数理解析研究所講究録
1048 巻 1998 年 37-52

37Graphical Methods for
Repair-Limit Replacement Policies$\mathrm{D}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}/\mathrm{E}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$OptimalTadashi DOHI, Kentaro TAKEITA and Shunji OSAKI
Department of Industrial and Systems Engineering, Hiroshima University
4-1 Kagamiyama 1 Chome, Higashi-Hiroshima 739-8527, Japan.
Abstract: In this paper, we consider two kinds of repair-limit replacement models and develop
the corresponding graphical methods to estimate the optimal repair-time limits which minimize
the expected costs per unit time in the steady-state. Then, both the total time on test statistics
and the Lorenz statistics play important roles. Some analytical results are derived to describe therelationship between two models. Numerical examples are devoted to illustrate the asymptotic
optimality of non-parametric estimators for the optimal repair-limit policies.

Keywords: repair-limit replacement policies, graphical methods, total time on test statistics,
Lorenz statistics, partial ordering, non-parametric estimation, stochastic optimization models.
1. INTRODUCTION
In general, system maintenance models may be classified into two categories; preventive maintenance and corrective maintenance. The preventive maintenance is executed in advance to avoid
a catastrophic failure state. On the other hand, the corrective maintenance is made to place
recovery actions efficiently after failures occur. The repair-limit replacement problems determine how to design the recovery mechanism of a system using two maintenance alternatives;
repair and replacement, in terms of cost minimization. First this problem was considered by
Hestings [1] for army vehicles and proposed three methods of optimizing the repair-limit policies
by simulation, hill-climbing and dynamic programming. Nakagawa and Osaki $[2, 3]$ , Okumoto
and Osaki [4], Nguyen and Murthy $[5, 6]$ and Kaio and Osaki [7] reformulated the Hastings’ original model from the viewpoint of renewal reward processes and discussed different repair-limit
replacement problems.
In the most repair-limit replacement problems [2-7], it is assumed that the distribution functionof the completion time to repair a failed unit is arbitrary but known. This assumption seems to
be rather strong in many practical situations. To this end, practitioners have to determine therepair-time limit under incomplete information on the repair-time distribution in most cases.
Applying a graphical idea by Bergman [8] and Bergman and Klefsj\"o [9-11], the authors [1215] analyzed several repair-limit replacement problems and proposed the corresponding nonto estimate the optimal repair-time limits from the complete sample ofparametricrepair-time data.
total time on test (TTT) concept (see, $e.g$ . Barlow and Campo [16] and Barlow [17])Thenis very useful to develop the estimation procedure. In other words, some kinds of repair-limit
andreplacement problems are reduced to the well-known age replacement-type )aHowever,suchconcept.
TTT
on
the
based
techniquegraphical
can be solved by the similar
instance,
cost
if
the
For
problems.
maintenanceforallavailable
alwaysspecific method does notcriterion such as the expected cost per unit time in the steady-state can not be represented by
two variables; the repair-time distribution function and the associated scaled TTT transform,
the method mentioned above will lose its usefulness. Hence, alternative devices instead of the
TTT statistics should be applied to the graphical and non-parametric estimation method for
different types of maintenance problems.
$1\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}_{\mathrm{S}}$$\mathrm{t}\mathrm{b}\mathrm{e}$$1$$\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}[8- 11]$

38
In this paper, we propose estimation methods based on the Lorenz curve as well as the TTT
statistics for two repair-limit replacement problems. The Lorenz curve was introduced first byLorenz [18] in Economics to describe income distributions. Since the Lorenz curve is essentiallyequivalent to the Pareto curve used in the quality control, it will be one of the most importantstatistics in every social science. The more general and tractable definition of the Lorenz curve
was made by Gastwirth [19]. Goldie [20] proved the strong consistency of the empirical Lorenz
curve and discovered several convergence properties of it. Chandra and Singpurwalla [21] andKlefsj\"o [22] investigated the relationship between the TTT statistics and the Lorenz statistics
and derived a few aging and partial ordering properties. Recently, Aly [23] developed the testingfor the Lorenz ordering.
It should be noted that the underlying repair-limit replacement problems have to be analyzed
by using different devices from their cost structure, respectively. In other words, both devices
are not used at the same time. The paper is organized as follows. In Section 2, we introduce
two repair-limit replacement problems considered by Nakagawa and Osaki $[2, 3]$ and develop
the corresponding graphical methods to derive the optimal repair-time limits which minimize
the expected costs per unit time in the steady-state. In Section 3, we obtain the comparative
results to describe the relationship between two models. In Section 4, the statistical estimation
problems are discussed. We show analytically and numerically that estimators of the optimal
repair-time limits have strong consistency. Finally, the paper is concluded with some remarks.

2. REPAIR-LIMIT REPLACEMENT MODELS

2.1 Model 1

Consider a single unit system, where each spare is provided only by an order after a leadtime $L(>0)$ and each failed unit is repairable. The original unit begins operating at time .
the unit has failed, the repair is started
The mean lifetime for each unit is $1/\lambda(>0)$ .
immediately. If the repair is completed up to the time limit for repair $t_{0}\in[0, \infty)$ , then theunit is installed at that time. It is assumed that the unit once repaired is presumed as good as$i.e$ . the repair is not completed after thenew. However, if the repair time is greater thantime , the repair is retired and the failed unit is scrapped. Then, the spare unit is ordered
immediately and delivered after the lead time . The time required for replacement is negligible
for convenience. The repair time for each unit has an arbitrary distribution $G(t)$ with density $g(t)$is assumed to have an inverse function, $i.e$ .
and finite mean $1/\mu(>0)$ , where the function, and to be absolutely continuous and strictly increasing. Without any loss of generality,
we assume $G(\mathrm{O})=0$ and $\lim_{tarrow\infty^{G}}(t)=1$ . Under these model assumptions, we define the timeinterval from the start of the operation to the following start as one cycle. The configuration ofModel 1 is depicted in Fig. 1.
Next, we consider the cost structure. The costs considered in this paper are the following;$0$$\backslash \mathbb{R}\mathrm{e}\mathrm{n}$$t_{0},$$t_{0}$$L$$G(\cdot)$$G^{-1}(\cdot)$$k_{r}(>0)$: a cost per unit repair time$k_{s}(>0)$: a cost per shortage period$c(>0)$ :an ordering cost for each spare unit.

