A performance-based warranty for products subject to competing hard and soft failures

https://doi.org/10.1016/j.ijpe.2020.107974Get rights and content

Highlights

  • Study a performance-based warranty with competing hard and soft failure modes.

  • Propose three compensation policies when the performance guarantee is violated.

  • Derive the warranty cost models for the three compensation policies.

  • Compare the three compensation policies in a profit-maximization manner.

Abstract

This article studies a performance-based warranty for products subject to competing hard and soft failures. The two failure modes are competing in the sense that either one, on a “whichever-comes-first” basis, can cause the product to fail. A performance-based warranty not only covers the repair or replacement of any defect, but also guarantees the minimum performance level throughout the warranty period. In this article, we propose three compensation policies—that is, free replacement, penalty, and full refund, when a product’s performance fails to meet the guaranteed level. The expected warranty servicing costs for the three policies are derived, based on the competing risks concept. A warranty design problem is further formulated to simultaneously determine the optimal product price, warranty length, and performance guarantee level so as to maximize the manufacturer’s total profit. Numerical studies are conducted to demonstrate and compare the three performance-based compensation policies. It is shown that the full refund policy always leads to the lowest total profit, whereas neither of the other two policies can dominate each other in all scenarios. In particular, the free replacement policy results in a higher total profit than the penalty policy when the replacement cost is low, the penalty cost coefficient is high, and/or the product reliability is high.

Introduction

Many durable products have specific key performance characteristics (e.g., capacity of batteries, energy efficiency of refrigerators, rated power output of solar panels) that deteriorate with time and usage. Such performance-critical products are subject to two failure modes—hard and soft failures. Hard failures are usually caused by manufacturing defects, wearout/aging, or even external shocks, whereas soft failures occur when the product performance becomes unsatisfactory—precisely, when the performance degradation exceeds a pre-set threshold. In principle, hard and soft failure processes are competing, meaning that either of the two processes can cause the product to fail.

In today’s highly competitive market, it is a common practice for manufacturers to provide attractive product warranties along with the sales of their products, in order to protect consumers against premature failures and signal product quality and reliability (Murthy and Djamaludin, 2002, Xie et al., 2017). In addition to product failures, consumers are increasingly concerned about product performance deterioration during the use period (Jin et al., 2015, Kim et al., 2007). Driven by consumer desires as well as technological advances, more and more manufacturers are offering performance warranties (Koschnick and Hartman, 2020). Unlike product warranties that provide protection against functional failures, performance warranties focus on the degradation of key performance characteristics and provide guarantee on the minimum performance level(s) over the warranty period. In general, performance warranties and product warranties are offered simultaneously, although the lengths of their protection periods might differ. In this article, we refer to the combination of product and performance warranties as performance-based warranty, and our aim is to study this new type of warranty for products subject to competing hard and soft failure processes.

Performance-based warranties have received a few applications. Two typical examples are presented below.

Lithium-ion battery warranties: Lithium-ion batteries experience gradual energy or power loss with time and usage, which results in capacity reduction. Hard failures of lithium-ion batteries may take place because of manufacturing defects or wearout/aging, etc. A typical example of performance-based warranties is the mid-range battery warranty for Tesla Model 3 electric cars: It covers the repair or replacement of any malfunctioning or defective battery for 8 years or 100 000 miles (whichever comes first), with minimum 70% retention of battery capacity over the warranty period.1 In this case, the protection periods of the product and performance warranties coincide.

Solar panel warranties: The performance of photovoltaic panels is also subject to stochastic degradation which is dependent on operational and environmental conditions. Brand-new solar panels are usually protected by both performance and product warranties. Take Canadian Solar’s warranty policy2 as an example. The performance warranty guarantees that the solar panels’ actual power output should be no less than 97.5% of the rated power output during the first year, and the actual annual power decline should be no more than 0.5% from year 2 to year 30. That is, the actual power output should be no less than 83.0% of the rated power output by the end of 30 years of operations. In addition, the product warranty guarantees that the solar panels should be free from defects in materials and workmanship for 12 years. In this case, however, the protection lengths of the product and performance warranties are different.

In the literature, there are two streams of research that come closely to our work. The first one is product warranty modeling and analysis, and the second one is reliability assessment and maintenance planning for systems subject to competing hard and soft failures.

Product warranty modeling and analysis have long been a vibrant topic in the warranty management field (Murthy and Djamaludin, 2002). In recent years, quite a few novel warranty concepts and policies (in terms of protection duration, compensation mechanisms, and maintenance strategies, etc.) have been investigated in the literature. Ye and Murthy (2016) study the design of a two-dimensional warranty menu that contains a number of rectangular regions. Luo and Wu (2018) collectively price the warranty policies for a portfolio of different products through the mean–variance optimization approach. Wang et al. (2019) develop cost model for a new piece-wise renewing free replacement warranty policy. Lu and Shang (2019) develop a new warranty mechanism for online pre-owned tech products to encourage product quality information disclosure between e-tailers and online warranty provider. Wang et al. (2020b) study the design and pricing of extended warranty menus which offer multiple options with differentiated lengths and prices. Liu et al. (2020) investigate the profit and pricing strategy for a complimentary extended warranty. Cha et al. (2020) propose a new renewing warranty policy with inspection for heterogeneous, stochastically degrading items. In addition, some studies attempt to develop novel maintenance strategies for better warranty servicing. Su and Wang (2016) investigate customized preventive-maintenance warranty policies, where preventive maintenance strategies are tailored for different consumer categories. Wang et al. (2020a) propose an unpunctual preventive maintenance policy for repairable products under two-dimensional warranties, where consumers are entitled to slightly advance or postpone maintenance executions. Peng et al. (2020) investigate a dynamic preventive maintenance problem under two-dimensional warranties, and show that the optimal policy is a control limit policy with usage-dependent failure rate thresholds. Furthermore, Shang et al. (2018) make the first attempt to study condition-based warranties for products suffering from stochastic degradation, but the warranty in their work is a traditional policy (failure-based, not performance-based).

To our knowledge, Su and Cheng (2018) and Koschnick and Hartman (2020) are the most closely related to our work. Su and Cheng (2018) propose an availability-based warranty under which the manufacturer not only provides free repair or replacement upon any failure, but also ensures that the operational availability over the warranty period meets a negotiated level. Our work differs from theirs in two main aspects: (i) The performance measure of interest in their work is the operational availability, whereas we focus on the stochastic deterioration of key performance characteristics; and (ii) the products in their work are subject to only one failure mode (i.e., hard failure), whereas the products in our work exhibit two competing failure modes—hard and soft failures. Recently, Koschnick and Hartman (2020) also introduce a performance-based warranty policy, where the manufacturer may offer to cap the amount of operating costs the consumer will pay each period for a certain amount of time. That is, if a consumer’s operating cost exceeds the guaranteed level, then the manufacturer has to offer a compensation. As Koschnick and Hartman (2020) merely focus on performance warranties against excessive operational costs, their problem setting is clearly different from ours.

The second stream of research focuses on reliability analysis and maintenance planning for systems with competing hard and soft failures. Most existing studies in this stream consider (in)dependent competing degradation processes and random shocks. In general, they assume that hard failures (resp. soft failures) are induced by fatal shocks (resp. stochastic degradation of key performance characteristics), and nonfatal shocks contribute to instantaneous increments in degradation levels or rates. Peng et al. (2010) develop reliability models and preventive maintenance policies for complex systems with dependent competing failure processes. Wang and Pham (2011) propose a s-dependent competing risk model for systems subject to degradation processes and random shocks. Huynh et al. (2012) study maintenance strategies for single-unit repairable systems subject to dependent competing failures due to degradation and shocks. Rafiee et al. (2014) model the reliability of devices experiencing dependent competing failure processes of random shocks and degradation with a changing rate. Jiang et al. (2015) explicitly consider zoned shock effects on stochastic degradation when modeling dependent degradation and shock processes. Song et al. (2016) develop new reliability models for series systems subject to competing hard and soft failures with dependent shock effects. Qiu et al. (2018) propose a new maintenance model under power purchase agreements for complex energy generation systems that experience competing hard and soft failure processes. Hao and Yang (2018) conduct reliability assessment for dependent competing failure processes with changing degradation rate and hard failure thresholds. Yousefi et al. (2019) attempt to optimize on-condition failure thresholds and inspection intervals for multi-component systems with each component undergoing degradation and shock processes. Gao et al. (2019) develop reliability models for systems subject to competing risks with degradation-shock dependency.

