Abstract
Interval linear programming is used for tackling interval uncertainties in real-world systems. An arbitrary point is a feasible point to the interval linear programming model if it lies in the largest feasible region of the interval linear programming model, and it is optimal if it is an optimal solution to a characteristic model. The optimal solution set to the interval linear programming is the union of all solutions that are optimal for a characteristic model. In this paper, we review some existing methods for solving interval linear programming problems. Using these methods the interval linear programming model is transformed into two sub-models. The optimal solutions of these sub-models form the solution space of these solving methods. A part of the solution space of some of these methods may be infeasible. To eliminate the infeasible part of the solution space of above methods, several methods have been proposed. The solution space of these modified methods may contain non-optimal solutions. Two improvement methods have been proposed to remove the non-optimal solutions of the solution space of above modified methods. Finally, we introduce an improved method and its sub-models. The solution space of our method is absolutely both feasible and optimal.
Similar content being viewed by others
References
Allahdadi M, Mishmast Nehi H (2013) The optimal solutions set of the interval linear programming problems. Optim Lett 7:893–1911
Allahdadi M, Mishmast Nehi H (2015) The optimal value bounds of the objective function in the interval linear programming problem. Chiang Mai J Sci 42(2):501–511
Allahdadi M, Mishmast Nehi H, Ashayerinasab HA, Javanmard M (2016) Improving the modified interval linear programming method by new techniques. Inf Sci 339:224–236
Ashayerinasab HA, Mishmast Nehi H, Allahdadi M (2018) Solving the interval linear programming problem: a new algorithm for a general case. Expt Syst Appl 93:39–49
Beeck H (1978) Linear programming with inexact data. Technical report TUM-ISU-7830, Technical University of Munich, Munich
Cai Y, Huang G, Nie X, Li Y, Tan Q (2007) Municipal solid waste management under uncertainty: a mixed interval parameter fuzzy-stochastic robust programming approach. Environ Eng Sci 24:338–352
Chinneck JW, Ramadan K (2000) Linear programming with interval coefficients. J Oper Res Soc 51:209–220
Fan YR, Huang GH (2012) A robust two-step method for solving interval linear programming problems within an environmental management context. J Environ Inf 19:1–12
Guo P, Huang GH (2009a) Inexact fuzzy-chance-constrained two-stage mixed integer programming approach for long-term planning of municipal solid waste management Part A: methodology. J Environ Manag 91:461–470
Guo P, Huang GH (2009b) Inexact fuzzy-chance-constrained two-stage mixed integer programming approach for long-term planning of municipal solid waste management Part B: case study. J Environ Manag 91:441–460
He L, Huang GH, Zeng GM, Lu HW (2009) An interval mixed-integer semi-infinite programming method for municipal solid waste management. J Air Wast Manag Assoc 59:236–246
Hladik M (2014) How to determine basis stability in interval linear programming. Optim Lett 8:375–389
Huang GH (1996) An interval parameter water quality management model. Eng Optim 26:79–103
Huang GH, Cao MF (2011) Analysis of solution methods for interval linear programming. J Environ Inf 17:54–64
Huang GH, Moore RD (1993) Grey linear programming, its solving approach, and its application. Int J Syst Sci 24:159–172
Huang GH, Baetz BW, Patry GG (1994) Capacity planning for solid waste management systems under uncertainty. Civil Eng Syst 11:43–73
Huang GH, Baetz BW, Patry GG (1995) Grey integer programming: an application to waste management planning under uncertainty. Eur J Oper Res 83:594–620
Jansson C (1988) 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. Comput Suppl 6:33–45
Jansson C, Rump SM (1991) Rigorous solution of linear programming problems with uncertain data. Z Oper Res 35(2):87–111
Konickova J (2001) Sufficient condition of basis stability of an interval linear programming problem. ZAMM Z Angew Math Mech 81(3):677–678
Lu HW, Huang GH, Liu L, He L (2008) An interval-parameter fuzzy-stochastic programming approach for air quality management under uncertainty. Environ Eng Sci 25:895–909
Lu HW, Cao MF, Wang Y, Fan X, He L (2014) Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Appl Math Modell 38:1092–1100
Luo B, Maqsood I, Huang GH, Yin YY, Han DJ (2005) IFTSP: an inexact modeling approach for management of nonpoint source pollution through euent trading under uncertainty. Sci Total Environ 347:21–34
Machost B (1970) Numerische Behandlung des Simplexverfahrens mitintervallanalytischen Methoden. Tech. Rep. 30, Berichte der Gesellschaft fr Mathematic and Datenverarbeitung, 54 pages, Bonn
Mraz F (1996) On infimum of optimal objective function values in interval linear programming. Technical report KAM Series, Department of Applied Mathematics, Charles University, Prague, pp. 96–337
Rex J, Rohn J (1998) Sufficient conditions for regularity and singularity of interval matrices. SIAM J Matrix Anal Appl 20(2):437–445
Rohn J (1993a) Cheap and tight bound: the recent result by E. Hansen can be made more efficient. Interv Comput 4:13–21
Rohn J (1993b) Stability of the optimal basis of a linear program under uncertainty. Oper Res Lett 13(1):9–12
Tong SC (1994) Interval number, fuzzy number linear programming. Fuzzy Sets Syst 66:301–306
Wang X, Huang G (2014) Violation analysis on two-step method for interval linear programming. Inf Sci 281:85–96
Zhou F, Huang GH, Chen G, Guo H (2009) Enhanced-interval linear programming. Eur J Oper Res 199:323–333
Acknowledgements
The authors would like to thank to the anonymous referees for their constructive comments and suggestions that have helped to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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, 1205–1229 (2020). https://doi.org/10.1007/s12351-018-0383-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12351-018-0383-4