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2021 | OriginalPaper | Chapter

1. Introduction

Authors : Chao Jiang, Xu Han, Huichao Xie

Published in: Nonlinear Interval Optimization for Uncertain Problems

Publisher: Springer Singapore

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Abstract

This chapter introduces the engineering background and research significance of uncertain optimization and analyzes the research status of several mainstream uncertain optimization methods, in which the research status and main technical problems of interval optimization method are emphasized. Finally, this chapter gives the framework for this book.

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next chapter The Basic Principles of Interval Analysis
Literature
1.
Himmelblau DM (1972) Applied nonlinear programming. McGraw-Hill, New YorkMATH
2.
Haftka RT, Gürdal Z (1992) Elements of structural optimization. Kluwer Academic Publishers, NetherlandsMATHCrossRef
3.
Liu WX (2002) Mechanical optimization design. Tsinghua University Press, Beijing
4.
Zhu J, Zhang W, Beckers P (2009) Integrated layout design of multi-component system. Int J Numer Meth Eng 78(6):631–651MATHCrossRef
5.
Cheng GD (2012) Introduction to optimum design of engineering structures. Dalian University of Technology Press, Dalian
6.
7.
Qiu ZP (1994) Interval analysis method of static response of structures with uncertain parameters and the eigenvalue problem. PhD thesis, Jilin Industrial University, Changchun
8.
Yang XW (2000) Dynamic and static analysis for structures with interval parameters. PhD thesis, Jilin University of Technology, Changchun
9.
Lian HD (2002) The interval FEM for structural analysis. PhD thesis, Jilin University, Changchun
10.
Li RF, Wang JJ (1997) Geared system dynamic-vibration, shock and noise. Science Press, Beijing
11.
Zhang JB, Chen ZY (1990) A novel method of stiffness identification for gears. In Proceedings of the 4th national conference on vibration theory and application, Zhengzhou, China, May 1990
12.
Tong ZF, Zhang J (1992) Research on the dynamic characteristic and its identification of the joint between column and bed of a machining center. J Vib Shock 2(3):13–19
13.
Zhu DP (1993) The influence of vibration environment on structural characteristics. In Proceedings of the 5th national conference on vibration theory and application, Tunxi, Huangshan, China, Oct 1993
14.
Zhang J (1996) Research on dynamic modelling and parameter identification of joint between complex mechanical structures. J Mech Strength 18(2):1–5
15.
Sun AR (1995) Uncertainty analysis of structural and systematic parameters of nuclear power plant. J Northeast For Univ 23(1):108–115
16.
Chang CC, Lo JG, Wang JL (2001) Assessment of reducing ozone forming potential for vehicles using liquefied petroleum gas as an alternative fuel. Atmos Environ 35(35):6201–6211CrossRef
17.
Yuan Q, Li YB (2005) Influence of parametrical uncertainties on traffic accident reappearance of vehicle side impact. Trans Chin Soc Agric Mach 36(5):16–19
18.
Wang HY, Liu HJ (2006) Stochastic simulation method of performance reliability estimation on liquid propellant rocket engine. J Rocket Propuls 32(4):26–30MathSciNet
19.
Kazemi Zanjani M, Nourelfath M, Ait-Kadi D (2010) A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand. Int J Prod Res 48(16):4701–4723
20.
Liu BD, Zhao RQ, Wang G (2003) Uncertain programming with applications. Tsinghua University Press, Beijing
21.
Alfieri A, Tolio T, Urgo M (2012) A two-stage stochastic programming project scheduling approach to production planning. Int J Adv Manuf Technol 62(1):279–290CrossRef
22.
Goli A, Tirkolaee EB, Malmir B, Bian GB, Sangaiah AK (2019) A multi-objective invasive weed optimization algorithm for robust aggregate production planning under uncertain seasonal demand. Computing 101(6):499–529MathSciNetCrossRef
23.
Liu BD (2001) Uncertain programming: a unifying optimization theory in various uncertain environments. Appl Math Comput 120(1–3):227–234MathSciNetMATH
24.
Kall P (2001) Stochastic programming: achievements and open problems. In: Kischka P, Möhring RH, Leopold-Wildburger U, Radermacher FJ (eds) Models, methods and decision support for management: essays in Honor of Paul Stähly. Physica-Verlag HD, Heidelberg, pp 285–302
25.
