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Joint determination of lease period and preventive maintenance policy for leased equipment with residual value
Computers & Industrial Engineering 61 (2011) 489–496
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Joint determination of lease period and preventive maintenance policy for leased equipment with residual value Wen Liang Chang a,⇑, Hui-Chiung Lo b a b
Department of Information Management, Cardinal Tien College of Healthcare & Management, Taipei, Taiwan Department of Business Administration, Tamkang University, Tamsui, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 14 September 2010 Received in revised form 2 February 2011 Accepted 12 April 2011 Available online 15 April 2011 Keywords: Preventive maintenance Lease period Residual value Controlled-limit Minimal repair Lease contract
a b s t r a c t This paper investigates the influence of the length of the lease period on the maintenance policy for leased equipment with residual value. The length of the lease period increases, however, the lessor’s income increases, and the maintenance cost of the equipment rises as well. Therefore, the lease payment and maintenance service of the equipment are crucial items in the lease contract for the lessor’s profit. If the equipment breaks down within the lease period, minimal repairs will be performed on the equipment and the lessor may incur a penalty cost if the repair time exceeds a pre-specified tolerable time. The imperfect preventive maintenance (PM) actions are carried out when the age of the equipment reaches a controlled-limit during the lease period. Under this maintenance scheme, the mathematical model of profit is constructed and then the optimal maintenance policy and the length of the lease period are obtained such that the expected total profit is maximized. Finally, numerical examples are given to illustrate the effects of the optimal length of the lease period and the maintenance policy for profit model. 2011 Elsevier Ltd. All rights reserved.
1. Introduction Facing highly competitive markets, most enterprises may need complex or multi-functional equipment to provide a variety of products and services to meet the divergent needs of customers. In addition, due to diversification of financing methods, the concept of tax saving is gradually valued by enterprises (lessees). Therefore, many enterprises begin to use finance leasing (that is, capital leasing) to reduce investment risk and working capital. Finance leasing means that enterprises pay rent to a leasing company (lessor) for the leased equipment, and then the lessor offers the equipment and the maintenance service. Hence, a finance leasing contract usually contains the specific length of the lease period, the rent and the penalty clauses. The penalty clauses state that the lessor would be penalized when the leased equipment could not carry out its intended performance according to some specified requests by the lessee. For example, a penalty incurs when the number of failures of the equipment during the lease period or the repair time of the equipment exceeds the specified tolerance limits. Therefore, a well-designed structure of the lease contract and the maintenance plan of the leased equipment are very important issues for the lessor to obtain high profit.
⇑ Corresponding author. Tel.: +886 2 22191131; fax: +886 2 22198074. E-mail addresses: [email protected] (W.L. Chang), [email protected] (H.-C. Lo). 0360-8352/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.04.003
In general, the rent usually depends on the length of the leased period and the specified maintenance service required by the lessees. A longer length of the leased period is expected to increase the revenue of lessors, and the maintenance cost of the equipment rises as well. For reducing the repair cost and the penalty cost, some lessors undertake preventive maintenance (PM) actions to reduce the number of the equipment failures within the leased period. In addition, because of various reasons such as environmental concerns, shortage of material, economic benefit, and legislative pressure, re-use has been receiving growing attention and the reclaiming channels are gradually increasing. Therefore, some lessors take the residual value of the equipment into account when they are considering the length of the leased period. For instance, car rental companies sell their used cars to the second-hand markets and most of these cars have been sold by the time they are 2 years old (Pongpech & Murthy, 2006). As mentioned above, this paper proposes a leasing model of the durable and repairable equipment for lessors. The motives of the leasing model are to provide a PM scheme to reduce the total cost within the lease period and to consider the income from the residual value of the equipment at the end of the lease period. Therefore, taking the factor of the residual value of the equipment into account, this paper aims simultaneously to determine the length of the lease period and the PM scheme in the lease contract for the lessor so that the expected profit is maximized. The remainder of this paper is as follows. Section 2 includes a literature review about leasing and maintenance policies. The
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proposed model of this paper is described in Section 3. In Section 4, the optimal PM policy and the length of the lease period are obtained and an algorithm is constructed to organize the solving processes. Some numerical examples are offered to illustrate the leasing decisions and the results are compared with different situations in Section 5. Finally, conclusions are drawn. 2. Literature review In this section, we provide a brief literature review on leasing and maintenance policies, and mainly focus on papers with decision making of preventive maintenance policies in a leasing environment. As the usage and age of the equipment increase, the failure rate of the equipment often increases. Therefore, maintenance on the equipment is necessary to avoid the interruption of production processes of goods/services. There is a vast literature dealing with optimal maintenance policies (Valdez-Flores & Feldman, 1989; Wang, 2002, 2006; Wang & Pham, 2006). In general, maintenance planning can be first classified into corrective maintenance (CM) and preventive maintenance (PM). CM rectifies failed equipment back to its operational status, whereas PM improves the operational status of the equipment. In addition, according to the degree to which the operating condition of the equipment is restored by maintenance, each of the CM and the PM can also be classified into five grades (from highest to lowest): perfect, imperfect, minimal, worse, and worst (Wang & Pham, 2006). For CM, minimal repair is the most commonly used corrective maintenance. Nakagawa (Nakagawa, 1981) used minimal repair to restore failed equipment back to operational status and the failure rate of the equipment remains unchanged after performing a minimal repair. Various maintenance models involving minimal repair can be found in the literature (Barlow & Hunter, 1960; Boland & Proschan, 1982; Chiu & Chiu, 2006; Jhang, 2005; Nakagawa & Kowada, 1983). For repairable equipment, PM actions are widely employed to reduce the number of equipment failures since the cost for carrying out a planned PM action is usually less than the cost incurred by an equipment failure. Various PM models have been proposed for different situations such as periodical or sequential PM (Nakagawa & Kowada, 1983; Sheu, 1991; Valdez-Flores & Feldman, 1989), and perfect or imperfect PM (Jack & Dagpunar, 1994; Nakagawa, 1979; Pham & Wang, 1996). Furthermore, for the quantitative description of the effectiveness of the imperfect PM, three methods are considered in the maintenance model: (i) age-reduction method (ARM), in which the age of the equipment is restored to the one younger than the current age after each PM action, e.g. (Jack & Dagpunar, 1994; Yeh & Lo, 2001), (ii) failure-rate reduction method (FRRM), in which the failure rate of equipment is reduced after each PM action, e.g. (Jaturonnatee, Murthy, & Boondiskulchok, 2006; Yeh, Kao, & Chang, 2009), and (iii) hybrid method, the combinations of the above two methods, e.g. (Zhang & Jardine, 1998). However, Wu and Zuo (2010) attempted to review the existing PM models and explore the interrelationships of these models. They further categorized these models into three classes: linear, nonlinear, and a hybrid of both. Based on the reasons that newer and better equipment appears on the market and that the cost of owning the equipment is increasing, more businesses have started leasing equipment rather than owning it (Pongpech & Murthy, 2006). Therefore, there is also a growing literature in the decision making about the lease contract. Nisbet and Ward (2001) introduced the processes of purchasing and leasing radiotherapy equipment in Raigmore Hospital in UK. They then dealt with the choices of either purchasing or leasing radiotherapy equipment. In addition, there exists the literature focusing on decision making in the pricing of lease contracts. For
instance, Huang and Yang (2002) investigated the pricing of lease contracts for the durable equipment, such as cars, furniture, computers, and other electronic appliances. Aras, Güllü, and Yürülmez (2010) considered a business situation in which a company leases new equipment and sells the remanufactured one at the same time. Under the business model, they developed a dynamic program formulation to determine the optimal price of the remanufactured equipment and the optimal payment structure for the leased equipment to maximize the profit. On the other hand, some researchers studied various PM policies to reduce the cost for the lessors. Jaturonnatee et al. (2006) developed a sequential PM scheme using FRRM and derived the optimal number and degree of PM for leased equipment with minimal repairs. For practical needs, Pongpech and Murthy (2006) and Yeh et al. (2009) used different strategies to reduce Jaturonnatee’s scheme. Pongpech and Murthy (2006) considered a periodical PM scheme with various maintenance degrees, whereas Yeh et al. (2009) proposed a sequential PM scheme with the fixed maintenance degree. Yeh and Chang (2007) found the optimal threshold value of failure-rate and maintenance policy within the lease period. Hu and Zong (2008) relaxed the condition under which the length of the lease period is given in the model of Pongpech and Murthy (2006), and determined the optimal length of the lease period and optimal PM scheme to minimize the total cost per unit time. For used equipment, Pongpech, Murthy and Boondiskulchock (2006) investigated the optimal upgrade level before leasing and optimal PM scheme within the lease period. However, most of the studies mentioned above did not investigate the optimal length of the lease period in the lease contract; instead they only focused on determining the optimal PM policy for leased equipment with a specified leased period. Furthermore, all of these papers utilized FFRM to build cost models to determine the optimal PM scheme, but they failed to take into account the residual value of the equipment after the lease period. Therefore, based on the viewpoint of the lessor’s profitability, this paper adopts ARM to describe the degree of PM for leased equipment and builds a profit model by considering the residual value. Moreover, a mathematical model of the expected total profit within the lease period is constructed and the optimal maintenance policy and the length of lease period are so obtained such that the expected total profit is maximized. At the end of Section 5, an example is given to illustrate the practical applications of our model.
