# Reliability and Sensitivity Analysis of a Repairable System with Warranty and Administrative Delay in Repair

Authors: Mohamed S. El-Sherbeny; Zienab M. Hussien
Journal: Journal of Mathematics
 (2021)

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## Abstract

The purpose of this study is to analyze the behavior of some industrial systems in light of the cost-free warranty policy. According to this policy, we assume that the repairman is not always present in the system. When the active unit fails, the repairman will be called to visit the system; however, administrative procedures may delay the visit for some time. Once on the system, the repairman first inspects whether the fault is caused by the user or not and whether it is repairable or not. According to product warranty laws, the repairman carries out the repair or replacement of the faulty unit. The failure time, administrative delay time, inspection time, and repair time are assumed taken as a negative exponential distribution. The system model is analyzed by the supplementary variable technique and Laplace transform, as various performance metrics of system efficiency have been obtained. The sensitivity and relative sensitivity analyses for the system parameters have also been performed. Finally, an illustrative example is taken to illustrate the efficiency of the system.


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## Body

## 1. Introduction

In the present time, warranty is a necessary condition for selling many industrial products. Warranty is defined as the mandatory contract afforded by a manufacturer with regard to the sale of a product. In other words, the warranty is a legal responsibility in the following cases: early failure of the product or the disability of the product to perform its function. When selling a product, the warranty covers either free repair or free replacement. Rules of the warranty are classified as follows: (1) the manufacturer is not legally responsible for the product in case of failure due to incorrect and careless use or deterioration of the product by the user; this makes the customer responsible for the cost of the product repair; (2) the customer is not legally responsible for the product in the following cases: (a) manufacturing defects and (b) errors in software; this makes the manufacturer responsible for free repair or free replacement of the product.More recently, users have distinguished single-unit systems due to their ability to afford the cost and ingrained reliability. Therefore, reliability models of single-unit systems have been examined by researchers, such as Kumar et al. [1], Du et al. [2], Goel and Mumtaz [3], Ram and Kadyan [4], Attia et al. [5], and Malhotra and Taneja [6], under different conditions from failure and repair policies without interest paying much attention to any warranty policies and service agreement. The only way to ensure the efficiency of the product sold is after-sales services related to warranty policy. Therefore, the longer warranty leads to best reliability and provides convenience for the users. Kadyan and Ramniwas [7] calculated the cost benefit of the single-unit system and derived some reliability measures when applying warranty policy on the system. The profit function of industrial system with the presence of a cost-free warranty policy is obtained by Ram and Garg [8]. Effect warranty policies on remanufactured products are discussed by Alqahtani and Gupta [9]. Ram et al. [10] study system consists of a single unit with inspection for the feasibility of repair beyond the warranty. Ram et al. [11] presented the concept of preventive maintenance (PM) beyond warranty for a single-unit system. Providing warranty to the system for a certain period of operation is one of the effective ways to ensure the reliability of a system. Therefore, Mo et al. [12] presented a new warranty policy where the buyer invests in PM costs within the product life cycle to reduce losses due to production failures. Huang et al. [13] presented an approach using preventive maintenance with extended policy. Chopra and Ram [14] derived the reliability measures of the system subjected to two types of failure, and under this effect, two different types of repair are modeled using Gumbel–Hougaard family copula.In this article, we discussed the administrative delay, that is, the time associated with processes or tasks not directly related to the recovery or repair of the failed equipment, such as processing requests, short-term unavailability of repair facilities, or delays in calling the repairman due to higher priorities. Therefore, the idea of administrative delay is very important because the administrative delay of failed equipment leads to more delays in expected production and more complaints from customers, which makes it more difficult for organizations to serve their customers. However, the concept of a single-unit system with administrative delay in the repair under warranty has not previously appeared in the literature.In light of the previously mentioned observations, this article analyzes a single-unit repairable system with administrative repair delay under warranty using the supplementary variables technique. The cost of repairs during the warranty period will be borne by the manufacturer, but the warranty does not cover product failure due to damage caused by the user, such as misuse, accident, and physical damage, during the warranty period. If the active unit is down, the technician has to be called to the system, which takes some time to reach the system due to some administrative procedures. In such systems, there is a single server that decides whether the error is caused by the user or not, and whether it can be repaired or replaced after the inspection process. All times of the system follow a negative exponential distribution. The expressions for some economic measures, such as reliability, mean time to system failure, and availability, were derived to illustrate system behavior and evolution.


