64 The imperfect production systems have been investigated by many researchers in the literature under the relaxed assumption that the production proc

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1 212 2 Chinese Journal of Applied Probability and Statistics Vol.28 No.1 Feb. 212 Optimal Periodic Preventive Overhaul Policy for Degenerate Production Systems under a Free-Repair Warranty Hu Fei (Department of Mathematics, Tianjin University, Tianjin, 372) Deng Shimei (The Eighteenth Research Institute of China Electronics Technology Group Corporation, Tianjin, 3381) Abstract A periodic maintenance policy for a degenerate production system in which the products are sold with free minimal repair warranty is considered. The degenerate process of the system is characterized by three states: in-control state, out-of control state and failure state. The amount of time that the process stays in in-control state and out-of-control state are both assumed to be exponentially distributed. The production system is overhauled at fixed time t or at failure, whichever occurs first. The optimal periodic overhaul time t minimizing the expected cost per item per cycle is analytically discussed, and three special cases show the property of the optimal value t. In addition, sensitivity analysis and numerical examples concerning model parameters are carried out to illuminate the effects of the model parameters on the optimal periodic overhaul policy. Keywords: Periodic overhaul policy, degenerate production system, warranty, EMQ. AMS Subject Classification: 9B3. 1. Introduction The classical economic manufacturing quantity (EMQ) models assumes that a production system is failure free, and that all the items produced are of perfect quality. But in reality, production system continuously degenerates due to usage or age such as corrosion, fatigue, and cumulative wear. Therefore, without taking any maintenance action to the system, although the production process starts from an in-control state which produces high or perfect quality items, the system will eventually leave the in-control state and shift to the out-of-control state, and finally to the failure state. This work is supproted by the National Natural Science Foundation of China (71216) and the Independent Innovation Fund of Tianjin University (63229). Received November 7, 29. Revised January 12, 211.

2 64 The imperfect production systems have been investigated by many researchers in the literature under the relaxed assumption that the production process always produces perfect items in EMQ models. Rosenblatt and Lee (1987) initially studied the effects of degenerate process on the traditional EMQ model where the random degeneration from in-control state to out-of-control state is exponentially distributed. Porteus (1986) demonstrated that it is better to produce lot sizes smaller than the classical EMQ for a process that will go into an out-of-control state with a given probability each time. Hsieh and Lee (25) simultaneously determined the optimal production run length and number of standbys for an imperfect production system. Lee and Rosenblatt (1987) considered an inspection mechanism to monitor the imperfect production system and simultaneously determined the production cycle and the inspection schedule. Tseng (1996) replaced a preventive maintenance policy with an inspection policy for an imperfect EMQ model. On the basis of Tseng s model, Wang and Sheu (2) provided several useful properties for obtaining an optimal production/maintenance policy. In the present day of competitive strategies, warranty for customers is an important part of the market planning. In order to promote customers confidence, a manufacturer usually sells item with a warranty period. As a result, offering warranty implies an additional cost to the manufacturer and this cost depends on the servicing strategy. A variety of warranty policies have been studied, and a taxonomy of these can be found in Blischke and Murthy (1994). The review article of Murthy and Djamaludin (22) provided a list of 186 references. By taking the difference between warranty cost after sale and rework cost before sale into account, Lee and Park (1991) simultaneity determined the optimal production cycle and the inspection schedule. Djamaludin et al (1994) utilized lot size to control the warranty cost per item for products under free repair warranty (FRW), where the production system can go into an out-of-control state with a given probability each time that an item is produced. Yeh et al (2) reformulated the Djamaludin et al model to consider that the production process is subject to a random deterioration from an in-control state to an out-of-control state, taking both the restoration cost and the inventory holding cost into account and obtaining the bounds of the optimal production run length. Wang (24) extends Yeh et al (2) model to incorporate a more generalized assumption that an elapsed time until process shift is arbitrarily distributed. Chen and Lo (26) extended the work of Yeh et al (2) to consider allowable shortages for the imperfect production processes. In above literatures, the failure state of the production system are not considered. However, in real production, the process failure rate increases with the length of the production time, hence, the system will inevitably enter a failure state, which leads to production shutdown.