We make the$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{U}\mathrm{l}\mathrm{n}\mathrm{p}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n};}$(A-1) $k_{r}L<c$ .

This as sumption implies that the unit ordering cost is greater than the repair cost during theinterval $[0, L],\dot{i}.e$ . until the delivery of a new unit. For an infinite planning horizon, it will beappropriate to adopt an expected cost per unit time in the steady-state. Since the mean timecycle isofone-

$T_{1}$(to)$=$$\int_{0}^{t_{0}}(1/\lambda+t)dG(t)+\int_{t_{0}}^{\infty}(1/\lambda+t_{0}+L)dG(t)$

39
$=$

(1)

$1/ \lambda+\int_{0}^{t_{0}}\overline{G}(t)dt+L\overline{G}(t_{0})$and the total expected cost for one cycle is$V_{1}(t_{0})=(k_{r}+k_{s}) \int_{0}^{t}0_{\overline{G}(}t)dt+(k_{s}L+c)\overline{G}(t_{0})$,(2)where $\overline{G}(t)=1-G(t)$ , then the cost per unit time in the steady-state is, from the well-known
renewal reward argument [24, p. 52],
$C_{1}(t_{0})$$\equiv$$=$$\lim_{tarrow\infty}\frac{[\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}(0,t]]}{t}$(3)$V_{1}(t_{0})/T_{1}(t_{0})$and the problem is to determine the optimal repair-time limit$t_{0^{*}}$

such as(4)$C_{1}(t^{*} \mathrm{o})=\min_{<0\leq t0\infty}C1(t_{0)}.$

It is straightforward to seek by differentiatingwith respect to , but we here employ
the following graphical method. Define the scaled total time on test (TTT) transform of therepair-time distribution $p\equiv G(t)$ by$C_{1}(t_{0})$$t_{0}^{*}$$\phi_{1}(p)\equiv\mu\int_{0}^{G^{-1}(p})(\overline{G}t)dt$,$t_{0}$$(0\leq p\leq 1)$

,

(5)

where
$G^{-1}(p)= \inf\{t\geq 0:G(t)\geq p\}$.

(6)

The curve $p_{1}=(p, \phi_{1}(p))\in[0,1]\cross[0,1]$ is called the scaled $TTT$ transform or simply the scaled$TTT$ curve. We shall propose a graphical method to solve the problem inon the scaled
TTT curve.
The following result is due to Koshimae, Dohi, Kaio and Osaki [13].
$\mathrm{E}\mathrm{q}.(4)$

THEOREM 2.1: Suppose the assumption (A-1) for Model 1. The minimization problem inis equivalent to obtain $p^{*}(0\leq p^{*}\leq 1)$ satisfying$\mathrm{E}\mathrm{q}.(4)$$\min_{0\leq p\leq 1}:\Lambda\ell_{1}(p, \phi 1(p))\equiv\frac{\phi_{1}(p)+\xi}{p+\eta}$

,

(7)

where
(8)

$\xi\equiv\frac{(k_{S}L+c,)\mu}{(c-k_{\Gamma}L)\lambda}>0$

and$\eta\equiv-(1+\frac{(k_{\Gamma}+k_{S})}{(k_{r}L-c)\lambda})$.

(9)

From THEOREM 2.1, the optimal policy is $p^{*}=G(t^{*})0$ which minimizes the tangent slope fromto the curve .
More precisely, we characterize the optimal policy from the aging property of $G(t)$ .
$(-\eta_{1}, -\xi_{1})$$\ell_{1}$40
DEFINITION 2.2: (1) The repair-time distribution $G(t)$ is IHR (DHR) if and only if thehazard rate $r(t)=g(t)/\overline{G}(t)$ is increasing (decreasing).
(2) $G(t)$ is IHR (DHR) if and only if
is concave (convex) in $p\in[0,1]$ .
$\phi_{1}(p)$The relationship (2) between the aging and the scaled TTT transform was proved by Barlow
and Campo [16]. In the plane $(x, y)=(-\infty, +\infty)\cross(-\infty, +\infty)$ , define the following three p.oints$\mathrm{B}\equiv(x_{B,y_{B})}=(-\eta, -\xi)$

,

(10)

(11)

$\mathrm{Z}\equiv(xz, yz)=(\frac{(k_{S}L+c)r(\mathrm{o})}{(k_{r}L-c)\lambda},$ $-\xi)$and$\mathrm{I}\equiv(x_{I}, y_{I})=(-\eta,$ $1+ \frac{(k_{r}+k_{f^{)\mu}}}{(k_{r}L-c)\lambda r(\infty)})$THEOREM 2.3: (1) Suppose that the scaled TTT curve
assumption (A-1).

$\ell_{1}$

(12)

.

is strictly convex under the(i) If $x_{B}>x_{Z}$ and $y_{B}>y_{I}$ , then there exists a unique optimal solution $p^{*}=G(t^{*})0(0<t_{0}^{*}<$
minimizing the expected cost per unit time in the steady-state given by Eq. (3), whereis given by the -coordinate in the point of contact for the curvefrom the point ,$\infty)$$p^{*}$$x$$\mathrm{B}$$\ell_{1}$where$\max(\mathrm{O}, -\eta)<p^{*}<1$

.

