The impact of a free-repair warranty policy on EMQ model for imperfect production systems

Computers & Operations Research 31 (2004) 2021 – 2035

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The impact of a free-repair warranty policy on EMQ model for imperfect production systems Chih-Hsiung Wang∗ Department of Commerce Automation and Management, National Pingtung Institute of Commerce, 51 Min-Sheng E. Road, Pingtung, Taiwan, ROC

Abstract In this study, the economic production quantity problem in the presence of imperfect processes for products sold under a free-repair warranty policy is considered. In the literature, that a production facility may deteriorate with time is assumed, and the time to shift from an in-control state to an out-of-control state is assumed to be exponentially distributed, i.e., the process failure rate is a constant. However, in many practical situations, the process possesses an increasing failure rate due to cumulative wear in producing items. This study is extended to consider a process subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. A mathematical model representing the expected total cost per item is developed to determine the optimal production policy. The objective here is to obtain the optimal production run-length (lot size) so that the expected total cost per item is minimized. Di4erent conditions for optimality, properties, and bounds on the optimal production run-length are provided. A numerical example is used to see the adequacy of using the exponential distribution when the actual distribution is Weibull with an increasing failure rate. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Lot sizing; Imperfect production systems; Warranty

1. Introduction The classical economic manufacturing quantity (EMQ) model assumes that a production facility is failure free, and that all the items produced are of perfect quality. In real production, although the production process starts in an “in-control” state producing items of high or perfect quality, it may shift to an “out-of-control” state while in-process, thereby resulting in the production of defective items, since product quality is usually a function of the state of the production process. Therefore, ∗

Corresponding author. Tel.: +886-8-723-8700 ext 6117; fax: +886-8-723-7941. E-mail address: [email protected] (C.-H. Wang).

0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00161-8

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C.-H. Wang / Computers & Operations Research 31 (2004) 2021 – 2035

in much of the inventory analysis literature, the EMQ model has been analyzed under the relaxed assumption that the production process always produces items of acceptable quality. Rosenblatt and Lee [1] initially studied the e4ect of process deterioration on the optimal EMQ. They derived an EMQ model when the process deteriorated during the production process and, thus, produced defective items. They assumed that the shift-time from “in-control” to “out-of-control” is exponentially distributed. The optimal production run-length can be easily derived by using a MacLaurin series to approximate an exponential function to the second order. Porteus [2] demonstrated that, for a process that can go into an “out-of-control” state with a given probability each time an item is produced, it is better to produce lot sizes smaller than the classical EMQ, a conclusion which agrees with the result obtained by Rosenblatt and Lee [1]. Hariga and Ben-Daya [3] and Kim and Hong [4] studied the RL model by considering the general time required to shift distribution and an optimal production run-length shown to be unique. On the basis of the RL model, Lee and Rosenblatt [5,6] introduced some inspection mechanisms to monitor the production process during a production run. In this case, an optimal production run-length and an optimal inspection policy must be determined simultaneously. They explored the conditions for monitoring the production process at equally spaced intervals, which constitutes the optimal inspection policy under a shift in the production process following an exponential distribution. Using a technique similar to the one used in the RL model, they derived an approximated optimal production run-length and inspection policy simultaneously by using a MacLaurin series to approximate an exponential function to the second order. Makis [7] further examined Lee and Rosenblatt’s [5] solution structure in detail. Several properties for the optimal production/inspection policy were investigated. In order to enhance the reliability of the deteriorating production process, Tseng [8] incorporated a preventive maintenance policy instead of an inspection policy into an imperfect EMQ model. However, a two-dimensional search procedure was needed to obtain the optimal production run-length and maintenance schedules. On the basis of Tseng’s [8] model, Wang and Sheu [9] provided several properties useful for obtaining an optimal production/maintenance policy. The e4ect of warranty cost (i.e., the cost incurred by a defective item after its sale) on the optimal lot size has been studied by Lee and Park [10] and Djamaludin et al. [11]. Lee and Park [10] reformulated the work of Lee and Rosenblatt [5] to consider the possibility that a defective item after its sale will incur a warranty cost greater than the reworking cost of a defective item before its sale. Without considering any inspection and maintenance actions during a production run, Djamaludin et al. [11] utilized lot size to control the warranty cost per item for products under a free-repair warranty (FRW), where the production system can go into an “out-of-control” state with a given probability each time an item is produced. Moreover, Yeh and Lo [12] studied the e4ect of an FRW policy on the optimal production lot size and the optimal burn-in time. Recently, Yeh et al. [13] reformulated the Djamaludin et al. [11] model to consider that the production process is subject to a random deterioration from an “in-control” state to an “out-of-control” state, in which the shift time is exponentially distributed. The e4ect of di4erent constant failure rates on the optimal production run-time was discussed. However, in many production systems, the process failure rate increases with the length of the production time. Therefore, in this paper, the Yeh et al. [13] model is reconsidered under the assumption that an elapsed time until a shift is a general shift distribution. Di4erent conditions for optimality, properties, and bounds on the optimal production run-length are provided. The remainder of this study is organized as follows. In Section 2, the model is described in detail, and the formula is derived for the expected total cost per unit item. In Section 3, an

