Optimal replacement policy for a deteriorating system with increasing repair times

Applied Mathematical Modelling 37 (2013) 9768–9775

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Optimal replacement policy for a deteriorating system with increasing repair times Shengliang Zong, Guorong Chai, Zhe George Zhang ⇑, Lei Zhao School of Management, Lanzhou University, Lanzhou 730000, PR China

a r t i c l e

i n f o

Article history: Received 19 April 2012 Received in revised form 30 March 2013 Accepted 24 May 2013 Available online 6 June 2013 Keywords: Replacement Poisson process Geometric repair process d-Shock model Finite search

a b s t r a c t This paper considers an optimal maintenance policy for a practical and reparable deteriorating system subject to random shocks. Modeling the repair time by a geometric process and the failure mechanism by a generalized d-shock process, we develop an explicit expression of the long-term average cost per time unit for the system under a thresholdtype replacement policy. Based on this average cost function, we propose a finite search algorithm to locate the optimal replacement policy N⁄ to minimize the average cost rate. We further prove that the optimal policy N⁄ is unique and present some numerical examples. Many practical systems fit the model developed in this paper. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction A critical machine or facility failure may interrupt the production of a manufacturing firm. Such an interruption will have negative impacts on a firm’s performance such as the revenue and customer service. To minimize the negative effects of machine failures, practitioners are interested in finding the appropriate machine maintenance and replacement policies. This paper addresses this issue. Usually, a machine, also called a system, experience two stages consecutively before its replacement- an operating stage (productive) and a repair stage (non-productive). We model both stages and consider a threshold replacement policy for the system subject to random shocks. There are extensive studies on the maintenance problems for the system with operating and repair stages. This is mainly because that some classical assumptions are not realistic in modeling the real systems. These assumptions include repaired system becoming ‘‘as good as new’’ and a failed system being replaced by a new one immediately. Barlow and Proschan [1] introduced an imperfect repair model, where the repair is prefect with probability p and minimal with probability 1  p. Other studies along this line include Block et al. [2], Kijima [3], Makis and Jardine [4], Dekker [5], Moustafa et al. [6], Sheu et al. [7], Wang and Zhang [8], Zhang and Wang [9], and Yuan and Xu [10]. Recently, shock models were utilized to model the operating time. Conceptually, a system fails due to the shock effect on the system. While most of existing shock models were based on the accumulated or extreme damage causing a system failure, Li [11] first introduced the d-shock model to avoid measuring amount of damage which may not be easy in many situations. Therefore, the d-shock model focuses mainly on the frequency of shocks rather than the magnitude of shocks. In the d-shock model, if the time interval between two successive shocks is smaller than a threshold value d, the system fails. Thus the operating time is equal to the time to failure caused by a shock that follows the previous shock too closely. The ⇑ Corresponding author. Tel.: +1 360 650 2867. E-mail address: [email protected] (Z.G. Zhang). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.05.019