3. Proposed model We used the following notations in this paper: L V td Vd W f(t) h(t) H(t) Cr tr G(tr)
s Cs
lease period the purchase price of the equipment life cycle of the equipment residual value of the equipment at time td the lease payment per unit lease period of equipment probability density function of the lifetime of the leased equipment failure rate function of the leased equipment cumulative failure rate function of the leased equipment minimal repair cost time required for performing a minimal repair cumulative distribution function of tr pre-specified time limit for carrying out a minimal repair penalty cost if the minimal repair time exceeds the time limit s
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ha x Cp(x) n Ti TC TR TP E[TC] E[TR] E[TP]
controlled-limit of age for performing PM actions maintenance degree of a PM action PM cost function with maintenance degree x total number of PM actions within the lease period time epoch of performing the ith PM action disbursement cost total income of the leased equipment of the lessor total profit for the lessor when the lease period of equipment ends expected total disbursement within the lease period expected total income within the lease period expected total profit within the lease period
Consider that a new equipment is leased with the lease period L. Suppose that the purchase price and life cycle of the equipment are V and td, respectively. The residual value of the equipment at time td is Vd. In general, the residual value of the equipment is a decreasing function of time t. All failures within the lease period are repaired using the minimal repair method by the lessor with a fixed cost Cr > 0 and within in a random period of repair time tr, such period of repair time follows a general cumulative distribution function G(tr) with the tiny mean and variance over the length of lease period. Base on the customer equity, there is a penalty cost Cs > 0 with probability GðsÞ ¼ 1 GðsÞ to the lessor if the repair time of the failed equipment exceeds a predetermined time s. When the age of the equipment reaches a controlled-limit ha, each PM action is performed to the same maintenance degree x. After an PM action is carried out, the age of the equipment becomes less than before by x unit of time and therefore each PM cost is Cp(x). In general, the cost of a PM action is a non-negative and nondecreasing function of the maintenance degree x. Moreover, the time required for performing each PM action is negligible. Consider that the failure rate of the equipment h(t) is strictly increasing with h(0) = 0. The failure process of the equipment at each interval [Ti, Ti+1) is a non-homogeneous Poisson process with intensity function h(t ix) since the failure of the equipment is remedied by the minimal repair. PM actions are performed by the lessor at time epoches Ti for i = 1, 2, . . . , n. For the repairable leased equipment with residual value, the maintenance scheme is described in terms of the failure rate function as shown in Fig. 1. Since the failure rate of the equipment is h(t), the expected Rt number of failures within the interval [0, t] is HðtÞ ¼ 0 hðuÞdu. From Fig. 1, the relationship equations Ti+1 Ti = x, i = 1, 2, . . . , n, T1 = ha and the boundary conditions x 6 ha 6 L, (L ha)/n 6 x 6 L can be obtained. Furthermore, the repair time is extremely small over the lease period L and then it can be neglected in Fig. 1. The expected total numbers of failures within the lease period L are
h(t)
θa + 2 x
x
T0=0
T2
3.2. Expected maintenance cost The maintenance cost to the lessor is comprised of the following two costs: 3.2.1. Repair cost When equipment fails within the lease period, each failure incurs a fixed minimal repair cost Cr to the lessor. Furthermore, if the repair time exceeds a predetermined value s, then there is a
C r þ C s GðsÞ. From Fig. 1, since the expected total numbers of Pn R T iþ1 failures within the lease period L are hðt ixÞ dt ¼ i¼0 T i Pn ½HðT ixÞ HðT ixÞ, where T = 0 and T iþ1 i 0 n+1 = L and Ti = i¼0 T1 + (i 1)x, i = 1, 2, . . . , n, the repair cost of the equipment within the lease period is
½C r þ C s GðsÞ
n X ½HðT iþ1 ixÞ HðT i ixÞ i¼0
¼ ½C r þ C s GðsÞfn½HðT 1 Þ HðT 1 xÞ þ HðL nxÞg:
3.2.2. Preventive maintenance cost When the age of the equipment reaches a controlled-limit ha, the PM actions are performed to the same degree x and each PM cost is Cp(x). In Fig. 1, when PM actions are performed at time epochs Ti for i = 1, 2, . . . , n, the total PM cost within the lease period P L is given by ni¼1 C p ðxÞ ¼ nC p ðxÞ. From Sections 3.2.1 and 3.2.2, the expected maintenance cost of equipment within the lease period L is
Cðn; L; x; T 1 Þ ¼ ½C r þ C s GðsÞfn½HðT 1 Þ HðT 1 xÞ þ HðL nxÞg þ nC p ðxÞ:
ð1Þ
When substituting T1 = ha in Eq. (1), the expected total maintenance cost becomes
E½TC ¼ Cðn; L; x; ha Þ þ V ¼ ½C r þ C s GðsÞfn½Hðha Þ Hðha xÞ þ HðL nxÞg þ nC p ðxÞ þ V:
ð2Þ
3.3. Expected total profit The lease payment and the maintenance service of the equipment are important factors for the lessor. Suppose that the lease payment per unit time of the lease period of the equipment is W and the residual value of the equipment is a linear decreasing function of t. The expected total income within the lease period L is
ð3Þ
E½TP ¼ E½TR E½TC
… t
T1
ixÞ HðT i ixÞ, where T0 = 0
Combining the expected total maintenance cost (i.e. Eq. (2)) and income (i.e. Eq. (3)), when the lease period of the equipment expires, the expected total profit over the lease period is
x x
i¼0 ½HðT iþ1
V Vd E½TR ¼ LW þ V L : td
Life Cycle of the Equipment
θa + x
hðt ixÞdt ¼ i¼0 T i and Tn+1 = L.
Pn
penalty cost Cs with probability GðsÞ to the lessor for the delay in restoring the equipment back to the operational condition, which means, the total repair cost to the lessor for each failure is
3.1. Expected number of failures
θa
Pn R T iþ1
T3 … Tn
L
Fig. 1. Age-reduction method of PM.
td
¼ LW
V Vd L ½C r þ C s GðsÞfn½Hðha Þ Hðha xÞ td
þ HðL nxÞg nC p ðxÞ:
ð4Þ
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Our objective is to find an optimal policy ðn ; L ; x ; ha Þ such that the expected total profit in Eq. (4) is maximized.
h(t)
Life Cycle of the Equipment
4. Optimal policy and lease period According to the objective function Eq. (4), we derive some properties and results of the optimal policy and the length of the lease period, two cases are considered: (1) When h(t) is decreasing (DFR) and (2) h(t) is increasing (IFR) in t. Taking the first partial derivative of Eq. (4) with respect to ha, we have
@E½TP ¼ n½C r þ C s GðsÞ½hðha Þ hðha xÞ: @ha
…
4.2. IFR case When h(t) is increasing, the following theorems show that properties and results of the optimal policy and the length of the lease period. Theorem 1. Given n, L, x > 0, when h0 (t) > 0, "t > 0, the optimal controlled-limit ha is x.
td
L
iL/(n + 1), i = 1, 2, . . . , n. within the lease period L as shown in Fig. 2. When substituting x⁄ = L/(n + 1) in Eq. (6), the expected total profit can be rewritten as
V Vd L L ½C r þ C s GðsÞðn þ 1ÞH nþ1 td L nC p : nþ1
E½TP ¼ LW
@E½TP V Vd ¼W @L td 1 L L ½C r þ C s GðsÞðn þ 1Þh þ nC 0p nþ1 nþ1 nþ1 ð9Þ and 2
Proof. Given n, L, x > 0, when h0 (t) > 0, "t > 0 is increasing. We have @E[TP]/@ha < 0 in Eq. (5). This implies that E[TP] is a decreasing function of ha. Therefore, under the boundary condition x 6 ha 6 L, the optimal controlled-limit is ha ¼ x. h When substituting ha ¼ x in the boundary condition (L ha)/ n 6 x 6 L and Eq. (5), the boundary condition becomes L/ (n + 1) 6 x 6 L and the expected total profit can be rewritten as
V Vd L ½C r þ C s GðsÞfnHðxÞ þ HðL nxÞg td ð6Þ
Taking the first partial derivative of Eq. (6) with respect to x, we have
ð7Þ
Observing Eq. (7), the following theorem holds. Theorem 2. When h0 (t) > 0 for all t > 0 and C 0p ðxÞ P 0 for all x > 0, the optimal PM degree is x⁄ = L/(n + 1) for any n, L > 0.
Proof. When h0 (t) > 0 for all t > 0, h(t) is increasing. Since the boundary condition is L/(n + 1) 6 x 6 L and h(t) is increasing, we have h(x) h(L nx) P 0. When C 0p ðxÞ P 0 for all x > 0, we have @E[TP]/@x < 0 in Eq. (7), which means, E[TP] is a decreasing function of x. Therefore, the optimal PM degree within the lease period is x⁄ = L/(n + 1) for any n, L > 0. h From the results of Theorems 1 and 2, the PM should be performed on the equipment to the degree L/(n + 1) at time
ð8Þ
Taking the first and second partial derivatives of Eq. (8) with respect to L, we have
@ 2 E½TP
@E½TP ¼ nf½C r þ C s GðsÞ½hðxÞ hðL nxÞ þ C 0p ðxÞg: @x
t
nL n +1
Fig. 2. Optimal PM policy.
When h(t) is decreasing, we have @E[TP]/@h a > 0 in Eq. (5). That is, E[TP] is a increasing function of ha. Since the boundary condition is x 6 ha 6 L, the optimal controlled-limit is ha ¼ L. In this case, the optimal number of PM is n⁄ = 0 and the maximum profit is E½TP ¼ LW ðV V d ÞL=td ½C r þ C s GðsÞHðLÞ. This result indicates that preventive maintenance is not necessary when the failure rate function of the equipment is decreasing.
nC p ðxÞ:
…
ð5Þ
4.1. DFR case
E½TP ¼ LW
2L L n +1 n +1
0
@L
¼
1 ðn þ 1Þ
2
½C r þ C s GðsÞðn þ 1Þh
0
L L þ nC 00p : nþ1 nþ1 ð10Þ
Observing Eqs. (9) and (10), the following theorem holds. Theorem 3. Given n > 0, when h0 (t) > 0,"t > 0 and C 00p ðxÞ P 0; 8x > 0, the following results hold.
Table 1 Optimal policy n ; L ; x ; ha under different residual values (Vd). b
b = 1.5
b = 2.0
b = 2.5
Vd = 0
Vd = 20
Vd = 0
Vd = 20
Vd = 0
Vd = 20
100 200 300 400
n⁄ 3 5 7 12
4 6 9 16
9 10 12 16
9 11 14 18
11 12 14 16
12 13 15 18
100 200 300 400
ha ¼ x 4.25 3.16 2.37 1.53
3.79 2.85 1.89 1.05
1.89 1.72 1.53 1.11
1.79 1.66 1.26 0.94
1.66 1.53 1.26 1.11
1.53 1.42 1.18 0.94
100 200 300 400
maxE[TP] 3030 2957 2090 1338
3980 3473 2678 1747
5619 4402 3414 2525
5689 4865 3990 2828
6683 5479 4480 3219
7335 5987 4834 3580
100 200 300 400
L⁄ 17 19 19 20
19 20 19 18
19 19 20 19
18 20 19 18
20 20 19 19
20 20 19 18
493
W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496 n d (i) If W V V 6 nþ1 C 0p ð0Þ, then the optimal lease period L⁄ = 0. td n o VV d td td 1 þ nC 0p nþ1 , ½C r þ C s GðsÞðn þ 1Þh nþ1 (ii) If W t P nþ1 d ⁄ then the optimal lease period L = td. n td n 1 d þ C 0 ð0Þ < W VV < nþ1 ½C r þ C s GðsÞðn þ 1Þh nþ1 (iii) If nþ1 td p 0 td ⁄ nC p nþ1 g, then there exists a unique solution L 2 (0, td) such
that Eq. (9) equals zero.
@E½TP
V Vd ¼W @L L¼td td 1 td td ½C r þ C s GðsÞðn þ 1Þh þ nC 0p : nþ1 nþ1 nþ1 ð12Þ Since @E[TP]/@L is a decreasing function of L, the following results hold. (i) If W ðV V d Þ=t d 6 ½n=ðn þ 1ÞC 0p ð0Þ, then W ðV
Proof. When h0 (t) > 0 and C 00p ðxÞ P 0; 8x > 0, we have @ 2E[TP]/ @L2 < 0 in Eq. (10). This implies that E[TP] is a convex function of L and oE[TP]/@L is decreasing. According to the boundary condition 0 6 L 6 td, substituting L = 0 and L = td in Eq. (9), respectively, we have
@E½TP
V Vd n C 0 ð0Þ ¼W @L L¼0 nþ1 p td
ð11Þ
V d Þ=t d ½n=ðn þ 1ÞC 0p ð0Þ 6 0. This means that @E[TP]/@L 6 0 and then E[TP] is a decreasing function of L. Therefore, the optimal L⁄ = 0. (ii) If W ðV V d Þ=td P f½C r þ C s GðsÞðn þ 1Þhðt d =ðn þ 1ÞÞþ then W ðV V d Þ=t d f½C r þ C s GðsÞ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ, ðn þ 1Þhðt d =ðn þ 1ÞÞ þ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ P 0. This means that @E[TP]/@L P 0 and then E[TP] is an increasing function of L. Hence, the optimal L⁄ = td. (iii) If ½n=ðn þ 1ÞC 0p ð0Þ < W ðV V d Þ=t d <
and beta=1.5
20 beta=1.5
20
beta=2.0
beta=2.5 18
beta=2.5
12
11
8
9
14
16
12
12
12
nstar
16
16
nstar
beta=2.0
15 12
12
4
7
9
11
4
6
100
200
16
14 9
8
10 5
13
4 0
3
0
100
200
300
b
300
400
b
400
Fig. 6. Marginal cost versus optimal PM number (Vd = 20). Fig. 3. Marginal cost versus optimal PM number (no residual value).
beta=1.5
5
beta=2.0
beta=1.5
beta=2.5
4.25 3
3.16
2.85
3
xstar
xstar
beta=2.5
3.79
4
2.37 1.89
2
1.66 1
1.72
1.53
1.53
1.89
2
1.53
100
200
1.26
beta=2.0
1.53
1.42
100
8000
2957
3414 2090
2000
3219 2525
E [TP]
3030
4480
4000
0.94
300
b
400
beta=2.0
beta=2.5
7335 5987
6000
5479 4402
200
beta=1.5
beta=2.5
6683
5619
1.05
Fig. 7. Marginal cost versus optimal PM degree (Vd = 20).
8000
6000
1.26 1.18
400
Fig. 4. Marginal cost versus optimal PM degree (no residual value).
beta=1.5
1.66
1.11
300
b
1.79
1 0
0
E [TP]
beta=2.0
4
4834
5689 4865
4000 3980
3990 3473
3580 2828
2678
2000
1747
1338 0
0 100
200
300
400
b Fig. 5. Marginal cost versus profit (no residual value).
100
200
300
b Fig. 8. Marginal cost versus profit (Vd = 20).
400
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W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496
Table 2 Optimal policy n ; L ; x ; ha under different life cycles (td). b = 1.5
b = 2.0
b = 2.5
b
td = 20
td = 25
td = 30
td = 20
td = 25
td = 30
td = 20
td = 25
td = 30
100 200 300 400
n⁄ 4 6 9 16
5 7 11 20
6 9 14 23
9 11 14 18
12 14 17 22
15 17 21 27
12 13 15 18
15 16 19 22
18 20 23 27
100 200 300 400
ha ¼ x 3.79 2.85 1.89 1.05
3.66 2.87 1.91 1.04
3.85 2.79 1.79 1.20
1.79 1.66 1.26 0.94
1.76 1.66 1.38 1.08
1.87 1.66 1.31 1.03
1.53 1.42 1.18 0.94
1.56 1.35 1.14 1.08
1.57 1.42 1.20 1.03
100 200 300 400
Max E[TP] 3980 3473 2678 1747
4894 4091 3284 2241
5819 5183 4152 2731
5689 4865 3990 2828
7538 6231 4961 3587
9393 7597 6153 4431
7335 5987 4834 3580
9149 7359 6181 4528
11112 9304 7544 5575
100 200 300 400
L⁄ 19 20 19 18
22 23 23 22
27 28 27 29
18 20 19 18
23 25 25 25
30 30 29 29
20 20 19 18
25 23 23 25
30 30 29 29
f½C r þ C s GðsÞ ðn þ 1Þhðt d =ðn þ 1ÞÞ þ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ, then @E[TP]/@L changes its sign exactly once from positive to negative in the interval (0, td), and there exists a unique solution L⁄ such that @E½TP=@LjL¼L ¼ 0. h Theorem 3 shows that the optimal lease period L⁄ is unique when h0 (t) > 0, "t > 0 and C 00p ðxÞ P 0; 8x > 0. The optimal lease period L⁄ can be obtained by solving Eq. (13)
V Vd 1 L L W ½C r þ C s GðsÞðn þ 1Þh þ nC 0p nþ1 nþ1 nþ1 td ¼ 0:
equal to btd/sc. Using this upper bound, we can easily search for because n is an integer. the optimal PM number n⁄ between 0 and n Using the results obtained in Theorems 1–3, the following algorithm provides an efficient search procedure for the optimal PM policy and length of lease period. Algorithm. Step 1. Step 2.
ð13Þ
According to the results of Theorem 3, the expected total profit in Eq. (8) can be changed as
V Vd L E½TP ¼ L W L ½C r þ C s GðsÞðn þ 1ÞH td nþ1 L : nC p nþ1
Step 3.
¼ bt d =sc and max E[TP] = 0. Set n ¼ 1; n 0 n d If W VV t d 6 nþ1 C p ð0Þ, then the optimal lease
period L⁄ = 0, the optimal policy ðn ; x ; ha Þ ¼ ð0; 0; 0Þ; max E½TP ¼ 0 and STOP. n td 1 d If W VV t d P nþ1 ½C r þ C s GðsÞðn þ 1Þh nþ1 þ td g, then L = td and compute ha = x = td/ nC 0p nþ1 (n + 1). d Compute E½TP ¼ LW VV td L
ð14Þ
Finally, the rest of the decision variables in Eq. (14) is the number of PM n. Recall that the time required for performing a minimal repair should not exceed a tolerable time s and the life cycle of the equipment is td. From the boundary condition 0 6 ns 6 td, there is a trivial ¼ bt d =sc, which is the greatest integer less than or upper bound n
½C r þ C s GðsÞfnHðxÞþ HðL nxÞg nC p ðxÞ and go to Step 5. Step 4.
