# Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

## 2. Assumptions and Cost Analysis of {τ, N} Model

#### 2.1. General Assumptions

#### 2.2. Assumptions of $\{\tau ,N\}$ Model

- (1)
- Consider a repairable product sold with a warranty period ${W}_{1}$ and the warranty is renewed according to the $\{\tau ,N\}$ mechanism. That is, it will be replaced with a new and identical one, and warranty terms are renewed at no charge to customers if the repair time for a failure (regardless of the failure type) is more than $\tau $ or the number of visits to the failure states set $D$ over the warranty period exceeds $N$. Suppose that the times for visiting the operating states set are independent random variables having finite expectations and they are also independent of the times for repair actions.
- (2)
- Let $\{{Y}_{i},i=1,2,\cdots \}$ be the sequence of repair times. Suppose that ${Y}_{1},{Y}_{2},\cdots $ are independent and identically distributed random variables with a probability distribution $G(t)$ and finite expectation.
- (3)
- At the expiration of the renewable warranty terms, if a repair is in progress, it will be continued at no charge to the customer. Repair times are not included in the warranty period ([13]). Repair times, unless otherwise specified, are not part of the time that the product goes through in this paper.

#### 2.3. Probability Analysis of $\{\tau ,N\}$ Model

#### 2.3.1. Renewable Warranty Service Analysis of $\{\tau ,N\}$ Model

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

#### 2.3.2. Nonrenewable Warranty Service Analysis of $\{\tau ,N\}$ Model

**Proposition**

**3.**

**Proof.**

#### 2.4. Cost Analysis of $\{\tau ,N\}$ Model

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Assumptions and Probability Analysis of {τ, N_{1}, 1} Model

#### 3.1. Assumptions of $\{\tau ,{N}_{1},1\}$ Model

#### 3.2. Probability Analysis of $\{\tau ,{N}_{1},1\}$ Model

#### 3.2.1. Renewable Warranty Analysis of $\{\tau ,{N}_{1},1\}$ Model

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proposition**

**6.**

#### 3.2.2. Nonrenewable Warranty Service Analysis of $\{\tau ,{N}_{1},1\}$ Model

**Proposition**

**7.**

**Proposition**

**8.**

#### 3.3. Cost Analysis of $\{\tau ,{N}_{1},1\}$ Model

**Theorem**

**3.**

**Theorem**

**4.**

## 4. Numerical Example

#### 4.1. Numerical Example for $\{\tau ,N\}$ Model

#### 4.2. Numerical Example for $\{\tau ,{N}_{1},1\}$ Model

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, L.; Song, Y.; Qiu, Q.; Yang, L.
Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws. *Appl. Sci.* **2023**, *13*, 1541.
https://doi.org/10.3390/app13031541

**AMA Style**

Wang L, Song Y, Qiu Q, Yang L.
Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws. *Applied Sciences*. 2023; 13(3):1541.
https://doi.org/10.3390/app13031541

**Chicago/Turabian Style**

Wang, Liying, Yushuang Song, Qingan Qiu, and Li Yang.
2023. "Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws" *Applied Sciences* 13, no. 3: 1541.
https://doi.org/10.3390/app13031541