# Warranty Cost Analysis for Multi-State Products Protected by Lemon Laws
## 1 Introduction
### Literature Review  
Few works have addressed warranty modeling based on lemon laws and warranty cost analysis ([5,6]). For instance, under the piece-wise renewing free replacement warranty policy proposed in [5], the entire warranty period is divided into two subperiods, and once an item fails within a specific subperiod, it is replaced with a new identical item, and the warranty period is either fully or partially renewed. In addition, when the number of product failures during the warranty period exceeds a predetermined level, the producer must return the product's purchase price. This policy's anticipated warranty cost and warranty duration are derived from the manufacturer's standpoint. [PERSON]. [2] categorized the components as critical and noncritical and examined the warranty cost of a multicomponent device protected by lemon laws. They assumed that a product becomes a lemon when a certain number of critical or noncritical component failures occur. The extended warranty cost of used products covered by lemon laws was determined from Ref. [7]. The optimization of warranty policies for products governed by lemon laws is also discussed. For instance, [PERSON]. [8,9] discussed the optimization of a post-warranty policy and periodic preventive maintenance policy for products with a repair time threshold. From the views of customers and manufacturers, [PERSON]. [10] optimized the length of the extended warranty period by minimizing the expected cost rate incurred over the whole warranty coverage. Ref. [11] developed a maintenance cost model based on Korean lemon laws and optimized the length of the warranty period and the warranty premium price for vehicles from the manufacturer's perspective. Ref. [12] considered the optimal maintenance strategies for warranty products with limited repair time and limited repair number.  
In engineering practice, numerous products have multiple working states, and Markov processes are commonly employed to characterize their operational processes. The degradation processes of offshore wind turbine gears were modeled using non-homogeneous Markov processes; preventive maintenance optimization and reliability analysis are detailed in Refs. [13,14], respectively. Modeling the operation of systems with neglected failures, redundant dependencies, and failure dependency using aggregated Markov processes is conducted in Refs. [15, 16, 17, 18, 19]. Using a two-dimensional Markov chain, Ref. [20] dealt with imperfect monitoring and forecasting models. Ref. [21] suggested a Markov decision process framework to optimize repair and replacement plans for medical equipment.  
Many products have multiple failure states. During the rail degradation process, two types of failures are observed: sudden failure without symptoms and degradation-based failure (see [22, 23, 24, 25, 26, 27]). The monitoring of ultrasonic waves reveals four failure modes on the railroad track ([23]). In Ref. [24], both a catastrophic failure and a two-stage delayed failure were simultaneously considered. Typically, when multiple-failure-state products are covered by lemon laws, they are deemed defective based on the number of failures (regardless of failure type) and the length of time required for one repair activity. To the best of our knowledge, few studies have discussed the warranty service cost issue of productsprotected by lemon laws in a targeted way. To fill the theoretical gap and promote practical application, warranty cost analysis of multi-state products protected by lemon laws is discussed using Markov process theory in this paper.  
The remainder of this paper is organized as follows. Section 2 gives detailed model assumptions and derives the warranty cost under the \(\{\tau,N\}\) model. Assumptions and warranty cost under the \(\{\tau,N_{1},1\}\) model are discussed in Section 3. Section 4 gives numerical examples to illustrate the effectiveness of the model. Finally, conclusions are provided in Section 5.