Joint Optimal Upgrade level and Imperfect Preventive

Joint Optimal Upgrade level and Imperfect Preventive Maintenance Strategy for Refreshed End-of-life Systems A. Khatab1 , C. Diallo2 , D. Ait-Kadi3 1 Ecole

Nationale d’Ingenieurs de Metz. Metz, France. 2 Dalhousie University. Canada 3 Universite Laval, Quebec (Qc) Canada. E-mail addresses: [email protected], [email protected], [email protected]

Abstract This paper investigates the relationship between rejunenation/upgrade decisions of recovered end-of-life systems and the subsequent maintenance costs incurred during their second life as refreshed products. A mathematical model for the joint determination of the optimal upgrade level and imperfect preventive maintenance strategy is developed. A numerical example is provided to illustrate the validity of the model.

1

Introduction

The introduction of legislation to force the collection and recovery of end-of-life products, and electronics in particular, has generated an increasingly sustainable source for parts and products that can be reconditioned/refreshed to be reused on assembly lines or in maintenance activities. This new source for components raises many interesting research problems some of which have been extensively studied by researchers. Closed-loop supply chain design and management models have flourished during the past decade. The area of optimal inventory control for returned products has been well covered (see (Fleischmann, van Nunen & Grave 2003), (Fleischmann, Galbreth & Tagaras 2010), and (Srivastava 2007)). Remanufacturing production planning and supply chain decision-making issues have also been extensively covered (see (Ait-Kadi, Chouinard, Marcotte & Riopel 2012), (Richter & Sombrutzki 2000), and (Vlachos, Georgiadisa & Iakovoua 2007)). However, there are key areas that have been neglected and these cover the particularly important areas of quality, reliability, maintenance engineering and warranty policies for remanufactured systems that will be reintroduced into the market as second-hand or refreshed products.

In order to generate demand for reconditioned or second-hand products, manufacturers or dealers/brokers have had to resort to a combination of initiatives to promote and infer the quality of their products. These initiatives include significant price reductions, generous warranty coverage (same coverage length as new systems, free preventive maintenance in the first year of the refreshed system), upgrades of recovered systems before resale.

(Yeh, Lo & Yu 2011) propose two periodical age reduction PM models for a secondhand product with known age and a pre-specified length of usage. Their objective is to obtain the optimal number of times of PM action and the optimal degree of each PM action such that the total expected maintenance cost is minimized. (Pongpech, Murthy & Boondiskulchock 2006) propose a mathematical model to determine the optimal upgrade and preventive maintenance actions that minimize the total expected cost (maintenance costs+penalty costs) for used equipment under lease. In recent years, the literature on warranty models for second-hand products have started to emerge. (Shafiee & Chukova 2013) provide a current summary of warranty models developed in this area. They propose a three-parameter optimization model to determine the optimal upgrade strategy, warranty policy and sale price of a second-hand product to maximize the dealer’s expected profit. (Chari, Diallo & Venkatadri 2013) develop a mathematical model for the optimal one-dimensional unlimited free-replacement warranty policy with replacements carried out with reconditioned products. (Lo & Yu 2013) develop a mathematical model for the determination of the optimal upgrade level and warranty length to maximize the expected profits for used products. They provide a practical application to used cars. (Naini & Shafiee 2011) propose a joint optimal price and upgrade level model for a warranted second-hand product. They present an application of their model to solve a second-hand electric drill remanufacturing problem.

After end-of-life (EOL) or end-of-use (EOU) products are recovered and dismantled to generate good components, upgrade activities are usually carried out to bring the recovered components to better condition and thus effectively reducing their age. The cost of this rejuvenating/refreshing action is proportional to the upgrade level carried out and is an expense that can increase the sale price of the reconditioned system. However, the age reduction provides better reliability which can translate into better performance during operational lifetime. In this paper, we propose a mathematical model to investigate the interactions between the upgrade level decisions, the optimal maintenance policy decisions and the total costs incurred during the lifetimes of these refreshed systems. The rest of the paper is organized as follows. The proposed mathematical model is presented in Section 2. All the cost components are defined and the total expected cost per unit of time for acquiring, rejuvenating and maintaining the system is developed. Numerical experiments are presented in Section 3. Conclusions and future 2

work are drawn in Section 4.

