Optimal warranty policy for repairable products with a three-dimensional renewable combination warranty
Introduction
Warranty is a contractual obligation between the manufacturer and the consumer (Chien, 2008), which plays important role in commercial transactions. It can not only provide protection and product quality signals to consumers, but also help manufacturers promote sales and avoid unreasonable claims/warranty fraud (Blischke, 1995, Shafiee and Chukova, 2013a). In the literature, various types of warranty policies have been designed and proposed (Blischke and Murthy, 1992, Khanna et al., 2020, Cha et al., 2021). According to the number of variables specified in warranty terms, warranty policies are divided into one-dimensional (1-D) and multi-dimensional (Blischke, 1995, Liu et al., 2020). Usually, the 1-D warranty is defined by only one variable, such as age (e.g., one year) or usage (e.g., miles driven), while the multi-dimensional warranty involves more than one variable (Blischke, 1995). Nowadays, with fierce competition and increasing customer requirements, more and more cost-effective and generous warranty terms have been designed and provided. In such a trend, the dimensionality of warranty is extended, and multi-dimensional warranty has received more and more attention.
Among multi-dimensional warranty policies, the two-dimensional (2-D) warranty is the most common. The classical 2-D warranty policy is characterized as a region denoted by age limit and usage limit, which has been provided for many products in real life, such as automobiles, printers, excavators, etc. (Blischke et al., 2011, Ye and Murthy, 2016). Due to the advantages of controlling the warranty cost caused by the product age or usage and helping manufacturers consider the consumers’ heterogeneity, this classical 2-D warranty policy has attracted a lot of attention (Zhang et al., 2019, Taleizadeh and Mokhtarzadeh, 2020, Lin and Chen, 2021). However, in some cases, it also has some limitations. For example, the ages of some electrical appliances and facilities, such as washing machines, refrigerators, nuclear reactors, etc., may be easily obtained, but their usage may not be easily known (Park & Pham, 2012a). Nonetheless, if these products have experienced failures and repairs, their failure time, repair time, and repair number can be easily collected by manufacturers. Furthermore, a warranty regulation named “lemon laws”, enacted in the USA, stipulates that automobile manufacturers must replace the defective vehicle or refund customers unconditionally if the vehicle cannot be repaired within a time limit or has more than a certain number of repairs (Park et al., 2020, Liu et al., 2021a). Under such background, Park and Pham (2012a) first proposed a new 2-D warranty policy considering repair time limit, which uses failure time and repair time as two factors. Under this new 2-D warranty, the failed product will be replaced if the repair time exceeds the limit, otherwise, it will be minimally repaired.
In recent years, with the applications of the “lemon laws” in other countries (such as Canada, Europe, Singapore, etc.) and for other products (such as all durable products) (Husniah, Pasaribu, & Iskandar, 2020), the 2-D warranty considering repair time limit has attracted the attention of scholars. For instance, Park and Pham (2012b) studied the optimal such 2-D warranty policy and post-warranty preventive maintenance (PM) strategy for the k-out-of-n systems. Park, Jung, and Park (2018) investigated the optimal periodic PM strategy after the expiry of this 2-D warranty. Wang, Pei, Zhu, and Liu (2018) studied an optimal extended warranty strategy for products sold with such a 2-D warranty. Since minimal repair and replacement are considered simultaneously, this new 2-D warranty policy is also known as the minimal repair-replacement warranty (MRRW), such as Park et al., 2013, Park and Pham, 2016, Jiao and Zhu, 2018 as well as Liu et al. (2021a). Moreover, similar to such a 2-D warranty policy, the rebate warranty considering repair time limit has also been proposed, in which if a failure cannot be repaired within the time limit, instead of replacement, the consumer will receive a full or proportional refund, and the warranty contract will terminate. Related researches can be seen in Park et al., 2020, Liu et al., 2021a.
In addition to the repair time limit, to protect the interest of customers, the repair number limit is also often considered in the product warranty. For example, Hooti, Ahmadi, and Longobardi (2020) proposed an extended 2-D warranty which includes limitations on service duration and repair number. Shang, Cai, Chen, and Zhang, (2019) studied a renewable repair-limit pro-rata replacement warranty (RRLPRRW), where the replacement depends on the repair number limit and pro-rata replacement cost is considered. Su and Yang (2020) introduced a novel two-stage renewing warranty in which the failed product will be replaced if the repair number exceeds the limit in the first stage. Meanwhile, some scholars have also considered similar rebate warranty. For instance, Wang, He, He, Li, and Xie (2019) considered a failure-limit piece-wise renewing warranty, where if the number of failures exceeds the limit, the consumer will receive a refund and the warranty will cease. Husniah et al. (2020) investigated refund for the multi-component warranty product with repair number limit.
The above-mentioned studies only considered one of the limitations on repair time or repair number. However, in practice, during the warranty period, too long repair time and too large repair number may happen simultaneously (Su & Yang, 2020). Thus, it is more realistic to study the warranty policies considering both the two limits. Recently, Liu, Wang, and Su (2021b) first proposed a rebate warranty with repair time and number limits, where if a failure cannot be minimally repaired within the time limit or the number of failures exceeds the limit, the customer will receive a proportional refund and the warranty will terminate. However, as shown in the literature above, besides refund, replacement is another commonly used rectification action for the cases of overtime repairs and frequent failures. But currently, it has not been well studied for warranty products with limited repair time and repair number.