Our work differs from the second stream of research in two main perspectives: (i) None of these studies takes product/performance warranties into consideration; and (ii) our work considers a slightly different setting of competing risks: Hard failures are triggered by manufacturing defects, wearout/aging, or external shocks and can be described by a lifetime-based reliability model; whereas soft failures are induced by the stochastic degradation of key performance characteristics and can be characterized by a degradation-based reliability model.

This article makes an early attempt to model and optimize the performance-based warranty for products subject to competing hard and soft failure processes. Under this performance-based warranty, the manufacturer not only provides free replacement upon any failure but also guarantees the minimum performance level throughout the warranty period. We propose three types of compensation mechanisms when a unit’s actual performance fails to meet the guaranteed level over the warranty period: (i) the unit will be replaced with a new one; (ii) a penalty cost will be induced; and (iii) a full refund will be issued and the warranty terminates. We then derive the expected warranty servicing expenses for the three compensation policies, based on the well-known competing risks model. An optimization problem is further formulated to simultaneously determine the optimal product price, warranty length, and performance guarantee level of the product so as to maximize the manufacturer’s total profit. Numerical experiments are carried out to demonstrate and compare the three performance-based warranty policies, as well as answering the following questions: In terms of total expected profit, does the performance-based warranty has a better outcome than traditional product warranties? If yes, which compensation policy is the most beneficial to the manufacturer?

The rest of the article is organized as follows. Section 2 defines the three performance-based warranty policies, formulates hard and soft failure processes, and then develops the associated warranty cost models. On this basis, Section 3 develops and discusses a profit-maximization optimization problem. Section 4 illustrates and compares the proposed compensation policies through numerical experiments. Finally, Section 5 concludes the article with some suggestions for future research.

Section snippets

Model formulation

Before formally defining the performance-based warranty, the following assumptions are made to facilitate policy definition.

  • (i)

    The performance characteristic of interest is the-higher-the-better, e.g., battery capacity and solar panel’s power output, so that we normalize the initial performance level to 100%.

  • (ii)

    The performance characteristic is gradually deteriorating. Mathematically, the actual performance level is continuously decreasing.

  • (iii)

    The product performance is continuously monitored.

A profit-maximization optimization problem

In this section, an optimization problem is developed to simultaneously determine the optimal product price, warranty length, and performance guarantee level, so as to maximize the manufacturer’s total profit. The optimization of this problem is briefly discussed, although there is no closed-form solutions to most decision variables.

Product price and warranty length are two typical factors that have a significant effect on the product demand or sales volume (Glickman and Berger, 1976, Xie et

Numerical experiments

In this section, numerical examples are presented to illustrate and compare the three compensation policies of performance-based warranty. Comprehensive sensitivity analyzes and policy comparisons are conducted with respect to key model parameters. The managerial insights would be of importance to manufacturers who seek the maximum profit to be generated from new durable products.

Consider that a firm produces and sells a hypothetical lithium-ion battery model, with unit manufacturing cost C0=$

Conclusions and future research topics

This article proposes the concept of performance-based warranty for products subject to competing hard and soft failure modes. Unlike product warranties that protect consumers solely from premature failures, the performance-based warranty not only covers the repair or replacement of any failure, but also guarantees the minimum performance level over the warranty period. This warranty policy would compensate consumers if the product’s actual performance throughout the warranty period fails to

Acknowledgments

This work was supported by the Departmental Start-up Fund of The Hong Kong Polytechnic University (grant number P0034462) and the National Natural Science Foundation of China (grant numbers 71532008, 71971181, 72002149). We are grateful to the two anonymous referees for their helpful comments and suggestions on an earlier version of the paper.

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