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upper bound of the optimal production run-length is derived so that it can be utilized for eLciently searching for the optimal solution. In particular, the conditions of the uniqueness of the optimal production run-time for a nondecreasing failure rate in the shift distribution are explored. In Section 4, the model considered by Yeh et al. [13] is shown to be a special case of our model. Moreover, for the optimal production run-length, a tighter bound than Yeh et al. [13] is provided. In Section 5, numerical examples are given to illustrate the use of this solution procedure. Concluding remarks are presented in Section 6. 2. Mathematical model Consider a deteriorating production system that produces a single product. The problem of optimal lot sizing for products sold under a FRW policy is studied. Suppose that k is the setup cost, cm is the manufacturing cost of an item, and h is the inventory holding cost of carrying a product per unit of time. The demand for the product is d units per unit time. The product can be produced at the rate of p units per unit time, where p ¿ d ¿ 0. Given production run-length t, the inventory buildup with rate p − d continues until the time t, after which the inventory is depleted at a rate of d until the end of the production cycle, and where the length of the production cycle is pt=d. It is obvious that the maximum inventory level is (p − d)t. The inventory holding-cost per item can be easily derived as h(p − d)t=2d. At any point in time, the process is classiNed as either an “in-control” state or an “out-of-control” state. It is assumed that the production process is in-control at the beginning of the production run. Let X denote the elapsed time of the process in the in-control O state that has a cumulative distribution F(x), density distribution f(x), survival function F(x) and O ’(x) = f(x)= F(x) is the failure rate function. Note that X has a nondecreasing failure rate (NDFR) if ’
(x) ¿ 0, for all x ¿ 0, and an increasing failure rate (IFR) if ’
(x) ¿ 0, for all x ¿ 0. Without considering any inspection and maintenance actions during a production run, once the process enters the out-of-control state, it remains in that state until the end of the production run. Inspection and preventive maintenance (PM) actions are performed at the end of the production run with a Nxed cost . If the system is found to be out-of-control, then it is restored to the in-control condition with an additional restoration cost r. On the other hand, if the process inspected is identiNed to be in the in-control state, only a PM is performed. Therefore, the process is in-control when it starts at the beginning of each production run. Due to variability in the manufacturing process, the quality of items produced varies. For simplicity, it is assumed that an item produced may be either conforming or nonconforming, depending on whether its performance meets the speciNcations of the product. If it is assumed that when the process is in the “in-control” (or “out-of-control”) state, an item produced is nonconforming with probability 1 (or 2 ), where 2 ¿ 1 . That is, an item produced is more likely to be nonconforming if the process state is out-of-control. Given production run-length t, the pre-sale cost per item represents the costs incurred before the product is sold, such as manufacturing cost, inventory-holding cost, setup cost, joint cost of process inspection and PM, and restoration cost. These costs are mathematically formulated as follows. M (t) = total pre-sale cost per unit = cm +

