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Article
Publication date: 12 September 2020

Niveditha A and Ravichandran Joghee

While Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for…

Abstract

Purpose

While Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for two-parameter Weibull distribution that is useful in many life test data analyses.

Design/methodology/approach

In the theory of Six Sigma, most of the processes are assumed normal and Six Sigma metrics are determined for such a process of interest. In reliability studies non-normal distributions are more appropriate for life tests. In this paper, a theoretical procedure is developed for determining Six Sigma metrics when the underlying process follows two-parameter Weibull distribution. Numerical evaluations are also considered to study the proposed method.

Findings

In this paper, by matching the probabilities under different normal process-based sigma quality levels (SQLs), we first determined the Six Sigma specification limits (Lower and Upper Six Sigma Limits- LSSL and USSL) for the two-parameter Weibull distribution by setting different values for the shape parameter and the scaling parameter. Then, the lower SQL (LSQL) and upper SQL (USQL) values are obtained for the Weibull distribution with centered and shifted cases. We presented numerical results for Six Sigma metrics of Weibull distribution with different parameter settings. We also simulated a set of 1,000 values from this Weibull distribution for both centered and shifted cases to evaluate the Six Sigma performance metrics. It is found that the SQLs under two-parameter Weibull distribution are slightly lesser than those when the process is assumed normal.

Originality/value

The theoretical approach proposed for determining Six Sigma metrics for Weibull distribution is new to the Six Sigma Quality practitioners who commonly deal with normal process or normal approximation to non-normal processes. The procedure developed here is, in fact, used to first determine LSSL and USSL followed by which LSQL and USQL are obtained. This in turn has helped to compute the Six Sigma metrics such as defects per million opportunities (DPMOs) and the parts that are extremely good per million opportunities (EGPMOs) under two-parameter Weibull distribution for lower-the-better (LTB) and higher-the-better (HTB) quality characteristics. We believe that this approach is quite new to the practitioners, and it is not only useful to the practitioners but will also serve to motivate the researchers to do more work in this field of research.

Details

International Journal of Quality & Reliability Management, vol. 38 no. 4
Type: Research Article
ISSN: 0265-671X

Keywords

Article
Publication date: 1 February 1998

Claudio Ruggieri and Robert H. Dodds

Describes a probabilistic methodology for fracture assessments of flawed structures constructed of ferritic steels using the research code WSTRESS. The probabilistic formulation…

Abstract

Describes a probabilistic methodology for fracture assessments of flawed structures constructed of ferritic steels using the research code WSTRESS. The probabilistic formulation for cleavage fracture implements a multiaxial form of the weakest link model which couples the macroscopic fracture behavior with a micromechanics model based on the statistics of microcracks. The Weibull stress, σw, emerges as a suitable near‐tip parameter to provide a connection between the microregime of failure and remote loading (J). WSTRESS builds on an iterative procedure to incorporate a 3‐D finite element description of the crack‐tip stress field and measured values of fracture toughness to calibrate the Weibull modulus, m, and the scale parameter, σu. Specific features of the code include statistical inference of Weibull parameters based on uncensored and censored models (with maximum likelihood method), construction of confidence intervals, several definitions for the near‐tip fracture process zone and other general facilities such as spatial integration of the stress field (to incorporate the random orientation of microcracks) and stochastic simulation of fracture data using the Monte Carlo method. The code also includes a convenient free‐form command language and a seamless interface with finite element results files stored in Patran binary or ASCII format.

Details

Engineering Computations, vol. 15 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 12 February 2021

Ahmet Esat Suzer and Aziz Kaba

The purpose of this study is to describe precisely the wind speed regime and characteristics of a runway of an International Airport, the north-western part of Turkey.

Abstract

Purpose

The purpose of this study is to describe precisely the wind speed regime and characteristics of a runway of an International Airport, the north-western part of Turkey.

Design methodology approach

Three different probability distributions, namely, Inverse Gaussian (IG), widely used two-parameter Weibull and Rayleigh distributions in the literature, are used to represent wind regime and characteristics of the runway. The parameters of each distribution are estimated by the pattern search (PS)-based heuristic algorithm. The results are compared with the other three methods-based numerical computation, including maximum-likelihood method, moment method (MoM) and power density method, respectively. To evaluate the fitting performance of the proposed method, several statistical goodness tests including the mostly used root mean square error (RMSE) and chi-squared (X2) are conducted.