S. Zong et al. / Applied Mathematical Modelling 37 (2013) 9768–9775

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threshold d is usually set to be a constant. In this paper, we adopt a general d-shock model by letting d be an exponentially distributed random variable with parameter varying with number of repairs. It is with noting that Lam and Zhang [12] studied a general d-shock model for both repairable improving and deteriorating systems and found the optimal N⁄ replacement policy. We also consider the non-zero repair and replacement times in contrast to a classical assumption that that repairs are instantaneous, see [13,14]. In most practical situations, to reflect the aging process of the system, the consecutive repair times are assumed to become longer and longer till the system is replaced with a new one according to some replacement rule. Lam [15,16] first introduced the geometric processes (GP) to study the maintenance for such a deteriorating system. Finkelstein [17] generalized Lam’s work based on a scale transformation after each repair. Zhang and Wang [18] applied the GP repair model to analyze a system consisting of two components (working and standby) and one repairman. They further found the optimal replacement policy N⁄ for component 1. Other works modeling the repair times based on the GP include Stanley [19], Francis [20], Lam and Tse [21], Zhang and Wang [22], Chen et al. [23], Leung et al. [24], Yuan and Xu [10]. In this paper, we also model the repair times by the GP. This paper is organized as follows. Section 2 presents the assumptions and definitions. Section 3 develops the long-run average cost per unit time. In Section 4, we discuss the optimal replacement policy. Numerical examples are provided in Section 5. 2. Model definitions and assumptions In this section, we make the assumptions for our model. To formulate the d-shock model, we need the geometric process first introduced in Lam [14,15]. Definition A stochastic process fnn ; n ¼ 1; 2; . . .g is called a geometric increasing or decreasing process if there exists a real number a (0 < a 6 1 or a > 1), thus, fan1 nn ; n ¼ 1; 2; . . .g forms a new renewal process. The real a is called the ratio of the geometric process (GP). Letting E(n1) = s and Var(n1) = r2, we have E(nn) = s/an1 and Var(nn) = r2/a2(n1). Therefore, a GP has three parameters of a,

s and r2. Our model is based on the following assumptions. Assumptions: (1) At t = 0, a new system is installed. Whenever the system fails, it is either repaired or replaced with a new one. (2) Shocks arrive according to a Poisson process with rate k1 or EX i ¼ 1=k1 , where Xi is the ith inter-arrival time of two consecutive shocks. Let di be another exponentially distributed random variable associated with Xi. We assume that the sequence fdi ; i ¼ 1; 2; . . .g forms an increasing geometric process with 0 < a 6 1. Then di has cumulative distribution function Q(ai1x), where Q(x) is the cumulative distribution function of d1. {Xi, di} follows a d-shock model if the system fails at i th shock which satisfies X i 6 di , and then the life time or equivalently the operating time is the sum of all Xi until the one satisfying the above condition. Further, we assume that Xi is independent of di. (3) Let Tn be the operating time after the (n  1) th repair. fT n ; n ¼ 1; 2; . . .g is a stochastically decreasing random variable sequence induced by the d-shock model. (4) Let Yn be the repair time after the n th failure and forms an increasing geometric process with 0 < b 6 1. Then Yn has cumulative distribution function G(bn1y), where G(y) is the cumulative distribution function of Y1 with EY1 = l > 0. (5) Tn and Yn, n ¼ 1; 2; . . . are two independent sequences. (6) Assume that the repair cost rate is c, the operating reward rate is r, and the replacement cost consists of fixed cost R and variable cost v = rpZ, where Z is the replacement time and rp is the rate of cost per time unit during replacement. Let E(Z) = t. (7) A threshold N replacement policy is adopted. Under such a policy, the system will be replaced with a new one after it fails for N times. Assumptions (2)–(4) are quite realistic for modeling a deteriorating system in practice. Due to aging process and accumulated wearing, it is reasonable to assume that the successive operating times after repairs will become shorter and shorter, and the corresponding repair times of the system will become longer and longer. Such a behavior is also supported by the empirical studies. Lam [25], and Lam and Chan [26] applied GP models to fit real date sets by using nonparametric and parametric methods, respectively. Furthermore, Lam et al. [27], Chan and Leung [28] and Aydogdu et al. [29] conducted more empirical studies in this area. Lam [15] considered two kinds of replacement policies for a general repair replacement model. One is the T-replacement policy, under which the system is replaced whenever its total operating time reaches T since the last replacement, and the other one is the N-replacement policy, under which the system is replaced at the N th failure instant since the last replacement. For minimizing the long-run average cost or equivalently maximizing the long-run average reward case, Stadje and Zuckerman [30] and Lam [31] showed that under some mild conditions, the optimal N⁄ policy is at least as good as the optimal T⁄ policy. Furthermore, Lam[31] proved that an optimal N⁄ policy is easier to implement than an optimal T⁄ policy. Therefore, we focus on the N replacement policy in our model. In addition, Assumption (6) makes our model more general and allows us to evaluate the economic impact of new system delivery and replacement time on the maintenance of the system.

