Annals of Operations Research

, Volume 210, Issue 1, pp 309–331

# Estimating the parameters of a fatigue model using Benders’ decomposition

• Enrique Castillo
• Roberto Mínguez
• Antonio J. Conejo
• Beatriz Pérez
• Oscar Fontenla
Article

## 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.

## Keywords

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

## Notation

A

Constant

M

Constant

B

B0

Fixed value of the threshold value of log-lifetime

C

Endurance limit (logarithm of Δσ)

C0

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

(⋅)

Likelihood function

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

Upper bound for

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

Lower bound for

m

Number of pieces at a given level for Δσ

N

N

Ni

N0

n

Sample size

P

Probability

pi

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

Vi

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

zP

Objective function value

α

Variable used in the Benders decomposition master problem

β

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

Δσ

Stress range or amplitude

Δσ

Dimensionless stress range or amplitude

Δσi

Stress range of sample item i

Δσst

Ultimate strength

Δσ0

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

σM

Maximum stress

σm

Minimum stress

η

Vector of dual variables associated with equality constraints

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## 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