# The Optimal Warranty and Preventive Maintenance Policy for the Four-State System

Authors: Ahmed F. Attia; Eman D. Abou Elela; Hany A. Hosham
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

A complete view for the multistate system considering the four-state system is here introduced. The exponential distribution for failure times and repair times is considered. The steady state availability is established via the Markov process. Different warranty and preventive maintenance policies are introduced, and also the cost of these policies for the manufacturer and the buyer is evaluated.


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## Body

## 1. Introduction

All products are unreliable in the sense that they fail. A failure can occur early in an item’s life due to manufacturing defects or late in its life due to degradation that is dependent on age and usage.Most equipment offers some level of warranty to gain some advantages in the highly competitive markets. The warranty assures the buyer that a faulty item will either be repaired or replaced at no cost or at reduced cost.To the seller, offering a warranty implies additional costs including PM, repair, and replacement that depend on the transaction contract and product characteristics. From the buyer’s perspective, however, PM can either lengthen the lifetime of product or effectively reduce its failure rate; well-performing PM during the warranty period may provide the consumer better product service in the postwarranty period and reduce the cost of repairing the deteriorated product.Kim et al. [1] gave a brief review of the literature linking warranty and preventive maintenance. Furthermore Sheu and Chien [2] obtained the expected total cost (i.e., manufacturing plus warranty costs) under various warranty policies (failure-free policies with and without renewing and rebate policy). Jain and Maheshwari [3] considered renewing pro rata warranty policy to determine the expected maintenance cost for different life-time distributions. Chen and Chien [4] investigated the effect of preventive maintenance carried out by the buyer on items sold under a free-replacement renewing warranty policy. Jung et al. [5] considered a replacement model following the expiration of warranty: renewing warranty and nonrenewing warranty. Huang and Yen [6] analyzed a two-dimensional warranty of both time and usage for deteriorating products; they presented the warranty policy that maximizes the manufacturers profits. Jung et al. [7] studied the system maintenance policy under the renewing warranty, defined the life cycle anew from the user’s perspective, and discussed the optimal maintenance policy after the renewing warranty is expired. Yedida and Munavar [8] considered a geometric process maintenance model with preventive repair and warranty using an objective function defined as the average cost per unit of working time.Some systems are able to perform their task with partial performance; failure of their components leads to degradation of performance. Such systems or components have one working state and two or more different failure states, These systems are called multistate systems (MSSs). There are many examples of MSS-like electrical power generation systems, domestic hot water systems, telecommunication networks systems, and radar systems; see the work of Lindqvist and Doksum in [9] and of Nourelfath and Ait-Kadi in [10].In our study, Attia et al. [11] introduced the preventive maintenance problem for the multistate system in the case of the three-state system, since we evaluated the optimal numbers of preventive maintenance that maximize the expected profit values.In this paper, we continue to introduce a complete view for the multistate system considering the four-state system. Also, we assume the exponential distribution for failure times and repair times. We evaluate the steady state availability, introduce different warranty and preventive maintenance policies, and evaluate the cost of these policies for the manufacturer and for the buyer in the multistate system case.The structure of this study is as follows. In Section2 the four-state system description is shown and the steady state availability is evaluated using the Markov process technique. In Section 3 different warranty and preventive maintenance policies and the cost of these policies for the multistate system are evaluated. The proposed model is illustrated through two numerical examples in Section 4.


