An improved three-step method for solving the interval linear programming problems

Feasibility condition, which ensures that the solution space does not violate any constraints, and optimality condition, which guarantees that all points of the solution space are optimal, are very significant conditions for the solution space of interval linear programming (ILP) problems. Among existing methods for ILP problems, the best-worst cases (BWC) method and two-step method (TSM) do not ensure feasibility condition, while the modified ILP (MILP), robust TSM (RTSM), improved TSM (ITSM) and three-step method (ThSM) guarantee feasibility condition, their solution spaces may not be completely optimal. Based on analyses of the above-mentioned methods on the point of view of feasibility and optimality conditions, we propose the improved ThSM (IThSM), which ensures both feasibility and optimality conditions, for ILP problems via introducing an extra step to optimality.

[1]  Jiri Rohn,et al.  Linear Programming with Inexact Data is NP‐Hard , 1998 .

[2]  J. Rohn Forty necessary and sufficient conditions for regularity of interval matrices: A survey , 2009 .

[3]  Jana Koníckocá,et al.  Sufficient condition of basis stability of an interval linear programming problem , 2001 .

[4]  Guohe Huang,et al.  Analysis of Solution Methods for Interval Linear Programming , 2011 .

[5]  G. Huang,et al.  A Robust Two-Step Method for Solving Interval Linear Programming Problems within an Environmental Management Context , 2012 .

[6]  G. Huang,et al.  Grey integer programming: An application to waste management planning under uncertainty , 1995 .

[7]  John W. Chinneck,et al.  Linear programming with interval coefficients , 2000, J. Oper. Res. Soc..

[8]  Guohe Huang,et al.  Violation analysis on two-step method for interval linear programming , 2014, Inf. Sci..

[9]  Guohe Huang,et al.  A GREY LINEAR PROGRAMMING APPROACH FOR MUNICIPAL SOLID WASTE MANAGEMENT PLANNING UNDER UNCERTAINTY , 1992 .

[10]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[11]  Jiri Rohn,et al.  Stability of the optimal basis of a linear program under uncertainty , 1993, Oper. Res. Lett..

[13]  Mehdi Allahdadi,et al.  Improving the modified interval linear programming method by new techniques , 2016, Inf. Sci..

[14]  Gordon H. Huang,et al.  Enhanced-interval linear programming , 2009, Eur. J. Oper. Res..

[15]  Milan Hladík,et al.  How to determine basis stability in interval linear programming , 2012, Optimization Letters.

[16]  Son-Lin Nie,et al.  A dual-interval vertex analysis method and its application to environmental decision making under uncertainty , 2010, Eur. J. Oper. Res..

[17]  M. Fiedler,et al.  Linear Optimization Problems with Inexact Data , 2006 .

[18]  Guohe Huang,et al.  Grey linear programming, its solving approach, and its application , 1993 .

[19]  Christian Jansson,et al.  A self-validating method for solving linear programming problems with interval input data , 1988 .

[20]  Mehdi Allahdadi,et al.  Solving the interval linear programming problem: A new algorithm for a general case , 2018, Expert Syst. Appl..

[21]  C. Jansson,et al.  Rigorous solution of linear programming problems with uncertain data , 1991, ZOR Methods Model. Oper. Res..

[22]  Mehdi Allahdadi,et al.  The optimal solution set of the interval linear programming problems , 2012, Optimization Letters.

[23]  Ching-Pin Tung,et al.  Interval number fuzzy linear programming for climate change impact assessments of reservoir active storage , 2009, Paddy and Water Environment.

[24]  Jiri Rohn,et al.  Sufficient Conditions for Regularity and Singularity of Interval Matrices , 1999, SIAM J. Matrix Anal. Appl..