2

Model Development

An end-of-life system of age u, recovered from consumer markets, is purchased from a used-products broker for a unit fee CA (u). The system is then refreshed (upgraded, rejuvenated) at a cost Cup (α, u) that is a function of the age u of the system and the improvement factor in effective age α. The rejuvenated system is then put into operation and maintained over an exploitation horizon at the end of which it is replaced with a similar refreshed system. During its operation, the system is subjected to random failures. Thus, in order to extend its useful life, it undergoes imperfect maintenance actions. The system is preventively maintained (PM) whenever its reliability falls to a threshold level R0 and replaced with a refreshed one at the instant of the N th PM. Whenever a failure occurs between PM instants, minimal repair is performed. The length of the k th PM cycle is denoted by Xk with k = 1, 2, · · · , N . All PM actions are imperfect and modelled using the hybrid hazard rate approach proposed by (Lin, Zuo & Yam 2000).

2.1

Cost components of the optimization model

The objective is to find the optimal values of the decision variables (R0 , u, α, N )∗ which minimize the expected total cost per unit of time C(R0 , u, α, N ) over an infinite time horizon. This total cost is the sum of purchase and rejuvenation costs together with minimal repair, PM costs.

2.1.1

Acquisition cost

To model the acquisition cost of a second hand system, it is quite reasonable to assume that such a cost is a decreasing function with the age u of the system. In this paper, the acquisition cost CA (u) is defined to be: CA (u) =

C0 (1 + u)θ

(1)

where C0 = CA (0) is the cost of a new system and θ is a positive parameter.

2.1.2

Upgrade Cost

To improve the reliability of the second-hand system, an upgrade action is performed. After an upgrade operation, the age u of the system is reduced to αu, 3

where α ∈ [0, 1] is the improvement factor in effective age. An improvement factor in effective age α ((Malik 1979)) is equivalent to an upgrade level 1 − α ( (Shafiee & Chukova 2013)). The Cup (α, u) is defined to be a function of both α and u such that: (2) Cup (α, u) = Cup0 (1 − α)ε1 uε2 where parameters Cup0 , ε1 and ε2 are positive valued parameters and usually estimated from historical data. From Equation (2), one may observe that the upgrade cost increases as the upgrade level 1 − α increases. Indeed, it is more costly to bring a used system to a younger condition (low values of α) because more parts have to be replaced and more efforts are needed to access the parts and perform finishing activities. As a result, α = 1 implies that no upgrade operation is performed and consequently Cup (α, u) = 0, while α = 0 implies that the upgrade operation restores the system to a "as good as new" operating state. In this case, Cup (α, u) = Cup0 uε2 . Note also that the upgrade cost increases with the initial age u. This means that the older the system, the more expensive the upgrade operation is.

2.1.3

Maintenance cost

The system is designed to experience N maintenance cycles. At the end of the N th cycle, the system is replaced. The system undergoes PM whenever its reliability reaches a threshold level R0 . In the case where the system fails before reaching the threshold R0 , a minimal repair is then carried out. PM actions are imperfect and modelled on the basis of the hybrid hazard rate model proposed by (Lin et al. 2000). Durations of PM, minimal repair and replacement are assumed to be negligible. Costs of PM and minimal repair are denoted by Cpm and Cmr respectively. Let us denote by h(t) the initial failure rate of the system when it is new. Assume that at the end of a given interval, imperfect PM are carried out on the system. If the length of the k th time interval is denoted by Xk (k = 1, 2, . . .), then the failure rate hk+1 (t) of the system after the k th PM is defined for t ∈ [0, Xk+1 [ as: hk+1 (t) = ak hk (t + bk Yk )

(3)

where ak and bk stand, respectively, for the hazard rate increase coefficient (adjustment factor) and the age reduction coefficient such that 1 ≤ a1 < a2 < · · · and 0 ≤ b1 < b2 < · · · ≤ 1. For t ∈ [0, X1 [, h0 (t) = h(t) is the initial hazard rate of the system. Variable Yk represents the effective age of the system right before the k th PM. From the above equation, the hazard rate function hk (t), for k = 1, 2, . . . , and t ∈ [0, Xk [, can be rewritten as: hk (t) = Ak h (t + bk−1 Yk−1 ) , 4

(4)

where Ak =

k−1 Q

ai such that A1 = 1.

i=1

From Equation (4), the system reliability Rk (t) for the k th PM cycle is:

Rk (t) = exp −Ak 

= exp −Ak

Z t+bk−1 Yk−1

!

h(x)dx

(5)

bk−1 Yk−1

Z t 0



h (x + bk−1 Yk−1 ) dx .

If PM times Xk (k = 1, . . .) corresponds to instants where the system reliability reaches the threshold level R0 , it follows that:

R0 = exp −Ak = exp −Ak = exp −Ak

Z Xk 0

!

h (x + bk−1 Yk−1 ) dx

Z Xk +bk−1 Yk−1

(6)

!

h(x)dx

bk−1 Yk−1

!