## 2. The Four-State System Model

We introduce here the multistate system considering the four-state system, and we assume that the system failure times and repair times are exponentially distributed.The system has four states,S={0,…,3} such as  state 0 represents the system before any failure occurs when the system operates. The intermediate states 1, 2 are ordered to reflect their relative degree of failures, meanwhile in state 3 the system fails completely and cannot operate.The failure rates areλi,i=1,…,6, and the repair rates are μi,i=1,…,6.Figure1 shows the transition rates.Figure 1
Transition rates.The transition probabilities of the Markov process are given by(2.1)P0(t+Δt)=P0(t)[1-(λ1+λ4+λ6)Δt]+P1(t)μ1Δt+P2(t)μ4Δt+P3(t)μ6Δt,P1(t+Δt)=P0(t)λ1Δt+P1(t)[1-(λ2+λ5+μ1)Δt]+P2(t)μ2Δt+P3(t)μ5Δt,P2(t+Δt)=P0(t)λ4Δt+P1(t)λ2Δt+P2(t)[1-(λ3+μ2+μ4)Δt]+P3(t)μ3Δt,P3(t+Δt)=P0(t)λ6Δt+P1(t)λ5Δt+P2(t)λ3Δt+P3(t)[1-(μ3+μ5+μ6)Δt],
where Pj(t+Δt)  is the probability that the system is in state j (j=0,…,3), at time (t+Δt).Lisnianski and Levitin [12] established a system of differential equations for the general case of k states. In the following, we take the four-state system as a special case and use the Laplace transformation technique to have the exact solution.The differential equations for our system are as follows:(2.2)P0′(t)=-(λ1+λ4+λ6)P0(t)+μ1P1(t)+μ4P2(t)+μ6P3(t),P1′(t)=λ1P0(t)-(λ2+λ5+μ1)P1(t)+μ2P2(t)+μ5P3(t),P2′(t)=λ4P0(t)+λ2P1(t)-(λ3+μ2+μ4)P2(t)+μ3P3(t),P3′(t)=λ6P0(t)+λ5P1(t)+λ3P2(t)-(μ3+μ5+μ6)P3(t).
Taking the Laplace transformations of (2.2) yields the following linear equations in terms of Pj*(s) for j=0,1,2,3:(2.3)sP0*(s)-P0(0)=-(λ1+λ4+λ6)P0*(s)+μ1P1*(s)+μ4P2*(s)+μ6P3*(s),sP1*(s)-P1(0)=λ1P0(s)-(λ2+λ5+μ1)P1*(s)+μ2P2*(s)+μ5P3*(s),sP2*(s)-P2(0)=λ4P0*(s)+λ2P1*(s)-(λ3+μ2+μ4)P2*(s)+μ3P3*(s),sP3*(s)-P3(0)=λ6P0*(s)+λ5P1*(s)+λ3P2*(s)-(μ3+μ5+μ6)P3*(s).
Under the initial conditions, P0(0)=1,P1(0)=0,P2(0)=0, and P3(0)=0, these means that the system is initially assumed to be in the working state, and we can rewrite the equations (2.3) in the following form:(2.4)P0*(s)=s3+a1s2+b1s+c1s(s3+ds2+es+f),P1*(s)=a2s2+b2s+c2s(s3+ds2+es+f),P2*(s)=a3s2+b3s+c3s(s3+ds2+es+f),P3*(s)=a4s2+b4s+c4s(s3+ds2+es+f),
where
