Quality and reliability oriented maintenance for multistage manufacturing systems subject to condition monitoring

Journal of Manufacturing Systems 52 (2019) 76–85

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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical Paper

Quality and reliability oriented maintenance for multistage manufacturing systems subject to condition monitoring Biao Lu, Xiaojun Zhou

T

⁎

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

ARTICLE INFO

ABSTRACT

Keywords: Multistage manufacturing system Quality improvement Condition-based maintenance Importance measure Cost optimization

For manufacturing systems, product quality and machine reliability are two key health indicators, which usually deteriorate as a result of machine deterioration. Maintenance can mitigate the machine deterioration and consequently improve product quality and machine reliability. In this context, we propose a quality and reliability oriented condition-based maintenance (CBM) policy for the serial multistage manufacturing systems. In the policy, the conditions of quality-related components are monitored to evaluate the quality loss of final products and the system failure rate, which are further compared with the corresponding thresholds to decide whether to trigger preventive maintenance (PM). When PM is triggered, a cost-based improvement factor is introduced to identify the importance ranking of PM groups and then determine a group of machines for PM. This factor is developed based on importance measure and jointly considers the improvement of quality and reliability and the reduction of maintenance costs. A multi-level CBM decision-making process is developed to evaluate the total cost over the planning horizon for decision optimization. The effectiveness and superior performance of the proposed CBM policy is demonstrated through a case study of a serial four-stage machining system producing shaft sleeves.

1. Introduction A multistage manufacturing system (MMS) refers to a system involving multiple stages to complete a final product. Such systems are widely applied in industrial practice, such as automobile assembling, engine head machining, and semiconductor manufacturing, etc. [1]. Product quality and machine reliability are two key health indicators for the MMSs. In the MMSs, the machines are usually subject to deterioration with age and usage. The deterioration of machines will consequently result in the deterioration of product quality and machine reliability. Maintenance can mitigate the machine deterioration and consequently improve the product quality and the machine reliability. However, maintenance activities will consume resources and interrupt normal production and consequently incur costs. It is thus desirable to develop effective maintenance policies for the MMSs to ensure high product quality and machine reliability with the minimum possible maintenance costs. A MMS usually contains multiple machines, and thus the maintenance problem of a MMS is a multicomponent maintenance problem. The multicomponent maintenance problem is a research issue of practical significance, and many maintenance policies have been proposed on this issue, see review articles [2–5]. In most of these maintenance ⁎

policies, the role of maintenance is to improve the component reliability, and the improvement of product quality is not considered. As these maintenance policies do not consider the improvement of product quality, they are generally not effective for the MMSs subject to quality and reliability deteriorations. In fact, there are a few studies which optimal maintenance policies for the MMSs with considering the improvement of product qualtiy through maintenance [6–12]. For instance, Lu and Zhou [6] propose an opportunistic maintenance policy for the serial-parallel MMSs with multiple streams of deterioration, where preventive maintenance (PM) is performed to improve both product quality and machine reliability. The stream of deterioration is a phenomenon describing how the deterioration of quality-related components (QRCs) results in product quality deterioration. The QRCs refer to the machine components having significant impacts on product quality [6]. Ji-wen, et al. [7] propose a maintenance optimization model for tools in the MMSs, where an interaction model is developed to mathematically describe the impact of tool wear on product quality. These studies [6–9] have the common viewpoint that the deterioration of QRCs or the tool wear is the systematic factor resulting in product quality deterioration, and proper maintenance of QRCs or tools is an effective way for quality improvement. In these efforts, the maintenance decisions are made

Corresponding author. E-mail address: [email protected] (X. Zhou).

https://doi.org/10.1016/j.jmsy.2019.04.003 Received 12 June 2018; Received in revised form 7 December 2018; Accepted 15 April 2019 0278-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

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based on the operating time of QRCs or tools instead of the actual conditions of them. With the development of sensor technology, it becomes more economic and convenient to obtain the condition monitoring information of systems/components. This promotes the researches on prognostic and health management, which refers to the scheme in which condition monitoring information is used to predict the health statuses of systems and then facilitate efficient maintenance decision-making [13,14]. The development of prognostic and health management concept motivates us to have the idea of using condition monitoring information of QRCs to assess and predict the health statuses of MMSs and then make efficient maintenance decisions. This leads to two problems having to be solved. One is the assessment and prognostic of the health statues of MMSs. The other is the development of an effective maintenance policy. For the first problem, a lot of prognostic approaches have been proposed in past decades. Prognostic techniques, such as vibration signature analysis, temperature analysis, oil analysis, and acoustic emission, have been widely used to measure statuses of systems [15]. Many prognostic approaches have thus been proposed to obtain a rational estimation of the reliability or remaining useful life of systems [16,17]. A fact is that the commonly used prognostic approaches are mainly devoted to assess and predict the reliability performance of systems. For the MMSs, product quality and machine reliability are both important health indicators. Hence, there is also a need to assess and predict the quality performance for the MMSs. For this end, the streamof-deterioration model proposed by Lu and Zhou [6] is adopted. This model mathematically describes how the deterioration of QRCs results in product quality deterioration in a MMS. With this model, the quality of final products can be assessed and predicted when the condition monitoring information of QRCs are obtained at inspections. For the second problem, an increasing number of condition-based maintenance (CBM) policies have been developed for multi-component systems in recent years [18–24]. For instance, Tian and Liao [19] propose a CBM optimization model for multi-component systems using proportional hazard model. If the hazard rate of a component exceeds a level-1 threshold, the component will be preventively replaced, and other components whose hazard rates exceed a level-2 threshold will be replaced together. Zhu, et al. [20] propose a CBM policy for multicomponent systems with a high maintenance setup cost. A periodic maintenance interval is adopted for the system, during which the components whose degradation levels exceed the corresponding limits are maintained together to save the setup cost. Poppe, et al. [21] propose a hybrid CBM policy for a multi-component system where the condition of a critical component is monitored. Two thresholds are implemented for the degradation level to decide when to maintain the monitored component. One is used to decide when to opportunistically maintain the critical component together with other components; the other is used to decide when to maintain the component for preventing a failure. Overall, the CBM policies for multi-component systems mainly address two issues. One is when to trigger PM, and the other is how to exploit the economic dependence between components to save maintenance costs. The economic dependence implies that maintaining multiple components together can be cheaper than maintaining them separately [25]. It can be seen that existing CBM policies generally implement thresholds for health indicators to trigger PM and to combine PM operations for saving PM setup cost and system downtime cost. This paper also implements thresholds for the health indicators to trigger PM. Specifically, a quality loss threshold (QLT) and a failure rate threshold (FRT) are implemented for the quality loss of final products and the system failure rate, respectively. As for determining a group of machines for PM, the method of implementing thresholds mainly concerns about the reduction of maintenance costs by combining PM operations and does not consider the improvement of product quality and machine reliability. For the MMSs, it is important to consider the improvement of product quality and machine reliability in PM decision-

making. Therefore, it is desirable to develop a method which jointly considers the improvement of quality and reliability and the reduction of maintenance costs. For this purpose, we propose a cost-based improvement factor based on the idea of importance measure [26–28]. This factor identifies the importance ranking of PM groups relating to ability on the improvement of quality and reliability as well as the reduction of maintenance costs. The group with the highest importance ranking is selected for PM. Ultimately, a novel quality and reliability oriented CBM policy is developed for the MMSs, where PM is performed to improve product quality and machine reliability, and the conditions of QRCs are periodically inspected for PM decision-making. At each inspection, the quality loss of final products and the system failure rate are assessed and then compared with the QTL and the FRT respectively to decide whether to trigger PM. When PM is triggered, the cost-based improvement factor is used to determine a group of machines for PM. A multi-level CBM decision-making process is developed to evaluate the total cost over the planning horizon for decision optimization. In summary, this paper makes contributions to the prognostic and health management techniques on both health prognostic and maintenance management. On the health prognostic, this paper extends the existing health indicators by introducing product quality as a new health indicator. On the maintenance management, this paper proposes a quality and reliability assessment based PM-triggering decision rule as well as a novel cost-based improvement factor for determining a group of machines for PM, which jointly considers the improvement of quality and reliability and the reduction of maintenance costs. Besides, it should be noted that although this paper and the paper by Lu and Zhou [6] both deal with the PM decision problem of MMSs, the PM decision methods of this two papers are basically different. In the paper [6], PM decisions are made based on the operating times of QRCs, while in this paper, PM decisions are made based on condition monitoring information of QRCs. Moreover, the paper [6] implements thresholds for the machine failure rates to determine a group of machines for PM, while this paper proposes a cost-based improvement factor to solve this problem. The rest of the paper is organized as follows. Section 2 presents the mathematical description of quality and reliability deteriorations in the MMSs. Section 3 models the improvement effects of PM on product quality and machine reliability. In Section 4, the quality and reliability oriented CBM policy is developed. In Section 5, a case study is conducted to illustrate the effectiveness of the proposed CBM policy. Section 6 gives some concluding remarks and future work direction. 2. Mathematical description of quality and reliability deteriorations Consider a serial MMS which is dedicated to producing a type of products at a high volume. In the serial MMS, each stage is allocated with one machine. The serial MMS has N machines, and the number of variables measuring the deterioration states of QRCs of machine k (k = 1, …, N) is nk. The QRCs of the machines deteriorate progressively, which leads to the deterioration of product quality. Besides, components of the machines suffer inevitable wear, corrosion or fatigue, which makes the machines suffer increasing failure risks. Lu and Zhou [6] have developed models to mathematically describe the deterioration of product quality and machine reliability in the MMSs. This paper adopts the models by Lu and Zhou [6] to describe the quality and reliability deteriorations in the serial MMS. Some assumptions are given as follows. (1) For the serial MMS, the production rate of each stage is the same. The serial MMS is dedicated to producing a type of products at a high volume. A common example of such a system is the flow production line. In a flow production line, the production rate of 77

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Fig. 1. Stream of deterioration in the serial MMS.

each stage is the same. As the production rate of each stage is the same, the production rate of the system equals to that of a stage. (2) In the serial MMS, when one machine stops, the whole system has to be stopped. The serial MMS is dedicate to producing a type of products at a high volume. A common example of such a system is the flow production line. In the flow production lines (i.e. production lines of automobile engine cylinder head or cylinder block), the situation that when one machine stops, the whole system has to be stopped is relatively common.

2.2. Mathematical description of quality deterioration Product quality is measured by key product characteristics (KPCs). The KPCs of a final product are the results of a sequence of operations performed in all the stages. In each stage, the machine perform an operation to form KPCs, and the deterioration of QRCs will result in the deviations of these KPCs. It is observed that the KPCs formed at different stages are usually interrelated. The interrelation between two KPCs is that the deviation of one KPC formed at an upstream stage will affect the deviation of the other formed at a downstream stage. Due to the interrelations between KPCs, the deterioration of QRCs in a stage will not only cause the deviations of the KPCs formed at that stage but also propagate to downstream stages by the KPCs and then affect the deviations of the KPCs formed there. This propagation of deterioration of QRCs among a sequence of stages is referred to as stream of deterioration in Lu and Zhou [6]. The stream-of-deterioration phenomenon describes how machine deteriorations result in product quality deterioration in a MMS. Fig. 1 illustrates the stream of deterioration phenomenon in the serial MMS. In Fig. 1, Zk (t ) = [Zk1 (t ), ,Zkok (t )]T denotes the uncontrollable process variables in stage k; Yk (t ) = [Y1k (t ), ..., Ymk (t )]T denote the deviations of KPCs of the products coming out of stage k, where Ysk (t ) (s = 1, …, m) denotes the deviation of the sth KPC, and m denotes the number of KPCs of a final product. The uncontrollable process variable is a general designation of the process variables which have significant impact on product quality and vary randomly and cannot be controlled by maintenance [6]. Examples of uncontrollable process variables include human factors, raw materials and environmental variations, etc. In a stage, the deviations of KPCs of outgoing products are contributed by three factors: (i) the deviations of the interrelated KPCs formed in upstream stages; (ii) the deterioration of QRCs of the machine; and (iii) the uncontrollable process variables in the stage. From Fig. 1, it can be seen that to model the stream of deterioration is to mathematically describe the update of deviations of KPCs when the products flow along the stages. There are three situations to consider for modeling the update of Yk (t ) . Situation 1. If the sth KPC is neither formed in stage k nor has been formed in an upstream stage, then we have

2.1. Mathematical description of deterioration processes of QRCs The QRCs refer to the machine components having significant impacts on product quality [6,29]. For different manufacturing processes, the QRCs are usually different. For example, in machining processes, the QRCs can be cutting tools (i.e. the wear of cutting tools can affect dimensional quality), spindles (i.e. the vibration on spindles can affect surface quality), or fixture locators; in assembly process, the QRCs can be locators (the wear or displacement of locators can affect assembly accuracy); in sheet metal stamping process, the QRCs can be dies (the wear of dies can affect the dimensions of products). For a manufacturing process, the QRCs can be identified through analysis of physical characteristics of the manufacturing process along with design of experiments. Engineers can firstly choose some machine components that may have impacts on product quality based on engineering analysis, and then they design experiments to test the significance of the chosen factors (components) and determine the significant factors as the QRCs. The deterioration processes of QRCs (e.g. the wear of cutting tools or dies) are generally stochastic processes with independent and nonnegative increments, which can be properly described by the gamma process [30]. Assume that the deterioration processes of QRCs of a machine are statistically independent of each other. For instance, in machining processes, the war of cutting tools and the wear of locators are independent; in sheet metal stamping processes, the wear of inner die and the wear of outer die are independent [8]. Let Wk (t ) = [Wk1 (t ), ..., Wknk (t )]T , t > 0 denote the deterioration processes of QRCs of machine k between two consecutive maintenance interventions, where t is a time index. Then Wki (t ), t 0 (k = 1, …, N; i = 1, …, nk) can be mathematically described by a gamma degradation process Ga (µki , ki ) with shape parameter µki and scale parameter ki :

Ysk (t ) = 0, k = 1, …, N

Situation 2. If the sth KPC has been formed in an upstream stage before stage k, then we have

• W (0) 0, • W (t ), t > 0 has independent increments, • For t > b > 0, W (t ) W (b) follows a gamma distribution: ki

Ysk (t ) = Ys, k

ki

ki

fki (w ) =

ki

µki (t b) µ (t b) 1 w ki exp( ki

[µki (t

b)]

(2)

1

1 (t ),

(3)

k = 2, …, N

Situation 3. If the sth KPC is formed in stage k, then Ysk (t ) is the result of combined effects of Yk 1 (t ) , Wk (t ) and Zk (t ) . To mathematically describe this impact mechanism, the stream-of-deterioration model developed by Lu and Zhou [6] is adopted. This model is given as

ki w )

(1)

where µki (t b) and ki are the scale parameter and shape parameter of the gamma distribution, respectively, and ( ) is a gamma function. The gamma process is a Markov process. Thus, the degradation model Ga (µki , ki ) can be used to predict the future deterioration of QRCs from the time when they are preventively maintained, or their actual conditions are revealed by inspections.

Ysk (t ) =

sk

Wk (t ) + [ aTsk eTsk gTsk ] Yk 1 (t ) Zk

+ [ Xk (t )T Yk

78

T 1 (t )

0 Bsk Fsk ZTk ] 0 0 Usk 0 0 0

Wk (t ) Yk 1 (t ) + Zk

sk

(4)

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where sk is a scalar constant, a sk , esk , and gsk are vectors defining the linear effects of Wk (t ) , Yk 1 (t ) , and Zk , respectively; Bsk , Fsk , and Usk are matrixes defining the effects of interactions between Wk (t ) and Yk 1 (t ) , Wk (t ) and Zk , Yk 1 (t ) and Zk , respectively; sk is the model error and is assumed to follow a normal distribution with zero mean N (0, sk ) . The uncontrollable process variables Zk is assumed to obey a multivariate normal distribution with zero means N (0 , k ). The parameters of the stream-of-deterioration model can be estimated based on data of products (defined in Section 3.2.1 of [6]) collated from the manufacturing processes. Based on the stream-of-deterioration model, a recursion formula describing the transformation of deviations of KPCs from Yk 1 (t ) to Yk (t ) can be derived, i.e.

Yk (t ) =

k (t ) Yk 1 (t )

+

k (t ),

(PM). Whenever a machine fails, minimal repair is conducted to restore it back to the normal operational state. PM is performed proactively to mitigate the machine deterioration and consequently improve the product quality and machine reliability. Some assumptions are given as follows. (1) Minimal repair changes neither the machine condition, nor the deterioration states of QRCs. The duration of minimal repair is neglected. (2) PM restores the machine condition to somewhere between as good as new and as bad as old, and restores the states of QRCs to as good as new. In a PM activity, not all components of the machine, but those components that are judged to be in bad conditions, are repaired or replaced. As a result, the machine is restored to a better but not as good as new state. Meanwhile, all the QRCs of a machine are calibrated, repaired or replaced. Consequently, the states of QRCs can be restored to as good as new.

(5)

k = 1, …, N

+ + where k (t ) = [ 1k (t ), ..., mk with sk (t ) = T and with k (t ) = [ 1k (t ), ..., mk (t )] sk (t ) = sk + aTsk Wk (t ) + gTsk Zk + Wk (t ) Fsk Zk + sk , and Y0 (t ) 0 . Based on the recursion formula, the deviations of KPCs of final products YN [t|W1(t ), ..., WN (t )] can be obtained. It can be seen that YN [t|W1(t ), ..., WN (t )] is the result of combined effects of deterioration of QRCs of all machines. The deviations of KPCs will lead to quality loss, which is measured by the Taguchi’s loss function [31]. Then, the quality loss of final products can be calculated by