Findings

In the light of the statistical goodness tests, the results of the IG-based PS attain better performance than the classical Weibull and Rayleigh functions. Both the RMSE and X2 values achieved by the IG-based PS method lower than that of Weibull and Rayleigh distributions. It exhibits a better fitting performance with 0.0074 for RMSE and 0.58 × 10−4 for X2 for probability density function (PDF) in 2012 and with RMSE of 0.0084 and X2 of 0.74 × 10−4 for PDF in 2013. As regard the cumulative density function of the measured wind data, the best results are found to be Weibull-based PS with RMSE of 0.0175 and X2 of 3.25 × 10−4 in 2012. However, Weibull-based MoM shows more excellent ability in 2013, with RMSE of 0.0166 and X2 of 2.94 × 10−4. Consequently, it is considered that the results of this study confirm that IG-based PS with the lowest error value can a good choice to model more accurately and characterize the wind speed profile of the airport.

Practical implications

This paper presents a realistic point of view regarding the wind regime and characteristics of an airport. This study may cast the light on researchers, policymakers, policy analysts and airport designers intending to investigate the wind profile of a runway at the airport in the world and also provide a significant pathway on how to determine the wind distribution of the runway.

Originality value

Instead of the well-known Weibull distribution for the representing of wind distribution in the literature, in this paper, IG distribution is used. Furthermore, the suitability of IG to represent the wind distribution is validated when compared with two-parameter Weibull and Rayleigh distributions. Besides, the performance and efficiency of PS have been evaluated by comparing it with other methods.

Details

Aircraft Engineering and Aerospace Technology, vol. 93 no. 2
Type: Research Article
ISSN: 1748-8842

Keywords

Article
Publication date: 3 January 2017

Reza Ghavijorbozeh and Ali Zeinal Hamadani

The purpose of this paper is to understand the consequence of the use of mixed Weibull distribution in the cell formation problem. In reliability theory, a mixed distribution is…

Abstract

Purpose

The purpose of this paper is to understand the consequence of the use of mixed Weibull distribution in the cell formation problem. In reliability theory, a mixed distribution is used for more than one hazard cause, and the Weibull distribution can be used for ascendant, monotonous and descendant failure rate. Here, the authors mixed these two theme and use it in a common problem in group technology.

Design/methodology/approach

In this paper, the authors made a non-polynomial-hard mathematical model based on past research and solved it with an exact algorithm. The algorithm is coded and solved in GAMS to illustrate the model, and the authors use simulation. A common numerical example is solved with the model, and the results are compared.

Findings

Reliability analysis model based on the mixed Weibull distribution approach will give options to a user to select the suitable failure rate and modes for a specific situation. If the user uses the exponential or Weibull distribution instead of the mixed Weibull distribution, the calculated cost and reliability are wrong; therefore, it leads to user making wrong decisions.

Originality/value

The model the authors use is the one used in past research, but in the past, researchers did not use the mixed distribution for explaining failure time. Therefore, the model can be considered as a new and more complete model.

Article
Publication date: 5 September 2017

Ijjou Tizgui, Fatima El Guezar, Hassane Bouzahir and Brahim Benaid

The purpose of this study is to select the most accurate and the most efficient method in estimating Weibull parameters at Agadir region in Morocco.

Abstract

Purpose

The purpose of this study is to select the most accurate and the most efficient method in estimating Weibull parameters at Agadir region in Morocco.

Design/methodology/approach

In this paper, Weibull distribution is used to model the wind speed in hourly time series format. Since several methods are used to adjust the Weibull distribution to the measured data, in reporting and analyzing the easiest and the most effective method, seven methods have been investigated, namely, the graphical method (GM), the maximum likelihood method (MLM), the empirical method of Justus (EMJ), the empirical method of Lysen (EML), the energy pattern factor method (EPFM), Mabchour’s method (MMab) and the method of moments (MM).