(2.5)a1=λ2+λ3+λ5+μ1+μ2+μ3+μ4+μ5+μ6,b1=μ4λ5+μ5λ3+μ5μ2+μ5μ4+μ6λ3+μ6μ2+μ6μ4+μ3μ1+μ3λ5+λ2μ5+μ3λ2+μ3μ2+μ6λ5+μ6λ2+μ5μ1+μ6μ1+λ3λ2+λ3μ1+μ2μ1+μ2λ5+λ3λ5+μ4μ1+μ4λ2+μ3μ4,c1=λ2μ5μ4+μ6λ2μ4+μ6λ5λ3+μ6λ5μ2+μ6λ5μ4+μ6λ2λ3+μ3μ4λ5+μ3μ4λ2+μ1μ5λ3+μ1μ5μ2+μ1μ5μ4+μ1μ6λ3+μ1μ6μ2+μ1μ6μ4+μ1μ3μ2+μ1μ3μ4,d=λ2+μ6+μ3+μ2+λ6+λ4+λ3+μ4+λ1+μ1+μ5+λ5,e=λ5λ1+λ5λ4+λ5λ6+μ1λ6+μ1λ4+λ2λ1+λ2λ6+μ4λ5+μ5λ3+μ5μ2+μ5μ4+μ6λ3+μ6μ2+λ3λ1+μ2λ4+μ2λ6+μ2λ1+μ4λ6+μ6μ4+μ5λ1+μ5λ6+μ6λ1+μ4λ1+λ3λ4+λ3λ6+μ3μ1+μ3λ5+λ2μ5+μ3λ2+λ2λ4+μ3μ2+λ4μ6+μ3λ4+μ3λ6+μ3λ1+μ6λ5+μ6λ2+μ5μ1+μ6μ1+λ4μ5+λ3λ2+λ3μ1+μ2μ1+μ2λ5+λ3λ5+μ4μ1+μ4λ2+μ3μ4,f=μ2μ1λ6+μ4λ2λ6+μ4λ5λ1+μ5λ3λ4+λ2μ5μ4+μ6λ2μ4+μ6λ5λ3+μ6λ5μ2+μ6λ5μ4+μ4λ5λ6+μ4μ1λ6+λ3λ2λ1+λ3λ2λ4+λ3λ2λ6+λ3λ5λ1+λ3λ5λ4+λ3λ5λ6+λ3μ1λ4+λ3μ1λ6+μ6λ2λ3+μ5λ1μ4+μ5λ1λ3+μ5λ1μ2+μ3λ4μ2+λ2μ5λ1+μ3λ2λ1+μ3λ2λ4+μ3λ2λ6+μ3λ5λ1+μ3λ5λ4+μ3λ5λ6+μ3μ1λ4+μ3μ1λ6+λ4μ6λ2+λ4μ6λ5+λ4μ6μ1+λ2μ5λ4+λ4μ6μ2+μ6λ1λ3+μ6λ1μ2+μ6λ1μ4+μ3μ4λ5+μ3μ2λ1+μ3μ4λ1+μ3μ4λ2+μ5λ3λ6+μ3λ6μ2+μ5μ4λ6+μ2λ5λ1+μ2λ5λ4+μ2λ5λ6+λ2μ5λ6+λ2μ6λ1+μ1λ4μ5+μ5μ2λ6+λ4μ5μ2+μ1μ5λ3+μ1μ5μ2+μ1μ5μ4+μ1μ6λ3+μ1μ6μ2+μ1μ6μ4+μ1μ3μ2+μ1μ3μ4,a2=λ1,b2=λ3λ1+μ2λ4+μ2λ1+μ5λ1+μ5λ6+μ6λ1+μ4λ1+μ3λ1,c2=μ5λ3λ4+μ5λ1μ4+μ5λ1λ3+μ5λ1μ2+μ3λ4μ2+λ4μ6μ2+μ6λ1λ3+μ6λ1μ2+μ6λ1μ4+μ3μ2λ1+μ3μ4λ1+μ5λ3λ6+μ3λ6μ2+μ5μ4λ6+μ5μ2λ6+μ4μ5μ2,a3=λ4,b3=λ5λ4+μ1λ4+λ2λ1+λ2λ4+λ4μ6+μ3λ4+μ3λ6+λ4μ5,c3=λ2μ5λ1+μ3λ2λ1+μ3λ2λ4+μ3λ2λ6+μ3λ5λ1+μ3λ5λ4+μ3λ5λ6+μ3μ1λ4+μ3μ1λ6+λ4μ6λ2+λ4μ6λ5+λ4μ6μ1+λ2μ5λ4+λ2μ5λ6+λ2μ6λ1+μ1λ4μ5,a4=λ6,b4=λ5λ1+λ5λ6+μ1λ6+λ2λ6+μ2λ6+μ4λ6+λ3λ4+λ3λ6,c4=μ2μ1λ6+μ4λ2λ6+μ4λ5λ1+μ4λ5λ6+μ4μ1λ6+λ3λ2λ1+λ3λ2λ4+λ3λ2λ6+λ3λ5λ1+λ3λ5λ4+λ3λ5λ6+λ3μ1λ4+λ3μ1λ6+μ2λ5λ1+μ2λ5λ4+μ2λ5λ6.The probability that the system is in working state at timetis called the point availability of the system.The system point availability is given by(2.6)A(t)=P0(t)+P1(t)+P2(t).
The Laplace transform of A(t) is given by(2.7)A*(s)=P0*(s)+P1*(s)+P2*(s),A*(s)=s3+a1s2+b1s+c1s(s3+ds2+es+f)+a2s2+b2s+c2s(s3+ds2+es+f)+a3s2+b3s+c3s(s3+ds2+es+f).
The steady state availability of the system is given by(2.8)A=lim⁡t→∞A(t)=lim⁡s→0sA*(s),(2.9)A=c1+c2+c3f,
where c1,c2,c3,andf are defined as before.Kim et al. [1] introduced different choices for the warranty and preventive maintenance. In the next section we develop these policies and apply them to the multistate system, present different warranty and preventive maintenance policies, and continue to evaluate the cost of these policies for the manufacturer and the buyer and the total cost.