(t )]T

eTsk

WTk (t ) Bsk

Q (t ) = E {YTN [t|W1(t ), ..., WN (t )] qYN [t|W1 (t ), ..., WN (t )]}

ZTk UTsk ,

PM restores the states of QRCs of machines to as good as new. Assume that when the QRCs are new their deterioration levels are zero. This means after each PM, the deterioration levels of QRCs are reset to 0. And thus the deterioration processes of QRCs are the same within each PM cycle (the duration between two consecutive PM operations). Let X k (t ) = [Xk1 (t ), ..., Xknk (t )]T denote the deterioration states of QRCs of machine k at time t within the planning horizon, and let tklpk (lk = 1, 2, …) denote the time when the lkth PM operation is conducted on machine k. Then we have X k (tklpk ) = 0 (lk = 0, 1, 2, …). The deterioration states of QRCs within each PM cycle can be described by the gamma degradation process Ga ( ki, ki ) given in Section 2.1. Then the deterioration states of QRCs of machine k at time t (t∈[0, PH]) can be given by

(6)

where q = diag (q1, ..., qm) is a matrix of quality loss coefficients. Q (t ) can be derived as a function of t. For the derivation method, please refer to Section 3.4.2 of [29]. 2.3. Mathematical description of reliability deterioration The deterioration of QRCs may affect the failure risk of the machine [6], and the deterioration states of QRCs are inspected. To more accurately assess the machine failure rate, it is desirable to integrate the deterioration states of QRCs into the machine failure rate model and estimate their effects. To this end, the proportional hazard model (PHM) is adopted to model the machine failure rate. Based on the PHM, the deterioration states of QRCs can be integrated into the machine failure rate model as covariates. The method of using PHM to develop the machine failure rate model is detailedly explained in Section 3.1.1 of the paper [6]. Thus, the failure rate function for machine k is given by

hk (t ) =

k k

t

k 1

k

E { exp [

T k Wk (t )]},

k = 1, …, N

X k (t ) =

Vklk = Vk, lk (7)

hk (t )

tkp, lk 1), if t

(tkp, lk 1, tklp k ), lk = 1, 2, ..

(9)

+ bklk (tklp k

1

tkp, lk 1), lk = 1, 2, …, Mk

(10)

where Vk0 0 , Mk is the number of PM operations for machine k over the planning horizon and bklk (0 < bklk < 1) is the age reduction factor. Note that bkj = 0 means that PM restores the machine condition to as good as new, and bkj = 1 means that PM does not change the machine condition. Based on Eqs. (7) and (10), the evolution of failure rate of machine k under PM interventions can be obtained, i.e. k k

k

h k (t ) =

k k

k

V k,klk 11, if t = tklpk , lk = 1, 2, … (t

tkp, lk

1

+ Vk, lk

1)

k 1E { exp [ T X

k

k (t

tkp, lk 1)]}, if t

(tkp, lk 1, tklp k ), lk = 1, 2, …

N k=1

Wk (t

p where tk0 0. PM restores the machine condition to somewhere between as good as new and as bad as old. This implies that the failure rate of the machine will be reduced by a certain degree after PM. The failure rate of a machine is dependent on its age, thus the virtual age method [32] is adopted to model the PM effect. Let Vk, lk 1 denote the virtual age of machine k after the (lk-1)th PM operation. Then, the virtual age of machine k after the lkth PM operation can be calculated by

where k k (t k ) k 1 is a Weibull distribution describing the baseline hazard rate (depending on the operating age of the machine only), k ( k > 1) , k is the shape and scale parameter of the Weibull distribution, respectively, and k = [ k1, ..., knk ]T is a vector of regression parameters defining the effects of deterioration of QRCs on machine failure rate. hk (t ) can be derived as a function of t. For the derivation method, please refer to section 3.3 of [29]. Assume that the deterioration processes of different machines are independent. For a series system with independently deteriorating machines, the failure rate of the system hsys (t ) is given by

hsys (t ) =

0, if t = tklpk , lk = 1, 2, ..

(11)

(8)

4. Development of quality and reliability oriented CBM policy

3. Modeling of preventive maintenance (PM) effect

A novel quality and reliability oriented CBM policy is proposed to facilitate the PM decision-making. In the policy, the deterioration states of QRCs are inspected periodically to evaluate the quality loss of final products and the system failure rate. Then, the evaluated quality loss of

Maintenance is employed to counter the deterioration of product quality and machine reliability. Two kinds of maintenance operations are considered, which are minimal repair and preventive maintenance 79

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final products and system failure rate are respectively compared with the quality loss threshold (QLT) and the failure rate threshold (FRT) to decide whether to trigger PM. When PM is triggered, a cost-based improvement factor is introduced to identify the importance ranking of PM groups and then determine a group of machines for PM.

reduction of quality loss of final products due to PM. The improvement of machine reliability is measured by the reduction of minimal repair cost due to PM. Note that when the reliability of a machine is improved, the machine will be less likely to fail and consequently need fewer minimal repairs. The cost-based improvement factor for a PM group Gu is denoted by EGu (t ) . It is given as follows:

4.1. PM-triggering decision based on quality and reliability assessment The QRCs are a special kind of machine components. For a machine, the deterioration of QRCs will lead to the deterioration of product quality and may affect the failure rate of the machine. Thus the deterioration states of QRCs are inspected to evaluate the quality loss of final products and the failure rates of the machines. The QRCs of all machines are inspected periodically. That is, the QRCs of all machine are inspected with a fixed time interval T, and each inspection incurs an inspection cost c i . It should be noted that in some cases, it is necessary to implement sensors for inspection, due to, for example, the inaccessibility of the QRCs or the convenience of detection. At each inspection epoch Tj = jT (j = 1, 2, 3, …), the deterioration states of QRCs of all machines x1j , x2j , ..., xNj are inspected, where x kj denotes the deterioration states of QRCs of machine k observed at the jth inspection. The observed deterioration states of QRCs of all machines x1j , x2j , ..., xNj can be used to evaluate the quality loss of final products and the failure rates of the machines. We can use x1j , x2j , ..., xNj to evaluate the deviations of KPCs of final products YN [Tj |x1j , ..., xNj ] by Eq. (5) (replacing Wk (t ) with x kj ) and then use YN [Tj |x1j , ..., xNj ] to calculate the quality loss of final products Q (Tj |x1j , ..., xNj ) by Eq. (6). Besides, we can use x kj to evaluate the failure rate of each machine hk (Tj | xkj ) by Eq. (7) and then use hk (Tj | xkj ) (k = 1, …, N) to calculate the system failure rate hsys (Tj |x1j , ..., xNj ) by Eq. (8). The evaluated quality loss of final products and system failure rate are compared with the predetermined QLT Dq (Dq > 0) and FRT Df (Df > 0) respectively to decide whether to trigger PM. If the quality loss of final products reaches the QLT, or the system failure rate reaches the FRT, PM is triggered. When a PM is triggered, there is a need to determine which machines should be maintained. We develop a cost-based improvement factor to solve this issue.