## 2. Model and Assumptions

The system under study has a single active unit with a single service repair facility. We, specifically, have the following assumptions.(1)
The system contains a single active unit for working(2)
There is a single repairer that is not always available with the system to perform repair or replacement and inspection of a failing unit(3)
When the active unit failed, the repairman must be called to visit the system that takes some time to come to the system due to some administrative actions(4)
As soon as the repairman arrives for the failed unit, he first inspects that appliance and decides whether or not the fault was caused by the user and can be repaired or replaced(5)
The cost of repair or replacement for the failed unit during warranty is borne by the manufacturer, provided malfunctions are not due to the users(6)
The distributions of all times in the system are negative exponential(7)
The system works as new after repair or replacement


## 3. Model Development

### 3.1. State Specification

To discuss the behavior of the system at any timet, we can describe this system according to the following states:S0/S1: the unit is operative during the working period within/out of the warrantyS2/S6: the unit is failure mode and waiting for repairman due to administrative delay within/out of the warrantyS3/S7: the failed unit under inspection within/out of the warrantyS4/S8: the failed unit under replacement within/out of the warrantyS5/S9: the failed unit under repair within/out of the warranty


### 3.2. Notations

λ1/λ2: the constant failure rate of the unit within the warranty period/out of the warranty periodα: constant rate of accomplishment of the warranty periodμ1/μ4: repair rate of the unit in warranty period/out of the warranty periodμ2/μ3: replacement rate of the unit in warranty period/out of the warranty periodβ1/β2: rate of administrative delay time for getting the repairman available within warranty period/out of the warranty periodγ/η: inspection rate of the failed unit within warranty period/out of the warranty periodθ1: the probability that the failed unit will be replaced within the warranty period, according to the inspector's decisionθ2: the probability that the failed unit will be repaired within the warranty period, according to the inspector's decisionθ3: the probability that the failed unit will be replaced outside the warranty period, due to customer misuseθ4: the probability that the failed unit will be repaired outside the warranty period, due to customer misuseφ1: the probability that the failed unit will be repaired outside the warranty period, according to the decision of the inspector at the completion of the warranty periodφ2: the probability that the failed unit will be replaced outside the warranty period, according to the decision of the inspector at the completion of the warranty periodπ0t/π1t: the probability that at time t the system is in good state in warranty period/beyond warranty periodπ2t/π6t: the probability that at time t the system is in a failed state and getting the repairman available in warranty period/beyond the warranty periodπ3t/π7t: the probability that the system will undergo an inspection process at time t to determine whether a failed unit should be replaced or repaired within/beyond the warranty periodπ4t/π8t: the probability of replacement of the failed unit by the decision of the inspector within/outside the warranty periodπ5t/π9t: the probability of repair of the failed unit by the decision of the inspector within/outside the warranty period


## 3.1. State Specification

To discuss the behavior of the system at any timet, we can describe this system according to the following states:S0/S1: the unit is operative during the working period within/out of the warrantyS2/S6: the unit is failure mode and waiting for repairman due to administrative delay within/out of the warrantyS3/S7: the failed unit under inspection within/out of the warrantyS4/S8: the failed unit under replacement within/out of the warrantyS5/S9: the failed unit under repair within/out of the warranty


## 3.2. Notations

λ1/λ2: the constant failure rate of the unit within the warranty period/out of the warranty periodα: constant rate of accomplishment of the warranty periodμ1/μ4: repair rate of the unit in warranty period/out of the warranty periodμ2/μ3: replacement rate of the unit in warranty period/out of the warranty periodβ1/β2: rate of administrative delay time for getting the repairman available within warranty period/out of the warranty periodγ/η: inspection rate of the failed unit within warranty period/out of the warranty periodθ1: the probability that the failed unit will be replaced within the warranty period, according to the inspector's decisionθ2: the probability that the failed unit will be repaired within the warranty period, according to the inspector's decisionθ3: the probability that the failed unit will be replaced outside the warranty period, due to customer misuseθ4: the probability that the failed unit will be repaired outside the warranty period, due to customer misuseφ1: the probability that the failed unit will be repaired outside the warranty period, according to the decision of the inspector at the completion of the warranty periodφ2: the probability that the failed unit will be replaced outside the warranty period, according to the decision of the inspector at the completion of the warranty periodπ0t/π1t: the probability that at time t the system is in good state in warranty period/beyond warranty periodπ2t/π6t: the probability that at time t the system is in a failed state and getting the repairman available in warranty period/beyond the warranty periodπ3t/π7t: the probability that the system will undergo an inspection process at time t to determine whether a failed unit should be replaced or repaired within/beyond the warranty periodπ4t/π8t: the probability of replacement of the failed unit by the decision of the inspector within/outside the warranty periodπ5t/π9t: the probability of repair of the failed unit by the decision of the inspector within/outside the warranty period