(13)

(ii) If $x_{B}\leq x_{Z}$ , then the optimal repair-limit policy is $p^{*}=G(0)=0$ .
(iii) If$y_{B}\leq y_{l}$, then the optimal repair-limit policy is $p^{*}=G(\infty)=1$ .

(2) Suppose that the scaled TTT curve
optimal solution is $p^{*}=0$ or $p^{*}=1$ .

PROOF: Differentiating$\Lambda l_{1}(p, \phi_{1}(p))$$p_{1}$

is concave under the assumption (A-1). Then, thewith respect to$p$and setting it equal to zero implies$q_{1}(p)\equiv\phi_{1}(p)’(p+\eta)-(\phi_{1()\xi)}p+=0,$(14)$\phi_{1}(p)’=\frac{\mu}{r(G^{-1}(p))}$

(15)

where

and tlle symbol ’ denotes the differentiation. Further, we have$q1(p)’=\phi 1(p)\prime\prime(p+\eta)$

(16)

.

XVhen the scaled TTT curve is strictly colllrex. then $q_{1}(p)’>0$ and theis strictly convex in .
In the plane $(x, y)\in(-\infty, +\infty)\cross(-\infty, +\infty)$ , we define the poirlt
tangent line for the pointon the curve is$\ell_{1}$$\mathrm{f}\mathrm{i}_{111}\mathrm{c}\mathrm{t}\mathrm{i}_{0\mathfrak{U}}$A (p.
$I_{1}$$\phi_{1}(p)$)$p$$\mathrm{B}=(x_{B}, y_{B})$$(p^{*}, \phi_{1}(p^{*}))$. Since the$\ell_{1}$$y= \frac{\mu}{r(G^{-1}(p^{*}))}(p-p^{*})+\phi 1(p^{*})$

,

(17)

the condition that the point is over the above tangent line is $q_{1}(p^{*})=0$ . Define the intersection$\mathrm{Z}=(x_{Z}, yz)$ of the tangent line for the origin $\mathrm{O}=(0,0)$ on the curveand $y=-\xi$ . If the$\mathrm{B}$$p_{1}$

41-coordinate of is strictly greater than the -coordinate of $q_{1}(0)<0$ , otherwise, $q_{1}(0)\geq 0$of the tangent line
under the assumption (A-1). Similarly, define the intersection$\mathrm{U}=(1,1)$
strictlyisofgreater than-coordinate
.
If
the
on
and
the
point
the
$x=-\eta$forthe -coordinate of $q_{1}(1)>0$ , otherwise, $q_{1}(1)\leq 0$ under (A-1). From these, we obtain the
results (1).
becomes a concave function
Secondly, consider the case where $G(t)$ is IHR. In this case,of . If the -coordinate of is strictly negative and if the slope of the straight line BO is, we havestrictly smaller than that of the line$x$$\mathrm{B}$$x$$\mathrm{Z},$$\mathrm{I}=(x_{I)}y_{I})$$\mathrm{B}$$p_{1}$$y$$\mathrm{Z},$$y$$\phi_{1}(p)$$\mathrm{B}$

$x$

$p$

$\mathrm{B}\mathrm{U}$(18)$(k_{S}L+c)/\lambda-\{k_{r}L-c+(k_{r}+ks)\}/\mu<0$ ,. Conversely, if $x_{B}<0$
which is equivalent to the condition of $M_{1}(0, \phi_{1}(0))<\mathbb{J}I_{1}(1,$$M_{1}(0, \phi_{1}(0))\geq$
and if the slope of the straight line BO is not small than that of the line
$M_{1}(0,\phi_{1}(0))\geq$$x_{B}\geq
0$implies$M_{1}(1, \phi_{1}(1))$ is satisfied.
On the other hand, the condition. Thus the proof is completed.
$\phi_{1())}1$$\mathrm{B}\mathrm{U},$$\lambda l_{1}(1, \phi_{1}(1))$

EXAMPLE 2.4: Nguyen and Murthy [6] and Dohi, Matsushima, Kaio and Osaki [14] considered the repair-limit replacement models with imperfect repair. Suppose that the repair isimperfect. The mean lifetime when the repair is completed is $1/\lambda_{1}(>0)$ . Also, a new unit delivered after ordering fails for an infinite time horizon and then the mean lifetime is $1/\lambda_{2}(>0)$ .
Defining the time interval from the start of repair to the following start of repair as one cycle,
the mean time of one cycle is$T_{1}(t_{0})=1/ \lambda_{1}+\int_{0}^{t0_{\overline{G}}}(t)dt+(L+1/\lambda_{2}-1/\lambda 1)\overline{G}(t_{0})$

.

(19)

Replacing the assumption (A-1) to $k_{r}L+(k_{r}+k_{s})(1/\lambda_{2}-1/\lambda_{1})<c$ , we have$M_{i}(p, \phi 1(p))\equiv(\phi_{1}(p)+\xi_{i})/(p+\eta_{i})$ , where.

$\min_{0\leq p\leq 1}\sim$(20)$\xi_{i}\equiv\frac{(k_{S}L+cJ)\mu}{\{c-k_{\mathrm{r}}L-(k_{r}+k_{S})(1/\lambda_{2}-1/\lambda_{1})\}\lambda_{1}}>0$

and$\eta_{i}\equiv-(1+\frac{(k_{r}+k_{s})}{\{k_{\gamma}L-C+(kr+k_{S})(1/\lambda 2-1/\lambda 1)\}\lambda})$

.

(21)

Therefore, the repair-limit replacement model with imperfect repair is reduced to the similar

problem with a different point$\mathrm{B}_{i}=(-\eta_{i}, -\xi_{i})$.