3 : 65 In this paper, a failure state is considered in a degenerate production system for items sold with FRW policy. The production is accidentally broken down after the production system goes into the failure state, and failure-repair actions are taken for restoring the system to the initial state. In order to reduce the failure occurrence we take a periodic overhaul policy for the production process. The rest of this paper is organized as follows: Mathematical notations, basic assumptions and model environments are presented in Section 2. The properties of the optimal periodic overhaul policy and searching procedure are given in Section 3. Special cases and sensitivity analysis are given in Section 4. Numerical examples are provided to illustrate the effects of our model parameters on optimal period preventive overhaul time in Section 5. Finally, concluding comments are presented in Section Model Description 2.1 Notation and Assumptions We considerer a degenerate production system, and assume that the degeneration of the system at any time point can be classified into one of three states: in-control, outof-control and failure state. The system cannot improve on its own. At the beginning of each production cycle, the production process is always in the in-control state. After staying some time in the in-control state, the production process shifts to the out-ofcontrol state, and eventually leaves the out-of-control state and go into the failure state. The elapse time of the system in the in-control state, X, and in the out-of-control state, Y, are independent each other and follow exponential distributions with finite means 1/λ and 1/µ, respectively. Since the system is more easier to enter the failure state with the deterioration of the production system, it is assumed that the system will stay less expected time in the out-of-control state than that in the in-control state, that is to say λ < µ. The probability density functions for X and Y are denoted by f X (x) and f Y (y). The production cycle, T, is over either at fixed time t or at failure, whichever occurs first, and the corresponding maintenance actions are taken so that the system can restore to the initial in-control state. After the completion of a production cycle, the system is setup with cost k > and is inspected to reveal the state of the system. If the system is out-ofcontrol, then it is brought back to the in-control state with an additional restoration cost c p > for the next production run. If the system is in the failure state, then failure repair actions are taken with a cost c r such that the system can restore to the initial in-control state.

4 66 Suppose that the production rate of the system and the demand rate of the product are p and d, respectively, where p > d >. The manufacturing cost is c o per item and the inventory holding cost for carrying a product is h > per unit time. We also assume that all the items produced can be classified as conforming or nonconforming according to whether its performance meets the products specifications or not. Let r 1 (t) and r 2 (t) denote the hazard rates of a conforming and a nonconforming item, respectively. Since a nonconforming item is more likely to fail than a conforming item, it is assumed that r 1 (t) < r 2 (t). Due to manufacturing variability, an item is nonconforming with probability θ 1 (or θ 2 ) when the production process is in-control (or out-of-control), where θ 1 < θ 2. All the items produced are sold with a free minimal repair warranty. That is, the product sold will be freely rectified by minimal repair instantaneously with a cost c m to the manufacturer when it occurs failure within the warranty period w, and the hazard rate of an item remains the same as that just before failure after a minimal repair. 2.2 Mathematical Formulation For the degenerate production system mentioned above, we investigate a optimal periodic overhaul policy. Our objective is to find the periodic overhaul time t such that the expected total cost per item in a production cycle is minimized. The total cost for our problem includes manufacturing cost, setup cost, inventory holding cost, restoration cost and warranty cost. The formulations of the latter three costs are described in detail in the rest of this section. Firstly, using convolution formula, the probability density function f Z (z) of the system operating time Z (= X + Y ) in a production cycle is given by by f Z (z) = z f X (x)f Y (z x)dx = λµ(e λz e µz ), z >. (2.1) µ λ For a fixed preventive overhaul time t, the operation run length per cycle T is given Z, for Z < t; T = t, for Z t and by (2.1), its expected value, denoted by u(t), is u(t) = E(T ) = t tf Z (z)dz + t zf Z (z)dz = 1 e λt λ Differentiating u(t) given in (2.3) with respect to t yields + 1 e µt µ (2.2) + e µt e λt. (2.3) µ λ u (t) = µe λt λe µt, t >. (2.4) µ λ

