The Lorenz transform approach to the optimal repair-cost limit replacement policy with imperfect repair
RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 1, pp. 21-36.

In this paper, we consider a repair-cost limit replacement problem with imperfect repair and develop a graphical method to determine the optimal repair-cost limit which minimizes the expected cost per unit time in the steady-state, using the Lorenz transform of the underlying repair-cost distribution function. The method proposed can be applied to an estimation problem of the optimal repair-cost limit from empirical repair-cost data. Numerical examples are devoted to examine asymptotic properties of the non-parametric estimator for the optimal repair-cost limit.

Classification : 90B25,  62N05,  62N02
Mots clés : repair-cost limit, imperfect repair, Lorenz transform, nonparametric estimation, maintenance policy
@article{RO_2001__35_1_21_0,
     author = {Dohi, T. and Othman, F. S. and Kaio, N. and Osaki, Sunji},
     title = {The {Lorenz} transform approach to the optimal repair-cost limit replacement policy with imperfect repair},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {21--36},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {1},
     year = {2001},
     zbl = {0993.90031},
     mrnumber = {1841812},
     language = {en},
     url = {http://www.numdam.org/item/RO_2001__35_1_21_0/}
}
TY  - JOUR
AU  - Dohi, T.
AU  - Othman, F. S.
AU  - Kaio, N.
AU  - Osaki, Sunji
TI  - The Lorenz transform approach to the optimal repair-cost limit replacement policy with imperfect repair
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2001
DA  - 2001///
SP  - 21
EP  - 36
VL  - 35
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/RO_2001__35_1_21_0/
UR  - https://zbmath.org/?q=an%3A0993.90031
UR  - https://www.ams.org/mathscinet-getitem?mr=1841812
LA  - en
ID  - RO_2001__35_1_21_0
ER  - 
Dohi, T.; Othman, F. S.; Kaio, N.; Osaki, Sunji. The Lorenz transform approach to the optimal repair-cost limit replacement policy with imperfect repair. RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 1, pp. 21-36. http://www.numdam.org/item/RO_2001__35_1_21_0/

[1] B. Bergman, On Age Replacement and the Total Time on Test Concept. Scand. J. Statist. 6 (1979) 161-168. | MR 555079 | Zbl 0422.62084

[2] B. Bergman and B. Klefsjö, The Total Time on Test Concept and Its Use in Reliability Theory. Oper. Res. 32 (1984) 596-607. | MR 756006 | Zbl 0546.62070

[3] M. Chandra and N.D. Singpurwalla, Relationship between Some Notions Which Are Common to Reliability and Economics. Math. Oper. Res. 6 (1981) 113-121. | MR 618968 | Zbl 0496.90041

[4] T. Dohi, M. Matsushima, N. Kaio and S. Osaki, Nonparametric Repair-Limit Replacement Policies with Imperfect Repair. European J. Oper. Res. 96 (1996) 206-273. | Zbl 0917.90138

[5] T. Dohi, H. Koshimae, N. Kaio and S. Osaki, Geometrical Interpretations of Repair-Cost Limit Replacement Policies. International Journal of Reliability, Quality and Safety Engineering 4 (1997) 309-333.

[6] R.W. Drinkwater and N.A.J. Hastings, An Economical Replacement Model. Oper. Res. Quarterly 18 (1967) 121-138.

[7] J.L. Gastwirth, A General Definition of the Lorenz Curve. Econometrica 39 (1971) 1037-1039. | Zbl 0245.62082

[8] C.M. Goldie, Convergence Theorems for Empirical Lorenz Curves and Their Inverses. Adv. in Appl. Probab. 9 (1977) 765-791. | MR 478267 | Zbl 0383.60003

[9] N.A.J. Hastings, The Repair Limit Replacement Method. Oper. Res. Quarterly 20 (1969) 337-349.

[10] N. Kaio and S. Osaki, Optimum Repair Limit Policies with Cost Constraint. Microelectronics and Reliability 21 (1981) 597-599.

[11] B. Klefsjö, Reliability Interpretations of Some Concepts from Economics. Naval Res. Logist. Quarterly 31 (1984) 301-308. | Zbl 0538.90027

[12] L.M. Lee, Reliability: Probabilistic Models and Statistical Methods. Prentice-Hall, New Jersey (1985). | MR 1320503

[13] M.O. Lorenz, Methods of Measuring the Concentration of Wealth. J. Amer. Statist. Assoc. 9 (1893) 209-219.

[14] C.E. Love, R. Rodger and G. Blazenko, Repair Limit Policies for Vehicle Replacement. INFOR 20 (1982) 226-237. | Zbl 0501.90042

[15] C.E. Love and R. Guo, Utilizing Weibull Failure Rates in Repair Limit Analysis for Equipment Replacement/Preventive Maintenance Decisions. J. Oper. Res. Soc. 47 (1996) 1366-1376. | Zbl 0863.90069

[16] T. Nakagawa and S. Osaki, Optimum Ordering Policies with Lead Time for an Operating Unit. RAIRO Oper. Res. 12 (1978) 383-393. | Numdam | MR 516516 | Zbl 0402.90047

[17] D.G. Nguyen and D.N.P. Murthy, A Note on the Repair Limit Replacement Policy. J. Oper. Res. Soc. 31 (1980) 1103-1104. | MR 603209 | Zbl 0444.90042

[18] D.G. Nguyen and D.N.P. Murthy, Optimal Repair Limit Replacement Policies with Imperfect Repair. J. Oper. Res. Soc. 32 (1981) 409-416. | MR 612693 | Zbl 0452.90031

[19] K.S. Park, Cost Limit Replacement Policy under Minimal Repair. Microelectronics and Reliability 23 (1983) 347-349.

[20] R. Perez-Ocon, M.L. Gamiz-Perez and J.E. Ruiz-Castro, A Study of Different Aging Classes via Total Time on Test Transform and Lorenz Curves. Appl. Stochastic Models Data Anal. 13 (1998) 241-248. | Zbl 0913.60076

[21] T.G. Pham and N. Turkkan, The Lorenz and the Scaled Total-Time-on-Test Transform Curves: A Unified Approach. IEEE Trans. Reliability R-43 (1994) 76-84.