EXAMPLE 2.5: Suppose that the repair-time distribution is the Weibull distribution withthe shape parameter $\alpha=0.8$ and the scale parameter $\beta=2.0$ . The other model paranleters0.8000, $L=$ 0.2000, $c=$ 6.5000, $k_{r}=$ 4.0000 and $k_{s}=$ 6.0000. The determinationareof the optimal repair-time limit for Model 1 is presented ill Fig. 2. In this case, we haveis $(p^{*}.\phi 1(p^{*}))=$
slope from$(-0.4035, - 0.4768)$ and the optimal point with$t_{0}^{*}=G^{-\mathrm{l}}(0.4280)=0.9659$.
(0.4280, 0.3162). Tllus. the optimal repair-time limit is$1/\lambda=$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\ln\iota \mathrm{l}\mathrm{n}\mathrm{l}$$\mathrm{B}=$$\mathrm{B}$2.2 Model 2
consider the similar, but somewhat different model from Model 1. The original unitLet
begins operating at time . When the unit has failed, the decision maker has to select repair or
replacement. Suppose that the decision maker has a subjective probability distribution function
on the repair-completion time $G(t)$ with finite mean $1/\mu(>0)$ . If he or she estimates that the, then the repair is immediately started at
repair is completed up to the time limit
the failure time. However, if he or she estimates that the repair time is greater than , then thefailed unit is scrapped at the failure time, the spare unit is ordered immediately and delivered
after the lead time . The configuration of Model 2 is illustrated in Fig. 3.
$11\mathrm{S}$$0$$t_{0}\in[L, \infty)$$t_{0}$$L$42
If we evaluate the expected cost per unit time in the steady-state objectively, it will be$C_{2(t_{0)}}\equiv V2(t_{0})/T_{2}(t_{0})$

(22)

,

where
(23)

$V_{2}(t_{0})=(k_{S}+k_{r}) \int_{0}^{t_{0}}tdG(t)+(k_{s}L+c)\overline{G}(t\mathrm{o})$and$T_{2}(t_{0})=1/ \lambda+\int_{0}^{t_{0}}tdG(t)+L\overline{G}(t_{0})$.

Hence the problem is to determine the optimal repair-time limit$C_{2}(t^{*} \mathrm{o})=0\leq t_{0}<\min_{\infty}C_{1}(t_{0})$(24)$t_{0^{*}}$

such as.

(25)

In addition to the assumption (A-1), we need(A-2) $(k_{r}+k_{s})>C_{2}(t_{0})$ for $t_{0}\in[0, \infty)$ .

The assumption (A-2) will be required in order to avoid an unrealistic case. To this end, if thereverse inequality holds, the repair and shortage have to always occur at the same time in thesteady-state with probability one.
To develop the similar graphical technique to Model 1, we define the Lorenz transform of therepair-time distribution $G(t)$ by$\phi_{2}(p)\equiv\mu\int_{0}^{G^{-1}()}p)xdG(x, (0\leq p\leq 1)$

.

(26)

The definition above of the Lorenz transform is essentially equivalent to$\phi_{2}(p)=\mu\int_{0}^{p}G^{-1}(t)dt$

,

(27)

$(0\leq p\leq 1)$(see [19-23]). Then the curve $p_{2}=(p, \phi_{2}(p))\in[0,1]\cross[0,1]$ is called the Lorenz curve. From thesimple algebraic manipulation, we have

THEOREM 2.6: Suppose that the assumptions (A-1) and (A-2) hold for Model 2. The
is equivalent tominimization problem in$\mathrm{E}\mathrm{q}.(25)$$\min_{0\leq p\leq 1}$:$\mathbb{J}I_{2}(p, \phi 2(p))\equiv\frac{\phi_{2}(p)+\xi}{p+\eta}$

(28)

.

Consequently, the optimal policy is determined by $p^{*}=G(t^{*})0$ which lninimizes the tangent slope
from $\mathrm{B}=(-\eta, -\xi)\in(-\infty, 0)\cross(-\infty, 0)$ to the curve .
$\ell_{2}$PROOF: The expected cost per unit time in the steady-state in$(k_{s}+k_{r})-K(t_{0})$$\mathrm{E}\mathrm{q}.(22)$becolnes $C_{2}(t_{0)}=$, where.

(29)$K(t_{0}) \equiv\frac{(k_{S}+k_{r})\mu/\lambda+\mu(krL-c)\overline{c}(t_{0})}{\mu/\lambda+\mu L\overline{G}(t\mathrm{o})+\int_{0}\iota_{0}tdG(t)}$from (A-2), it is found that $(k_{s}+k_{r})/\lambda>c-k_{r}L$ from (A-1)
Since $K(t_{0})>0$ for.
and yields
and then $\eta>0$ . Hence the underlying problem is$t_{0}\in[0, \infty)$$\min_{0\leq\iota_{0}<\infty}1/K(t_{0})$$\mathrm{E}\mathrm{q}.(28)$43
THEOREM 2.7: (1) Suppose that the assumptions (A-1) and (A-2) hold. Then there exists a
unique optimal solution $p^{*}=G(t_{0}*)(0<t_{0}^{*}<\infty)$ minimizing the expected cost per unit time inthe steady-state given by Eq. (28), where is given by the -coordinate in the point of contact
for the curve from the point B.
$x$$p^{*}$$\ell_{2}$PROOF: Differentiating$M_{2}(p, \phi_{2}(p))$

with respect to$p$and setting it equal to zero implies(30)$q_{2}(p)\equiv\phi_{2}(p)’(p+\eta)-(\phi_{2()\xi)}p+,$where $\phi_{2}(p)’=\mu G-1(p)$ . Further, we have(31)

$q_{2}(p)^{;}=\phi 2(p)’’(p+\eta)>0$

and the function$\Lambda I_{2}(p, \phi_{2}(p))$

is strictly convex in , since $\phi_{2}(p)\prime\prime=\mu/g(G^{-1}(p))>0$ . From$p$$q_{2}(0)=-\xi<0$ and $q_{2}(1)arrow\infty$ , the result is proved.