5 : 67 Since µ > λ, we have u (t) > for all t >. In addition, from (2.3)-(2.4), we have lim u(t) =, lim t u(t) = λ + µ t λµ (2.5) and lim t u (t) = 1, lim u (t) =. (2.6) t In order to determine the warranty costs, we need to find out the expected number of nonconforming items, denoted by E(N), and the fraction of nonconforming items in a production cycle. The number of nonconforming items N per cycle can be expressed as follows and its expected value is θ 1 pt, if X t; N = θ 1 px + θ 2 py, if X < t and X + Y < t; θ 1 px + θ 2 p(t X), if X < t and X + Y t E(N) = t + θ 1 pxλe λx dx + t x t { t x (θ 1 px + θ 2 py)µe µy dy } (θ 1 px + θ 2 p(t x))µe µy dy λe λx dx (2.7) = θ 2 pu(t) p(θ 2 θ 1 )(1 e λt ). (2.8) λ Furthermore, the fraction of nonconforming items in a production cycle, q(t), is given by Lemma 2.1 q(t) = E(N) pu(t) = θ 2 (θ 2 θ 1 )(1 e λt ), t >. (2.9) λu(t) For all t >, q(t) given by (2.9), is an increasing function of t and satisfies lim q(t) = θ 1 and lim q(t) = (µθ 1 + λθ 2 )/(λ + µ). t t Proof the first derivative of q(t) with respect to t is given by Let q (t) = (θ 2 θ 1 )[(1 e λt )u (t) λe λt u(t)] λu 2. (2.1) (t) d(t) = (1 e λt )u (t) λe λt u(t) = u (t) e λt + λe λt (1 e µt ), (2.11) µ since u (t) > e λt and 1 e µt > for all t >, we have d(t) >. Hence q (t) has the same sign as d(t) and q(t) is an increasing function of t.

6 68 By L Hospital rule, we have lim (θ lim q(t) = θ 2 θ 1 )e λt t 2. t lim t u (t) Using (2.5) and (2.6), we can obtain lim t q(t) = θ 2 (θ 2 θ 1 ) = θ 1 and lim (θ lim q(t) = θ 2 θ 1 )(1 e λt ) t 2 t lim λu(t) = θ 2 µ(θ 2 θ 1 ) = µθ 1 + λθ 2 λ + µ λ + µ. t Since the production run length in a production cycle is T = min(z, t), the maximum inventory level is equal to P T. Therefore, the inventory cost in a production cycle is (p d)ht, if Z > t; I = 2d (2.12) (p d)hz, if Z t 2d and its expected value, denoted by E(I), is E(I) = (p d)h 2d where a = (p d)h/(2d). t tf Z (z)dz + (p d)h 2d t zf Z (z)dz = au(t), (2.13) If the system goes into the failure state before time t, the failure-repair action is taken with a fixed cost c r for restoring the system to the in-control state. If not, at the end of period time t, preventive overhaul action is performed and it takes an additional cost c p for restoring the system to the in-control state if the system is in the out-of-control state. Hence, the expected cost for restoring the system to the initial in-control condition per item in a production cycle can be obtained as follows M(t) = where ξ = µc r λc p, η = λ(c r c p ). = t t ( [c r f Z (z)dz + c p t x ) /pu(t) µe µy dy λe dx] λx c r pu(t) ξe λt ηe µt p(µ λ)u(t), (2.14) Under the free-repair warranty, the failure process of conforming and nonconforming items are nonhomogeneous process with intensity r 1 (t) and r 2 (t), respectively. Therefore, the corresponding mean numbers of failures of conforming and nonconforming items within [, w] are w o r 1 (τ)dτ and a lot size pt is given by W (t) = c r [(1 q(t)) w o w o r 2 (τ)dτ, and the expected total post-sale cost per item for w ] w r 1 (τ)dτ + q(t) r 2 (τ)dτ = c r r 1 (τ)dτ + bq(t), (2.15) o o