Findings

According to the statistical analysis tools (the coefficient of determination, root mean square error and chi-square test), it was found that for five months, the MLM presents more efficiency, and for four months, EMJ is ranked first and it is ranked second for February. To select only one method, the selected methods (MLM and EMJ) were compared by calculating the error in estimating the power density using Weibull distribution adjusted by those methods. The average error was found to be −0.51 and −4.56 per cent for MLM and EMJ, respectively.

Originality/value

This investigation is the first of its kind for the studied region. To estimate the available wind power at Agadir in Morocco, investors can directly use MLM to determine the Weibull parameters at this site.

Details

International Journal of Energy Sector Management, vol. 11 no. 4
Type: Research Article
ISSN: 1750-6220

Keywords

Article
Publication date: 28 August 2019

Fatemeh FaghihKhorasani, Mohammad Zaman Kabir, Mehdi AhmadiNajafabad and Khosrow Ghavami

The purpose of this paper is to provide a method to predict the situation of a loaded element in the compressive stress curve to prevent failure of crucial elements in…

Abstract

Purpose

The purpose of this paper is to provide a method to predict the situation of a loaded element in the compressive stress curve to prevent failure of crucial elements in load-bearing masonry walls and to propose a material model to simulate a compressive element successfully in Abaqus software to study the structural safety by using non-linear finite element analysis.

Design/methodology/approach

A Weibull distribution function was rewritten to relate between failure probability function and axial strain during uniaxial compressive loading. Weibull distribution parameters (shape and scale parameters) were defined by detected acoustic emission (AE) events with a linear regression. It was shown that the shape parameter of Weibull distribution was able to illustrate the effects of the added fibers on increasing or decreasing the specimens’ brittleness. Since both Weibull function and compressive stress are functions of compressive strain, a relation between compressive stress and normalized cumulative AE hits was calculated when the compressive strain was available. By suggested procedures, it was possible to monitor pretested plain or random distributed short fibers reinforced adobe elements (with AE sensor and strain detector) in a masonry building under uniaxial compression loading to predict the situation of element in the compressive stress‒strain curve, hence predicting the time to element collapse by an AE sensor and a strain detector. In the predicted compressive stress‒strain curve, the peak stress and its corresponding strain, the stress and strain point with maximum elastic modulus and the maximum elastic modulus were predicted successfully. With a proposed material model, it was illustrated that the needed parameters for simulating a specimen in Abaqus software with concrete damage plasticity were peak stress and its corresponding strain, the stress and strain point with maximum elastic modulus and the maximum elastic modulus.

Findings

The AE cumulative hits versus strain plots corresponding to the stress‒strain curves can be divided into four stages: inactivity period, discontinuous growth period, continuous growth period and constant period, which can predict the densifying, linear, non-linear and residual stress part of the stress‒strain relationship. By supposing that the relation between cumulative AE hits and compressive strain complies with a Weibull distribution function, a linear analysis was conducted to calibrate the parameters of Weibull distribution by AE cumulative hits for predicting the failure probability as a function of compressive strain. Parameters of m and θ were able to predict the brittleness of the plain and tire fibers reinforced adobe elements successfully. The calibrated failure probability function showed sufficient representation of the cumulative AE hit curve. A mathematical model for the stress–strain relationship prediction of the specimens after detecting the first AE hit was developed by the relationship between compressive stress versus the Weibull failure probability function, which was validated against the experimental data and gave good predictions for both plain and short fibers reinforced adobe specimens. Then, the authors were able to monitor and predict the situation of an element in the compressive stress‒strain curve, hence predicting the time to its collapse for pretested plain or random distributed short fibers reinforced adobe (with AE sensor and strain detector) in a masonry building under uniaxial compression loading by an AE sensor and a strain detector. The proposed model was successfully able to predict the main mechanical properties of different adobe specimens which are necessary for material modeling with concrete damage plasticity in Abaqus. These properties include peak compressive strength and its corresponding axial strain, the compressive strength and its corresponding axial strain at the point with maximum compressive Young’s modulus and the maximum compressive Young’s modulus.

Research limitations/implications

The authors were not able to decide about the effects of the specimens’ shape, as only cubic specimens were chosen; by testing different shape and different size specimens, the authors would be able to generalize the results.

Practical implications

The paper includes implications for monitoring techniques and predicting the time to the collapse of pretested elements (with AE sensor and strain detector) in a masonry structure.

Originality/value

This paper proposes a new method to monitor and predict the situation of a loaded element in the compressive stress‒strain curve, hence predicting the time to its collapse for pretested plain or random distributed short fibers reinforced adobe (with AE sensor and strain detector) in a masonry building under uniaxial compression load by an AE sensor and a strain detector.

Article
Publication date: 2 February 2015

Edilson M. Assis, Ernesto P. Borges, Silvio A.B. Vieira de Melo and Leizer Schnitman

The purpose of this paper is to compare four life data models, namely the exponential and the Weibull models, and their corresponding generalized versions, q-exponential and q

Abstract

Purpose

The purpose of this paper is to compare four life data models, namely the exponential and the Weibull models, and their corresponding generalized versions, q-exponential and q-Weibull models, by means of one practical application.

Design/methodology/approach

Application of the models to a practical example (a welding station), with estimation of parameters by the use of the least squares method, and the Akaike Information Criterion (AIC).

Findings

The data of the example considered in this paper is divided into three regimes, decreasing, constant and increasing failure rate, and the q-Weibull model describes the bathtub curve displayed by the data with a single set of parameters.

Practical implications

The simplicity and flexibility of the q-Weibull model may be very useful for practitioners of reliability analysis, and its benefits surpasses the inconvenience of the additional parameter, as AIC shows.

Originality/value

The q-Weibull model is compared in detail with other three models, through the analysis of one example that clearly exhibits a bathtub curve, and it is shown that it can describe the whole time range with a single set of parameters.

Details

International Journal of Quality & Reliability Management, vol. 32 no. 2
Type: Research Article
ISSN: 0265-671X

Keywords

Article
Publication date: 26 August 2014

Bruce J. Sherrick, Christopher A. Lanoue, Joshua Woodard, Gary D. Schnitkey and Nicholas D. Paulson

The purpose of this paper is to contribute to the empirical evidence about crop yield distributions that are often used in practical models evaluating crop yield risk and…

Abstract

Purpose

The purpose of this paper is to contribute to the empirical evidence about crop yield distributions that are often used in practical models evaluating crop yield risk and insurance. Additionally, a simulation approach is used to compare the performance of alternative specifications when the underlying form is not known, to identify implications for the choice of parameterization of yield distributions in modeling contexts.

Design/methodology/approach

Using a unique high-quality farm-level corn yield data set, commonly used parametric, semi-parametric, and non-parametric distributions are examined against widely used in-sample goodness-of-fit (GOF) measures. Then, a simulation framework is used to assess the out-of-sample characteristics by using known distributions to generate samples that are assessed in an insurance valuation context under alternative specifications of the yield distribution.

Findings

Bias and efficiency trade-offs are identified for both in- and out-of-sample contexts, including a simple insurance rating application. Use of GOF measures in small samples can lead to inappropriate selection of candidate distributions that perform poorly in straightforward economic applications. The β distribution consistently overstates rates even when fitted to data generated from a β distribution, while the Weibull consistently understates rates; though small sample features slightly favor Weibull. The TCMN and kernel density estimators are least biased in-sample, but can perform very badly out-of-sample due to overfitting issues. The TCMN performs reasonably well across sample sizes and initial conditions.

Practical implications

Economic applications should consider the consequence of bias vs efficiency in the selection of characterizations of yield risk. Parsimonious specifications often outperform more complex characterizations of yield distributions in small sample settings, and in cases where more demanding uses of extreme-event probabilities are required.

Originality/value

The study helps provide guidance on the selection of distributions used to characterize yield risk and provides an extensive empirical demonstration of yield risk measures across a high-quality set of actual farm experiences. The out-of-sample examination provides evidence of the impact of sample size, underlying variability, and region of the probability measure used on the performance of candidate distributions.

Details

Agricultural Finance Review, vol. 74 no. 3
Type: Research Article
ISSN: 0002-1466

Keywords

Article
Publication date: 1 October 2018

Mahesh Narayan Dhawalikar, V. Mariappan, P.K. Srividhya and Vishal Kurtikar

Degraded failures and sudden critical failures are quite prevalent in industries. Degradation processes commonly belong to Weibull family and critical failures are found to follow…

Abstract

Purpose

Degraded failures and sudden critical failures are quite prevalent in industries. Degradation processes commonly belong to Weibull family and critical failures are found to follow exponential distribution. Therefore, it becomes important to carry out reliability and availability analysis of such systems. From the reported literature, it is learnt that models are available for the situations where the degraded failures as well as critical failures follow exponential distribution. The purpose of this paper is to present models suitable for reliability and availability analysis of systems where the degradation process follows Weibull distribution and critical failures follow exponential distribution.

