Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty

doi:10.1016/j.ejor.2015.01.030

European Journal of Operational Research

Volume 243, Issue 3, 16 June 2015, Pages 874-882

https://doi.org/10.1016/j.ejor.2015.01.030 Get rights and content

Highlights

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We model warranty cost for multi-component systems.
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We analyze the impact of failure interaction on warranty cost.
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For series system, the impact of failure interaction is minor.
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For parallel system, the impact of failure interaction is significant.
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Extra resource should be allocated to compensate failure interactions.

Abstract

In a multi-component system, the assumption of failure independence among components is seldom valid, especially for those complex systems with complicated failure mechanism. For such systems, warranty cost is subject to all the factors including system configuration, quality of each component and the extent of failure dependence among components. In this paper, a model is developed based on renewing free-replacement warranty by considering failure interaction among components. It is assumed that whenever a component (subsystem) fails, it can induce a failure of one or more of the remaining components (subsystems). Cost models for series and parallel system configurations are presented, followed by numerical examples with sensitivity analysis. The results show that, compared with series systems, warranty cost for parallel systems is more sensitive to failure interaction.

Introduction

A warranty is an assurance by the manufacturer (vendors, sellers, or third parties) to the buyer which requires manufacturers to offer pre-defined compensation to buyers if the product or service fails to meet the standards under normal usage within warranty duration (Vahdani, Mahlooji and Eshraghnia Jahromi, 2013, van der Heijden and Iskandar, 2013, Xie, Liao and Zhu, 2014). Different types of warranties are offered by manufacturers based on the characteristics of products, e.g., the products’ complexity, reparability, and reliability (Lo & Yu, 2013). These characteristics are closely related to the key target of warranty research – warranty cost modeling.

There are numerous warranty policies adopted in cost modeling. Detailed studies can be found in Sheu and Chien (2005), Bouguerra, Chelbi, and Rezg (2012), Park, Mun Jung, and Park (2013), Shafiee and Chukova (2013), Xie and Liao (2013). Specifically, cost modeling under renewing free-replacement policy (RFRW) is extensively studied in literature (Chien, 2008, Chien, 2012, Darghouth, Chelbi and Ait-Kadi, 2012, Vahdani, Chukova and Mahlooji, 2011, Yeh, Chen and Chen, 2005). Under the RFRW policy, a product failed within the warranty period is replaced by a new one with a full warranty. Bai and Pham (2006) proposed a renewing ‘full-service’ warranty for multi-component system, which assumes an extra perfect maintenance upon system failure after the normal free replacement under RFRW.

Related research has been focused increasingly on the warranty studies for complex systems. The main reason is due to the need for an accurate estimation of warranty cost for those systems as the warranty cost could be very high. When a complex system is treated as a ‘single-component’ system or a ‘black box’, the inner structure information is ignored (Wu and Xie, 2008, Ye, Murthy, Xie and Tang, 2013). It is desirable to consider a different warranty policy for multi-component system when failure of several units instead of one unit results in the warranty cost (Bai and Pham, 2006, Scarf and Majid, 2011). When shifting the reliability analysis from single item to multiple items, failure dependence is a common phenomenon that cannot be neglected (see Peng, Coit and Feng, 2012, Tsoukalas and Agrafiotis, 2013; Yu, Chu, Châtelet, & Yalaoui, 2007). Murthy and Nguyen (1985) formulated three different types (Types I–III) of failure interaction for a two-component system. Type I failure interaction assumes that whenever a component fails, it can induce a simultaneous failure of one or more of the remaining components of the system. They define this simultaneous failure of the remaining components as ‘induced failure’, compared with ‘natural failure’ described by components’ lifetime distribution without failure interaction. Type II failure interaction is known as failure rate interaction, which assumes a change on failure rate of components whenever a component fails. It is further discussed by Zequeira and Bérenguer (2005), Lai (2008), and Golmakani and Moakedi (2012). A combination model of Type I and Type II is the Type III failure interaction (Murthy & Nguyen, 1985). So far, failure interaction is mainly discussed in maintenance models. Warranty cost models under failure interaction are seldom explored.