7 : 69 where b = c m [ ω ω r 2 (τ)dτ ] r 1 (τ)dτ. By considering pre-sale and post-sale costs, from (2.13)-(2.15), the expected total cost per item per cycle, denoted by C(t), is given by C(t) = c o + c m w o r 1 (τ)dτ + k + c r pu(t) + au(t) ξe λt ηe µt + bq(t). (2.16) p(µ λ)u(t) Our object here is to find the optimal periodic maintenance time t, which minimizes C(t) given in (2.16). The properties of the optimal periodic maintenance time are investigated in detail in the following Section Optimal Periodic Overhaul Policy 3.1 Model Analysis In this section, the uniqueness property of the optimal periodic maintenance time is explored. In order to solve the cost minimization problem in (2.16), we first take the first derivative of C(t) with respect to t as follows C (t) = (k + c r)u (t) pu 2 (t) + au (t) + λξe λt µηe µt p(µ λ)u(t) + (ξe λt ηe µt )u (t) p(µ λ)u 2 (t) + bq (t). (3.1) If the optimal t is finite, it is well known that the necessary condition for t to be optimal is C (t ) =. following Theorem 3.1. The uniqueness of the optimal periodic overhaul policy is given in the Theorem 3.1 There exists a unique t > that minimizes C(t) in (2.16). Proof Let G(t) = u 2 (t)c (t)/u (t), then from (3.1), we have G(t) = k + c r p + au 2 (t) + ξe λt ηe µt p(µ λ) + b(θ 2 θ 1 )(1 e λt ) λ b(θ 2 θ 1 )e λt u(t) u (t) + (λξe λt µηe µt )u(t) p(µ λ)u (t) (3.2) and lim t G(t) = k/p <. Since u(t) > and u (t) > for all t >, it is obviously that G(t) has the same sign as C (t) for t >. For discussing the property of G(t) in (3.2), the first derivative of G(t) is given by Let [ G (t) = u(t)u (t) 2a + λ(µc r (λ + µ)c p ) + λbp(θ 2 θ 1 ) ] p[u (t)] 3 e (λ+µ)t. (3.3) K(t) = [u (t)] 3 e (λ+µ)t = µ3 e (2λ µ)t 3λµ 2 e λt + 3λ 2 µe µt λ 3 e (2µ λ)t (µ λ) 3, (3.4)

8 7 then the first derivative of K(t) is K (t) = (u (t)) 2 e (λ+µ)t [λ(2µ λ)e µt + µ(µ 2λ)e λt ]. (3.5) µ λ It follows from (3.4) that (3.3) can be rewritten as G (t) = u(t)u (t) pk(t) [2apK(t) + λ(µc r (λ + µ)c p ) + λbp(θ 2 θ 1 )]. (3.6) The unique property of the optimal t is shown for the following two cases: (i) If µc r (λ + µ)c p + bp(θ 2 θ 1 ), then it is obviously that G (t) > for t >. Therefore, G(t) is an increasing function of t. Furthermore, if a(λ + µ)2 ξ(λ + µ) G( ) = lim G(t) = t λ 2 µ 2 + pµ 2 + bλ(θ 2 θ 1 ) µ 2 k + c r p we can easily prove that there exists a finite and unique positive t satisfying the equation G(t ) = and so does C (t ) =, which minimizes C(t). If G( ), then C (t) for all t >. Hence the optimal value t =. (ii) If µc r (λ + µ)c p + bp(θ 2 θ 1 ) <, then (a) if µ 2λ, then K(t) is an increasing function with lim t K(t) =. Consequently, we can obtain that either there existing a finite nonnegative threshold t o, which satisfying G (t) for t t o and G (t) > for t > t o, or G (t) > for t. Using the same argument with part (i), we can complete the proof in this case. >, (b) if µ < 2λ, then it follows from (3.4) that K(t) has a unique maximum point v = 1 λ(2µ λ) ln µ λ µ(2λ µ) satisfying K (v) =. From (3.6), if G (v), then we have that G(t) is a decreasing function for t >, since lim G(t) <, we have G(t) < and so does C (t) < for t >, t and the optimal t =. if G (v) >, then, since lim K(t) =, it is easy to verify that t there exists a unique maximum point v 1 [v, v r ] maximizing G(t) given in (3.2), where v r = 3 [ µ λ λ 2µ ln 3 µ In this case, if G(v 1 ), then t =. λ[(λ + µ)c p µc r bp(θ 2 θ 1 )]/(2ap) Otherwise if G(v 1 ) >, then there exists a unique finite solution t = v 2 v 1 that minimizes C(t) since lim t G(t) <. Furthermore, comparing C(v 2 ) with a(λ + µ) C( ) = lim C(t) = c o + + λµ(k + c r) + bp(µθ 1 + λθ 2 ) w + c m r 1 (τ)dτ, t λµ p(λ + µ) we have t = v 2 if C(v 2 ) C( ) and t = if C(v 2 ) > C( ). Theorem 3.1 shows that the unique optimal periodic maintenance time t is likely to, and even if it is an finite positive solution t, we cannot obtain the analytic solution of t, and it can only be obtained by a nonlinear search procedure as follows. ].

