Mixture reliability analysis of a product consisting of new and remanufactured components Open Access

Remanufacturing is perhaps the well-known and extensively used product recovery option that transforms the used products into “as new as” the original product in terms of quality standard, warranty and work content (Thierry et al., 1995). It encompasses activities like disassembly, cleaning, inspection or testing, salvaging and reassembly. A remanufactured product can reduce the manufacturing cost up to 50%, energy consumption up to 60% and air pollution up to 80% (Jiang et al., 2016). In remanufacturing, after complete disassembly of the collected cores, the parts or components undergo the cleaning process. The cleaned parts/components are inspected and tested whether they go for another life cycle. Damaged parts or components are replaced by the new ones and components requiring repair go for salvaging or reconditioning process. These remanufactured components along with the new components are reassembled to build the remanufactured product. According to BSI standard BS8887-2:2009, remanufacturing is a process that provides warranty to the customers. Generally, consumers are commonly perceived remanufactured systems as lower quality product in comparison to new product. However, this is not the case. To increase the demand of remanufacturing, remanufacturers use different initiatives to promote the better quality and warranty of remanufactured products. Remanufacturers have to provide better performance and warranty period because the second hand product or reused product are becoming more common. Moreover, customers also want to lowering the ownership cost. Hence, the purchase of second-hand products has increased.

The quality and reliability of reused components are the important decision making area in remanufacturing. Furthermore, a remanufactured product is an assembly of both remanufactured and new components or parts, the success of this business lies on the fact that the percentage of used components or parts that can be recovered and can undergo another product life cycle, or in other words, the remanufacturability of the product. Remanufacturability of a product is highly dependent on the reliability of the parts or components, as it affects different criteria such as amount of recovery, recovery cost and warranty cost in remanufacturing (Diallo et al., 2017). The warranty period offered by manufacturers during the second life is also based on the reliability of the used product. Moreover, reliability is often used to represent the quality aspect of a product. Reliability itself is defined as the probability of an item to perform its initial functions successfully during a certain period of time (Anityasari and Kaebernick, 2008) and reliability of a used item must be examined based on the probability of its survival during the second life.

The reliability of a remanufactured product can be analysed from the viewpoint of mixture reliability concept. The heterogeneous populations of new and remanufactured components have different time-to-failure values. They may either follow different failure distributions or same distribution with different parameter values. The reliability of the remanufactured product depends on the mixture proportions of the heterogeneous components, that is new and remanufactured components. Increase in the reliability of the remanufactured product leads to increase in remanufacturing cost. Thus, an optimum mixture proportion of new and remanufactured components is needed which bring a trade-off between the remanufacturing cost and a desired level of reliability of the remanufactured product. The aim of this paper is to explore the time-to-failure characteristics of a remanufactured product under different mixture proportions and to evaluate the optimum mixture proportion of new and remanufactured components in order to get maximum reliability with respect to minimum remanufacturing cost. Weibull distribution is considered in this paper to analyse the time-to failure characteristics.

The organization of the article is structured as follows. Literature review is associated with reliability analysis of the remanufactured product and mixture reliability analysis is discussed in section 2. Methodology is discussed in section 3 which is followed by numerical illustrations discussed in section 4. Results and Discussions are discussed in section 4. In section 6, the managerial implications of the study have been explained. Finally, the conclusion of this study are discussed in section 7.

Review of literature has been carried on two different areas, one on analysis of the remanufactured product and secondly on the concept of mixture reliability analysis.

Shu and Flowers (1998) developed a stochastic reliability model of a remanufacturing process and Jiang et al. (1999) extended the work of Shu and Flowers (1998) by considering the replacement rate in the remanufacturing process. Parkinson and Thompson (2003) used a systematic approach to analysis the remanufacturing process planning. Researchers such as Kara et al. (2005) and Mazhar et al. (2007) focused on the evaluation of the remaining useful life aspect in reliability. Kara et al. (2005) developed a two-stage decision making model to evaluate the remaining useful life of a product, whereas, Mazhar et al. (2007) used artificial intelligence to calculate the remaining useful life of a product. Kleyner and Sandborn (2008) developed a model to minimize the life cycle cost of product by considering the reliability of the product. Anityasari and Kaebernick (2008) developed a reliability evaluation framework for remanufacturing and reuse of a product. To assess the performance of the remanufactured product, Pandey and Thurston (2009) combined value degradation with failure data of a component. Agarwal et al. (2012) used product and component reliability to evaluate the incentive which is provided to the customers. Hu et al. (2014) used machine learning approach to calculate the aforesaid aspect of reliability. Performance assessment of a remanufactured product is very critical due to presence of new and remanufactured parts. Jiang et al. (2016) developed an optimized remanufacturing process planning. In their model, they considered quality of the used product as a major factor in reliability analysis. Diallo et al. (2014) explored the condition under which the mixture of increasing failure rate of new and remanufactured components behave like a modified bathtub failure rate. Ghayebloo et al. (2015) proposed a model where a special attention is given to the reliability and the greenness aspect of a product to enhance the profitability. On the other hand, Liao et al. (2015) analysed the role of warranty to overcome the uncertainty in the minds of the customer. They optimize the warranty period of the remanufactured products and observed that incorporating warranty service in the product increases the price and manufacturer’s profit. Kang et al. (2016) did a comparison among the quality standards in remanufacturing which is used in various countries to increase the reliability. Yazdian et al. (2016) developed a mathematical and statistical method to make decisions regarding acquisition price of core, selling price and the warranty time period of a remanufactured product. Darghouth and Chelbi (2018) developed a decision model to identify the optimal upgrade level, warranty period and preventive maintenance effort to maximize the total expected profit and sales. Wang et al. (2021) proposed some possible strategies for companies to increase product reliability. They addressed reward and dependability considerations for a product composed of two components from two separate suppliers. Du et al. (2022) analysed the reliability of a remanufactured machine tool where they had represented the reliability as a function of individual parts. They aimed to improve the reliability allocation of remanufactured machine tool by using neural network method. Kuik et al. (2023) used optimisation models to analyse and investigate the sale of warranty plans for remanufactured products. They considered a quadratic programming solution strategy to determine the appropriate warranty period.

Jiang and Murthy (1998) discussed various failure characteristics of mixed Weibull distribution. Different shapes of the failure characteristics are due to different parameter values. They considered two-parameter Weibull distribution in their analysis. Wondmagegnehu (2004) analysed the characteristics of the mixture distribution with an increasing failure rate. They considered two cases, in the first case, the shape parameters of the two Weibull distribution are same and in the second case the shape parameters are different. Block et al. (2003) considered a heterogeneous population of defective parts which have small life and new parts which have larger life. They also determined the both the shape and behaviour of the mixture failure rate. In their study, both the sub populations have increasing linear failure rate. Razali et al. (2008) estimated the parameters of the Weibull distribution such as shape parameter, scale parameter and location parameter from the mixture distribution. They used Maximum Likelihood Estimation Method (MLE) to estimate the parameters. Based on the above, a comprehensive methodology to assess remaining life of products, adopted by various researchers is presented in Table 1.

From the above literature review, it has been observed that most of the papers focused on quality of returned products and how the quality of the recovered items affects and influence the reliability of the remanufactured products. More emphasize was given on how to screen or sort the used items in terms of quality so that better performance can be derived. Furthermore, literature related to reliability models in remanufacturing mainly related to maintenance and failure model analysis and cost minimization. From Table 1, it can be observed that a very few papers exist to address the characteristics of the mixture reliability of the new and remanufactured products. The multicomponent nature of remanufactured product is largely overlooked. Thus, using the concept of mixture reliability, an optimum mixture proportion can be proposed which increases the reliability of the remanufactured product and reduces the remanufacturing cost. The following research issues are addressed in this paper.

• A1.

  Identification of the changes in the behavioural pattern of the mixture failure rate of a heterogeneous population, that is mixture of new and remanufactured components with respect to different mixture proportions. The failure rate of the new and remanufactured products is assumed to follow Weibull distribution with different parameter value.

• A2.

  Determination of the optimum mixture proportion of new and remanufactured components in order to get maximum reliability with respect to minimum remanufacturing cost.