If
0 n nþ1 C p ð0Þ
1 nþ1
n
d < W V V td <
td td þ nC 0p nþ1 g, ½C r þ C s GðsÞðn þ 1Þh nþ1
then find L 2 (0, td) satisfying @E[TP]/@L = 0 and compute ha = x = L/(n + 1). d Compute E½TP ¼ LW VV td L
b=100
25
b=200
b=300
b=400 23
20
nstar
16
Step 5.
20
15
14
Step 6.
½C r þ C s GðsÞfnHðxÞþ HðL nxÞg nC p ðxÞ. If maxE[TP] < E[TP] then set max E½TP ¼ E½TP; n ; L ; x ; ha ¼ ðn; L; x; ha Þ and n = n + 1. Otherwise, set n = n + 1. then STOP. Otherwise, go to Step 3. If n > n
11
10 5
9
7
6
9 6
4
5
0 20
25
30
td Fig. 9. Marginal cost versus optimal PM number (b = 1.5).
5. Numerical examples Consider that the lifetime distribution of the photocopier follow b a two-parameter Weibull distribution f ðtÞ ¼ kbðktÞb1 eðktÞ for t P 0, where k is the scale parameter and b is the shape parameter, the expected life time is l = 1/kC(1 + 1/b). Moreover, we suppose
W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496
b=100
b=200
b=300
3. When the marginal cost b increases, the number of PM increases, but maximum profit, PM degree and controlled-limit both decrease, under the life cycle of equipment given.
b=400
8000 5819
E [TP]
6000
4000
4894
5183
4091
4152
2678
3284
2731
1747
2241
3980 3473
2000
495
0 20
25
30
td Fig. 10. Marginal cost versus profit (b = 1.5).
the parameter k = 0.6 and the shape parameter b = 1.5, 2, 2.5, respectively. Following definition of a failure rate function, the failure rate function of the Weibull distribution is h(t) = kb(kt)b1. Let us consider that photocopier is leased with a length of lease period L (years) and the lease payment of the photocopier and the initial purchase prices of the photocopier are W = 800($) and V = 1000($), respectively. The life cycle of the photocopier is td (years) and residual value of the photocopier is Vd at time td. Suppose that the minimal repair cost Cr = 280($) and the repair time tr follows a Weibull(2, 1). There is a penalty cost Cs = 450($) with probability GðsÞ 0:49 to the lessor if the repair time exceeds a predetermined time s = 0.0083 (year) = 3 (days). Each failure of the photocopier within the lease period incurs an expected disbursement cost C r þ C s GðsÞ 500ð$Þ (including the minimal repair cost and possible penalty cost) and the cost of performing a PM action with maintenance degree x is Cp(x) = 50 + bx($). 5.1. Different residual value Suppose that the residual values of the leased equipment are Vd = 0($) and Vd = 20($) at time td = 20 (years). Table 1 summarizes the optimal maintenance policy ðn ; L ; x ; ha Þ according to different residual values. From Figs. 3–8, we derive following results: 1. When the marginal PM cost b increases, the number of PM increases, but the PM degree, controlled-limit, and maximum profit E[TP] all decrease. 2. When b increases, the number of PM and maximum profit both increase, but the PM degree and controlled-limit decrease. 3. The number of PM and maximum profit under the considered residual value are higher than under the non-considered residual value. 4. The PM degree and controlled-limit under the considered residual value are lower than under the non-considered residual value. 5.2. Different life cycle Suppose that the residual value of the leased equipment and the life cycle of equipment are Vd = 20($) and td = 20, 25, 30 (years), respectively. Table 2 summarizes the optimal maintenance policy and maximum profit under different life cycle. From Figs. 9, 10 and Table 2, we have some results as follows. 1. When the life cycle of the equipment td increases, the number of PM and maximum profit E[TP] both increase. 2. Under the life cycle of equipment given, when b increases, the number of PM and maximum profit both increase, but PM degree and controlled-limit both decrease.
6. Conclusions In this paper, the age-reduction method is adopted to describe the degree of preventive maintenance and a mathematical model of profit is constructed for the leased equipment within the lease period. From the model of profit, we show that there exists a unique optimal length of lease period and maintenance policy within the lease period such that the expected total profit is maximized. Within the lease period, the equipment should be restored to its original state after each preventive maintenance action. That is, preventive maintenance actions should be performed at time epochs L/(n + 1), 2L/(n + 1), 3L/(n + 1), . . . , nL/(n + 1) with the same maintenance degree L/(n + 1). When the preventive maintenance cost b or b increases, the number of preventive maintenance actions n increases but preventive maintenance degree, controlledlimit and maximum profit decrease. The preventive maintenance number and maximum profit under the considered residual value are higher than the non-considered residual value. Under the equipment with residual value, the expected total profit is maximized. Further, two-phase maintenance of equipment and the interest rate of the company within the lease period are investigated for various maintenance plans. References Aras, N., Güllü, R., & Yürülmez, S. (2010). Optimal inventory and pricing policies for remanufacturable leased products. International Journal of Production Economics. Published online. Barlow, R. E., & Hunter, L. C. (1960). Optimum preventive mathematical policies. Operations Research, 8, 90–100. Boland, P. J., & Proschan, F. (1982). Periodic replacement with increasing minimal repair costs at failure. Operations Research, 30, 1183–1189. Chiu, S. W., & Chiu, Y. S. (2006). Mathematical modeling for production system with backlogging and failure in repair. Journal of Scientific & Industrial Research, 65, 499–506. Huang, S., & Yang, Y. (2002). Pricing leased contracts with options in imperfect markets of durable goods. Technical Report, Ford Research Laboratory, Dearborn. Hu, F., & Zong, Q. (2008). Optimal periodic preventive maintenance policy and lease period for leased equipment. Journal of Tianjin University Science and Technology, 41, 248–253. Jack, N., & Dagpunar, J. S. (1994). An optimal imperfect maintenance policy over a warranty period. Microelectronics and Reliability, 34, 529–534. Jaturonnatee, J., Murthy, D. N. P., & Boondiskulchok, R. (2006). Optimal preventive maintenance of leased equipment with corrective minimal repairs. European Journal of Operational Research, 174, 201–215. Jhang, J. P. (2005). A study of the optimal use period and number of minimal repairs of a repairable product after the warranty expires. International Journal of Systems Science, 36, 697–704. Nakagawa, T. (1979). Imperfect preventive-maintenance. Journal of the Operations Research Society of Japan, 24, 213–227. Nakagawa, T. (1981). A summary of periodic replacement with minimal repair at failure. Journal of the Operations Research Society of Japan, 24, 213–227. Nakagawa, T., & Kowada, M. (1983). Analysis of a system with minimal repair and its application to replacement policy. European Journal of Operational Research, 12, 176–182. Nisbet, A., & Ward, A. (2001). Radiotherapy equipment – Purchase or lease? The British Journal of Radiology, 74, 735–744. Pham, H., & Wang, H. (1996). Imperfect maintenance. European Journal of Operational Research, 94, 425–438. Pongpech, J., & Murthy, D. N. P. (2006). Optimal periodic preventive maintenance policy for leased equipment. Reliability Engineering & System Safety, 91, 772–777. Pongpech, J., Murthy, D. N. P., & Boondiskulchock, R. M. (2006). Maintenance strategies for used equipment under lease. Journal of Quality in Maintenance Engineering, 12, 52–67. Sheu, S. H. (1991). Periodic replacement with minimal repair at failure and general random repair cost for a multi-unit system. Microelectronics Reliability, 31, 1019–1025. Valdez-Flores, C., & Feldman, R. M. (1989). A survey of preventive maintenance models for stochastically deteriorating single-unit systems. Naval Research Logistics, 36, 419–446.
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Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal of Operational Research, 139, 469–489. Wang, C. H. (2006). Optimal production and maintenance policy for imperfect production systems. Naval Research Logistics, 53, 152–156. Wang, H., & Pham, H. (2006). Reliability and optimal maintenance. Springer series in reliability engineering series. London: Springer-Verlag. Wu, S., & Zuo, M. J. (2010). Linear and nonlinear preventive maintenance models. IEEE Transactions on Reliability, 59, 242–249. Yeh, R. H., & Chang, W. L. (2007). Optimal threshold value of failure-rate for leased products with preventive maintenance actions. Mathematical and Computer Modelling, 46, 730–737. Yeh, R. H., Kao, K. C., & Chang, W. L. (2009). Optimal preventive maintenance policy for leased equipment using failure rate reduction. Computers & Industrial Engineering, 57, 304–309. Yeh, R. H., & Lo, H. C. (2001). Optimal preventive-maintenance warranty policy for repairable products. European Journal of Operational Research, 134, 59–69. Zhang, F., & Jardine, A. K. S. (1998). Optimal maintenance models with minimal repair, periodic overhaul and complete renewal. IIE Transactions, 30, 1109–1119.