9 : Searching Procedure Step 1: Step 2: go to Step 4. Step 3: Calculate G( ) and C( ). If µc r (λ + µ)c p + bp(θ 2 θ 1 ), then if G( ) <, go to Step 8; otherwise, If µc r (λ + µ)c p + bp(θ 2 θ 1 ) <, then Step 3.1: if µ 2λ, then if G( ), go to Step 8; otherwise, go to Step 4. Step 3.2: if µ < 2λ, compute v and G (v). Further, if G (v), go to Step 8; otherwise compute v and v r, find v 1 [v, v r ) using any nonlinear search method, then if G(v 1 ), go to Step 8. Otherwise if G(v 1 ) >, find v 2 (, v 1 ] using a nonlinear search method and calculate C(v 2 ). Furthermore, if C(v 2 ) > C( ), go to Step 8. Otherwise, output t = v 2 and C(v 2 ), stop. Step 4: Initialization t, k =. Step 5: Let k = k + 1 and t = kt. Step 6: Step 7: and C(t ). Stop. Step 8: if G(t ) <, then go to Step 5; otherwise go to next. Search t [(k 1)t, kt ] satisfying G(t ) = and compute C(t ). Output t Output t = and C( ). Stop. 4. Special Cases and Sensitivity Analysis In this section, three special cases and sensitivity analysis of model parameters are discussed. Case 1 (µ = ): when the transition rate µ =, the production system shifts to the failure state with probability 1 after leaving the in-control state. In this case, the fraction of nonconforming items is equal to θ 1, and the expected cost per items per cycle and its first derivative, denoted by C 1 (t) and C 1 (t) respectively, are given by and C 1 (t) = c o + λc r p C 1(t) = + bθ 1 + c m w r 1 (τ)dτ + a(1 e λt ) λ + λk p(1 e λt ) (4.1) 1 p(1 e λt ) 2 e λt [ap(1 e λt ) 2 kλ 2 ]. (4.2) From (4.1)-(4.2), the optimal preventive overhaul time t 1 for this case is given in the following Corollary 4.1. Corollary 4.1 If ap > λ 2 k, then t 1 = ln(1 λ k/(ap) )/λ; otherwise t 1 =.

10 72 Case 2 (µ λ + ): when the transition rate µ approaches λ +, where λ + denotes approaching λ from its right, the production system will stay almost the same expected time as in that in-control state after shifting to the out-of-control state. In this case, the expected preventive overhaul time is given by and its first derivative is given by 2 (2 + λt)e λt lim u(t) = µ λ + λ From (2.16), the expected total cost, denoted by C 2 (t), is From (3.4), we have Let C 2 (t) = lim µ λ + C(t) = c o + c m w lim K(t) = (1 + µ λ λt)3 e λt, + G 2 (t) = lim µ λ + G(t) (4.3) lim µ λ u (t) = (1 + λt)e λt. (4.4) + o r 1 (τ)dτ + bθ 2 + a[2 (2 + λt)e λt ] λ + λ(k + c r) λ(c r + ηt)e λt bp(θ 2 θ 1 )(1 e λt ) p(2 (2 + λt)e λt. (4.5) ) lim µ λ K (t) = λ(2 λt)(1 + λt) 2 e λt. (4.6) + = a2 [2 (2 + λt)e λt ] 2 λ 2 + b(θ 2 θ 1 )(1 e λt ) + [(1 + λt)c r λc p ]e λt λ p + [2 (2 + λt)][λc p + λµ(c r c p )t bp(θ 2 θ 1 )]e λt λp(1 + λt) then we obtain that the first derivative of G 2 (t) is G 2(t) = lim µ λ + G (t) k + c r, (4.7) p = (1+λt)(2 (2+λt))e 2λt λp(1+λt) 3 e λt [2ap(1+λt) 3 e λt +λ 2 (c r 2c p )+λbp(θ 2 θ 1 )]. (4.8) Under the above arguments, we have the following Corollary 4.2. Corollary 4.2 Assume that µ λ +, (i) if λ 2 (c r 2c p ) + bpλ(θ 2 θ 1 ) max (, kλ 2 4ap), then there exists a finite unique t 2 such that G(t 2 ) = ;

