development and optimization of a condition-based maintenance policy … · 2019. 7. 30. ·...

Research ArticleDevelopment and Optimization of a Condition-BasedMaintenance Policy with Sustainability Requirementsfor Production System

Aiping Jiang ,1 Ning Dong,1 Kwok Leung Tam,2 and Chonghao Lyu1

1SHU-UTS SILC Business School, Shanghai University, 20 Chengzhong Road, Jiading District, Shanghai 201899, China2School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia

Correspondence should be addressed to Aiping Jiang; [email protected]

Received 18 August 2017; Revised 25 November 2017; Accepted 21 December 2017; Published 28 February 2018

Academic Editor: J.-C. Cortés

Copyright © 2018 Aiping Jiang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the field of condition-based maintenance, maintenance costs and system reliability criteria are the primary considerationsfor traditional maintenance management. These methods lack consideration of the environmental impact caused by equipmentdegradation, such as excessive emissions and energy consumption. In addition, because equipment degradation has variousimpacts on the ecological environment, companies with excessive emissions and energy consumption can receive huge fines,making it of great value to study ecoconscious maintenance strategies. In this paper, we propose a condition-based maintenancestrategy considering energy consumption and carbon dioxide emissions. The major objective of the research is to extend a modelwhich integrates ecological aspects with maintenance decision-making and optimization. The simulation and sensitivity analysesconducted verify that the model proposed can minimize total costs, as well as the environmental impact.

1. Introduction

Maintenance refers to maintaining and repairing equipmentand systems to ensure they are in good condition. It isan indispensable activity related to machines, ranging fromgrinding or replacing a small unit to precisely moderat-ing or repairing the whole system. When it comes to thelengthy operation of systems, such as social infrastructure,military equipment, production system, household electricalappliances, and vehicles, maintenance plays a crucial rolein ensuring their safety and reliability. If the equipmentand system cannot be maintained, its continuous use willlead to deterioration (e.g., aging, wear, corrosion, and creep)resulting in declining performance, reduced reliability, andeven a shortened life. Within this context, condition-basedmaintenance strategies have been studied and widely appliedby organizations as they propose an optimal maintenancepolicy based on the deterioration state of the system.

At present, most organizations focus solely on the relativemaintenance cost when establishing a maintenance strategyfor normal operation and system reliability. Minimizing cost

is the primary task, while the impact of the system onthe environment is seldom considered. Take, for example,a ship company that finds belt relaxation in various motorboilers, blowers, and belt conveyors or inefficiency of dooradjustment in the induced draft fan, where shutdownwill nothappen. The company will not undertake immediate main-tenance considering the cost criteria; however, its continueduse will seriously affect the critical value of combustionconditions (i.e., the excess coefficient of air). Generally, whenthe critical value is too large, the exhaust heat loss andelectricity consumption of the draft fan increases. Conversely,when the value is too small, combustion is not complete andharmful gas emissions increase, causing serious environmen-tal pollution.

As energy and environmental issues become increasinglysevere, awareness of ecological impacts is progressing in theinternational community. Since the late 1980s human societyhas been advocating a “green era,” where protecting theenvironment, embracing nature, and promoting sustainabledevelopment are the center of all activities. In this context,green maintenance models have become the new trend

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 4187575, 19 pageshttps://doi.org/10.1155/2018/4187575

http://orcid.org/0000-0002-4822-9627https://doi.org/10.1155/2018/4187575

2 Mathematical Problems in Engineering

for saving energy and resources, implementing environ-mental protection, and achieving sustainable development.The questions of how to achieve environmentally-friendlyand efficient production and maintenance and of how toimprove resource utilization rate and competitive power tocreate a green image have now become main concerns ofmany companies. In the field of maintenance managementresearch, traditional, condition-based maintenance is mainlyconcerned with maintenance costs and system reliability-optimization, while ignoring the subsequent excess wastedgas emissions and excess energy consumption. This focusdoes not consider the efficient use of resources, impactson the environment, and sustainable development. Energy,resources and the environment are all in the scope ofecology [1]; hence, studying condition-based maintenancewhile taking ecological factors into account can be of greatvalue.

2. Literature Review

The traditional research on CBM focused on the opti-mization of maintenance itself but lacked consideration ofecological aspects. With the increasingly serious energy andenvironmental problems over the past decade, literature onecologically aware maintenance optimization has graduallyemerged. More scholars are beginning to integrate energyconservation and emission reduction into CBM manage-ment.

Van Horenbeek et al. proposed a CBM model in 2014with the first integration of ecological factors into decision-making and optimization [1]. The model is an economic andecological analysis tool that incorporates severalmaintenancestrategies (e.g., failure-based, block-based, and use-basedmaintenance).Using this analysis tool, we can obtain the opti-mal maintenance interval considering economic, ecological,and integrated aspects. The author verified the model andvalidated that it can be effectively applied to the cutting toolof a turning machine. The results also reveal that the eco-logical optimal maintenance interval is much smaller thanthe economic optimal maintenance interval. Integrated costis mainly determined by economic costs, but the ecologicalimpact cannot be ignored. In addition, a Pareto analysisshowed that, by varying the preventive maintenance interval,a slight increase in the economic cost of maintenance willsignificantly reduce the ecological impact.

Xu and Cao [2] proposed to use energy efficiency perfor-mance as an objective for maintenance by updating periodicmaintenance on energy efficiency evolution, but mainly onmastering the deterioration of machine tool (a Markovprocess application). Shafiee et al. [3] use the deteriorationlevel of multibladed wind turbine system to rise/trigger theunplanned maintenance action for any failure items, whileelectrical energy as the useful output of this system is notfully taken into account. So energy efficiency is seen onlyas the auxiliary result of these previous CBM plans. Inaddition, Mora et al. [4] point out that energy can be savedby optimal periodic preventive maintenance. Yildirim andNezami [5] also think that energy can impact preventivemaintenance by considering energy consumption in the case

ofminimal repair. To summarize, energy consumption in linkwith maintenance is mainly considered as an item to adapt apreventive maintenance plan in terms of scheduled one butnot in case of CBM strategy.

Therefore, Hoang et al. [6] presented an optimizationmodel for the CBM strategy in 2016 based on the stochasticprocess theory. This model assumes that the degradationincrement is subject to the Gamma process and that thedeterioration state, along with available output, determinesthe Energy Efficiency Index (EEI). If the EEI exceeds thethreshold level in a periodic inspection, preventive main-tenance action will be undertaken. The number of differ-ent maintenance actions in the objective function of thisoptimization model is affected by the EEI threshold, whichfully integrates the energy consumption problem into themaintenance activity. Moreover, the total cost is divided intomaintenance cost and energy consumption cost so that bothof them are considered whenmaking maintenance decisions.Compared with the traditional CBM strategy based on thedeterioration of the system, the CBM strategy based on EEIproposed by the literature [6] can more efficiently reducecosts and save energy.

In addition to energy consumption, some scholars alsointegrate environmental degradation into the CBM strat-egy. Martorell et al. [7] proposed integrated multicriteriadecision-making (IMCDM) in order to achieve better scoresin reliability, availability, maintainability, safety, and cost(RAMS+C) at nuclear power plants. The proposed method-ology integrates, among others, the risk to environment. TheIMCDM is formulated as a multiobjective optimizationwhere the decision variables are preventive maintenanceinterval, surveillance test interval, and maximum allowedoutage time. Vassiliadis and Pistikopoulos [8] developedan optimization framework which takes into account theenvironmental risks and the operability characteristics atthe early stage of process design. This framework permitsthe identification of optimal preventive maintenance sched-ules.

The excess of environmental damage generated by pro-duction systems is, in numerous situations, caused indirectlyby the deterioration of those systems (e.g., degradationand/or corrosion yielding leakages and wear causing excessof energy consumption). For instance, in a nuclear powerplant, the degradation of the mechanical shat seal of therefrigeration compressor induces toxic refrigerant leakages[9]. Yan andHua [10] showed that the degradation ofmachinetools causes an increase of energy consumption which canbe converted to carbon emission by referring to the standarddeveloped by the Climate Registry.

Therefore, when it is technically possible, condition-based monitoring, repair, and maintenance are the mostappropriate activities to be adopted. It allows monitoring thedegradation level and the resulting environmental damagein order to take the appropriate preventive actions and limitthe risk of penalties and sometimes catastrophes. Chouikhiet al. [11] pointed out that, with the global enhancement ofenvironmental awareness, many countries have introducedpolicies of severely punishing companies with excess emis-sions, which is overlooked by the traditional CBM strategy.

Mathematical Problems in Engineering 3

Thismeans the exhaust emissionmay have exceeded the stan-dard before maintenance. In order to remedy this problem,Chouikhi et al. proposed a modeling method of CBM in 2012based on probabilistic statistical theory to integrate environ-mental issues into maintenance actions. The two parameters(the time interval from equipment in good condition to thetime point at which the exhaust gas leaks go beyond thewarning line, in addition to the time interval from exceedingthe warning line to the downtime) are assumed to be subjectto a joint probability density function, so then the proba-bility of preventive maintenance (when the leak exceeds thecritical line), corrective maintenance, and inspection can beobtained; thus, we get the expected total cost per time unitduring the renewal cycle.

Tlili et al. [12] made some improvement on the method[11] and proposed a threshold lower than the critical levelfixed by legislation as a criterion for judging whether toperform preventive maintenance, thereby avoiding the hugeamount of economic penalties for excess exhaust gas. Fur-thermore, Tlili et al. [12] integrated the penalty cost for excessemissions into the cost function, making the total cost inthe objective functionmore comprehensive. In addition, theycompared periodic inspection with nonperiodic inspection,with the results showing that the nonperiodic inspectionpolicy can save more costs and induce less environmentalpollution.

Traditional CBM does not take into account the envi-ronmental attributes of maintenance activities. In view ofthis problem, although some scholars integrate environmen-tal issues into CBM strategy, the factors involved are notcomprehensive. The literature [11, 12] quantified the mutualeffects of exhaust emissions and maintenance activities butwas limited to the emissions. In reality,maintenance activitiesalso induce energy consumption, so the decision optimiza-tion should also consider this ecological factor to reflecthow maintenance activities and ecology impact each other.This paper proposes a CBM strategy that combines energyconsumption and CO2 emissions with the deterioration stateof the system, establishing a single-unit system maintenanceoptimizationmodel based on the probabilistic characteristicsof the combination of energy consumption and CO2 emis-sions. The model makes maintenance decisions based on thethreshold of these two ecological factors; in this way, the eco-factors are integrated into maintenance activities, enablingthe maintenance activities both to save economic costs andto effectively conserve energy and protect the environment,which can ultimately achieve the goal of sustainable develop-ment.

3. Integrative Condition-BasedMaintenance Model for Single-Unit SystemsBased on Sustainability Requirements

3.1. Problem Description and Model Assumptions

3.1.1. Problem Description. This paper considers the eco-logical impact of a production system assimilated to asingle-unit system as it continuously degrades; that is, it

considers excessive carbon dioxide emissions and excessiveenergy consumption. Since in many cases environmentaldegradation generated by the production system has a direct(or indirect) relationshipwith the systemdegradation, we canstudy carbon dioxide emissions and energy consumption inreplacement of the deterioration state of the system [11]. Itis assumed that the system’s carbon dioxide emissions andenergy consumption are, respectively, subject to a Wienerprocess and that the degradation level of the system canonly be known through inspection. This paper concerns asystem that is periodically inspected. If inspection resultsshow that any one of the system’s CO2 emissions or energyconsumption exceeds the critical value, 𝑈𝑖 (𝑈1 refers tothe critical level of CO2 emissions fixed by legislation and𝑈2 refers to the critical level of energy consumption fixedby legislation), the enterprise has to pay a heavy penalty.Meanwhile, a tremendous impact on the environment willbe averted, along with a waste of resources, as when a CMactionmust be performed. In order to lower the possibility ofgetting into such an unfavorable situation, a threshold level𝐿 𝑖 lower than𝑈𝑖 is set to trigger a PM action (𝐿1 refers to PMthreshold for CO2 emission and 𝐿2 refers to PM threshold forenergy consumption). In case an inspection shows that any ofthe environmental indicators exceeds the PM threshold, aPM action should be undertaken; otherwise, nomaintenanceaction will be taken before the next inspection.

3.1.2. Model Assumptions. Notations section explains thenotations related to the model of the CBM strategy consid-ering carbon dioxide emissions and energy consumption, forease of understanding.

For the proposed model, the following assumptions aremade.

(1) CO2 emissions and energy consumption of the sys-tem are subject to two independentWiener processes.

(2) The time of CO2 emissions and energy consumptionfrom the moment when the degradation hits the PMthreshold level 𝐿 to the instant at which it reachesthe critical level are subject to two independent IGprocesses.

(3) The inspection is periodic and the period is 𝜏 =𝑘Δ𝑡, 𝑘 = 1, 2, 3, . . ., where Δ𝑡 is unit of time.(4) The inspection is perfect and time spent is negligible.(5) The actions of preventive maintenance and corrective

maintenance return the equipment to its original stateand the maintenance time is negligible.

(6) According to the inspection results, only one of thethree following events is possible: undertake a PMaction, undertake a CM action, or do nothing beforenext inspection.

(7) All maintenance costs, inspection costs, and penaltycosts are included in the average cost and assumed tobe constants.

(8) A renewal cycle is from current time to the end ofmaintenance.


(9) The time from current time to the instant at which thedegradation hits the PM threshold and the time fromthe instant at which it reaches the critical level followindependent normal distributions.

3.2. Model Establishment. Tlili et al. [12] assume the environ-mental degradation process of the equipment (namely, thecarbon dioxide emission of the equipment) obeys a Wienerprocess:

𝑋 (𝑡) = 𝜆𝑡 + 𝜎𝑊 (𝑡) , 𝑡 > 0, (1)where𝑋(𝑡) denotes standard Brownian motion, 𝜆 is the driftcoefficient, 𝜎 is the diffusion coefficient, and the mean andvariance 𝑋(𝑡) are expressed by 𝜇𝑡 and 𝜎2𝑡, respectively. 𝑋(𝑡)has the following properties:

(a)𝑋(0) = 0.(b) For any time sequence 𝑡𝑖, 𝑖 = 1, 2, 3, . . . , 𝑛 (0 0, 𝑟 > 0 follows anormal distribution𝑁(0, 𝜎2|𝑡 − 𝑟|).

(c)The paths of𝑋(𝑡) are continuous with probability one.The time 𝑇𝑈 from current time to the moment when the

degradation reaches the critical level, 𝑈, follows the inverseGaussian distribution and can be defined as

𝑇𝑈 = inf {𝑡 ≥ 0 | 𝑋 (𝑡) ≥ 𝑈} . (2)The pdf of 𝑇𝑈 is expressed as

𝑓𝑇𝑈 (𝑡) = 𝑈√2𝜋𝑡3𝜎2 exp(−(𝑈 − 𝜆𝑡)22𝜎2𝑡 ) (3)

and the cdf of 𝑇𝑈 is given by𝐹𝑇𝑈 (𝑡) = Φ(−𝑈 + 𝜆𝑡𝜎√𝑡 )

+ exp(2𝜆𝑈𝜎2 )Φ(−𝑈 − 𝜆𝑡𝜎√𝑡 ) .(4)

The time𝑇𝑆 from the instant at which the degradation hitsthe PM threshold level 𝐿 to the moment when it reaches thecritical level, 𝑈, can be defined as

Δ𝑇𝑆 = 𝑇𝑈 − 𝑇𝐿 = inf {𝑠 ≥ 0 | 𝑋 (𝑡 + 𝑠) ≥ 𝑈} . (5)If 𝐿 < 𝑈, otherwise Δ𝑇𝑆 = 0.The degradation increments are independent based on

property (b) of the Wiener process; therefore, if 𝐿 < 𝑈, 𝑇𝑆can be expressed as follows:

𝑇𝑆 = 𝑇𝑈 − 𝑇𝐿 = inf {𝑠 ≥ 0 | 𝑋 (𝑡 + 𝑠) ≥ 𝑈}= inf {𝑠 ≥ 0 | 𝑋 (𝑡 + 𝑠) ≥ 𝑈 − 𝐿} . (6)The pdf of 𝑇𝑆 is written as follows:

𝑓𝑇𝑈−𝑇𝐿 (𝑡) = 𝑈 − 𝐿√2𝜋𝑡3𝜎2 exp(−(𝑈 − 𝐿 − 𝜆𝑡)22𝜎2𝑡 ) . (7)

And the cdf of 𝑇𝑆 is given by𝐹𝑇𝑈−𝑇𝐿 (𝑡)= Φ(− (𝑈 − 𝐿) + 𝜆𝑡𝜎√𝑡 )+ exp(2𝜆 (𝑈 − 𝐿)𝜎2 )Φ(− (𝑈 − 𝐿) − 𝜆𝑡𝜎√𝑡 ) .

(8)

This paper will take into account energy consumptionon the basis of Tlili et al.’s literature. Assuming the CO2emissions and energy consumption of the system followtwo independent Wiener processes, the threshold level ofCO2 emissions and energy consumption are 𝐿1 and 𝐿2,respectively, and the critical level of CO2 emissions andenergy consumption fixed by legislation are 𝑈1 and 𝑈2,respectively.

TheWiener process can be expressed as

𝑋𝑖 (𝑡) = 𝜆𝑖𝑡 + 𝜎𝑖𝑊(𝑡) , 𝑡 > 0, (9)where 𝑋(𝑡) denotes standard Brownian motion, 𝜆𝑖 is thedrift coefficient, 𝜎𝑖 is the diffusion coefficient, and the meanand variance of 𝑋(𝑡) are expressed as 𝜇𝑡 and 𝜎2𝑡, respec-tively.

The time 𝑇𝑈𝑖 from current time to the instant at whichthe degradation hits the critical level, 𝑈𝑖, follows the inverseGaussian distribution and can be defined as

𝑇𝑈𝑖 = inf {𝑡 ≥ 0 | 𝑋𝑖 (𝑡) ≥ 𝑈𝑖} , 𝑖 = 1, 2. (10)The pdf of 𝑇𝑈𝑖 is expressed as𝑓𝑇𝑈𝑖 (𝑡) = 𝑈𝑖√2𝜋𝑡3𝜎2𝑖 exp(−

(𝑈𝑖 − 𝜆𝑖𝑡)22𝜎2𝑖 𝑡 ) , 𝑖 = 1, 2. (11)And the cdf of 𝑇𝑈𝑖 is given by

𝐹𝑇𝑈𝑖 (𝑡) = Φ(−𝑈𝑖 + 𝜆𝑖𝑡𝜎𝑖√𝑡 )+ exp(2𝜆𝑖𝑈𝑖𝜎2𝑖 )Φ(−𝑈𝑖 − 𝜆𝑖𝑡𝜎𝑖√𝑡 ) .

(12)

The time 𝑇𝑆𝑖 from the instant at which the degradationhits the PM threshold level 𝐿 to the moment when it reachesthe critical level 𝑈 can be defined as


𝑇𝑆𝑖 = 𝑇𝑈𝑖 − 𝑇𝐿 𝑖 = inf {𝑠𝑖 ≥ 0 | 𝑋 (𝑡 + 𝑠𝑖) ≥ 𝑈𝑖} . (13)If 𝐿 𝑖 < 𝑈𝑖, otherwise 𝑇𝑆𝑖 = 0.Based on property (b) of the Wiener process, the degra-

dation increments are independent; hence, if 𝐿 < 𝑈, 𝑇𝑆𝑖 canbe written as follows:𝑇𝑆𝑖 = 𝑇𝑈𝑖 − 𝑇𝐿 𝑖 = inf {𝑆𝑖 ≥ 0 | 𝑋 (𝑡 + 𝑆𝑖) ≥ 𝑈𝑖}= inf {𝑆𝑖 ≥ 0 | 𝑋 (𝑡 + 𝑆𝑖) ≥ 𝑈𝑖 − 𝐿 𝑖} , 𝑖 = 1, 2. (14)

The pdf of 𝑇𝑆𝑖 is expressed as𝑓𝑇𝑈𝑖−𝑇𝐿𝑖 (𝑡) = 𝑈𝑖 − 𝐿 𝑖√2𝜋𝑡3𝜎2𝑖 exp(−

(𝑈𝑖 − 𝐿 𝑖 − 𝜆𝑖𝑡)22𝜎2𝑖 𝑡 ) . (15)And the cdf of 𝑇𝑆𝑖 is given by𝐹𝑇𝑈𝑖−𝑇𝐿𝑖 (𝑡)= Φ(− (𝑈𝑖 − 𝐿 𝑖) + 𝜆𝑖𝑡𝜎𝑖√𝑡 )+ exp(2𝜆𝑖 (𝑈𝑖 − 𝐿 𝑖)𝜎2𝑖 )Φ(− (𝑈𝑖 − 𝐿 𝑖) − 𝜆𝑖𝑡𝜎𝑖√𝑡 ) .

(16)

We suppose that periodic inspections are conducted attimes 𝑖𝜋, 𝑖 = 1, 2, 3, . . .. The objective is to determine theoptimal PM threshold level of CO2 emission 𝐿∗1, optimalPM threshold level of energy consumption 𝐿∗2, and theinterinspection period 𝜏∗.

According to classical renewal theory, the average long-run cost rate per time unit can be expressed over a renewalcycle 𝑆, defined as the interval between successive mainte-nance actions.

Hence, the expression of the average long-run cost rate isgiven by

𝐸𝐶 (𝐿1, 𝐿2, 𝜏) = 𝐸 [𝐶𝑇 (𝐿1, 𝐿2, 𝜏)]𝐸 [𝐶cycle (𝐿1, 𝐿2, 𝜏)] , (17)where 𝐸[𝐶𝑇] denotes the expected cost during the cycle and𝐸[𝐶cycle] denotes the expected renewal cycle length.3.2.1. The Expected Renewal Cycle Length. Suppose therenewal period of the system denotes 𝑇 = 𝑖𝜋 , 𝑖 =1, 2, 3, . . .. Since the system is considered to be as goodas new after both maintenance actions (PM or CM),there are three possible events determined by the system’sCO2 emissions and energy consumption, which are nowdescribed.

(a)The time𝑇𝐿1 from current time to the instant at whichthe CO2 emission 𝑋1(𝑡) reaches the PM threshold level isfound (𝑖 − 1)𝜏 < 𝑇𝐿1 ≤ 𝑖𝜏, while the time 𝑇𝐿2 from currenttime to the instant at which the energy consumption 𝑋2(𝑡)reaches the PM threshold level is found 𝑇𝐿2 ≥ 𝑖𝜏. At 𝑖𝜏, PMor CM action is performed since themeasured CO2 emission𝑋1(𝑡) has exceeded the PM threshold level 𝐿1. Express thisevent as 𝐴1(𝑖, 𝜏) = {(𝑖 − 1)𝜏 < 𝑇𝐿1 ≤ 𝑖𝜏, 𝑇𝐿2 ≥ 𝑖𝜏},where 𝑖 = 1, 2, 3, . . ., and its probability of occurrence is givenby

𝑃 (𝐴1 (𝑖, 𝜏)) = ∫𝑖𝜏(𝑖−1)𝜏

∫∞𝑖𝜏𝐹 (𝑢, V) 𝑑𝑢 𝑑V = ∫𝑖𝜏

(𝑖−1)𝜏∫∞𝑖𝜏𝑓𝑇𝐿1 (𝑢) 𝑓𝑇𝐿2 (V) 𝑑𝑢 𝑑V

= ∫𝑖𝜏(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

∞

𝑖𝜏

𝐿2√2𝜋V3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V.

(18)

(b)The time𝑇𝐿2 from current time to the instant at whichthe energy consumption 𝑋2(𝑡) reaches the PM threshold isfound, (𝑖 − 1)𝜏 < 𝑇𝐿2 ≤ 𝑖𝜏, while the time 𝑇𝐿1 from currenttime to the instant at which the CO2 emission 𝑋1(𝑡) reachesthe PM threshold is found, 𝑇𝐿1 ≥ 𝑖𝜏. At 𝑖𝜏, PM or CM

action is performed since the measured energy consumption𝑋2(𝑡) has exceeded the PM threshold level 𝐿2. Express thisevent as 𝐴2(𝑖, 𝜏) = {(𝑖 − 1)𝜏 < 𝑇𝐿2 ≤ 𝑖𝜏, 𝑇𝐿1 ≥ 𝑖𝜏},where 𝑖 = 1, 2, 3, . . ., and its probability of occurrence is givenby

𝑃 (𝐴2 (𝑖, 𝜏)) = ∫∞𝑖𝜏∫𝑖𝜏(𝑖−1)𝜏

𝐹 (𝑢, V) 𝑑𝑢 𝑑V = ∫∞𝑖𝜏∫𝑖𝜏(𝑖−1)𝜏

𝑓𝑇𝐿1 (𝑢) 𝑓𝑇𝐿2 (V) 𝑑𝑢 𝑑V= ∫∞𝑖𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

𝑖𝜏

(𝑖−1)𝜏

𝐿2√2𝜋V3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V.

(19)


(c) The time from current time to the instant at which𝑋1(𝑡) and𝑋2(𝑡) reaches the threshold level are found (𝑖−1)𝜏


(b) At 𝑖𝜏, the energy consumption 𝑋2(𝑡) was higher thanthe PM threshold level 𝐿2, but lower than the critical level𝑈2, while the CO2 emissions 𝑋1(𝑡) has not reached the PM

threshold level 𝐿1. Express this event as 𝐵2(𝑖, 𝜏) = {(𝑖 − 1)𝜏


Therefore, the probability 𝑃𝑃 that the cycle ends with apreventive maintenance action is expressed as

𝑃𝑃 (𝐿1, 𝐿2, 𝜏) = ∞∑𝑖=1

[𝑃 (𝐵1 (𝑖, 𝜏)) + 𝑃 (𝐵2 (𝑖, 𝜏)) + 𝑃 (𝐵3 (𝑖, 𝜏))] = ∞∑𝑖=1

{{{{{∫𝑖𝜏

(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )

⋅ [1 − Φ(− (𝑈1 − 𝐿1) + 𝜆1𝑢𝜎1√𝑢 ) − exp(2𝜆1 (𝑈1 − 𝐿1)𝜎21 )Φ(− (𝑈1 − 𝐿1) − 𝜆1𝑢𝜎1√𝑢 )]𝑑𝑢∫∞

𝑖𝜏

𝐿2√2𝜋V3𝜎22⋅ exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V + ∫

∞

𝑖𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

𝑖𝜏

(𝑖−1)𝜏

𝐿2√2𝜋𝑢3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )

⋅ [1 − Φ(− (𝑈2 − 𝐿2) + 𝜆2V𝜎1√V ) − exp(2𝜆2 (𝑈2 − 𝐿2)𝜎22 )Φ(− (𝑈2 − 𝐿2) − 𝜆2V𝜎2√V )]𝑑V+ ∫𝑖𝜏(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )

⋅ [1 − Φ(− (𝑈1 − 𝐿1) + 𝜆1𝑢𝜎1√𝑢 ) − exp(2𝜆1 (𝑈1 − 𝐿1)𝜎21 )Φ(− (𝑈1 − 𝐿1) − 𝜆1𝑢𝜎1√𝑢 )]𝑑𝑢⋅ ∫𝑖𝜏(𝑖−1)𝜏

𝐿2√2𝜋𝑢3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )

⋅ [1 − Φ(− (𝑈2 − 𝐿2) + 𝜆2V𝜎1√V ) − exp(2𝜆2 (𝑈2 − 𝐿2)𝜎22 )Φ(− (𝑈2 − 𝐿2) − 𝜆2V𝜎2√V )]𝑑V}}}}} .

(26)

The sum of 𝑃𝑃 and 𝑃𝐶 equals 1 as follows:𝑃𝑃 + 𝑃𝐶 = 1 (27)

Probability 𝑃𝐶 that the cycle ends with a correctivemaintenance action is

𝑃𝐶 (𝐿1, 𝐿2, 𝜏) = 1 − 𝑃𝑃 (𝐿1, 𝐿2, 𝜏) . (28)(2) The Expected Number of Inspections during a Cycle. Theexpected renewal cycle length can be obtained from (1),which is here considered to be the interval between consecu-tive maintenance actions. It is assumed that the maintenanceaction to be performed is determined after the inspectionusing CO2 emissions or energy consumption and that themaintenance time is negligible. Since the inspection is peri-odic and the time interval is 𝜏, while the renewal period is anintegral multiple of 𝜏, the expected number of inspectionsduring a cycle can be expressed by

𝐸 [𝑁] = 𝐸cycle [(𝐿1, 𝐿2, 𝜏)]𝜏 . (29)(3) The Average Time for Paying a Penalty during a Cycle. At𝑖𝜏, in case an inspection result shows that CO2 emissions orenergy consumption has exceeded the critical level, thecompany has to pay a heavy penalty for the excess amount.The discussion here is divided into two circumstances.

(a) The Average Time for Generation of an Excess Amount ofCO2 Emissions. At 𝑖𝜏, CO2 emissions 𝑋1(𝑡) have exceeded𝑈1; that is, (𝑖 − 1)𝜏 < 𝑇𝐿1 ≤ 𝑇𝑈1 < 𝑖𝜏. 𝑇𝜉(𝑖) = 𝑖𝜏 − 𝑇𝑈1represents the time from the moment when the amount ofCO2 emitted exceeds the critical level to the instant of theinspection. Express this event as

𝐶1 (𝑖, 𝜏)= {(𝑖 − 1) 𝜏 < 𝑇𝐿1 ≤ 𝑇𝑈1 < 𝑖𝜏 | 𝑇𝜉 (𝑖) = 𝑖𝜏 − 𝑇𝑈1} . (30)Then the expected time for paying a penalty is given by

𝐸 [𝑇𝜉] = ∞∑𝑖=1

𝑃 {(𝑖 − 1) 𝜏 < 𝑇𝐿1 ≤ 𝑇𝑈1 < 𝑖𝜏 | 𝑇𝜉 (𝑖) = 𝑖𝜏 − 𝑇𝑈1} = ∞∑𝑖=1

∫𝑖𝜏(𝑖−1)𝜏

∫𝑧−𝑢0

∫𝑧(𝑖−1)𝜏

𝐹 (𝑢, 𝑠) ⋅ (𝑖𝜏 − 𝑧) 𝑑𝑢 𝑑𝑠 𝑑𝑧= ∞∑𝑖=1


∫𝑧−𝑢0

∫𝑧(𝑖−1)𝜏

𝑓𝑇𝐿1 (𝑢) 𝑓𝑇𝑈1−𝑇𝐿1 (𝑠) ⋅ (𝑖𝜏 − 𝑧) 𝑑𝑢 𝑑𝑠 𝑑𝑧 = ∞∑𝑖=1


(𝑖𝜏 − 𝑧) ∫𝑧(𝑖−1)𝜏

𝑓𝑇𝐿1 (𝑢) 𝐹𝑇𝑈1−𝑇𝐿1 (𝑧 − 𝑢) 𝑑𝑢 𝑑𝑧


= ∞∑𝑖=1


(𝑖𝜏 − 𝑧) ∫𝑧(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )

⋅ [Φ(− (𝑈1 − 𝐿1) + 𝜆1𝑢𝜎1√𝑢 ) + exp(2𝜆1 (𝑈1 − 𝐿1)𝜎21 )Φ(− (𝑈1 − 𝐿1) − 𝜆1𝑢𝜎1√𝑢 )]𝑑𝑢𝑑𝑧.(31)

(b) The Average Time for Excess Amount of Energy Consump-tion. At 𝑖𝜏, the energy consumption 𝑋2(𝑡) has exceeded 𝑈2;that is, (𝑖−1)𝜏 < 𝑇𝐿2 ≤ 𝑇𝑈2 < 𝑖𝜏.𝑇𝛿(𝑖) = 𝑖𝜏−𝑇𝑈2 represents theaverage time between the instant when the amount of energyconsumption exceeds the critical level and themoment of theinspection. Express this event as

𝐶2 (𝑖, 𝜏)= {(𝑖 − 1) 𝜏 < 𝑇𝐿2 ≤ 𝑇𝑈2 < 𝑖𝜏 | 𝑇𝛿 (𝑖) = 𝑖𝜏 − 𝑇𝑈2} . (32)So the expected time of paying a penalty is given by

𝐸 [𝑇𝛿] = ∞∑𝑖=1

𝑃 {(𝑖 − 1) 𝜏 < 𝑇𝐿2 ≤ 𝑇𝑈2 < 𝑖𝜏 | 𝑇𝛿 (𝑖) = 𝑖𝜏 − 𝑇𝑈2} = ∞∑𝑖=1


∫𝑤−V0

∫𝑤(𝑖−1)𝜏

𝐹 (V, 𝑡) ⋅ (𝑖𝜏 − 𝑤)= ∞∑𝑖=1


∫𝑤−V0

∫𝑤(𝑖−1)𝜏

𝑓𝑇𝐿2 (V) 𝑓𝑇𝑈2−𝑇𝐿2 (𝑡) ⋅ (𝑖𝜏 − 𝑤) 𝑑V 𝑑𝑡 𝑑𝑤 = ∞∑𝑖=1


(𝑖𝜏 − 𝑤)⋅ ∫𝑤(𝑖−1)𝜏

𝑓𝑇𝐿2 (V) 𝐹𝑇𝑈2−𝑇𝐿2 (𝑤 − V) 𝑑V 𝑑𝑤 = ∞∑𝑖=1


(𝑖𝜏 − 𝑤)∫𝑤(𝑖−1)𝜏

𝐿2√2𝜋𝑢3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )

⋅ [Φ(− (𝑈2 − 𝐿2) + 𝜆2V𝜎2√V ) + exp(2𝜆2 (𝑈2 − 𝐿2)𝜎22 )Φ(− (𝑈2 − 𝐿2) − 𝜆2V𝜎2√V )]𝑑V 𝑑𝑤.

(33)

3.2.3. Expected Average Cost per TimeUnit during the RenewalCycle. Based on the formula above, the expected average costper time unit during the renewal cycle is given by

𝐸𝐶 (𝐿1, 𝐿2, 𝜏) = 𝐸 [𝐶𝑇 (𝐿1, 𝐿2, 𝜏)]𝐸 [𝐶cycle (𝐿1, 𝐿2, 𝜏)]= 𝐶𝐶𝑃𝑃 + 𝐶𝑃𝑃𝑃 + 𝐶𝐼𝐸 [𝑁] + 𝐶CO2𝐸 [𝑇𝜉] + 𝐶energy𝐸 [𝑇𝛿]𝐸 [𝐶cycle (𝐿1, 𝐿2, 𝜏)] ,

(34)

where the numerator is expressed as

𝐶𝑃 ∞∑𝑖=1

[𝑃 (𝐵1 (𝑖, 𝜏)) + 𝑃 (𝐵2 (𝑖, 𝜏)) + 𝑃 (𝐵3 (𝑖, 𝜏))] + 𝐶𝐶(1− ∞∑𝑖=1

[𝑃 (𝐵1 (𝑖, 𝜏)) + 𝑃 (𝐵2 (𝑖, 𝜏)) + 𝑃 (𝐵3 (𝑖, 𝜏))]) + 𝐶𝐼⋅ ∑∞𝑖=1 [(𝑃 (𝐴1 (𝑖, 𝜏)) + 𝑃 (𝐴2 (𝑖, 𝜏)) + 𝑃 (𝐴3 (𝑖, 𝜏))) 𝑖𝜏]𝜏

+ 𝐶CO2 ∞∑𝑖=1


(𝑖𝜏 − 𝑧)⋅ ∫𝑧(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )

⋅ [Φ(− (𝑈1 − 𝐿1) + 𝜆1𝑢𝜎1√𝑢 ) + exp(2𝜆1 (𝑈1 − 𝐿1)𝜎21 )⋅ Φ(− (𝑈1 − 𝐿1) − 𝜆1𝑢𝜎1√𝑢 )]𝑑𝑢𝑑𝑧+ 𝐶energy∞∑

𝑖=1


(𝑖𝜏 − 𝑤)⋅ ∫𝑤(𝑖−1)𝜏

𝐿2√2𝜋𝑢3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )

⋅ [Φ(− (𝑈2 − 𝐿2) + 𝜆2V𝜎2√V ) + exp(2𝜆2 (𝑈2 − 𝐿2)𝜎22 )⋅ Φ(− (𝑈2 − 𝐿2) − 𝜆2V𝜎2√V )]𝑑V 𝑑𝑤.(35)

The denominator is given by


∞∑𝑖=1

[(𝑃 (𝐴1 (𝑖, 𝜏)) + 𝑃 (𝐴2 (𝑖, 𝜏)) + 𝑃 (𝐴3 (𝑖, 𝜏))) 𝑖𝜏] . (36) The denominator can be further developed as follows:

∞∑𝑖=1

[[[(∫𝑖𝜏

(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

∞

𝑖𝜏

𝐿2√2𝜋V3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V + ∫

∞

𝑖𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

𝑖𝜏

(𝑖−1)𝜏

𝐿2√2𝜋V3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V

+ ∫𝑖𝜏(𝑖−1)𝜏

𝐿1√2𝜋𝑢3𝜎21 exp(−(𝐿1 − 𝜆1𝑢)22𝜎21𝑢 )𝑑𝑢∫

𝑖𝜏

(𝑖−1)𝜏

𝐿2√2𝜋V3𝜎22 exp(−(𝐿2 − 𝜆2V)22𝜎22V )𝑑V)𝑖𝜏]]] .

(37)

3.3. Numerical Procedure. To integrate the calculations illus-trated above, after inputting the parameters Δ𝐿1, Δ𝐿2, Δ𝜏,𝑇𝑢, 𝐶𝐶, 𝐶𝑃, 𝐶𝑖, 𝐶CO2 , and 𝐶energy, the expected average costin the renewal cycle can be computed through the numericalprocedure in Figure 1.

The detailed steps of the algorithm are described asfollows.

Step 1. Initialize 𝐿1 = 0, 𝐿2 = 0, 𝑡 = 0, 𝑖 = 0 and let 𝐿1 =𝐿1 + Δ𝐿1, 𝐿2 = 𝐿2 + Δ𝐿2, and 𝑡 = 𝑡 + Δ𝜏.Step 2. Inspect amount of CO2 emissions 𝑋1(𝑖𝑡) and energyconsumption𝑋2(𝑖𝑡) at 𝑖𝑡.Step 3. If 𝑋1(𝑖𝑡) < 𝐿1 and 𝑋2(𝑖𝑡) < 𝐿2, let 𝑖 = 𝑖 + 1 and go toStep 2; otherwise, go to Step 4.

Step 4. If 𝑡 < 𝑇𝑢, let 𝑡 = 𝑡 + Δ𝜏 and go to Step 2; otherwise, goto Step 5.

Step 5. Compute 𝐸𝐶(𝐿1, 𝐿2, 𝑖𝑡) for each 𝑖𝑡 and store the min-imum 𝐸𝐶1(𝐿1, 𝐿2, 𝑖𝑡).Step 6. If 𝐿1 < 𝑈1, let 𝐿1 = 𝐿1 + Δ𝐿1 and go to Step 2;otherwise, go to Step 7.

Step 7. Compute 𝐸𝐶1(𝐿1, 𝐿2, 𝑖𝑡) for each 𝐿1 and store theminimum 𝐸𝐶2(𝐿1, 𝐿2, 𝑖𝑡).Step 8. If 𝐿2 < 𝑈2, let 𝐿2 = 𝐿2 + Δ𝐿2 and go to Step 2;otherwise, go to Step 9.

Step 9. Compute 𝐸𝐶2(𝐿1, 𝐿2, 𝑖𝑡) for each 𝐿2 and store theminimum 𝐸𝐶min(𝐿1, 𝐿2, 𝑖𝑡).Step 10. Output 𝐸𝐶min(𝐿1, 𝐿2, 𝑖𝑡) and the corresponding𝐿1, 𝐿2, 𝑖𝑡.Step 11. End.

The above is the fully unfolded formula for the averageexpected cost per time unit within the renewal cycle. Next,we use the simulation method to obtain the optimal intervalof periodic inspection, the PM threshold level of CO2emissions, and energy consumption and then give the mostappropriate maintenance strategy. In addition, sensitivityanalyses will be conducted on the penalty cost for excess CO2emissions, the penalty cost for excess energy consumption,the inspection cost, the PM cost, and the CM cost.

4. Simulation Analysis and Sensitivity Analysis

4.1. Basic Assumption of the Model. The model assumes thatthe critical level of CO2 emissions and energy consumptionfixed by legislation are known constants. The critical level ofCO2 emissions will be set at 10 tons with reference to Tlili etal.’s study [12], which are presented in Table 1.

In terms of the threshold of energy consumption, it isimportant to note that since much equipment now usesmixed fuels to reduce environmental pollution (e.g., themixed boiler which, apart from coal, also consumes dieseland natural gas), the consumption of a single energy source isnot proportional to the overall CO2 emissions of the system.In other words, energy consumption and CO2 emissions areindependent and thus call for an integrative consideration inthis study. Notice each threshold cannot be obtained fromthe other directly. Here, one energy source will be separatedfrom mixed energy and studied, which is assumed to becoal. Given that China sets different coal limits on variouscompanies, we refer to the coal limit of one specific companyas the benchmark and assume the critical level of energyconsumption is 4 tons. Again, in reference to Tlili et al.’sstudy, the critical level of CO2 emissions will be set to 10 tonsfor the simulation. Other values (PM cost 𝐶𝑃, CM cost 𝐶𝐶,inspection cost 𝐶𝐼, and penalty cost for excess CO2 emission𝐶CO2 , 𝜆1, 𝜎1) also refer to Tlili et al.’s study.

The pdf of the time from current time to the momentwhen CO2 emission reaches the critical value is given by

𝑓𝑇𝑈1 (𝑡) = 10√2𝜋𝑡30.352 exp(−(10 − 1.3𝑡)22 ⋅ 0.352𝑡 ) . (38)

Its cdf is expressed as

𝐹𝑇𝑈1 (𝑡) = Φ(−10 + 1.3𝑡0.35√𝑡 )+ exp (2 ⋅ 1.3 ⋅ 100.352 )Φ(−10 − 1.3𝑡0.35√𝑡 ) .

(39)

The pdf of the time from the instant at which CO2emissions hit the threshold to the moment when it reachesthe critical value is given by𝑓𝑇𝑈1−𝑇𝐿1 (𝑡)

= 10 − 𝐿1√2𝜋𝑡30.352 exp(−(10 − 𝐿1 − 1.3𝑡)2

2 ⋅ 0.352𝑡 ) . (40)


Table 1: Known data.𝐶𝐶 (USD) 𝐶𝑃 (USD) 𝐶𝐼 (USD) 𝐶CO2 (USD/week) 𝐶energy (USD/week) 𝑈1 𝑈2 𝜆1 𝜎1 𝜆2 𝜎2900 500 100 10000 3500 10 4 1.3 0.35 0.416 0.112

Start

End

Yes

No

No

Yes

Yes No

No

Yes

No

Yes

Yes

CM actionNo

No

PM action

Yes

End the circle and compute Ec

InitializeL1 = 0, L2 = 0,

t = 0, i = 0

L2 = L2 + ΔL2

L1 = L1 + ΔL1

t = t + Δt

i = i + 1

Inspect X1(it) andX2(it) at it

X1(it) < L1

X2(it) < L2

Compute EC2(L1, L2, it) for each L2 andstore the minimum ECＧＣＨ(L1, L2, it) . OutputECＧＣＨ(L1, L2, it) and the corresponding L1, L2, it

L2 < U2Compute EC1(L1, L2, it) for each L1 and

store the minimum EC2(L1, L2, it)

L1 < U1

Compute EC(L1, L2, it) for each itand store the minimum EC1(L1, L2, it)

X1(it) < U1

X2(it) < U2

t < Tu

Figure 1: The numerical procedure for the expected average cost.

Its cdf is expressed as𝐹𝑇𝑈1−𝑇𝐿1 (𝑡)= Φ(𝐿1 − 10 + 1.3𝑡0.35√𝑡 )+ exp(2 ⋅ 1.3 (10 − 𝐿1)0.352 )Φ(𝐿1 − 10 − 1.3𝑡0.35√𝑡 ) .

(41)

Therefore, the pdf of the time from current time to themoment when energy consumption reaches the critical valueis given by

𝑓𝑇𝑈2 (𝑡) = 4√2𝜋𝑡30.1122 exp(−(4 − 0.416𝑡)22 ⋅ 0.1122𝑡 ) . (42)


Its cdf is expressed as

𝐹𝑇𝑈2 (𝑡) = Φ(−4 + 0.416𝑡0.112√𝑡 )+ exp(2 ⋅ 0.416 ⋅ 40.1122 )Φ(−4 − 0.416𝑡0.112√𝑡 ) .(43)

The pdf of the time from the instant at which energyconsumption hits the threshold to the moment when itreaches the critical value is given by

𝑓𝑇𝑈2−𝑇𝐿2 (𝑡)= 4 − 𝐿2√2𝜋𝑡30.1122 exp(−(4 − 𝐿2 − 0.416𝑡)

2

2 ⋅ 0.1122𝑡 ) . (44)Its cdf is expressed as

𝐹𝑇𝑈2−𝑇𝐿2 (𝑡)= Φ(𝐿2 − 4 + 0.416𝑡0.112√𝑡 )+ exp(0.832 (4 − 𝐿2)0.1122 )Φ(𝐿2 − 4 − 0.416𝑡0.112√𝑡 ) .