## 3. The Warranty and Preventive Maintenance Policy for the Multistate System

All malfunctions of a product in the warranty period [0,W) are repairable by manufacturer with a free warranty policy and charges no fee to the buyer. The lifetime of each product is L. In the postwarranty period [W, L), the cost of repairing deteriorated product is borne by the buyer. In the warranty period, we assume that the cost of PM should be borne by the seller. In the postwarranty period [W, L), the cost of PM is borne by the buyer. We present here different options of warranty and PM policies as follows.


### 3.1. Policy A1

We assume that this policy have no PM action over the whole useful life of the system. Suppose that the useful life of the system isL and the warranty period [0, W) includes the first state, in this case we compute the warranty cost for the manufacturer and for the buyer as follows. (i)
The warranty cost for the manufacturer can be represented by(3.1)CMA1=CR∫0W(failure  rate  for  the  first  state)dt=CR∫0Wλ1dt=CRλ1W,
where CMA1 is the cost for the manufacturer in policy A1 and CR is the repair cost.(ii)
The warranty cost for the buyer can be represented by(3.2)CBA1=CR∫W1/λ2(failure  rate  for  the  second  state)dt+CR∫1/λ2L(failure  rate  for  the  third  state)dt,CBA1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt=CR(1-λ2W+λ3L-λ3λ2),
where CBA1 is the cost for the buyer in policy A1.


### 3.2. Policy A2

This policy also assumes no PM action over the whole useful life of the system, and the warranty includes the first and the second states.(i)
The warranty cost for the manufacturer can be represented by(3.3)CMA2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt=CR(1+λ2W-λ2λ1).(ii)
The warranty cost for the buyer can be represented by(3.4)CBA2=CR∫WLλ3dt=CRλ3(L-W).


### 3.3. Policy B1

We assume that this policy have continuous PM action over the whole useful life of the system, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.5)CMB1=CR∫0Wλ1dt+CPMW=CRλ1W+CPMW,
where CPM is the cost for the PM.(ii)
The warranty cost for the buyer is computed as  follows:(3.6)CBB1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt+CPM(L-W)=CR(1-λ2W+λ3L-λ3λ2)+CPM(L-W).


### 3.4. Policy B2

We assume that this policy have continuous PM action over the whole useful life of the system, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer(3.7)CMB2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt+CPMW=CR(1+λ2W-λ2λ1)+CPMW.(ii)
The warranty cost for the buyer(3.8)CBB2=CR∫WLλ3dt+CPM(L-W)=CRλ3(L-W)+CPM(L-W).


### 3.5. Policy C1

We assume that this policy have no PM action over the warranty period and continuous PM over the postwarranty period, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.9)CMC1=CR∫0Wλ1dt=CRλ1W.(ii)
The warranty cost for the buyer is computed as  follows:(3.10)CBC1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt+CPM(L-W)=CR(1-λ2W+λ3L-λ3λ2)+CPM(L-W).


### 3.6. Policy C2

We assume that this policy have no PM action over the warranty period and continuous PM over the postwarranty period, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer is given as:(3.11)CMC2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt=CR(1+λ2W-λ2λ1).(ii)
The warranty cost for the buyer is given as:(3.12)CBC2=CR∫WLλ3dt+CPM(L-W)=CRλ3(L-W)+CPM(L-W).


### 3.7. Policy D1

We assume that this policy have continuous PM action over the warranty period and no PM over the postwarranty period, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.13)CMD1=CR∫0Wλ1dt+CPMW=CRλ1W+CPMW.(ii)
The warranty cost for the buyer is computed as  follows:(3.14)CBD1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt=CR(1-λ2W+λ3L-λ3λ2).