Mousavi SM, Vahdani B, Tavakkoli-Moghaddam R, Hashemi H (2014) Location of cross-docking centers and vehicle routing scheduling under uncertainty: a fuzzy possibilistic–stochastic programming model. Appl Math Model 38(7–8):2249–2264MathSciNetMATHCrossRef
26.
Sun L, Lin L, Li HJ, Gen M (2018) Hybrid cooperative co-evolution algorithm for uncertain vehicle scheduling. IEEE Access 6:71732–71742CrossRef
27.
Chen C, Wang F, Zhou B, Chan KW, Cao Y, Tan Y (2015) An interval optimization based day-ahead scheduling scheme for renewable energy management in smart distribution systems. Energy Convers Manag 106:584–596CrossRef
28.
Wei F, Wu QH, Jing ZX, Chen JJ, Zhou XX (2016) Optimal unit sizing for small-scale integrated energy systems using multi-objective interval optimization and evidential reasoning approach. Energy 111:933–946CrossRef
29.
Zhou J, Liu BD (2003) New stochastic models for capacitated location-allocation problem. Comput Ind Eng 45(1):111–125MathSciNetCrossRef
30.
Alizadeh M, Mahdavi I, Mahdavi-Amiri N, Shiripour S (2015) A capacitated location-allocation problem with stochastic demands using sub-sources: an empirical study. Appl Soft Comput 34:551–571CrossRef
31.
Elishakoff I, Haftka RT, Fang J (1994) Structural design under bounded uncertainty-optimization with anti-optimization. Comput Struct 53(6):1401–1405MATHCrossRef
32.
Doltsinis I, Kang Z (2004) Robust design of structures using optimization methods. Comput Methods Appl Mech Eng 193(23–26):2221–2237MATHCrossRef
33.
Mrabet E, Guedri M, Ichchou MN, Ghanmi S (2015) Stochastic structural and reliability based optimization of tuned mass damper. Mech Syst Signal Process 60–61:437–451CrossRef
34.
Bhattacharjya S, Chakraborty S (2018) An improved robust multi-objective optimization of structure with random parameters. Adv Struct Eng 21(11):1597–1607CrossRef
35.
Liu BS, Jiang C, Li GY, Huang XD (2020) Topology optimization of structures considering local material uncertainties in additive manufacturing. Comput Methods Appl Mech Eng 360 Article 112786 
36.
Bellman RE (1957) Dynamic programming. Princeton University Press, New JerseyMATH
37.
38.
39.
40.
Ferguson AR, Dantzig GB (1956) The allocation of aircraft to routes—an example of linear programming under uncertain demand. Manag Sci 3(1):45–73MathSciNetMATHCrossRef
41.
Beale EM (1955) On minimizing a convex function subject to linear inequalities. J R Stat Soc Ser B (Methodological) 173–184
42.
43.
44.
Charnes A, Cooper WW, Symonds GH (1958) Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag Sci 4(3):235–263CrossRef
45.
46.
Prékopa A (1980) Logarithmic concave measures and related topics. In: Stochastic programming
47.
Markowitz HM, Todd GP (2000) Mean-variance analysis in portfolio choice and capital markets. Wiley, New Hope, Pennsylvania
48.
Levy H, Markowitz HM (1979) Approximating expected utility by a function of mean and variance. Am Econ Rev 69(3):308–317
49.
Dupačová J (1976) Minimax stochastic programs with nonconvex nonseparable penalty functions. Progress Oper Res. North-Holland, Amsterdam
50.
Pollak RA (1967) Additive von Neumann-Morgenstern utility functions. Econometrica 35(3/4):485–494MATHCrossRef
51.
52.
Rong XX (2005) Research on model and algorithm about uncertain optimization problems. PhD thesis, Shandong University, Jinan
53.
Gartska SJ (1980) The economic equivalence of several stochastic programming models. In: Dempster MAH (ed) Stochastic programming. Academic Press, New York
54.
Ziemba WT (1974) Stochastic programs with simple recourse. Mathematical programming in theory and practice. Amsterdam, North-Holland, June 1972
55.
Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, New YorkMATH
56.
Leövey H, Römisch W (2015) Quasi-Monte Carlo methods for linear two-stage stochastic programming problems. Math Program 151(1):315–345MathSciNetMATHCrossRef
57.