(45)

4.2. Comparative Analysis. The CO2 emissions and energyconsumption of the system are simulated using the MonteCarlo method and the Particle Swarm Optimization (PSO)algorithm is used to accelerate the operation. The optimalthresholds of CO2 emissions and energy consumption andthe time interval of periodic inspection are obtained byminimizing the expected cost per time unit during therenewal cycle.

Results obtained, which are shown in Table 2, give theoptimal threshold of CO2 emissions as 6 tons and the oneof energy consumption as 3 tons and the interval for periodicinspection as 7 week, when the renewal period is 7.002 week.With the limitation of energy consumption, the time ofgeneration of excess CO2 emissions is relatively low (0.0181week), compared with Tlili et al. [12], whose time of gen-eration of it is 0.02. The average expected cost per week is$109.406, which is much lower than the one from Tlili et al.[12], $123.94. Therefore, the method of this paper performsbetter in terms of average expected cost.

4.3. Comparative Sensitivity Analysis. This section will varythe penalty cost for excess CO2 emissions 𝐶CO2 , penalty costfor excess energy consumption 𝐶energy, inspection cost 𝐶𝐼,and CM cost 𝐶𝐶, respectively, compared with the resultsof Tlili et al.’s [12] research, to study their influence on theoptimal solution and figure out which method performsbetter.

4.3.1. Penalty Cost for Excess CO2 Emissions 𝐶CO2 . The opti-mum values are obtained under different circumstances byvarying the penalty cost of excess CO2 emissions. Moreover,the comparison between the results of the paper and Tlili etal. [12] is shown in Table 3.

First, with the increase of𝐶CO2 , its threshold 𝐿1 decreasesand the probability of executing a PM action and thefrequency of inspection increases, while the average renewalcycle length shrinks, possibly because the increase in penaltyfor CO2 emissions forces companies to tightly monitor theemission of CO2. Companies try to minimize the possibilitythat CO2 emissions exceed the critical level by reducing itsthreshold along with the interval of periodic inspections, soas to avoid the heavy penalty induced. Moreover, it is worthnoticing that, in every pair, the time of generation excess CO2emission is less than the Tlili et al.’s [12], which may resultsfrom the limitation of energy consumption. Furthermore,compared to the method from Tlili et al. [12], in every pair,the expected average cost is much lower.

4.3.2. Penalty Cost for Excess Energy Consumption 𝐶energy.The optimum values are obtained under different circum-stances by varying the value of the penalty for excessiveenergy consumption, and the comparison between the resultsof the paper and Tlili et al. [12] is shown in Table 4.

It can be inferred that, as the penalty cost for excessenergy consumption increases, its threshold 𝐿2 decreasesand the probability of executing PM action decreases. It isworth noting that the results keep relatively still in case thepenalty of excess energy consumption is 0 and $3500. Thisprobably is because when energy consumption reaches itscritical level, the CO2 emissions have already exceeded itscritical level, whose induced penalty could counteract theadvantage from the reduction of penalty for excess energyconsumption. Meanwhile, the expected time of generation ofexcess CO2 emissions increases, probably because the penaltyof $35000 for excess energy consumption weighs way morethan the one for excess CO2 emissions, which makes it worthof taking the cost from the penalty for excess CO2 emissions.Besides, the table also shows that when the penalty of excessenergy consumption is 0 and $3500, the average expected costis relatively lower than the result of Tlili et al. [12]. However,when the penalty increases to $35000, the cost increases sorapidly that exceeds the Tlili et al.’s. We think this is becausethe penalty of energy consumption contributes too muchcompared the one of CO2 emissions.

4.3.3. Inspection Cost 𝐶𝐼. The optimum values are obtainedunder different circumstances by varying the value of inspec-tion costs. The comparison between the results of the paperand Tlili et al. [12] is shown in Table 5.

This result shows that, as the inspection cost increases,the threshold of CO2 emissions 𝐿2 decreases and the intervalof periodic inspection increases, whereas the average renewalcycle length increases. Since the company extends the intervalof periodic inspection to reduce the total cost, less frequentinspections increase the average time of generation of excessCO2 emissions and energy consumption; therefore, thepenalty cost for excess energy consumption contributes sub-stantially to the average cost of the firm.Moreover, comparedto Tlili et al. [12], without the limitation of threshold of energyconsumption, the penalty of this paper caused by excessCO2 emissions is much lower than that of Tlili et al.’s [12].Therefore, the average expected cost of this paper is lower


Table2:Com

paris

onof

theo

btainedop

timalsolutio

nforC

BMpo

licyconsideringCO

2em

issionandenergy

consum

ption.

𝐿 1𝐿 2

𝜏𝑃𝑃(%

)𝑃 𝐶(%

)E[N]

Inspectio

ncost

𝐸[𝑇 𝜉] weekPenalty

fore

xcess

CO2em

issions

𝐸[𝑇 𝛿] weekPenalty

fore

xcess

energy

consum

ption

𝐸[𝐶 cycle]

week

𝐸𝐶(USD

)

Thispaper

63

782.21

17.79

1.0003

100.03

0.0181

181

00

7.002

109.4

06Tlili

etal.[12]

2/

782.24

17.76

1.08

108

0.02

200

//

7.1123.94


Table3:Eff

ecto

f𝐶 CO2.

𝐶 CO 2𝐿 1

𝐿 2𝜏

𝑃 𝑃 (%)𝑃 𝐶 (%)

E[N]

Inspectio

ncost


fore

xcess

CO2em

ission


for

excessenergy

consum

ption

𝐸[𝐶 cycle]

week

𝐸𝐶(USD

)

0Th

ispaper

73

831.91

68.09

1.00

100

0.2472

00.001

3.5

866.9282

Tlili

etal.[12]

9/

738.57

61.43

1.44

144

7.10

//

10.06

88.47

1000

0Th

ispaper

63

782.21

17.79

1.0003

100.03

0.0181

181

00

7.002

109.4

06Tlili

etal.[12]

2/

782.24

17.76

1.08

108

0.02

200

//

7.1123.94

1000

00Th

ispaper

53

395.75

4.25

1.92

192

00

00

5.758

119.9967

Tlili

etal.[12]

2/

599.57

0.43

1.00

100

0.007

700

//

5260.37


Table4:Eff

ecto

f𝐶 energy.

𝐶 energy

𝐿 1𝐿 2

𝜏𝑃 𝑃(%

)𝑃 𝐶(%

)E[N]

Inspectio

ncost


for

excessCO

2em

ission


for

excessenergy

consum

ption

𝐸[𝐶 cycle]

week

𝐸𝐶 (USD)

This

paper

03

47

82.23

17.77

1100

0.0181

181

00

7109.0

243500

63

782.21

17.79

1.0003

100.03

0.0189

189

00

7.002

109.4

063500

010

24

29.88

70.12

1.85

185

0.2472

2472

00

7.4024

417.8

5Tlili

etal.[12]

\2

\7

82.24

17.76

1.08

108

0.02

200

\\

7.1123.94


Table5:Eff

ecto

f𝐶 𝐼.𝐶 𝐼

𝐿 1𝐿 2

𝜏𝑃 𝑃(%

)𝑃 𝐶(%

)E[N]

Inspectio

ncost

𝐸[𝑇 𝜉]week

Penalty

for

excessCO

2em

ission


for

excessenergy

consum

ption

𝐸[𝐶 cycle]

week

𝐸𝐶(USD

)

0Th

ispaper

101

797.75

2.25

20

0.001

100

014

72.637

Tlili

etal.[12]

8/

199.79

0.21

6.65

08×10−6

0.08

//

6.65

75.35

100

Thispaper

63

782.21

17.79

1.0003

100.03

0.0181

181

00

7.002

109.4

06Tlili

etal.[12]

2/

782.24

17.76

1.08

108

0.02

200

//

7.1123.94

1000

Thispaper

32

831.92

68.08

11000

0.2472

2472

0.001

3.5

8488.44

07Tlili

etal.[12]

2/

94.75

95.25

11000

1.13

11300

//

91464

.56


than the one of Tlili et al.’s [12], especially when the inspectioncost increases to $1000.

4.3.4. Corrective Maintenance Cost 𝐶𝐶. The optimum val-ues are obtained under different circumstances by vary-ing the value of the CM cost. The comparison betweenthe results of the paper and Tlili et al. [12] is shown inTable 6.

The results indicate that as the CM cost increases, thecompany will lower the two thresholds so as to reduce theprobability of executing CM action. Meanwhile, the intervalof periodic inspections and the expected average time ofexcess CO2 emissions are relatively stable and the inspectionskeep almost similar frequency, which we think this may bebecause the balance point for minimizing the expectedaverage cost and its contribution is relative smaller than thatof penalty. Since the CM cost increases, as mentioned above,the probability of PM will increase due to its lower cost,which effectively control the time of excess CO2 emissionsand energy consumption. Therefore, the penalty will alsodecrease, which is lower than the Tlili et al.’s [12] results.Thus,this paper’s method performs better.

5. Conclusion and Future Directions

The original contribution of this paper is to propose a CBMpolicy for single-unit systems that integrates ecological fac-tors (CO2 emissions and energy consumption) into a previ-ous model to obtain the optimal thresholds of CO2 emissionand energy consumption, as well as the interval of periodicinspection, with the objective function ofminimizing averageexpected cost during the cycle. This paper assumes that thesystem’s CO2 emissions and energy consumption follow twoindependent normal distributions and that the time fromthe instant at which one of them hits its correspondingthreshold to the moment when it reaches the critical levelis independent of the two ecological factors. Based on thesetwo assumptions, the joint pdf of three parameters wasestablished to compute the objective function. We conductedsimulations using the Monte Carlo method and algorithmto obtain the optimal threshold of CO2 emissions, optimalthreshold of energy consumption, and the interval of periodicinspections. After a sensitivity study for each variable, therewere three discoveries. First, the penalty for excess CO2emissions and excess energy consumption will affect thevalues of the company’s thresholds. Secondly, the PM costsand CM costs will also influence the values of the twothresholds. Lastly, the inspection cost will affect the timeinterval of periodic inspections. Meanwhile, the comparisonbetween results whether to take energy consumption or notindicates that only considering CO2 emissions still leads tosome extra penalty, while adding energy consumption effec-tively reduces the penalty and probability of CM action,which could result in lower maintenance cost. Therefore,it can be concluded that this paper’s method performsbetter.

This paper leaves much room for further study in twopotential directions: (1) to extendCBMpolicy under periodicinspection to one under nonperiodic inspection; (2) to

extend the research object of this article to a multiunitsystem; and (3) to establish a model of the relationshipbetween CO2 consumption and mixed energy consumptionfor a generalized model of optimization of maintenancecost.

Notations𝜏: The interval of periodic inspection𝑋1(𝑡): The Wiener process carbon dioxideemission is subject to𝑋2(𝑡): The Wiener process energy consumptionis subject to𝐿1: PM threshold level of CO2 emission𝐿2: PM threshold level of energy consumption𝑈1: Critical level of CO2 emission fixed bylegislation𝑈2: Critical level of energy consumption fixedby legislation𝑇𝑈: The time 𝑇𝑈 from present time to theinstant one at which the degradation firsthits the critical level 𝑈𝑇𝑆: The time 𝑇𝑆 from the moment when thedegradation hits the PM threshold 𝐿 to theinstant at which it reaches the critical level𝑈𝑓𝑇𝑈(𝑡): Probability density function associatedwith 𝑇𝑈𝐹𝑇𝑈(𝑡): Cumulative distribution functionassociated with 𝑇𝑈𝑓𝑇𝑈−𝑇𝐿(𝑡): Probability density function associated 𝑇𝑆𝐹𝑇𝑈−𝑇𝐿(𝑡): Cumulative distribution associated 𝑇𝑆𝑆: The renewal cycle length (A cycle is fromcurrent time to the end of maintenance)𝐸𝐶(𝐿1, 𝐿2, 𝜏): The average cycle cost per time unit𝐸[𝐶𝑇]: The expected total cost within a cycle𝐸[𝐶cycle]: The expected renewal cycle length𝐶𝐶: Corrective maintenance (CM) action cost𝐶𝑃: Preventive maintenance (PM) action cost𝐶𝐼: Inspection cost𝐶CO2 : Penalty cost per time unit for excess CO2emission incurred once the critical level isexceeded𝐶energy: Penalty cost per time unit for excessenergy consumption incurred once thecritical level is exceeded𝑃𝐶: Probability that the cycle ends with a CMaction𝑃𝑃: Probability that the cycle ends with a PMaction𝐸[𝑁]: The expected number of inspectionsduring a cycle𝐸[𝑇𝜉]: The expected average time of generation ofexcess amount of CO2 emissions during acycle𝐸[𝑇𝛿]: The expected average time of generation ofexcess amount of energy consumptionduring a cycle.


Table6:Eff

ecto

f𝐶 𝐶.𝐶 𝐶

𝐿 1𝐿 2

𝜏𝑃 𝑃(%

)𝑃 𝐶(%

)E[N]

Inspectio

ncost


for

excessCO

2em

ission


for

excessenergy

consum

ption

𝐸[𝐶 cycle]

week

𝐸𝐶 (USD)

600

Thispaper

84

771.55

28.45

0.89

890.0181

181

00

6.252

105

Tlili

etal.[12]

4/

832.53

67.47

1100

0.0262

262

//

8116.25

900

Thispaper

63

782.21

17.79

1.0003

100.03

0.0181

181

00

7.002

109.4

06Tlili

etal.[12]

2/

782.24

17.76

1.08

108

0.02

200

//

7.1123.94

1100

Thispaper

41

782.23

17.77

1100

0.0181

181

00

7114

.1002

Tlili

etal.[12]

2/

599.57

0.43

1100

0.007

70/

/5

134.37


Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

The research described in this paper has been funded by theNational Natural Science Foundation of China (Grant no.71302053).

References

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[8] C. G. Vassiliadis and E. N. Pistikopoulos, “Maintenance-basedstrategies for environmental risk minimization in the processindustries,” Journal of Hazardous Materials, vol. 71, no. 1–3, pp.481–501, 2000.

[9] H. Chouikhi, A. Khatab, and N. Rezg, “A condition-basedmaintenance policy for a production system under excessiveenvironmental degradation,” Journal of Intelligent Manufactur-ing, vol. 25, no. 4, pp. 727–737, 2014.

[10] J. Yan and D. Hua, “Energy consumptionmodeling for machinetools after preventive maintenance,” in Proceedings of the IEEEInternational Conference on Industrial Engineering and Engi-neering Management (IEEM ’10), pp. 2201–2205, IEEE, Macao,China, December 2010.

[11] H. Chouikhi, S. Dellagi, and N. Rezg, “Development andoptimisation of a maintenance policy under environmentalconstraints,” International Journal of Production Research, vol.50, no. 13, pp. 3612–3620, 2012.

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