### Optimal values range of interval polynomial programming problems

Document Type : Research Paper

Authors

Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

Uncertainty exists in many real-life engineering and mechanical problems. Here, we assume that uncertainties are caused by intervals of real numbers. In this paper, we consider the interval nonlinear programming (INLP) problems where the objective function and constraints include interval coefficients. So that the variables are deterministic and sign-restricted. Additionally, the constraints are considered in the form of inequalities. A basic task in INLP is calculating the optimal values range of objective function, which may be computationally very expensive. However, if the boundary functions are available, the problems become much easier to solve. By making these assumptions, an efficient method is proposed to compute the optimal values range using two classic nonlinear problems. Then, the optimal values range are obtained by direct inspection for a special kind of interval polynomial programming (IPP) problems. Two numerical examples are given to verify the effectiveness of the proposed method.

Keywords

[1] E. Garajov´a, M. Hlad´ık and M. Rada, Interval linear programming under transformations: Optimal solutions and
optimal value range, Cent. Eur. J. Oper. Res. 27 (2019) 601–614.
[2] M. Hlad´ık, Optimal value bounds in nonlinear programming with interval data, TOP 19 (2011) 93–106.
[3] C. Jiang, X. Han, GR. Liu and GP. Liu, A nonlinear interval number programming method for uncertain
optimization problems, Eur. J. Oper. Res. 188(1) (2008) 1–13.
[4] C. Jiang, ZG. Zhang, QF. Zhang, X. Han, HC. Xie and J. Liu, A new nonlinear interval programming method
for uncertain problems with dependent interval variables, Eur. J. Oper. Res., 238 (2014) 245-253.
[5] P. Kumar and G. Panda, Solving nonlinear interval optimization problem using stochastic programming technique,
Opsearch 54 (2017) 752–765.
[6] VI. Levin, Nonlinear optimization under interval uncertainty, Cybern. Syst. Anal. 35(2) (1999) 297–306.
[7] VI. Levin, Optimization in terms of interval uncertainty: The determinization method, Autom. Control Comput.
Sci. 46(4) (2012) 157-163.
[8] X. Liua, Z. Zhanga and L. Yina, A multi-objective optimization method for uncertain structures based on nonlinear
interval number programming method, Mech. Based Des. Struct. Mach. 45(1) (2017) 25–42.
[9] Y. Li and YL. Xu, Increasing accuracy in the interval analysis by the improved format of interval extension based
on the first order Taylor series, Mech. Syst. Sig. Process. 104 (2018) 744–757.
[10] A. Mostafaee and M. Hlad´ık, Optimal value bounds in interval fractional linear programming and revenue efficiency measuring, Cent. Eur. J. Oper. Res. 28 (2020) 963–981.
[11] R.E. Moore, Methods and Applications of Interval Analysis, Philadelphia, 1979.
###### Volume 13, Issue 1March 2022Pages 1917-1929
• Receive Date: 26 June 2021
• Accept Date: 10 November 2021
• First Publish Date: 18 November 2021