobe, Japan

    Tomoe Entani

  3. Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani, Thailand

    Chawalit Jeenanunta

  4. Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan

    Masahiro Inuiguchi

  5. Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani, Thailand

    Pisal Yenradee

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Garajová, E., Rada, M., Hladík, M. (2020). Outcome Range Problem in Interval Linear Programming: An Exact Approach. In: Huynh, VN., Entani, T., Jeenanunta, C., Inuiguchi, M., Yenradee, P. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2020. Lecture Notes in Computer Science(), vol 12482. Springer, Cham. https://doi.org/10.1007/978-3-030-62509-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-62509-2_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-62508-5

  • Online ISBN: 978-3-030-62509-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation