International Journal of Performability Engineering, 2018, 14(12): 3014-3024 doi: 10.23940/ijpe.18.12.p11.30143024

Multi-Criteria Decision Model for Imperfect Maintenance using Multi-Attribute Utility Theory

Xiao Zhaoa,b,c, Jianhua Yang,a, and Xin Shib,c

a Donlinks School of Economics and Management, University of Science and Technology Beijing, Beijing, 100083, China

b Business School, Manchester Metropolitan University, Manchester, M15 6BH, UK

c School of Management, Shanghai University, 200400, China

*Corresponding Author(s): * E-mail address: yangjh@ustb.edu.cn

First author contact:

Xiao Zhao is a Ph.D. student in the Donlinks School of Economics and Management at the University of Science and Technology Beijing. She also is a scholarship student in the Business School at Manchester Metropolitan University.
Jianhua Yang is a professor in the Donlinks School of Economics and Management at the University of Science and Technology Beijing.
Xin Shi is a professor in the Business School at Manchester Metropolitan University.

Accepted:  Published:   

Abstract

Many research works have been conducted in the preventive maintenance area since maintenance strategies have become more and more significant in industry and supply chain services. However, previous studies are mainly based on age maintenance policies and other multi-criteria approaches instead of Multi-Attribute Utility Theory (MAUT). There are some studies proposed by using MAUT, but they assume that the maintenances are perfect. This paper presents an imperfect maintenance model of a one-unit system to obtain an optimal inspection interval based on MAUT. The proposed model is designed to identify the systems’ status by making a trade-off between cost attribute and reliability attribute. By taking decision makers’ preferences into account, with the assumption of imperfect maintenance, the model receives a result of an optimal inspection interval. This model is applicable for equipment or systems that suffer graded failures and can be repaired at any time during the operation. A numerical application is given to illustrate that with consideration of decision-makers’ priorities, the model can provide new solutions for imperfect maintenance interval optimization, which is a suggestion for future research.

Keywords: imperfect maintenance; optimal intervals; reliability; cost; multi-attribute utility theory

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Xiao Zhao, Jianhua Yang, Xin Shi. Multi-Criteria Decision Model for Imperfect Maintenance using Multi-Attribute Utility Theory. International Journal of Performability Engineering, 2018, 14(12): 3014-3024 doi:10.23940/ijpe.18.12.p11.30143024

1. Introduction

Systems used in commodity production and delivery services are subject to deterioration with usage and age. This causes higher production costs and lower product output with deterioration, erosion, and so on [1]. Preventive maintenance is an effective way to prevent failures of systems as failure rates increase with service time. Moreover, it can prevent productivity from decreasing. Many models based on preventive maintenance policies assumed that systems become as good as new after preventive maintenance. In Yang’s paper, a perfect preventive maintenance model is proposed based on age-based replacement policy to obtain an optimization. In his research, a three-stage failure process is considered dividing the lifetime of the system into four states. When a defective stage is identified or at failure, the system is renewed. Moreover, the system is replaced once a certain age T is reached [2]. In Wang’s paper, a joint optimization for spare parts inventory and preventive maintenance inspection interval is presented. The model develops a delay-time concept in the situation of perfect maintenance [3]. Wang also proposes several perfect maintenance models using the delay-time concept, such as a block-based inspection model that uses a recursive algorithm for determining a limiting distribution in order to optimize the inspection interval [4].

However, perfect maintenance is not true in practice. In fact, systems may be as bad as before or become a little worse than new after preventive maintenance due to wrong adjustments, bad parts, and damage done during preventive maintenance [5]. This kind of maintenance is known as imperfect maintenance. A precise definition of imperfect preventive maintenance is as follows: any maintenance action that makes a system “younger” and results in an improved system operating condition[1]. Recently, issues of imperfect maintenance have drawn a large amount of attention, and much research has been conducted based on them. Some researchers focus on imperfect preventive maintenance problems under the situation of delay-time concept. Yang modifies Wang’s three-stage delay-time failure process and puts imperfect maintenance into his model by assuming minor defects to be imperfect but perfect at severe defects. The renew action is only taken at failure [6]. With regard to imperfect maintenance, Li presents a new imperfect-maintenance model for delay-time concept and accumulative age with the assumptions that failure modes are independent from each other and all faults will be repaired in each maintenance action [7]. In another paper, taking into consideration the process of corrective and condition-based preventive maintenance, two classes of generalized competing risks models are studied: one is the generalized random sign model and the other is the generalized alert delay model in which imperfect preventive maintenance is expressed [8]. In other research, issues of imperfect preventive maintenance are discussed from other aspects rather than the view of delay-time concept. In one study, a new analytical model of a manufacturing system subject to an increasing random failure rate is established based on an improved imperfect preventive maintenance policy with minimal repair at failures. The number of produced batches before performing the imperfect preventive maintenance, the number of imperfect maintenance actions, and the total expected budget are the main factors in his research [9]. In Mercier’s paper, two imperfect preventive maintenance models for a degrading system are explored and compared. The deterioration level is expressed by a non-homogeneous gamma process [10].

Multi-criteria decision making, as one of the most widely applied methods in the decision-making area, is spread over many fields nowadays. A few works focus on preventive maintenance issues based on methods of multi-criteria decision making. For example, with regard to decision-making, Shan proposes a multi-objective maintenance model and uses the Dempster-Shafer evidence theory. The change of distribution system reliability after maintenance, load loss, and maintenance cost are three indexes in the model [11]. Still, there exist some studies researching decision-making model based on imperfect maintenance. In one study, Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE), one method of decision-making, is used to select the optimal maintenance interval by improving the equipment reliability and minimizing total maintenance costs [12]. According to Ana’s paper, unavailability and cost are two conflicting decision criteria for determining imperfect maintenance optimization [13].

Multi-attribute utility theory (MAUT) is a branch of multi-criteria decision making that concerns modeling utility functions with multiple attribute outcomes and obtaining the best choice among different options [14]. However, there have not been many articles taking decision makers’ preferences into account based on MAUT. Even though some papers explore maintenance interval optimization problems based on MAUT approaches, they do not consider imperfect preventive maintenance situations [15]. According to Garmabaki and Ahmadi, maintenance decision making may occur in various complex systems subject to technology, maintainability, reliability and availability requirements, etc. In their study, three attributes, cost, reliability, and availability, are considered in the model. They assume that performing preventive maintenance or repair will take some time. However, they only consider the situation of perfect preventive maintenance [16]. In Adiel’s paper, some constraints are reduced but perfect maintenance is still assumed.

In this paper, we propose a multi-criteria decision model for imperfect maintenance based on MAUT. Dealing with two uncertain variables, decision-makers’ preferences are taken into account and the inspection interval is optimized. Different preference uncertain attributes, cost and reliability, are considered in our utility functions. This paper aims to determine an optimal imperfect preventive maintenance interval by improving reliability and reducing total cost during the operation of the system. We then illustrate an application to analyze a practical problem and make a discussion based on it.

The remaining parts of this paper are organized as follows. Section 2 provides the problem description and assumptions. In this section, the imperfect maintenance policy is presentedand the cost model is proposed. Moreover, the decision model is formulated based on multi-attribute utility theory. Then, Section 3 gives a numerical example to illustrate the applicability of the model, and the obtained results are analyzed. Finally, Section 4 provides a conclusion that the multi-criteria decision model is effective in solving problems for imperfect maintenance.

2. Model Assumptions and Description

2.1. Assumptions

As we know, failures can be classified into catastrophic failures, which mean systems fail all of sudden, and degraded failures, which refer to systems failing due to performance deterioration. The proposed model is based on degraded failures in this paper. Some systems can only accept maintenance actions before given tasks, that is, defects may be tested while the systems are in storage. This includes rocket engines, missiles, and so on [17]. Here, we only study the situation that maintenance actions can be taken at any time. A one-unit system is used in our case. It can be representative of equipment, a component, or multi-unit systems in which failures can be detected by inspections. In many papers, researchers define action space as a continuous set and generate a continuous consequence space in order to obtain smooth curves. However, this is impossible in real life. Preventive maintenance is always carried out in a period, such as one day, two weeks, three months, etc. Taking the reality into account, the action space and consequence space in our paper are defined as discrete sets of alternatives.

The periodical inspections to detect defects or failures may not be perfect. Preventive maintenance is also imperfect. In this paper, we assume the system becomes as good as new only after perfect preventive maintenance or after repairing, which means that maintenance is carried out just after failure occurs. Utilities are considered additive independent. We also assume the unit has the same failure rate before or after preventive maintenance. We define a gamma distribution as our failure distribution. According to assumptions of our imperfect maintenance method, all maintenance actions take negligible time.

The notations used in the following text are listed below:

$T$ Time at which the operating unit is repaired at failure or is preventively maintained

$P$ $0\le p<1$, is the probability that the unit has the same failure rate after preventive maintenance as it had before maintenance

$q$ $q=1-p$, which refers to the unit becoming as good as new after preventive maintenance

$f(t)$ Probability density function of failure

$F(t)$ Cumulative distribution function of $f(t)$ with mean value $\mu $

$R(t)$ Reliability function, namely $R(t)=1-F(t)$

${{c}_{f}}$ Cost of each corrective maintenance

${{c}_{p}}$ Cost of each preventive maintenance

$E(c)$ Expected cost rate

${{w}_{c}}$ Weight parameters for cost attribute

${{w}_{R}}$ Weight parameters for reliability attribute

$r$ Random variable of cost

$c$ Random variable of cost

${{k}_{i}}$ Constants that keep numerical values $U(x)$ ranging from 0 to 1. i=1, 2, 3,

$U(C)$ Single utility functions for cost

$U(R)$ Single utility functions for cost

$U({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})$ Namely $U(C,R)$, multi-attribute utility function

2.2. Multi-Attribute Utility Theory (MAUT)

Generally, MAUT is defined as

$U({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})=f[{{u}_{1}}({{x}_{1}}),{{u}_{2}}({{x}_{2}}),\cdots ,{{u}_{n}}({{x}_{n}})]=\sum\limits_{i=1}^{n}{{{w}_{i}}{{u}_{i}}({{x}_{i}})}$

Where $\sum\limits_{i=1}^{n}{{{w}_{i}}=1}$.

In this paper, we consider two attributes: cost attribute and reliability attribute. According to Jansen’s paper [18], we assume utility functions are additive independent. Then, the function based on MAUT is given by

$Max:U(C,R)={{w}_{c}}U(C)+{{w}_{R}}U(R)$

Where ${{w}_{c}}+{{w}_{R}}=1$.

By maximizing this multi-attribute utility function, the optimal inspection ${{T}^{*}}$ will be obtained by maximizing the function above.

2.3. The Cost Model & Cost Attribute

First, we define a cycle as a period of time that begins from a perfect situation of the unit and ends in a failure. Then, the average cost in a cycle based on optimal preventive replacement is given by

$C(T,p)=\frac{E(c)}{E(T)}$

According to the assumptions above, the probability of repairing at failure is

$\sum\limits_{j=1}^{\infty }{{{p}^{j-1}}\int_{(j-1)T}^{jT}{\text{d}F(t)=1-q\sum\limits_{j=1}^{\infty }{{{p}^{j-1}}R(jT)}}}$

Where $j=1,2,3,\cdots $

The expected preventive maintenance number in one cycle is

$\sum\limits_{j=1}^{\infty }{(j-1){{p}^{j-1}}\int_{(j-1)T}^{jT}{\text{d}F(t)+}}q\sum\limits_{j=1}^{\infty }{j{{p}^{j-1}}R(jT)}=\sum\limits_{j=1}^{\infty }{{{p}^{j-1}}R(jT)}$

Moreover, we can easily obtain the function of meantime of a single cycle, that is

$\sum\limits_{j=1}^{\infty }{{{p}^{j-1}}\int_{(j-1)T}^{jT}{t\text{d}F(t)+}}q\sum\limits_{j=1}^{\infty }{(jT){{p}^{j-1}}R(jT)}=\sum\limits_{j=1}^{\infty }{{{p}^{j-1}}\int_{(j-1)T}^{jT}{R(jT)}}\text{d}t$

Above all, $C(T,p)$ is given by

$C(T,p)=\frac{{{c}_{f}}[1-q\sum\nolimits_{j=1}^{\infty }{{{p}^{j-1}}R(jT)}]+{{c}_{p}}\sum\nolimits_{j=1}^{\infty }{{{p}^{j-1}}R(jT)}}{\sum\nolimits_{j=1}^{\infty }{{{p}^{j-1}}\int_{(j-1)T}^{jT}{R(t)\text{d}t}}}$

From Equation (5), a minimum ${{C}_{\min }}$ can be calculated. According to Garmabaki’s paper, the cost attribute function is

${{U}_{Cost}}=\frac{{{C}_{\min }}}{C(T,p)}$

2.4. Reliability Attribute

A reliability function is obtained since we know the probability density function of failure $f(t)$.

$R(t)=\Pr (T\ge t)=1-F(t)$

From the properties of reliability distribution, we know that the larger the value of $R(t)$, the better the obtained results. The reliability attribute is given by

${{U}_{R}}=\frac{R(T)}{{{R}_{Max}}}$

2.5. Decision Modeling of Single Utility Function

Decision makers’ preferences are taken into account to analyze our model according to decision makers’ behaviors towards risk. From Almeida’s paper, utility functions can be classified into three forms: linear form and two exponential forms.

The linear utility function is as follows:

$U(x)={{k}_{1}}x+{{k}_{2}}$

The exponential utility functions are as follows:

$U(x)={{k}_{3}}\exp (-\frac{{{k}_{4}}}{x})$

and

$U(x)={{k}_{5}}\exp (-{{k}_{6}}x)$

It can be seeneasily that when the decision maker is risk neutral, the linear utility function is suitable for the model; when the decision maker is risk averse, the exponential utility function is recommended. For the cost attribute in this paper, the linear utility function is applied, and the logistic utility function is applied for the reliability attribute.

That is, for the cost attribute:

U(c)={{k}_{1}}c+{{k}_{2}}

For the reliability attribute:

$U(r)={{k}_{3}}\exp (-\frac{{{k}_{4}}}{r})$

3. Numerical Examples

In order to demonstrate the practicability of the proposed model, a numerical application is illustrated to show details about how the model works. We assume the failure distribution of one unit system obeys the gamma distribution with parameters $\alpha$ and $\lambda $. Therefore, it can be indicated by

$p(T;\alpha ,\lambda )=\left\{ \begin{matrix} & \frac{{{\lambda }^{\alpha }}}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\lambda x}}, \\ & 0, \\ \end{matrix} \right.\begin{matrix} & T\ge 0 \\ & \\ & T<0 \\ \end{matrix}$

Then, we establish the action space. According to the assumptions we made above, the action space is defined as five days to satisfy a discrete condition. That is,

${{t}_{1}}=5,\text{ }{{t}_{2}}=10,\text{ }\cdots ,\text{ }{{t}_{i}}=5i$, $i=1,2,3,\cdots $

We also define $H(t;p)$ as

$H(t;p)\equiv \frac{\sum\nolimits_{j=1}^{\infty }{{{p}^{j-1}}jf(jt)}}{\sum\nolimits_{j=1}^{\infty }{{{p}^{j-1}}j(1-F(jt))}}$

In order to generate a unique finite optimal inspection interval ${{T}^{*}}$ according to unique solution conditions of imperfect preventive maintenance [3], values of the parameters should satisfy:

${{c}_{f}}q>{{c}_{p}}$

and

$H(\infty ;p)>{{c}_{f}}q/[\mu ({{c}_{f}}q-{{c}_{p}})]$

Considering the conditions above and data of real examples in other papers [13,15], we set simulated data to parameters as below. They are given in Table 1.

Table 1.   Estimated parameters and attributes

$\alpha $2
$\lambda $1
${{c}_{p}}$600
${{c}_{f}}$3000
$q$0.95

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The failure cumulative distribution function can be obtained by given $\alpha $ and $\lambda $, as drawn in Figure 1.

Figure 1

Figure 1.   Cumulative distribution of failure


It can be easily verified that the given data above satisfies Equation (19) and Equation (20) by taking the specific values into these two equations.

After verification, we put the simulated data into Equation (7) to get the values of costs versus time. Thus, a modified cost function $C(T,p)$ can be obtained as below.

$C(T,0.05)=\frac{3000-2250\sum\limits_{j=1}^{\infty }{{{(0.95)}^{j-1}}(1+jT){{e}^{-jT}}}}{\sum\limits_{j=1}^{\infty }{{{(0.95)}^{j-1}}\int_{(j-1)T}^{jT}{(1+jt){{e}^{-jt}}\text{d}t}}}$

Thus, a graph through the curve of cost versus time is easily obtained as Figure 2 by running MATLAB.

Figure 2

Figure 2.   Cost versus time


It can be seen that from the first four days, the cost rate has a tendency to decline rapidly. Then, it declines gradually and almost approximates a specific number as the value of time increases. The values are expressed in Table 2.

Table 2.   Values of $C(T,0.05)$

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From Table 2, it can be seen that the values of the cost reach the minimum when i is greater than or equal to 11. This is because we set the reservation arrives decimally hind four in the calculation of $C(T,0.05)$.

From the description of the limiting conditions above, we know that the model satisfies one unique solution condition in Equations (19) and (20). Therefore, there must be one unique result in our case study. In order to see clearly, we extend the time of interval T and plot the figure again. At the same time, we set the reservation arrives decimally hind four and increase the values by multiplying by 1000. Then, the results arrived at in Figure 3 show that an optimal inspection interval T* is obtained as expected.

Figure 3

Figure 3.   Calculation of the minimum


Though it seems that cost rate approaches a specific value as time goes byinFigure 3, this curve still has its lowest point by calculation. As illustratedat the top of Figure 3, the optimal cost rate is approximately equal to 983 at the optimal inspection interval ${{i}^{\text{*}}}$= 39. That is, ${{T}^{\text{*}}}=195$. Moreover, we assume the action space is discrete, namely, the assume inspection is carried out infive days. Still, the result approaches a smooth curve in the figure and keeps the integrity of its tendency.

As we calculated above, the highest cost rate is approximately equal to 2849 when i equals 0 and the lowest is 982.72 when i equals 39. Thus, from Equation (14), we conduct an elicitation procedure of cost rate.Then, we can easily obtain ${{k}_{1}}=-5.36\times {{10}^{-4}}$ and ${{k}_{2}}=1.53$.

According to Equation (9), we can observe the expression of $R(t)$ based on the expression of the failure cumulative distribution. Figure 4 displays the tendency of $R(t)$.

Figure 4

Figure 4.   Reliability versus time


Then, according to Equation (15), we conduct an elicitation procedure and obtain the results ${{k}_{3}}=1.01$ and ${{k}_{4}}=0.01$.

From the above, we get

$\left\{ \begin{matrix} & U(c)=-5.36\times {{10}^{-4}}c+1.53 \\ & U(r)=1.01\exp (-0.01/r) \\ \end{matrix} \right.$

Then, the Equation (22) can be easily drawn as Figure 5 and Figure 6.

Figure 5

Figure 5.   Utility function of reliability


Figure 6

Figure 6.   Utility function of cost


As illustrated in Figure 5, the higher the reliability of the system or equipment, the higher the utility and the more satisfaction the decision makers will get. If the reliability equals 1, the utility reaches its maximum value of 1. If the reliability of the system or the equipment is equal to 0, the corresponding utility is 0. From the other side, Figure 6 shows that the higher the system or equipment cost, the less utility it will obtain.

Considering a case where the reliability is 1.5 times more important than the cost rate, then combine $U(C)$ and $U(R)$ with ${{W}_{\text{c}}}\text{=}0.6$ and ${{W}_{R}}=0.4$. We can obtain Equation (23) as below.

$U(C,R)=-3.22\times {{10}^{-4}}c+0.404{{e}^{(-0.01/r)}}+0.92$

Then, the graphof $U(C,R)$ is illustrated as below.

As displayed in Figure 7, a unique optimal inspection interval exists and received at $i\in [2,5]$, that is, $T\in [10,25]$. By calculation, the optimal utility equals 0.9593.It arrives at the maximum value when i approximates to 4, which means that the optimal maintenance plan of the system or equipment is to conduct a maintenance action about every 20 days.

Figure 7

Figure 7.   Utility function of U(C,R)


4. Sensitivity Analyses of Parameters

According to Table 1, the calculations above are based on settings where $\alpha \text{=}2,$ $\lambda \text{=}1,$ ${{C}_{P}}=600,$ ${{C}_{f}}=3000,$ and $q=0.95$. Furthermore, we set weight parameters ${{w}_{c}}=0.6$ and ${{w}_{R}}=0.4$. If we change the values of any parameters, the results may be changed. Here, we give some examples to illustrate the sensitivity of these parameters.

For example, if we reset weight parameters ${{w}_{c}}=0.8$ and ${{w}_{R}}=0.2$, thenthe optimal inspection time is $T\in [15,25]$ and the optimal utility equals 0.9682.

The results of changed weights are graphed by Figure 8. It is apparent that the tendency is still the same as that seen in Figure 7. However, the values of utility tend to different balance points.

Figure 8

Figure 8.   Utility function with changing weights


From Equation (1), it is implied that the utility functions are additive independent. If we weaken the condition, the utility function can be built as Equation (24) or other equation forms.

$Max:U(C,R)={{w}_{c}}U(C)+{{w}_{R}}U(R)+{{w}_{CR}}U(C)U(R)$

With other conditions unchanged, if we set the weight parameters as ${{w}_{c}}=0.3$, ${{w}_{R}}=0.2$, and ${{w}_{CR}}=0.5$, we obtain the optimal value of utility as 0.9448 by using Equation (22).

From the other aspect, risk preferences of decision makers may be different from assumptions in our case study. For example, a decision maker is risk averse to cost attributes, and then the utility function of cost attribute should be expressed as an exponential form.

It can be assumedas the function below.

${U}'(c)={{k}_{5}}\exp (-\frac{{{k}_{6}}}{c})$

According to theanalysis in Section 3, following the elicitation procedure, a modified cost utility function is obtained. If we keep the utility function of reliability the same as in the previous assumption, they can be written as

$\left\{ \begin{matrix} & {U}'(c)=2.3\times {{10}^{-20}}\exp (4.44\times {{10}^{4}}/c) \\ & U(r)=1.01\exp (-0.01/r) \\ \end{matrix} \right.$

According to the equations set, we set weight parameters ${{w}_{c}}=0.6$ and ${{w}_{R}}=0.4$, and then ${U}'(c,r)$ can be obtained, that is

${U}'(c,r)=1.38\times {{10}^{-20}}{{e}^{4.44\times {{10}^{4}}/c}}+0.404{{e}^{-0.01/r}}$

The graph is shown below in Figure 9. The maximum value of utility is 0.6743 from calculation, which is lower than the situation of a linear utility function. This is because it is easier for risk neutral to be satisfied than risk averter.

Figure 9

Figure 9.   Modified utility function


Above all, the comparison of calculated functions is shownin Figure 10.

Figure 10

Figure 10.   Compared values of utility functions


As can be seen from the graph, at the beginning, all utility values of cost risk neutral sharply increase as the interval time gets larger. Then,they respectively reachtheir maximum values. Though different maximum values are obtained, they follow the same tend. From the perspective of cost risk aversion, though the tendency at the beginning is different from risk neutral ones, there is still only one maximum point.

In general, the model and structure are based on real situations of complex systems or equipment. The model is applicable to systems that suffer graded failures and can take maintenance actions at any time during the operation. Therefore, it can be applied to factories that focus on manufacturing production, companies that provide service production, and so on.

5. Conclusions

In this paper, we build a multi-attribute utility function and find a unique optimal inspection interval of imperfect preventive maintenance. The cost attribute and the reliability attribute are two main indexes in our model. We estimate the cost rate using Equation (7). As discussed above, it is assumed that the failure of the system or equipment follows the gamma distribution. In order to conduct a unique optimal interval, parameters have to satisfy the limit conditions of Equation (19) and Equation (20). Then, we use a linear function and an exponential function respectively representing cost and reliability utility functions. We assume that the utility functions mentioned are additive independent, which means that they can be estimated by using Equation (1).

From the numerical examples in Section 3, a unique optimal inspection interval is obtained. Moreover, values of some parameters are changed to analyze the sensitivity of the proposed model. This numerical application has illustrated that the use of MAUT in imperfect preventive maintenance is applicable. In conclusion, this model determines a new solution to optimize an optimal interval of imperfect preventive maintenance with the consideration of decision-makers’ preferences.

Further studies may be carried out on the analysis of a modified model that contains more considered attributes. More complex but accurate methods maybe used to express actions of imperfect preventive maintenance. It may also be conducted from the aspect of whole supply chain management.

Acknowledgements

The research was partially supported by the National Natural Science Foundation (No. 71231001) and the Fundamental Research Funds for the Central Universities (NTUT-USTB-107-01, No. TW2018009). Xiao Zhao is also funded by CSC (China Scholarship Council) and BC (British Council) (No.201603780048).

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,” International Journal of System Assurance Engineering & Management, Vol.6, No. 4, pp. 479-486, 2015

DOI:10.1007/s13198-014-0306-6      URL     [Cited within: 1]

Components with multiple failure modes may encounter imperfect preventive maintenance triggered by inspections during their two-stage failure. Employing delay-time concepts and accumulative age, a new imperfect-maintenance model for these components is presented under the assumptions that failure modes are independent of each other and all kinds of defects will be dealt with in each maintenance task. Reliability and cost models are derived. Their characteristics are analyzed in numerical simulations. The results show components with more failure modes and worse imperfect inspection maintenance will have lower reliability, lower expected cycle cost and higher average cost per unit time. With an increasing inspection interval, the cycle cost will monotonically decrease while the average cost per unit time may have a local minimum. If inspection maintenance is insufficient, the minimum of the average cost per unit time will disappear, as does the best inspection interval.

Y. Dijoux and O. Gaudoin, “

Generalized Random Sign and Alert Delay Models for Imperfect Maintenance

,” Lifetime Data Analysis, Vol. 20, pp. 185-209, 2014

DOI:10.1007/s10985-013-9249-5      URL     PMID:23460491      [Cited within: 1]

This paper considers the modelling of the process of Corrective and condition-based Preventive Maintenance, for complex repairable systems. In order to take into account the dependency between both types of maintenance and the possibility of imperfect maintenance, Generalized Competing Risks models have been introduced in “Doyen and Gaudoin (J Appl Probab 43:825–839, 2006 )”. In this paper, we study two classes of these models, the Generalized Random Sign and Generalized Alert Delay models. A Generalized Competing Risks model can be built as a generalization of a particular Usual Competing Risks model, either by using a virtual age framework or not. The models properties are studied and their parameterizations are discussed. Finally, simulation results and an application to real data are presented.

A. Gouiaa-Mtibaa, S. Dellagi, Z. Achour, W. Erray , “

Integrated Maintenance-Quality Policy with Rework Process under Improved Imperfect Preventive Maintenance

,” Reliability Engineering and System Safety, Vol. 173, pp. 1-11, 2018

DOI:10.1016/j.ress.2017.12.020      URL     [Cited within: 1]

We consider a manufacturing system subject to an increasing random failure rate and producing conforming and non-conforming items. Two types of non-defective products are considered: high quality items noted first-rate products and substandard quality items noted second-rate products. Rework activities are proposed for second-rate and non-conforming products in order to improve their quality conditions and sell them at the best price. In order to slow down the degradation process of the manufacturing system and reduce its impact on the quality of output products, a new improved imperfect preventive maintenance (PM) policy with minimal repair at failures is established. It consists in performing a given number of imperfect PM actions before undertaking a perfect one. This study consists in developing an analytical model to determine simultaneously the optimal values of two decision variables: the number of produced batches before performing the imperfect PM, and the number of imperfect PM actions to undertake before applying a perfect one by maximizing the total expected profit taking into account the selling price and production, maintenance and reworking costs. Numerical examples and a sensitivity study are presented to illustrate the use of the proposed model and the obtained results.

S. Mercier and I. T. Castro, “

Stochastic Comparisons of Imperfect Maintenance Models for a Gamma Deteriorating System

,” European Journal of Operational Research, Vol. 273, No. 1, pp. 237-248, 2019

DOI:10.1016/j.ejor.2018.06.020      URL     [Cited within: 2]

This paper compares two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process. Both models consider instantaneous and periodic repairs. The first model assumes that a repair reduces the degradation of the system accumulated from the last maintenance action. The second model considers a virtual age model and assumes that a repair reduces the age accumulated by the system since the last maintenance action. Stochastic comparison results between the two resulting processes are obtained. Furthermore, a specific case is analyzed, where the two repair models provide identical expected deterioration levels at maintenance times. Finally, two optimal maintenance strategies are explored, considering the two models of repair.

R. F. Yang, Z. W. Yan, J. S. Kang , “

An Inspection Maintenance Model based on a Three-stage Failure Process with Imperfect Maintenance via Monte Carlo Simulation

,” International Journal of System Assurance Engineering & Management, Vol.6, No. 3, pp. 231-237, 2015

DOI:10.1007/s13198-014-0292-8      URL     [Cited within: 1]

Inspection is widely used in industry to identify the status of plant and make maintenance decisions. The maintenance models using the delay time concept for optimizing the inspection intervals have been researched for years. However, the three-stage failure process proposed by Wang is closer to reality and provides more modeling options. Imperfect maintenance is common in practice, but not considered based on the three-stage failure process before. An inspection maintenance model based on a three-stage failure process with imperfect maintenance is proposed. The maintenance at minor defect is assumed to be imperfect, but perfect at severe defect. Replacement is implemented at failure. Age reduction concept is utilized to describe the effect of imperfect maintenance. The expected renewed cycle downtime per unit time is derived. Finally, Monte Carlo simulation is presented to show the efficiency of the proposed model.

E. M. P. Hidalgo and G.F.M. Souza. , “A Multi-Criteria Decision Model to Determine Intervals of Preventive Maintenance with Equipment Reliability Degradation Due to Imperfect Maintenance,” CRC Press-Taylor & Francis Group, 2014

[Cited within: 1]

A. Sanchez and S. D. Carlos, “

Addressing Imperfect Maintenance Modelling Uncertainty in Unavailability and Cost based Optimization

,” Reliability Engineering & System Safety, Vol. 94, pp. 22-32, 2009

DOI:10.1016/j.ress.2007.03.022      URL     [Cited within: 2]

Optimization of testing and maintenance activities performed in the different systems of a complex industrial plant is of great interest as the plant availability and economy strongly depend on the maintenance activities planned. Traditionally, two types of models, i.e. deterministic and probabilistic, have been considered to simulate the impact of testing and maintenance activities on equipment unavailability and the cost involved. Both models present uncertainties that are often categorized as either aleatory or epistemic uncertainties. The second group applies when there is limited knowledge on the proper model to represent a problem, and/or the values associated to the model parameters, so the results of the calculation performed with them incorporate uncertainty. This paper addresses the problem of testing and maintenance optimization based on unavailability and cost criteria and considering epistemic uncertainty in the imperfect maintenance modelling. It is framed as a multiple criteria decision making problem where unavailability and cost act as uncertain and conflicting decision criteria. A tolerance interval based approach is used to address uncertainty with regard to effectiveness parameter and imperfect maintenance model embedded within a multiple-objective genetic algorithm. A case of application for a stand-by safety related system of a nuclear power plant is presented. The results obtained in this application show the importance of considering uncertainties in the modelling of imperfect maintenance, as the optimal solutions found are associated with a large uncertainty that influences the final decision making depending on, for example, if the decision maker is risk averse or risk neutral.

R.L. Keeney and H. Raiffa , “Decisions with Multiple Objectives: Preferences and Value Trade-offs,” Cambridge university press, Cambridge, 1993

[Cited within: 1]

A. T. Almeida , “

Multicriteria Model for Selection of Preventive Maintenance Intervals

,” Quality & Reliability Engineering International,Vol. 28, pp. 585-593, 2012

DOI:10.1002/qre.1415      URL     [Cited within: 3]

Selecting preventive maintenance intervals is a problem addressed in the literature in which several different models are used according to the context and other issues. This article presents a multicriteria decision model to support decision makers in choosing the best maintenance interval based on the combination of conflicting criteria, such as reliability and cost. A procedure is proposed for using the model, which is based on multiattribute utility theory (MAUT). Moreover, a numerical application illustrates the use of the procedure, based on a real case study. The discussion of the results addresses theoretical issues and practical managerial implications related to the model proposed. This study shows that a multicriteria decision model is important for the maintenance and reliability community when a context of service production systems is to be accounted for due to interruptions caused by failures in this kind of production system. Copyright 2012 John Wiley & Sons, Ltd.

A. H.S. Garmabaki, A. Ahmadi, and M. Ahmadi, “

Maintenance Optimization Using Multi-Attribute Utility Theory

,” Current Trends in Reliability, Availability, Maintainability and Safety, Lecture Notes in Mechanical Engineering, Springer, Cham, pp. 13-37, 2016

DOI:10.1007/978-3-319-23597-4_2      URL     [Cited within: 1]

Several factors such as reliability, availability, and cost may consider in the maintenance modeling. In order to develop an optimal inspection program, it is necessary to consider the simultaneous ef

H. Wang and H. Pham, “

Optimal Preparedness Maintenance of Multi-Unit Systems with Imperfect Maintenance and Economic Dependence

,” Reliability and Optimal Maintenance, pp. 135-150, 2006

DOI:10.1142/9789812811868_0006      [Cited within: 1]

The following sections are included:IntroductionSystem Reliability and Cost MeasuresSystem Maintenance Cost Rate and "Availability"Other Operating CharacteristicsOptimization ModelsConcluding RemarkReferences Introduction System Reliability and Cost MeasuresSystem Maintenance Cost Rate and "Availability"Other Operating Characteristics System Maintenance Cost Rate and "Availability" Other Operating Characteristics Optimization Models Concluding Remark References

S. J.T. Jansen, “

The Multi-Attribute Utility Method

,” Measurement & Analysis of Housing Preference & Choice, pp. 101-125, 2011

DOI:10.1007/978-90-481-8894-9_5      URL     [Cited within: 1]

In this chapter the methodology and techniques behind Multi-Attribute Utility Theory are introduced. The basic assumption underlying this theory is that a decision-maker chooses the alternative (for e

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