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

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.


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.

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.
Figure 1 shows the transition rates.The transition probabilities of the Markov process are given by where P j 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
Taking the Laplace transformations of 2.2 yields the following linear equations in terms of P * j s for j 0, 1, 2, 3:

2.3
Under the initial conditions, P 0 0 1, P 1 0 0, P 2 0 0, and P 3 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: where

2.5
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 A t is given by

2.7
The steady state availability of the system is given by where c 1 , c 2 , c 3 , and f 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.

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.

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 C MA1 is the cost for the manufacturer in policy A1 and C R is the repair cost.
ii The warranty cost for the buyer can be represented by where C BA1 is the cost for the buyer in policy A1.

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 3.4

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 C PM is the cost for the PM.
ii The warranty cost for the buyer is computed as follows: 3.6

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
ii The warranty cost for the buyer 3.8

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: 3.10

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:

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
ii The warranty cost for the buyer is computed as follows: 3.14

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
ii The warranty cost for the buyer is given as:

3.16
In the next section, we will show the cost of these polices by two numerical examples.

Example 1
We assume that the exponential distribution for the failure time and repair time for the fourstate 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.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 leastcosting 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.

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.
From Table 2, comparing the suggestion policies by regarding the expected total cost, we find that policy A2 is the optimal policy.
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 .
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.