### 3.8. Policy D2

We assume that this policy have continuous PM action over the warranty period and no PM over the postwarranty period, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer is given as:(3.15)CMD2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt+CPMW=CR(1+λ2W-λ2λ1)+CPMW.(ii)
The warranty cost for the buyer is given as:(3.16)CBD2=CR∫WLλ3dt=CRλ3(L-W).
In the next section, we will show the cost of these polices by two numerical examples.


## 3.1. Policy A1

We assume that this policy have no PM action over the whole useful life of the system. Suppose that the useful life of the system isL and the warranty period [0, W) includes the first state, in this case we compute the warranty cost for the manufacturer and for the buyer as follows. (i)
The warranty cost for the manufacturer can be represented by(3.1)CMA1=CR∫0W(failure  rate  for  the  first  state)dt=CR∫0Wλ1dt=CRλ1W,
where CMA1 is the cost for the manufacturer in policy A1 and CR is the repair cost.(ii)
The warranty cost for the buyer can be represented by(3.2)CBA1=CR∫W1/λ2(failure  rate  for  the  second  state)dt+CR∫1/λ2L(failure  rate  for  the  third  state)dt,CBA1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt=CR(1-λ2W+λ3L-λ3λ2),
where CBA1 is the cost for the buyer in policy A1.


## 3.2. Policy A2

This policy also assumes no PM action over the whole useful life of the system, and the warranty includes the first and the second states.(i)
The warranty cost for the manufacturer can be represented by(3.3)CMA2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt=CR(1+λ2W-λ2λ1).(ii)
The warranty cost for the buyer can be represented by(3.4)CBA2=CR∫WLλ3dt=CRλ3(L-W).


## 3.3. Policy B1

We assume that this policy have continuous PM action over the whole useful life of the system, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.5)CMB1=CR∫0Wλ1dt+CPMW=CRλ1W+CPMW,
where CPM is the cost for the PM.(ii)
The warranty cost for the buyer is computed as  follows:(3.6)CBB1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt+CPM(L-W)=CR(1-λ2W+λ3L-λ3λ2)+CPM(L-W).


## 3.4. Policy B2

We assume that this policy have continuous PM action over the whole useful life of the system, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer(3.7)CMB2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt+CPMW=CR(1+λ2W-λ2λ1)+CPMW.(ii)
The warranty cost for the buyer(3.8)CBB2=CR∫WLλ3dt+CPM(L-W)=CRλ3(L-W)+CPM(L-W).


## 3.5. Policy C1

We assume that this policy have no PM action over the warranty period and continuous PM over the postwarranty period, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.9)CMC1=CR∫0Wλ1dt=CRλ1W.(ii)
The warranty cost for the buyer is computed as  follows:(3.10)CBC1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt+CPM(L-W)=CR(1-λ2W+λ3L-λ3λ2)+CPM(L-W).


## 3.6. Policy C2

We assume that this policy have no PM action over the warranty period and continuous PM over the postwarranty period, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer is given as:(3.11)CMC2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt=CR(1+λ2W-λ2λ1).(ii)
The warranty cost for the buyer is given as:(3.12)CBC2=CR∫WLλ3dt+CPM(L-W)=CRλ3(L-W)+CPM(L-W).


## 3.7. Policy D1

We assume that this policy have continuous PM action over the warranty period and no PM over the postwarranty period, and the warranty includes the first state.(i)
The warranty cost for the manufacturer is computed as  follows:(3.13)CMD1=CR∫0Wλ1dt+CPMW=CRλ1W+CPMW.(ii)
The warranty cost for the buyer is computed as  follows:(3.14)CBD1=CR∫W1/λ2λ2dt+CR∫1/λ2Lλ3dt=CR(1-λ2W+λ3L-λ3λ2).


## 3.8. Policy D2

We assume that this policy have continuous PM action over the warranty period and no PM over the postwarranty period, and the warranty includes the first state and the second state.(i)
The warranty cost for the manufacturer is given as:(3.15)CMD2=CR∫01/λ1λ1dt+CR∫1/λ1Wλ2dt+CPMW=CR(1+λ2W-λ2λ1)+CPMW.(ii)
The warranty cost for the buyer is given as:(3.16)CBD2=CR∫WLλ3dt=CRλ3(L-W).
In the next section, we will show the cost of these polices by two numerical examples.