K r (p − d)ht + + F(t); 2d pt pt

(1)

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C.-H. Wang / Computers & Operations Research 31 (2004) 2021 – 2035

K = k + . Next, one needs to determine the expected number of nonconforming items produced after any setup so that the post-sale (warranty) costs can be determined. Let N be the number of nonconforming items in lot of size pt. Then
1 pt; X ¿ t; N= X ¡t 1 pX + 2 p(t − X ); and its expected value is



E(N ) = 2 pt + (1 − 2 )p

0

t

O F(x) d x:

(2)

See the appendix for a derivation of (2). The fraction of nonconforming items produced in a lot of a certain size pt is given by  t O F(x) d x=t: q(t) = E(N )=pt = 2 + (1 − 2 ) 0

It is easy to verify that q(t) is an increasing function of t and limt →0 q(t) = 1 and limt →∞ q(t) = 2 . These imply that 1 6 q(t) 6 2 , ∀t ¿ 0. Suppose that ’1 () and ’2 () are the failure-rate functions for a conforming and nonconforming item, respectively. Under the free-repair warranty policy, where all failures occurring during warranty period w are rectiNed through minimal  wrepair, the mean number  w of failures of conforming and nonconforming items within [0; w] is R1 = 0 ’1 () d and R2 = 0 ’2 () d, respectively. Furthermore, the mean number of failures under warranty for items produced in lot of size pt is given by (1 − q(t))R1 + q(t)R2 . Let cr be the cost at which an item failing within warranty period w is instantaneously rectiNed by minimal repair. Hence, the expected total post-sale cost per item for a lot with size pt is W (t) = cr (q(t)(R2 − R1 ) + R1 ):

(3)

Upon investigating Eq. (3), it is clear that W (t) is an increasing function of t given w ¿ 0. That is, as the production run-length increases, the expected warranty cost per item increases, due to the premise that a production system is more likely to go to an out-of-control state for a long production run. As results from (1) and (3), the expected total cost per unit item, including pre-sale and post-sale costs for a lot of size pt, is given by K r (p − d)ht + + F(t) + cr (q(t)(R2 − R1 ) + R1 ); C(t) = cm + (4) 2d pt pt t O where q(t)=2 +(1 −2 ) 0 F(x) d x=t. The objective here is to Nnd the optimal production run-length ∗ t that minimizes C(t) given in (4). 3. Optimal production run-length In this section, the properties of the production run-length problem for products sold under a free minimal repair policy are studied. In the following theorem, we provide an upper bound for the optimal production run-length. Furthermore, the condition of uniqueness of the optimal production run-length is explored when the production facility possesses a nondecreasing failure rate.

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Theorem 1. (i) Assume that limt →0 tf(t) = 0, limt →∞ tf(t) = 0 and f(t) ¿ 0 for all 0 ¡ t ¡ ∞. Thereexists t ∗ ∈ (0; tU ] that minimizes the expected total cost per item C(t) given in (4), where 2d(K+r) tU = p(p −d)h . (ii) Furthermore, if X has an NDFR and   2 A1 A1 p(p − d)h + + ’(tU ) ¡ ; (5) 2 rd 2 where A1 = cr p(2 − 1 )(R2 − R1 )=r, then C(t) has unique minimizer t ∗ in the interval (0; tU ], and the optimal value of t ∗ can be found by solving the following nonlinear equation:  t  (p − d)ht 2 K r r O O [F(x) − F(t)] d x = 0: (6) − − F(t) + tf(t) + cr (2 − 1 )(R2 − R1 ) 2d p p p 0 Note that, the conditions required in Theorem 1(i), limt →0 tf(t)=0, limt →∞ tf(t)=0 and f(t) ¿ 0 for all 0 ¡ t ¡ ∞, are held when the shift distribution is Weibull, which is the most widely used in reliability modeling (see Elsayed [14]). Therefore, they can be assumed to be hold reasonably. From Theorem 1, the Nnite interval (0; tU ] can be searched to obtain the optimal production run-length t ∗ by using a numerical procedure. Furthermore, when the process has an NDRF, the condition of the uniqueness of the optimal production run-length is explored. At this moment, an eLcient algorithm commonly known as the bisection algorithm can be used to search the optimal production run-length t ∗ . On the basis of the fact that t ∗ is the solution to dC(t)=dt =R(t)=t 2 =0, where R(t) is equal to the left-hand side of (6) that is an increasing function of t, which has been demonstrated in parts of the proof of Theorem 1(ii), from Eq. (6) the inRuences of model parameters on the optimal run-length t ∗ , it can be seen that the optimal production run-length is an increasing function of d, K, R1 and 1 and a decreasing function of cr , R2 , 2 and h. 4. Special case In this section, extensions in the investigations to the exponential failure distributions case is given. The case of the exponential is shown to be an extension of previously reported result [13]. A tighter bound for the optimal production run-length than [13] is provided. Using f(t) =  exp(−t) in expression (4), one has K r (p − d)h C(t) = cm + t+ + (1 − exp(−t)) + cr (q(t)(R2 − R1 ) + R1 ); (7) 2d pt pt which is the formula obtained by Yeh et al. [13]. When the transition rate  is small, Yeh et al. [13] used an approximation of exp(−t) ≈ 1 − t + (t)2 =2 to obtain the approximately optimal solution to (7) as  2Kd ∗ ; t˜ = p[(p − d)h − A2 d]

