Two-dimensional replacement-repair warranty policy
M.RABBANI, A.A.ZAMANI
Department of Industrial Engineering
Faculty of Engineering, University of Tehran
Tehran, IRAN
Abstract: In this paper, a method is developed to minimize the expected warranty servicing cost per item sold. In this method, to be decided to repair or replace a failed item. These kinds of items work in multi-state deteriorating and they may fail in every state. Zuo, Liu and Murthy (2000), presented a paper to minimize the expected warranty servicing cost under one-dimensional (Time) and nonrenewable warranty policy. But here, the warranty policy has two-dimensions (Time & Usage). The decision to repair or replace a failed item depends on two parameters, the deterioration degree of the item and the length of the residual warranty period. The optimality of these two parameters for minimizing the expected warranty servicing cost are examined by two methods, analytically and simulation. Then an offered algorithm is used as an approximating method for performing the warranty policy.
Key-Words:Warranty, Fail, Repair, Replace, Maintenance
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1 Introduction
A warranty is a contractual obligation incurred by a manufacturer (vendor or seller) in connection with the sale of product. The warranty normally specifies that the manufacturer agrees to replace or repair a failed item with in a predetermined warranty period [1]. The most commonly used warranty policies include the free replacement warranty (FRW), the pro-rata warranty, and the combined warranty [2]. The total expected cost to the manufacturer in order to determine the warranty policy is very important during the warranty period. The manufacturer can determine the suitable warranty policy in order to minimize the expected warranty cost. It depends on characteristics of product and usage conditions.
Nguyen and Murthy [3] present models for evaluation of expected warranty cost and the variance of the warranty cost considering general product lifetime distribution and time depended repair cost. In another paper [4] for the FRW with a fixed warranty period T, they study the warranty servicing policy to minimize the expected warranty servicing cost. When an item fails, the decision that the manufacturer has to make is that, replace a failed item by a new one or by a repaired item from the stock of repaired items. Nguyen and Murthy [5] examine the combined FRW with fixed and renewal periods of T and W. Rao [6] develops an algorithm for evaluating the cost to the manufacturer for products with the phase-type lifetime distributions under FRW. Bohoris and Yun [7] provide equations for calculation of the expected value and variance of the manufacturer’s total warranty costs under the combined warranty policy and weibull lifetime distributions. Chun and Tang [8] determine the warranty price for the FRW policy assuming constant failure rate and constant repair cost through out the warranty period. In these papers, they have been considered only two situations for an item, working state and failure state. They don’t model multi-state equipment or items due to natural deterioration.
Derman, Lieberman and Ross [9] study the optimal replacement problem of a component where there are n types of replacements available differing only in price and the failure rates of exponential life distributions. Zuo, Liu and Murthy [10] present a model for a class of multi-state deteriorating with N steps and repairable products. In all steps, the items had exponential lifetime distribution with differing failure rates. They present a replacement-repair policy to minimize the expected warranty servicing cost under FRW with fixed period of T. fig.1 shows all the possible states of the item and the possible transition among the states.
In this paper, we study the model of fig.1 in two dimensions, Time and Usage. The goal of this paper is determination of optimal replacement-repair policy for minimizing the expected warranty servicing cost.
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Fig.1: State transition diagram of an item.
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2 Description of model
2.1 Notations
Nnumber of working states = number of failure states,
Tlength of time for warranty period,
Uamount of usage for warranty period,
ka decision variable (1kn-1),
θa decision variable (0θU),
cricost of repairing given that the item is fixed in state i (i=1,2,…,N),
cmicost of replacing given that the item is fixed in state i (i=1,2,…,N),
μirate of usage exponential distribution in state i (i=1,2,…,N),
piprobability that the item enters working state i+1 given that it has made a transition out of working state i (i=1,2,…,N-1),
1-piprobability that the item enters failure state i given that it has made a transition out of working state i (i=1,2,…,N-1),
yta random variable that indicates the customer usage during t times,
Zthe usage along the warranty period (Z=min{U,yt}),
uthe amount of residual usage to the end of the warranty period when an item goes on failure state (u=Z- the amount of usage item up to it goes on failure state),
CWthe expected warranty servicing cost per item sold,
CW1the expected warranty servicing cost per item sold when ytU,
CW2the expected warranty servicing cost per item sold when ytθ,
CW3the expected warranty servicing cost per item sold when θyt<U.
2.2 Assumptions
(A1)(1-p1)μ1<(1-p2)μ2<...<(1-pn-1)μn-1μn.
(A2)cm1cm2 ...cmn.
(A3)cr1cr2<...<crn.
(A1) states that as the item deteriorates, it becomes more likely to make a transition to a failure state. Assumptions (A2) and (A3) imply that the item becomes more costly to replace and repair as the deterioration increases [10].
2.3 Multi-state FRW policy in two dimensions and nonrenewable warranty period
This policy will be used for a failed item when the usage of item is less than U and the length of expired time from beginning of the warranty period is less than T. Otherwise, the warranty period is expired. In this policy, the manufacturer has to repair or replace the failed item. The replacement-repair policy for minimization of the expected warranty servicing cost will be defined as follow:
If the failed item is in failure state i, then it is replaced by a new one if and only if k+1iN and θu; otherwise, it is repaired.
However, the problem is the determination of optimal k and θ for minimizing the CW.
3 Special case (N=2)
According to the policy is presented in the last section, k will be equal to 1 if there are two working states and two failure states.