## 4. Numerical Examples

### 4.1. Example  1

We assume that the exponential distribution for the failure time and repair time for the four-state system.The parameters values of the failure rates and repair rates were given as (Lisnianski and Levitin [12]): (4.1)λ1=2,λ2=1,λ3=0.7,λ4=0.3,λ5=0.4,λ6=0.1,μ1=100,μ2=80,μ3=50,μ4=45,μ5=40,μ6=32.
Then the steady state availability will be 0.9991.Now to compare between the different policies, we introduce four values for repair cost: 100, 200, 1000, and 3000 and two values for preventive maintenance cost: 20, 10. Table1 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.Table 1Choices12345678CR1001002002001000100030003000CPM2010201020102010CMA160060012001200600060001800018000CBA1430430860860430043001290012900Tot.CA1103010302060206010300103003090030900CMA2350350700700350035001050010500CBA2490490980980490049001470014700Tot.CA284084016801680840084002520025200CMB166063012601230606060301806018030CBB15705001000930444043701304012970Tot.CB1123011302260216010500104003110031000CMB2410380760730356035301056010530CBB263056011201050504049701484014770Tot.CB2104094018801780860085002540025300CMC160060012001200600060001800018000CBC15705001000930444043701304012970Tot.CC1117011002200213010440103703104030970CMC2350350700700350035001050010500CBC263056011201050504049701484014770Tot.CC298091018201750854084702534025270CMD166063012601230606060301806018030CBD1430430860860430043001290012900Tot.CD1109010602120209010360103303096030930CMD2410380760730356035301056010530CBD2490490980980490049001470014700Tot.CD290087017401710846084302526025230From Table1, comparing the suggestion policies by regarding the expected total cost, we find that policy A2 is the optimal policy, where no PM action over the whole useful life of the system is assumed and the warranty includes the first and the second states.If we order these policies according to the least total cost, we find that A2 policy is the least-costing policy for choice 1, followed by policies D2, C2, A1, B2, D1, C1, B1 in an ascending order regarding the cost. But the rest of choices (2–8), although A2 is still the least-costing policy for each, are of a bit different sequence (D2, C2, B2, A1, D1, C1, B1) also in an ascending order regarding the cost.But from the manufacturer point of view, if we order these policies  according to the least cost, we find that the policies can be ordered in an ascending order regarding the cost as follows, (A2 or C2), (B2 or D2), (A1 or C1), (B1 or D1). So we find that the least-costing policy for the manufacturer is A2 or C2.From the buyer point of view, we find that for the choices (1–4), the policies can be ordered as follows: (A1 or D1), (A2 or D2), (B1 or C1), (B2 or C2). But for the choices (5–8) the policies can be ordered as follows: (A1 or D1), (B1 or C1), (A2 or D2), (B2 or C2) in an ascending order regarding the cost. So the least-costing policy for the buyer is A1 or D1.


