Statistical estimation algorithms for repairs-time limit replacement scheduling under earning rate criteria

K e y w o r d s M a i n t e n a n c e , Repair-time limit, Imperfect repair, Earning rate, Lorenz statistics, Nonparametric. This research was partially supported by the Ministry Education, Culture, Sports, Science and Technology, Grantin-Aid for Exploratory Research; Grant No. 15651076 (2003-2005), Scientific Research (B); Grant No~ 16310116 (2004-2006), Scientific Research (C); Grant No. 16510128 (2004-2005), Nanzan University Pache Research Subsidy for 2005, and Research Program 2005 under the Institute for Advanced Studies of Hiroshima Shudo University. 0898-1221/06/$ see front matter @ 2006 Elsevier Ltd. All rights reserved. doi: 10. 1016/j.camwa.2005.11.004 Typeset by .43,~-TEX 346 T. DOHI et al. 1. I N T R O D U C T I O N Since the seminal contribution by Hastings [1], a large number of repair limit replacement problems were considered in literature [2-6]. Nakagawa [2] considered a simple repair-time limit replacement problem under an earning rate criterion. Nguyen and Murthy [4] analyzed a different repair-time limit policy with imperfect repair. In this paper, we focus on a mixed model of Nakagawa [2] and Nguyen and Murthy [4]. Consider a single-unit system where each spare unit is provided only by an order after a lead time, and each failed unit is repairable. When the unit fails, the decision maker (DM) estimates the completion time distribution of repair, which may be a possibly subjective one. If DM estimates that the repair is completed up to a prespecified time-limit, the repair is started immediately, otherwise, the spare unit is ordered with a lead time. Since the repair is imperfect, the unit repaired or even ordered can fail again during a finite time horizon. The problem for DM is to determine the optimal repair-time limit which maximizes any earning rate criterion. Dohi et al. [7,8] considered the above models with subjective repair time distribution under the expected cost criteria, and developed estimators of the optimal repair-time limits, by applying the Lorenz statistics or the Lorenz curve. Since the knowledge on the repair-time distribution is incomplete in general, such a statistical estimation method for the optimal repair-time limit will bc useful in the practical maintenance situation [9]. The Lorenz curve was first introduced by Lorenz [10] to describe income distributions. Since the Lorenz curve is essentially equivalent to the Pareto curve used in the quality control, it will be one of the most important statistics applied in every social sciences. The more general and tractable definition of the Lorenz curve was made by Gastwirth [11]. Goldie [12] proved the strong consistency of the empirical Lorenz curve and discovered its several convergence properties. Chandra and Singpurwalla [13] investigated the relationship between the total time on test statistics [9] and the Lorenz statistics, and derived a few aging and partial ordering properties. In this paper, we formulate the repair-time limit replacement model with imperfect repair [8] under earning rate criteria with and without discounting. First, after describing the notation and assumptions used here, the optimal repair-time limits which maximize the long-run average profit rate and the expected total discounted profit over an infinite time horizon are analytically derived. Next, we develop the nonparametric algorithms for estimating the optimal repair-time limits, provided that the complete sample data of repair time are given. The basic idea is to apply the Lorenz statistics and to transform the underlying algebraic problems to the graphical ones. Finally, we present some numerical examples to show that the proposed algorithms can bc useful to estimate the profit-based repair limit replacement schedule. 2. R E P A I R T I M E L I M I T R E P L A C E M E N T M O D E L 2.1. N o t a t i o n The repair time X for each unit is a nonnegative i.i.d, random variable. The decision maker (DM) has a subjective probability distribution function Pr{X < t} = G(t) on the repair time, with density g(t) (> 0) and finite mean 1/~ (> 0). Suppose that the distribution function G(t) E (0, 1) is arbitrary, continuous and strictly increasing in t E (0, c~), and, in addition, has an inverse function Gl ( . ) . Further, we define to E [0, oo): repair-time limit (decision variable), Fl(t), f l( t) , 1/pl (> 0): c.d.f., p.d.f., and mean of time to failure for a repaired unit, F2(t), f2(t), 1/p2 (> 0): c.d.f., p.d.f., and mean of time to failure for a new (spare) unit, k (> 0): penalty cost per unit time when the system is in down state, e0 (> 0): earning rate per unit operation time, el (> 0): repair cost per unit time, c (> 0): fixed cost associated with the ordering of a new unit, Statistical Estimation Algorithms " " ~ ~ , t i m e X l ip: q-. -q 347 . . . . . . . . i . . . . . . . . . 1/~; time X : failure (renewal point) 9 : recovery point for a unit : operation period m : repair period 9 -: lead time Figure 1. Configuration of repair-time limit replacement with imperfect repair. L (> 0): lead t ime for delivery of a new unit, (> 0): discount rate, /2{r = f o e x p ( ~ t ) ' O ( t ) d t for an a rb i t r a ry continuous function y)(.) (Laplace t ransform of ~( . ) ) , ~(-): = 1 r (survivor function). 2.2. M o d e l D e s c r i p t i o n Consider a single-unit repairable system, where each spare is provided only by an order after a lead t ime L and each failed unit is repairable. When the unit has failed at t ime t = 0, the DM wishes to de termine whether he or she should repair it or order a new spare. If DM est imates tha t the repair is completed within a prespecified t ime limit to E [0, co), then the repair is s ta r ted immedia te ly at t = 0 and completes at t ime t = X. After the complet ion of repair , the unit is s ta r ted to opera te again, bu t can fail again for a finite t ime span since the repair is imperfect. Then, the mean failure t ime is l / p 1 . On the other hand, if DM es t imates t ha t the repair t ime exceeds the t ime l imit to, then ti le failed unit is scrapped at t ime t = 0 and a new spare unit is ordered immediately. A new unit is delivered after the lead t ime L. Fur ther , the new unit can also fail for a finite t ime span and then the mean failure t ime is 1/#2. Wi thou t any loss of generality, it is assumed tha t the t ime required for replacement of a failed unit can be negligible. Under these model set t ing, we define the interval from the failure t ime to the following failure t ime as one cycle. Figure 1 depicts the configuration of the repa i r t ime limit replacement problem with imperfect repair under considerat ion.