In the methodology section, before going to the mixture reliability characteristics, a brief description about the Weibull distribution is highlighted. Weibull distribution is a powerful tool in reliability analysis of a product which is used for analysing the characteristics and classification of the failure (Mazhar et al., 2007). Weibull distribution works extremely well when the sample size is small, that is less than 20. A two parameters Weibull distribution is represented as

where, f (x) is the time-to-failure distribution, $β$ and $η$ is the shape and scale parameter respectively. The shape parameter represents the different shape of the time-to-failure density curve. For example, the value of shape parameter equals to one or less than one represents the exponential nature of the density curve. Whereas, the value greater than one shows the dominance of the polynomial part. The scale parameter signifies the characteristics life or age to failure. The classification of failure depends upon the value of shape parameter that is $β$ which also represents the slope of the Weibull distribution. If the $β$ value is less than one, then it represents the infant mortality, whereas, $β$ value equals to one and greater than one represents random failure and wear out failure respectively (Mazhar et al., 2007).

A remanufactured product consists of heterogeneous mixture of new and remanufactured/used components. The used or remanufactured components need to be treated differently as they are dependent on several factors including age of the component, usage rate and maintenance history. The distribution parameters of the used components are different from the new components. Consider a mixture contains of $p$ proportion of new components because of nonavailability of used component when needed and $(1−p)$ proportion of remanufactured/used components when needed. The failure density of the new components is represented as $∼We(η1,β1$⁠) and the failure density of the remanufactured components is represented as $∼We(η2,β2)$⁠. The mixture failure distribution function can be represented as the linear combination of individual’s failure distribution function which is shown in Equation (1).

where,

$f1(t)$ = failure density function of new components

$f2(t)=$ failure density of the remanufactured components

According to the mixture reliability distribution concept, the mixture reliability function can also be represented as the linear combination of the reliability of new and remanufactured component which is shown in Equation (2).

where,

From, Equation (1) and (2), we get the equation of mixture failure rate which is shown in Equation (3).

where,

$β1,η1$ and $β2,η2$ are the shape and scale parameter of the new and remanufactured product respectively. In this study, we considered only random failure for the new components and both random and wear out failure for the remanufactured components. Also, we considered the scale parameter value of the new product is greater than the remanufactured product. According to these assumptions, we formulate two cases to identify the behavioural pattern of the mixture failure rate, case 1: random failure for new and remanufactured components are considered and case 2: random failure for new product and wear-out failure for the remanufactured product.

We consider both the shape parameters of new and remanufactured components are same and scale parameter of the new components is greater than the scale parameter of remanufactured components. According to Equation (3) and (3.a), the mixture failure rate can be written as

Substitute the value of $h1(t)$⁠, $h2(t)$⁠, $R1(t)$ and $R2(t)$ in Equation (4), we get

We consider $p(1−p)=P$ and Equation (6) is represented as

To know the characteristic or behaviour of the mixture failure rate, first-order derivative of Equation (6.a) with respect to t is shown in Equation (7).

Consider$,exp{tη2−tη1}=Z,then$

From Equation (8), it is shown that $hmix′(Z)<0$ because ${1η2−1η1}>0,Z>0$ and $(P*Z+1)2>0$⁠. Thus, $hmix(t)$ is a decreasing function.

Here, we consider the shape parameters of new and remanufactured components are different and scale parameter of the new components is greater than the scale parameter of remanufactured components. $β1,η1$ and $β2,η2$ are the shape and scale parameter of the new and remanufactured product. from Equation (3), we can write the mixture failure rate in a different way which is shown in Equation (9).

From Equation (3.a), $w(t)$ can be written in a different way, shown in Equation (9.a)

Substitute, $(tηi)βi=Ci∀i=1,2$ in the above equation.

On the other hand,

Again,

As, t trends to infinity then Equation (9.c) trends to zero because of $exp{C1−C2}$ trends to infinity in much faster rate than $h2(t)h1(t)$⁠. The value of $hmix(t)$ is equal to $1η1$ when t trends to infinity because $w(t)$ trends to one. Similarly, at t equals to zero the value of $hmix(t)$ is equal to $pη1$⁠. Thus, the value of the mixture failure rate lies between $pη1$ and $1η1$⁠. To know the characteristic or behaviour of the mixture failure rate, the first order derivative of Equation (3) is done. The first order derivative of the mixture failure rate of Equation (3) is shown in Equation (10).