Ivanov SV, Kibzun AI, Mladenovic N, Urosevic D (2019) Variable neighborhood search for stochastic linear programming problem with quantile criterion. J Global Optim 74(3):549–564MathSciNetMATHCrossRef
58.
Sherali HD, Fraticelli BMP (2002) A modification of Benders’ decomposition algorithm for discrete subproblems: an approach for stochastic programs with integer recourse. J Global Optim 22(1):319–342MathSciNetMATHCrossRef
59.
Schrijver A (1998) Theory of linear and integer programming. Wiley, AmsterdamMATH
60.
Ahmed S, Sahinidis NV (2003) An approximation scheme for stochastic integer programs arising in capacity expansion. Oper Res 51(3):461–471MathSciNetMATHCrossRef
61.
Bastin F (2004) Nonlinear stochastic programming. PhD Thesis, University of Namur, Namur
62.
Dumskis V, Sakalauskas L (2015) Nonlinear stochastic programming involving CVaR in the objective and constraints. Informatica 26(4):569–591MathSciNetMATHCrossRef
63.
Doagooei AR (2020) Generalized cutting plane method for solving nonlinear stochastic programming problems. Optimization 69(7–8):1751–1771
64.
Krasko V, Rebennack S (2017) Two-stage stochastic mixed-integer nonlinear programming model for post-wildfire debris flow hazard management: mitigation and emergency evacuation. Eur J Oper Res 263(1):265–282MathSciNetMATHCrossRef
65.
66.
Liu C, Lee C, Chen H, Mehrotra S (2016) Stochastic robust mathematical programming model for power system optimization. IEEE Trans Power Syst 31(1):821–822CrossRef
67.
Rahimian H, Bayraksan G, Homem-de-Mello T (2019) Identifying effective scenarios in distributionally robust stochastic programs with total variation distance. Math Program 173(1–2):393–430
68.
Enevoldsen I, Sørensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15(3):169–196CrossRef
69.
Du XP, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. ASME J Mech Des 126(2):225–233CrossRef
70.
Liang J, Mourelatos ZP, Tu J (2004) A single-loop method for reliability-based design optimization. In: ASME 2004 international design engineering technical conferences and computers and information in engineering conference, Salt Lake City, UT, Sept/Oct 2004
71.
Cheng GD, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84(21):1353–1367CrossRef
72.
Zhang YM (2015a) Reliability-based robust design optimization of vehicle components, part I: theory. Front Mech Eng 10(2):138–144CrossRef
73.
Zhang YM (2015b) Reliability-based robust design optimization of vehicle components, part II: case studies. Front Mech Eng 10(2):145–153CrossRef
74.
Chen ZZ, Qiu HB, Gao L, Li P (2013) An optimal shifting vector approach for efficient probabilistic design. Struct Multidiscip Optim 47(6):905–920CrossRef
75.
Shan SQ, Wang GG (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93(8):1218–1230CrossRef
76.
Zhang Z, Deng W, Jiang C (2020) Sequential approximate reliability-based design optimization for structures with multimodal random variables. Struct Multidiscip Optim 62(2):511–528
77.
Cui D (2005) Some problems on stochastic programming. Master thesis, Shandong University of Science and Technology, Qingdao
78.
Feng YJ, Wei QL (1982) The general form of multi-objective fuzzy programming solution. Fuzzy Math 2(2):29–35MathSciNet
79.
Inuiguchi M, Ramı́k J (2000) Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst 111(1):3–28
80.
81.
82.
Tanaka H, Okuda T, Asai K (1973) On fuzzy-mathematical programming. J Cybern 3(4):37–46
83.
84.
Buckley JJ, Feuring T (2000) Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets Syst 109(1):35–53MathSciNetMATHCrossRef
85.
Liu QM, Shi FG (2015) Stratified simplex method for solving fuzzy multi-objective linear programming problem. J Intell Fuzzy Syst 29(6):2357–2364MATHCrossRef
86.
87.
Inuiguchi M, Ichihashi H, Kume Y (1992) Relationships between modality constrained programming problems and various fuzzy mathematical programming problems. Fuzzy Sets Syst 49(3):243–259MathSciNetMATHCrossRef
88.
Hong ZY (2001) The stability of fuzzy multi-objective nonlinear programming problem and a solving method based on the order structure of numbers. Master thesis, Harbin Institute of Technology, Harbin
89.