Wen Liang Chang is currently an assistant professor in the Department of Information Management at Cardinal Tien College of Healthcare & Management, Taipei, Taiwan. He received his Ph.D. in the Department of Industrial Management from National Taiwan University of Science and Technology in 2007. His research interests include reliability theory, warranty cost analysis, applied probability and maintenance policy of leased equipment. He has published academic articles in Mathematical and Computer Modelling, Computers & Industrial Engineering, Annals of Operations Research, Journal of Quality of Chinese Society for Quality. Hui-Chiung Lo is currently an associate professor in the Department of Business Administration at the Tamkang University. She received her Ph.D. Department of Industrial Management from National Taiwan University of Science and Technology. Her research interests include reliability theory, warranty cost analysis, and applied probability. She has published academic articles in European Journal of Operational Research, Computers & Industrial Engineering, Annals of Operations Research.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Joint determination of lease period and preventive maintenance policy for leased equipment with residual value Wen Liang Chang a,⇑, Hui-Chiung Lo b a b
Department of Information Management, Cardinal Tien College of Healthcare & Management, Taipei, Taiwan Department of Business Administration, Tamkang University, Tamsui, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 14 September 2010 Received in revised form 2 February 2011 Accepted 12 April 2011 Available online 15 April 2011 Keywords: Preventive maintenance Lease period Residual value Controlled-limit Minimal repair Lease contract
a b s t r a c t This paper investigates the influence of the length of the lease period on the maintenance policy for leased equipment with residual value. The length of the lease period increases, however, the lessor’s income increases, and the maintenance cost of the equipment rises as well. Therefore, the lease payment and maintenance service of the equipment are crucial items in the lease contract for the lessor’s profit. If the equipment breaks down within the lease period, minimal repairs will be performed on the equipment and the lessor may incur a penalty cost if the repair time exceeds a pre-specified tolerable time. The imperfect preventive maintenance (PM) actions are carried out when the age of the equipment reaches a controlled-limit during the lease period. Under this maintenance scheme, the mathematical model of profit is constructed and then the optimal maintenance policy and the length of the lease period are obtained such that the expected total profit is maximized. Finally, numerical examples are given to illustrate the effects of the optimal length of the lease period and the maintenance policy for profit model. 2011 Elsevier Ltd. All rights reserved.
1. Introduction Facing highly competitive markets, most enterprises may need complex or multi-functional equipment to provide a variety of products and services to meet the divergent needs of customers. In addition, due to diversification of financing methods, the concept of tax saving is gradually valued by enterprises (lessees). Therefore, many enterprises begin to use finance leasing (that is, capital leasing) to reduce investment risk and working capital. Finance leasing means that enterprises pay rent to a leasing company (lessor) for the leased equipment, and then the lessor offers the equipment and the maintenance service. Hence, a finance leasing contract usually contains the specific length of the lease period, the rent and the penalty clauses. The penalty clauses state that the lessor would be penalized when the leased equipment could not carry out its intended performance according to some specified requests by the lessee. For example, a penalty incurs when the number of failures of the equipment during the lease period or the repair time of the equipment exceeds the specified tolerance limits. Therefore, a well-designed structure of the lease contract and the maintenance plan of the leased equipment are very important issues for the lessor to obtain high profit.
⇑ Corresponding author. Tel.: +886 2 22191131; fax: +886 2 22198074. E-mail addresses: [email protected] (W.L. Chang), [email protected] (H.-C. Lo). 0360-8352/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.04.003
In general, the rent usually depends on the length of the leased period and the specified maintenance service required by the lessees. A longer length of the leased period is expected to increase the revenue of lessors, and the maintenance cost of the equipment rises as well. For reducing the repair cost and the penalty cost, some lessors undertake preventive maintenance (PM) actions to reduce the number of the equipment failures within the leased period. In addition, because of various reasons such as environmental concerns, shortage of material, economic benefit, and legislative pressure, re-use has been receiving growing attention and the reclaiming channels are gradually increasing. Therefore, some lessors take the residual value of the equipment into account when they are considering the length of the leased period. For instance, car rental companies sell their used cars to the second-hand markets and most of these cars have been sold by the time they are 2 years old (Pongpech & Murthy, 2006). As mentioned above, this paper proposes a leasing model of the durable and repairable equipment for lessors. The motives of the leasing model are to provide a PM scheme to reduce the total cost within the lease period and to consider the income from the residual value of the equipment at the end of the lease period. Therefore, taking the factor of the residual value of the equipment into account, this paper aims simultaneously to determine the length of the lease period and the PM scheme in the lease contract for the lessor so that the expected profit is maximized. The remainder of this paper is as follows. Section 2 includes a literature review about leasing and maintenance policies. The
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proposed model of this paper is described in Section 3. In Section 4, the optimal PM policy and the length of the lease period are obtained and an algorithm is constructed to organize the solving processes. Some numerical examples are offered to illustrate the leasing decisions and the results are compared with different situations in Section 5. Finally, conclusions are drawn. 2. Literature review In this section, we provide a brief literature review on leasing and maintenance policies, and mainly focus on papers with decision making of preventive maintenance policies in a leasing environment. As the usage and age of the equipment increase, the failure rate of the equipment often increases. Therefore, maintenance on the equipment is necessary to avoid the interruption of production processes of goods/services. There is a vast literature dealing with optimal maintenance policies (Valdez-Flores & Feldman, 1989; Wang, 2002, 2006; Wang & Pham, 2006). In general, maintenance planning can be first classified into corrective maintenance (CM) and preventive maintenance (PM). CM rectifies failed equipment back to its operational status, whereas PM improves the operational status of the equipment. In addition, according to the degree to which the operating condition of the equipment is restored by maintenance, each of the CM and the PM can also be classified into five grades (from highest to lowest): perfect, imperfect, minimal, worse, and worst (Wang & Pham, 2006). For CM, minimal repair is the most commonly used corrective maintenance. Nakagawa (Nakagawa, 1981) used minimal repair to restore failed equipment back to operational status and the failure rate of the equipment remains unchanged after performing a minimal repair. Various maintenance models involving minimal repair can be found in the literature (Barlow & Hunter, 1960; Boland & Proschan, 1982; Chiu & Chiu, 2006; Jhang, 2005; Nakagawa & Kowada, 1983). For repairable equipment, PM actions are widely employed to reduce the number of equipment failures since the cost for carrying out a planned PM action is usually less than the cost incurred by an equipment failure. Various PM models have been proposed for different situations such as periodical or sequential PM (Nakagawa & Kowada, 1983; Sheu, 1991; Valdez-Flores & Feldman, 1989), and perfect or imperfect PM (Jack & Dagpunar, 1994; Nakagawa, 1979; Pham & Wang, 1996). Furthermore, for the quantitative description of the effectiveness of the imperfect PM, three methods are considered in the maintenance model: (i) age-reduction method (ARM), in which the age of the equipment is restored to the one younger than the current age after each PM action, e.g. (Jack & Dagpunar, 1994; Yeh & Lo, 2001), (ii) failure-rate reduction method (FRRM), in which the failure rate of equipment is reduced after each PM action, e.g. (Jaturonnatee, Murthy, & Boondiskulchok, 2006; Yeh, Kao, & Chang, 2009), and (iii) hybrid method, the combinations of the above two methods, e.g. (Zhang & Jardine, 1998). However, Wu and Zuo (2010) attempted to review the existing PM models and explore the interrelationships of these models. They further categorized these models into three classes: linear, nonlinear, and a hybrid of both. Based on the reasons that newer and better equipment appears on the market and that the cost of owning the equipment is increasing, more businesses have started leasing equipment rather than owning it (Pongpech & Murthy, 2006). Therefore, there is also a growing literature in the decision making about the lease contract. Nisbet and Ward (2001) introduced the processes of purchasing and leasing radiotherapy equipment in Raigmore Hospital in UK. They then dealt with the choices of either purchasing or leasing radiotherapy equipment. In addition, there exists the literature focusing on decision making in the pricing of lease contracts. For
instance, Huang and Yang (2002) investigated the pricing of lease contracts for the durable equipment, such as cars, furniture, computers, and other electronic appliances. Aras, Güllü, and Yürülmez (2010) considered a business situation in which a company leases new equipment and sells the remanufactured one at the same time. Under the business model, they developed a dynamic program formulation to determine the optimal price of the remanufactured equipment and the optimal payment structure for the leased equipment to maximize the profit. On the other hand, some researchers studied various PM policies to reduce the cost for the lessors. Jaturonnatee et al. (2006) developed a sequential PM scheme using FRRM and derived the optimal number and degree of PM for leased equipment with minimal repairs. For practical needs, Pongpech and Murthy (2006) and Yeh et al. (2009) used different strategies to reduce Jaturonnatee’s scheme. Pongpech and Murthy (2006) considered a periodical PM scheme with various maintenance degrees, whereas Yeh et al. (2009) proposed a sequential PM scheme with the fixed maintenance degree. Yeh and Chang (2007) found the optimal threshold value of failure-rate and maintenance policy within the lease period. Hu and Zong (2008) relaxed the condition under which the length of the lease period is given in the model of Pongpech and Murthy (2006), and determined the optimal length of the lease period and optimal PM scheme to minimize the total cost per unit time. For used equipment, Pongpech, Murthy and Boondiskulchock (2006) investigated the optimal upgrade level before leasing and optimal PM scheme within the lease period. However, most of the studies mentioned above did not investigate the optimal length of the lease period in the lease contract; instead they only focused on determining the optimal PM policy for leased equipment with a specified leased period. Furthermore, all of these papers utilized FFRM to build cost models to determine the optimal PM scheme, but they failed to take into account the residual value of the equipment after the lease period. Therefore, based on the viewpoint of the lessor’s profitability, this paper adopts ARM to describe the degree of PM for leased equipment and builds a profit model by considering the residual value. Moreover, a mathematical model of the expected total profit within the lease period is constructed and the optimal maintenance policy and the length of lease period are so obtained such that the expected total profit is maximized. At the end of Section 5, an example is given to illustrate the practical applications of our model.
