Annals of Operations Research

, Volume 210, Issue 1, pp 309–331 | Cite as

Estimating the parameters of a fatigue model using Benders’ decomposition

  • Enrique CastilloEmail author
  • Roberto Mínguez
  • Antonio J. Conejo
  • Beatriz Pérez
  • Oscar Fontenla


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.


Linear optimization Least-squares Maximum likelihood Sensitivity analysis Benders’ decomposition Fatigue 







Threshold value of log-lifetime


Fixed value of the threshold value of log-lifetime


Endurance limit (logarithm of Δσ)


Fixed value of the endurance limit (logarithm of Δσ)


Cumulative distribution function


Cumulative distribution function




Positive constant used to bound B and C


Likelihood function


Upper bound for


Lower bound for


Number of pieces at a given level for Δσ


Lifetime or number of cycles


Dimensionless lifetime


Lifetime of sample item i


Threshold value for lifetime


Sample size




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


Functional equation


Upper bound for Q


Lower bound for Q


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


Objective function value


Variable used in the Benders decomposition master problem


Weibull shape parameter of the cumulative distribution function in the SN field


Stress range or amplitude


Dimensionless stress range or amplitude


Stress range of sample item i


Ultimate strength


Threshold value for stress range


Weibull scale factor


Parameter defining the position of the corresponding zero-percentile hyperbola




Vector of dual variables associated with inequality constraints


Counter used for Benders’ cuts


Maximum stress


Minimum stress


Vector of dual variables associated with equality constraints


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Enrique Castillo
    • 1
    • 2
    Email author
  • Roberto Mínguez
    • 3
  • Antonio J. Conejo
    • 4
  • Beatriz Pérez
    • 5
  • Oscar Fontenla
    • 5
  1. 1.Department of Applied Mathematics and Computational SciencesUniversidad de CantabriaSantanderSpain
  2. 2.University of Castilla-La ManchaCiudad RealSpain
  3. 3.Environmental Hydraulics Institute “IH Cantabria”Universidad de CantabriaSantanderSpain
  4. 4.Department of Electrical EngineeringUniversity of Castilla-La ManchaCiudad RealSpain
  5. 5.LIDIAUniversity of A CoruñaA CoruñaSpain

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