Sakawa M, Yano H (1989) Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters. Fuzzy Sets Syst 29(3):315–326MathSciNetMATHCrossRef
90.
Huang HZ (1997) Fuzzy multi-objective optimization decision-making of reliability of series system. Microelectron Reliab 37(3):447–449CrossRef
91.
Tang JF, Wang DW (1997) An interactive approach based on a genetic algorithm for a type of quadratic programming problems with fuzzy objective and resources. Comput Oper Res 24(5):413–422MathSciNetMATHCrossRef
92.
Tang JF, Wang DW (1998) Model and method for a type of nonlinear programming problems with fuzzy resources. Fuzzy Syst Math 12(3):58–67MATH
93.
Liu BD, Zhao RQ (1998) Stochastic programming and fuzzy programming. Tsinghua University Press, Beijing
94.
Wu CW, Liao MY (2014) Fuzzy nonlinear programming approach for evaluating and ranking process yields with imprecise data. Fuzzy Sets Syst 246:142–155MathSciNetMATHCrossRef
95.
Mansoori A, Effati S (2019) An efficient neurodynamic model to solve nonlinear programming problems with fuzzy parameters. Neurocomputing 334:125–133CrossRef
96.
Jiang Z (2005) Optimization method of uncertain system with the interval parameter and its research on the applies about gasoline blending. PhD thesis, Zhejiang University, Hangzhou
97.
Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainty in applied mechanics. Elsevier Science Publisher, AmsterdamMATH
98.
Salinetti G (1983) Approximations for chance-constrained programming problems. Stoch Int J Probab Stoch Process 10(3–4):157–179
99.
Ermoliev Y (1983) Stochastic quasigradient methods and their application to system optimization. Stoch Int J Probab Stoch Process (1–2):1–36
100.
Guo SX (2002) Non-stochastic reliability and optimization of uncertain structural systems. PhD thesis, Northwestern Polytechnical University, Xi'an
101.
102.
Ben-Haim Y (1993) Convex models of uncertainty in radial pulse buckling of shells. ASME J Appl Mech 60(3):683–688MATHCrossRef
103.
Ben-Haim Y (1995) A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Struct Saf 17(2):91–109CrossRef
104.
Kang Z, Luo YJ (2009a) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198(41–44):3228–3238MathSciNetMATHCrossRef
105.
Jiang C, Zhang QF, Han X, Liu J, Hu DA (2015) Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis. Int J Numer Meth Eng 103(1):31–59MathSciNetMATHCrossRef
106.
Elishakoff I, Bekel Y (2013) Application of Lamé’s super ellipsoids to model initial imperfections. ASME J Appl Mech 80(6):061006CrossRef
107.
Jiang C, Ni BY, Han X, Tao YR (2014) Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems. Comput Methods Appl Mech Eng 268:656–676MathSciNetMATHCrossRef
108.
Lombardi M (1998) Optimization of uncertain structures using non-probabilistic models. Comput Struct 67(1–3):99–103MATHCrossRef
109.
Pantelides CP, Ganzerli S (1998) Design of trusses under uncertain loads using convex models. J Struct Eng 124(3):318–329CrossRef
110.
Ganzerli S, Pantelides CP (1999) Load and resistance convex models for optimum design. Struct Optim 17(4):259–268CrossRef
111.
Ganzerli S, Pantelides CP (2000) Optimum structural design via convex model superposition. Comput Struct 74(6):639–647CrossRef
112.
Pantelides CP, Ganzerli S (2001) Comparison of fuzzy set and convex model theories in structural design. Mech Syst Signal Process 15(3):499–511CrossRef
113.
Qiu ZP (2003) Comparison of static response of structures using convex models and interval analysis method. Int J Numer Meth Eng 56(12):1735–1753MATHCrossRef
114.
Qiu ZP, Wang XJ (2003) Comparison of dynamic response of structures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach. Int J Solids Struct 40(20):5423–5439MATHCrossRef
115.
Qiu ZP (2005) Convex method based on non-probabilistic set-theory and its application. National Defence Industry Press, Beijing
116.
Au FTK, Cheng YS, Tham LG, Zeng GW (2003) Robust design of structures using convex models. Comput Struct 81(28–29):2611–2619CrossRef
117.