(8)

(9)

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C.-H. Wang / Computers & Operations Research 31 (2004) 2021 – 2035

where A = r=p − (cr (2 − 1 )=)(R2 − R1 ). Obviously, the necessary and suLcient condition for t˜∗ to exist is (p − d)h − A2 d ¿ 0:

(10)

It is not diLcult to verify that the necessary and suLcient condition for the existence of t˜∗ given in (10) holds if and only if, (5), the condition in Theorem 1(ii) also holds. In this case, one can apply Theorem 1(ii) to derive a unique t ∗ ∈ (0; tU ] instead of usingan approximate solution given in (9). Furthermore, in the following theorem, we use t˜∗ and tE = 2Kd=p(p − d)h, the classical economic production time, to provide a range for the optimal production run-length. Theorem 2. Assume that (10) holds. (i) If A ¡ 0, then t˜∗ ¡ t ∗ ¡ tE . (ii) If A = 0, then t˜∗ = t ∗ = tE (also see [13]). (iii) If A ¿ 0, then tE ¡ t ∗ ¡ t˜∗ . Yeh et al. [13] have shown that if A ¡ 0 then 0 ¡ t ∗ ¡ tE . Thus, Theorem 2(i) provides a tighter lower bound than Yeh et al. [13]. On the other hands, Yeh et al. [13] provide an interval [tE ; t˜∗3 =  2Kd=(p(p − d)h − r2 d)] over which to search the optimal solution t ∗ when A ¿ 0. It is easy to check that t˜∗ ¡ t˜∗3 . Therefore, Theorem 2(iii) provides a tighter upper bound than Yeh et al. [13]. Besides, it is noteworthy that t˜∗3 may not exist if p(p − d)h − r2 d 6 0, even if t˜∗ exists. 5. Numerical examples In this section, we illustrate the developed EMQ model under a free-repair warranty policy, a deteriorating production system with Weibull shift distribution F(t) = 1 − exp(−(t) ) for t ¿ 0 is considered, where  ¿ 0 and ¿ 1. Note that when = 1, it is the exponential distribution case which is commonly used in the literature of deteriorating production systems; however, it is not always accurate. To study the adequacy of using the exponential distribution, the following nominal values for the parameters are considered: cm =5 ($/unit), d=900 (units/month), K =250 (k +=250) ($), h=1 ($/unit/month), p=1500 (units/month), cr =1 ($), 1 =0:05, 2 =0:95, w=12 (months) and r = 200 ($). For the moment, assume that the lifetime distribution of conforming and nonconforming are Weibull distributed, F1 (t) = 1 − exp(−(t=6)2 ) and F2 (t) = 1 − exp(−(t=2)2 ), respectively. The optimal production run-length is found by minimizing the expected total cost per unit item given in (4). The percentage of penalty (PCP) for implementing the exponential solution when Weibull is the actual distribution is computed as PCP = 100 × [C(t ∗ ) − C(te∗ )]=C(t ∗ ), where te∗ is obtained by minimizing (7). In the author’s calculations, the values of parameters  and were varied in order to determine their inRuence on the optimal solution, t ∗ and C(t ∗ ), and PCP. The results are given in Table 1. For the purpose of illustration, Figs. 1(a) and (b) have been devised to show the behavior of the optimal production run-length and its associated optimal expected total cost per item, respectively. From Table 1, where 0:1 6  6 1 and 1 6 6 5:5, it can observed that the expected total cost per item increases when is Nxed and  increases (or decreases and  is Nxed). On the other