## 4. Analysis of the System

### 4.1. Reliability

The system reliabilityRt is the probability that the product or system functions adequately for a speciﬁed period (0, t]. To deduce an expression for the reliability of the system, we using the Markov process to the up states, namely, S0 and S1, as shown in Figure 1. We derive the following differential equations for reliability:(1)ddt+λ1+απ0t=0,(2)ddt+λ2π1t=απ0t.Figure 1
State-transition diagram of the system.The initial conditions are as follows:(3)πi0=1,i=0,0,i≠0.Using the Laplace transform of (1) and (2) and using also the initial condition, we get(4)s+λ1+απ0s=1,(5)s+λ2π1s=απ0s.From (4), we get(6)π0s=1s+λ1+α.By substituting (6) into (5), we get(7)π1s=αs+λ1+αs+λ2.Then, the Laplace transformation formula of the reliabilityRs of the system is(8)Rs=π0s+π1s=s+λ2+αs+λ1+αs+λ2.Taking the inverse Laplace transform for (8), we get(9)Rt=L−1Rs=λ1−λ2λ1−λ2+αexp−λ1+αt+αλ1−λ2+αexp−λ2t.


### 4.2. Mean Time to System Failure (MTTF)

MTTF is a measurement of reliability for nonrepairable systems. It is the mean time predictable until the first fail of the system. By using (9), we get(10)MTTF=∫0∞Rtdt=λ2λ1−λ22+αλ1+α2λ2λ1−λ2+αλ1+α.


### 4.3. System Availability

AvailabilityAvt is the probability that the system (or unit) will be inoperative in the long term. Starting from the states of the system as in Figure 1, we can describe the differential equations in each state of this system and are summarized as follows:(11)ddt+λ1+απ0t=μ2π4t+μ1π5t,ddt+λ2π1t=απ0t+μ3π8t+μ4π9t,ddt+β1π2t=λ1π0t,ddt+γπ3t=β1π2t,ddt+μ2π4t=θ1γπ3t,ddt+μ1π5t=θ2γπ3t,ddt+β2π6t=λ2π1t,ddt+ηπ7t=β2π6t,ddt+μ3π8t=θ3γπ3t+φ2ηπ7t,ddt+μ4π9t=θ4γπ3t+φ1ηπ7t.The initial conditions are as follows:(12)πi0=1,i=0,0,i≠0.Taking Laplace transform of all the previous equations, we obtain the following:(13)s+λ1+απ0s=1+μ2π4s+μ1π5s,(14)s+λ2π1s=απ0s+μ3π8s+μ4π9s(15)s+β1π2s=λ1π0s,(16)s+γπ3s=β1π2s,(17)s+μ2π4s=θ1γπ3s,(18)s+μ1π5s=θ2γπ3s,(19)s+β2π6s=λ2π1s,(20)s+ηπ7s=β2π6s,(21)s+μ3π8s=θ3γπ3s+φ2ηπ7s,(22)s+μ4π9s=θ4γπ3s+φ1ηπ7s.From (13)–(22), we obtain(23)π0s=Cs,(24)π1s=C1s,(25)π2s=λ1s+β1Cs,(26)π3s=β1λ1s+γs+β1Cs,(27)π4s=θ1γβ1λ1s+γs+β1s+μ2Cs,(28)π5s=θ2γβ1λ1s+γs+β1s+μ1Cs,(29)π6s=λ2s+β2C1s,(30)π7s=β2λ2s+ηs+β2C1s,(31)π8s=As+A1sC1s,(32)π9s=A2s+A3sC1s,where,(33)Cs=s+μ1s+μ2s+γs+β1s+λ1+αs+μ1s+μ2s+γs+β1−μ2θ1s+μ1+μ1θ2s+μ2γβ1λ1,As=θ3γβ1λ1Css+μ3s+γs+β1,A1s=φ2ηβ2λ2s+μ3s+ηs+β2,A2s=θ4γβ1λ1Css+μ4s+γs+β1,A3s=φ1ηβ2λ2s+μ4s+ηs+β2,C1s=αCs+μ3As+μ1A2ss+λ2−μ3A1s−μ1A3s.It is worth mentioning that(34)∑i=09πis=1s.Using (23)–(34), the Laplace transformation formula of the availability Avs is given by(35)Avs=π0s+π1s=Cs+C1s.Hence, the steady-state availabilityAv∞ of the system is(36)Av∞=Lims⟶0sπ0s+π1s.