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3. Long-run average cost per unit time Now we develop the average cost function of the N -replacement policy under the imposed cost structure. Let C(N) be the long-run average cost per unit time of the system. According to renewal reward theorem (see [32]).

CðNÞ ¼

Expected cost incurred in a cycle Expected length in a cycle

ð1Þ

Now, let W be the length of a renewal cycle under N-replacement policy. Thus, we have

W¼

N N1 X X Tj þ Y j þ Z: j¼1

ð2Þ

j¼1

To evaluate the expected cost in a cycle, we first calculate E(Tn), the expected operating time of the system after the (n  1)th failure. Let lni be the inter-arrival time between the (i  1)th and i th shock following the (n  1)th repair, where i ¼ 1; 2; . . . Define

 Mn ¼ minfmln1 > an1 d1 ; . . . ; lnðm1Þ > an1 d1 ; lnm < an1 d1 g and

Tn ¼

Mn X

lni :

i¼1

Thus Mn denotes the number of shocks till the first deadly shock occurs. Obviously, Mn has a geometric distribution, with

PðM n ¼ kÞ ¼ qk1 n pn ; k ¼ 1; 2; . . . ; where pn is the probability of a shock, following the (n  1)th repair and qn = 1  pn. Therefore, we have EMn = 1/pn. As Mn is a stopping time with respect to the random sequence flni ; i ¼ 1; 2; . . .g, which are independent identically distributed random variables. Using Wald equation [32], we have Mn X EðT n Þ ¼ E lni

! ¼ Eln1 EMn ¼

i¼1

Eln1 pn

According to Assumption (2), as F(x) and Q(x) are all exponentially distributed, we have

FðxÞ ¼ 1  ek1 x ; x P 0;

Qðan1 xÞ ¼ 1  ea

n1 k

2x

; xP0

ð3Þ

and

Eln1 ¼

Z

1

xdFðxÞ ¼

Z

0

1

xdð1  ek1 x Þ ¼

0

1 : k1

Furthermore, as lni and dn (an1d1) are independent and have the marginal exponential distributions with means of 1=k1 and 1=an1 k2 , respectively. Therefore, we obtain

pn ¼ Pðlni < dn Þ ¼

Z

1

ea

n1 k x 2

k1 ek1 x dx ¼ k1

0

Z 0

1

eða

n1 k þk Þx 2 1

dx ¼

k1 : k1 þ an1 k2

ð4Þ

and

fn ¼ EðT n Þ ¼

k1 þ an1 k2 k21

:

ð5Þ

Consequently, N X E Tn

! ¼

n¼1

N N X X k1 þ an1 k2 EðT n Þ ¼ : k21 n¼1 n¼1

ð6Þ

On the other hand, since Y n ; n ¼ 1; 2; . . . is an increasing GP process with ratio 0 < b 6 1, we have

EðY n Þ ¼

l b

n1

:

Then, by Eq. (1), the long-run average cost C(N) of the system under the policy N is given by

ð7Þ

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S. Zong et al. / Applied Mathematical Modelling 37 (2013) 9768–9775

 P  PN P PN E c N1 n¼1 Y n  r n¼1 T n þ R þ r p Z c N1 n¼1 EðY n Þ  r n¼1 EðT n Þ þ ER þ Eðr p ZÞ P  ¼ CðNÞ ¼ PN PN1 PN1 N EðY Þ þ n E n¼1 n¼1 EðT n Þ þ EðZÞ n¼1 T n þ n¼1 Y n þ Z P PN k1 þan1 k2 l c N1 þ R þ rp t n¼1 bn1  r n¼1 k21 : ¼ PN k1 þan1 k2 PN1 l þ n¼1 bn1 þ t n¼1 k2

ð8Þ

1

The optimal replacement policy N⁄ can be determined by minimizing C(N). In the following section, we will show theoretically that the optimal N⁄ is unique. 4. The optimal replacement policy N⁄ To determine the optimal N⁄, Eq. (8) can be re-written as.

