Both structural sizes and dimensional tolerances strongly influence the manufacturing cost and the functional performance of a practical product. This paper presents an optimization method to simultaneously find the optimal combination of structural sizes and dimensional tolerances. Based on a probability-interval mixed reliability model, the imprecision of design parameters is modeled as interval uncertainties fluctuating within allowable tolerance bounds. The optimization model is defined as to minimize the total manufacturing cost under mixed reliability index constraints, which are further transformed into their equivalent formulations by using the performance measure approach. The optimization problem is then solved with the sequential approximate programming. Meanwhile, a numerically stable algorithm based on the trust region method is proposed to efficiently update the target performance points (TPPs) and the worst case points (WCPs), which shows better performance than traditional approaches for highly nonlinear problems. Numerical results reveal that reasonable dimensions and tolerances can be suggested for the minimum manufacturing cost and a desirable structural safety.
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Ahn J, Kim S, Kim JH (2005) Reliability-based wing design optimization using trust region-sequential quadratic programming framework. J Aircraft 42(5):1331–1336
Byrd RH, Schnabel RB, Schultz (1985) A trust region algorithm for nonlinearly constrained optimization. University of Colorado, Ph.D. thesis.
Chase KW, Greenwood WH, Loosli BG, Hauglund LF (1990) Least cost tolerance allocation for mechanical assemblies with automated process selection. Manuf Rev 3(1):49–59
Chen T, Fischer GW (2000) A GA-based search method for the tolerance allocation problem. Artific Intell Eng 14(2):133–141
Chen X, Neill DJ (1997) Reliability based structural design optimization for practical applications. In 38th Structures, Structural Dynamics, and Materials Conference. AIAA-97-1403, pp2724–2732.
Cheng G, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84(21):1353–1367
Cho TM, Lee BC (2010) Reliability-based design optimization using a family of methods of moving asymptotes. Struct Multidiscip Optim 42:255–268
Dong Z, Hu W, Xue D (1994) New production cost-tolerance models for tolerance synthesis. J Eng Ind 116:199–206
Du X (2007) Interval reliability analysis. In: ASME 2007 Design Engineering Technical Conference & Computers and Information in Engineering Conference (DETC2007). Las Vegas, Nevada, USA
Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. ASME J Mech Des 126(2):225–233
Dupinet E, Balazinski M, Czogala E (1996) Tolerance allocation based on fuzzy logic and simulated annealing. J Intell Manuf 7(6):487–497
Elishakoff I, Colombi P (1993) Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Comput Methods Appl Mech Eng 104(2):187–209
Forouraghi B (2009) Optimal tolerance allocation using a multiobjective particle swarm optimizer. Int J Adv Manuf Tech 44(7–8):710–724
Geetha K, Ravindran D, Kumar MS, Islam MN (2013) Multi-objective optimization for optimum tolerance synthesis with process and machine selection using a genetic algorithm. Int J Adv Manuf Tech 67:2439–2457
Huang YM, Shiau CS (2006) Optimal tolerance allocation for a sliding vane compressor. J Mech Des 128(1):98–107
Jiang C, Li W, Han X, Liu L, Le PH (2011) Structural reliability analysis based on random distributions with interval parameters. Comput Struct 89(23–24):2292–2302
Jiang C, Lu G, Han X, Liu L (2012) A new reliability analysis method for uncertain structures with random and interval variables. Int J Mech Mater Des 8:169–182
Jiang C, Long X, Han X, Tao Y, Liu J (2013) Probability-interval hybrid reliability analysis for cracked structures existing epistemic uncertainty. Eng Fract Mech 112–113:148–164
Kang Z, Luo Y (2010) Reliability-based structural optimization with probability and convex set hybrid models. Struct Multidiscip Optim 42:89–102
Kharmanda G, Mohamed A, Lemaire M (2002) Efficient reliability-based design optimization using a hybrid space with application to finite element analysis. Struct Multidiscip Optim 24(3):233–245
Kusiak A, Feng C (1995) Deterministic tolerance synthesis: a comparative study. Comput-Aided Des 27(10):759–768
Lee KH, Park GJ (2001) Robust optimization considering tolerances of design variables. Comput Struct 79(1):77–86
Lee JO, Yang YS, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80(3):257–269
Luo Y, Kang Z, Li A (2009) Structural reliability assessment based on probability and convex set mixed model. Comput Struct 87(21):1408–1415
Muthu P, Dhanalakshmi V, Sankaranarayanasamy K (2009) Optimal tolerance design of assembly for minimum quality loss and manufacturing cost using metaheuristic algorithms. Int J Adv Manuf Tech 44(11–12):1154–1164
Penmetsa RC, Grandhi RV (2002) Efficient estimation of structural reliability for problems with uncertain intervals. Comput Struct 80(12):1103–1112
Prabhaharan G, Asokan P, Ramesh P, Rajendran S (2004) Genetic-algorithm-based optimal tolerance allocation using a least-cost model. Int J Adv Manuf Tech 24(9–10):647–660
Qiu Z, Wang J (2010) The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system. Eng Fail Anal 17(5):1142–1154
Rao SS, Wu W (2005) Optimum tolerance allocation in mechanical assemblies using an interval method. Eng Optim 37(3):237–257
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23(3):470–472
Sivakumar K, Balamurugan C, Ramabalan S (2011a) Simultaneous optimal selection of design and manufacturing tolerances with alternative manufacturing process selection. Comput-Aided Des 43(2):207–218
Sivakumar K, Balamurugan C, Ramabalan S (2011b) Concurrent multi-objective tolerance allocation of mechanical assemblies considering alternative manufacturing process selection. Int J Adv Manuf Tech 53(5–8):711–732
Spotts MF (1973) Allocation of tolerances to minimize cost of assembly. J Eng Ind 95:762–764
Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. J Mech Des 121(4):557–564
Wu F, Dantan JY, Etienne A, Siadat A, Martin P (2009) Improved algorithm for tolerance allocation based on Monte Carlo simulation and discrete optimization. Comput Ind Eng 56(4):1402–1413
Yi P, Cheng G, Jiang L (2008) A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization. Struct Saf 30(2):91–109
Youn BD, Choi KK, Du L (2005a) Enriched performance measure approach for reliability-based design optimization. AIAA J 43(4):874–884
Youn BD, Choi KK, Du L (2005b) Adaptive probability analysis using an enriched hybrid mean value method. Struct Multidiscip Optim 29(2):134–148
Zhang C, Wang HPB (1993) Integrated tolerance optimisation with simulated annealing. Int J Adv Manuf Tech 8(3):167–174
Zou T, Mahadevan S (2006) A direct decoupling approach for efficient reliability-based design optimization. Struct Multidiscip Optim 31(3):190–200
This co-authored work had been finished when the authors worked at The Chinese University of Hong Kong. The financial support from the National Natural Science Foundation of China (11472215), the NCET Program (12–0462) and the Fundamental Research Funds for the Central Universities (3102014JCQ01034) is gratefully acknowledged.
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Luo, Y., Wu, X., Zhou, M. et al. Simultaneous parameter and tolerance optimization of structures via probability-interval mixed reliability model. Struct Multidisc Optim 51, 705–719 (2015). https://doi.org/10.1007/s00158-014-1167-y
- Trust region method