## 4.1. Reliability

The system reliabilityRt is the probability that the product or system functions adequately for a speciﬁed period (0, t]. To deduce an expression for the reliability of the system, we using the Markov process to the up states, namely, S0 and S1, as shown in Figure 1. We derive the following differential equations for reliability:(1)ddt+λ1+απ0t=0,(2)ddt+λ2π1t=απ0t.Figure 1
State-transition diagram of the system.The initial conditions are as follows:(3)πi0=1,i=0,0,i≠0.Using the Laplace transform of (1) and (2) and using also the initial condition, we get(4)s+λ1+απ0s=1,(5)s+λ2π1s=απ0s.From (4), we get(6)π0s=1s+λ1+α.By substituting (6) into (5), we get(7)π1s=αs+λ1+αs+λ2.Then, the Laplace transformation formula of the reliabilityRs of the system is(8)Rs=π0s+π1s=s+λ2+αs+λ1+αs+λ2.Taking the inverse Laplace transform for (8), we get(9)Rt=L−1Rs=λ1−λ2λ1−λ2+αexp−λ1+αt+αλ1−λ2+αexp−λ2t.


## 4.2. Mean Time to System Failure (MTTF)

MTTF is a measurement of reliability for nonrepairable systems. It is the mean time predictable until the first fail of the system. By using (9), we get(10)MTTF=∫0∞Rtdt=λ2λ1−λ22+αλ1+α2λ2λ1−λ2+αλ1+α.


## 4.3. System Availability

AvailabilityAvt is the probability that the system (or unit) will be inoperative in the long term. Starting from the states of the system as in Figure 1, we can describe the differential equations in each state of this system and are summarized as follows:(11)ddt+λ1+απ0t=μ2π4t+μ1π5t,ddt+λ2π1t=απ0t+μ3π8t+μ4π9t,ddt+β1π2t=λ1π0t,ddt+γπ3t=β1π2t,ddt+μ2π4t=θ1γπ3t,ddt+μ1π5t=θ2γπ3t,ddt+β2π6t=λ2π1t,ddt+ηπ7t=β2π6t,ddt+μ3π8t=θ3γπ3t+φ2ηπ7t,ddt+μ4π9t=θ4γπ3t+φ1ηπ7t.The initial conditions are as follows:(12)πi0=1,i=0,0,i≠0.Taking Laplace transform of all the previous equations, we obtain the following:(13)s+λ1+απ0s=1+μ2π4s+μ1π5s,(14)s+λ2π1s=απ0s+μ3π8s+μ4π9s(15)s+β1π2s=λ1π0s,(16)s+γπ3s=β1π2s,(17)s+μ2π4s=θ1γπ3s,(18)s+μ1π5s=θ2γπ3s,(19)s+β2π6s=λ2π1s,(20)s+ηπ7s=β2π6s,(21)s+μ3π8s=θ3γπ3s+φ2ηπ7s,(22)s+μ4π9s=θ4γπ3s+φ1ηπ7s.From (13)–(22), we obtain(23)π0s=Cs,(24)π1s=C1s,(25)π2s=λ1s+β1Cs,(26)π3s=β1λ1s+γs+β1Cs,(27)π4s=θ1γβ1λ1s+γs+β1s+μ2Cs,(28)π5s=θ2γβ1λ1s+γs+β1s+μ1Cs,(29)π6s=λ2s+β2C1s,(30)π7s=β2λ2s+ηs+β2C1s,(31)π8s=As+A1sC1s,(32)π9s=A2s+A3sC1s,where,(33)Cs=s+μ1s+μ2s+γs+β1s+λ1+αs+μ1s+μ2s+γs+β1−μ2θ1s+μ1+μ1θ2s+μ2γβ1λ1,As=θ3γβ1λ1Css+μ3s+γs+β1,A1s=φ2ηβ2λ2s+μ3s+ηs+β2,A2s=θ4γβ1λ1Css+μ4s+γs+β1,A3s=φ1ηβ2λ2s+μ4s+ηs+β2,C1s=αCs+μ3As+μ1A2ss+λ2−μ3A1s−μ1A3s.It is worth mentioning that(34)∑i=09πis=1s.Using (23)–(34), the Laplace transformation formula of the availability Avs is given by(35)Avs=π0s+π1s=Cs+C1s.Hence, the steady-state availabilityAv∞ of the system is(36)Av∞=Lims⟶0sπ0s+π1s.