P l ðc þ rÞ N1 n¼1 n1 þ R þ ðr p þ rÞt CðNÞ ¼ PN k þan1bk  r: P l 1 2 þ N1 n¼1 n¼1 bn1 þ t k2

ð9Þ

1

Thus, to minimize C(N) is equivalent to minimize the first term of (9) denoted by B(N)

P l ðc þ rÞ N1 n¼1 n1 þ R þ ðr p þ rÞt BðNÞ ¼ PN k þan1bk : P l 1 2 þ N1 n¼1 n¼1 bn1 þ t k2

ð10Þ

1

Now, we study the difference between B(N + 1) and B(N). Let

f ðNÞ ¼

N X k1 þ an1 k2 n¼1

k21

þ

N1 X

l

þ t;

n1

n¼1 b

then

BðN þ 1Þ  BðNÞ ¼

1 b

N1

f ðN þ 1Þf ðNÞ

N N1 X X Nn fn  fNþ1 b þt

fðc þ rÞl

n¼1

! N1

 ½R þ ðr p þ rÞtðfNþ1 b

þ lÞg

ð11Þ

n¼1

See Appendix A.1 for the detailed calculation of B(N + 1)  B(N). Define the auxiliary function, A(N) as follow,

AðNÞ ¼

P  PN1 Nn N ðc þ rÞl þt n¼1 fn  fNþ1 n¼1 b N1

½R þ ðr p þ rÞtðfNþ1 b

þ lÞ

ð12Þ

:

As the denominator of B(N + 1)  B(N) is always positive, it is clear that the sign of B(N + 1)  B(N) is the same as that of its numerator. Thus, we have

BðN þ 1ÞSBðNÞ () AðNÞS1: Furthermore, it is clearly that,

AðN þ 1Þ  AðNÞ ¼

ðc þ rÞl

P Nþ1

n¼1 fn

 fNþ2

PN

n¼1 b

Nþ1n

þt



P  PN1 Nn N ðc þ rÞl þt n¼1 fn  fNþ1 n¼1 b

 N N1 ðR þ ðr p þ rÞtÞðfNþ2 b þ lÞ þ lÞ ðR þ ðr p þ rÞtÞðfNþ1 b P  P 1n Nþ1 N ðc þ rÞlðfNþ1  bfNþ2 Þ þt n¼1 fn þ l n¼1 b ¼ : N N1 þ lÞ ðR þ ðrp þ rÞtÞðfNþ2 b þ lÞðfNþ1 b

ð13Þ

See Appendix A.2 for detailed calculation of A(N + 1)  A(N). 0 < a 6 1 and Eq. (5) imply that fn is a decreasing function (or non-increasing function) in n. Meanwhile, 0 < b 6 1, so fNþ1 P fNþ2 P bfNþ2 .Then, we have AðN þ 1Þ P AðNÞ. Therefore, for any integer N, this indicates that A(N) is an increasing function (or non-decreasing function). Then the optimal N⁄ can be determined by

N ¼ minfNjAðNÞ P 1g: ⁄

ð14Þ ⁄

⁄

Furthermore, if A(N ) > 1 for certain N , then the optimal N is unique. Because A(N) is non-decreasing in N, there exists an integer N⁄, thus

AðNÞ P 1

()

N P N

AðNÞ < 1

()

N < N :

and

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S. Zong et al. / Applied Mathematical Modelling 37 (2013) 9768–9775

Fig. 1. The plot of A(N) against N. Table 1 Results of A(N) Obtained from Eq. (12). N

A(N)

N

A(N)

N

A(N)

N

A(N)

N

A(N)

