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April  2012, 8(2): 493-505. doi: 10.3934/jimo.2012.8.493

Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive

Shaojun Zhang 1, and  Zhong Wan 2,

1. 

School of Mathematical Sciences and Computing Technology, Central South University, Hunan Changsha, 410083, China
2. 

School of Mathematics Sciences and Computing Technology, Central South University, Hunan Changsha, 410083

Received  February 2011 Revised  December 2011 Published  April 2012

In this paper, a polymorphic uncertain nonlinear programming (PUNP) model is constructed to formulate the problem of maximizing the V-belt's fatigue life according to the practical engineering design conditions. The model is converted into an equivalent interval programming only involved with interval parameters for any given degree of membership and confidence level. Then, a deterministic equivalent formulation (DEF) for the original model is obtained based on the concept of possibility degree for the order of two interval numbers. An algorithm, called sampling based algorithm, is developed to find a robust optimal design scheme for maximizing the fatigue life of the V-belt. Case study is employed to demonstrate the validity and the practicability of the constructed model and the algorithm.
Keywords: approximation-based optimal design., design optimization, Belt drives, polymorphic uncertainty modeling, fatigue life.
Mathematics Subject Classification: Primary: 90B50, 90C15; Secondary: 60K1.
Citation: Shaojun Zhang, Zhong Wan. Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive. Journal of Industrial & Management Optimization, 2012, 8 (2) : 493-505. doi: 10.3934/jimo.2012.8.493
References:
[1]

C. Carlsson and R. Fullér, "Fuzzy Reasoning in Decision Making and Optimization,", Physica-Verlag, (2002).   Google Scholar

[2]

X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar

[3]

G. Facchinetti, R. G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers,, International Journal of Intelligent Systems, 13 (1998), 613.   Google Scholar

[4]

B. Q. Hu and S. Wang, A novel approach in uncertain programming. I: New arithmetic and order relation for interval numbers,, Journal of Industrial and Management Optimization, 2 (2006), 351.  doi: 10.3934/jimo.2006.2.351.  Google Scholar

[5]

C. Jiang, "Theories and Algorithms of Uncertain Optimization Based on Interval,'', Ph.D thesis, (2008).   Google Scholar

[6]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems,, European Journal of Operational Research, 188 (2008), 1.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[7]

A. Kumar, J. Kaur and P. Singh, A new method for solving fully fuzzy linear programming problems,, Applied Mathematical Modelling, 35 (2011), 817.  doi: 10.1016/j.apm.2010.07.037.  Google Scholar

[8]

J. Li, J. P. Xu and M. S. Gen, A class of multiobjective linear programming model with fuzzy random coefficients,, Mathematical and Computer Modelling, 44 (2006), 1097.  doi: 10.1016/j.mcm.2006.03.013.  Google Scholar

[9]

T. F. Liang and H. W. Cheng, Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method,, Journal of Industrial and Management Optimization, 7 (2011), 365.  doi: 10.3934/jimo.2011.7.365.  Google Scholar

[10]

Q. G. Lin, G. H. Huang, B. Bass and X. S. Qin, IFTEM: An interval-fuzzy two-stage stochastic optimization model for regional energy systems planning under uncertainty,, Energy Policy, 37 (2009), 868.  doi: 10.1016/j.enpol.2008.10.038.  Google Scholar

[11]

Y. D. Liu, Calculation of V-belt life,, Journal of Hubei Automotive Industries Institute, 21 (1997), 1.   Google Scholar

[12]

S. M. Luo, Y. D. Yu and Y. F. Guo, et al., "Theory on Belt Transmission and New Types of Belt Transmission,", National Defence Industry Press, (2006).   Google Scholar

[13]

X. S. Qin, G. H. Huang, G. M. Zeng, A. Chakma, and Y. F. Huang, An interval-parameter fuzzy nonlinear optimization model for stream water quality management under uncertainty,, European Journal of Operational Research, 180 (2007), 1331.  doi: 10.1016/j.ejor.2006.03.053.  Google Scholar

[14]

Z. Ren and S. Glode, Computational service life estimation of contacting mechanical elements in regard to pitting,, Computers & Structures, 80 (2002), 2209.  doi: 10.1016/S0045-7949(02)00263-8.  Google Scholar

[15]

M. F. Spotts, "Design of Machine Elements," 6th edition,, Englewood Prentice-Hall Inc., (1985).   Google Scholar

[16]

Z. Wan, A. Y. Hao, F. Z. Meng and C. M. Hu, Hybrid method for a class of stochastic bi-criteria optimization problems,, Journal of Inequalities and Applications, 2010 (2010).  doi: 10.1155/2010/745162.  Google Scholar

[17]

Z. Wan, F. Z. Meng, A. Y. Hao and Y. L. Wang, Fuzzy and stochastic parameters-based prediction method for the components of alkali in the sintering process of aluminium,, Fuzzy System and Mathematics, 25 (2011), 163.   Google Scholar

[18]

Z. Wan, K. L. Teo, L. S. Kong and C. Yang, A class of mix design problems: Formulation, solution methods and applications,, ANZIAM Journal, 50 (2009), 455.  doi: 10.1017/S1446181109000145.  Google Scholar

[19]

M. Z. Wang, M. Montaz Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks,, Journal of Industrial and Management Optimization, 7 (2011), 317.  doi: 10.3934/jimo.2011.7.317.  Google Scholar

[20]

Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application,, Journal of Systems Engineering, 18 (2003), 67.   Google Scholar

[21]

H. B. Yan, S. C Yuan and W. X. Ji, Design optimization of V-belt applying genetic algorithm and MATLAB toolbox,, Machinery, 35 (2008), 23.   Google Scholar

[22]

C. S. Yang, Design optimization of belt transmission by intelligent algorithm,, in, (2009), 1.   Google Scholar

[23]

S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design method for maximizing the capacity of V-belt drive,, SCINCE CHINA: Technological Sciences, 54 (2011), 140.  doi: 10.1007/s11431-010-4193-z.  Google Scholar

[24]

S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design of V-belt fatigue life,, China Mechanical Engineering, 22 (2011), 403.   Google Scholar

show all references

References:
[1]

C. Carlsson and R. Fullér, "Fuzzy Reasoning in Decision Making and Optimization,", Physica-Verlag, (2002).   Google Scholar

[2]

X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar

[3]

G. Facchinetti, R. G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers,, International Journal of Intelligent Systems, 13 (1998), 613.   Google Scholar

[4]

B. Q. Hu and S. Wang, A novel approach in uncertain programming. I: New arithmetic and order relation for interval numbers,, Journal of Industrial and Management Optimization, 2 (2006), 351.  doi: 10.3934/jimo.2006.2.351.  Google Scholar

[5]

C. Jiang, "Theories and Algorithms of Uncertain Optimization Based on Interval,'', Ph.D thesis, (2008).   Google Scholar

[6]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems,, European Journal of Operational Research, 188 (2008), 1.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[7]

A. Kumar, J. Kaur and P. Singh, A new method for solving fully fuzzy linear programming problems,, Applied Mathematical Modelling, 35 (2011), 817.  doi: 10.1016/j.apm.2010.07.037.  Google Scholar

[8]

J. Li, J. P. Xu and M. S. Gen, A class of multiobjective linear programming model with fuzzy random coefficients,, Mathematical and Computer Modelling, 44 (2006), 1097.  doi: 10.1016/j.mcm.2006.03.013.  Google Scholar

[9]

T. F. Liang and H. W. Cheng, Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method,, Journal of Industrial and Management Optimization, 7 (2011), 365.  doi: 10.3934/jimo.2011.7.365.  Google Scholar

[10]

Q. G. Lin, G. H. Huang, B. Bass and X. S. Qin, IFTEM: An interval-fuzzy two-stage stochastic optimization model for regional energy systems planning under uncertainty,, Energy Policy, 37 (2009), 868.  doi: 10.1016/j.enpol.2008.10.038.  Google Scholar

[11]

Y. D. Liu, Calculation of V-belt life,, Journal of Hubei Automotive Industries Institute, 21 (1997), 1.   Google Scholar

[12]

S. M. Luo, Y. D. Yu and Y. F. Guo, et al., "Theory on Belt Transmission and New Types of Belt Transmission,", National Defence Industry Press, (2006).   Google Scholar

[13]

X. S. Qin, G. H. Huang, G. M. Zeng, A. Chakma, and Y. F. Huang, An interval-parameter fuzzy nonlinear optimization model for stream water quality management under uncertainty,, European Journal of Operational Research, 180 (2007), 1331.  doi: 10.1016/j.ejor.2006.03.053.  Google Scholar

[14]

Z. Ren and S. Glode, Computational service life estimation of contacting mechanical elements in regard to pitting,, Computers & Structures, 80 (2002), 2209.  doi: 10.1016/S0045-7949(02)00263-8.  Google Scholar

[15]

M. F. Spotts, "Design of Machine Elements," 6th edition,, Englewood Prentice-Hall Inc., (1985).   Google Scholar

[16]

Z. Wan, A. Y. Hao, F. Z. Meng and C. M. Hu, Hybrid method for a class of stochastic bi-criteria optimization problems,, Journal of Inequalities and Applications, 2010 (2010).  doi: 10.1155/2010/745162.  Google Scholar

[17]

Z. Wan, F. Z. Meng, A. Y. Hao and Y. L. Wang, Fuzzy and stochastic parameters-based prediction method for the components of alkali in the sintering process of aluminium,, Fuzzy System and Mathematics, 25 (2011), 163.   Google Scholar

[18]

Z. Wan, K. L. Teo, L. S. Kong and C. Yang, A class of mix design problems: Formulation, solution methods and applications,, ANZIAM Journal, 50 (2009), 455.  doi: 10.1017/S1446181109000145.  Google Scholar

[19]

M. Z. Wang, M. Montaz Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks,, Journal of Industrial and Management Optimization, 7 (2011), 317.  doi: 10.3934/jimo.2011.7.317.  Google Scholar

[20]

Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application,, Journal of Systems Engineering, 18 (2003), 67.   Google Scholar

[21]

H. B. Yan, S. C Yuan and W. X. Ji, Design optimization of V-belt applying genetic algorithm and MATLAB toolbox,, Machinery, 35 (2008), 23.   Google Scholar

[22]

C. S. Yang, Design optimization of belt transmission by intelligent algorithm,, in, (2009), 1.   Google Scholar

[23]

S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design method for maximizing the capacity of V-belt drive,, SCINCE CHINA: Technological Sciences, 54 (2011), 140.  doi: 10.1007/s11431-010-4193-z.  Google Scholar

[24]

S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design of V-belt fatigue life,, China Mechanical Engineering, 22 (2011), 403.   Google Scholar

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