## 5. Sensitivity and Relative Sensitivity Analysis

In this section, we achieve sensitivity analysis and relative sensitivity analysis on MTTF andRt with regard to one of the system parameters θ, where θ=λ1,λ2,β1,β2,μ1,μ2,μ3,μ4,α,η,γ.


### 5.1. Sensitivity and Relative Sensitivity Analysis for MTTF

Differentiating (10) with regard toκ, we obtain(37)ηκ=∂MTTF∂κ.We can define the relative sensitivity (R.S) of MTTF as the percentage (P.C) change resulting from the percentage (P.C) change in one of the system parametersκ.(38)ωκ=ηκκMTTF,where κ=λ1,λ2,α


### 5.2. Sensitivity and Relative Sensitivity Analysis forRt

We performed the sensitivity analysis of changes inRt of one of the system parameters κ.Differentiating (9) with regard to κ, we obtain(39)σκ=∂Rt∂κ=∂∂κλ1−λ2λ1−λ2+αexp−λ1+αt+αλ1−λ2+αexp−λ2t.The relative sensitivity (R. S) ofRt is as follows:(40)ψκ=σκκRt.


## 5.1. Sensitivity and Relative Sensitivity Analysis for MTTF

Differentiating (10) with regard toκ, we obtain(37)ηκ=∂MTTF∂κ.We can define the relative sensitivity (R.S) of MTTF as the percentage (P.C) change resulting from the percentage (P.C) change in one of the system parametersκ.(38)ωκ=ηκκMTTF,where κ=λ1,λ2,α


## 5.2. Sensitivity and Relative Sensitivity Analysis forRt

We performed the sensitivity analysis of changes inRt of one of the system parameters κ.Differentiating (9) with regard to κ, we obtain(39)σκ=∂Rt∂κ=∂∂κλ1−λ2λ1−λ2+αexp−λ1+αt+αλ1−λ2+αexp−λ2t.The relative sensitivity (R. S) ofRt is as follows:(40)ψκ=σκκRt.


## 6. Numerical Results for Related Analysis

In this section, we discuss some numerical results to perform several related analyses. In Subsection5.1, we present how system reliability Rt, mean time to system failure (MTTF), and steady-state availability are impacted by changing the values of the several system parameters. Moreover, in Subsection 5.2, some numerical results for sensitivity and relative sensitivity analysis for the proposed system are discussed.


