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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

 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.
Citation: Shaojun Zhang, Zhong Wan. Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive. Journal of Industrial and 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, Heidelberg, 2002. [2] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 51-80. doi: 10.1007/s10107-007-0163-z. [3] G. Facchinetti, R. G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers, International Journal of Intelligent Systems, 13 (1998), 613-622. [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), 351-371. doi: 10.3934/jimo.2006.2.351. [5] C. Jiang, "Theories and Algorithms of Uncertain Optimization Based on Interval,'' Ph.D thesis, Hunan University, Changsha, 2008. [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), 1-13. doi: 10.1016/j.ejor.2007.03.031. [7] A. Kumar, J. Kaur and P. Singh, A new method for solving fully fuzzy linear programming problems, Applied Mathematical Modelling, 35 (2011), 817-823. doi: 10.1016/j.apm.2010.07.037. [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), 1097-1113. doi: 10.1016/j.mcm.2006.03.013. [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), 365-383. doi: 10.3934/jimo.2011.7.365. [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), 868-878. doi: 10.1016/j.enpol.2008.10.038. [11] Y. D. Liu, Calculation of V-belt life, Journal of Hubei Automotive Industries Institute, 21 (1997), 1-4. [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, Beijing, 2006. [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), 1331-1357. doi: 10.1016/j.ejor.2006.03.053. [14] Z. Ren and S. Glode, Computational service life estimation of contacting mechanical elements in regard to pitting, Computers & Structures, 80 (2002), 2209-2216. doi: 10.1016/S0045-7949(02)00263-8. [15] M. F. Spotts, "Design of Machine Elements," 6th edition, Englewood Prentice-Hall Inc., Cliffs, 1985. [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. doi: 10.1155/2010/745162. [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), 163-167. [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), 455-474. doi: 10.1017/S1446181109000145. [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), 317-345. doi: 10.3934/jimo.2011.7.317. [20] Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70. [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), 23-25. [22] C. S. Yang, Design optimization of belt transmission by intelligent algorithm, in "2009 International Conference on Computational Intelligence and Software Engineering" (CiSE 2009), (2009), 1-4. [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), 140-147. doi: 10.1007/s11431-010-4193-z. [24] S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design of V-belt fatigue life, China Mechanical Engineering, 22 (2011), 403-407.

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##### References:
 [1] C. Carlsson and R. Fullér, "Fuzzy Reasoning in Decision Making and Optimization," Physica-Verlag, Heidelberg, 2002. [2] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 51-80. doi: 10.1007/s10107-007-0163-z. [3] G. Facchinetti, R. G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers, International Journal of Intelligent Systems, 13 (1998), 613-622. [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), 351-371. doi: 10.3934/jimo.2006.2.351. [5] C. Jiang, "Theories and Algorithms of Uncertain Optimization Based on Interval,'' Ph.D thesis, Hunan University, Changsha, 2008. [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), 1-13. doi: 10.1016/j.ejor.2007.03.031. [7] A. Kumar, J. Kaur and P. Singh, A new method for solving fully fuzzy linear programming problems, Applied Mathematical Modelling, 35 (2011), 817-823. doi: 10.1016/j.apm.2010.07.037. [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), 1097-1113. doi: 10.1016/j.mcm.2006.03.013. [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), 365-383. doi: 10.3934/jimo.2011.7.365. [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), 868-878. doi: 10.1016/j.enpol.2008.10.038. [11] Y. D. Liu, Calculation of V-belt life, Journal of Hubei Automotive Industries Institute, 21 (1997), 1-4. [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, Beijing, 2006. [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), 1331-1357. doi: 10.1016/j.ejor.2006.03.053. [14] Z. Ren and S. Glode, Computational service life estimation of contacting mechanical elements in regard to pitting, Computers & Structures, 80 (2002), 2209-2216. doi: 10.1016/S0045-7949(02)00263-8. [15] M. F. Spotts, "Design of Machine Elements," 6th edition, Englewood Prentice-Hall Inc., Cliffs, 1985. [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. doi: 10.1155/2010/745162. [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), 163-167. [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), 455-474. doi: 10.1017/S1446181109000145. [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), 317-345. doi: 10.3934/jimo.2011.7.317. [20] Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70. [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), 23-25. [22] C. S. Yang, Design optimization of belt transmission by intelligent algorithm, in "2009 International Conference on Computational Intelligence and Software Engineering" (CiSE 2009), (2009), 1-4. [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), 140-147. doi: 10.1007/s11431-010-4193-z. [24] S. J. Zhang, Z. Wan and G. L. Liu, Global optimization design of V-belt fatigue life, China Mechanical Engineering, 22 (2011), 403-407.
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