3. Proposed model We used the following notations in this paper: L V td Vd W f(t) h(t) H(t) Cr tr G(tr)
s Cs
lease period the purchase price of the equipment life cycle of the equipment residual value of the equipment at time td the lease payment per unit lease period of equipment probability density function of the lifetime of the leased equipment failure rate function of the leased equipment cumulative failure rate function of the leased equipment minimal repair cost time required for performing a minimal repair cumulative distribution function of tr pre-specified time limit for carrying out a minimal repair penalty cost if the minimal repair time exceeds the time limit s
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ha x Cp(x) n Ti TC TR TP E[TC] E[TR] E[TP]
controlled-limit of age for performing PM actions maintenance degree of a PM action PM cost function with maintenance degree x total number of PM actions within the lease period time epoch of performing the ith PM action disbursement cost total income of the leased equipment of the lessor total profit for the lessor when the lease period of equipment ends expected total disbursement within the lease period expected total income within the lease period expected total profit within the lease period
Consider that a new equipment is leased with the lease period L. Suppose that the purchase price and life cycle of the equipment are V and td, respectively. The residual value of the equipment at time td is Vd. In general, the residual value of the equipment is a decreasing function of time t. All failures within the lease period are repaired using the minimal repair method by the lessor with a fixed cost Cr > 0 and within in a random period of repair time tr, such period of repair time follows a general cumulative distribution function G(tr) with the tiny mean and variance over the length of lease period. Base on the customer equity, there is a penalty cost Cs > 0 with probability GðsÞ ¼ 1 GðsÞ to the lessor if the repair time of the failed equipment exceeds a predetermined time s. When the age of the equipment reaches a controlled-limit ha, each PM action is performed to the same maintenance degree x. After an PM action is carried out, the age of the equipment becomes less than before by x unit of time and therefore each PM cost is Cp(x). In general, the cost of a PM action is a non-negative and nondecreasing function of the maintenance degree x. Moreover, the time required for performing each PM action is negligible. Consider that the failure rate of the equipment h(t) is strictly increasing with h(0) = 0. The failure process of the equipment at each interval [Ti, Ti+1) is a non-homogeneous Poisson process with intensity function h(t ix) since the failure of the equipment is remedied by the minimal repair. PM actions are performed by the lessor at time epoches Ti for i = 1, 2, . . . , n. For the repairable leased equipment with residual value, the maintenance scheme is described in terms of the failure rate function as shown in Fig. 1. Since the failure rate of the equipment is h(t), the expected Rt number of failures within the interval [0, t] is HðtÞ ¼ 0 hðuÞdu. From Fig. 1, the relationship equations Ti+1 Ti = x, i = 1, 2, . . . , n, T1 = ha and the boundary conditions x 6 ha 6 L, (L ha)/n 6 x 6 L can be obtained. Furthermore, the repair time is extremely small over the lease period L and then it can be neglected in Fig. 1. The expected total numbers of failures within the lease period L are
h(t)
θa + 2 x
x
T0=0
T2
3.2. Expected maintenance cost The maintenance cost to the lessor is comprised of the following two costs: 3.2.1. Repair cost When equipment fails within the lease period, each failure incurs a fixed minimal repair cost Cr to the lessor. Furthermore, if the repair time exceeds a predetermined value s, then there is a
C r þ C s GðsÞ. From Fig. 1, since the expected total numbers of Pn R T iþ1 failures within the lease period L are hðt ixÞ dt ¼ i¼0 T i Pn ½HðT ixÞ HðT ixÞ, where T = 0 and T iþ1 i 0 n+1 = L and Ti = i¼0 T1 + (i 1)x, i = 1, 2, . . . , n, the repair cost of the equipment within the lease period is
½C r þ C s GðsÞ
n X ½HðT iþ1 ixÞ HðT i ixÞ i¼0
¼ ½C r þ C s GðsÞfn½HðT 1 Þ HðT 1 xÞ þ HðL nxÞg:
3.2.2. Preventive maintenance cost When the age of the equipment reaches a controlled-limit ha, the PM actions are performed to the same degree x and each PM cost is Cp(x). In Fig. 1, when PM actions are performed at time epochs Ti for i = 1, 2, . . . , n, the total PM cost within the lease period P L is given by ni¼1 C p ðxÞ ¼ nC p ðxÞ. From Sections 3.2.1 and 3.2.2, the expected maintenance cost of equipment within the lease period L is
Cðn; L; x; T 1 Þ ¼ ½C r þ C s GðsÞfn½HðT 1 Þ HðT 1 xÞ þ HðL nxÞg þ nC p ðxÞ:
ð1Þ
When substituting T1 = ha in Eq. (1), the expected total maintenance cost becomes
E½TC ¼ Cðn; L; x; ha Þ þ V ¼ ½C r þ C s GðsÞfn½Hðha Þ Hðha xÞ þ HðL nxÞg þ nC p ðxÞ þ V:
ð2Þ
3.3. Expected total profit The lease payment and the maintenance service of the equipment are important factors for the lessor. Suppose that the lease payment per unit time of the lease period of the equipment is W and the residual value of the equipment is a linear decreasing function of t. The expected total income within the lease period L is
ð3Þ
E½TP ¼ E½TR E½TC
… t
T1
ixÞ HðT i ixÞ, where T0 = 0
Combining the expected total maintenance cost (i.e. Eq. (2)) and income (i.e. Eq. (3)), when the lease period of the equipment expires, the expected total profit over the lease period is
x x
i¼0 ½HðT iþ1
V Vd E½TR ¼ LW þ V L : td
Life Cycle of the Equipment
θa + x
hðt ixÞdt ¼ i¼0 T i and Tn+1 = L.
Pn
penalty cost Cs with probability GðsÞ to the lessor for the delay in restoring the equipment back to the operational condition, which means, the total repair cost to the lessor for each failure is
3.1. Expected number of failures
θa
Pn R T iþ1
T3 … Tn
L
Fig. 1. Age-reduction method of PM.
td
¼ LW
V Vd L ½C r þ C s GðsÞfn½Hðha Þ Hðha xÞ td
þ HðL nxÞg nC p ðxÞ:
ð4Þ
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Our objective is to find an optimal policy ðn ; L ; x ; ha Þ such that the expected total profit in Eq. (4) is maximized.
h(t)
Life Cycle of the Equipment
4. Optimal policy and lease period According to the objective function Eq. (4), we derive some properties and results of the optimal policy and the length of the lease period, two cases are considered: (1) When h(t) is decreasing (DFR) and (2) h(t) is increasing (IFR) in t. Taking the first partial derivative of Eq. (4) with respect to ha, we have
@E½TP ¼ n½C r þ C s GðsÞ½hðha Þ hðha xÞ: @ha
…
4.2. IFR case When h(t) is increasing, the following theorems show that properties and results of the optimal policy and the length of the lease period. Theorem 1. Given n, L, x > 0, when h0 (t) > 0, "t > 0, the optimal controlled-limit ha is x.
td
L
iL/(n + 1), i = 1, 2, . . . , n. within the lease period L as shown in Fig. 2. When substituting x⁄ = L/(n + 1) in Eq. (6), the expected total profit can be rewritten as
V Vd L L ½C r þ C s GðsÞðn þ 1ÞH nþ1 td L nC p : nþ1
E½TP ¼ LW
@E½TP V Vd ¼W @L td 1 L L ½C r þ C s GðsÞðn þ 1Þh þ nC 0p nþ1 nþ1 nþ1 ð9Þ and 2
Proof. Given n, L, x > 0, when h0 (t) > 0, "t > 0 is increasing. We have @E[TP]/@ha < 0 in Eq. (5). This implies that E[TP] is a decreasing function of ha. Therefore, under the boundary condition x 6 ha 6 L, the optimal controlled-limit is ha ¼ x. h When substituting ha ¼ x in the boundary condition (L ha)/ n 6 x 6 L and Eq. (5), the boundary condition becomes L/ (n + 1) 6 x 6 L and the expected total profit can be rewritten as
V Vd L ½C r þ C s GðsÞfnHðxÞ þ HðL nxÞg td ð6Þ
Taking the first partial derivative of Eq. (6) with respect to x, we have
ð7Þ
Observing Eq. (7), the following theorem holds. Theorem 2. When h0 (t) > 0 for all t > 0 and C 0p ðxÞ P 0 for all x > 0, the optimal PM degree is x⁄ = L/(n + 1) for any n, L > 0.
Proof. When h0 (t) > 0 for all t > 0, h(t) is increasing. Since the boundary condition is L/(n + 1) 6 x 6 L and h(t) is increasing, we have h(x) h(L nx) P 0. When C 0p ðxÞ P 0 for all x > 0, we have @E[TP]/@x < 0 in Eq. (7), which means, E[TP] is a decreasing function of x. Therefore, the optimal PM degree within the lease period is x⁄ = L/(n + 1) for any n, L > 0. h From the results of Theorems 1 and 2, the PM should be performed on the equipment to the degree L/(n + 1) at time
ð8Þ
Taking the first and second partial derivatives of Eq. (8) with respect to L, we have
@ 2 E½TP
@E½TP ¼ nf½C r þ C s GðsÞ½hðxÞ hðL nxÞ þ C 0p ðxÞg: @x
t
nL n +1
Fig. 2. Optimal PM policy.
