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An optimal repair policy for systems with a limited number of repairs
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European Journal of Operational Research 187 (2008) 84–97 www.elsevier.com/locate/ejor
Stochastics and Statistics
An optimal repair policy for systems with a limited number of repairs I.T. Castro
a,*
, E.L. Sanjua´n
b
a
b
Departamento de Matema´ticas, Escuela Polite´cnica, Universidad de Extremadura, Avenida de la Universidad, s/n, 10071 Ca´ceres, Spain Departamento de Matema´ticas, Facultad de Veterinaria, Universidad de Extremadura, Avenida de la Universidad, s/n, 10071 Ca´ceres, Spain Received 18 December 2005; accepted 7 March 2007 Available online 24 April 2007
Abstract Traditionally in reliability literature, the repair facilities are always available. This work considers a more general case in which the repair facilities are not always available, but are available only until a fixed number of repairs have been completed. Different assumptions are made to analytically determine an optimal repair policy maximizing the expected reward. 2007 Elsevier B.V. All rights reserved. Keywords: Repairable systems; Optimal policy
1. Introduction There is a extensive literature on the design of new strategies for the replacement/repair of repairable systems, mainly because of the practical applicability to modeling problems of complex electronics, communication equipment, etc. Indeed, thousands of replacement policies have been created. Ascher and Feingold [1], Barlow and Proschan [2], Nakagawa [5], and Osaki [6] are landmark works presenting the mathematical theory of maintenance. There have also been many useful reviews of the history of research in this field, including Pham and Wang [7], Valdez-Flores and Feldman [8], and, most recently, Wang [9]. In general, replacement policies depend on the cost that they involve and the required availability and reliability of the systems. Frequently the reliability literature has made the general assumption that, after a failure detection, the system can immediately go to a repair facility and that after a certain waiting time a repairman is immediately available. This assumption, however, does not reflect practical situations when access to the repairman may be restricted (for example, when the failure of the system occurs out of business hours). For that reason, in the present communication we assume that the repairman is not always available. If the system fails and the repair facilities are available the repair begins. On the other hand, if the system fails and the repair facilities *
Corresponding author. Tel.: +34 927257444. E-mail addresses: [email protected] (I.T. Castro), [email protected] (E.L. Sanjua´n).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.03.027
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are not available the repair of the failure is delayed until the repair facilities can be used again. This idea can be implemented in two models that depend on the accessibility of the repair facilities. 1. The first model assumes that the repair facilities are available until T *, 0 6 T * 6 T where ½0; T represents an entire working period. If the system fails in ½0; T (0 6 T * 6 T) then it undergoes repair at failure, whereas if it fails in ðT ; T , then the repair is not made and it is delayed until the beginning of a new working period when a planned replacement is performed [4]. 2. The second model assumes that the repair facilities are available until SK where SK represents the instant when K repairs have been completed by the repairman. If the system fails during ð0; S K then it undergoes repair at failure, whereas if it fails in ðS K ; T , then the repair is not made and it is delayed until the beginning of a new working period when a planned replacement is carried out. We here analyze the second model. Thus, the repair facilities are available in ð0; S K and are unavailable in ðS K ; T . Without loss of generality, we assume that the system holds N unrepaired failures, 1 6 N, in ðS K ; T . This means that, after a failure, the system may be used to complete some other task for which repair of the fault that has occurred is not required. However, these unrepaired failures are deleterious to the operating time of the system, i.e., the successive operating times after an unrepaired failure, become progressively shorter. This situation is modeled using a deteriorating process. Let Xn be the random variable that represents the operating time of the system after the nth unrepaired failure. We say that fX n ; n P 0g is a deteriorating process if F i ðtÞ P F j ðtÞ;
8i; j P 0; j > i; t > 0;
ð1Þ
where F i and F j denote the survival functions of Xi and Xj, respectively. If Xi and Xj satisfy (1) we write Xi Pst Xj. In consequence of this damage, if the number of unrepaired failures per working period reaches a certain fixed limit, N, then the system remains failed until the planned replacement at the beginning of the next working period. The case N = 1 corresponds to a catastrophic failure, and in this case, if the system fails after SK, then it remains failed until the next planned replacement time. A practical example of this model can be found in a military logistics problem with a central depot facility and forward locations (or bases). The bases are capable of repairing parts of the military material and the central depot serves all of the bases at the beginning of each working period. An unlimited number of repairs at the bases would require an extraordinary amount of replacement parts and a very high cost. The central depot therefore serves only parts sufficient to carry out at most K repairs per working period at the bases. After K repairs have been carried out, the bases are not capable of repairing the material from that moment to the next working period (when they receive the necessary parts from the central depot). The problem studied in the present communication is concerned with the optimal stocking of the replacement parts at the bases. Notice that, this model corresponds to a combined replacement model. For K = 0, the model corresponds to the situation where the system is always replaced at times kT ðk ¼ 1; 2; . . .Þ but it is not repaired at failures, and hence, it remains in failed state for the time interval from a failure to its detection. This particular case can apply to systems not monitored continuously whose failures can be detected only at times kT ðk ¼ 1; 2; . . .Þ, see [5] for more details. For K = 1, the model is reduced to the classical repair policy (that is, if the system fails it is repaired and when the repair is completed the system starts to operate again) at the beginning of the working period. Costs are associated with the repair and down times and reward is associated with the operating times, and the objective is to determine an optimal number of repairs. Optimal means a value of K maximizing the expected long-run reward. The combined model and the expected reward of the system are specified in detail in Section 2. In Section 3, we search for an optimal replacement policy. Section 4 presents some numerical examples of the model and concludes. 2. Problem definition and formulation The system is considered to be a single unit which suffers repairable failures. It is activated during an interval of time ½0; T which represents an entire working period. We study the repair strategy by making the following assumptions.
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1. Let X1,i be the operating time of the system after the (i 1)th repair. We assume that X1,i has an exponential distribution with mean 1/k1, "i. Let Xr,i be the repair time of the system after the ith failure. Assume that Xr,i has an exponential distribution with mean 1/l, "i (k1 5 l). We define, n n X X X 1;i þ X r;i ; n P 1; S 0 0: ð2Þ Sn i¼1
i¼1
Let A0 be the reward rate per unit time of the system when it is operating and Ar be the repair cost rate per unit time. 2. The repair facilities are available until K failures have occurred and have been repaired in a working period, i.e. SK. The system is replaced by a new one at the beginning of each working period. It is assumed that the time for replacement is negligible. 3. Let X i ; i ¼ 1; 2; . . ., be the operating time after the i 1th unrepaired failure. fX i ; i ¼ 1; 2; . . .g forms a sequence of exponential random variables with means E(Xi) = 1/ki, i ¼ 1; 2; . . . that verify k1 < k2 < k3 < and ki 5 l, "i. If N failures have not been repaired in a working period, 1 6 N, the system remains in a failed state for the time interval from the Nth unrepaired failure to the planned replacement, with Cs being the downtime cost. Under these assumptions, we shall now derive the expected reward in a working period. The following definition helps to simplify the procedure. Definition 1. Given a positive random variable X with distribution F, the expectation of X truncated at T, T > 0, is defined as E½X T ¼
Z
T
F ðuÞ du ¼
Z
0
T
u dF ðuÞ þ T F ðT Þ:
ð3Þ
0
Notation. • • • • • •
Fi (fi) is the distribution function (density function) of Xi, 1 6 i 6 N. Fr (fr) is the distribution function (density function) of Xr. F iþþj ðfiþþj Þ is the distribution function (density function) of X i þ þ X j , 1 6 i 6 j 6 N. F rþiþþj ðfrþiþþj Þ is the distribution function (density function) of X r þ X i þ þ X j , 1 6 i 6 j 6 N. F S K þiþþj ðfS K þiþþj Þ is the distribution function (density function) of S K þ X i þ þ X j , 0 6 i 6 j 6 N. F S K þrþiþþj ðfS K þrþiþþj ) is the distribution function (density function) of S K þ X r þ X i þ þ X j .
To evaluate the expected reward in a working period, we shall first calculate W1(K), the expected operating time when the repair facilities are available (i.e., before SK). One can deduce that " # K K K Z T X X X W 1 ðKÞ ¼ E X 1i I fS i1 þX 1i 6T g þ ðT S i1 ÞI fS i1
i¼1
from conditional expectation properties as we show next. For i ¼ 1; . . . ; K Z T Z E½X 1i I fS i1 þX 1i 6T g ¼ E½X 1i I fS i1 þX 1i 6T g jS i1 ¼ u dF S i1 ðuÞ ¼ u¼0
u¼0
i¼1
T
u¼0
Z
T u
v dF 1 ðvÞ dF S i1 ðuÞ:
v¼0
Also E½ðT S i1 ÞI fS i1
Z
T
E½ðT S i1 ÞI fS i1
u¼0 Z T u¼0
ðT uÞF 1 ðT uÞ dF S i1 ðuÞ:
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And finally W 1 ðKÞ ¼
K Z X
T
Z
u¼0
i¼1
T u
K Z X v dF 1 ðvÞ þ ðT uÞF 1 ðT uÞ dF S i1 ðuÞ ¼
v¼0
i¼1
T
u¼0
E½X 1 T u dF S i1 ðuÞ:
Analogously, let Wr(K) be the expected repair time when the repair facilities are available. Then " # K K X X W r ðKÞ ¼ E X ri I fS i 6T g þ ðT S i1 X 1i ÞI fS i1 þX 1i
¼
K X
Z
i¼1 T
u¼0
i¼1
Z
T u
E½X r T uv dF 1 ðvÞ dF S i1 ðuÞ:
ð5Þ
v¼0
Equality (5) follows by a calculation similar to that of (4): Z T Z T Z E½X ri I fX 1i þX ri 6T ug dF S i1 ðuÞ ¼ E½X ri I fS i 6T g ¼ 0
0
T u
Z
v¼0
T uv
w dF r ðwÞ dF 1 ðvÞ dF S i1 ðuÞ
w¼0
and E½ðT S i1 X 1i ÞI fS i1 þX 1i
Z
T
u¼0 Z T
Z
T u
E½ðT u vÞI fuþv
v¼0 Z T u
u¼0
ðT u vÞF r ðT u vÞ dF 1 ðvÞ dF S i1 ðuÞ:
v¼0
Finally, let W2(K) be the expected operating time when the repair facilities are unavailable. Then, in a similar form to the above, W 2 ðKÞ ¼ E½ðX 1 þ X 2 þ þ X N ÞI fS K þX 1 þþX N 6T g þ E½ðT S K ÞI fS K 6T
ð6Þ
u¼0
We now give a derivation of the expression for the expected reward. For any K 2 f0; 1; . . .g, denote by RN(K) the expected reward per working period if the repair facilities are unavailable after SK and the system remains failed until the planned time if it fails N-times in (SK, T]. Using (4)–(6), RN(K) can be expressed as follows: K Z T K Z T Z T u X X RN ðKÞ ¼ A0 E½X 1 T u dF S i1 ðuÞ Ar E½X r T uv dF 1 ðvÞ dF S i1 ðuÞ u¼0
i¼1
þ A0
Z
i¼1
u¼0
v¼0
T u¼0
E½X 1 þ X 2 þ þ X N T u dF S K ðuÞ C s F S K þ1þ2þþN ðT Þ:
ð7Þ
First, a simple but important lemma is derived. Lemma 1. If X and Y are positive random variables such that Y[exp (k), then fX þY ðxÞ ¼ k½F X ðxÞ F X þY ðxÞ;
x P 0;
where fX+Y (FX+Y) denotes the density (distribution) function of X + Y and FX denotes the distribution of X. Proof. The result is evident considering that fY ðxÞ ¼ kF Y ðxÞ where fY (FY) denotes the density (distribution) function of Y. h Expression (7) is simplified with the use of the following lemma. Lemma 2. If Xi exp(ki), for all i 2 f1; . . . ; N g, then E½X 1 þ þ X N T ¼
N X i¼1
F 1þþi ðT ÞE½X i ;
N 2 f1; 2; . . .g; T > 0:
ð8Þ
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Proof. The result is obtained using induction on N. For N = 1, considering X1 exp (k1), E½X 1 T ¼ F 1 ðT ÞE½X 1 : Assume that (8) is true for N 1. Using Lemma 1, one has that F 1þþN ðxÞ ¼ F 1þþðN 1Þ ðxÞ þ
1 f1þþN ðxÞ; kN
x P 0:
Finally, Z
T
Z
T
1 F 1þþN ðuÞ du ¼ F 1þþðN 1Þ ðuÞ du þ E½X 1 þ þ X N T ¼ k N 0 0 N 1 N X X F 1þþi ðT Þ F 1þþN ðT Þ F 1þþi ðT Þ þ ¼ : ¼ k k ki i N i¼1 i¼1
Z
T
f1þþN ðuÞ du 0
Using Lemma 2, we rewrite (7) as follows. Proposition 1. For T > 0 and under the assumptions of the model, RN(K) is given by RN ð0Þ ¼
N X A0 F 1þþi ðT Þ C s F 1þþN ðT Þ; ki i¼1
ð9Þ
and for K 2 f1; 2; . . .g, RN ðKÞ ¼
K K N X A0 X Ar X A0 F S i1 þ1 ðT Þ F S i ðT Þ þ F S þ1þþi ðT Þ C s F S K þ1þþN ðT Þ: k1 i¼1 l i¼1 ki K i¼1
ð10Þ
Proof. The proof follows from (7) with the aid of Lemma 2. h From (10), we can obtain important measures of system performance such as the availability of the system " # K N X 1 1 X F S K þ1þ2þþi ðT Þ A¼ F S þ1 ðT Þ þ ; T k1 i¼1 i1 ki i¼1 and the average system up-time " # K N X 1 X F S K þ1þ2þþi ðT Þ W 1 ðKÞ þ W 2 ðKÞ ¼ F S þ1 ðT Þ þ : k1 i¼1 i1 ki i¼1 3. Optimization In this section, we deal with the problem of obtaining the optimal value of K, denoted by Kopt, that maximizes RN(K) given by (9) and (10), in other words, finding a value Kopt such that RN ðK opt Þ ¼ supfRN ðKÞ : K 2 f0; 1; 2; . . .gg; where N is fixed. For K 2 f0; 1; . . .g and N P 1, straightforward calculation yields DRN ðKÞ ¼ RN ðK þ 1Þ RN ðKÞ P 0 () DK;N ðT Þ P
Ar ; l
ð11Þ
where, for x > 0 and K 2 f0; 1; . . .g, DK;1 ðxÞ ¼
F S þ1 ðxÞ F S Kþ1 þ1 ðxÞ A0 F S Kþ1 þ1 ðxÞ þ Cs K ; F S Kþ1 ðxÞ k1 F S Kþ1 ðxÞ
ð12Þ
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A0 F S Kþ1 þ1 ðxÞ A0 F S K þ1þ2 ðxÞ F S Kþ1 þ1þ2 ðxÞ þ Cs ; DK;2 ðxÞ ¼ k1 F S Kþ1 ðxÞ k2 F S Kþ1 ðxÞ A0 F S Kþ1 þ1 ðxÞ A0 F S K þ1þþN ðxÞ F S Kþ1 þ1þþN ðxÞ DK;N ðxÞ ¼ þ Cs F S Kþ1 ðxÞ k1 F S Kþ1 ðxÞ kN N 1 X A0 F S þ1þþi ðxÞ F S þ1þþi ðxÞ K Kþ1 ; N > 2: F k ðxÞ i S Kþ1 i¼2
89
ð13Þ
ð14Þ
To obtain Kopt, we analyze the monotonicity in K of DK,N. To this end, we shall show a useful result in Lemma 4 that is obtained as a consequence of a result of Barlow and Proschan [3]. Lemma 4 is used to prove results about the sign of the convolution of two functions. Indeed, the key to the present work is to determine the sign of convolutions of the type Z x F ðuÞgðx uÞ du; 8x P 0; 0
where F(u) is the distribution function of a positive random variable and g(x u) is a not necessarily positive function. First, we repeat the result of Barlow and Proschan for ease of reference. R1 Lemma 3. Let W(u) be a Lebesgue-Stieltjes measure not necessarily positive and let h(u) P 0. If t dW ðuÞ P 0 for all t and if h is non-decreasing, then Z
1
hðuÞ dW ðuÞ P 0;
8t:
t
The following lemma is very important for later analysis and it is obtained as a consequence of Lemma 3. Lemma 4. Consider the convolution Z
x
F ðuÞgðx uÞ du;
x P 0;
0
where F(u) is R xa distribution function and g(x u) is not necessarily positive. If G(x t) P 0, for 0 6 t 6 x, where Gðx tÞ ¼ t gðx uÞ du, then Z x F ðuÞgðx uÞ P 0; 8x P 0: 0
Proof. Using the notation of Lemma 3, take dW ðuÞ ¼ gðx uÞ;
0 6 u 6 x:
Then notice that Z t
1
8Rx t < 0; > < R0 gðx uÞ du ¼ GðxÞ x gðx uÞ du ¼ gðx uÞ du ¼ Gðx tÞ 0 6 t 6 x; t > : 0 t > x:
Therefore, as G(x t) P 0, for 0 6 t 6 x, from Lemma 3 one has that Z 1 F ðuÞgðx uÞ P 0; 8t: t
In particular, for t = 0, Z 1 Z x F ðuÞgðx uÞ du ¼ F ðuÞgðx uÞ du P 0; 0
0
8x P 0:
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The following results (Lemmas 5 and 6, and Proposition 2) show that the problem of obtaining Kopt (and in consequence the monotonicity of DK,N in K) reduces to the study of the monotonicity in x of the functions given by A0 F S 1 þ1 ðxÞ F 1 ðxÞ F S 1 þ1 ðxÞ þ Cs ; ð15Þ For N ¼ 1 D0;1 ðxÞ ¼ F S 1 ðxÞ k1 F S 1 ðxÞ F S 1 ðxÞ A0 F 2 ðxÞ F S 1 þ2 ðxÞ For N ¼ 2 B1;2 ðxÞ ¼ A0 þ Cs ; ð16Þ k1 F r ðxÞ F r ðxÞ k2 F S ðxÞ A0 F 2þþN ðxÞ F S 1 þ2þþN ðxÞ For N > 2 B1;N ðxÞ ¼ A0 1 þ Cs k1 F r ðxÞ F r ðxÞ kN N 1 X A0 F 2þþi ðxÞ F S 1 þ2þþi ðxÞ k1 : ð17Þ F r ðxÞ ki i¼2 This fact is deduced using Lemmas 5 and 6 and Proposition 2. Lemma 5 shows that, fixed N, if the function B1,N(x) is non-decreasing (non-increasing) in x, then D0,N (x) is non-decreasing (non-increasing) in x. The proof of Lemma 5 is given in the appendix. Lemma 5. For x > 0 fixed N P 2, if B1,N(x) given in (16) and (17) is a non-decreasing (non-increasing) function of x, then D0,N(x) given setting K = 0 in (13) and (14) respectively is a non-decreasing (non-increasing) function of x. Lemma 6 shows that, fixed N, if D0,N(x) is non-decreasing (non-increasing) in x then DK,N(x) is non-decreasing (non-increasing) in x for any K 2 f1; 2; . . .g. The proof of this lemma also is given in the appendix. Lemma 6. Given x > 0 and fixed N P 1, if D0,N(x) is non-decreasing (non-increasing) in x, then DK,N(x) is nondecreasing (non-increasing) in x for any K 2 f1; 2; . . .g. The combination of Lemmas 5 and 6 yields the following proposition. Given x, this result assures the monotonicity in K of DK,N(x). Proposition 2. Fixed N P 1, if DK,N(x) given in (12), (13) and (14) respectively is non-decreasing (nonincreasing) in x, then DK,N(x) is non-increasing (non-decreasing) in K for K 2 f0; 1; 2; . . .g. Proof. Consider the differences DDK1;N ðxÞ ¼ DK;N ðxÞ DK1;N ðxÞ and for DK,N given in (14), these differences are expressed as Z x F S 1 ðuÞF S K ðxÞhðuÞ du; DDK1;N ðxÞF S Kþ1 ðxÞF S K ðxÞ ¼ 0
where hðuÞ ¼
Since
Z
A0 A0 fS K þ1 ðx uÞ þ C s ðfS K1 þ1þþN ðx uÞ fS K þ1þþN ðx uÞÞ k1 kN N 1 X A0 ðfS K1 þ1þþi ðx uÞ fS K þ1þþi ðx uÞÞ DK1;N ðxÞfS K ðx uÞ: ki i¼2
x
hðuÞ du ¼ DK1;N ðx tÞF S K ðx tÞ DK1;N ðxÞF S K ðx tÞ;
t
we have DDK1;N ðxÞ P ð6Þ0 () D0K1;N ðxÞ 6 ðPÞ0; applying Lemma 4. The proof for DK1 and DK,2 is similar and hence omitted.
