• Title, Summary, Keyword: Mean Value Function

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MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS

  • Matkowski, Janusz
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.263-273
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    • 2013
  • A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation $$\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y$$, where M is a given mean and $f$, F, $g$, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.

A Comparative Study of Software Reliability Model Considering Log Type Mean Value Function (로그형 평균값함수를 고려한 소프트웨어 신뢰성모형에 대한 비교연구)

  • Shin, Hyun Cheul;Kim, Hee Cheul
    • Journal of the Korea Society of Digital Industry and Information Management
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    • v.10 no.4
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    • pp.19-27
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    • 2014
  • Software reliability in the software development process is an important issue. Software process improvement helps in finishing with reliable software product. Infinite failure NHPP software reliability models presented in the literature exhibit either constant, monotonic increasing or monotonic decreasing failure occurrence rates per fault. In this paper, proposes the reliability model with log type mean value function (Musa-Okumoto and log power model), which made out efficiency application for software reliability. Algorithm to estimate the parameters used to maximum likelihood estimator and bisection method, model selection based on mean square error (MSE) and coefficient of determination($R^2$), for the sake of efficient model, was employed. Analysis of failure using real data set for the sake of proposing log type mean value function was employed. This analysis of failure data compared with log type mean value function. In order to insurance for the reliability of data, Laplace trend test was employed. In this study, the log type model is also efficient in terms of reliability because it (the coefficient of determination is 70% or more) in the field of the conventional model can be used as an alternative could be confirmed. From this paper, software developers have to consider the growth model by prior knowledge of the software to identify failure modes which can be able to help.

A Study for Robustness of Objective Function and Constraints in Robust Design Optimization

  • Lee Tae-Won
    • Journal of Mechanical Science and Technology
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    • v.20 no.10
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    • pp.1662-1669
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    • 2006
  • Since randomness and uncertainties of design parameters are inherent, the robust design has gained an ever increasing importance in mechanical engineering. The robustness is assessed by the measure of performance variability around mean value, which is called as standard deviation. Hence, constraints in robust optimization problem can be approached as probability constraints in reliability based optimization. Then, the FOSM (first order second moment) method or the AFOSM (advanced first order second moment) method can be used to calculate the mean values and the standard deviations of functions describing constraints and object. Among two methods, AFOSM method has some advantage over FOSM method in evaluation of probability. Nevertheless, it is difficult to obtain the mean value and the standard deviation of objective function using AFOSM method, because it requires that the mean value of function is always positive. This paper presented a special technique to overcome this weakness of AFOSM method. The mean value and the standard deviation of objective function by the proposed method are reliable as shown in examples compared with results by FOSM method.

Parameter Estimation and Prediction for NHPP Software Reliability Model and Time Series Regression in Software Failure Data

  • Song, Kwang-Yoon;Chang, In-Hong
    • Journal of the Chosun Natural Science
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    • v.7 no.1
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    • pp.67-73
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    • 2014
  • We consider the mean value function for NHPP software reliability model and time series regression model in software failure data. We estimate parameters for the proposed models from two data sets. The values of SSE and MSE is presented from two data sets. We compare the predicted number of faults with the actual two data sets using the mean value function and regression curve.

Predictions of MLE and LSE in NHPP Software Reliability Model

  • Song, Kwang-Yoon;Chang, In-Hong;Lee, Seung-Woo
    • Journal of the Chosun Natural Science
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    • v.6 no.2
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    • pp.111-117
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    • 2013
  • We propose a mean value function for software failures in NHPP software reliability model. And we deal with the maximum likelihood estimation and the least squares estimation in the proposed mean value function. The explicit mean value function solution for the proposed model is presented by MLE and LSE in two data sets. The values of SSE and MSE is presented in two data sets by MLE and LSE. We compare the predicted number of faults with the actual two data sets using the proposed mean value function.

A Coverage-Based Software Reliability Growth Model for Imperfect Fault Detection and Repeated Construct Execution (불완전 결함 발견과 구문 반복 실행을 고려한 커버리지 기반 신뢰성 성장 모형)

  • Park, Joong-Yang;Park, Jae-Heung;Kim, Young-Soon
    • The KIPS Transactions:PartD
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    • v.11D no.6
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    • pp.1287-1294
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    • 2004
  • Recently relationships between reliability measures and the coverage have been developed for evaluation of software reliability. Particularly the mean value function of the coverage-based software reliability growth model is important because of its key role in rep-resenting the software reliability growth. In this paper, we first review the problems of the existing mean value functions with respect to the assumptions on which they are based. Then a new mean value function is proposed. The new mean value function is developed for a general testing environment in which imperfect fault detection and repeated construct execution are allowed. Finally performance of the proposed model is empirically evaluated by applying it to a real data set.