When h(t) is decreasing, we have @E[TP]/@h a > 0 in Eq. (5). That is, E[TP] is a increasing function of ha. Since the boundary condition is x 6 ha 6 L, the optimal controlled-limit is ha ¼ L. In this case, the optimal number of PM is n⁄ = 0 and the maximum profit is E½TP ¼ LW ðV V d ÞL=td ½C r þ C s GðsÞHðLÞ. This result indicates that preventive maintenance is not necessary when the failure rate function of the equipment is decreasing.
nC p ðxÞ:
…
ð5Þ
4.1. DFR case
E½TP ¼ LW
2L L n +1 n +1
0
@L
¼
1 ðn þ 1Þ
2
½C r þ C s GðsÞðn þ 1Þh
0
L L þ nC 00p : nþ1 nþ1 ð10Þ
Observing Eqs. (9) and (10), the following theorem holds. Theorem 3. Given n > 0, when h0 (t) > 0,"t > 0 and C 00p ðxÞ P 0; 8x > 0, the following results hold.
Table 1 Optimal policy n ; L ; x ; ha under different residual values (Vd). b
b = 1.5
b = 2.0
b = 2.5
Vd = 0
Vd = 20
Vd = 0
Vd = 20
Vd = 0
Vd = 20
100 200 300 400
n⁄ 3 5 7 12
4 6 9 16
9 10 12 16
9 11 14 18
11 12 14 16
12 13 15 18
100 200 300 400
ha ¼ x 4.25 3.16 2.37 1.53
3.79 2.85 1.89 1.05
1.89 1.72 1.53 1.11
1.79 1.66 1.26 0.94
1.66 1.53 1.26 1.11
1.53 1.42 1.18 0.94
100 200 300 400
maxE[TP] 3030 2957 2090 1338
3980 3473 2678 1747
5619 4402 3414 2525
5689 4865 3990 2828
6683 5479 4480 3219
7335 5987 4834 3580
100 200 300 400
L⁄ 17 19 19 20
19 20 19 18
19 19 20 19
18 20 19 18
20 20 19 19
20 20 19 18
493
W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496 n d (i) If W V V 6 nþ1 C 0p ð0Þ, then the optimal lease period L⁄ = 0. td n o VV d td td 1 þ nC 0p nþ1 , ½C r þ C s GðsÞðn þ 1Þh nþ1 (ii) If W t P nþ1 d ⁄ then the optimal lease period L = td. n td n 1 d þ C 0 ð0Þ < W VV < nþ1 ½C r þ C s GðsÞðn þ 1Þh nþ1 (iii) If nþ1 td p 0 td ⁄ nC p nþ1 g, then there exists a unique solution L 2 (0, td) such
that Eq. (9) equals zero.
@E½TP
V Vd ¼W @L L¼td td 1 td td ½C r þ C s GðsÞðn þ 1Þh þ nC 0p : nþ1 nþ1 nþ1 ð12Þ Since @E[TP]/@L is a decreasing function of L, the following results hold. (i) If W ðV V d Þ=t d 6 ½n=ðn þ 1ÞC 0p ð0Þ, then W ðV
Proof. When h0 (t) > 0 and C 00p ðxÞ P 0; 8x > 0, we have @ 2E[TP]/ @L2 < 0 in Eq. (10). This implies that E[TP] is a convex function of L and oE[TP]/@L is decreasing. According to the boundary condition 0 6 L 6 td, substituting L = 0 and L = td in Eq. (9), respectively, we have
@E½TP
V Vd n C 0 ð0Þ ¼W @L L¼0 nþ1 p td
ð11Þ
V d Þ=t d ½n=ðn þ 1ÞC 0p ð0Þ 6 0. This means that @E[TP]/@L 6 0 and then E[TP] is a decreasing function of L. Therefore, the optimal L⁄ = 0. (ii) If W ðV V d Þ=td P f½C r þ C s GðsÞðn þ 1Þhðt d =ðn þ 1ÞÞþ then W ðV V d Þ=t d f½C r þ C s GðsÞ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ, ðn þ 1Þhðt d =ðn þ 1ÞÞ þ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ P 0. This means that @E[TP]/@L P 0 and then E[TP] is an increasing function of L. Hence, the optimal L⁄ = td. (iii) If ½n=ðn þ 1ÞC 0p ð0Þ < W ðV V d Þ=t d <
and beta=1.5
20 beta=1.5
20
beta=2.0
beta=2.5 18
beta=2.5
12
11
8
9
14
16
12
12
12
nstar
16
16
nstar
beta=2.0
15 12
12
4
7
9
11
4
6
100
200
16
14 9
8
10 5
13
4 0
3
0
100
200
300
b
300
400
b
400
Fig. 6. Marginal cost versus optimal PM number (Vd = 20). Fig. 3. Marginal cost versus optimal PM number (no residual value).
beta=1.5
5
beta=2.0
beta=1.5
beta=2.5
4.25 3
3.16
2.85
3
xstar
xstar
beta=2.5
3.79
4
2.37 1.89
2
1.66 1
1.72
1.53
1.53
1.89
2
1.53
100
200
1.26
beta=2.0
1.53
1.42
100
8000
2957
3414 2090
2000
3219 2525
E [TP]
3030
4480
4000
0.94
300
b
400
beta=2.0
beta=2.5
7335 5987
6000
5479 4402
200
beta=1.5
beta=2.5
6683
5619
1.05
Fig. 7. Marginal cost versus optimal PM degree (Vd = 20).
8000
6000
1.26 1.18
400
Fig. 4. Marginal cost versus optimal PM degree (no residual value).
beta=1.5
1.66
1.11
300
b
1.79
1 0
0
E [TP]
beta=2.0
4
4834
5689 4865
4000 3980
3990 3473
3580 2828
2678
2000
1747
1338 0
0 100
200
300
400
b Fig. 5. Marginal cost versus profit (no residual value).
100
200
300
b Fig. 8. Marginal cost versus profit (Vd = 20).
400
494
W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496
Table 2 Optimal policy n ; L ; x ; ha under different life cycles (td). b = 1.5
b = 2.0
b = 2.5
b
td = 20
td = 25
td = 30
td = 20
td = 25
td = 30
td = 20
td = 25
td = 30
100 200 300 400
n⁄ 4 6 9 16
5 7 11 20
6 9 14 23
9 11 14 18
12 14 17 22
15 17 21 27
12 13 15 18
15 16 19 22
18 20 23 27
100 200 300 400
ha ¼ x 3.79 2.85 1.89 1.05
3.66 2.87 1.91 1.04
3.85 2.79 1.79 1.20
1.79 1.66 1.26 0.94
1.76 1.66 1.38 1.08
1.87 1.66 1.31 1.03
1.53 1.42 1.18 0.94
1.56 1.35 1.14 1.08
1.57 1.42 1.20 1.03
100 200 300 400
Max E[TP] 3980 3473 2678 1747
4894 4091 3284 2241
5819 5183 4152 2731
5689 4865 3990 2828
7538 6231 4961 3587
9393 7597 6153 4431
7335 5987 4834 3580
9149 7359 6181 4528
11112 9304 7544 5575
100 200 300 400
L⁄ 19 20 19 18
22 23 23 22
27 28 27 29
18 20 19 18
23 25 25 25
30 30 29 29
20 20 19 18
25 23 23 25
30 30 29 29
f½C r þ C s GðsÞ ðn þ 1Þhðt d =ðn þ 1ÞÞ þ nC 0p ðt d =ðn þ 1ÞÞg=ðn þ 1Þ, then @E[TP]/@L changes its sign exactly once from positive to negative in the interval (0, td), and there exists a unique solution L⁄ such that @E½TP=@LjL¼L ¼ 0. h Theorem 3 shows that the optimal lease period L⁄ is unique when h0 (t) > 0, "t > 0 and C 00p ðxÞ P 0; 8x > 0. The optimal lease period L⁄ can be obtained by solving Eq. (13)
V Vd 1 L L W ½C r þ C s GðsÞðn þ 1Þh þ nC 0p nþ1 nþ1 nþ1 td ¼ 0:
equal to btd/sc. Using this upper bound, we can easily search for because n is an integer. the optimal PM number n⁄ between 0 and n Using the results obtained in Theorems 1–3, the following algorithm provides an efficient search procedure for the optimal PM policy and length of lease period. Algorithm. Step 1. Step 2.
ð13Þ
According to the results of Theorem 3, the expected total profit in Eq. (8) can be changed as
V Vd L E½TP ¼ L W L ½C r þ C s GðsÞðn þ 1ÞH td nþ1 L : nC p nþ1
Step 3.