EGu (t ) =

QGu (t )] +

[Q (t )

c m [hk (t ) k Gu k

hk0 (t )]

CGu

,

(12)

denotes the quality loss of final products if the machines where in Gu is preventively maintained at time t, hk0 (t ) denotes the failure rate of machine k if it is preventively maintained at time t, ckm denotes the cost of a minimal repair for machine k, and CGu denotes the PM-related cost for the PM group Gu . Thus, Q (t ) QGu (t ) represents the reduction of quality loss of final products if the machines in Gu is preventively maintained at time t. ckm [hk (t ) hk0 (t )] represents the reduction of minimal repair cost if machine k is preventively maintained at time t, c m [hk (t ) hk0 (t )] represents the sum of reductions of and k Gu k minimal repair costs if the machines in Gu is preventively maintained at time t. According to assumption (2), in the serial MMS, when one machine stops the whole system has to be stopped. Hence, maintaining multiple machines together incurs a common setup cost and downtime cost (production loss). If these machines are maintained separately, multiple setup costs and production losses will be incurred. This implies that joint maintenance of a group of machines can reduce costs compared with maintaining them separately, which is the economic dependence [25]. Due to the economic dependence, it is profitable to maintain a group of machines together. To promote the combination of PM operations, the save of setup cost and production loss brought by combining PM operations is considered in the development of PM-related cost CGu . Let |Gu| denote the number of machines in the PM group Gu . If the machines in Gu are maintained together, (|Gu| -1) times setup cost and production loss will be saved. Then, the PM-related cost for the group Gu can be represented by

QGu (t )

ckp

CGu =

(|Gu|

k Gu

4.2. PM group determination using a cost-based improvement factor

1)(c s + c er

Gu )

Re duction of setup cost and production loss

Direct maintenance cost

(13)

where denotes the direct maintenance cost for machine k, c s denotes the setup cost for a PM activity, and c er Gu represents the production loss during the PM of the group Gu . Specifically, c e denotes the revenue of a product, r denotes the production rate of the system and Gu is the duration for the PM of the group Gu . For the serial MMS, when one machine stops for PM, the whole system has to be stopped. Thus, during the PM of the group Gu , the whole system stops operation. In this case, r Gu represents the amount of products that can be produced during the PM of the group Gu , and thus c er Gu represents the production loss during the PM of the group Gu . Based on Eq. (13), it can be seen that the bigger |Gu| is, the smaller the PM-related cost will be. This implies that the ability of a PM group on the reduction of PM-related cost is integrated into the importance measurement of the PM group. Besides, we assume that enough resources are available to ensure that all machines in the same group can be maintained in parallel. Usually, manufacturing companies can use maintenance resources inside and/or buy maintenance services from outside to ensure that all machines in the same group can be maintained in parallel. Then, the PM duration of the group Gu equals to the maximum of PM durations of the machines in the group Gu . Let k denote the PM duration of machine k. Then we have

ckp

The PM of machines will improve the product quality and machine reliability, but it will also incur costs. The costs incurred by a PM operation includes setup cost (i.e. the cost for transportation of maintenance crews and tools), downtime cost (production loss) and direct maintenance cost (i.e. the cost of replacing machine components). These costs are summarized as PM-related cost. From a practical point of view, a high improvement of quality and reliability with a very high PM-related cost may not be a proper choice for PM optimization. A “reasonable” improvement of quality and reliability with a lower PMrelated cost may be more appropriate. Hence, it is necessary to comprehensively consider the improvement of quality and reliability and the PM-related cost in determine a group of machines for PM. Importance measure is a method commonly used in reliability improvement activity to identify the importance ranking of components relating to the improvement ability on the system reliability [26–28]. Based on the idea of importance measure, we develop a cost-based improvement factor to identify the importance ranking of PM groups relating to the improvement ability on quality and reliability and the reduction ability on PM-related cost. The group with the highest importance ranking is selected for PM. The cost-based improvement factor is defined for a PM group as the ratio of improvement of product quality and machine reliability to its PM-related cost. To make the unit of quality and reliability improvement consistent with PM-related cost, we use cost reduction to measure the improvement of quality and reliability. The improvement of product quality is measured by the

Gu

= max { k : k

Gu}

(14)

4.3. Optimization of decision parameters The QLT, the FRT and the inspection interval are the three decision 80

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Fig. 2. The multi-level CBM decision-making process.

parameters of the CBM policy, which can be optimized by minimizing the total cost over the planning horizon [0, PH]. The total cost includes inspection cost, quality loss, minimal repair cost, and PM-related cost. To evaluate the total cost over the planning horizon, a multi-level CBM decision-making process is proposed. The multi-level decision-making process consists of system-level and machine-level: the system-level is to decide whether to trigger PM, and the machine-level is to determine a group of machines for PM when PM is triggered. Fig. 2 shows the multi-level CBM decision-making process. System level. At each inspection epoch Tj=jT (j=1, 2, …), the quality loss of final products Q (Tj |x1j , ..., xNj ) and the system failure rate hsys (Tj |x1j , ..., xNj ) are firstly evaluated and then respectively compared with the QLT Dq and the FRT Df to decide whether to trigger PM. If Q (Tj |x1j , ..., xNj ) Dq , or hsys (Tj |x1j , ..., xNj ) Df , PM is triggered. Otherwise, PM is not triggered, and the system keeps operation until the next inspection. From the measure x kj (k = 1, …, N), the future deterioration of QRCs of machine k after the jth inspection can be predicted using the gamma degradation process Ga ( ki, ki ) given in Section 2.1, i.e.

X k (t|xkj )t Tj

=

Wk (t

Tj ), if Tj =

x kj + Wk (t

EGL* (tLp ) = max {EGu (tLp | x1j, ..., xNj )}

Based on the PM schedule obtained from the maintenance decisionmaking process, the total cost over the planning horizon can be calculated by PH

TC =

N

+

T

Tj + 1

+

rQ (t ) dt j=0

Tj Quality loss Tj + 1

Mk

p k = 1 lk = 0 tkp, l 1 Tj < tkl k k

ckm

hk (t ) dt Tj

Minimal repair cost Ms

+

c set + c er

GL*

ckp ,

+

L =1

k GL*

(18)

PM-related cost

where c i is the cost of a single inspection, PH T denotes the number of inspections conducted during the planning horizon,

(tklpk 1, tkp, lk )

(15)

1}

Tj + 1 Tj

rQ (t ) dt re-

presents the cumulative quality loss of final products between the jth

When PM is triggered, there is a needed to determine a group of machines for PM, which is addressed in the machine level. Machine level. For the N-stage system, the number of alternative groups is 2N-1. A group is said to be admissible if the PM of all machines in the group can reduce both the quality loss of final products and the system failure rate to below the corresponding threshold. Let Uadm denote the set of all the admissible groups. Uadm can be derived by Gu p Uadm = {u| QGu (tLp ) < Dq , hsys (tL ) < Df : u = 1, ..., 2 N

PH T

ci

Inspection cost

tklpk

Tj ), if Tj

(17)

u Uadm

and (j+1)th inspections,

p tkl

k

cm Tj < tkp, l + 1 k k

Tj + 1 Tj

hk (t ) dt represents the

expected minimal repair cost for machine k between the lkth and (lk+1) th PM operations, Mk denotes the number of PM operations for machine k over the planning horizon, (c set + c er GL* + k G * ckp) represents the L PM-related cost of the Lth PM activity for the system, and Ms denotes the number of PM activities for the system over the planning horizon. It should be noted that the formula of PM-related cost in Eq. (18) is different from that in Eq. (13). The reason for their difference is that they are developed for different purposes. The formula of PM-related cost in Eq. (18) is to evaluate the actual PM-related cost for a PM group, while CGu in Eq. (13) is to evaluate the ability of a PM group on cost reduction. Stochastic Monte Carlo simulation is used for decision optimization. For each value of the decision parameters (Dq , Df , T ) , multiple simulations are done to evaluate the average total cost. To ensure that the

(16)

where is the time of the Lth PM activity for the system. Based on the set of all the admissible groups, an optimal group is determined as the one having the highest value of the cost-based improvement factor E• (tLp ) . Let GL* denote the optimal group of machines that are selected for PM. It satisfies

tLp

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convergence of the average total cost is reached, a large number of simulations must be done. The average total cost is given by

chain. The QRCs of the machines and uncontrollable process variables in each machine are shown in Table 2. Parameter values are obtained based on the data of products, the failure data of machines and the maintenance records from the machining process. The parameters of the stream-of-deterioration model are shown in Eqs. (21)–(24). The parameters for the model errors are: σ22 = 0.0008, σ32 = 0.00085, σ43 = 0.0009, σ11 = 0.00082, σ54 = 0.00086, σ64 = 0.0008. The parameters for the uncontrollable process variables are: Σ1 = 0.00171 mm2, Σ2=diag (0.00142, 0.00126) mm2, Σ3 = 0.0029 mm2, Σ4=diag (0.0013, 0.00236) mm2. Based on the parameters of stream-of-deterioration model, the deviations of KPCs of final products can be derived by Eq. (5). Then the quality loss of final products can be derived by Eq. (6). The values for other parameters are shown in Table 3. In addition, r = 330 unit/day, q=diag (2, 4, 16, 20, 5, 10) $/mm2, and PH = 200 day.

K

Ca (Dq , Df , T ) =

TCi

K

(19)

i=1

where TCi is the total cost of for ith simulation. By varying different values of (Dq , Df , T ) , the minimum value for the average total cost can be obtained. The optimal values for (Dq , Df , T ) are derived when the minimum average total cost is reached, i.e.

Ca (Dq , Df , T ) = min {Ca (Dq , Df , T ): 0 < Dq Dfmax , 0 < T

Dqmax , 0 < Df (20)

PH }

where is a threshold value for Dq that when Dq is greater than this value, PM will be less likely to be triggered by Dq , and Dfmax is a threshold value for Df that when Df is greater than this value, PM will be less likely to be triggered by Df .

Dqmax

Stage 1: Y11 (t ) = 0.00036 + 0.987X11 (t ) + 0.935Z11 0.0082X11 (t ) Z11 + Y21 (t ) = 0, Y31 (t ) = 0, Y41 (t ) = 0, Y51 (t ) = 0, Y61 (t ) = 0

5. Case study

11

(21)

Stage 2:

5.1. Case overview

Y22 (t ) = 0.00034

The purpose of this section is to demonstrate the effectiveness of the proposed CBM policy through a case study. The case study is about a serial four-stage machining system employed to produce shaft sleeves illustrated in Fig. 3. The shaft sleeve is completed through four stages of machining operations, as described in Table 1. For the machining process, the 1 st KPC and 2nd KPC are interrelated, as they form a dimensional chain. The 3rd KPC and 5th KPC are interrelated, as the Outer cylinder D is used as a locating datum for broaching the Keyway F. The 4th KPC and 5th KPC are interrelated, as they form a dimensional

+ 0.0074X21 (t ) Z21 + 22 Y32 (t ) = 0.0004 + 2.012X22 (t ) + 1.923Z22 + 0.0085X22 (t ) Z22 + Y12 (t ) = Y11 (t ), Y42 (t ) = 0, Y52 (t ) = 0, Y62 (t ) = 0

0.985X21 (t ) + 0.991Y11 (t )

0.926Z21 32

(22)

stage 3: Y43 (t ) = 0.00032 1.984X31 (t ) + 0.843Z31 0.0382X31 (t ) Z31 + 43 Y13 (t ) = Y12 (t ), Y23 (t ) = Y22 (t ), Y33 (t ) = Y32 (t ), Y53 (t ) = 0, Y63 (t ) = 0 (23)

Fig. 3. Schematic illustration of the shaft sleeve.

Table 1 Description of the operations. Stage

Machine

Operation

KPCs

1 2

Milling machine Lathe

Mill Face B Turn Outer cylinder D

3 4

Boring machine Broaching machine

Bore Hole E Broach Keyway F

1st KPC: distance between Face B and Face A 2nd KPC: distance between Face C and Face B 3rd KPC: diameter of Outer cylinder D 4th KPC: diameter of Hole E 5th KPC: distance between the top of Keyway F and bottom of Hole E; 6th KPC: width of Keyway F

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Table 2 The QRCs of machines and the uncontrollable process variables in each stage. Stage

QRCs of machines

Uncontrollable process variables

1 2

X11(t): the wear of the mill cutter X21(t): the wear on the major frank of the turning tool X22(t): the wear on the minor frank of the turning tool X31(t): the wear of the boring tool X41(t): the wear along the height direction of the broach X42(t): the wear along the width direction of the broach

Z11: the locating error Z21: the locating error Z22: the tool setting error

3 4

Z31: the tool setting error Z32: the vibration of the boring tool Z41: the locating error Z42: the expansion of the broach

Table 3 Parameter values. k

1 2 3 4

µk

k

0.00142 [0.00163, 0.00166] 0.00163 [0.00149, 0.00318]

Notes: µk = [µk1 , ..., µknk ],

=[

k1,

...,

k

(day)

knk ],

0.015 [0.01, 0.02] 0.019 [0.016, 0.011] k

=[

k1 ,

...,

2.3 2.5 2.8 2.6

120 88 126 135

0.08 0.085 0.083 0.079

0.00052

0.988X 41 (t )

0.5 0.5 0.5 0.5

ckp

($)

ckm ($)

cs ($)

ci ($)

ci ($/unit)

3800 3530 3610 3840

3700 3680 3550 3600

230

380

15

knk ].

(cycles) during the planning horizon. In the first PM activity, machine 4 is schedule a PM operation, and thus the PM group contains machine 4. For the four PM activities, the number of machines in the PM group is 1, 1, 1, and 2, respectively. This implies the average number of machines in a PM group is 1.25. For the 10,000 simulations, the average number of machines in a PM group is obtained as 1.18.

stage 4: Y54 (t ) =

p k

bk

(day)

3.5 [3.3, 4.5] 4.2 [4.4, 5.2] k

k

k

( mm−1)

0.707Y33 (t ) + 0.501Y43 (t ) + 0.945Z41

0.0081X 41 (t ) Z41 + 54 Y64 (t ) = 0.0003 0.993X 42 (t ) + 0.674Z42 + 0.0085X 42 (t ) Z42 + Y14 (t ) = Y13 (t ), Y24 (t ) = Y23 (t ), Y34 (t ) = Y33 (t ), Y44 (t ) = Y43 (t )

64

(24)

5.3. Sensitivity analysis 5.2. PM decision results

Sensitivity analysis is carried out for the parameters that may significantly impact the PM decisions. These parameters include minimal repair cost rate ckm , quality loss coefficient q, and PM cost rate ckp . As there are four machines in the system, the change of ckm or ckp for one machine may have little impact on the PM decision results. For this sake, ckm and ckp are amplified or reduced for all machines by the same ratio simultaneously. Table 5 shows the PM decision results under different ckm or q . Table 6 shows the PM decision results under different ckp . V is the scaling ratio for ckm , q or ckp . M is the average number of PM activities for the system. From Table 5 and Table 6, it can be seen that

For the machining system, stochastic Monte Carlo simulation is used to evaluate the average total cost. For each value of the decision parameters (Dq , Df , T ) , 10,000 simulations are done to ensure that the convergence of the average total cost is reached. To find the optimal values for the decision parameters (Dq , Df , T ) , the average total cost C (Dq , Df , T ) under different values of (Dq , Df , T ) is calculated, where the varying range for them is 0 < Dq < 2.4 , 0 < Df < 0.09 and 1 T 20 , respectively. The minimum average total cost is obtained as C (Dq*, Df*, T *) = 84976 , and the corresponding optimum values for the decisions parameters are Dq* = 0.5, Df* = 0.064 , and T* = 8. The results imply that when the proposed CBM policy is applied for the machining system, the actual quality loss of final products can be controlled below 0.5, and the system failure rate can be kept below 0.064. Thus, it can be summarized that under the proposed CBM policy, both the final-product quality and the system reliability can be ensured at a high level. For the optimal decision parameters, we run a simulation to see the PM schedule for the system during the planning horizon. Table 4 shows the PM schedule for the system under the optimal decision parameters. From Table 4, it can be seen that there are totally 4 PM activities

Table 5 The PM decision results under different ckm or q . V

0.5 1 2 3 4

ckm

q

D* q

Df*

T*

M

D* q

Df*

T*

M

0.6 0.5 0.5 0.6 0.6

0.078 0.064 0.058 0.056 0.054

8 8 8 8 8

3.18 4.16 4.55 4.66 6.43

– 0.5 0.7 0.8 0.9

– 0.064 0.064 0.064 0.064

– 8 8 8 8

– 4.16 4.17 4.24 4.29

Table 4 The PM schedule for the system under the optimal decision parameters. PM cycle PM time Machine Machine Machine Machine

(day) 1 2 3 4

1

2

3

4

75

104

120

176 ■ ■

■

Table 6 The PM decision results under different ckp .

■ ■

Notes: ‘■’ indicates that the machine is scheduled a PM action in the corresponding PM cycle. 83

V

D* q

Df*

T*

M

0.5 1 2 3 4

0.5 0.5 0.7 0.9 1.0

0.060 0.064 0.076 0.078 0.084

8 8 8 8 8

4.74 4.16 3.83 3.51 3.25

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Table 7 Comparison of the proposed CBM policy and CBM policy 1 in terms of cost performance. Policies /Costs

Inspection cost

Minimal repair cost

Quality loss

PM-related cost

Total cost

CBM Policy 1 Proposed CBM policy

9120 9120

28400 26699

16783 14698

36169 34406

90472 84923

Table 8 Comparison of the proposed CBM policy and CBM policy 2 in terms of cost performance. Policies /Costs

Inspection cost

Minimal repair cost

Quality loss

PM-related cost

Total cost

CBM Policy 2 Proposed CBM policy

9120 9120

27207 26699

15220 14698

34944 34406

86491 84923

(1) With the increase of minimal repair cost rate ckm , the optimal QLT remains almost unchanged, the optimal FRT decreases constantly, and the optimal inspection interval remains unchanged. When ckm increases, there is a need to reduce the number of machine failures. Reducing the number of machine failures can be achieved by setting smaller FRT, because smaller FRT means more PM operations will be triggered to prevent machine failure. Meanwhile, as the change of QLT or inspection interval has little influence on the number of machine failures, the optimal values for them remain unchanged when ckm increases. (2) With the increase of quality loss coefficient q , the optimal QLT shows an increasing trend, the optimal FRT and the optimal inspection interval both remain unchanged. The increase of QLT is resulted from the increase of quality loss coefficient q . From Table 5, it can be seen that the increasing rate of QLT is smaller than that of q . This means more PM operations will be triggered to decrease the quality deterioration when q increases. In fact, the average number of PM activities increases with the increase of q , as shown in Table 4. Besides, as the change of FRT or inspection interval has little impact on the quality deterioration, the optimal values for them remain unchanged when q increases. (3) With the increase of PM cost rate ckp , the optimal QLT and the optimal FRT both increase constantly, and the optimal inspection interval remains unchanged. When ckp increases, there is a need to reduce the number of PM activities. Reducing the number of PM activities can be achieved by setting greater FRT and QLT. This is because, greater FRT and QLT means less PM operations will be triggered. In fact, the average number of PM activities decreases when ckp increases, as shown in Table 5. In addition, as the change of inspection interval has little impact on the number of PM activities, the optimal inspection interval remains unchanged when ckp increases.

operations is not considered in the development of PM-related cost. The PM-related cost CGu (Eq. (13)) is given by k Gu Direct maintenance cost

c set + Setup cost

c er

Gu

Downtime cos t

, (26)

6. Conclusions This paper proposes a novel quality and reliability oriented CBM policy for the serial MMSs where the conditions of QRCs are inspected. At each inspection, the revealed deterioration states of QRCs are used to evaluate the quality loss of final products and the system failure rate, which are further compared with the QLT and the FRT respectively to decide whether to trigger PM. The utilization of QLT and FRT is to ensure that the final product quality and the system failure rate can be ensured at desired levels. When PM is triggered, the cost-based improvement factor is introduced to determine a group of machines for PM. This factor jointly considers the improvement of quality and reliability and the reduction of PM costs. The proposed CBM policy is applied to a serial four-stage machining system producing shaft sleeves. The results show that the proposed CBM policy can ensure a high level of product quality and machine reliability with the minimum total cost. Sensitivity analyses show that the proposed CBM policy can adaptively and effectively adjust the PM decisions when the cost parameters change. Moreover, the results also

To highlight the effectiveness of the proposed quality and reliability oriented CBM policy, it is compared with two relevant CBM policies which are variations of the proposed CBM policy.

• In CBM policy 1, the improvement of product quality and machine

reliability due to PM is not considered in the detremination of the optimal PM group using the cost-based improvement factor. The cost-based improvement factor (Eq. (12)) is given by

1 , CGu

+

Under CBM policy 1, the optimal values for the decision parameters are obtained as Dq* = 0.5, Df* = 0.064 , T* = 8. Table 7 shows the comparison of proposed CBM policy and CBM policy 1 in terms of cost performance. From Table 7, it can be seen that the minimal repair cost, the quality loss as well as the PM-related cost under the proposed CBM policy is obviously smaller than that under CBM policy 1. This implies that compared with CBM policy 1, the proposed policy can ensure higher product quality and machine reliability with a lower PM-related cost. It means that it is beneficial to consider the improvement of quality and reliability due to PM in determine the opitmal PM group. Under CBM policy 2, the optimal values for the decision parameters are obtained as Dq* = 0.5, Df* = 0.064 , T* = 8. Under CBM policy 2, the average number of machines in a PM group is 1.06, while it is 1.18 under the proposed CBM policy. It can be seen that the average number of machines in a PM group under the proposed CBM policy is greater than that under CBM policy 2. This is because, the proposed CBM policy considers the reduction of maintenance costs due to the combination of PM operations and thus it encourages the combination of PM operations. Table 8 shows the comparison of proposed CBM policy and CBM policy 2 in terms of cost performance. From Table 8, it can be seen that the minimal repair cost, the quality loss or the PM-related cost under the proposed CBM policy is slightly smaller than that under CBM policy 2. This implies that it is profitable to considering the reduction of maintenance costs by combining PM operations in the determination of the optimal PM group.

5.4. Effectiveness of proposed CBM policy

EGu (t ) =

ckp

CGu =

(25)

• In CBM policy 2, the reduction of maintenance costs by combining

PM operations is not considered in the determination of the optimal PM group using the cost-based improvement factor. To be specific, the reduction of PM setup cost and production loss by combining PM 84

Journal of Manufacturing Systems 52 (2019) 76–85

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show that compared with two relevant CBM policies, the proposed CBM policy has a better performance in improving quality and reliability as well as reducing PM costs. This paper is intended for dedicated manufacturing systems for mass production. In the future work, we will consider investigating optimal quality and reliability oriented maintenance policies for flexible manufacturing systems or reconfigurable manufacturing systems. Besides, this paper assumes that the deterioration processes of QRCs are statistically independent. For different MMSs, the QRCs are usually different. There are certainly situations where the deterioration processes of QRCs are dependent. There situations can be a good direction for our future research.

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