In this paper, we present a warranty cost model with Type I failure interaction under a type of renewing free-replacement warranty (RFRW). The target system is composed of multiple components and is repairable. Compared with previous works, the following two important extensions are made in this paper. (1) Instead of assuming independent failures among components, we derive the expected warranty cost based on certain failure interaction model. (2) We consider failure interaction between each two components instead of the failure interaction between one component and the whole remaining components. Warranty cost models for system configurations such as series and parallel are discussed separately, which provides a basis for future study of even more complex system configurations, such as parallel-series, series-parallel, hierarchical, and k-out-of-n. Since failure dependence is a common problem in complex system which affects both system reliability and service cost, this paper can help decision-makers better evaluate system reliability and reduce risk in estimating future warranty cost.

The rest of this paper is organized as follows. Section 2 introduces the model and the assumptions for multi-component system subject to failure interaction, maintenance strategy and warranty policy. Sections 3 and 4 formulate the specific warranty cost models under series and parallel structure. Section 5 gives numerical examples for a 3-component parallel system; comparison with series system is made to illustrate the result. Conclusions and potential extensions are made in Section 6.

Section snippets

Assumptions and specifications

This section provides preliminary assumptions and model specifications in RFRW and failure interaction.

RFRW for series systems with failure interaction

In this section, a warranty cost model is derived for a series system under failure interaction. It is obvious that $F_{S} (w) = 1 - \prod_{i = 1}^{n} (1 - F_{i} (w)) = 1 - \prod_{i = 1}^{n} R_{i} (w)$

We further define N_i (i ∈ Ω) as the number of natural failures of component i within warranty cycle T_S, and N_ij as the number of induced failures of component j caused by the natural failures of component i.

RFRW for parallel systems with failure interaction

For a parallel system, system fails when all the components have failed, and the one-time warranty cost can be expressed as $\sum_{i = 1}^{n} c_{i} + c_{M}$ . Both the system lifetime distribution and the warranty cost are affected by failure interaction under parallel structure.

As the number of warranty renewals N_S(w) satisfy a geometric distribution (Bai & Pham, 2006), N_S ∼ G(R_S(w)), i.e., $p (N_{S} = i) = {(1 - R_{S} (w))}^{i} R_{S} (w)$

We can obtain the expected warranty cost per cycle as $E [CR] = (\sum_{i = 1}^{n} c_{i} + c_{M}) \frac{F_{S} (w)}{R_{S} (w)}$

In order to calculate

Numerical examples

In this section, we perform two numerical examples to illustrate the cost model. In Section 5.1, we consider the case where the components follow exponential lifetime distribution. In Section 5.2, we analyze the warranty cost for components suffering degradation process.

Conclusion and future work

In this study, we derived a warranty cost model for multi-component system considering failure interaction among components. It was assumed that when the system fails, failed components are replaced and perfect maintenance is used for the remaining components. Based on these assumptions, we derived an analytical expression of warranty cost under both series and parallel structures and a recursive algorithm was proposed to compute the reliability of parallel system. Specifically, we illustrated

Acknowledgments

We are grateful to the editor and the three anonymous referees for their helpful comments and constructive suggestions. The work described in this paper was partially supported by a grant from City University of Hong Kong (project no. 9380058).

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Stochastics and StatisticsCost analysis for multi-component system with failure interaction under renewing free-replacement warranty

Highlights

Abstract

Introduction

Section snippets

Assumptions and specifications

RFRW for series systems with failure interaction

RFRW for parallel systems with failure interaction

Numerical examples

Conclusion and future work

Acknowledgments

European Journal of Operational Research

European Journal of Operational Research

International Journal of Production Economics

European Journal of Operational Research

International Journal of Production Economics

Computers & Industrial Engineering

Reliability Engineering & System Safety

Reliability Engineering & System Safety

Reliability Engineering & System Safety

European Journal of Operational Research

European Journal of Operational Research

Computers & Industrial Engineering

Computers & Mathematics with Applications

Computers & Industrial Engineering

European Journal of Operational Research

European journal of operational research

Stochastics and Statistics
Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty