[논문]설비 생존곡선 추정을 위한 혼합형 Weibull 함수의 활용

이한교; 김경택; 오현승

설비 생존곡선 추정을 위한 혼합형 Weibull 함수의 활용
A Study on the Application of Mixed Weibull Function to Estimate Survivor Curves of Industrial Property 원문보기

산업경영시스템학회지 = Journal of society of korea industrial and systems engineering, v.30 no.1, 2007년, pp.66 - 73

이한교 (한남대학교 공과대학 산업경영공학과) , 김경택 (한남대학교 공과대학 산업경영공학과) , 오현승 (한남대학교 공과대학 산업경영공학과)

초록
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일반 투자안의 의사결정에서와 마찬가지로 산업설비의 경제성 분석에서도 가장 중요한 결정 요소 중의 하나가 설비의 생존곡선 추정이다. 설비의 자산 가치가 감소하는 원인은 여러 가지가 있으나, 여러 원인 중 물리적 훼손이 과거의 산업설비에서는 가장 중요한 원인이었으므로 기존의 생존모형 분석에서는 lows 생존곡선을 이용하여 설비의 생존곡선을 추정하였다. 그러나 새로운 기술상의 변화로 인한 첨단 생산시스템의 설비교체 분석 시에는 적합지 않다. 따라서, 본 연구에서 제안된 혼합형 Weibull 함수를 이용하여 설비의 폐기 형태를 추정함으로써 설비들의 실제적인 생존곡선을 정확하게 파악할 수 있다.

주제어

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문제 정의

The object of this study is to analyze the ability of the Weibull distribution to describe industrial property mortality characteristics. Thirty three sets of data were used in this study.
Kurtz [18]. This publication was the original one in a series from Iowa State College (now Iowa State University) dealing with the life character istics of physical property and their relationships to valuation and depreciation. Between 1931 and 1935, Winfrey con tinued data collection and added 111 property group curves to the 65 used 1931.

제안 방법

As outlined in the previous section of this study, statistical test were performed on the data that resulted from fitting the 25 test curves to the Weibull fittings and the Iowa type curves. The results for each of these statistical tests are sum marized in <Table 3>.
In this study, the survivor function is the medium for esti mating the parameters and an iterative process is investigated. The graphical method gives fast but rough estimate, and con sequently is not explored in detail.
The study undertaken in this study was prompted because of the search for a relatively simple mathematical expression which would describe industrial property mortality character istics. While there are undoubtedly many expressions which could be tried, for theoretical reasons the Weibull distribution was chosen.
The test involved computer fitting of the 25 test curves to both the Weibull fitting array and Iowa type curves to determine which produced the best fit for each teat curve. The test curves were held constant in each case and the Weibull fitting and Iowa type curves adjusted until the clos est fit were found.
A total of 25 test accounts, all with 0% surviving, were pre pared for testing. The tests involved fitting of each test curve to each set of curves to determine which set produced the best fit based on area difference between the test curve and the fit curves.
This test concentrated on and compared mean optimal areas of the Weibull fittings and Iowa curves and had some what better discriminating powers than the other two tests. This test also failed to adduce evidence the Iowa type curves produced better fits than the Weibull fitting, since the T val ue calculated was 1.
Three statistical procedures were chosen for analysis of the 25 test curves fit to Weibull and the Iowa type curves. The procedures were chosen so that conclusions could be drawn as to which curve set best fit the test curves.

대상 데이터

Thirty three sets of data were used in this study. These data were collected by Russo [15] to revalidate the Iowa curves in use under current economic, techno logical, and managerial conditions.
The data from totally separate companies for testing were as follows: electric utilities, gas utilities, telephone company. Thus, a total of 25 test curves, all 0% surviving, were prepared for testing. Preparation involved formulation of survivor curves by the retirement rate method so that graphic com parisons with Weibull fitting and Iowa type curves could be accomplished.

성능/효과

1. Based on the area difference and visual review of the fits, the Weibull fitting produced 10 best fits, the Iowa type curves 15 best fits. This is reflected in the remarks column of <Table 2>.
2. Based on the area difference and visual review of the fits, the Weibull fittings produced 10 best fits, the Iowa curves 15 best fits.
3. Of the 15 best curves fit by the Iowa type curve, 2 were origin modal, 6 left modal, 1 symmetrical modal, and 6 right modal.
4. Of the total array of the 31 Iowa type curves, 15 curves did not produce any fits, no matter whether best fit or not. These non-fit curves included 9 symmetrical modal, 1 left modal, 2 right modal, 2 origin modal, and the square curve.
tests. This test also failed to adduce evidence the Iowa type curves produced better fits than the Weibull fitting, since the T val ue calculated was 1.96. The value of T would have to have been substantially greater than 2.
This test failed to adduce evidence that the Iowa type curves produced better fits than the Weibull fitting, since Z value calculated was 1.0. The value of Z would have to have been substantially greater than 1.
68. Thus, the test failed to adduce evidence the Iowa type curves produced better fits than the Weibull fitting. The value of Z would have to have been substantially greater than 1.

참고문헌 (19)

Couch, F. V. B, Jr.; 'Classification of Type O Retirement Characteristics of Industrial Property,' M. S. thesis, Iowa State University of Science and Technology, Ames, Iowa, U.S.A., 1957
Cox, D. R.; 'The Analysis of Exponentially Distributed Life-times with Two Types of Failure,' Journal of Research Statistical Society B, 21 : 411-421, 1959
Elanat, R. C. and Jonson, N. L.; Survival Models and Data Analysis, John Wiley and Sons, Inc., New York, 2000
Gross, A. J. and Clark, V. A.; Survival Distributions: Reliability Applications in the Biomedical Science, John Wiley and Sons, Inc., New York, 1995
Hager, H. W. and Bain, L. J.; 'Inferential Procedures for the Generalized Gamma Distribution,' Journal of American Statistical Association, 65 : 1601-1609, 1967

상세보기
Harter, H. L.; 'Maximum Likelihood of the Parameters of Four Parameter Generalized Gamma Population from Complete and Censored Samples,' Technometrics, 9 : 159-165, 1965
Harley, H. O.; 'The Modified Gauss-Newton Method for Fitting of Nonlinear Regression Function by Least Squares,' Technometrics, 3(2) : 269-280, 1961

상세보기
Jonson, N. I. and Kortz, S.; Distributions in Statistics: Continuous Univariate Distribution, Vol. 1., Boston : Houghton Mifflin, 1970
Mann, N. R., Schafer, R. E., and Singpurwalla, N. D.; Methods for Statistical Analysis of Reliability and Life Data, John Wiley and Sons, Inc., New York, 1994
Marston, A., Winfrey, R., and Hemstead, J. C.; Engineering Valuation and Depreciation, Iowa State University Press, Ames, Iowa, 1979
Oh, H. S.; 'A Study on the Application of Weibull Survivor Curves to Estimate Mortality Characteristics of Industrial Property,' Journal of Society of Korea Industrial and Systems Engineering, 23 : 113-122, 2000
Oh, H. S., Kim, C. S., and Cho, J. H.; 'Estimation of Retirement Rate on Domestic Industrial Property,' Journal of Society of Korea Industrial and Systems Engineering, 25(4) : 79-85, 2002
Parr, V. B. and Webster, J. T.; 'A Method for Discriminating Between Failure Density Functions Used in Reliability Predictions,' Technometrics, 9 : 1-10, 1965
Prentice, R. L.; 'Discrimination Among Some Parametric Models,' Biometrika, 62 : 607-614, 1975

상세보기
Russo, J. G.; 'Revalidation of Iowa Type Survivor Curves,' Ph.D. Dissertation, Iowa State University of Science and Technology, Ames, Iowa, U.S.A., 1978
Stacy, E. W.; 'A Generalization of the Gamma Distribution,' Annual Mathematical Statistics, 33 : 1187-1192, 1962

상세보기
Weibull. W. A.; 'A Statistical Distribution Function of Wide Applicability,' Journal of Applied Mechanics, 18 : 293-297, 1951
Winfrey, R. and Kurtz, E. B.; Life Characteristics of Physical Property, ERI Bulletin 103, Iowa State University of Science and Technology, Ames, Iowa, U.S.A., 1931
Winfrey, R.; Statistical Analysis of Industrial Property Retirement, Revised edition: ERI Bulletin 125, Iowa State University of Science and Technology, Ames, Iowa, U.S.A., 1967

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