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Table 1 Optimal production-inventory policy for products sold with free minimal repair warrantya  0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1:0

0.3092 11.6959 0

0.2310 12.0797 0

0.1928 12.3847 0

0.1691 12.6460 0

0.1525 12.8785 0

0.1402 13.0903 0

0.1305 13.2861 0

0.1226 13.4692 0

0.1160 13.6418 0

0.1105 13.8055 0

1:5

0.4834 11.2309 0.6774

0.3633 11.4113 0.9370

0.2981 11.5759 1.0535

0.2565 11.7269 1.1129

0.2274 11.8673 1.1427

0.2057 11.9992 1.1526

0.1887 12.1242 1.1513

0.1750 12.2434 1.1442

0.1637 12.3577 1.1347

0.1541 12.4677 1.1188

2:0

0.6078 11.1130 1.2474

0.4850 11.1980 1.9840

0.4016 11.2924 2.3967

0.3449 11.3872 2.6555

0.3039 11.4799 2.8271

0.2730 11.5699 2.9402

0.2488 11.6574 3.0157

0.2292 11.7423 3.0668

0.2130 11.8250 3.1021

0.1993 11.9056 3.1182

2:5

0.6717 11.0819 1.4588

0.5788 11.1194 2.5456

0.4919 11.1737 3.2408

0.4256 11.2352 3.7167

0.3753 11.2995 4.0595

0.3364 11.3646 4.3108

0.3056 11.4295 4.5003

0.2805 11.4938 4.6473

0.2598 11.5574 4.7644

0.2423 11.6201 4.8498

3:0

0.6962 11.0740 1.5203

0.6411 11.0893 2.7946

0.5647 11.1195 3.6935

0.4953 11.1595 4.3463

0.4389 11.2051 4.8380

0.3940 11.2535 5.2157

0.3575 11.3035 5.5140

0.3276 11.3542 5.7561

0.3028 11.4051 5.9574

0.2818 11.4559 6.1174

3:5

0.7041 11.0720 1.5364

0.6765 11.0779 2.8948

0.6189 11.0938 3.9225

0.5531 11.1195 4.7053

0.4943 11.1519 5.3140

0.4449 11.1886 5.7947

0.4042 11.2279 6.1841

0.3702 11.2688 6.5075

0.3417 11.3107 6.7822

0.3176 11.3531 7.0087

4:0

0.7064 11.0716 1.5404

0.6939 11.0737 2.9325

0.6559 11.0817 4.0341

0.5994 11.0976 4.9076

0.5415 11.1205 5.6052

0.4897 11.1484 6.1671

0.4456 11.1798 6.6300

0.4084 11.2134 7.0198

0.3768 11.2485 7.3551

0.3500 11.2846 7.6372

4:5

0.7069 11.0714 1.5413

0.7018 11.0722 2.9462

0.6793 11.0760 4.0868

0.6348 11.0856 5.0208

0.5809 11.1015 5.7846

0.5288 11.1227 6.4103

0.4824 11.1479 6.9320

0.4426 11.1758 7.3758

0.4084 11.2056 7.7607

0.3791 11.2368 8.0886

5:0

0.7069 11.0714 1.5416

0.7050 11.0717 2.9510

0.6930 11.0734 4.1109

0.6608 11.0789 5.0833

0.6133 11.0898 5.8959

0.5624 11.1058 6.5716

0.5151 11.1261 7.1409

0.4732 11.1494 7.6287

0.4369 11.1750 8.0544

0.4056 11.2023 8.4202

5:5

0.7069 11.0714 1.5416

0.7064 11.0715 2.9527

0.6999 11.0723 4.1216

0.6786 11.0754 5.1172

0.6392 11.0826 5.9649

0.5913 11.0945 6.6799

0.5438 11.1108 7.2878

0.5006 11.1304 7.8121

0.4625 11.1525 8.2717

0.4294 11.1765 8.6691

a

Each cell in each column shows the calculated results of (t ∗ ; C(t ∗ ); PCP).

hand, the optimal production run-length (or lot size) decreases when is Nxed and  increases (or decreases and  is Nxed). This is true because the Nrst derivative of the process failure rate ’(t; ; ) = (t) −1 with respect to  is d’(t; ; )=d = 2 (t) −1 ¿ 0 for ¿ 1, i.e., ’(t; ; )

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C.-H. Wang / Computers & Operations Research 31 (2004) 2021 – 2035 0.8 0.6 0.4 0.2 0 10 10 5  = 0.1~1.0 (a)

5 0

0
= 1.0~5.5

14 13 12 11 10 10 5

5 0

(b)