Z Yk

h(x)dx ,

bk−1 Yk−1

where Yk = Xk + bk−1 Yk−1 is the effective age of the system at the end of the k th PM cycle. From the above equation, the time instant Xk when to perform the k th PM can be obtained from !

Xk = H

−1

where the function H(t) =

ln(R0 ) − bk−1 Yk−1 , H(bk−1 Yk−1 ) − Ak Z t

(7)

h(x)dx is the cumulative hazard rate and H −1 (t)

0

denotes the inverse function of H(t). Dealing with the particular case where the system lifetime is assumed to be Weibull distributed with shape and scale parameters denoted, respectively, by β and η, PM time Xk given by Equation (7) now becomes: 

bk−1 Yk−1 Xk = η  η

!β

1

ln(R0 )  β − − bk−1 Yk−1 . Bk

(8)

The expected total maintenance cost CM (N, R0 ) induced by both PM and minimal repair costs during N PM cycles is:

CM (N, R0 ) = (N − 1)Cpm + Cmr

N X k=1

Ak

Z Yk bk−1 Yk−1

= (N − 1)Cpm − N Cmr ln (R0 ) 5

!

h(x)dx

(9)

where Ak

Z Yk

h(x)dx is the expected number of failure occurring during the

bk−1 Yk−1

k th PM cycle. Since the system is assumed to be of a real initial age u which is reduced by an upgrade action to αu, we can therefore set b0 = α and Y0 = u. The optimization model is formulated as a cost function, hereafter denoted by C, given on the basis of the above cost components and consisting of four decision variables, the reliability level R0 , the initial age u, the upgrade level α and the number N of PM cycles. The cost function C is then defined as:



C0 C(R0 , u, α, N ) =

1 1+u

θ

+ Cup0 (1 − α)ε1 uε2 + Cpm (N − 1) − N Cmr2 ln (R0 ) N −1 X

(1 − bk ) Yk + YN

k=1

(10) By minimizing the cost function C(R0 , u, α, N ), an optimal solution can be determined by means of optimal values R0∗ , u∗ , α∗ and N ∗ obtained by solving a system of equations composed of four partial derivatives corresponding to each decision variable. Unfortunately, the optimal solutions that minimize Equation (10) are in general difficult to obtain analytically even for simple lifetime distributions. Consequently, a numerical procedure is developed.

3

Numerical example

In this section, a numerical example is provided for a system whose lifetime follows a Weibull distribution with shape parameter β = 1.8 and scale parameter η = 4 (given in unit of time). Without loss of generality, we set cost C0 corresponding to the price of a new system equal to 1 and thus one only need to know the ratios Cmr Cup0 Cpm δ1 = , δ2 = and δ 3 = . Accordingly, the expected total cost is C0 Cpm C0 equivalently written as: 

C(R0 , u, α, N ) = C0

1 1+u

θ

+ δ 3 (1 − α)ε1 uε2 + δ 1 [(N − 1) − δ 2 N ln (R0 )] N −1 X

(1 − bk ) Yk + YN

k=1

(11) The other parameter values are arbitrary and considered for illustration purposes. Ratios δ 1 and δ 3 are chosen to be of values δ 1 = 250 and δ 3 = 0.5 and remain unchanged, while ratio δ 2 is made variable such that δ 2 ∈ {50, 100, 150, 200}. 6

Figure 1. Expected total cost versus the number of maintenance cycles: case of δ 2 = 50, and R0 = 75%.

Parameter θ of the acquisition cost function and parameters ε1 and ε2 of the upgrade cost function are set to θ = 0.2, ε1 = 0.01 and ε2 = 0.22. Finally, parameter values of the imperfect PM model, namely values of the adjustment factor ak and the age reduction factor bk (k = 1, . . .) are derived from the following formula: ak =

6k + 1 5k + 1

and

bk =

k . 10k + 1

(12)

First, let us assume that the manufacturer purchases a second hand system which must be operating with a level of reliability not less than 75%. This means that the system must undergo PM whenever its reliability reaches the threshold value R0 = 75%. The objective of the manufacturer is then to find the optimal values of the other three decision variables, namely the age u, the upgrade factor α and the number N of PM cycles. Figure (1) depicts the average total cost per unit of time versus the number N of PM cycles for δ 2 = 50, . The optimal acquisition age is found to be u∗75 = 3.5 (time unit) while the optimal upgrade factor is found to be ∗ ∗ of PM cycles is found to be N75 = 15. Thus, α∗75 = 0.44. The optimal number N75 the system is replaced after 14 PM actions.