11 : 73 (ii) if λ 2 (c r 2c p ) + bpλ(θ 2 θ 1 ) kλ 2 4ap, then t 2 =. (iii) if 54ap/e 2 < λ 2 (c r 2c p ) + bpλ(θ 2 θ 1 ) <, then if G 2 (s), the optimal t = s; if G 2 (s) >, then if G 2(s 1 ), t = ; if not, then t = on condition that C 2 (s 2 ) > C 2 ( ) and t = s 2 on condition that C 2 (s 2 ) C 2 ( ), where C 2 ( ) = lim C 2 (t) = c o + 2a t λ + λ(k + c r) + b(θ 1 + θ 2 ) w + c m r 1 (τ)dτ, 2p 2 s 1 satisfying (1 + λs 1 ) 3 e λs 1 = [λ 2 (2c p c r ) bpλ(θ 2 θ 1 )]/(2ap) and s 1 2/λ, and s 2 satisfying G 2 (s 2 ) = and s 2 s 1. (iv) if λ 2 (c r 2c p ) + bpλ(θ 2 θ 1 ) 54ap/e 2, then t 2 =. Case 3 (w = ): when the warranty period w =, the product are sold without warranty policy, In this case, C(t) and G(t), which corresponding given by (2.16) and (3.2), denoted by G 3 (t) and C 3 (t), respectively. Since G 3 (t) < G(t) for all w > and lim G 3 (t) = lim G(t) <. Therefore, if there t t exists a finite solution t 3 satisfying G 3(t 3 ) and minimizing C 3(t), then we have C 3 (t 3 ) < lim C 3(t) and G(t t 3 ) >. We can find a unique t (, t 3 ) satisfying G(t ) =. Since w C(t) = C 3 (t) + bq(t) + c m r 1 (τ)dτ and q(t) > for t >, we have C(t ) < C(t 3 ) < C 3 (t 3 ). Under the above discussion, we have the following Corollary 4.3. Corollary 4.3 bigger than C 3 (t 3 ). The optimal t is smaller than t 3, and the corresponding C(t ) is In our EMQ model, the optimal periodic maintenance time t is a function of the model parameters. Therefore, the optimal t will vary with the variation of the model parameters. The following Theorem 4.1 summarizes some sensitivity analysis of model parameters on optimal t. Theorem 4.1 The optimal t is a decreasing function of p, c r, c m, w, and θ 2 θ 1, and is an increasing function of c p and d. Proof Let p 1 and p 2 are two different production rates that satisfy p 2 > p 1, suppose that the corresponding optimal periodic maintenance time are t (p 1 ) and t (p 2 ). Let G(t p i ) denotes G(t) given in (3.1) on condition that p = p i, i = 1, 2. if t (p 1 ) and t (p 2 ) are two finite optimal value, we use reduction to absurdity to prove t (p 1 ) > t (p 2 ) as follows. Given that t (p 1 ) t (p 2 ), then from Theorem 3.1, we can obtain that p 2 G(t (p 2 ) p 2 ) > p 1 G(t (p 2 ) p 1 ) p 1 G(t (p 1 ) p 1 ) =. (4.9)

12 74 It follows from (4.9) that G(t (p 2 ) p 2 ) >, this is contrary to G(t (p 2 ) p 2 ) =, therefore, the initiation supposition is not correct, that is to say t (p 1 ) > t (p 2 ). Further, we can prove that there must be t (p 1 ) = if t (p 2 ) = with analogously method. Using the same argument as p, we can have how the other model parameters take effect on t. Hence we omit their proofs. 5. Numerical Example In this section, we provide some numerical examples to illustrate the features of the proposed model. Suppose that the life time distribution of both conforming and nonconforming items are Weibull distributions with hazard rate functions r 1 (t) = α β 1 1 β 1t β 1 1 and r 2 (t) = α β 2 2 β 2t β 2 1, respectively. Without loss of generality, we assume that the shape parameters are β 1 = β 2 = 2 and the scale parameters are α 1 = 1/36 and α 2 = 1/12. The remaining model parameters are λ =.1, µ =.3, k = 2, p = 6, d = 4, h =.1, c o = 1, c r = 3, c p = 2, θ 1 =.15 and θ 2 =.65. For this numerical example, we compute t, t 1, t 2 and t 3 as the following Table 1. From Table 1, we can observe that optimal periodic maintenance time decreases and corresponding expected total cost per item increases as w increases, and c m has the same effect on them. This is true because longer warranty period and larger minimal repair cost results in higher warranty cost. Table 1 Optimal periodic overhaul policy under various warranty period w c m t C(t ) t 1 C(t 1) t 2 C(t 2) t 3 C(t 3) Given that the rest model parameters are given by λ =.2, µ =.4, p = 6, d = 5, c o = 1, c m =.1, c r = 3, c p = 2, θ 1 =.15, θ 2 =.65 and w = 12, we obtain Table