N

A(N)

1 2 3 4 5 6 7 8 9 10 11

0.0025 0.0029 0.0035 0.0044 0.0056 0.0071 0.0090 0.0113 0.0141 0.0174 0.0213

12 13 14 15 16 17 18 19 20 21 22

0.0258 0.0311 0.0372 0.0441 0.0520 0.0610 0.0711 0.0825 0.0954 0.1098 0.1260

23 24 25 26 27 28 29 30 31 32 33

0.1440 0.1641 0.1865 0.2114 0.2389 0.2694 0.3031 0.3404 0.3814 0.4267 0.4765

34 35 36 37 38 39 40 41 42 43 44

0.5313 0.5915 0.6575 0.7299 0.8092 0.8960 0.9910 1.0948 1.2082 1.3320 1.4670

45 46 47 48 49 50 51 52 53 54 55

1.6142 1.7745 1.9491 2.1391 2.3458 2.5704 2.8144 3.0794 3.3669 3.6788 4.0169

56 57 58 59 60 62 64 65 66 68 69

4.3833 4.7801 5.2096 5.6743 6.1768 7.3068 8.6242 9.3619 10.1572 11.9373 12.9310

Table 2 Result of C(N) obtained from Eq. (8). N

C(N)

N

C(N)

N

C(N)

N

C(N)

N

C(N)

N

C(N)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

2.4594 2.7271 2.8163 2.8609 2.8877 2.9055 2.9182 2.9277 2.9351 2.9410 2.9458 2.9498 2.9531 2.9560

15 16 17 18 19 20 21 22 23 24 25 26 27 28

2.9585 2.9606 2.9625 2.9641 2.9656 2.9669 2.9680 2.9691 2.9700 2.9708 2.9716 2.9723 2.9729 2.9734

29 30 31 32 33 34 35 36 37 38 39 40 41 42

2.9739 2.9743 2.9747 2.9751 2.9753 2.9756 2.9758 2.9760 2.9761 2.9762 2.9763 2.97631 2.97634 2.97631

43 44 45 46 47 48 49 50 51 52 53 54 55 56

2.97624 2.97615 2.9760 2.9759 2.9757 2.9754 2.9752 2.9749 2.9745 2.9742 2.9737 2.9733 2.9728 2.9722

57 58 59 60 61 62 63 66 69 75 90 100 150 200

2.9717 2.9710 2.9703 2.9696 2.9688 2.9680 2.9671 2.9640 2.9602 2.9503 2.9039 2.8429 1.2942 1.5177

250 280 290 300 310 320 330 340 350 355 360 362 400 450

1.9718 1.9952 1.9973 1.9985 1.9992 1.9996 1.9998 1.9999 1.9999 1.9999 2.0000 2.0000 2.0000 2.0000

Then, N⁄ is the minimum integer that satisfies Eq. (14). In particular, if AðNÞ P 1 for all N, then we have N⁄ = 1. This means that the optimal replacement policy is to replace the machine immediately when it fails. If A(1) exist and Að1Þ 6 1, then we have N⁄ = 1. This means that the system will never be replaced. 5. Numerical example In this section, we present a numerical example to demonstrate the determination of the optimal replacement policy. Consider a system with the following parameters: a = 0.98, b = 0.94, c = 2, r = 3, R = 8000, t = 20, rp = 3, l = 4, k1 =0.0002 and k2 =0.0004. Using Eq. (12), we plot A(N) in Fig. 1. From this non-decreasing function, we find that A (41) = 1.0948 and

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S. Zong et al. / Applied Mathematical Modelling 37 (2013) 9768–9775

Fig. 2. The plot of C(N) against N.