h
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91
Fixed N P 2, notice that if B1,N(x) given in (16) and (17) is non-decreasing in x, then from Lemmas 5 and 6 and Proposition 2, the optimum value of K is obtained for Ar K opt ¼ min DK;N ðT Þ 6 : KP0 l If B1,N(x) is non-decreasing in x and Ar lim DK;N ðT Þ P ; K!1 l then Kopt = 1. Analogously, if B1,N(x) is non-decreasing in x and Ar D0;N ðT Þ 6 ; l then Kopt = 0. Fixed N, we shall now analyze the limit of DK,N(T), given by (13) and (14), when K ! 1. Note that K = 1 corresponds to the classical repair policy modeled by an alternating renewal process with planned replacement at T. Propositions 3 and 4 show that, under some conditions, the limit of DK,N(T) is zero. For the proofs of these propositions, see the appendix. Proposition 3. For any x > 0, lim
K!1
F S Kþ1 þ1 ðxÞ ¼ 0: F S Kþ1 ðxÞ
Proposition 4. For any x > 0 and l P k2, lim
K!1
F S K þ1þ2 ðxÞ ¼ 0: F S K þ1þr ðxÞ
Fixed N P 2, from Propositions 3 and 4 for l 6 k2 it follows lim DK;N ðT Þ ¼ 0
K!1
and therefore if B1,N(x) is non-decreasing in x and l 6 k2 then Kopt < 1. Note that K = 1 corresponds to the classical repair policy. If B1,N(x) is non-increasing in x, then the optimum value of K is obtained for K opt ¼ 0;
if RN ð0Þ > RN ð1Þ;
K opt ¼ 1
if RN ð0Þ < RN ð1Þ:
In consequence, the analysis of the optimal repair policy is reduced to the analysis of the monotonicity of B1,N(x) given in (17). The next section shows an application of this repair policy for N = 1 and N = 2. 3.1. Optimal repair policies for N = 1 and N = 2 For N = 1, from Lemma 6, the problem of obtaining Kopt reduces to the study of the monotonicity in x of the function D0,1(x) given in (15) and the next result follows. Theorem 1. If A0 6 Cs/k1, then an optimal repair policy Kopt for N = 1 is given by K opt ¼ 0
if R1 ð0Þ > R1 ð1Þ;
K opt ¼ 1
if R1 ð0Þ < R1 ð1Þ:
If R1(0) = R1(1) then both K = 0 and K = 1 maximize R1(K) given in (9) and (10) setting N = 1. Proof. Since F S 1 ðxÞF S 1 ðxÞ F S 1 þ1 ðxÞF r ðxÞ P 0;
8x;
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92
one has that D0,1(x) is non-increasing in x. Then from Lemma 6 and Proposition 2, DK,1(x) is non-decreasing in K, and the result holds. For N = 2, from Lemmas 5 and 6 and Proposition 2, the problem of obtaining Kopt is reduced to the study of the monotonicity in x of B1, 2(x). Lemma 7 provides conditions for the monotonicity of B12. Lemma 7. If either • Ak20 6 C s 6 Ak10 and k2 6 l + k1 or • Csk2 6 A0 and l + k1 6 k2, then B1,2(x) is non-decreasing in x. Analogously, if A0/k1 6 Cs and l + k1 6 k2 then B1,2(x) is non-increasing in x. Proof. As X1, Xr, and X2 are exponentially distributed, B1,2(x) becomes F 1 ðxÞ l k2 ðA0 C s k1 Þ A0 F 2 ðxÞ k2 ðk2 l k1 Þ þ Cs B1;2 ðxÞ ¼ þ C; F r ðxÞ l k1 k2 k1 k2 F r ðxÞ ðl k2 Þðk1 k2 Þ where C is a constant. Notice that F 1 ðxÞ 1 F 2 ðxÞ 1 ; F r ðxÞ l k1 F r ðxÞ l k2 are non-decreasing functions of x, and therefore the result holds. h Now that we have Lemma 7, we are ready to determine Kopt explicitly. The result is shown in the following theorems. Theorem 2. If A0 P Csk2 and k2 P l + k1 then an optimal repair policy Kopt for N = 2 is given by Ar K opt ¼ min DK;2 ðT Þ 6 ; KP0 l where DK,2 is given in (13). The optimal policy Kopt is unique if and only if DK opt ðT Þ < Ar =l. If there exists K* such that Ar DK ðT Þ ¼ ; l * then both K and K* + 1 maximize R2(K). Theorem 3. If A0 A0 6 Cs 6 k2 k1
and
k2 6 l þ k1 ;
then an optimal repair policy Kopt is given by Ar K opt ¼ min DK;2 ðT Þ 6 ; KP0 l
ð18Þ
where DK,2(T) is given in (13). The optimal policy Kopt is unique if and only if DK opt ðT Þ < Ar =l. If there exists a K* such that DK ðT Þ ¼
Ar ; l
then both K* and K* + 1 maximize R2(K). Clearly, if A0 A0 6 Cs 6 k2 k1 then Kopt < 1.
and
k2 6 l;
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Theorem 4. If A0 6 Csk1 and k2 P l + k1, then an optimal repair policy Kopt is given by K opt ¼ 0
if R2 ð0Þ > R2 ð1Þ;
K opt ¼ 1
if R2 ð0Þ < R2 ð1Þ:
If R2(0) = R2(1) then both K = 0 and K = 1 maximize R2(K). 4. Numerical examples In this section, three numerical examples are studied. These examples correspond to particular cases of Theorems 2–4. 4.1. Example 1 In this case, the parameters are k1 = 1/6 failures per unit time (u.t.), k2 = 1/4 failures per u.t., l = 1/2 repairs per u.t., Ar = 280 000 monetary units (m.u.) per u.t., Cs = 1 000 000 m.u., A0 = 200 000 m.u. per u.t., and T = 10 u.t. The values of DK,2(T) and R2(K) are listed in Table 1. For this example, notice that A0 A0 < Cs < ; k2 k1
k2 < l þ k1 ;
and, using Theorem 3, one obtains the value of Kopt as K opt ¼ minfDK;2 ð10Þ 6 560 000g ¼ 3; KP0
which agrees with the results in Table 1. As one observes in Table 1, DK,2(10) is non-increasing in K. 4.2. Example 2 In this case, the parameters are k1 = 1/10, k2 = 1/4, l = 1/20, Ar = 15 000, Cs = 750 000, A0 = 200 000, and T = 10. The values of DK,2(T) and R2(K) are listed in Table 2. For this example, notice that Cs <
A0 ; k2
l þ k1 < k2 ;
and, using Theorem 2, one obtains the value of Kopt as K opt ¼ minfDK;2 ð10Þ 6 300 000g ¼ 2; KP0
which agrees with the results in Table 2. As one observes in Table 2, DK,2(10) is non-increasing in K.
Table 1 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
1
0
1
2
3
4
5
750 224 853 841
712 545 990 812
590 223 1.01088 · 106
509 984 1.01127 · 106
453 770 1.01123 · 106
288 789 1.01123 · 106
0 9.79 · 105
Table 2 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
0
1
2
3
4
5
450 353 1.28623 · 106
399 481 1.3096 · 106
296 144 1.31379 · 106
229 840 1.31376 · 106
185 077 1.31376 · 106
455 465 1.3090 · 106
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Table 3 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
0
1
2
3
4
5
539 115 370 508
625 824 381 246
740 351 313 648
851 401 188 686
946 946 78 585
1.02632 · 106 18 705.2
4.3. Example 3 In this case, the parameters are k1 = 1/2, k2 = 2, l = 1, Ar = 550 000, Cs = 106, A0 = 250 000 and T = 10. The values of DK,2(T) and R2(K) are listed in Table 3. For this example, notice that A0 < Csk1 and k2 > l + k1, so that K opt ¼ 1
if R2 ð0Þ < R2 ð1Þ:
Acknowledgement This research was supported by the Ministerio de Educacio´n y Ciencia, Spain, under grant MTM200601973. Appendix Proof of Lemma 5. Fixed N P 3, first assume that B1,N(x) given in (17) is a non-decreasing function of x. Differentiating D0,N(x), we shall express Z x 2 D00;N ðxÞF S 1 ðxÞ ¼ F 1 ðuÞF r ðxÞhðuÞ du; 0
where hðuÞ ¼ fr ðx uÞB1;N ðxÞ A0 fS 1 ðx uÞ ðC s A0 =kN Þk1 ðf2þþN ðx uÞ fS 1 þ2þþN ðx uÞÞ þ Also,
Z
N 1 X A0 k1 ðf2þþi ðx uÞ fS 1 þ2þþi ðx uÞÞ: ki i¼2
x
hðuÞ du ¼ F r ðx tÞB1;N ðxÞ F r ðx tÞB1;N ðx tÞ P 0;
t
applying the monotonicity of B1,N(x). From Lemma 4, D00;N ðxÞ P 0 and thus D0,N(x) is non-decreasing in x and the result holds. If B1,N(x) is non-increasing, the proof is similar and hence omitted. One obtains the result for B1,2(x) given in (16) analogously. h Proof of Lemma 6. For the proof of this lemma, we shall use the inductive method for K. Setting K = 1 in (14), A0 F S 2 þ1 ðxÞ A0 F S 1 þ1þþN ðxÞ F S 2 þ1þþN ðxÞ D1;N ðxÞ ¼ þ Cs F S 2 ðxÞ k1 F S 2 ðxÞ kN N 1 X A0 F S 1 þ1þþi ðxÞ F S 2 þ1þþi ðxÞ x > 0: ð19Þ F S 2 ðxÞ ki i¼2 First, if D0,N(x) is non-decreasing in x, we shall prove that B2,N(x) given by F S 2 ðxÞ A0 F S þ2þþN ðxÞ F S 2 þ2þþN ðxÞ þ Cs B2;N ðxÞ ¼ A0 k1 1 F S 1 þr ðxÞ F S 1 þr ðxÞ kN N 1 X A0 F S 1 þ2þþi ðxÞ F S 2 þ2þþi ðxÞ k1 F S 1 þr ðxÞ ki i¼2
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is non-decreasing in x. Differentiating with respect to x, it follows Z x 2 F S 1 ðxÞF r ðuÞhðuÞ du; B02;N ðxÞF S 1 þr ðxÞ ¼ l 0
where hðuÞ ¼ k1 fS 1 ðx uÞD0;N ðxÞ A0 fS 1 þ1 ðx uÞ þ
N 1 X A0 k1 ðf1þ2þþi ðx uÞ fS 1 þ1þ2þþi ðx uÞÞ ki i¼2
ðC s A0 =kN Þk1 ðf1þ2þþN ðx uÞ fS 1 þ1þ2þþN ðx uÞÞ: Also one obtains that Z x hðuÞ ¼ k1 F S 1 ðx tÞD0;N ðxÞ k1 F S 1 ðx tÞD0;N ðx tÞ P 0; t
by using D0(x) is non-decreasing in x, and, from Lemma 4, B02;N ðxÞ P 0. Second, we show that the monotonicity of B2,N implies the monotonicity of D1,N(x) given in (19). Differentiating D1,N(x) with respect to x, one obtains that Z x 2 D01;N ðxÞF S 2 ðxÞ ¼ F 1 ðuÞF S 1 þr ðxÞhðuÞ; 0
where hðuÞ ¼ fS 1 þr ðx uÞB2;N ðxÞ A0 fS 2 ðx uÞ k1 ðC s A0 =kN ÞðfS 1 þ2þþN ðx uÞ fS 2 þ2þþN ðx uÞÞ N 1 X A0 þ k1 ðfS 1 þ2þþi ðx uÞ fS 2 þ2þþi ðx uÞÞ: ki i¼2 Also,
Z
x
hðuÞ du ¼ F S 1 þr ðx tÞB2;N ðxÞ F S 1 þr ðx tÞB2;N ðx tÞ P 0;
t
applying B2,N(x) is non-decreasing in x, and, from Lemma 4, D01;N ðxÞ P 0. Now, we assume that DK2,N(x) is non-decreasing in x and we show that the function F S K ðxÞ A0 F S þ2þþN ðxÞ F S K þ2þþN ðxÞ BK;N ðxÞ ¼ A0 þ Cs k1 K1 F S K1 þr ðxÞ F S K1 þr ðxÞ kN N 1 X A0 F S K1 þ2þþi ðxÞ F S K þ2þþi ðxÞ ; x>0 k1 F S K1 þr ðxÞ ki i¼2 is non-decreasing in x. The derivative of BK,N(x) is satisfies Z x 2 0 BK;N ðxÞF S K1 þr ðxÞ ¼ l F S K1 ðxÞF r ðuÞhðuÞ du; 0
where hðuÞ ¼ k1 fS K1 ðx uÞDK2;N ðxÞ þ
N 1 X A0 k1 ðfS K2 þ1þ2þþi ðx uÞ fS K1 þ1þ2þþi ðx uÞÞ ki i¼2
A0 Cs k1 ðfS K2 þ1þ2þþN ðx uÞ fS K1 þ1þ2þþN ðx uÞÞ A0 fS K1 þ1 ðx uÞ: kN As before, Z x hðuÞ du ¼ F S K1 ðx tÞDK2;N ðxÞ F S K1 ðx tÞDK2;N ðx tÞ; t
and, if DK2,N(x) is non-decreasing in x, applying Lemma 4, BK,N(x) is non-decreasing in x.
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Also, the derivative of DK1,N(x) verifies satisfies Z x 2 D0K1;N ðxÞF S K ðxÞ ¼ F 1 ðuÞF S K1 þr ðxÞhðuÞ du; 0
where
A0 hðuÞ ¼ fS K1 þr ðx uÞBK;N ðxÞ A0 fS K ðx uÞ C s k1 ðfS K1 þ2þþN ðx uÞ fS K þ2þþN ðx uÞÞ kN N 1 X A0 þ k1 ðfS K1 þ2þþi ðx uÞ fS K þ2þþi ðx uÞÞ: ki i¼2
Since
Z
x
hðuÞ ¼ F S K1 þr ðx tÞBK;N ðxÞ F S K1 þr ðx tÞBK;N ðx tÞ P 0;
t
using that BK,N(x) is non-decreasing in x and applying Lemma 4, we obtain that DK,N(x) is non-decreasing in x, and the result holds. For D0,N(x) non-increasing, the proof is similar and hence omitted. The result for DK,1(x) and DK,2(x) is obtained analogously. h Proof of Proposition 3. First, we consider the function F ðKþ1Þ1 ðxÞ ; x > 0; C K ðxÞ ¼ F ðKþ2Þ1 ðxÞ where F ðKþ1Þ1 ðxÞ ðfðKþ1Þ1 ðxÞÞ and F ðKþ2Þ1 ðxÞ ðF ðKþ2Þ1 ðxÞÞ are Erlang distributions (densities) of parameters ðk1 ; K þ 1Þ and ðk1 ; K þ 2Þ, respectively. The derivative of CK(x) is given by fðKþ1Þ1 ðxÞ k1 x F ðxÞ F ðxÞ C 0K ðxÞ ¼ ðKþ2Þ1 ðKþ1Þ1 K þ1 ðF ðKþ2Þ1 ðxÞÞ2 k1 x and C 0K ðxÞ 6 0 follows from the fact that F ðKþ2Þ1 ðxÞ F ðKþ1Þ1 ðxÞ Kþ1 is non-increasing in x. Second, we show that
F S Kþ1 þ1 ðxÞ F ðKþ2Þ1 ðxÞ 6 ; F S Kþ1 ðxÞ F ðKþ1Þ1 ðxÞ
K 2 f0; 1; 2; . . . ; g; 8x > 0:
To prove (20), notice that F ðKþ2Þ1 ðxÞF S Kþ1 ðxÞ F S Kþ1 þ1 ðxÞF ðKþ1Þ1 ðxÞ ¼
Z
ð20Þ
x
F ðKþ1Þr ðxÞhðuÞ du;
0
where F(K+1)r(x) is an Erlang distribution of parameters ðl; K þ 1Þ and hðuÞ ¼ F ðKþ2Þ1 ðxÞfðKþ1Þ1 ðx uÞ fðKþ2Þ1 ðx uÞF ðKþ1Þ1 ðxÞ: Also
Z
x
hðuÞ du ¼ F ðKþ2Þ1 ðxÞF ðKþ1Þ1 ðx tÞ F ðKþ2Þ1 ðx tÞF ðKþ1Þ1 ðxÞ P 0;
t
using that CK is non-increasing in x. Finally, inequality (20) follows from Lemma 4. Third, for x > 0 lim
K!1
F ðKþ2Þ1 ðxÞ ¼ 0; F ðKþ1Þ1 ðxÞ
using the Stolz rule. Therefore, from (20), the desired result is obtained. Proof of Proposition 4. For K 2 f0; 1; . . .g consider M K ðxÞ ¼
F Krþ2 ðxÞ ; F ðKþ1Þr ðxÞ
x > 0;
h
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97
where F(K+1)r(x) is an Erlang distribution of parameters ðl; K þ 1Þ and FKr+2(x) represents the convolution Z x f2 ðuÞF Kr ðx uÞ du; F Krþ2 ðxÞ ¼ 0
where FKr(x) is an Erlang distribution of parameters ðl; KÞ. First, we prove that MK(x) is non-decreasing in x for x > 0 and for all K 2 f0; 1; 2; . . .g. For K = 0, M0(x) is non-decreasing in x for k2 6 l. We show that MK(x) non-decreasing in x implies M Kþ1 ðxÞ non-decreasing in x. It holds Z x 2 ðF ðKþ2Þr ðxÞÞ M 0Kþ1 ðxÞ ¼ l F r ðuÞhðuÞ du; 0
where hðuÞ ¼ F Krþ2 ðxÞfðKþ1Þr ðx uÞ fKrþ2 ðx uÞF ðKþ1Þr ðxÞ: Also,
Z
x
hðuÞ du ¼ F Krþ2 ðxÞF ðKþ1Þr ðx tÞ F Krþ2 ðx tÞF ðKþ1Þr ðxÞ P 0;
t
applying the monotonicity of MK(x). Second, we show that F S K þ1þ2 ðxÞ F Krþ2 ðxÞ 6 ; F S K þ1þr ðxÞ F ðKþ1Þr ðxÞ
ð21Þ
for x > 0 and for all K 2 f0; 1; 2; . . .g. We rewrite the difference Z x F Krþ2 ðxÞF S K þ1þr ðxÞ F S K þ1þ2 ðxÞF ðKþ1Þr ðxÞ ¼ F ðKþ1Þ1 ðuÞhðuÞ du; 0
where hðuÞ ¼ F Krþ2 ðxÞfðKþ1Þr ðx uÞ fKrþ2 ðx uÞF ðKþ1Þr ðxÞ: Also,
Z
x
hðuÞ du ¼ F Krþ2 ðxÞF ðKþ1Þr ðx tÞ F Krþ2 ðx tÞF ðKþ1Þr ðxÞ P 0;
t
using MK(x) is non-decreasing in x for all K 2 f0; 1; 2; . . .g. Therefore (21) is verified applying Lemma 4. Third, applying the Stolz rule one has that lim
K!1
F Krþ2 ðxÞ ¼ 0; F ðKþ1Þr ðxÞ
and the desired result is obtained.
h
References [1] [2] [3] [4] [5] [6] [7] [8]
H. Ascher, H. Feingold, Repairable Systems Reliability, Marcel Dekker, New York, 1984. R.E. Barlow, F. Proschan, Mathematical Theory of Reliability, John Wiley and Sons, New York, 1965. R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, To Begin With, Silver Spring, MD, 1981. I.T. Castro, E.L. Sanjua´n, The type I delayed repair policy, Asia-Pacific Journal of Operational Research 24 (2007). T. Nakagawa, Maintenance Theory of Reliability, Springer-Verlag, Berlin, 2005. S. Osaki, Applied Stochastic System Modeling, Springer-Verlag, Berlin, 1992. H. Pham, H. Wang, Imperfect maintenance, European Journal of Operational Research 94 (1996) 425–438. C. Valdez-Flores, R.M. Feldman, A survey of preventive maintenance models for stochastically deteriorating single-unit systems, Naval Research Logistics Quarterly 36 (1989) 419–446. [9] H. Wang, A survey of maintenance policies of deteriorating systems, European Journal of Operational Research 139 (2002) 469–489.
European Journal of Operational Research 187 (2008) 84–97 www.elsevier.com/locate/ejor
Stochastics and Statistics
An optimal repair policy for systems with a limited number of repairs I.T. Castro
a,*
, E.L. Sanjua´n
b
a
b
Departamento de Matema´ticas, Escuela Polite´cnica, Universidad de Extremadura, Avenida de la Universidad, s/n, 10071 Ca´ceres, Spain Departamento de Matema´ticas, Facultad de Veterinaria, Universidad de Extremadura, Avenida de la Universidad, s/n, 10071 Ca´ceres, Spain Received 18 December 2005; accepted 7 March 2007 Available online 24 April 2007
Abstract Traditionally in reliability literature, the repair facilities are always available. This work considers a more general case in which the repair facilities are not always available, but are available only until a fixed number of repairs have been completed. Different assumptions are made to analytically determine an optimal repair policy maximizing the expected reward. 2007 Elsevier B.V. All rights reserved. Keywords: Repairable systems; Optimal policy
1. Introduction There is a extensive literature on the design of new strategies for the replacement/repair of repairable systems, mainly because of the practical applicability to modeling problems of complex electronics, communication equipment, etc. Indeed, thousands of replacement policies have been created. Ascher and Feingold [1], Barlow and Proschan [2], Nakagawa [5], and Osaki [6] are landmark works presenting the mathematical theory of maintenance. There have also been many useful reviews of the history of research in this field, including Pham and Wang [7], Valdez-Flores and Feldman [8], and, most recently, Wang [9]. In general, replacement policies depend on the cost that they involve and the required availability and reliability of the systems. Frequently the reliability literature has made the general assumption that, after a failure detection, the system can immediately go to a repair facility and that after a certain waiting time a repairman is immediately available. This assumption, however, does not reflect practical situations when access to the repairman may be restricted (for example, when the failure of the system occurs out of business hours). For that reason, in the present communication we assume that the repairman is not always available. If the system fails and the repair facilities are available the repair begins. On the other hand, if the system fails and the repair facilities *
Corresponding author. Tel.: +34 927257444. E-mail addresses: [email protected] (I.T. Castro), [email protected] (E.L. Sanjua´n).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.03.027
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85
are not available the repair of the failure is delayed until the repair facilities can be used again. This idea can be implemented in two models that depend on the accessibility of the repair facilities. 1. The first model assumes that the repair facilities are available until T *, 0 6 T * 6 T where ½0; T represents an entire working period. If the system fails in ½0; T (0 6 T * 6 T) then it undergoes repair at failure, whereas if it fails in ðT ; T , then the repair is not made and it is delayed until the beginning of a new working period when a planned replacement is performed [4]. 2. The second model assumes that the repair facilities are available until SK where SK represents the instant when K repairs have been completed by the repairman. If the system fails during ð0; S K then it undergoes repair at failure, whereas if it fails in ðS K ; T , then the repair is not made and it is delayed until the beginning of a new working period when a planned replacement is carried out. We here analyze the second model. Thus, the repair facilities are available in ð0; S K and are unavailable in ðS K ; T . Without loss of generality, we assume that the system holds N unrepaired failures, 1 6 N, in ðS K ; T . This means that, after a failure, the system may be used to complete some other task for which repair of the fault that has occurred is not required. However, these unrepaired failures are deleterious to the operating time of the system, i.e., the successive operating times after an unrepaired failure, become progressively shorter. This situation is modeled using a deteriorating process. Let Xn be the random variable that represents the operating time of the system after the nth unrepaired failure. We say that fX n ; n P 0g is a deteriorating process if F i ðtÞ P F j ðtÞ;
8i; j P 0; j > i; t > 0;
ð1Þ
where F i and F j denote the survival functions of Xi and Xj, respectively. If Xi and Xj satisfy (1) we write Xi Pst Xj. In consequence of this damage, if the number of unrepaired failures per working period reaches a certain fixed limit, N, then the system remains failed until the planned replacement at the beginning of the next working period. The case N = 1 corresponds to a catastrophic failure, and in this case, if the system fails after SK, then it remains failed until the next planned replacement time. A practical example of this model can be found in a military logistics problem with a central depot facility and forward locations (or bases). The bases are capable of repairing parts of the military material and the central depot serves all of the bases at the beginning of each working period. An unlimited number of repairs at the bases would require an extraordinary amount of replacement parts and a very high cost. The central depot therefore serves only parts sufficient to carry out at most K repairs per working period at the bases. After K repairs have been carried out, the bases are not capable of repairing the material from that moment to the next working period (when they receive the necessary parts from the central depot). The problem studied in the present communication is concerned with the optimal stocking of the replacement parts at the bases. Notice that, this model corresponds to a combined replacement model. For K = 0, the model corresponds to the situation where the system is always replaced at times kT ðk ¼ 1; 2; . . .Þ but it is not repaired at failures, and hence, it remains in failed state for the time interval from a failure to its detection. This particular case can apply to systems not monitored continuously whose failures can be detected only at times kT ðk ¼ 1; 2; . . .Þ, see [5] for more details. For K = 1, the model is reduced to the classical repair policy (that is, if the system fails it is repaired and when the repair is completed the system starts to operate again) at the beginning of the working period. Costs are associated with the repair and down times and reward is associated with the operating times, and the objective is to determine an optimal number of repairs. Optimal means a value of K maximizing the expected long-run reward. The combined model and the expected reward of the system are specified in detail in Section 2. In Section 3, we search for an optimal replacement policy. Section 4 presents some numerical examples of the model and concludes. 2. Problem definition and formulation The system is considered to be a single unit which suffers repairable failures. It is activated during an interval of time ½0; T which represents an entire working period. We study the repair strategy by making the following assumptions.
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1. Let X1,i be the operating time of the system after the (i 1)th repair. We assume that X1,i has an exponential distribution with mean 1/k1, "i. Let Xr,i be the repair time of the system after the ith failure. Assume that Xr,i has an exponential distribution with mean 1/l, "i (k1 5 l). We define, n n X X X 1;i þ X r;i ; n P 1; S 0 0: ð2Þ Sn i¼1
i¼1
Let A0 be the reward rate per unit time of the system when it is operating and Ar be the repair cost rate per unit time. 2. The repair facilities are available until K failures have occurred and have been repaired in a working period, i.e. SK. The system is replaced by a new one at the beginning of each working period. It is assumed that the time for replacement is negligible. 3. Let X i ; i ¼ 1; 2; . . ., be the operating time after the i 1th unrepaired failure. fX i ; i ¼ 1; 2; . . .g forms a sequence of exponential random variables with means E(Xi) = 1/ki, i ¼ 1; 2; . . . that verify k1 < k2 < k3 < and ki 5 l, "i. If N failures have not been repaired in a working period, 1 6 N, the system remains in a failed state for the time interval from the Nth unrepaired failure to the planned replacement, with Cs being the downtime cost. Under these assumptions, we shall now derive the expected reward in a working period. The following definition helps to simplify the procedure. Definition 1. Given a positive random variable X with distribution F, the expectation of X truncated at T, T > 0, is defined as E½X T ¼
Z
T
F ðuÞ du ¼
Z
0
T
u dF ðuÞ þ T F ðT Þ:
ð3Þ
0
Notation. • • • • • •
Fi (fi) is the distribution function (density function) of Xi, 1 6 i 6 N. Fr (fr) is the distribution function (density function) of Xr. F iþþj ðfiþþj Þ is the distribution function (density function) of X i þ þ X j , 1 6 i 6 j 6 N. F rþiþþj ðfrþiþþj Þ is the distribution function (density function) of X r þ X i þ þ X j , 1 6 i 6 j 6 N. F S K þiþþj ðfS K þiþþj Þ is the distribution function (density function) of S K þ X i þ þ X j , 0 6 i 6 j 6 N. F S K þrþiþþj ðfS K þrþiþþj ) is the distribution function (density function) of S K þ X r þ X i þ þ X j .
To evaluate the expected reward in a working period, we shall first calculate W1(K), the expected operating time when the repair facilities are available (i.e., before SK). One can deduce that " # K K K Z T X X X W 1 ðKÞ ¼ E X 1i I fS i1 þX 1i 6T g þ ðT S i1 ÞI fS i1
i¼1
from conditional expectation properties as we show next. For i ¼ 1; . . . ; K Z T Z E½X 1i I fS i1 þX 1i 6T g ¼ E½X 1i I fS i1 þX 1i 6T g jS i1 ¼ u dF S i1 ðuÞ ¼ u¼0
u¼0
i¼1
T
u¼0
Z
T u
v dF 1 ðvÞ dF S i1 ðuÞ:
v¼0
Also E½ðT S i1 ÞI fS i1
Z
T
E½ðT S i1 ÞI fS i1
u¼0 Z T u¼0
ðT uÞF 1 ðT uÞ dF S i1 ðuÞ:
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And finally W 1 ðKÞ ¼
K Z X
T
Z
u¼0
i¼1
T u
K Z X v dF 1 ðvÞ þ ðT uÞF 1 ðT uÞ dF S i1 ðuÞ ¼
v¼0
i¼1
T
u¼0
E½X 1 T u dF S i1 ðuÞ:
Analogously, let Wr(K) be the expected repair time when the repair facilities are available. Then " # K K X X W r ðKÞ ¼ E X ri I fS i 6T g þ ðT S i1 X 1i ÞI fS i1 þX 1i
¼
K X
Z
i¼1 T
u¼0
i¼1
Z
T u
E½X r T uv dF 1 ðvÞ dF S i1 ðuÞ:
ð5Þ
v¼0
Equality (5) follows by a calculation similar to that of (4): Z T Z T Z E½X ri I fX 1i þX ri 6T ug dF S i1 ðuÞ ¼ E½X ri I fS i 6T g ¼ 0
0
T u
Z
v¼0
T uv
w dF r ðwÞ dF 1 ðvÞ dF S i1 ðuÞ
w¼0
and E½ðT S i1 X 1i ÞI fS i1 þX 1i
Z
T
u¼0 Z T
Z
T u
E½ðT u vÞI fuþv
v¼0 Z T u
u¼0
ðT u vÞF r ðT u vÞ dF 1 ðvÞ dF S i1 ðuÞ:
v¼0
Finally, let W2(K) be the expected operating time when the repair facilities are unavailable. Then, in a similar form to the above, W 2 ðKÞ ¼ E½ðX 1 þ X 2 þ þ X N ÞI fS K þX 1 þþX N 6T g þ E½ðT S K ÞI fS K 6T
ð6Þ
u¼0
We now give a derivation of the expression for the expected reward. For any K 2 f0; 1; . . .g, denote by RN(K) the expected reward per working period if the repair facilities are unavailable after SK and the system remains failed until the planned time if it fails N-times in (SK, T]. Using (4)–(6), RN(K) can be expressed as follows: K Z T K Z T Z T u X X RN ðKÞ ¼ A0 E½X 1 T u dF S i1 ðuÞ Ar E½X r T uv dF 1 ðvÞ dF S i1 ðuÞ u¼0
i¼1
þ A0
Z
i¼1
u¼0
v¼0
T u¼0
E½X 1 þ X 2 þ þ X N T u dF S K ðuÞ C s F S K þ1þ2þþN ðT Þ:
ð7Þ
First, a simple but important lemma is derived. Lemma 1. If X and Y are positive random variables such that Y[exp (k), then fX þY ðxÞ ¼ k½F X ðxÞ F X þY ðxÞ;
x P 0;
where fX+Y (FX+Y) denotes the density (distribution) function of X + Y and FX denotes the distribution of X. Proof. The result is evident considering that fY ðxÞ ¼ kF Y ðxÞ where fY (FY) denotes the density (distribution) function of Y. h Expression (7) is simplified with the use of the following lemma. Lemma 2. If Xi exp(ki), for all i 2 f1; . . . ; N g, then E½X 1 þ þ X N T ¼
N X i¼1
F 1þþi ðT ÞE½X i ;
N 2 f1; 2; . . .g; T > 0:
ð8Þ
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88
Proof. The result is obtained using induction on N. For N = 1, considering X1 exp (k1), E½X 1 T ¼ F 1 ðT ÞE½X 1 : Assume that (8) is true for N 1. Using Lemma 1, one has that F 1þþN ðxÞ ¼ F 1þþðN 1Þ ðxÞ þ
1 f1þþN ðxÞ; kN
x P 0:
Finally, Z
T
Z
T
1 F 1þþN ðuÞ du ¼ F 1þþðN 1Þ ðuÞ du þ E½X 1 þ þ X N T ¼ k N 0 0 N 1 N X X F 1þþi ðT Þ F 1þþN ðT Þ F 1þþi ðT Þ þ ¼ : ¼ k k ki i N i¼1 i¼1
Z
T
f1þþN ðuÞ du 0
Using Lemma 2, we rewrite (7) as follows. Proposition 1. For T > 0 and under the assumptions of the model, RN(K) is given by RN ð0Þ ¼
N X A0 F 1þþi ðT Þ C s F 1þþN ðT Þ; ki i¼1
ð9Þ
and for K 2 f1; 2; . . .g, RN ðKÞ ¼
K K N X A0 X Ar X A0 F S i1 þ1 ðT Þ F S i ðT Þ þ F S þ1þþi ðT Þ C s F S K þ1þþN ðT Þ: k1 i¼1 l i¼1 ki K i¼1
ð10Þ
Proof. The proof follows from (7) with the aid of Lemma 2. h From (10), we can obtain important measures of system performance such as the availability of the system " # K N X 1 1 X F S K þ1þ2þþi ðT Þ A¼ F S þ1 ðT Þ þ ; T k1 i¼1 i1 ki i¼1 and the average system up-time " # K N X 1 X F S K þ1þ2þþi ðT Þ W 1 ðKÞ þ W 2 ðKÞ ¼ F S þ1 ðT Þ þ : k1 i¼1 i1 ki i¼1 3. Optimization In this section, we deal with the problem of obtaining the optimal value of K, denoted by Kopt, that maximizes RN(K) given by (9) and (10), in other words, finding a value Kopt such that RN ðK opt Þ ¼ supfRN ðKÞ : K 2 f0; 1; 2; . . .gg; where N is fixed. For K 2 f0; 1; . . .g and N P 1, straightforward calculation yields DRN ðKÞ ¼ RN ðK þ 1Þ RN ðKÞ P 0 () DK;N ðT Þ P
Ar ; l
ð11Þ
where, for x > 0 and K 2 f0; 1; . . .g, DK;1 ðxÞ ¼
F S þ1 ðxÞ F S Kþ1 þ1 ðxÞ A0 F S Kþ1 þ1 ðxÞ þ Cs K ; F S Kþ1 ðxÞ k1 F S Kþ1 ðxÞ
ð12Þ
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A0 F S Kþ1 þ1 ðxÞ A0 F S K þ1þ2 ðxÞ F S Kþ1 þ1þ2 ðxÞ þ Cs ; DK;2 ðxÞ ¼ k1 F S Kþ1 ðxÞ k2 F S Kþ1 ðxÞ A0 F S Kþ1 þ1 ðxÞ A0 F S K þ1þþN ðxÞ F S Kþ1 þ1þþN ðxÞ DK;N ðxÞ ¼ þ Cs F S Kþ1 ðxÞ k1 F S Kþ1 ðxÞ kN N 1 X A0 F S þ1þþi ðxÞ F S þ1þþi ðxÞ K Kþ1 ; N > 2: F k ðxÞ i S Kþ1 i¼2
89
ð13Þ
ð14Þ
To obtain Kopt, we analyze the monotonicity in K of DK,N. To this end, we shall show a useful result in Lemma 4 that is obtained as a consequence of a result of Barlow and Proschan [3]. Lemma 4 is used to prove results about the sign of the convolution of two functions. Indeed, the key to the present work is to determine the sign of convolutions of the type Z x F ðuÞgðx uÞ du; 8x P 0; 0
where F(u) is the distribution function of a positive random variable and g(x u) is a not necessarily positive function. First, we repeat the result of Barlow and Proschan for ease of reference. R1 Lemma 3. Let W(u) be a Lebesgue-Stieltjes measure not necessarily positive and let h(u) P 0. If t dW ðuÞ P 0 for all t and if h is non-decreasing, then Z
1
hðuÞ dW ðuÞ P 0;
8t:
t
The following lemma is very important for later analysis and it is obtained as a consequence of Lemma 3. Lemma 4. Consider the convolution Z
x
F ðuÞgðx uÞ du;
x P 0;
0
where F(u) is R xa distribution function and g(x u) is not necessarily positive. If G(x t) P 0, for 0 6 t 6 x, where Gðx tÞ ¼ t gðx uÞ du, then Z x F ðuÞgðx uÞ P 0; 8x P 0: 0
Proof. Using the notation of Lemma 3, take dW ðuÞ ¼ gðx uÞ;
0 6 u 6 x:
Then notice that Z t
1
8Rx t < 0; > < R0 gðx uÞ du ¼ GðxÞ x gðx uÞ du ¼ gðx uÞ du ¼ Gðx tÞ 0 6 t 6 x; t > : 0 t > x:
Therefore, as G(x t) P 0, for 0 6 t 6 x, from Lemma 3 one has that Z 1 F ðuÞgðx uÞ P 0; 8t: t
In particular, for t = 0, Z 1 Z x F ðuÞgðx uÞ du ¼ F ðuÞgðx uÞ du P 0; 0
0
8x P 0:
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The following results (Lemmas 5 and 6, and Proposition 2) show that the problem of obtaining Kopt (and in consequence the monotonicity of DK,N in K) reduces to the study of the monotonicity in x of the functions given by A0 F S 1 þ1 ðxÞ F 1 ðxÞ F S 1 þ1 ðxÞ þ Cs ; ð15Þ For N ¼ 1 D0;1 ðxÞ ¼ F S 1 ðxÞ k1 F S 1 ðxÞ F S 1 ðxÞ A0 F 2 ðxÞ F S 1 þ2 ðxÞ For N ¼ 2 B1;2 ðxÞ ¼ A0 þ Cs ; ð16Þ k1 F r ðxÞ F r ðxÞ k2 F S ðxÞ A0 F 2þþN ðxÞ F S 1 þ2þþN ðxÞ For N > 2 B1;N ðxÞ ¼ A0 1 þ Cs k1 F r ðxÞ F r ðxÞ kN N 1 X A0 F 2þþi ðxÞ F S 1 þ2þþi ðxÞ k1 : ð17Þ F r ðxÞ ki i¼2 This fact is deduced using Lemmas 5 and 6 and Proposition 2. Lemma 5 shows that, fixed N, if the function B1,N(x) is non-decreasing (non-increasing) in x, then D0,N (x) is non-decreasing (non-increasing) in x. The proof of Lemma 5 is given in the appendix. Lemma 5. For x > 0 fixed N P 2, if B1,N(x) given in (16) and (17) is a non-decreasing (non-increasing) function of x, then D0,N(x) given setting K = 0 in (13) and (14) respectively is a non-decreasing (non-increasing) function of x. Lemma 6 shows that, fixed N, if D0,N(x) is non-decreasing (non-increasing) in x then DK,N(x) is non-decreasing (non-increasing) in x for any K 2 f1; 2; . . .g. The proof of this lemma also is given in the appendix. Lemma 6. Given x > 0 and fixed N P 1, if D0,N(x) is non-decreasing (non-increasing) in x, then DK,N(x) is nondecreasing (non-increasing) in x for any K 2 f1; 2; . . .g. The combination of Lemmas 5 and 6 yields the following proposition. Given x, this result assures the monotonicity in K of DK,N(x). Proposition 2. Fixed N P 1, if DK,N(x) given in (12), (13) and (14) respectively is non-decreasing (nonincreasing) in x, then DK,N(x) is non-increasing (non-decreasing) in K for K 2 f0; 1; 2; . . .g. Proof. Consider the differences DDK1;N ðxÞ ¼ DK;N ðxÞ DK1;N ðxÞ and for DK,N given in (14), these differences are expressed as Z x F S 1 ðuÞF S K ðxÞhðuÞ du; DDK1;N ðxÞF S Kþ1 ðxÞF S K ðxÞ ¼ 0
where hðuÞ ¼
Since
Z
A0 A0 fS K þ1 ðx uÞ þ C s ðfS K1 þ1þþN ðx uÞ fS K þ1þþN ðx uÞÞ k1 kN N 1 X A0 ðfS K1 þ1þþi ðx uÞ fS K þ1þþi ðx uÞÞ DK1;N ðxÞfS K ðx uÞ: ki i¼2
x
hðuÞ du ¼ DK1;N ðx tÞF S K ðx tÞ DK1;N ðxÞF S K ðx tÞ;
t
we have DDK1;N ðxÞ P ð6Þ0 () D0K1;N ðxÞ 6 ðPÞ0; applying Lemma 4. The proof for DK1 and DK,2 is similar and hence omitted.