Fig. 1. (a) Plots of t ∗ vs. ! and

 = 0.1~1.0

0
= 1.0~5.5

for numerical example. (b) Plots of C(t ∗ ) vs. ! and

for numerical example.

is an increasing function of . However, for a given scale parameter , the failure rate function ’(t; ; ) does not exhibit monotone behavior with the shape parameter . Moreover, an increase in ( ¿ 1) results in an increase in PCP, while the PCP does not exhibit monotone behavior when  increases; however, an increasing tendency is indicated. In Figs. 2–11, sensitivity analysis of the model controlled by the optimal production run-length (lot sizes) with respect to the input parameters is implemented by varying one or two parameters at a time. The remaining parameters have the values selected at the beginning of this section, where the time to shift distribution is Weibull and possesses parameters  = 0:3 and = 2:5. Figs. 2–4 indicate that smaller lot sizes are required to ensure that the expected total cost per item is minimized when one or more of these factors w, cr and 2 , which determine quality related costs, are large. This is true because smaller lots can produce fewer nonconforming items, resulting in a lower warranty cost. The e4ect of 1 on the optimal production run time is the opposite of 2 as shown in Fig. 5. Fig. 6 indicates that when the costs of setup, inspection or preventive maintenance are large, larger lot sizes are required to compensate for these higher costs so that a lower expected total cost per item can be incurred in a production cycle.

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Optimal production run time

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0

5

10

15

20

25

Fig. 2. Optimal production run-time vs. warranty periods for numerical example.

Optimal production run time

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0

0.5

1

1.5

2

2.5

Fig. 3. Optimal production run-time vs. for numerical example.

Fig. 7 indicates that an increase in the value of demand rate d leads to an increase in the optimal production run-length, as was expected. This increase is due to a higher demand rate making a lower inventory holding cost per item in a production cycle. Figs. 8 and 9 indicate the e4ects of the inventory holding cost h and production rate p on t ∗ , respectively, which are the opposite of the demand rate d as shown in Fig. 7. When the cost of process restoration increases (Fig. 10), the optimal production run-length does not exhibit monotone behavior; however, a decreasing tendency is indicated. This violates the result obtained by Yeh et al. [13], the optimal production run-length t ∗ is an increasing function of r. The reason is that they do not properly consider an IFR process. Fig. 11(a) shows that when both cr and h are small, t ∗ is large and sensitive. Fig. 11(b) indicates that C(t ∗ ) is more sensitive to cr than h.

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Optimal production run time

0.498 0.496 0.494 0.492 0.49 0.488 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

Fig. 4. Optimal production run-time vs. cr 2 for numerical example.

