Skip to main content
SpringerLink
Log in

Estimating the parameters of a fatigue model using Benders’ decomposition

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

This paper shows how Benders decomposition can be used for estimating the parameters of a fatigue model. The objective function of such model depends on five parameters of different nature. This makes the parameter estimation problem of the fatigue model suitable for the Benders decomposition, which allows us to use well-behaved and robust parameter estimation methods for the different subproblems. To build the Benders cuts, explicit formulas for the sensitivities (partial derivatives) are obtained. This permits building the classical iterative method, in which upper and lower bounds of the optimal value of the objective function are obtained until convergence. Two alternative objective functions to be optimized are the likelihood and the sum of squares error functions, which relate to the maximum likelihood and the minimum error principles, respectively. The method is illustrated by its application to a real-world problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A :

Constant

M :

Constant

B :

Threshold value of log-lifetime

B 0 :

Fixed value of the threshold value of log-lifetime

C :

Endurance limit (logarithm of Δσ)

C 0 :

Fixed value of the endurance limit (logarithm of Δσ)

E(⋅):

Cumulative distribution function

F(⋅):

Cumulative distribution function

K :

Constant

k :

Positive constant used to bound B and C

ℓ(⋅):

Likelihood function

\({\ell}^{(\nu)}_{\mathrm{up}}\) :

Upper bound for ℓ

\({\ell}^{(\nu)}_{\mathrm{down}}\) :

Lower bound for ℓ

m :

Number of pieces at a given level for Δσ

N :

Lifetime or number of cycles

N ∗ :

Dimensionless lifetime

N i :

Lifetime of sample item i

N 0 :

Threshold value for lifetime

n :

Sample size

P :

Probability

p i :

Plotting point value (p i =i/(m+1))

q(⋅):

Functional equation

\({Q}^{(\nu)}_{\mathrm{up}}\) :

Upper bound for Q

\({Q}^{(\nu)}_{\mathrm{down}}\) :

Lower bound for Q

V i :

\(V_{i}=(\log N_{i}-B)(\log\Delta\sigma _{i}-C)=N^{*}_{i}\Delta \sigma^{*}_{i}\)

z P :

Objective function value

α :

Variable used in the Benders decomposition master problem

β :

Weibull shape parameter of the cumulative distribution function in the S−N field

Δσ :

Stress range or amplitude

Δσ ∗ :

Dimensionless stress range or amplitude

Δσ i :

Stress range of sample item i

Δσ st :

Ultimate strength

Δσ 0 :

Threshold value for stress range

δ :

Weibull scale factor

λ :

Parameter defining the position of the corresponding zero-percentile hyperbola

μ :

Mean

μ :

Vector of dual variables associated with inequality constraints

ν :

Counter used for Benders’ cuts

σ M :

Maximum stress

σ m :

Minimum stress

η :

Vector of dual variables associated with equality constraints

References

  • Aczél, J. (1966). Mathematics in science and engineering: Vol. 19. Lectures on functional equations and their applications. New York: Academic Press.

    Google Scholar 

  • Aczél, J. (1984). On history, applications and theory of functional equations (introduction). In J. Aczél (Ed.), Functional equations: history, applications and theory (pp. 3–12). Dordrecht: Reidel. Chap. 1.

    Chapter  Google Scholar 

  • ASTM (1981). Statistical analysis of fatigue data. American Society for Testing and Materials STP 744, Philadelphia.

  • Basquin, O. (1910). The exponential law of endurance tests (Technical Report). ASTM, Philadelphia.

  • Bastenaire, FA (1972). New method for the statistical evaluation of constant stress amplitude fatigue-test results (Technical Report). ASTM, Philadelphia.

  • Benders, J. F. (1962). Partitioning procedures for solving mixed-variable programming problems. Numerische Mathematik, 4, 238–252.

    Article  Google Scholar 

  • Bolotin, V. V. (1998). Mechanics of fatigue. Boca Raton: CRC Press.

    Google Scholar 

  • Buckingham, E. (1915). The principle of similitude. Nature, 96, 396–397.

    Article  Google Scholar 

  • Cai, X., McKinney, D. C., Lasdon, L. S., & Watkins, D. W. (2001). Solving large nonconvex water resources management models using generalized Benders decomposition. Operations Research, 49, 235–245.

    Article  Google Scholar 

  • Castillo, E. (1988). Extreme value theory in engineering. San Diego: Academic Press.

    Google Scholar 

  • Castillo, E., & Fernández-Canteli, A. (2001). A general regression model for lifetime evaluation and prediction. International Journal of Fracture, 107, 117–137.

    Article  Google Scholar 

  • Castillo, E., & Galambos, J. (1987). Lifetime regression models based on a functional equation of physical nature. Journal of Applied Probability, 24, 160–169.

    Article  Google Scholar 

  • Castillo, E., & Hadi, A. S. (1994). Parameter and quantile estimation for the generalized extreme-value distribution. Environmetrics, 5, 417–432.

    Article  Google Scholar 

  • Castillo, E., & Hadi, A. S. (1995). Modeling lifetime data with application to fatigue models. Journal of the American Statistical Association, 90, 1041–1054.

    Article  Google Scholar 

  • Castillo, E., & Ruiz Cobo, M. R. (1992). Monographs and textbooks in pure and applied mathematics: Vol. 161. Functional equations and modelling in science and engineering. New York: Marcel Dekker.

    Google Scholar 

  • Castillo, E., Fernández-Canteli, A., Esslinger, V., & Thürlimann (1985). Statistical models for fatigue analysis of wires, strands and cables. In IABSE proceedings, vol. 82 (pp. 1–140).

    Google Scholar 

  • Castillo, E., Fernández Canteli, A., & Hadi, A. (1999). On fitting a fatigue model to data. International Journal of Fatigue, 21, 97–106.

    Article  Google Scholar 

  • Castillo, E., Conejo, A., Pedregal, P., García, R., & Alguacil, N. (2001). Building and solving mathematical programming models in engineering and science. New York: Wiley.

    Book  Google Scholar 

  • Castillo, E., Hadi, A. S., Conejo, A., & Fernández-Canteli, A. (2004). A general method for local sensitivity analysis with application to regression models and other optimization problems. Technometrics, 46(4), 430–445.

    Article  Google Scholar 

  • Castillo, E., Hadi, A. S., Balakrishnan, N., & Sarabia, J. M. (2005). Extreme value and related models with applications in engineering and science. New York: Wiley.

    Google Scholar 

  • Castillo, E., Conejo, A. J., Mínguez, R., & Ortigosa, D. (2006a). Perturbation approach to sensitivity analysis in nonlinear programming. Journal of Optimization Theory and Applications, 128(1), 49–74.

    Article  Google Scholar 

  • Castillo, E., Conejo, A. J., Mínguez, R., & Castillo, C. (2006b). A closed formula for local sensitivity analysis in mathematical programming. Engineering Optimization, 38(1), 93–112.

    Article  Google Scholar 

  • Castillo, E., Conejo, A. J., Castillo, C., & Mínguez, R. (2007a). Closed formulas in local sensitivity analysis for some classes of linear and non-linear problems. TOP, 15(2), 355–371.

    Article  Google Scholar 

  • Castillo, E., Fernández-Canteli, A., Hadi, A. S., & López-Aenlle, M. (2007b). A fatigue model with local sensitivity analysis. Fatigue & Fracture of Engineering Materials & Structures, 30(2), 149–168.

    Article  Google Scholar 

  • Chernick, M. R. (1999). Wiley series in probability and statistics. Bootstrap methods. New York: Wiley.

    Google Scholar 

  • Choi, J., & Choi, C. (1992). Sensitivity analysis of multilayer perceptron with differentiable activation functions. IEEE Transactions on Neural Networks, 3(1), 101–107.

    Article  Google Scholar 

  • Coleman, B. D. (1958). On the strength of classical fibres bundles. International Journal of Mechanics of Solids, 7, 60–70.

    Article  Google Scholar 

  • Coleman, T. F., & Li, Y. (1994). On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds. Mathematical Programming, 67(2), 189–224.

    Article  Google Scholar 

  • Coleman, T. F., & Li, Y. (1996). An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization, 6, 418–445.

    Article  Google Scholar 

  • Conejo, A., Castillo, E., Mínguez, R., & García-Bertrand, R. (2006). Decomposition techniques in mathematical programming. Engineering and science applications. Berlin: Springer.

    Google Scholar 

  • Cordeau, J. F., Soumis, F., & Desrosiers, J. (2000). A benders decomposition approach for the locomotive and car assignment problem. Transportation Science, 34, 133–149.

    Article  Google Scholar 

  • Cordeau, J. F., Soumis, F., & Desrosiers, J. (2001). Simultaneous assignment of locomotives and cars to passenger trains. Operations Research, 49, 531–548.

    Article  Google Scholar 

  • Cordeau, J. F., Pasin, F., & Solomon, M. (2006). An integrated model for logistics network design. Annals of Operation Research, 144(24), 59–82.

    Article  Google Scholar 

  • Costa, A. M. (2005). A survey on Benders decomposition applied to fixed-charge network design problems. Computers & Operations Research, 32, 1429–1450.

    Article  Google Scholar 

  • Davison, A. C., & Hinkley, D. V. (1997). Cambridge series in statistical and probabilistic mathematics. Bootstrap methods and their applications.

    Google Scholar 

  • Diaconis, P., & Efron, B. (1983). Computer intensive methods in statistics. Scientific American, 248, 116–130.

    Article  Google Scholar 

  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of Statistics, 7, 1–26.

    Article  Google Scholar 

  • Efron, B., & Tibshirami, R. J. (1993). Introduction to the bootstrap. London: Chapman and Hall-CRC.

    Google Scholar 

  • Florian, M., Bushell, G., Ferland, J., Guérin, G., & Nastansky, L. (1976). The engine scheduling problem in a railway network. INFOR, 14(2), 121–138.

    Google Scholar 

  • Galambos, J. (1987). The asymptotic theory of extreme order statistics, 2nd edn. Malabar: Robert E. Krieger.

    Google Scholar 

  • Geoffrion, AM (1972). Generalized benders decomposition. Journal of Optimization Theory and Applications, 10(4), 237–260.

    Article  Google Scholar 

  • Geoffrion, AM, & Graves, G. W. (1974). Multicommodity distribution system-design by benders decomposition. Management Science Series A, Theory, 20(5), 822–844.

    Article  Google Scholar 

  • Greenwood, JA, Landwehr, J. M., Matalas, N. C., & Wallis, J. R. (1979). Probability weighted moments: Definition and relation to parameters of several distributions expressible in inverse form. Water Resources Research, 15, 1049–1054.

    Article  Google Scholar 

  • Holmen, JO (1979). Fatigue of concrete by constant and variable amplitude loading. PhD thesis, University of Trondheim.

  • Hosking, J. R. M., Wallis, J. R., & Wood, E. F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27, 251–261.

    Article  Google Scholar 

  • Jenkinson, A. F. (1969). Statistics of extreme (Technical Report). World Meteorological Office, Geneva.

  • Kohout, J. (1999). A new function describing fatigue crack growth curves. International Journal of Fatigue, 21, 813–821.

    Article  Google Scholar 

  • Lawless, J. F. (1982). Statistical models and methods for lifetime data. New York: Wiley.

    Google Scholar 

  • Mínguez, R., Milano, F., Zárate-Miñano, R., & Conejo, A. J. (2007). Optimal network placement of SVC devices. IEEE Transactions on Power Systems, 22(4), 1851–1860.

    Article  Google Scholar 

  • Palmgren, A. (1924). Die Lebensdauer von Kugellagern. Verein deutscher Ingenieure, 68(14), 339–3413.

    Google Scholar 

  • Pascual, F. G., & Meeker, W. Q. (1999). Estimating fatigue curves with the random fatigue-limit model. Technometrics, 41, 89–94.

    Article  Google Scholar 

  • Prescott, P., & Walden, A. T. (1980). Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika, 67, 723–724.

    Article  Google Scholar 

  • Prescott, P., & Walden, A. T. (1983). Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Journal of Statistical Computing and Simulation, 16, 241–250.

    Article  Google Scholar 

  • Saharidis, G. K., & Ierapetritou, M. G. (2009). Resolution method for mixed integer bievel linear problems based on decomposition technique. Journal of Global Optimization, 44, 29–51.

    Article  Google Scholar 

  • Saharidis, G. K. D., & Ierapetritou, M. G. (2010). Improving benders decomposition using maximum feasible subsystem (MFS) cut generation strategy. Computers & Chemical Engineering, 34(8), 1237–1245.

    Article  Google Scholar 

  • Saharidis, G. K. D., Michel, M., & Ierapetritou, M. G. (2010). Accelerating benders method using covering cut bundle generation. International Transactions in Operational Research, 17(2), 221–237.

    Article  Google Scholar 

  • Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72, 67–90.

    Article  Google Scholar 

  • Spindel, J. E., & Haibach, E. (1981). Some considerations in the statistical determination of the shape of S-N curves (Technical Report). Philadelphia.

  • Strohmeyer, C. E. (1914). The determination of fatigue limits under alternating stress conditions. Proceedings of the Royal Society of London. Series A, 90(620), 411–425.

    Article  Google Scholar 

  • Stüssi, F. (1955). Die Theorie der Dauerfestigkeit und die Versuche von August Wöhler. Mitt. TKVSB no. 13.

  • Weibull, W. (1949). A statistical report of fatigue failure in solids. Transactions no. 27. Stockholm: Royal Institute of Technology of Sweden.

    Google Scholar 

  • Wöhler, A. (1870). Über die Festigkeits-Versuche mit Eisen und Stahl. Zeitschrift für Bauwesen (20), 73.

  • Zhao, G. (2001). A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs. Mathematical Programming, 90, 507–536.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, Universidad de Cantabria, Santander, Spain

    Enrique Castillo

  2. University of Castilla-La Mancha, Ciudad Real, Spain

    Enrique Castillo

  3. Environmental Hydraulics Institute “IH Cantabria”, Universidad de Cantabria, Santander, Spain

    Roberto Mínguez

  4. Department of Electrical Engineering, University of Castilla-La Mancha, Ciudad Real, Spain

    Antonio J. Conejo

  5. LIDIA, University of A Coruña, A Coruña, Spain

    Beatriz Pérez & Oscar Fontenla

Authors
  1. Enrique Castillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Mínguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio J. Conejo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Beatriz Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Oscar Fontenla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Enrique Castillo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Castillo, E., Mínguez, R., Conejo, A.J. et al. Estimating the parameters of a fatigue model using Benders’ decomposition. Ann Oper Res 210, 309–331 (2013). https://doi.org/10.1007/s10479-011-0891-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-011-0891-6

Keywords

Search

Navigation