Replace the value of $h1(t)$ and $h2(t)$ by $1η1$ and $β2η2(tη2)β2−1$ respectively in Equation (10). After substituting these values in Equation (10), the first-order derivative of the mixture failure rate is shown in Equation (11).

Equation (11) can also be written as

And, the mixture failure rate is shown in Equation (12.a)

If we consider $(tη2)β2=S$⁠. In the S domain, Equation (12) can be represented as a combination of a straight line and a curve. If we take the derivative of the curve equation in S domain, then we get different values of $β2$ on which the characteristic of the mixture failure rate varies.

In this section, the aforesaid methodology is illustrated by an example. To know the behaviour of the mixture reliability when both the shape parameters of new and remanufactured components are same, that is both are one and the characteristics life, that is the scale parameter of the new parts (⁠$η1)$ and remanufactured parts $(η2)$ is different (⁠$η1>η2)$⁠, simulation is carried out in the MATLAB environment under different mixture proportion values. Consider, the failure density function of new components is $f1(t)=(11800)*exp(−{t1800})$ and the remanufactured components is $f2(t)=(11500)*exp(−{t1500})$⁠. Where, 1800 h and 1,500 h are the characteristics life or scale parameter value of the new and remanufactured components. In the above equations, the shape parameter of new and remanufactured components is 1. $t$ is randomly generated. Equation 1 illustrates the mixed failure distribution.

In case 2, value of the shape parameter of new components is not same as the remanufactured components. Here, the failure density function of new component is $f1(t)=(11800)*exp(−{t1800})$ and the failure density of remanufactured component is $f2(t)=(β21500)*(t1500)β2−1*exp(−{t1500}β2).$ From Equation (12) in case 2, it is observed that the behaviour of the mixture failure rate depends upon the shape parameter of the remanufactured components. Again, a simulation analysis is conducted under different value of the shape parameter, that is $β2$ and mixture proportion. The nature of Equation (12) and the mixed failure rate under the different mixture proportion (p = 0.2, 0.4 and 0.6) is shown in Figures 2–4 respectively.

If $f1(t)=(11800)*exp(−{t1800})$⁠, then the mean time-to-failure of the new components, that is $MTTF1$ is 1800 h and if $f2(t)=(11500)*exp(−{t1500})$⁠, then the mean time-to-failure of the remanufactured components, that is $MTTF2$ is 1,500 h. Thus, the ratio between the $MTTF2$ and $MTTF1$ is $56$⁠. Whereas, in case 2 the mean time-to-failure of the remanufactured component varies when the shape parameter varies.