Determination of the Resetting Time to the Process Mean Shift based on the Cpm+ (Cpm+ 기준에서의 공정평균이동에 대한 재조정 기간 결정)

  • Lee, Do-Kyung
    • Journal of the Society of Korea Industrial and Systems Engineering
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    • v.41 no.1
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    • pp.110-117
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    • 2018
  • Machines and facilities are physically or chemically degenerated by continuous usage. One of the results of this degeneration is the process mean shift. By the result of degeneration, non-conforming products and malfunction of machine occur. Therefore a periodic preventive resetting the process is necessary. This type of preventive action is called 'preventive maintenance policy.' Preventive maintenance presupposes that the preventive (resetting the process) cost is smaller than the cost of failure caused by the malfunction of machine. The process mean shift problem is a field of preventive maintenance. This field deals the interrelationship between the quality cost and the process resetting cost before machine breaks down. Quality cost is the sum of the non-conforming item cost and quality loss cost. Quality loss cost is due to the deviation between the quality characteristics from the target value. Under the process mean shift, the quality cost is increasing continuously whereas the process resetting cost is constant value. The objective function is total costs per unit wear, the decision variables are the wear limit (resetting period) and the initial process mean. Comparing the previous studies, we set the process variance as an increasing concave function and set the quality loss function as Cpm+ simultaneously. In the Cpm+, loss function has different cost coefficients according to the direction of the quality characteristics from target value. A numerical example is presented.

Determination of Optimal Mean Value and Screening Limit for a Production Process with Logistic Function (로지스틱 함수를 갖는 생산공정에 대한 최적공정평균 및 스크리닝 한계선의 결정)

  • Hong, Sung Hoon
    • Journal of Korean Institute of Industrial Engineers
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    • v.29 no.3
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    • pp.239-246
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    • 2003
  • Individual items are produced continuously from an industrial process. Each item is checked to determine whether it satisfies a lower screening limit for the quality characteristic which is the weight of an expensive ingredient. If it does, it is sold at a regular price; if it does not, it is reprocessed or sold at a reduced price. The process mean may be adjusted to a higher value in order to reduce the proportion of the nonconforming items. Using a higher process mean, however, may result in a higher production cost. In this paper, the optimal process mean and lower screening limit are determined in situations where the probability that an item functions well is given by a logistic function of the quality characteristic. Profit models are constructed which involve four price/cost components; selling prices, cost from an accepted nonconforming item, and reprocessing and inspection costs. Methods of finding the optimal process mean and lower screening limit are presented and numerical examples are given.

Determination of the Resetting Time to the Process Mean Shift by the Loss Function (손실함수를 적용한 공정평균 이동에 대한 조정시기 결정)

  • Lee, Do-Kyung
    • Journal of the Society of Korea Industrial and Systems Engineering
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    • v.40 no.1
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    • pp.165-172
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    • 2017
  • Machines are physically or chemically degenerated by continuous usage. One of the results of this degeneration is the process mean shift. Under the process mean shift, production cost, failure cost and quality loss function cost are increasing continuously. Therefore a periodic preventive resetting the process is necessary. We suppose that the wear level is observable. In this case, process mean shift problem has similar characteristics to the maintenance policy model. In the previous studies, process mean shift problem has been studied in several fields such as 'Tool wear limit', 'Canning Process' and 'Quality Loss Function' separately or partially integrated form. This paper proposes an integrated cost model which involves production cost by the material, failure cost by the nonconforming items, quality loss function cost by the deviation between the quality characteristics from the target value and resetting the process cost. We expand this process mean shift problem a little more by dealing the process variance as a function, not a constant value. We suggested a multiplier function model to the process variance according to the analysis result with practical data. We adopted two-side specification to our model. The initial process mean is generally set somewhat above the lower specification. The objective function is total integrated costs per unit wear and independent variables are wear limit and initial setting process mean. The optimum is derived from numerical analysis because the integral form of the objective function is not possible. A numerical example is presented.

Frameworks for NHPP Software Reliability Growth Models

  • Park, J.Y.;Park, J.H.;Fujiwara, T.
    • International Journal of Reliability and Applications
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    • v.7 no.2
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    • pp.155-166
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    • 2006
  • Many software reliability growth models (SRGMs) based on nonhomogeneous Poisson process (NHPP) have been developed and applied in practice. NHPP SRGMs are characterized by their mean value functions. Mean value functions are usually derived from differential equations representing the fault detection/removal process during testing. In this paper such differential equations are regarded as frameworks for generating mean value functions. Currently available frameworks are theoretically discussed with respect to capability of representing the fault detection/removal process. Then two general frameworks are proposed.

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