In the literature, for replacement, two types of popular warranty policies are renewable free replacement warranty (RFRW) and renewable pro-rata replacement warranty (RPRRW) (Park et al., 2018). During the warranty period, both policies renew the warranty period whenever the replacement occurs. Under the RFRW, the manufacturer replaces the failed product at no cost to the consumer (Vahdani, Mahlooji, & Jahromi, 2013), while the manufacturer charges the consumer a pro-rata replacement cost under the RPRRW (Park et al., 2018). Obviously, there is a conflict between the manufacturer and the consumer, where RFRW is more favorable to the consumer since there is no replacement cost to pay, while RPRRW is more beneficial to the manufacturer since less replacement cost is shouldered (Shang, Si, Cai, Zhu, & Zou, 2021). To balance the interests of both sides, therefore, it is necessary to study renewable combination replacement warranty (RCRW) policies, which integrate characteristics of RFRW and RPRRW. However, there are only a few studies in this area, and they all focused on one type of RCRW policy, in which RFRW and RPRRW are used at different stages of the warranty period (Chien, 2010, Shang et al., 2021). For warranty products with repair time or number limit, RFRW and RPRRW are usually investigated separately, and no work considering RCRW policies so far.
In the market, a better warranty policy not only provides better warranty terms to consumers but also implies higher product quality and reliability. The better the warranty terms, the higher the selling price. Also, better warranty terms will result in a higher warranty cost. Obviously, there are interactions between product warranty and price. Therefore, to optimize profit, manufacturers should consider both factors simultaneously. While most of the above studies focused only on warranty cost analysis, which didn’t consider product price and profit. Motivated by these research gaps, in this paper, we incorporate the repair number limit into the 2-D warranty with repair time limit to consider minimal repair and replacement simultaneously. Due to failure time, repair time, and repair number being taken into account, such a warranty policy is referred to as a “three-dimensional (3-D) warranty” throughout this paper. Based on the types of replacements that may occur during the warranty period, further, a new RCRW policy is designed to reduce the conflict between the manufacturer and the consumer. From the manufacturer’s perspective, by integrating the impacts of the product price and warranty, the optimal warranty policy is obtained to maximize the expected profit.
The remainder of this paper is organized as follows. Section 2 describes the proposed warranty policy and the optimization problem in detail. Section 3 develops the sale price model for the warranty product. In Section 4, the warranty cost model is derived correspondingly, and some special cases are presented. Then, a numerical example is presented in Section 5, and sensitivity analysis and policy comparisons are carried out. Finally, the conclusions and future research directions are summarized in Section 6.
Section snippets
Warranty policy
The proposed warranty policy is a 3-D RCRW with an original warranty period length of . Under such a warranty policy, the product failure is either minimal repaired or replaced by the manufacturer. Assume the time and number limits of minimal repairs are and respectively. During the warranty period , the following two types of replacements may occur:
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unplanned replacement: if a failure cannot be repaired within , the failed product will be replaced.
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planned replacement: if the product
Sale price
The sale price, as one of the most powerful marketing tools, undoubtedly affects the manufacturers’ profit. In the literature, a variety of pricing models have been reported, such as the simple linear model (Chien, Zhang, Wang, & Sheu, 2020), exponential model (Shang, Si, Sun, & Jin, 2018), and Cobb-Douglas-type logarithmic linear model (Glickman & Berger, 1976). Among them, the Cobb-Douglas-type logarithmic linear model is widely applied due to its good ability of fitting and flexibility (Su
Warranty cost model
Let denote the warranty cost per product undertaken by the manufacturer. In this section, from the perspective of the manufacturer, we derive the expected warranty cost per product . From section 2, minimal repair and replacement are executed during the warranty period, and three types of costs are charged to the manufacturer for per unit product: the replacement cost , the minimal repair cost , and the failure cost . Then, . Thus, the
Numerical example
The main purpose of this study is to find the optimal combination of , , and that maximizes from the manufacturer’s perspective. If from Eq. (8) and from Eq. (31) are taken into Eq. (2), we may find that the profit function is very complex, which involves integral and series expressions. The application of the standard optimization approaches is quite involved or impossible to obtain the general analytical solution. Thus, the numerical method is
Concluding remarks
In today’s market, how to come up with more novel ideas to design very attractive warranty policies and achieve the balance between the interests of manufacturers and consumers has become more and more important. In this paper, a new 3-D RCRW policy, as a generalized warranty policy, was proposed. Under this policy, the repair time limit and repair number limit are considered simultaneously. The product was replaced at the occurrence time of the failure or when the repair time exceeds
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 71931006, 71871119, and 71801128) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX21_0363).
Peirui Qiao is currently a PhD student in the Department of Management Science and Engineering at Nanjing University of Science and Technology. Her research interests include warranty, maintenance, reliability and stochastic modeling.
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Peirui Qiao is currently a PhD student in the Department of Management Science and Engineering at Nanjing University of Science and Technology. Her research interests include warranty, maintenance, reliability and stochastic modeling.
Jingyuan Shen is an associate professor in the School of Economics & Management at Nanjing University of Science & Technology. She received her Ph.D. degree in the School of Management & Economics from Beijing Institute of Technology of China in 2018. Her research interests include system reliability, importance measures, stochastic modeling, and applications of probability.
Fengxia Zhang is currently a PhD student in the Department of Management Science and Engineering at Nanjing University of Science and Technology. Her research interests include reliability and maintenance modeling.
Yizhong Ma is a professor in the Department of Management Science and Engineering at Nanjing University of Science and Technology. He received his PhD in Control Science from Northwestern Polytechnical University, Xi’ an, China. He is also assigned as the Director of Quality Society of China, and the Expert Member of Six Sigma Promotion Committee in China. His research interest includes quality and reliability engineering.