EXAMPLE 2.8: Suppose that the repair-time distribution is the Weibull distribution withthe shape parameter $\alpha=0.8$ and the scale parameter $\beta=2.0$ . The other model parameters0.8000, $L=$ 0.2000, $c=$ 6.5000, $k_{r}=$ 4.0000 and $k_{s}=$ 6.0000. The determinationareof the optimal repair-time limit for Model 2 is presented in Fig. 4. In this case, we have
$1/\lambda=$$\mathrm{B}=$$(-0.4035, - 0.4768)$ and the optimal point with minimum slope from$\mathrm{B}$is$(p^{*}, \phi_{2}(p^{*}))=$(0.5530, 0.1407). Thus, the optimal repair-time limit is $t_{0}^{*}=G^{-1}(0.5530)=1.5255$ .

3. COMPARATIVE RESULTS
We compare two repair-limit replacement models. From Eqs. (3) and (22), it can be seen
and $V_{1}(t_{0})=V_{2}(t_{0})+(k_{r}+k_{s})t0\overline{G}(t_{0})$ for a fixed . When $t_{0}=0$
that$(to)=C_{2}$ (to). Thus, we pay our attention to the case of
that
,
it
is
obviousand. Now, define $C_{1}(t_{1}^{*})= \min_{0<t_{0}<1}\infty C(t_{0}),$ $C_{2}(t_{2}^{*})= \min 0<t_{0}<\infty C2(t_{0)}$ , A.
$\min_{0<p1}<$ All $(p, \phi_{1}(p))$ and$T_{1}(t_{0})=T_{2}(t_{0})+t_{0}\overline{G}(t0)$$t_{0}$$t_{0}\in$$C_{1}$$t_{0}arrow\infty$$f_{1}(p_{1}^{*}, \phi_{1}(p_{1}^{*}))=$$(0, \infty)$$\Lambda l_{2}(p^{*}2’\phi 2(p_{2}^{*}))=\mathrm{m}\mathrm{i}\mathrm{n}0<p<1\mathbb{J}I2(p, \phi 2(\overline{p}))$.
THEOREM 3.1: (i) Under the assumption (A-2), $C_{1}(t_{0})>C_{2}(t_{0})$ holds for a fixed(ii) Under the assumptions (A-1) and (A-2), if $\phi_{j}(p)(j=1,2)$ is monotonically increasing and.
andstrictly convex in $p\in(0,1)$ , then$t_{0}\in[0, \infty)$$p_{1}^{*}<p_{2}^{*}$$t_{1}^{*}<t_{2}^{*}$(to). Since(to)
PROOF: (i) $C_{1}(t_{0})>C_{2}(t_{0})$ if and only ifand (A-2), the result is derived. (ii) Chandra and Singpurwalla
for a fixed[21] proved the following relation;$(k_{r}+k_{S})t_{0}\overline{c}(t0)T2$$>t_{0}\overline{G}(t\mathrm{o})V_{2}$$t_{0}\in(0, \infty)$$t_{0}\overline{G}(t_{0})>0$$\phi_{1}(p)=\phi_{2}(p)+\mu(1-p)c^{-1}(p)$

(32)

.

for $p\in(0,1)$ . Since $\phi_{j}(p)(j=1,2)$ is monotonically increasing
Hence, we. The proof isandand strictly convex in $p\in(0.1)$ , it is straightforward to see$\mathrm{h}\mathrm{a}\backslash r\mathrm{c}\phi_{1}(p)>\phi_{2}(p)$$t_{1}^{*}<t_{2}^{*}$$p_{1}^{*}<p_{2}^{*}$

completed.

results on respective repair-linlit replacement models. SupNext, we develop thepose that there are two repairmen with different repair abilities. We classify two repairmen into
are non-negative
andRepairman 1 and Repairman 2, respectively. Their repair times
mean $1/\mu$ ,
same
the
and
$G_{j}(t)(j=1,2)$
variables with distribution functions.
[25].
ordering
stochastic
theonrequire the following definition
respectively.
$\mathrm{c}\mathrm{o}\ln_{\mathrm{I}^{)}}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\backslash r\mathrm{c}$$X_{2}$$X_{1}$$\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}_{r}\mathrm{e}$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}_{0}$$\mathrm{m}$$1^{1}\dagger^{\mathrm{r}_{\mathrm{e}}}$DEFINITION 3.2: (1)$X_{2})$

if$\overline{c_{1}}(t)\leq\overline{G_{2}}(t)$$X_{1}$

is usually stochastic-ordered with respect to.

is star-shaped stochastic-ordered with respect to(2).
is increasing in$X_{1}$$t\in(\mathrm{O}, G_{1}^{-1}(1))$$X_{2}$(denoted as$\lambda_{2}’$(denoted as$X_{1}\leq_{*}X_{2}$$X_{1}\leq_{\mathrm{s}\mathrm{t}}$) if $G_{2}^{-1}(G_{1}(t))/t$44
THEOREM 3.3: In Model 1, define the optimal repair-time limits for two repairmen with, respectively, whereandas
the same mean repair time
is monotonically increasing
are the solutions for Eq. (7) with $G_{j}(t)(j=1,2)$ . Ifandand strictly convex in $p\in[0,1]$ and if the repair time for Repairman 1 is smaller than that for.
Repairman 2 in the usual stochastic ordering, then
PROOF: From the well-known result by Barlow [17], we find that
if. and,and. Immediately, we have
only if$p\in[0,1]$.
stochastic
the
usual
in
Since
strictly
convexandincreasing
monotonicallyissinceand $G_{1}(t)\geq G_{2}(t)$ yield
includes the star-shaped orderingordering$1/\mu$$t_{12}^{*}=G_{2}^{-1}(p_{1}^{*}2)$$t_{11}^{*}=G_{1}^{-1}(p_{11}^{*})$$p_{11}^{*}$$\phi_{1}(p)$$p_{12}^{*}$$t_{11}^{*}\leq t_{12}^{*}$$\phi_{1}(G_{1}(t))\geq\phi_{1}(c_{2}(t))$$X_{1}\leq_{*}X_{2}$$p_{11}^{*}\leq p_{12}^{*},$$G_{2}^{-1}(G_{1}(t_{11}^{*}))\leq t_{12}^{*}$$G_{1}(t_{1}^{*})1\leq G_{2}(t_{12}^{*})$$\phi_{1}(p)$$\leq*’ c_{2}^{-1}(c_{1}(t^{*})11)\leq t_{12}^{*}$$\leq_{\mathrm{s}\mathrm{t}}$$G_{1}(t_{1}^{*})1\leq G_{2}(t_{1}^{*})2$

