ISRN Applied MathematicsVolume 2011 (2011), Article ID 407457, 12 pagesdoi:10.5402/2011/407457

Research Article

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

Ahmed F. Attia,¹ Eman D. Abou Elela,² and Hany A. Hosham³

¹Department of Mathematical Statistics, Institute of Statistical Studies & Research, Cairo University, Giza 12613, Egypt²Department of Statistics, Faculty of Commerce, Al-Azhar University (Assiut), Assiut, Egypt³Mathematical Institute, University of Cologne, 50923 Cologne, Germany

Received 28 March 2011; Accepted 28 April 2011

Academic Editor: A. Bellouquid

Copyright © 2011 Ahmed F. Attia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 Section 2 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, 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 are , and the repair rates are .

Figure 1 shows the transition rates.

Figure 1: Transition rates.

The transition probabilities of the Markov process are given by where   is the probability that the system is in state j (), at time .

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: Taking the Laplace transformations of (2.2) yields the following linear equations in terms of for : Under the initial conditions, , and , 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: where

The probability that the system is in working state at time t is called the point availability of the system.

The system point availability is given by The Laplace transform of is given by The steady state availability of the system is given by where 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 is L 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 where 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 where 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(ii)The warranty cost for the buyer can be represented by

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: where PM is the cost for the PM.(ii)The warranty cost for the buyer is computed as follows:

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 (ii)The warranty cost for the buyer

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:(ii)The warranty cost for the buyer is computed as follows:

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:(ii)The warranty cost for the buyer is given as:

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:(ii)The warranty cost for the buyer is computed as follows:

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:(ii)The warranty cost for the buyer is given as: 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]): 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. Table 1 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.

Table 1

Choices
1
2
3
4
5
6
7
8

100
100
200
200
1000
1000
3000
3000

PM
20
10
20
10
20
10
20
10

600
600
1200
1200
6000
6000
18000
18000

430
430
860
860
4300
4300
12900
12900

Tot.
1030
1030
2060
2060
10300
10300
30900
30900

350
350
700
700
3500
3500
10500
10500

490
490
980
980
4900
4900
14700
14700

Tot.
840
840
1680
1680
8400
8400
25200
25200

660
630
1260
1230
6060
6030
18060
18030

570
500
1000
930
4440
4370
13040
12970

Tot.
1230
1130
2260
2160
10500
10400
31100
31000

410
380
760
730
3560
3530
10560
10530

630
560
1120
1050
5040
4970
14840
14770

Tot.
1040
940
1880
1780
8600
8500
25400
25300

600
600
1200
1200
6000
6000
18000
18000

570
500
1000
930
4440
4370
13040
12970

Tot.
1170
1100
2200
2130
10440
10370
31040
30970

350
350
700
700
3500
3500
10500
10500

630
560
1120
1050
5040
4970
14840
14770

Tot.
980
910
1820
1750
8540
8470
25340
25270

660
630
1260
1230
6060
6030
18060
18030

430
430
860
860
4300
4300
12900
12900

Tot.
1090
1060
2120
2090
10360
10330
30960
30930

410
380
760
730
3560
3530
10560
10530

490
490
980
980
4900
4900
14700
14700

Tot.
900
870
1740
1710
8460
8430
25260
25230

From Table 1, 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: 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. Table 2 illustrates the total cost, the manufacturer cost, and buyer cost resulting from using these policies.

Table 2

Choices
1
2
3
4
5
6
7
8

C R
100
100
200
200
1000
1000
3000
3000

40
20
40
20
40
20
40
20

150
150
300
300
1500
1500
4500
4500

283.1
283.1
566.2
566.2
2831.2
2831.2
8493.7
8493.7

Tot. A1
433.1
433.1
866.2
866.2
4331.2
4331.2
12994
12993.7

122.2
122.2
244.3
244.3
1221.7
1221.7
3665
3665

300
300
600
600
3000
3000
9000
9000

Tot. A2
422.2
422.2
844.3
844.3
4221.7
4221.7
12665
12665

350
250
500
400
1700
1600
4700
4600

883.1
583.1
1166.2
866.2
3431.2
3131.2
9093.7
8793.7

Tot. B1
1233.1
833.1
1666.2
1266.2
5131.2
4731.2
13793.7
13393.7

322.2
222.2
444.3
344.3
1421.7
1321.7
3865
3765

900
600
1200
900
3600
3300
9600
9300

Tot. B2
1222.2
822.2
1644.3
1244.3
5021.7
4621.7
13465
13065

150
150
300
300
1500
1500
4500
4500

883.1
583.1
1166.2
866.2
3431.2
3131.2
9093.7
8793.7

Tot. C1
1033.1
733.1
1466.2
1166.2
4931.2
4631.2
13593.7
13293.7

122.2
122.2
244.3
244.3
1221.7
1221.7
3665
3665

900
600
1200
900
3600
3300
9600
9300

Tot. C2
1022.2
722.2
1444.3
1144.3
4821.7
4521.7
13265
12965

350
250
500
400
1700
1600
4700
4600

283.1
283.1
566.2
566.2
2831.2
2831.2
8493.7
8493.7

Tot. D1
633.1
533.1
1066.2
966.2
4531.2
4431.2
13193.7
13093.7

322.2
222.2
444.3
344.3
1421.7
1321.7
3865
3765

300
300
600
600
3000
3000
9000
9000

Tot. D2
622.2
522.2
1044.3
944.3
4421.7
4321.7
12865
12765

From Table 2, 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.

References

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