¼ bt d =sc and max E[TP] = 0. Set n ¼ 1; n 0 n d If W VV t d 6 nþ1 C p ð0Þ, then the optimal lease
period L⁄ = 0, the optimal policy ðn ; x ; ha Þ ¼ ð0; 0; 0Þ; max E½TP ¼ 0 and STOP. n td 1 d If W VV t d P nþ1 ½C r þ C s GðsÞðn þ 1Þh nþ1 þ td g, then L = td and compute ha = x = td/ nC 0p nþ1 (n + 1). d Compute E½TP ¼ LW VV td L
ð14Þ
Finally, the rest of the decision variables in Eq. (14) is the number of PM n. Recall that the time required for performing a minimal repair should not exceed a tolerable time s and the life cycle of the equipment is td. From the boundary condition 0 6 ns 6 td, there is a trivial ¼ bt d =sc, which is the greatest integer less than or upper bound n
½C r þ C s GðsÞfnHðxÞþ HðL nxÞg nC p ðxÞ and go to Step 5. Step 4.
If
0 n nþ1 C p ð0Þ
1 nþ1
n
d < W V V td <
td td þ nC 0p nþ1 g, ½C r þ C s GðsÞðn þ 1Þh nþ1
then find L 2 (0, td) satisfying @E[TP]/@L = 0 and compute ha = x = L/(n + 1). d Compute E½TP ¼ LW VV td L
b=100
25
b=200
b=300
b=400 23
20
nstar
16
Step 5.
20
15
14
Step 6.
½C r þ C s GðsÞfnHðxÞþ HðL nxÞg nC p ðxÞ. If maxE[TP] < E[TP] then set max E½TP ¼ E½TP; n ; L ; x ; ha ¼ ðn; L; x; ha Þ and n = n + 1. Otherwise, set n = n + 1. then STOP. Otherwise, go to Step 3. If n > n
11
10 5
9
7
6
9 6
4
5
0 20
25
30
td Fig. 9. Marginal cost versus optimal PM number (b = 1.5).
5. Numerical examples Consider that the lifetime distribution of the photocopier follow b a two-parameter Weibull distribution f ðtÞ ¼ kbðktÞb1 eðktÞ for t P 0, where k is the scale parameter and b is the shape parameter, the expected life time is l = 1/kC(1 + 1/b). Moreover, we suppose
W.L. Chang, H.-C. Lo / Computers & Industrial Engineering 61 (2011) 489–496
b=100
b=200
b=300
3. When the marginal cost b increases, the number of PM increases, but maximum profit, PM degree and controlled-limit both decrease, under the life cycle of equipment given.
b=400
8000 5819
E [TP]
6000
4000
4894
5183
4091
4152
2678
3284
2731
1747
2241
3980 3473
2000
495
0 20
25
30
td Fig. 10. Marginal cost versus profit (b = 1.5).
the parameter k = 0.6 and the shape parameter b = 1.5, 2, 2.5, respectively. Following definition of a failure rate function, the failure rate function of the Weibull distribution is h(t) = kb(kt)b1. Let us consider that photocopier is leased with a length of lease period L (years) and the lease payment of the photocopier and the initial purchase prices of the photocopier are W = 800($) and V = 1000($), respectively. The life cycle of the photocopier is td (years) and residual value of the photocopier is Vd at time td. Suppose that the minimal repair cost Cr = 280($) and the repair time tr follows a Weibull(2, 1). There is a penalty cost Cs = 450($) with probability GðsÞ 0:49 to the lessor if the repair time exceeds a predetermined time s = 0.0083 (year) = 3 (days). Each failure of the photocopier within the lease period incurs an expected disbursement cost C r þ C s GðsÞ 500ð$Þ (including the minimal repair cost and possible penalty cost) and the cost of performing a PM action with maintenance degree x is Cp(x) = 50 + bx($). 5.1. Different residual value Suppose that the residual values of the leased equipment are Vd = 0($) and Vd = 20($) at time td = 20 (years). Table 1 summarizes the optimal maintenance policy ðn ; L ; x ; ha Þ according to different residual values. From Figs. 3–8, we derive following results: 1. When the marginal PM cost b increases, the number of PM increases, but the PM degree, controlled-limit, and maximum profit E[TP] all decrease. 2. When b increases, the number of PM and maximum profit both increase, but the PM degree and controlled-limit decrease. 3. The number of PM and maximum profit under the considered residual value are higher than under the non-considered residual value. 4. The PM degree and controlled-limit under the considered residual value are lower than under the non-considered residual value. 5.2. Different life cycle Suppose that the residual value of the leased equipment and the life cycle of equipment are Vd = 20($) and td = 20, 25, 30 (years), respectively. Table 2 summarizes the optimal maintenance policy and maximum profit under different life cycle. From Figs. 9, 10 and Table 2, we have some results as follows. 1. When the life cycle of the equipment td increases, the number of PM and maximum profit E[TP] both increase. 2. Under the life cycle of equipment given, when b increases, the number of PM and maximum profit both increase, but PM degree and controlled-limit both decrease.
6. Conclusions In this paper, the age-reduction method is adopted to describe the degree of preventive maintenance and a mathematical model of profit is constructed for the leased equipment within the lease period. From the model of profit, we show that there exists a unique optimal length of lease period and maintenance policy within the lease period such that the expected total profit is maximized. Within the lease period, the equipment should be restored to its original state after each preventive maintenance action. That is, preventive maintenance actions should be performed at time epochs L/(n + 1), 2L/(n + 1), 3L/(n + 1), . . . , nL/(n + 1) with the same maintenance degree L/(n + 1). When the preventive maintenance cost b or b increases, the number of preventive maintenance actions n increases but preventive maintenance degree, controlledlimit and maximum profit decrease. The preventive maintenance number and maximum profit under the considered residual value are higher than the non-considered residual value. Under the equipment with residual value, the expected total profit is maximized. Further, two-phase maintenance of equipment and the interest rate of the company within the lease period are investigated for various maintenance plans. References Aras, N., Güllü, R., & Yürülmez, S. (2010). Optimal inventory and pricing policies for remanufacturable leased products. International Journal of Production Economics. Published online. Barlow, R. E., & Hunter, L. C. (1960). Optimum preventive mathematical policies. Operations Research, 8, 90–100. Boland, P. J., & Proschan, F. (1982). Periodic replacement with increasing minimal repair costs at failure. Operations Research, 30, 1183–1189. Chiu, S. W., & Chiu, Y. S. (2006). Mathematical modeling for production system with backlogging and failure in repair. Journal of Scientific & Industrial Research, 65, 499–506. Huang, S., & Yang, Y. (2002). Pricing leased contracts with options in imperfect markets of durable goods. Technical Report, Ford Research Laboratory, Dearborn. Hu, F., & Zong, Q. (2008). Optimal periodic preventive maintenance policy and lease period for leased equipment. Journal of Tianjin University Science and Technology, 41, 248–253. Jack, N., & Dagpunar, J. S. (1994). An optimal imperfect maintenance policy over a warranty period. Microelectronics and Reliability, 34, 529–534. Jaturonnatee, J., Murthy, D. N. P., & Boondiskulchok, R. (2006). Optimal preventive maintenance of leased equipment with corrective minimal repairs. European Journal of Operational Research, 174, 201–215. Jhang, J. P. (2005). A study of the optimal use period and number of minimal repairs of a repairable product after the warranty expires. International Journal of Systems Science, 36, 697–704. Nakagawa, T. (1979). Imperfect preventive-maintenance. Journal of the Operations Research Society of Japan, 24, 213–227. Nakagawa, T. (1981). A summary of periodic replacement with minimal repair at failure. Journal of the Operations Research Society of Japan, 24, 213–227. Nakagawa, T., & Kowada, M. (1983). Analysis of a system with minimal repair and its application to replacement policy. European Journal of Operational Research, 12, 176–182. Nisbet, A., & Ward, A. (2001). Radiotherapy equipment – Purchase or lease? The British Journal of Radiology, 74, 735–744. Pham, H., & Wang, H. (1996). Imperfect maintenance. European Journal of Operational Research, 94, 425–438. Pongpech, J., & Murthy, D. N. P. (2006). Optimal periodic preventive maintenance policy for leased equipment. Reliability Engineering & System Safety, 91, 772–777. Pongpech, J., Murthy, D. N. P., & Boondiskulchock, R. M. (2006). Maintenance strategies for used equipment under lease. Journal of Quality in Maintenance Engineering, 12, 52–67. Sheu, S. H. (1991). Periodic replacement with minimal repair at failure and general random repair cost for a multi-unit system. Microelectronics Reliability, 31, 1019–1025. Valdez-Flores, C., & Feldman, R. M. (1989). A survey of preventive maintenance models for stochastically deteriorating single-unit systems. Naval Research Logistics, 36, 419–446.
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Wen Liang Chang is currently an assistant professor in the Department of Information Management at Cardinal Tien College of Healthcare & Management, Taipei, Taiwan. He received his Ph.D. in the Department of Industrial Management from National Taiwan University of Science and Technology in 2007. His research interests include reliability theory, warranty cost analysis, applied probability and maintenance policy of leased equipment. He has published academic articles in Mathematical and Computer Modelling, Computers & Industrial Engineering, Annals of Operations Research, Journal of Quality of Chinese Society for Quality. Hui-Chiung Lo is currently an associate professor in the Department of Business Administration at the Tamkang University. She received her Ph.D. Department of Industrial Management from National Taiwan University of Science and Technology. Her research interests include reliability theory, warranty cost analysis, and applied probability. She has published academic articles in European Journal of Operational Research, Computers & Industrial Engineering, Annals of Operations Research.