### 6.1. System Reliability Analysis

First, we analyze graphically the effects of various system parameters onRt, shown in Figures 2–5. In Figure 2, we can observe that the system reliability increases as λ1 decreases. Remember that the reliability is the probability that the system is working correctly over a period of time. Figure 4 shows that the system reliability increases when the failure rate of the active unit λ2 decreases. We observe that the system reliability depends on the change of the system parameters λ1, λ2, and α. Therefore, we conclude from Figure 3 that the system reliability increases when α>λ1. By following the same reason previously mentioned, the higher the system warranty completion rate α and the lower the failure rate λ2, the higher the system reliability, as shown in Figure 5.Figure 2
System reliability versus failure rateλ1.Figure 3
System reliability versus failure rateλ1 and rate of completion of warranty α.Figure 4
System reliability versus failure rateλ2.Figure 5
System reliability versus failure rateλ2 and rate of completion of warranty α.Next, we present some numerical results of the mean time to system failure MTTF to discuss the influence of various system parameters on the behavior of the mean time to system failure, which are shown in Tables1 and 2. In Table 1, the influence of parameter α on MTTF is shown as follows.(1)
In the first column of this table, the value of MTTF decreases whenλ1<λ2, and α varies from 0.2 to 1.4.(2)
In the second column of the same table, the value of MTTF is constant whenλ1=λ2, and α varies from 0.2 to 1.4.(3)
Finally, in the third column, we show that the value of MTTF increases whenλ1>λ2, and α varies from 0.2 to 1.4.Table 1
The mean time to system failure (MTTF) for different values ofα.
αλ1<λ2λ1=λ2λ1>λ2λ1=0.3,λ2=0.6λ1=0.3,λ2=0.3λ1=0.6,λ2=0.30.22.666673.333332.083330.42.380953.333332.333330.62.222223.333332.50.82.121213.333332.619051.02.051283.333332.708331.22.003.333332.777781.41.960783.333332.83333Table 2
The mean time to system failure (MTTF) for different values of parametersλ1 and λ1 when (α=0.6).
λ1λ2=0.3λ2λ1=0.3MTTFMTTF0.14.285710.17.777780.23.750.24.444440.33.333330.33.333330.43.00.42.777780.52.727270.52.444440.62.50.62.222220.72.307690.72.063490.82.142860.81.94444Table2 shows the effect of system parameters λ1 and λ2 on MTTF. It is observed that the MTTF decreases with increasing λ1 and λ2, and this is quite obvious at α=0.6.Finally, we display the numerical results for the steady-state availability. We fixedβ1=0.4;β2=0.6;η=0.4;θ3=0.25;θ4=0.25;μ4=0.6;μ5=0.4;φ2=0.4 as the base in this study. First, Table 3 shows how the steady-state availability of the system is affected by the change of α values as follows.(1)
In the first and second columns of this table, the value of the steady-state availability decreases whenλ1≤λ2, and α∈0.1,…,0.9.(2)
In the third column, we show that the value of the steady-state availability increases whenλ1≤λ2, and α∈0.1,…,0.9.Table 3
The steady-state availability for different values ofα when (β1=0.4,β2=0.6,η=0.4,θ3=0.25,θ4=0.25,μ3=0.4,μ4=0.6,φ2=0.4).
αλ1<λ2λ1=λ2λ1>λ2λ1=0.3,λ2=0.6λ1=0.3,λ2=0.3λ1=0.6,λ2=0.30.10.3191490.5263160.2105260.20.2836880.4678360.2339180.30.2659570.4385960.2506270.40.2553190.4210530.2631580.50.2482270.4093570.2729040.60.2431610.4010030.2807020.70.2393620.3947370.2870810.80.2364070.3898640.2923980.90.2340430.3859650.296896Next, the changes in the failure rate of an operating unit under warranty (out of warranty),λ1 (λ2), are calculated and the results are presented in Table 4. From Table 4, it can be seen that the steady-state availability of the system decreases when both λ1 and λ2 increase and α=0.6.Table 4
The steady-state availability for different values ofλ1 and λ1 when (α=0.6,β1=0.4,β2=0.6,η=0.4,θ3=0.25,θ4=0.25,μ3=0.4,μ4=0.6,φ2=0.4).
λ1λ2=0.3λ2λ1=0.3Av∞Av∞0.10.2990650.10.4664720.20.2492210.20.293040.30.2136180.30.2136180.40.1869160.40.1680670.50.1661470.50.1385280.60.1495330.60.117820.70.1359390.70.1024980.80.1246110.80.0907029


### 6.2. Numerical Results for Sensitivity Analysis and Relative Sensitivity Analysis

In this part, the numerical results are given for both the sensitivity and relative sensitivity analysis. First, the sensitivity analysis ofRt caused by various parameters is presented, and the results are displayed in Figure 6. From Figure 6, it can be seen that λ1 has a significant effect on Rt, the parameter λ2 has a moderate effect on Rt, and the parameter α has a small influence on Rt, because this curve is close to the horizontal axis. Moreover, the signs of the sensitivity of Rt for λ1, λ2, and α are negative, which implies that the incremental changes of these parameters worsen the system reliability Rt. The results of relative sensitivity on reliability are almost the same as those of sensitivity analysis, but the high levels of each curve are quite different from each other as shown in Figure 7.Figure 6
Sensitivity analysis on reliability functiont∈0,20.Figure 7
Relativity sensitivity analysis on reliability functiont∈0,1.Next, the sensitivity and relative sensitivity analyses of MTTF are performed with various parameters. The sensitivity and relative sensitivity analyses of MTTF with three parameters are tabulated in three tables. Tables5–7 show that the signs of the sensitivity of MTTF with parameters λ1, λ2, and α are negative, indicating that the incremental changes in these parameters degrade MTTF. Furthermore, by looking at the absolute values in these tables, the order of impact on the MTTF with each parameter can be determined. It can be observed that the order of sensitivity on MTTF is λ2>λ1>α and the order of relative sensitivity on MTTF is λ2>λ1>α.Table 5
Sensitivity and relativity sensitivity analysis of MTTF withα.
αλ1=0.4,λ2=0.6ηαωα0.1−1.33333−0.05714290.3−0.680272−0.09523810.5−0.411523−0.101010.7−0.275482−0.09790210.9−0.197239−0.09230771.1−0.148148−0.0862745Table 6
Sensitivity and relativity sensitivity analysis of MTTF withλ1.
λ1α=0.4,λ2=0.6ηλ1ωλ10.1−6.66667−0.20.3−3.40136−0.4285710.5−2.05761−0.5555560.7−1.37741−0.6363640.9−0.986193−0.6923081.1−0.740741−0.733333Table 7
Sensitivity and relativity sensitivity analysis of MTTF withλ2.
λ2α=0.4,λ1=0.3ηλ2ωλ20.2−14.2857−0.6666670.4−3.57143−0.50.6−1.5873−0.40.8−0.892857−0.3333331.0−0.571429−0.2857141.2−0.396825−0.25