In Figure 1, it is observed that under the different proportion such as $p=0.2,0.4and0.6,$ the rate of change of the failure rate value is less than zero and it also satisfies Equation (8). From Figure 1, the value of the mixture failure rate for $p=0.2$ is followed by $p=0.4$ and $p=0.6$⁠. Increasing the value of mixture proportion leads to increase in the p value in the denominator of Equation (6.a). Thus, this decreases the overall mixture reliability. Also, if we notice in Figure 1, the failure rate value decreases with the increases of t. This happens because with the increasing in t value, the value of $exp{tη2−tη1}$ in Equation (6.a) increases exponentially which results in decreasing the failure rate. The mixture failure rate is decreasing in nature because both the failure density function of new and remanufactured components is decreasing type. The mixture failure rate at mixture proportion 0.2 is higher than mixture proportion 0.4 and 0.6. The reason behind it that the mixture proportion 0.2 means the mixture contains only 20% of new components and 80% of remanufactured components, so, mixture failure rate is increased. In case 2, value of the shape parameter of new components is not same as the remanufactured components. From the derivation of the case 2, it is observed that behaviour of the mixture failure rate depends upon the shape parameter of the remanufactured components. Figure 2a represents the behavioural pattern of the rate of change of mixture failure rate under the different shape parameter (of remanufactured product) value of 1.2, 1.4, 1.6, 2 and 3 under the mixture proportion value of 0.2. Whereas, Figure 2b represents the mixture failure rate under the above same condition. From Figure 2a, it shows that the behaviour of the failure rate is non-decreasing type. As the value of Equation (12) depends upon the shape parameter value of the remanufactured product, so increasing in value of the shape parameter in Figure 2a leads to increase the value of Equation (12). The mixture failure rate is also increasing with respect to the shape parameter value of the remanufactured components, shown in Figure 2b. From Figure 3a, the nature of the curves under different shape parameters are also non-decreasing in nature. But the magnitude of the value according to equation to Equation (12) in Figure 3a is less than from Figure 2a. This happens because of the mixture proportion in Figure 3a is 0.4. Thus, the increasing in new component amount in the mixture leads to reduction in failure rate. The failure rate characteristics in Figure 3b is also same as in Figure 2b but the vertical axis value in Figure 3b is less than Figure 2b due to the above reason, that is increasing the number of new components. Figure 4a represents the same characteristics as Figure 3a. But, in Figure 4a, the mixture proportion value is 0.6; this means 60% of new components and 40% of remanufactured components which results in decreasing of the rate of change of failure rate value. Figure 4b represents the mixture failure rate value under p equals to 0.6. If we notice Figures 2, 3, and 4b respectively, then we found that for small value of t the failure rate value under shape parameter value of 1.2 is followed by the shape parameter value of 1.4, 1.6, 2 and 3 respectively. The reason behind this phenomenon is that scale parameter of remanufactured components (⁠$η2)$ is less than the scale parameter value of new components (⁠$η1)$ that is $η1>η2$⁠. So, we can say $(tη2)>(tη1)$ and we can also say $(tη2)β2<(tη1),∀β2>1$⁠. Which means that for the small value of t (mixture proportion is constant) the mixture failure rate of Equation (12.a) depends upon the $β2$⁠. If we increase the $β2$ value from 1.2 to 3, we observe that $exp(tη1)$ value in the denominator increases exponentially which decreases the values of the denominator in Equation (12.a). However, the numerator value of Equation (12.a) also decreases with the increases of $β2$ value. So, overall value of the function decreases. For this reason, the mixture failure rate value reduces with the increases of shape parameter value (⁠$β2)$⁠, when t is small. Again, when t-value increases, that is when $t>η2$⁠, the numerator of Equation (12.a) increases. So, the value increases. On the other hand, if we increase the proportion value (⁠$β2$ and t is fixed) the numerator value of Equation (12.a) increases which results in decreasing the value of the function.

The condition under which the mean time-to-failure of the system is maximum when the ratio between the remanufacturing cost and production cost is equal to the mean time-to failure of the remanufacturing component and new components. If the shape parameter value of the new and remanufacturing component is same, then the ratio between the mean time-to-failure of remanufactured components and new components is 5/6. It signifies that ratio between the remanufacturing cost of the remanufactured components and production cost of the new components is 5/6. So, remanufacturing cost is near about eighty percent of the production cost. In case of when the shape parameter of the new and remanufacturing component is not same, then the ratio between the mean time-to-failure of remanufactured and new components varies.

This study proved the need of determining the appropriate percentage of new and reused product in a remanufactured product to improve overall reliability. The producer or remanufacturer carefully selects the optimum proportion to maximise profit. It has a substantial influence on both sustainability and waste reduction.

There are various implications from this study. Under a specified distribution, we proposed a solution for optimal mixture reliability. From the manufacturer’s standpoint, failure rate estimation and behaviour analysis help the manufacturer comprehend the direct impact on profit. Furthermore, producers aim to maximize the mean time to failure of the remanufactured product. This is only achievable if remanufactured components are of comparable quality to new components. From the buyer’s perspective, the buyer will get more confident as the optimum proportion of new and remanufactured components gives better warranty.

In this study, we introduce and investigate the characteristics or behaviour of the failure rate and the reliability aspect of the heterogeneous population of new and remanufactured components with same distribution but different parameters value. This type of heterogeneous population has a significant importance in the industries. From the analysis, we find that under the same value of the shape parameter, that is in case random failure occurs in both new and remanufactured components, the mixture reliability is always decreasing in nature. But the magnitude of the failure rate depends on the mixture proportion value. However, in the other case where the shape parameters value is different, that is new components follow random failure and remanufactured components follow the wear out failure, the mixture failure rate behaviour pattern depends upon the shape parameter or slope of the distribution of the remanufactured component. This study also evaluates the condition under which the system observes the maximum mean time-to-failure.