Consider that an equipment with Weibull lifetime distribution is leased at age A and intended to be leased for a period of L. The failure rate function of the Weibull distribution is given by

(1)

Where α is called the scale parameter and β is called the shape parameter. The inverse function of λ(t) is given by

(2)

Note that both λ(t) and its inverse function λ^-1(t) increase in t when the shape parameter β > 1, and decrease in t when β > 1. In this article, we focus on the case where failure rate function λ(t) continuously increases in t and λ^-1(t) exists (i.e. β > 1) within the lease period.

Within the lease period, any failure of the leased equipment is repaired using minimal repair by the lessor with a fixed repair cost C_m and the lessor incurs a penalty cost C_n (we call penalty 1) for each failure that occurs over the lease period. After minimal repair, the equipment is operational, but the failure rate of the equipment remains the same as that just before failure. It is assumed that each failure requires a random amount of repair time t which is subject to a general cumulative distribution function G(t). Furthermore, if the repair time exceeds a pre-specified time limit τ, there is a penalty cost C_τ (we call penalty 2) to the lessor.

To reduce the number of possible failures over the lease period, n PM actions are carried out at time epoch t_i, where A=t_O < t₁ < t₂ < ... < t_n < L + A. After the PM actions, the failure rate of the equipment is reduced by a fixed amount δ ≥ 0. In general, the cost to perform a PM action is a non-negative and non-decreasing function of the maintenance degree δ. In this paper, we consider the case where the PM cost C_p (δ) is a linearly increasing function of the maintenance degree δ; that is, C_p (δ)= a+b (δ) for any a > 0, b > 0. It is assumed that the time required for performing PM actions is negligible. After the ith PM action, the PM failure intensity becomes λ₀ (t_i-A)-iδ for all i=1,2,...,n as shown in Fig. (1).

Fig. (1) Optimal PM policy under fixed failure rate reduction.

The expected number of failures over the lease period, with the proposed PM actions, is given by

(3)

Define , then Eq. (3) is equivalent to

(4)

Where denotes the vector of time epochs to perform PM actions.

The expected total cost within the lease period L including minimal repair cost, penalty cost, and PM cost which can be easily obtained as follows:

(5)

Define , and substitute (4) into (5), then (5) can be rewritten as

(6)

Note that there are n=3 decision variables including the number of PM actions n, PM degree δ, and the time epochs t_i. Hence, our objective here is to find an optimal PM policy such that the expected total cost in Eq. (6) is minimized.

Now we will give a theorem that shows the relationship between the optimal time epoch and the inverse failure rate function in Eq. (2).

Theorem 1. Given any n > 0 and δ > 0, if λ_O(t) is a strictly increasing function of t when β > 1, then .

Given any n > 0 and δ > 0, we have , for all i=1,2,...,n. This result indicates that is an increasing function of δ, and implies that to minimize is equivalent to minimize , since C_p(δ), L and C^´ are all constants. Therefore, t_i should take the minimum value which is subject to the constraint λ (t_i-A)-iδ ≥ 0 for all i=1,2,...,n. Since λ₀ (t) is a strictly increasing function of t, we obtain . Hence, the minimal value of is obtained at for all i=1,2,...,n.

Theorem 1 shows that optimal value of is that , with this, the objective function becomes

(7)

There are only two decision variables n and δ to be determined. In order to find the optimal value of n^* and δ^*, we suppose that n is given and we find the optimal value δ^*. After we find the optimal value δ^*, the value n can be obtained by any search method directly.

In Eq. (7), if is an increasing function, there is no need to perform PM actions. Therefore, we will focus on the situation that . Given any n > 0, the theorem below shows that if it satisfies some reasonable conditions, there exists a unique closed-form solution of δ^* such that the expected total cost is minimized.

Theorem 2. Given any n > 0, if the failure intensity function is subject to Weibull distribution, then the following results hold:

(1) If , then δ^*=0;

(2) If , and ,then there exists .

Given any n > 0, the first and second partial derivatives of Eq. (7) with respect to δ, we have

(8)

(9)

We know that the inverse function is strictly increasing in (iδ), so for all i. Hence, when , Eq. (8) is positive for all δ which implies that is strictly increasing function of δ for any n > 0. In this case, the optimal maintenance degree δ^* is zero.

On the other hand, if , we obtain and . When , then Eq. (9) is positive for all δ. In this case, Eq. (8) is a strictly increasing function of δ and changes its sign at most once from negative to positive. Therefore, there exists a unique δ^* > 0, such that . Furthermore, given any n > 0, there is an upper bound of δ, which is . Hence, if , then ; otherwise,

For the Weibull case with any n > 0, Eq. (7) becomes

(10)

where . We take the first derivative of Eq. (10) with respect to δ we obtain

(11)

If , from the derivation above, we know that Eq. (10) is positive. Hence, the PM action is not necessary. When , and , let Eq. (11) equal to zero, we obtain the optimal δ:

Using the result of theorem 2 above, the optimal maintenance degree δ^* can be obtained easily. Now, there is only one decision variable n which represents the optimal number of the PM actions within the leased period to be determined. In order to find the optimal expected total cost, we use enumeration algorithm. Without loss of generality, we may set or any other large number and search the optimal value of n from 0 to . The following algorithm provides an efficient search procedure for deriving the optimal policy .

Step 1: if , then , and stop.

Step 2: set or any large number, and n=1.

step 3: , such that we obtain .

Step 4: if , then set , and .

Step 5: if , then stop; otherwise, set n=n+1 and go to step 3.

In this section, the performance of the proposed optimal PM policy is evaluated through numerical examples.

We assume that the failure distribution for the used equipment(for example: used cars, used computers) is given by the two-parameter Weibull distribution with scale parameter α=1 and the shape parameter β > 1 which implies that the failure rate is increasing. We suppose that repair time t is subject to a two-parameter Weibull distribution (2,0.5). If the repair time exceeds τ=2, then there is a penalty cost C_τ = 300per unit time to the lessor. Then, we obtain the total expected cost to the lessor at each failure is , and the cost for performing a PM action with maintenance degree δ is a linear function C_p(δ)=100+50δ. We consider used equipment with age A is planned to be leased for a period of L = 3 years. To evaluate the performance of the proposed optimal PM policy, we use C₀ denoted the expected total cost without PM actions, and define to denote the percentage of cost reduction, where C^* denotes the optimal expected total cost. (Table 1) summarizes the numerical results for various combinations of β,A,C_n,C_τ.

From (Table 1), we make some interesting observations:

1. When A=0, the results are the same as the maintenance model [15R.H. Yeh, K.C. Kao, and W.L. Chang, "Optimal preventive maintenance policy for leased equipment using failure rate reduction", Comput. Ind. Eng., vol. 57, no. 1, pp. 304-309, 2009.
   [http://dx.doi.org/10.1016/j.cie.2008.11.025] ], the model in this paper degenerates to the model in their paper.
2. As A increases, C₀ and C^* increase, but the percentage of cost reduction decreases. This result implies that the PM actions have little impact on the expected cost with the duration of the used equipment increasing.
3. As β increases, both the optimal number of PM actions n^* and the optimal maintenance degree δ^* increase.
4. As penalty 1 increases, the optimal expected total cost C^* increases, the optimal number of PM actions n^* increases, and the percentage of cost reduction also increases which implies that there is a significant impact in the PM actions on the expected cost when the lease period is relatively long. The situations of penalty 2, penalty 1 and penalty 2 combined are similar to penalty 1.

In this section, the performance of the optimal policies in this paper is compared with the model I in Yeh [14R.H. Yeh, and H.C. Lo, "H, and R. Y. Yu, “A study of maintenance policies for second-hand products", Comput. Ind. Eng., vol. 60, no. 3, pp. 438-444, 2011.
[http://dx.doi.org/10.1016/j.cie.2010.07.033] ] (policy Y).

We consider that used equipment will be leased at age A for a lease period L=5. We assume the each minimal repair cost C_m=100, C_n=C_τ=0, and the cost for performing a PM action with maintenance degree δ is a linear function C_p(δ)=100+50 δ. With loss of generality, we suppose α = 1. In comparison with the results of Yeh’s policy [13G.N. Chattopadhyay, and D.N. Murthy, "Warranty cost analysis for second-hand products", Math. Comput. Model., vol. 31, no. 10, pp. 81-88, 2000.
[http://dx.doi.org/10.1016/S0895-7177(00)00074-1] ], we set .

Table 1
Numerical results for various combinations of β,A,C_n,C_τ.

Table 2
Numerical results among different maintenance scheme.

(Table 2) lists and compares the two policies. From (Table 2), we make the following observations:

(1) When β = 1.2, there is no need to perform PM actions for policy Y, policy D has better performance at this time.

(2) When β = 1.2 and A with different values, the optimal number of PM actions n^* is the same between the two policies, but policy D also has better performance than policy Y.