Gurav SP, Goosen JFL, vanKeulen F (2005) Bounded-But-Unknown uncertainty optimization using design sensitivities and parallel computing: application to MEMS. Comput Struct 83(14):1134–1149CrossRef
118.
Guo X, Bai W, Zhang WS, Gao XX (2009) Confidence structural robust design and optimization under stiffness and load uncertainties. Comput Methods Appl Mech Eng 198(41–44):3378–3399MathSciNetMATHCrossRef
119.
Guo SX, Lv ZZ (2002) Optimization of uncertain structures based on non-probabilistic reliability model. Chin J Comput Mech 19(2):198–201
120.
Guo SX, Lv ZZ (2003) Comparison of possibilistic reliability and stochastic reliability methods for uncertain structures. Chin J Appl Mech 20(3):107–110
121.
Cao HJ, Duan BY (2005a) Approach to optimization of uncertain structures based on on-probabilistic reliability. Chin J Comput Mech 22(5):546–549
122.
Cao HJ, Duan BY (2005b) Approach to optimization of uncertain structures based on non-probabilistic reliability. Chin J Appl Mech 22(3):381–385
123.
Kang Z, Luo YJ (2006) On structural optimization for non-probabilistic reliability based on convex models. Chin J Theor Appl Mech 38(6):807–815
124.
Kang Z, Luo YJ (2009b) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198(41):3228–3238MathSciNetMATHCrossRef
125.
Li FY, Luo Z, Rong JH, Hu L (2013) A non-probabilistic reliability-based optimization of structures using convex models. Comput Model Eng Sci 95(6):453–482MathSciNet
126.
127.
Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48(2):219–225MATHCrossRef
128.
Ramik J, Rommelfanger H (1993) A single- and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients. Fuzzy Sets Syst 57(2):203–208MathSciNetMATHCrossRef
129.
130.
Ohta H, Yamaguchi T (1996) Linear fractional goal programming in consideration of fuzzy solution. Eur J Oper Res 92(1):157–165MATHCrossRef
131.
132.
Liu XW, Da QL (1999) A satisfactory solution for interval linear programming. J Syst Eng 14(2):123–128MathSciNet
133.
Zhang Q, Fan ZP, Pan DH (1999) A ranking approach for interval numbers in uncertain multiple attribute decision making problems. Syst Eng Theory Practice 19(5):129–133
134.
Xu ZS, Da QL (2003) Possibility degree method for ranking interval numbers and its application. J Syst Eng 18(1):67–70
135.
Chanas S, Kuchta D (1996) Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur J Oper Res 94(3):594–598MATHCrossRef
136.
137.
Sengupta A, Pal TK, Chakraborty D (2001) Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets Syst 119(1):129–138MATHCrossRef
138.
Lai KK, Wang SY, Xu JP, Zhu SS, Fang Y (2002) A class of linear interval programming problems and its application to portfolio selection. IEEE Trans Fuzzy Syst 10(6):698–704CrossRef
139.
Chen MZ, Wang SG, Wang PP, Ye XX (2016) A new equivalent transformation for interval inequality constraints of interval linear programming. Fuzzy Optim Decis Making 15(2):155–175MathSciNetMATHCrossRef
140.
Inuiguchi M, Kume Y (1992) Extensions of efficiency to possibilistic multiobjective linear programming problems. In: Proceeding of the tenth international conference on multiple criteria decision making
141.
Inuiguchi M, Sakawa M (1995) Minimax regret solution to linear programming problems with an interval objective function. Eur J Oper Res 86(3):526–536MATHCrossRef
142.
Shimizu K, Aiyoshi E (1980) Necessary conditions for min-max problems and algorithms by a relaxation procedure. IEEE Trans Autom Control 25(1):62–66MathSciNetMATHCrossRef
143.
Inuiguchi M, Sakawa M (1996) Maximum regret analysis in linear programs with an interval objective function. In: Proceedings of IWSCI'96
144.
Mausser HE, Laguna M (1998) A new mixed integer formulation for the maximum regret problem. Int Trans Oper Res 5(5):389–403CrossRef
145.
Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, BostonMATHCrossRef
146.
Mausser HE, Laguna M (1999) A heuristic to minimax absolute regret for linear programs with interval objective function coefficients. Eur J Oper Res 117(1):157–174MATHCrossRef
147.