Optimal production run time

0.498 0.496 0.494 0.492 0.49 0.488 0.486 0

0.02

0.04

0.06

0.08

0.1

Fig. 5. Optimal production run-time vs. 1 for numerical example. 0.56 Optimal production run time

2030

0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 150

200

250

300

350

Fig. 6. Optimal production run-time vs. K + V for numerical example.

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Optimal production run time

0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 200

400

600

800

1000

1200

1400

Fig. 7. Optimal production run-time vs. d for numerical example.

Optimal production run time

0.6

0.5

0.4

0

0.5

1

1.5

2

2.5

Fig. 8. Optimal production run-time vs. h for numerical example.

Optimal production run time

0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1000

1500

2000

2500

Fig. 9. Optimal production run-time vs. p for numerical example.

2031

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Optimal production run time

0.493

0.4925

0.492

0.4915

0.491 100

150

200

250

300

Fig. 10. Optimal production run-time vs. r for numerical example.

0.8 0.6 0.4 10 10 5

5 0

h = 0.2 ~ 2 (a)

0

cr = 0.2 ~ 2

18 16 14 12 10 8 6 10 10 5 h = 0.2 ~ 2

0

0

5 cr = 0.2 ~ 2

(b)

Fig. 11. (a) Plots of t ∗ vs. h and cr for numerical example. (b) Plots of C(t ∗ ) vs. h and cr for numerical example.

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6. Concluding remarks In this paper, an imperfect EMQ model for products which are repairable and sold under a free-repair warranty policy, discussed by Yeh et al. [13] has been extended to consider general shift distribution. In particular, a shift distribution with a nondecreasing failure rate has been investigated. In this case, a suLcient condition for a unique optimal production run-length has been explored, thereby extending the condition for the existence of the approximate solution obtained by Yeh et al. [13]. For the exponential case, a tighter bound than [13] for the optimal solution is provided. Numerical examples have been presented to illustrate the developed model. It can be noted that as the factor of the failure function improves, the number of nonconforming items is reduced, thereby resulting in a lower expected total cost per item. Thus, it seems worthwhile to deal with a model having an option for investing in process reliability improvement to reduce the failure rate, so that some improvement can be made in the minimum expected total cost per item. One future addition to this study is to consider a more realistic presentation where a proportion of the failing items has irreparable damage and must be replaced entirely with new ones. Acknowledgements The author thanks the referee for his helpful comments on an earlier version of this paper. Appendix. A derivation of Eq. (2):  t  E(N ) = (1 px + 2 p(t − x))f(x) d x + 0

∞

t



O + 2 pt F(t) O + (1 − 2 )p = −1 pt F(t)

1 ptf(x) d x  0

t



O O F(x) d x + 2 ptF(t) + 1 pt F(t):

The last  t equation above is obtained by integration by parts. Hence, one has E(N ) = 2 pt + (1 − O 2 )p 0 F(x) d x. A proof of Theorem 1(i). The optimal production run-length t ∗ is the solution of dC(t)=dt =R(t)=t 2 = 0, where r r (p − d)ht 2 K − − F(t) + tf(t) R(t) = 2d p p p  t  O O + cr (2 − 1 )(R2 − R1 ) [F(x) − F(t)] dx (11) 0

or, equivalently, t ∗ is the solution to R(t) = 0, where t ¿ 0. By using that limt →0 tf(t) = 0 and limt →∞ tf(t) = 0, it is clear that lim R(t) = −K=p ¡ 0; t →0

(A.1)

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C.-H. Wang / Computers & Operations Research 31 (2004) 2021 – 2035

lim R(t) → ∞:

(A.2)

t →∞

In addition, from Eq. (11) one has R(t) ¿

(p − d)h 2 K + r t − ; 2d p

(A.3)

 which implies that R(tU ) ¿ 0, ∀t ¿ tU , where tU = 2d(K + r)=p(p − d)h. The above facts in (A.1)– (A.3) collectively imply that there exists tj , j = 1; 2; : : : ; k; k + 1; : : : ; 2k − 1, k ¿ 1 so that R(tj ) = 0, R
(tj ) ¿ 0 for j = 1; 2; : : : ; k and R
(tj ) 6 0 for j = k + 1; k + 2; : : : ; 2k − 1, where tj 6 tU . Furthermore, since d 2 C(t)=dt 2 = −2R(t)=t 3 + R
(t)=2t 2 one has d 2 C(tj )=dt 2 = −2R(tj )=(tj )3 + R
(tj )=2(tj )2 = R
(tj )=2(tj )2 ¿ 0, for j = 1; 2; : : : ; k, which implies that C(t) is convex at tj , j = 1; 2; : : : ; k. In addition, limt →0 C(t) → ∞ and limt →∞ C(t) → ∞; thus, one has C(t ∗ ) = minj=1; 2; :::; k C(tj ), where t ∗ 6 tU . A proof of Theorem 1(ii). From Eq. (11), the Nrst derivative of R(t) with respect to t is given by

(p − d)h r r

R (t) = t + f (t) + A1 f(t) ; d p p where A1 = cr p(2 − 1 )(R2 − R1 )=r. From the proofs of Theorem 1(i), we know that the uniqueness property of the optimal production run-length holds if R
(t) ¿ 0, 0 ¡ t 6 tU , or, equivalently (p − d)h=d + (r=p)f
(t) + (r=p)A1 f(t) ¿ 0, 0 ¡ t 6 tU . To show the uniqueness property of the optimal production run-length, we Nrst note that (a) since X has a nondecreasing failure rate, we O have d=dt [ln(’(t))] ¿ 0 if and only if d=dt [ln(f(t))] ¿ d=dt [ln(F(t))]. This implies that ’(t) ¿ −
O f (t)=f(t), for all t ¿ 0. (b) It is obvious to see that 1=f(t) ¿  F(t)=f(t), for all t ¿ 0, which implies that 1=f(t) ¿ 1=’(t), for all t ¿ 0. Since ’(tU ) ¡A1 =2 + p(p − d)h=rd + (A1 =2)2 and X has a nondecreasing failure rate, we have ’(t) ¡ A1 =2 + p(p − d)h=rd + (A1 =2)2 , for 0 ¡ t 6 tU , which implies that (p(p − d)h=rd)=’(t) + A1 ¿ ’(t), for 0 ¡ t 6 tU . Moreover, from (a) and (b), we can derive that   p(p − d)h p(p − d)h f(t) + A1 ¿ ’(t) + A1 ¿ ’(t) ¿ − f
(t)=f(t) for 0 ¡ t 6 tU : rd rd This implies that (p − d)h=d + 0 ¡ t 6 tU .

r p

f
(t) +

r p

A1 f(t) ¿ 0, 0 ¡ t 6 tU , or, equivalently, R
(t) ¿ 0,

Proof of Theorem 2. Using (9) and f(t) =  exp(−t) in (11), we have R(t˜∗ ) = AV (t˜∗ ), where V (t˜∗ ) = 2 (t˜∗ )2 =2 + exp(−t˜∗ ) + t˜∗ exp(−t˜∗ ) − 1. It is not hard to verify that V (0) = 0 and V
(t) = 2 t(1 − exp(−t)) ¿ 0, for all t ¿ 0. These imply that V (t˜∗ ) is positive. Therefore, if A ¿ 0, then R(t˜∗ ) = AV (t˜∗ ) ¿ 0. Moreover, based on the result obtained in the proofs of Theorem 1(ii), R(0) ¡ 0, R(t) is strictly increasing in t, 0 ¡ t 6 tU and R(t) ¿ 0, for all t ¿ tU , thus, we have t˜∗ is an upper bound of t ∗ . Similarly, we have that t˜∗ is a lower bound of t ∗ for A ¡ 0. Obviously, if A = 0, then R(t˜∗ ) = 0, which implies that t ∗ = t˜∗ . Using the same argument, it is easy to verify the relationship among A, t ∗ and tE (or see Yeh et al. [13]), which is omitted here. It is noted that if A = 0, then t ∗ = t˜∗ = tE .

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