## 6.1. System Reliability Analysis

First, we analyze graphically the effects of various system parameters onRt, shown in Figures 2–5. In Figure 2, we can observe that the system reliability increases as λ1 decreases. Remember that the reliability is the probability that the system is working correctly over a period of time. Figure 4 shows that the system reliability increases when the failure rate of the active unit λ2 decreases. We observe that the system reliability depends on the change of the system parameters λ1, λ2, and α. Therefore, we conclude from Figure 3 that the system reliability increases when α>λ1. By following the same reason previously mentioned, the higher the system warranty completion rate α and the lower the failure rate λ2, the higher the system reliability, as shown in Figure 5.Figure 2
System reliability versus failure rateλ1.Figure 3
System reliability versus failure rateλ1 and rate of completion of warranty α.Figure 4
System reliability versus failure rateλ2.Figure 5
System reliability versus failure rateλ2 and rate of completion of warranty α.Next, we present some numerical results of the mean time to system failure MTTF to discuss the influence of various system parameters on the behavior of the mean time to system failure, which are shown in Tables1 and 2. In Table 1, the influence of parameter α on MTTF is shown as follows.(1)
In the first column of this table, the value of MTTF decreases whenλ1<λ2, and α varies from 0.2 to 1.4.(2)
In the second column of the same table, the value of MTTF is constant whenλ1=λ2, and α varies from 0.2 to 1.4.(3)
Finally, in the third column, we show that the value of MTTF increases whenλ1>λ2, and α varies from 0.2 to 1.4.Table 1
The mean time to system failure (MTTF) for different values ofα.
αλ1<λ2λ1=λ2λ1>λ2λ1=0.3,λ2=0.6λ1=0.3,λ2=0.3λ1=0.6,λ2=0.30.22.666673.333332.083330.42.380953.333332.333330.62.222223.333332.50.82.121213.333332.619051.02.051283.333332.708331.22.003.333332.777781.41.960783.333332.83333Table 2
The mean time to system failure (MTTF) for different values of parametersλ1 and λ1 when (α=0.6).
λ1λ2=0.3λ2λ1=0.3MTTFMTTF0.14.285710.17.777780.23.750.24.444440.33.333330.33.333330.43.00.42.777780.52.727270.52.444440.62.50.62.222220.72.307690.72.063490.82.142860.81.94444Table2 shows the effect of system parameters λ1 and λ2 on MTTF. It is observed that the MTTF decreases with increasing λ1 and λ2, and this is quite obvious at α=0.6.Finally, we display the numerical results for the steady-state availability. We fixedβ1=0.4;β2=0.6;η=0.4;θ3=0.25;θ4=0.25;μ4=0.6;μ5=0.4;φ2=0.4 as the base in this study. First, Table 3 shows how the steady-state availability of the system is affected by the change of α values as follows.(1)
In the first and second columns of this table, the value of the steady-state availability decreases whenλ1≤λ2, and α∈0.1,…,0.9.(2)
In the third column, we show that the value of the steady-state availability increases whenλ1≤λ2, and α∈0.1,…,0.9.Table 3
The steady-state availability for different values ofα when (β1=0.4,β2=0.6,η=0.4,θ3=0.25,θ4=0.25,μ3=0.4,μ4=0.6,φ2=0.4).
αλ1<λ2λ1=λ2λ1>λ2λ1=0.3,λ2=0.6λ1=0.3,λ2=0.3λ1=0.6,λ2=0.30.10.3191490.5263160.2105260.20.2836880.4678360.2339180.30.2659570.4385960.2506270.40.2553190.4210530.2631580.50.2482270.4093570.2729040.60.2431610.4010030.2807020.70.2393620.3947370.2870810.80.2364070.3898640.2923980.90.2340430.3859650.296896Next, the changes in the failure rate of an operating unit under warranty (out of warranty),λ1 (λ2), are calculated and the results are presented in Table 4. From Table 4, it can be seen that the steady-state availability of the system decreases when both λ1 and λ2 increase and α=0.6.Table 4
The steady-state availability for different values ofλ1 and λ1 when (α=0.6,β1=0.4,β2=0.6,η=0.4,θ3=0.25,θ4=0.25,μ3=0.4,μ4=0.6,φ2=0.4).
λ1λ2=0.3λ2λ1=0.3Av∞Av∞0.10.2990650.10.4664720.20.2492210.20.293040.30.2136180.30.2136180.40.1869160.40.1680670.50.1661470.50.1385280.60.1495330.60.117820.70.1359390.70.1024980.80.1246110.80.0907029