and$t_{11}^{*}\leq t_{12}^{*}$.

THEOREM 3.4: In Model 2, define the optimal repair-time limits for two repairmen with theand, respectively, whereandassame mean repair time
are the solutions for Eq. (28) with $G_{j}(t)(j=1,2)$ . If the repair time for Repairman 1 is.
smaller than that for Repairman 2 in the usual stochastic ordering, then
PROOF: Chandra and Singpurwalla [21] proved $\phi_{2}(G_{1}(t))\geq\phi_{2}(G_{2(}t))$ if $X_{1}\leq*X_{2}$ . The rest
part of the proof is similar to THEOREM 3.3.
$t_{22}^{*}=G_{2}^{-1}(p_{22}^{*})$$t_{21}^{*}=G_{1}^{-1}(p_{21}^{*})$$1/\mu$$p_{21}^{*}$$p_{22}^{*}$$t_{21}^{*}\leq t_{22}^{*}$

4. NON-PARAMETRIC ESTIMATION METHODS
Based on the graphical ideas in Section 2, we propose statistical methods to estimate theoptimal repair-limit policies for two models. Suppose that the optimal repair-time limit has tobe estimated from an ordered complete sample $0=x_{0}\leq x_{1}\leq x_{2}\leq\cdots\leq x_{n}$ of repair times
continuous repair-time distribution , which is unknown. The estimator offrom an$G(t)=p$ is the empirical distribution given by$G$$\mathrm{a}\mathrm{b}_{\mathrm{S}\mathrm{O}}1\acute{\mathrm{u}}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}$$i/n$$G_{n}(_{X)}=\{$

1

for $x_{i}\leq x<x_{i+1}$ , $(i=0,1,2, \cdots, n-1)$
for $x_{n}\leq x$ .

(33)

Then the scaled TTT statistics based on this sample are$\phi_{1i}\equiv S_{i}/S_{n}$

(34)

,

where
$S_{i} \equiv\sum_{i=1}^{i}(n-j+1)(x_{jj-1}-x)$,$(\dot{i}=1,2, \cdots, n;S_{0}=0)$

.

(35)

$(\dot{i}=0,1,2, \cdots, n)$ , and connecting them by line segments, we
By plotting the point
obtain the so-called scaled $TTT$ plot, $\ell_{1n}\in[0,1]\cross[0,1]$ . On the other hand, the empirical
Lorenz curve is$(\dot{i}/n, \phi_{1i}),$$\phi_{2i}\equiv\sum_{1i=}^{]}x_{i}/\sum_{i=1}^{n}X_{i}[np$

(36)

,

$(i=0.1,2, \cdots , n)$ ,ill
Silllilarl}r, plotting tlle point
where $[0]$ is the greatest$\ell_{2n}\in[0,1]\cross[0,1]$.
curve
Lorenzempirical
theand connecting them by line segments, we obtain
non-parametricpropose
2.6,
we
2.1
and
THEOREM
of
THEOREMempirical
counterpartsAs
estimators of the repair-time limits for respective models.
)$\mathrm{i}_{11\mathrm{t}(}$$\mathrm{g}\mathrm{e}\mathrm{r}$$(i/r\}, \phi_{\mathit{2}i}),$$a.$

THEOREM 4.1: The optimal repair-time limit for Model $j(=1,2)$ can be estimated by, where$t_{jn}\wedge=x_{i^{*}}$$\{i^{*}|\min_{\leq 0\leq in}\frac{\phi_{ji}+\xi}{i/n+\eta}\}$

.

(37)

45
The proof is omitted for brevity. We consider the following two examples for better understanding of the result above.

EXAMPLE 4.2: The repair-time data were made by the random number following the Weibull
distribution with shape parameter $\alpha=0.2$ and scale parameter $\beta=2.0$ . The other model parameters are $1/\lambda=0.8000,$ $L=0.2000,$ $c=6.5000,$ $k_{r}=4.0000$ and $k_{s}=6.0000$ . The scaled TTT$(-0.4035, - 0.4771)$ ,plot based on the 200 sample data for Model 1 is shown in Fig. 5. Since–
119/200,
becomesthe optimal point with minimum slope from(0.5980, 0.4873). Thus, the estimator of the optimal repair-time limit $\hat{t}_{1,20}0=x_{119}=1.7006$ .
$\mathrm{B}=$$(\dot{i}^{*}/n, \phi_{1i}*)=$$\mathrm{B}$$\phi_{1,119})$$($

EXAMPLE 4.3: Under the same model parameters with EXAMPLE 4.2 except for$0.8$ , the empirical Lorenz curve based on the 200 sample data for Model 2 is shown in Fig.
becomes$(-0.4035, - 0.4771)$ , the optimal point with minimum slope from
6. Since
110/200, $\phi_{2,110})=$ (0.5829, 0.1742). Thus, the estimator of the optimal repairtime limit $\hat{t}_{2,200=X_{11}}0=1.6492$ .
$\alpha=$$\mathrm{B}$$\mathrm{B}=$$(\dot{i}^{*}/n, \phi_{2i^{*}})=$$($$(j=1,2)$ in
Of our interest is the investigation of asymptotic properties of the estimators
THEOREM 4.1. The following theorem guarantees the asymptotic optimality of theabove.
$\hat{t}_{jn},$$\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}_{0}\mathrm{r}\mathrm{S}$THEOREM 4.4: (i) The expected cost

$C_{j}(t_{jn})\wedge,$$(j=1,2)$ of using the repair-time limit$\hat{t}_{jn}$may beestimated by$\hat{C}_{j}(t_{j}^{*})7\mathrm{t}--\frac{(k_{r}+kS)\phi_{j(p)}jn*/\mu n+(k_{S}L+c)(1-p_{j}^{*}n)}{1/\lambda+\phi_{j}(pjn)*/\mu n+L(1-p_{jn})}$

(38)

,

is empirical mean of the repair time. Then the estimator is strongly consistent.
where(iii) If a unique optimal repair-time limit exists then $\hat{t}_{jn}(j=1,2)$ is strongly consistent.
PROOF: See for $j=1$ (Model 1) Bergman [8]. For $j=2$ (Model 2), it is straightforward fromconvergence theorems for empirical Lorenz curve proved by Goldie [20].
$1/\mu_{n}$We examine numerically the strong consistency of estimators proposed for two repair-limit
replacement models. Since the real optimal repair-time limit can be calculated under the known
repair-time distribution, we investigate the convergence property of estimators to the real value.

EXAMPLE 4.5: For Model 1, suppose that the repair-time distribution is the Weibull disscale parameter $\beta=2.0$ . The othertribution with the shape parameter $\alpha=0.2$ and.
$\mathrm{t}_{)}\mathrm{h}\mathrm{e}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}_{1\mathrm{p}\mathrm{e}\mathrm{C}}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{l}/\lambda=0.8000,L=0.2000,c\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{C}\mathrm{a}\mathrm{i}\Gamma- \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{u}\mathrm{m}=6.5\mathrm{o}\mathrm{o}\mathrm{t}\exp \mathrm{e}\mathrm{d}0,k_{r,\mathrm{c}}=\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{b}\mathrm{e}4.0\mathrm{o}\mathrm{o}0\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{e}t_{0}*0=1ks.6=29^{\cdot}9\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{o}\mathrm{o}0\mathrm{d}$,her hand, the asymptotic behaviour of estimators forlninim\iota lln
costs are depicted in Figs.
associated
and
their
limit
repair-time
optimaltheconverge to the corresponding realthethese figures, we observe
7 and 8.
values around which the number of data is 50.
$C_{1}(t_{0}^{*})=0.2050,$. On the$\mathrm{o}\mathrm{t}$$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\text{ノ}\mathrm{i}\mathrm{l}\prime \mathrm{e}\mathrm{l}\mathrm{y}$$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{f}_{7}\mathrm{e}\mathrm{d}$$\mathrm{t}_{J}\mathrm{h}\mathrm{a}\mathrm{t}$$\mathrm{F}\mathrm{r}\mathrm{o}\ln$$\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{a}\mathrm{t}_{\mathrm{O}}\mathrm{r}\mathrm{S}$, suppose that tlle repair-tinlc distribution is tlle XVeibull
EXAMPLE 4.6: For
alld the scale parameter $\beta=2.0$ . The other model
bution with the shape )parameters are similar to EXAMPLE 3.5. Then the optimal repair-time limit and the mininuurnother hand, the0.7217, respectively. On
1.5255 andexpected cost become
theirassociated minand
limit
repair-time
optimalasymptotic behaviour of estimators for thethat theobserve
wefigures,
From
these
10.
and
Figs.
in
9
depicted
imum expected costs areis 80.
dataof
numberthewhicharoundvaluesreal
corresponding
thetoconverge
estimators$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{l}\cdot \mathrm{i}-$$\mathrm{I}\backslash \mathrm{I}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}2$$\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{G}\mathrm{t}\mathrm{e}\mathrm{r}\alpha=0.8$$1$$t_{0}^{*}=$

5. CONCLUDING REMARKS$C_{2}(t_{0}*)=$$\mathrm{t}_{\mathit{1}}\mathrm{h}\mathrm{e}$46
In this paper, we have proposed graphical methods to estimate the optimal repair-time limits
for two kinds of repair-limit replacement models, based on the total time on test statistics andthe Lorenz statistics. It has been shown that both estimators provided are non-parametric andhave strong consistency. These statistical properties will be useful for practitioners to determine
plan if they can obtain a sufficiently large number of repair-time data. If
thewe have to determine the optimal repair-limit policy from small sample, then any statisticaltechnique such as Jack-Knife method should be applied to make the number of given sample
increase or existing parametric methods should be used to specify the repair-time distribution
with some theoretical distribution functions.
$\mathrm{m}\dot{\mathrm{a}}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{C}\mathrm{e}$References

repair[1] Hastings, N. A. J.,
vol. 20, pp. 337-349 (1969).
$‘(\mathrm{T}\mathrm{h}\mathrm{e}$$\acute{\lim}$

it replacement method”, Operational Research Quarterly,

optimum repair limit replacement policies”, Operational[2] Nakagawa, T. and Osaki, S.,
Research Quarterly, vol. 25, pp. 311-317 (1974).
$‘(\mathrm{T}\mathrm{h}\mathrm{e}$[3] Nakagawa, T. and Osaki, S., “Optimum ordering policies with lead time for an operatingunit”, R.A.I.R. 0. Recherche $Op\dot{e}rat\dot{i}onnelle/o_{pera}t\dot{i}onS$ Research, vol. 12, pp. 383-393(1978).
[4] Okumoto, K. and Osaki, S., “Repair limit replacement policies with lead time”, Zeitschrift
fiir Operations Research, vol. 20, pp. 133-142 (1976).