h
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Fixed N P 2, notice that if B1,N(x) given in (16) and (17) is non-decreasing in x, then from Lemmas 5 and 6 and Proposition 2, the optimum value of K is obtained for Ar K opt ¼ min DK;N ðT Þ 6 : KP0 l If B1,N(x) is non-decreasing in x and Ar lim DK;N ðT Þ P ; K!1 l then Kopt = 1. Analogously, if B1,N(x) is non-decreasing in x and Ar D0;N ðT Þ 6 ; l then Kopt = 0. Fixed N, we shall now analyze the limit of DK,N(T), given by (13) and (14), when K ! 1. Note that K = 1 corresponds to the classical repair policy modeled by an alternating renewal process with planned replacement at T. Propositions 3 and 4 show that, under some conditions, the limit of DK,N(T) is zero. For the proofs of these propositions, see the appendix. Proposition 3. For any x > 0, lim
K!1
F S Kþ1 þ1 ðxÞ ¼ 0: F S Kþ1 ðxÞ
Proposition 4. For any x > 0 and l P k2, lim
K!1
F S K þ1þ2 ðxÞ ¼ 0: F S K þ1þr ðxÞ
Fixed N P 2, from Propositions 3 and 4 for l 6 k2 it follows lim DK;N ðT Þ ¼ 0
K!1
and therefore if B1,N(x) is non-decreasing in x and l 6 k2 then Kopt < 1. Note that K = 1 corresponds to the classical repair policy. If B1,N(x) is non-increasing in x, then the optimum value of K is obtained for K opt ¼ 0;
if RN ð0Þ > RN ð1Þ;
K opt ¼ 1
if RN ð0Þ < RN ð1Þ:
In consequence, the analysis of the optimal repair policy is reduced to the analysis of the monotonicity of B1,N(x) given in (17). The next section shows an application of this repair policy for N = 1 and N = 2. 3.1. Optimal repair policies for N = 1 and N = 2 For N = 1, from Lemma 6, the problem of obtaining Kopt reduces to the study of the monotonicity in x of the function D0,1(x) given in (15) and the next result follows. Theorem 1. If A0 6 Cs/k1, then an optimal repair policy Kopt for N = 1 is given by K opt ¼ 0
if R1 ð0Þ > R1 ð1Þ;
K opt ¼ 1
if R1 ð0Þ < R1 ð1Þ:
If R1(0) = R1(1) then both K = 0 and K = 1 maximize R1(K) given in (9) and (10) setting N = 1. Proof. Since F S 1 ðxÞF S 1 ðxÞ F S 1 þ1 ðxÞF r ðxÞ P 0;
8x;
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one has that D0,1(x) is non-increasing in x. Then from Lemma 6 and Proposition 2, DK,1(x) is non-decreasing in K, and the result holds. For N = 2, from Lemmas 5 and 6 and Proposition 2, the problem of obtaining Kopt is reduced to the study of the monotonicity in x of B1, 2(x). Lemma 7 provides conditions for the monotonicity of B12. Lemma 7. If either • Ak20 6 C s 6 Ak10 and k2 6 l + k1 or • Csk2 6 A0 and l + k1 6 k2, then B1,2(x) is non-decreasing in x. Analogously, if A0/k1 6 Cs and l + k1 6 k2 then B1,2(x) is non-increasing in x. Proof. As X1, Xr, and X2 are exponentially distributed, B1,2(x) becomes F 1 ðxÞ l k2 ðA0 C s k1 Þ A0 F 2 ðxÞ k2 ðk2 l k1 Þ þ Cs B1;2 ðxÞ ¼ þ C; F r ðxÞ l k1 k2 k1 k2 F r ðxÞ ðl k2 Þðk1 k2 Þ where C is a constant. Notice that F 1 ðxÞ 1 F 2 ðxÞ 1 ; F r ðxÞ l k1 F r ðxÞ l k2 are non-decreasing functions of x, and therefore the result holds. h Now that we have Lemma 7, we are ready to determine Kopt explicitly. The result is shown in the following theorems. Theorem 2. If A0 P Csk2 and k2 P l + k1 then an optimal repair policy Kopt for N = 2 is given by Ar K opt ¼ min DK;2 ðT Þ 6 ; KP0 l where DK,2 is given in (13). The optimal policy Kopt is unique if and only if DK opt ðT Þ < Ar =l. If there exists K* such that Ar DK ðT Þ ¼ ; l * then both K and K* + 1 maximize R2(K). Theorem 3. If A0 A0 6 Cs 6 k2 k1
and
k2 6 l þ k1 ;
then an optimal repair policy Kopt is given by Ar K opt ¼ min DK;2 ðT Þ 6 ; KP0 l
ð18Þ
where DK,2(T) is given in (13). The optimal policy Kopt is unique if and only if DK opt ðT Þ < Ar =l. If there exists a K* such that DK ðT Þ ¼
Ar ; l
then both K* and K* + 1 maximize R2(K). Clearly, if A0 A0 6 Cs 6 k2 k1 then Kopt < 1.
and
k2 6 l;
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Theorem 4. If A0 6 Csk1 and k2 P l + k1, then an optimal repair policy Kopt is given by K opt ¼ 0
if R2 ð0Þ > R2 ð1Þ;
K opt ¼ 1
if R2 ð0Þ < R2 ð1Þ:
If R2(0) = R2(1) then both K = 0 and K = 1 maximize R2(K). 4. Numerical examples In this section, three numerical examples are studied. These examples correspond to particular cases of Theorems 2–4. 4.1. Example 1 In this case, the parameters are k1 = 1/6 failures per unit time (u.t.), k2 = 1/4 failures per u.t., l = 1/2 repairs per u.t., Ar = 280 000 monetary units (m.u.) per u.t., Cs = 1 000 000 m.u., A0 = 200 000 m.u. per u.t., and T = 10 u.t. The values of DK,2(T) and R2(K) are listed in Table 1. For this example, notice that A0 A0 < Cs < ; k2 k1
k2 < l þ k1 ;
and, using Theorem 3, one obtains the value of Kopt as K opt ¼ minfDK;2 ð10Þ 6 560 000g ¼ 3; KP0
which agrees with the results in Table 1. As one observes in Table 1, DK,2(10) is non-increasing in K. 4.2. Example 2 In this case, the parameters are k1 = 1/10, k2 = 1/4, l = 1/20, Ar = 15 000, Cs = 750 000, A0 = 200 000, and T = 10. The values of DK,2(T) and R2(K) are listed in Table 2. For this example, notice that Cs <
A0 ; k2
l þ k1 < k2 ;
and, using Theorem 2, one obtains the value of Kopt as K opt ¼ minfDK;2 ð10Þ 6 300 000g ¼ 2; KP0
which agrees with the results in Table 2. As one observes in Table 2, DK,2(10) is non-increasing in K.