### 4.2. Example  2

In this example we assume the failure rates and repair rates are as follows:(4.2)λ1=0.3,λ2=0.133,λ3=0.2,λ4=0.333,λ5=0.15,λ6=0.25,μ1=0.467,μ2=0.326,μ3=0.256,μ4=0.167,μ5=0.526,μ6=0.356.
Then the steady state availability will be 0.8522.To compare between the different policies, we introduce four values for repair cost: 100, 200, 1000, and 3000 and two values for preventive maintenance cost: 40, 20. Table2 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.Table 2Choices12345678CR1001002002001000100030003000CPM4020402040204020CMA11501503003001500150045004500CBA1283.1283.1566.2566.22831.22831.28493.78493.7Tot.CA1433.1433.1866.2866.24331.24331.21299412993.7CMA2122.2122.2244.3244.31221.71221.736653665CBA23003006006003000300090009000Tot.CA2422.2422.2844.3844.34221.74221.71266512665CMB13502505004001700160047004600CBB1883.1583.11166.2866.23431.23131.29093.78793.7Tot.CB11233.1833.11666.21266.25131.24731.213793.713393.7CMB2322.2222.2444.3344.31421.71321.738653765CBB290060012009003600330096009300Tot.CB21222.2822.21644.31244.35021.74621.71346513065CMC11501503003001500150045004500CBC1883.1583.11166.2866.23431.23131.29093.78793.7Tot.CC11033.1733.11466.21166.24931.24631.213593.713293.7CMC2122.2122.2244.3244.31221.71221.736653665CBC290060012009003600330096009300Tot.CC21022.2722.21444.31144.34821.74521.71326512965CMD13502505004001700160047004600CBD1283.1283.1566.2566.22831.22831.28493.78493.7Tot.CD1633.1533.11066.2966.24531.24431.213193.713093.7CMD2322.2222.2444.3344.31421.71321.738653765CBD23003006006003000300090009000Tot.CD2622.2522.21044.3944.34421.74321.71286512765From Table2, comparing the suggestion policies by regarding the expected total cost, we find that policy A2 is the optimal policy.For choices (1–5) we find that the policies can be ordered as follows: A2, A1, D2, D1, C2, C1, B2, B1, arranged in an ascending order regarding the cost.But for choices (6, 7) the policies can be ordered as follows: A2, D2, A1, D1, C2, B2, C1, B1, and for the choice (8) the policies can be ordered as follows: A2, D2, C2, A1, B2, D1, C1, B1, also arranged in an ascending order regarding the cost.But from the manufacturer point of view, if we order these policies according to the least cost, we find that for choices (1–4) the policies can be arranged as follows: (A2 or C2), (A1 or C1), (B2 or D2), (B1 or D1).For choices (5–8) the policies can be ordered as follows: (A2 or C2), (B2 or D2), (A1 or C1), (B1 or D1).From the buyer point of view, we find that for choices (1–7) the policies can be ordered as follows: (A1 or D1), (A2 or D2), (B1 or C1), (B2 or C2). And for the choice (8) the policies can be ordered as follows: (A1 or D1), (B1 or C1), (A2 or D2), (B2 or C2).For both the manufacturer and the buyer, the policies are also arranged in an ascending order regarding the cost.


## 4.1. Example  1

We assume that the exponential distribution for the failure time and repair time for the four-state system.The parameters values of the failure rates and repair rates were given as (Lisnianski and Levitin [12]): (4.1)λ1=2,λ2=1,λ3=0.7,λ4=0.3,λ5=0.4,λ6=0.1,μ1=100,μ2=80,μ3=50,μ4=45,μ5=40,μ6=32.
Then the steady state availability will be 0.9991.Now to compare between the different policies, we introduce four values for repair cost: 100, 200, 1000, and 3000 and two values for preventive maintenance cost: 20, 10. Table1 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.Table 1Choices12345678CR1001002002001000100030003000CPM2010201020102010CMA160060012001200600060001800018000CBA1430430860860430043001290012900Tot.CA1103010302060206010300103003090030900CMA2350350700700350035001050010500CBA2490490980980490049001470014700Tot.CA284084016801680840084002520025200CMB166063012601230606060301806018030CBB15705001000930444043701304012970Tot.CB1123011302260216010500104003110031000CMB2410380760730356035301056010530CBB263056011201050504049701484014770Tot.CB2104094018801780860085002540025300CMC160060012001200600060001800018000CBC15705001000930444043701304012970Tot.CC1117011002200213010440103703104030970CMC2350350700700350035001050010500CBC263056011201050504049701484014770Tot.CC298091018201750854084702534025270CMD166063012601230606060301806018030CBD1430430860860430043001290012900Tot.CD1109010602120209010360103303096030930CMD2410380760730356035301056010530CBD2490490980980490049001470014700Tot.CD290087017401710846084302526025230From Table1, comparing the suggestion policies by regarding the expected total cost, we find that policy A2 is the optimal policy, where no PM action over the whole useful life of the system is assumed and the warranty includes the first and the second states.If we order these policies according to the least total cost, we find that A2 policy is the least-costing policy for choice 1, followed by policies D2, C2, A1, B2, D1, C1, B1 in an ascending order regarding the cost. But the rest of choices (2–8), although A2 is still the least-costing policy for each, are of a bit different sequence (D2, C2, B2, A1, D1, C1, B1) also in an ascending order regarding the cost.But from the manufacturer point of view, if we order these policies  according to the least cost, we find that the policies can be ordered in an ascending order regarding the cost as follows, (A2 or C2), (B2 or D2), (A1 or C1), (B1 or D1). So we find that the least-costing policy for the manufacturer is A2 or C2.From the buyer point of view, we find that for the choices (1–4), the policies can be ordered as follows: (A1 or D1), (A2 or D2), (B1 or C1), (B2 or C2). But for the choices (5–8) the policies can be ordered as follows: (A1 or D1), (B1 or C1), (A2 or D2), (B2 or C2) in an ascending order regarding the cost. So the least-costing policy for the buyer is A1 or D1.