3.1 yt with exponential distribution
(1)
(2)
(3)
(4)
(5)
Then:
(6)
(7)
Then:
(8)
According to [10], Eqs. (9), (10) and (11) would be concluded:
(9)
From Eq. (6):
(10)
From Eq. (8):
(11)
From Eqs. (2), (3), (4), (9), (10) and (11), CW is calculated in the following form:
(12)
For optimal policy, if the differentiating CW given by Eq. (12) with respect to θ is calculated, the following cases will be needed to consider.
Case 1: When
,
we have for all . This implies that CW is a non-increasing function in θ over interval [0,U]. Therefore the optimal θis U(θ*=U).
Case 2: When
,
by solving we have
and
Now we can summarize this as follows:
If , then optimal policy is to replace the item with a new one if it is in failure state 2 and the remaining warranty period is longer than θ*Є (0,U), where θ* is equal to , and to repair the item otherwise.
3.2 yt with uniform distribution
(13)
(14)
(15)
(16)
(17)
Then:
(18)
(19)
Then:
(20)
After calculation of CW, θ* could have been obtained just like as section 3.1.
3.3 Comment
With considering of 3.1 and 3.2, the manufacturer will replace the failed item when k=1 and θ*u. Otherwise, it will be repaired. Since yT is not determinable for a failed item, so the usage along the warranty period (Z) is unknown and u is not calculable.
If u΄ define as the residual warranty period from U (u΄=U- The amount of usage item up to it goes on failure state), then always it will calculable. u΄ can be written in the following form:
(21)
From Eq. (21), β is defined in Eq. (22).
(22)
3.3.1 yT with exponential distribution like Eq. (1).
(23)
From Eq. (6):
(24)
From Eqs. (23) and (24):
(25)
3.3.2 yT with uniform distribution like Eq. (13).
(26)
From Eq. (18):
(27)
From Eqs. (26) and (27):
(28)
According to Eqs. (25) and (28), the replacement-repair policy will be performed in the following form:
When an item fails and its warranty period has not been expired yet, if k=1 and u΄>E[β*], then the failed item will be replaced with new one. Otherwise, it will be repaired.
4 General case (N > 2)
In general case that N3, could be assigned more than one value to k. In this situation, there is no analytical solution for determination of optimal k and β. By using computerize simulation method (fig.2), CW can be calculated with all combinations of k and β. Then considering to minimum value of CW, optimal value of k and β will be determined.
One of advantage of using the simulation method is ability to termination of some warranty model assumptions. For example, every kind of distribution function could be used for the lifetime or the customer usage during t times.
4.1 Distribution function of yT is known
Example 1: If μi is rate of exponential distribution function and yt~Normal(2t,t), N=4, μ1=0.5, μ2=2, μ3=3, μ4=3.5, p1=0.9, p2=p3=0.6, cr1=40, cr2=50, cr3=300, cr4=400, cm1=300, cm2=500, cm3=600, cm4=800, U=3 and T=2, then the result of simulation is shown in table 1.
The model is simulated for three times by k=1, 2 and 3. The minimum value of CW is 397 with k*=2 and β*=0.34.
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Fig.2: Visual SLAM simulation model for calculation of CW.
Table 1: Results for simulation of example 1.
k / Min ( CW ) / β1 / 411 / 0.53
2 / 397 / 0.34
3 / 433 / 0.16
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4.2 Distribution function of yT is unknown
In order to minimizing the expected warranty servicing cost when the warranty period is not expired yet, is suggested to perform the following algorithm for a failed item.
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Fig.3: Visual SLAM simulation model for determination of the algorithm precision that is presented in section 4.2.
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Algorithm:
Step 1: Solve the problem in one dimension with U as a period of warranty. Then get k* and β* [10].
Step 2: Repair the failed item if ik*+1. Otherwise, go to next step.
Step 3: Calculate . If , go to step 4. Otherwise, go to step 5.
Step 4: If , then replace the failed item. Otherwise, repair it.
Step 5: Calculate . If , then repair the failed item. Otherwise, repair it.
Example 2: Consider the example 1 where the distribution of yT is not known. The failure item has these conditions: (u΄=2, t=1, i=3)
1. Step 1:
2. Step 2:
3. Step 3:
4. Step 5: Replace the failed item.
Example 3: Consider the example 2. The failure item has these conditions: (u΄=2, t=1.6, i=3)
1. Step 1:
2. Step 2:
3. Step 3:
4. Step 4: Replace the failed item.
The precision measure of this algorithm can be calculated by simulation method for a known distribution function of yT. If yT has a uniform distribution function, the expected warranty cost of the algorithm performing will be calculated by simulation model that is shown in fig.3.
The value of expected warranty cost is gotten from this method can be compared with the expected warranty cost when the distribution function of yT is known (section 4.1).
By a little change on the simulation model, could be measured the algorithm efficiency for other distribution function of yT.
5 Conclusion
In this paper has been presented a warranty policy for those items that have N working states and N failure states. In this policy has been presented a method for decision to repair or replace the failed item, in order to minimizing the expected warranty servicing cost. This method is developing of Zuo, Liu and Murthy (2000) [10] method with two dimensions. In this method, if N=2, decision variables (k*,β*) have been calculated analytically. Then the problem has been solved in general case (N3) by using computerize simulation method. At last, it has been presented the practical algorithm for performing the warranty policy.
The model reported in this paper can be extended in several ways. Some of the many issues that need further study are listed bellow.
(1)Determination of β* for other distribution function of yt, when N=2.
(2)Changing the deterioration of an item from discretely to continuously and developing the analytical and simulation methods.
(3)Changing the simulation model when an item goes from working state j to working state i for i=j+1,j+2,…,N.
(4)Using the other warranty policies.
(5)Releasing the independent and constant repair cost assumption so that repair cost is a function of the failed state.
(6)Presenting more efficient algorithms for performing the warranty policy.
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