## 6.2. Numerical Results for Sensitivity Analysis and Relative Sensitivity Analysis

In this part, the numerical results are given for both the sensitivity and relative sensitivity analysis. First, the sensitivity analysis ofRt caused by various parameters is presented, and the results are displayed in Figure 6. From Figure 6, it can be seen that λ1 has a significant effect on Rt, the parameter λ2 has a moderate effect on Rt, and the parameter α has a small influence on Rt, because this curve is close to the horizontal axis. Moreover, the signs of the sensitivity of Rt for λ1, λ2, and α are negative, which implies that the incremental changes of these parameters worsen the system reliability Rt. The results of relative sensitivity on reliability are almost the same as those of sensitivity analysis, but the high levels of each curve are quite different from each other as shown in Figure 7.Figure 6
Sensitivity analysis on reliability functiont∈0,20.Figure 7
Relativity sensitivity analysis on reliability functiont∈0,1.Next, the sensitivity and relative sensitivity analyses of MTTF are performed with various parameters. The sensitivity and relative sensitivity analyses of MTTF with three parameters are tabulated in three tables. Tables5–7 show that the signs of the sensitivity of MTTF with parameters λ1, λ2, and α are negative, indicating that the incremental changes in these parameters degrade MTTF. Furthermore, by looking at the absolute values in these tables, the order of impact on the MTTF with each parameter can be determined. It can be observed that the order of sensitivity on MTTF is λ2>λ1>α and the order of relative sensitivity on MTTF is λ2>λ1>α.Table 5
Sensitivity and relativity sensitivity analysis of MTTF withα.
αλ1=0.4,λ2=0.6ηαωα0.1−1.33333−0.05714290.3−0.680272−0.09523810.5−0.411523−0.101010.7−0.275482−0.09790210.9−0.197239−0.09230771.1−0.148148−0.0862745Table 6
Sensitivity and relativity sensitivity analysis of MTTF withλ1.
λ1α=0.4,λ2=0.6ηλ1ωλ10.1−6.66667−0.20.3−3.40136−0.4285710.5−2.05761−0.5555560.7−1.37741−0.6363640.9−0.986193−0.6923081.1−0.740741−0.733333Table 7
Sensitivity and relativity sensitivity analysis of MTTF withλ2.
λ2α=0.4,λ1=0.3ηλ2ωλ20.2−14.2857−0.6666670.4−3.57143−0.50.6−1.5873−0.40.8−0.892857−0.3333331.0−0.571429−0.2857141.2−0.396825−0.25


## 7. Conclusions

In this paper, we studied the behavior of the system, which consists of a single-unit operating under a free repair or replacement warranty. To solve the differential equations of the system under study, we used the Laplace transform technique. We provided the explicit expressions for system reliabilityRt, mean time to system failure (MTTF), and steady-state availability Av∞. The numerical results show that the influence of the failure rates λ1, λ2 cannot be neglected for a repairable system with changing completion of warranty rate α. Both the sensitivity and relative sensitivity analyses of the system reliability R (t) and mean time to system failure (MTTF) are also derived. The results show that the sensitivities of Rt are signiﬁcant only with λ1, λ2, and α. Finally, it can be shown that the order of sensitivity on (MTTF) is λ2>λ1>α and the order of relative sensitivity on (MTTF) is λ2>λ1>α. From the previous results, it can be seen that the parameter λ2 is the most important parameter affecting both performance measures, so its value should be well observed.



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