## 4.2. Example  2

In this example we assume the failure rates and repair rates are as follows:(4.2)λ1=0.3,λ2=0.133,λ3=0.2,λ4=0.333,λ5=0.15,λ6=0.25,μ1=0.467,μ2=0.326,μ3=0.256,μ4=0.167,μ5=0.526,μ6=0.356.
Then the steady state availability will be 0.8522.To compare between the different policies, we introduce four values for repair cost: 100, 200, 1000, and 3000 and two values for preventive maintenance cost: 40, 20. Table2 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.Table 2Choices12345678CR1001002002001000100030003000CPM4020402040204020CMA11501503003001500150045004500CBA1283.1283.1566.2566.22831.22831.28493.78493.7Tot.CA1433.1433.1866.2866.24331.24331.21299412993.7CMA2122.2122.2244.3244.31221.71221.736653665CBA23003006006003000300090009000Tot.CA2422.2422.2844.3844.34221.74221.71266512665CMB13502505004001700160047004600CBB1883.1583.11166.2866.23431.23131.29093.78793.7Tot.CB11233.1833.11666.21266.25131.24731.213793.713393.7CMB2322.2222.2444.3344.31421.71321.738653765CBB290060012009003600330096009300Tot.CB21222.2822.21644.31244.35021.74621.71346513065CMC11501503003001500150045004500CBC1883.1583.11166.2866.23431.23131.29093.78793.7Tot.CC11033.1733.11466.21166.24931.24631.213593.713293.7CMC2122.2122.2244.3244.31221.71221.736653665CBC290060012009003600330096009300Tot.CC21022.2722.21444.31144.34821.74521.71326512965CMD13502505004001700160047004600CBD1283.1283.1566.2566.22831.22831.28493.78493.7Tot.CD1633.1533.11066.2966.24531.24431.213193.713093.7CMD2322.2222.2444.3344.31421.71321.738653765CBD23003006006003000300090009000Tot.CD2622.2522.21044.3944.34421.74321.71286512765From Table2, comparing the suggestion policies by regarding the expected total cost, we find that policy A2 is the optimal policy.For choices (1–5) we find that the policies can be ordered as follows: A2, A1, D2, D1, C2, C1, B2, B1, arranged in an ascending order regarding the cost.But for choices (6, 7) the policies can be ordered as follows: A2, D2, A1, D1, C2, B2, C1, B1, and for the choice (8) the policies can be ordered as follows: A2, D2, C2, A1, B2, D1, C1, B1, also arranged in an ascending order regarding the cost.But from the manufacturer point of view, if we order these policies according to the least cost, we find that for choices (1–4) the policies can be arranged as follows: (A2 or C2), (A1 or C1), (B2 or D2), (B1 or D1).For choices (5–8) the policies can be ordered as follows: (A2 or C2), (B2 or D2), (A1 or C1), (B1 or D1).From the buyer point of view, we find that for choices (1–7) the policies can be ordered as follows: (A1 or D1), (A2 or D2), (B1 or C1), (B2 or C2). And for the choice (8) the policies can be ordered as follows: (A1 or D1), (B1 or C1), (A2 or D2), (B2 or C2).For both the manufacturer and the buyer, the policies are also arranged in an ascending order regarding the cost.



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*Source: 407457-2011-06-26.xml*