Table 1 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
1
0
1
2
3
4
5
750 224 853 841
712 545 990 812
590 223 1.01088 · 106
509 984 1.01127 · 106
453 770 1.01123 · 106
288 789 1.01123 · 106
0 9.79 · 105
Table 2 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
0
1
2
3
4
5
450 353 1.28623 · 106
399 481 1.3096 · 106
296 144 1.31379 · 106
229 840 1.31376 · 106
185 077 1.31376 · 106
455 465 1.3090 · 106
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Table 3 The values of the expected reward and DK,2(10) for different values of K DK,2(10) R2(10)
0
1
2
3
4
5
539 115 370 508
625 824 381 246
740 351 313 648
851 401 188 686
946 946 78 585
1.02632 · 106 18 705.2
4.3. Example 3 In this case, the parameters are k1 = 1/2, k2 = 2, l = 1, Ar = 550 000, Cs = 106, A0 = 250 000 and T = 10. The values of DK,2(T) and R2(K) are listed in Table 3. For this example, notice that A0 < Csk1 and k2 > l + k1, so that K opt ¼ 1
if R2 ð0Þ < R2 ð1Þ:
Acknowledgement This research was supported by the Ministerio de Educacio´n y Ciencia, Spain, under grant MTM200601973. Appendix Proof of Lemma 5. Fixed N P 3, first assume that B1,N(x) given in (17) is a non-decreasing function of x. Differentiating D0,N(x), we shall express Z x 2 D00;N ðxÞF S 1 ðxÞ ¼ F 1 ðuÞF r ðxÞhðuÞ du; 0
where hðuÞ ¼ fr ðx uÞB1;N ðxÞ A0 fS 1 ðx uÞ ðC s A0 =kN Þk1 ðf2þþN ðx uÞ fS 1 þ2þþN ðx uÞÞ þ Also,
Z
N 1 X A0 k1 ðf2þþi ðx uÞ fS 1 þ2þþi ðx uÞÞ: ki i¼2
x
hðuÞ du ¼ F r ðx tÞB1;N ðxÞ F r ðx tÞB1;N ðx tÞ P 0;
t
applying the monotonicity of B1,N(x). From Lemma 4, D00;N ðxÞ P 0 and thus D0,N(x) is non-decreasing in x and the result holds. If B1,N(x) is non-increasing, the proof is similar and hence omitted. One obtains the result for B1,2(x) given in (16) analogously. h Proof of Lemma 6. For the proof of this lemma, we shall use the inductive method for K. Setting K = 1 in (14), A0 F S 2 þ1 ðxÞ A0 F S 1 þ1þþN ðxÞ F S 2 þ1þþN ðxÞ D1;N ðxÞ ¼ þ Cs F S 2 ðxÞ k1 F S 2 ðxÞ kN N 1 X A0 F S 1 þ1þþi ðxÞ F S 2 þ1þþi ðxÞ x > 0: ð19Þ F S 2 ðxÞ ki i¼2 First, if D0,N(x) is non-decreasing in x, we shall prove that B2,N(x) given by F S 2 ðxÞ A0 F S þ2þþN ðxÞ F S 2 þ2þþN ðxÞ þ Cs B2;N ðxÞ ¼ A0 k1 1 F S 1 þr ðxÞ F S 1 þr ðxÞ kN N 1 X A0 F S 1 þ2þþi ðxÞ F S 2 þ2þþi ðxÞ k1 F S 1 þr ðxÞ ki i¼2
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is non-decreasing in x. Differentiating with respect to x, it follows Z x 2 F S 1 ðxÞF r ðuÞhðuÞ du; B02;N ðxÞF S 1 þr ðxÞ ¼ l 0
where hðuÞ ¼ k1 fS 1 ðx uÞD0;N ðxÞ A0 fS 1 þ1 ðx uÞ þ
N 1 X A0 k1 ðf1þ2þþi ðx uÞ fS 1 þ1þ2þþi ðx uÞÞ ki i¼2
ðC s A0 =kN Þk1 ðf1þ2þþN ðx uÞ fS 1 þ1þ2þþN ðx uÞÞ: Also one obtains that Z x hðuÞ ¼ k1 F S 1 ðx tÞD0;N ðxÞ k1 F S 1 ðx tÞD0;N ðx tÞ P 0; t
by using D0(x) is non-decreasing in x, and, from Lemma 4, B02;N ðxÞ P 0. Second, we show that the monotonicity of B2,N implies the monotonicity of D1,N(x) given in (19). Differentiating D1,N(x) with respect to x, one obtains that Z x 2 D01;N ðxÞF S 2 ðxÞ ¼ F 1 ðuÞF S 1 þr ðxÞhðuÞ; 0
where hðuÞ ¼ fS 1 þr ðx uÞB2;N ðxÞ A0 fS 2 ðx uÞ k1 ðC s A0 =kN ÞðfS 1 þ2þþN ðx uÞ fS 2 þ2þþN ðx uÞÞ N 1 X A0 þ k1 ðfS 1 þ2þþi ðx uÞ fS 2 þ2þþi ðx uÞÞ: ki i¼2 Also,
Z
x
hðuÞ du ¼ F S 1 þr ðx tÞB2;N ðxÞ F S 1 þr ðx tÞB2;N ðx tÞ P 0;
t
applying B2,N(x) is non-decreasing in x, and, from Lemma 4, D01;N ðxÞ P 0. Now, we assume that DK2,N(x) is non-decreasing in x and we show that the function F S K ðxÞ A0 F S þ2þþN ðxÞ F S K þ2þþN ðxÞ BK;N ðxÞ ¼ A0 þ Cs k1 K1 F S K1 þr ðxÞ F S K1 þr ðxÞ kN N 1 X A0 F S K1 þ2þþi ðxÞ F S K þ2þþi ðxÞ ; x>0 k1 F S K1 þr ðxÞ ki i¼2 is non-decreasing in x. The derivative of BK,N(x) is satisfies Z x 2 0 BK;N ðxÞF S K1 þr ðxÞ ¼ l F S K1 ðxÞF r ðuÞhðuÞ du; 0
where hðuÞ ¼ k1 fS K1 ðx uÞDK2;N ðxÞ þ
N 1 X A0 k1 ðfS K2 þ1þ2þþi ðx uÞ fS K1 þ1þ2þþi ðx uÞÞ ki i¼2
A0 Cs k1 ðfS K2 þ1þ2þþN ðx uÞ fS K1 þ1þ2þþN ðx uÞÞ A0 fS K1 þ1 ðx uÞ: kN As before, Z x hðuÞ du ¼ F S K1 ðx tÞDK2;N ðxÞ F S K1 ðx tÞDK2;N ðx tÞ; t
and, if DK2,N(x) is non-decreasing in x, applying Lemma 4, BK,N(x) is non-decreasing in x.
I.T. Castro, E.L. Sanjua´n / European Journal of Operational Research 187 (2008) 84–97
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Also, the derivative of DK1,N(x) verifies satisfies Z x 2 D0K1;N ðxÞF S K ðxÞ ¼ F 1 ðuÞF S K1 þr ðxÞhðuÞ du; 0
where
A0 hðuÞ ¼ fS K1 þr ðx uÞBK;N ðxÞ A0 fS K ðx uÞ C s k1 ðfS K1 þ2þþN ðx uÞ fS K þ2þþN ðx uÞÞ kN N 1 X A0 þ k1 ðfS K1 þ2þþi ðx uÞ fS K þ2þþi ðx uÞÞ: ki i¼2
Since
Z
x
hðuÞ ¼ F S K1 þr ðx tÞBK;N ðxÞ F S K1 þr ðx tÞBK;N ðx tÞ P 0;
t
using that BK,N(x) is non-decreasing in x and applying Lemma 4, we obtain that DK,N(x) is non-decreasing in x, and the result holds. For D0,N(x) non-increasing, the proof is similar and hence omitted. The result for DK,1(x) and DK,2(x) is obtained analogously. h Proof of Proposition 3. First, we consider the function F ðKþ1Þ1 ðxÞ ; x > 0; C K ðxÞ ¼ F ðKþ2Þ1 ðxÞ where F ðKþ1Þ1 ðxÞ ðfðKþ1Þ1 ðxÞÞ and F ðKþ2Þ1 ðxÞ ðF ðKþ2Þ1 ðxÞÞ are Erlang distributions (densities) of parameters ðk1 ; K þ 1Þ and ðk1 ; K þ 2Þ, respectively. The derivative of CK(x) is given by fðKþ1Þ1 ðxÞ k1 x F ðxÞ F ðxÞ C 0K ðxÞ ¼ ðKþ2Þ1 ðKþ1Þ1 K þ1 ðF ðKþ2Þ1 ðxÞÞ2 k1 x and C 0K ðxÞ 6 0 follows from the fact that F ðKþ2Þ1 ðxÞ F ðKþ1Þ1 ðxÞ Kþ1 is non-increasing in x. Second, we show that
F S Kþ1 þ1 ðxÞ F ðKþ2Þ1 ðxÞ 6 ; F S Kþ1 ðxÞ F ðKþ1Þ1 ðxÞ
K 2 f0; 1; 2; . . . ; g; 8x > 0:
To prove (20), notice that F ðKþ2Þ1 ðxÞF S Kþ1 ðxÞ F S Kþ1 þ1 ðxÞF ðKþ1Þ1 ðxÞ ¼
Z
ð20Þ
x
F ðKþ1Þr ðxÞhðuÞ du;
0
where F(K+1)r(x) is an Erlang distribution of parameters ðl; K þ 1Þ and hðuÞ ¼ F ðKþ2Þ1 ðxÞfðKþ1Þ1 ðx uÞ fðKþ2Þ1 ðx uÞF ðKþ1Þ1 ðxÞ: Also
Z
x
hðuÞ du ¼ F ðKþ2Þ1 ðxÞF ðKþ1Þ1 ðx tÞ F ðKþ2Þ1 ðx tÞF ðKþ1Þ1 ðxÞ P 0;
t
using that CK is non-increasing in x. Finally, inequality (20) follows from Lemma 4. Third, for x > 0 lim
K!1
F ðKþ2Þ1 ðxÞ ¼ 0; F ðKþ1Þ1 ðxÞ
using the Stolz rule. Therefore, from (20), the desired result is obtained. Proof of Proposition 4. For K 2 f0; 1; . . .g consider M K ðxÞ ¼
F Krþ2 ðxÞ ; F ðKþ1Þr ðxÞ
x > 0;
h
I.T. Castro, E.L. Sanjua´n / European Journal of Operational Research 187 (2008) 84–97
97
where F(K+1)r(x) is an Erlang distribution of parameters ðl; K þ 1Þ and FKr+2(x) represents the convolution Z x f2 ðuÞF Kr ðx uÞ du; F Krþ2 ðxÞ ¼ 0
where FKr(x) is an Erlang distribution of parameters ðl; KÞ. First, we prove that MK(x) is non-decreasing in x for x > 0 and for all K 2 f0; 1; 2; . . .g. For K = 0, M0(x) is non-decreasing in x for k2 6 l. We show that MK(x) non-decreasing in x implies M Kþ1 ðxÞ non-decreasing in x. It holds Z x 2 ðF ðKþ2Þr ðxÞÞ M 0Kþ1 ðxÞ ¼ l F r ðuÞhðuÞ du; 0
where hðuÞ ¼ F Krþ2 ðxÞfðKþ1Þr ðx uÞ fKrþ2 ðx uÞF ðKþ1Þr ðxÞ: Also,
Z
x
hðuÞ du ¼ F Krþ2 ðxÞF ðKþ1Þr ðx tÞ F Krþ2 ðx tÞF ðKþ1Þr ðxÞ P 0;
t
applying the monotonicity of MK(x). Second, we show that F S K þ1þ2 ðxÞ F Krþ2 ðxÞ 6 ; F S K þ1þr ðxÞ F ðKþ1Þr ðxÞ
ð21Þ
for x > 0 and for all K 2 f0; 1; 2; . . .g. We rewrite the difference Z x F Krþ2 ðxÞF S K þ1þr ðxÞ F S K þ1þ2 ðxÞF ðKþ1Þr ðxÞ ¼ F ðKþ1Þ1 ðuÞhðuÞ du; 0
where hðuÞ ¼ F Krþ2 ðxÞfðKþ1Þr ðx uÞ fKrþ2 ðx uÞF ðKþ1Þr ðxÞ: Also,
Z
x
hðuÞ du ¼ F Krþ2 ðxÞF ðKþ1Þr ðx tÞ F Krþ2 ðx tÞF ðKþ1Þr ðxÞ P 0;
t
using MK(x) is non-decreasing in x for all K 2 f0; 1; 2; . . .g. Therefore (21) is verified applying Lemma 4. Third, applying the Stolz rule one has that lim
K!1
F Krþ2 ðxÞ ¼ 0; F ðKþ1Þr ðxÞ
and the desired result is obtained.
h
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