[5] Nguyen, D. G. and Murthy, D. N. P., ((A note on the repair limit replacement policy”,
Journal of Operational Research Society, vol. 31, pp. 1103-1104 (1980).
[6] Nguyen, D. G. and Murthy, D. N. P., (‘Optimal repair limit replacement policies with imperfect repair”, Journal of Operational Research Society, vol. 32, pp. 409-416 (1981).

[7] Kaio, N. and Osaki, S., ((Optimum repair limit policies with a time constraint”, International
Journal of Systems Science, vol. 13, pp. 1345-1350 (1982).
[8] Bergman, B., “On age replacement and the total time on test concept”, Scandinavian Journal of Statistics, vol. 6, pp. 161-168 (1979).
[9] Bergman, B. and Klefsj\"o, B., “A graphical method applicable to age replacement problems”,
IEEE Transactions on Reliability, vol. R-31, pp. 478-481 (1982).

[10] Bergman, B. and Klefsj\"o, B., “TTT-transforms and age replacements with discounted
costs”, Naval Research Logistics Quarterly, vol. 30, pp. 631-639 (1983).
, B., “The total time on test concept and its use in reliability[11] Bergman, B. andResearch,
vol. 32, pp. 596-607 (1984).
Operations
theory”,$\mathrm{K}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{s}\mathrm{j}_{\ddot{\mathrm{O}}}$[12] Dohi, T., Kaio, N. and Osaki, S., “Solution procedure for a repair limit problem using
TTT-transform”. IMA Journal of Mathematics, Appli,ed Business and Industry, vol. 6,
pp. 101-111 (1995).

approach to repair[13] Koshimae, H., Dohi, T., Kaio, N. and Osaki, S.,
limit replacement problem”,
of the Operations Research Society of Japan, vol. 39,
pp. 230-246 (1996).
$‘\prime \mathrm{G}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{C}\mathrm{a}1/\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}\mathrm{t}}\mathrm{i}\mathrm{c}\mathrm{a}1$$Jou\gamma\cdot nal$

[14] Dohi, T., Matsushima, N., Kaio, N. and Osaki, S., “Non-parametric repair limit replacement policies with imperfect repair”, European Journal of Operational Research, vol. 96,
pp. 260-273 (1996).

47[15] Dohi, T., Koshimae, H., Kaio, N. and Osaki, S., “Geometrical interpretations of repair
cost limit replacement policies”, International Journal of Reliability, Quality and Safety
Engineering, vol. 4, pp. 309-333 (1997).
[16] Barlow, R. E. and Campo, R., “Total time on test processes and applications to failure
data analysis”, Reliability and Fault Tree Analysis (R. E. Barlow, J. Fussell and N. D.
Singpurwalla, eds.), pp. 451-481, SIAM, Philadelphia (1975).

[17] Barlow, R. E., “Geometry of the total time on test transform”, Naval Research Logistics,
vol. 26, pp. 393-402 (1979).
of measuring the concentration of wealth”) Journal of American[18] Lorenz, M. O.,
Statistical Association, vol. 9, pp. 209-219 (1893).
$‘\zeta \mathrm{M}\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}\mathrm{s}$[19] Gastwirth, J. L., “A general definition of the Lorenz curve”, Econometrica, vol. 39, pp.

1037-1039 (1971).

theorems for empirical Lorenz curves and their inverses”,[20] Goldie, C. M.,
Advances in Applied Probability, vol. 9, pp. 765-791 (1977).
($‘ \mathrm{C}_{\mathrm{o}\mathrm{n}\mathrm{V}\mathrm{e}\mathrm{r}}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$[21] Chandra, M. and Singpurwalla, N. D., “Relationship between some notions which arecommon to reliability and economics”, Mathematics of Operations Research, vol. 6, pp.
113-121 (1981).

[22] Klefsj\"o, B., “Reliability interpretations of some concepts from economics”, Naval Research
Logistics Quarterly, vol. 31, pp. 301-308 (1984).
[23] Aly, E. A. A., (‘On testing for Lorenz ordering”, Metrika, vol. 38, pp. 117-124 (1991).
[24] Ross, S. M., Applied Probability Models with Optimization Applications, Holden-Day, San
Francisco (1970).

48

X$\bullet$failureregenerative point–

operation period–

repair period–

lead time

Figure 1: Configuration of Model 1.

Figure 2: Determination of the optimal repair-time limit based on the scaled TTT transform(Model 1).

49

X$\bullet$failul epoint$1^{\cdot}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\iota \mathrm{i}\mathrm{V}\mathrm{e}$operation period–

repair$\mathrm{p}\mathrm{e}\mathrm{l}$iodlead time

Figure 3: Configuration of Model 2.

$d_{\mathrm{I}},/\mathrm{n}\mathrm{Y}$Figure 4: Determination of the optimal repair-time limit based on the Lorenz transform (Model
2).

50

Figure 5: Estimation of the optimal repair-time limit based on the scaled TTT plot (Model 1).

Figure 6: Estimation of the optimal repair-time limit based on the empirical Lorenz curve (AIodel
2).

51

Figure 7: Asymptotic property of the estimated repair-time limit (Model 1)’(t ハ

Figure 8: Asymptotic property of the estimated minimum expected (Model 1)52$\overline{*}$$t_{\cap}$

Figure 9: Asymptotic property of the estimated repair-time limit (Model 2)$(^{\neg}\vee(f_{\cap^{*}}\wedge)$

Figure 10: Asymptotic property of the estimated minimum expected (Model 2)