Design/methodology/approach

The research uses Semi-Markov modeling using the approach of method of stages which is suitable when the failure processes follow Weibull distribution. The paper considers various states of the system and uses state transition diagram to present the transition of the system among good state, degraded state and failed state. Method of stages is used to convert the semi-Markov model to Markov model. The number of stages calculated in Method of stages is usually not an integer value which needs to be round off. Method of stages thus suffers from the rounding off error. A unique approach is proposed to arrive at failure rates to reduce the error in method of stages. Periodic inspection and repairs of systems are commonly followed in industries to take care of system degradation. This paper presents models to carry out reliability and availability analysis of the systems including the case where degraded failures can be arrested by appropriate inspection and repair.

Findings

The proposed method for estimating the degraded failure rate can be used to reduce the error in method of stages. The models and the methodology are suitable for reliability and availability analysis of systems involving degradation which is very common in systems involving moving parts. These models are very suitable in accurately estimating the system reliability and availability which is very important in industry. The models conveniently cover the cases of degraded systems for which the model proposed by Hokstad and Frovig is not suitable.

Research limitations/implications

The models developed consider the systems where the repair phenomenon follows exponential and the failure mechanism follows Weibull with shape parameter greater than 1.

Practical implications

These models can be suitably used to deal with reliability and availability analysis of systems where the degradation process is non-exponential. Thus, the models can be practically used to meet the industrial requirement of accurately estimating the reliability and availability of degradable systems.

Originality/value

A unique approach is presented in this paper for estimating degraded failure rate in the method of stages which reduces the rounding error. The models presented for reliability and availability analyses can deal with degradable systems where the degradation process follows Weibull distribution, which is not possible with the model presented by Hokstad and Frovig.

Details

International Journal of Quality & Reliability Management, vol. 35 no. 9
Type: Research Article
ISSN: 0265-671X

Keywords

Article
Publication date: 25 May 2010

Chung‐Chu Pan and Liou Chu

In daily life, many products, such as light bulbs, fuses, dry batteries, fireworks, semiconductors, are non‐repairable. The non‐repairable products are usually referred to as…

Abstract

Purpose

In daily life, many products, such as light bulbs, fuses, dry batteries, fireworks, semiconductors, are non‐repairable. The non‐repairable products are usually referred to as one‐shot products, or as failed products that are not worth repairing. A one‐shot product is usually required to perform a function once only since its use is normally accompanied by an irreversible reaction or process, e.g. chemical reaction or physical destruction. However, most one‐shot products being stored or deployed are usually not under continuous surveillance. The failed products can only be found by inspection or at the beginning of operation. Therefore, this paper seeks to assess the reliability of one‐shot products.

Design/methodology/approach

The study considers a series system consisting of m components with lifetime following Weibull distribution, and applies a competing failure model to investigate the proposed series system for one‐shot products. The maximum likelihood estimators (MLEs) of parameters of the Weibull distribution based on the quantal‐response data in the proposed series system are derived. The model is illustrated with a two‐component series system, and the statistical properties of the MLEs are investigated by Monte Carlo simulation under the two‐stage inspection scheme and the three‐stage inspection scheme.

Findings

Simulation results reveal not only that the MLEs of Weibull parameters gradually approximate the true values of Weibull parameters under rising sample sizes, but also that the precision and accuracy of the MLEs of parameters increase with an increasing sample size. Furthermore, the standard deviations of MLEs of Weibull parameters for the two‐stage inspection scheme are smaller than those for the three‐stage inspection scheme.

Originality/value

The paper focuses on the reliability assessment of one‐shot products, e.g. firework, ammunition, airbag, injector, dry battery, with Weibull components lifetime distribution.

Details

International Journal of Quality & Reliability Management, vol. 27 no. 5
Type: Research Article
ISSN: 0265-671X

Keywords

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