Times to perform PM and the final replacement are given in Figure (2). The first PM is performed at time X1 = 2.63 time unit. From that time, the second PM is performed at X2 = 1.62, and so on. The replacement of the system is done X15 = 0.46 unit of time after the 14th PM. From Figure (2), one may conclude that time intervals between PM and replacement actions decreases as the number N increases. This classical result is due to the fact that a PM action consists not only in reducing the effective age of the system but also on increasing its failure rate. Thus, the more the system ages the more frequent the PM actions are. In this case, the overall acquisition, upgrade and maintenance costs induce an average total cost per time unit C(R0 , u∗ , α∗ , N ∗ ) = 0.15 (see Figure 1). 7

Figure 2. Duration of PM cycle: case of R0 = 75% and δ 2 = 50.

The present work seeks, however, to ensure a balance between four decision variables which are the reliability threshold R0 , the initial acquisition age u, the level of upgrade α and the number of PM actions to minimize the expected total cost per unit of time. For each value of δ 2 and by varying simultaneously these decision variables, Tables (1) and (2) give, respectively, the optimal values of decisions variables and the corresponding PM and replacement schedule. From Table (1), it is found that the reliability level at which PM actions are performed increase with the increasing of the ratio δ 2 . However, the upgrade level to be performed before putting the system into exploitation decreases with the increasing of the ratio δ 2 . This implies that to achieve high reliability thresholds, it is required to perform higher degree of upgrade. Finally, as one may expect, the average total cost per unit of time increases with the increasing of the ratio δ 2 . Table (2) gives the duration of PM and replacement cycles for a fixed value of the ratio δ 2 . One may observe that the PM cycle duration decrease with the increase of the PM cycle number. This is mainly due to both the system degradation and the imperfect effect of PM actions carried out on the system. For example, in the case where the ratio δ 2 = 50, results from Table (2) suggest to perform 8 PM after which the system is replaced. The first PM is performed after X1 = 5.52 time unit. From that time and after X2 = 3.33 time units the system undergoes the second PM. After performing the 8th PM, the system is replaced by a new one 1.72 time unit later.

4

Conclusion

This paper proposed a preventive maintenance optimization model for an end of life system subject to stochastic degradations. Imperfect preventive maintenance 8

δ2

50

100

150

200

R0∗

0.35

0.6

0.7

0.8

u∗

4.6

5.3

5.3

5.2

α∗

0.73

0.44

0.37

0.27

u∗ × α∗

3.35

2.3

1.96

1.4

N∗

9

9

9

10

C(R0∗ , u∗ , α∗ , N ∗ , )

0.13

0.19

0.24

0.28

Table 1 Optimal values of decision variables and their corresponding induced average total cost per unit of time. δ2

50

100

150

200

5.52

3.76

3.09

2.33

3.33

2.22

1.82

1.41

3.11

2.08

1.71

1.31

2.82

1.89

1.55

1.19

2.56

1.71

1.40

1.08

2.32

1.55

1.27

0.98

2.10

1.41

1.15

0.89

1.90

1.27

1.04

0.80

1.72

1.15

0.95

0.73 0.66

Table 2 PM cycle Durations corresponding to optimal solution.

actions are undertaken on the basis of hybrid hazard rate model. The system undergoes a PM whenever its reliability reach a given threshold. The hybrid hazard model offers the advantage to combine the effect of two coefficients, namely the age reduction coefficient and the hazard rate increase coefficient. Furthermore, it allows the representation of the system degradation process and the imperfect effect of preventive maintenance. A mathematical model is then proposed and numerically solved. The proposed model takes into account of acquisition cost and upgrade cost. Four decision variables are considered, namely the reliability threshold, initial acquisition age, upgrade level, in addition to the number of PM cycles. Optimal values of these decision variables which minimize the average total cost per unit of time are determined. The proposed PM model can also be exploited to determine maintenance policy in the case where the system is required to perform its missions with a fixed degree of reliability. In this case, reliability threshold is considered as an input data rather than a decision variable of the optimization problem. 9

Extensions being investigated include the modelling and optimization of the availability of these refreshed systems instead of minimizing a cost function, generalized repair processes instead of minimal repair when the system fails between PM instants.

References

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Yeh, R. H., Lo, H.-C. & Yu, R.-Y. (2011). A study of maintenance policies for secondhand products, Computers & Industrial Engineering 60(3): 438 – 444. Recent Developments in the Analysis of Manufacturing and Support Systems.

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