13 : From Table 2, we can find out that smaller lot sizes are required as h becomes larger, larger lot sizes are required as k increases, and the expected total cost per item increases as either k (or h) increases. Table 2 Effect of variations in k and h on optimal periodic overhaul policy k h t C(t ) t 1 C(t 1) t 2 C(t 2) t 3 C(t 3) Table 3 Optimal periodic overhaul policy under various λ, c r and c p λ c r c p t C(t ) t 3 C(t 3) t C(t ) t 3 C(t 3) t C(t ) t 3 C(t 3) Assume that some parameters are µ =.5, k = 25, p = 5, d = 4, h =.1, c o = 1, c m =.2, θ 1 =.2, θ 2 =.6 and w = 12, we obtain Table 3. From Table 3, it shows that the optimal period preventive overhaul time decreases for reducing the failure occurrence as c r increases and increases for reducing average maintenance cost with an

14 76 increase of c p, and the corresponding expected cost increases as c r (or c p ) increases for the reason that it results in higher failure repair (or restoration) cost. Both the optimal t (or t 3 ) and corresponding C(t ) (or C(t 3 )) increase as λ increases. In addition, without warranty cost, we noted that t 3 and C(t 3 ) become bigger and smaller, respectively. Table 4 Effect of variations in p, d, θ 1 and θ 2 on t and C(t ) p d (θ 1, θ 2 ) θ 2 θ 1 t C(t ) 5 35 (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55) (.2,.6) (.2,.55) (.3,.6) (.3,.55)

15 : 77 Suppose that some model parameters are λ =.25, µ =.5, k = 3, h =.15, c o = 1, c r = 4, c p = 25, c m =.2 and w = 18, we obtain Table 4. Table 4 indicates that an increase of demand rate d leads to the increase of optimal t and C(t ). This increase is due to a higher demand rate making a lower inventory holding cost per item per unit time in a production cycle. The production rate p, which leading to higher average inventory holding cost, has the opposite effect on them. Therefore, if demand rate is fixed, it is not wise to extend production scale. An increase in θ 1 results in an increase in t and the optimal t decreases as θ 2 increases. This is true because the item quality difference between in the in-control state and in the out-of-control state is reduced with the increase of θ 1 and augmented with the increase of θ 2. But optimal t decreases if θ 2 θ 1 increases for the reason that it is absolutely augmented quality difference. Further, because an increase of either θ 1 or θ 2 results in an increase of fraction of nonconforming item and so does warranty cost, both θ 1 and θ 2 increase the expected total cost C(t ) per item in a production cycle. 6. Conclusion In this paper, we propose an optimal periodic overhaul policy for a three-state degenerate production system, where the products are sold with FRW policy and system failure are considered. We explore the unique property of optimal periodic overhaul time. Unlike the conclusion drawn in [11 13], in which the optimal production run time is always a finite positive solution, the optimal value t may be infinite under certain condition in our EMQ model. From the optimality conditions, a searching procedure is developed to find the optimal solution. In order to show the effects of model parameters on the optimal periodic overhaul policy, sensitivity analysis are summarized. Finally, numerical examples are carried out to illustrate features of the model. Acknowledgements The authors are thankful to the editor and the anonymous referees for their valuable comments and suggestions. References [1] Blischke, W.R. and Murthy, D.N.P., Warranty Cost Analysis, New York: Marcel Dekker, [2] Chen, C.K. and Lo, C.C., Optimal production run length for products sold with warranty in an imperfect production system with allowable shortages, Math. Comput. Model, 44(3-4)(26),

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