Table 3 Optimal N⁄ and C(N⁄) for different values of a; b; l; k1 ; k2 and t. a

C(N⁄)

N⁄

b

C(N⁄)

N⁄

l

C(N⁄)

N⁄

k1

C(N⁄)

N⁄

k2

C(N⁄)

N⁄

t

C(N⁄)

N⁄

0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

2.9623 2.9635 2.9649 2.9664 2.9681 2.9699 2.9719 2.9740 2.9763 2.9788

38 38 38 38 38 38 39 40 41 42

0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

2.9691 2.9708 2.9726 2.9744 2.9763 2.9783 2.9804 2.9826 2.9851 2.9880

28 30 33 36 41 46 53 64 83 126

1.0 2.0 3.0 3.5 4.0 5.0 6.0 8.0 9.0 10.0

2.9832 2.9803 2.9782 2.9772 2.9763 2.9747 2.9733 2.9706 2.9694 2.9682

55 48 43 42 41 38 37 34 33 32

0.00001 0.00005 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0009 0.001

2.9999 2.9978 2.9925 2.9763 2.9561 2.9336 2.9098 2.8852 2.8089 2.7831

38 39 40 41 41 42 42 42 43 43

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

2.9583 2.9667 2.9723 2.9763 2.9793 2.9817 2.9835 2.9850 2.9863 2.9873

43 42 41 41 40 40 40 40 39 39

10 15 20 30 40 50 60 80 90 100

2.9765 2.9764 2.9763 2.9762 2.9761 2.9760 2.9758 2.9756 2.9755 2.9754

41 41 41 41 41 41 41 41 41 41

41 is the smallest integer such that AðNÞ P 1 as shown in Table 1. Therefore, Eq. (14) indicates that the optimal threshold is N⁄ = 41. Therefore, we should replace the system at the time of the 41th failure. This is consistent with the result by searching over N based on C(N) presented in Table 2 and Fig. 2. It shows that C (41) = 2.97634 is the unique minimum average cost rate of the system. Furthermore, we can also examine the impacts of the parameters a; b; l; k1 ; k2 and t on the optimal solution. We present the optimal replacement policy of N⁄ and the minimum average cost rate C(N⁄) for by changing one of parameters ‘‘ a; b; l; k1 ; k2 and t00 at a time in Table 3. Table 3 shows that N⁄ has a positive relationship with a; b and k1 , while a negative relationship with l and k2 . Meanwhile, the minimum average cost rate C(N⁄) has a positive relationship with l; t and k1 , and a negative relationship with a; b and k2 . Eq. (5) indicates that the system operating time Tn after (n  1)th repair is increasing with a, which also means that the operating reward is increasing with a. Therefore, N⁄ will increase with a. Similarly, we can quantify the relationships of N⁄ and C(N⁄) with b; l and k2 . For example, from Eq. (8), the expected cost and cycle length all increase in the replacement t. This information is useful when the manager evaluates the value of the fast delivery of the new machine from a supplier. Furthermore, it is obvious that k1 and k2 both have significant effects on C(N⁄) and N⁄. 6. Conclusions In this paper, we studied the optimal replacement policy for a reparable and deteriorating system. Using a d-shock model, we obtained the optimal replacement policy N⁄ by minimizing the average cost rate C(N). We showed the uniqueness of the optimal replacement policy N⁄ and also presented a numerical example. Furthermore, we conducted the sensitivity analysis for some parameters and investigate the effects of these parameters on the optimal replacement policy N⁄ and the average cost rate C(N⁄). Our model provides a useful quantitative tool for managers to evaluate the system performance and design an optimal maintenance policy. In this paper, the shock process is assumed to follow a Poisson process. Extending such a shock process into a more general non-homogeneous Poisson process or renewal process can be a good topic for future research.