148.
Dong C, Huang GH, Cai YP, Xu Y (2011) An interval-parameter minimax regret programming approach for power management systems planning under uncertainty. Appl Energy 88(8):2835–2845CrossRef
149.
Rivaz S, Yaghoobi MA (2013) Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients. CEJOR 21(3):625–649MathSciNetMATHCrossRef
150.
Ma LH (2002) Research on method and application of robust optimization for uncertain system. PhD thesis, Zhejiang University, Hangzhou
151.
Cheng ZQ, Dai LK, Sun YX (2004) Feasibility analysis for optimization of uncertain systems with interval parameters. Acta Autom Sin 30(3):455–459
152.
Jiang C (2008) Theories and algorithms of uncertain optimization based on interval. PhD thesis, Hunan University, Changsha
153.
Wu HC (2007) The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176(1):46–59MathSciNetMATHCrossRef
154.
Wu XY, Huang GH, Liu L, Li JB (2006) An interval nonlinear program for the planning of waste management systems with economies-of-scale effects—a case study for the region of Hamilton, Ontario, Canada. Eur J Oper Res 171(2):349–372MathSciNetMATHCrossRef
155.
Gong DW, Sun J (2013) Theories and applications of interval multi-objective evolutionary optimization. Science Press, Beijing
156.
Cheng J, Liu ZY, Wu ZY, Tang MY, Tan JR (2016) Direct optimization of uncertain structures based on degree of interval constraint violation. Comput Struct 164:83–94CrossRef
157.
Wu J, Zhou SN (2012) Robustness optimization method for frequency and decoupling ratio of powertrain mounting systems. J Vib Shock 31(4):1–7
158.
Wu JL, Luo Z, Zhang N, Zhang YQ (2015) A new interval uncertain optimization method for structures using Chebyshev surrogate models. Comput Struct 146:185–196CrossRef
159.
Xu B, Jin YJ (2014) Multiobjective dynamic topology optimization of truss with interval parameters based on interval possibility degree. J Vib Control 20(1):66–81MathSciNetMATHCrossRef
160.
Cheng J, Liu ZY, Qian YM, Wu D, Zhou ZD, Gao W, Zhang J, Tan JR (2019) Robust optimization of uncertain structures based on interval closeness coefficients and the 3D violation vectors of interval constraints. Struct Multidiscip Optim 60(1):17–33CrossRef
161.
Wang LQ, Yang GL, Xiao H, Sun QQ, Ge JL (2020) Interval optimization for structural dynamic responses of an artillery system under uncertainty. Eng Optim 52(2):343–366MathSciNetCrossRef
162.
Li YL, Wang XJ, Huang R, Qiu ZP (2015) Actuator placement robust optimization for vibration control system with interval parameters. Aerosp Sci Technol 45:88–98CrossRef
163.
Badri SA, Ghazanfari M, Shahanaghi K (2014) A multi-criteria decision-making approach to solve the product mix problem with interval parameters based on the theory of constraints. Int J Adv Manuf Technol 70(5):1073–1080CrossRef
164.
Li YZ, Wu QH, Jiang L, Yang JB, Xu DL (2016) Optimal power system dispatch with wind power integrated using nonlinear interval optimization and evidential reasoning approach. IEEE Trans Power Syst 31(3):2246–2254CrossRef
165.
Huang CX, Yue D, Deng S, Xie J (2017) Optimal scheduling of microgrid with multiple distributed resources using interval optimization. Energies 10(3):339
166.
Soltani M, Kerachian R, Nikoo MR, Noory H (2016) A Conditional value at risk-based model for planning agricultural water and return flow allocation in river systems. Water Resour Manag 30(1):427–443CrossRef
167.
Zarghami M, Safari N, Szidarovszky F, Islam S (2015) Nonlinear interval parameter programming combined with cooperative games: a tool for addressing uncertainty in water allocation using water diplomacy framework. Water Resour Manag 29(12):4285–4303CrossRef
Metadata
Title
Introduction
Authors
Chao Jiang
Xu Han
Huichao Xie
Copyright Year
2021
Publisher
Springer Singapore
Book
Nonlinear Interval Optimization for Uncertain Problems

Print ISBN: 978-981-15-8545-6

Electronic ISBN: 978-981-15-8546-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-981-15-8546-3_1