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Acknowledgments The authors are grateful to the editor and referees for their useful comments and constructive suggestions which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (Nos. 71072070, 70702013) and Fundamental Research Funds for the Central Universities of China (Nos. 11LZUJBWZY098, 09LZUJBWZY007). Appendix A Calculation. of Eq. (11)

BðN þ 1Þ  BðNÞ PN l PN1 l ðcþrÞ þRþðrp þrÞt ðcþrÞ þRþðrp þrÞt n1 n1  PN n¼1 bPN1 l ¼ PNþ1n¼1 bPN l f þ n¼1 n

n¼1 bn1

("

þt

f þ n¼1 n

n¼1 bn1

þt

N X l ðc þ rÞ þ R þ ðr p þ rÞt bn1

1 ¼ f ðNþ1Þf ðNÞ

#

N N1 X X l fn þ þt bn1

n¼1

n¼1

n¼1

N1 Nþ1 N X X X l l f þ þt ½ðc þ rÞ n1 þ R þ ðr p þ rÞt n b bn1 n¼1

n¼1

(

1 ðc þ rÞl ¼ f ðNþ1Þf ðNÞ

N X n¼1

þðc þ rÞlt

N X n¼1

ðc þ rÞl

n¼1

ðc þ rÞlt

n¼1

N N N1 X X X 1 2 1 1 þ ðc þ rÞ f l n1 n1 n b b bn1

N X

1 b

n1

n¼1

n¼1

n¼1

!

n¼1

1 bn1

N X n¼1

"

n¼1

1 fðc þ rÞl ¼ bN1 f ðNþ1Þf ðNÞ

PN

n¼1 fn  fNþ1

N1 X Nn b þt

N1 X

1 b

N X fn þ fNþ1

 ½R þ ðr p þ rÞt

!

n¼1

 ðc þ rÞl2

fn þ fNþ1

n¼1

N1 X n¼1

!)

N N1 X X l 1 f þ þt n1 þ ½R þ ðr p þ rÞt n b bn1

N1 X

!

n1

!

n¼1

1 bn1

N1 X l l þ bN1 þ bn1

!

#) þ tÞ

n¼1

!

 ½R þ ðr p þ rÞtðfNþ1 b

N1

þ lÞg

n¼1

Calculation. of Eq. (13)

AðN þ 1Þ  AðNÞ ¼

P  PN Nþ1n Nþ1 ðc þ rÞl þt n¼1 fn  fNþ2 n¼1 b N

ðR þ ðr p þ rÞtÞðfNþ2 b þ lÞ N



¼

n¼1 fn

 fNþ1

PN1

Nn n¼1 b

þt



þ lÞ ðR þ ðr p þ rÞtÞðfNþ1 b ! N X Nþ1n N1 fn  fNþ2 b þ t ðfNþ1 b þ lÞ

N þ1 X

N1

ðR þ ðr p þ rÞtÞðfNþ2 b þ lÞðfNþ1 b þ lÞ ! ) N N 1 X X Nn N fn  fNþ1 b þ t ðfNþ2 b þ lÞ  n¼1

P N

N1

(

ðc þ rÞl

¼

ðc þ rÞl

n¼1

n¼1

n¼1

ðc þ rÞl N

N1

ðR þ ðr p þ rÞtÞðfNþ2 b þ lÞðfNþ1 b þ lÞ ( Nþ1 Nþ1 N N X X X X N1 N1 Nþ1n Nþ1n N1 fn þ l fn  fNþ1 b fNþ2 b  lfNþ2 b þ tfNþ1 b  fNþ1 b n¼1

n¼1

n¼1

n¼1

n¼1

n¼1

) N N N1 N1 X X X X N Nn Nn N fn  l fn þ fNþ1 fNþ2 b b þ lfNþ1 b  tfNþ2 b  t l þtl  fNþ2 b N

¼

n¼1

P  PN 1n Nþ1 ðc þ rÞlðfNþ1  bfNþ2 Þ þt n¼1 fn þ l n¼1 b N

ðR þ ðrp þ rÞtÞðfNþ2 b þ lÞðfNþ1 b

N1

þ lÞ

n¼1

S. Zong et al. / Applied Mathematical Modelling 37 (2013) 9768–9775

9775

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