A Condition-Based Opportunistic Maintenance Policy Integrated with Energy Efficiency for Two-Component Parallel Systems

Purpose: This paper deals with the problem of traditional maintenance model ignoring energy consumption in two-component parallel systems. Thus, the aim of the article is to propose a new maintenance model with ecological consciousness for two-component parallel systems, which can improve the energy utilization and achieve sustainable development. The objective is to obtain the optimal maintenance policy by minimizing total cost. Design/methodology/approach: This paper integrates energy efficiency into condition-based maintenance (CBM) decision-making for two-component parallel systems. Based on energy efficiency, the paper considers the economic dependence between the two components to take opportunistic maintenance. Specifically, the objective function consists of traditional maintenance cost and energy cost incurred by energy consumption of components. In order to assess the performance of the proposed new maintenance policy, the paper uses Monte-Carlo method to evaluate the total cost and find the optimal maintenance policy. Findings: Simulation results indicate that the new maintenance policy is superior to the classical condition-based opportunistic maintenance policy in terms of total costs. Originality/value: For two-component parallel systems, previous researches usually simply establish a condition-based opportunistic maintenance model based on real deterioration data, but ignore energy consumption, energy efficiency (EE) and their contributions of sustainable development. This paper creatively takes energy efficiency into condition-based maintenance (CBM) decision-making process, and proposes a new condition-based opportunistic maintenance policy by using energy efficiency indicator (EEI).


Introduction
For fear of sudden failure, most companies are willing to repair/replace their components before breakdown.Garg, Rani, and Sharma (2013) simultaneously consider mechanical service, repair and replacement in periodic preventive maintenance.However, this maintenance policy only focuses on the fixed preventive maintenance interval, but ignore the real deterioration level of components.Nowadays, condition-based maintenance is used extensively in various industries, and companies always regard this method highly to ensure maintenance before breakdown.Although the corporations give high priority to the periodic preventive maintenance or condition-based maintenance, they ignore the fact that maintenance activities can strengthen or weaken the ecological burden exerted by system.Horenbeek, Kellens, Pintelon, and Duflou (2014) points out that energy, resources and environment all belong to the category of ecology.It came to be a common situation that companies take blind eyes to the machines in bad condition, since they will not maintain the components when they can still work, even if these components are gradually in poor state which will increase energy consumption.For instance, a ship company, when encounters motor boilers, blowers, belt conveyor belt relaxation and induced draft fan adjustment door open and closed out of work, will not conduct instant maintenance for the purpose of saving money, if these devices mentioned above incur small problems but they can still on operation.However, the negative point is that continuous uses without maintenance will seriously affect the key indicators alpha value of the combustion conditions (ie, air excess coefficient).In general, when the value is too large, the fan energy consumption increased sharply.
With the enhanced awareness of energy conservation, a new trend concerning about saving energy, protecting the environment, and achieving sustainable development is prevalent in modern society.In actual industrial production, a great amount of energy, such as electricity, etc. needs to be consumed to maintain machines' normal operation, and if the system cannot work in good condition, there would be much more energy loss in the manufacture process and thus pose a huge burden on the whole ecology.Thus, for most of the enterprises, it should be the long-term strategic focus how they can apply valid maintenance activities to the improvement of system's resource utilization and establishment of a green image.Traditional maintenance mainly focuses on controlling maintenance fee at a low level and keeping the reliability of the system at a high level, but ignores to avoid excessive energy consumption under operation.Therefore, it is imperative to consider the energy consumption of system under operation when developing a maintenance policy.
The rest of the paper is structured as follows: In section 2, several related research literatures are reviewed.Section 3 presents degradation model of individual component and introduces the energy efficiency indicator.Section 4 proposes the new condition-based opportunistic maintenance policy integrated with energy efficiency (EE).Section 5 conducts numerical experiments to testify the advantage of new proposed policy by comparing new proposed and classical maintenance policy.Finally, conclusions and future work are stated in Section 6.

Literature Review
The reliability and maintainability are the critical factors for maintenance activities, because their purpose is to estimate the probability of failure.Therefore, some scholars are interested in studying the reliability and failure function.For example, Garg and Sharma (2012) propose a two-phase approach to get more precise distribution parameters of reliability, failure rate and repair rate.Maintenance plays a significant role in an industrial system to ensure normal operation.However, most industrial systems are rather complicated and they have some subsystems with components.In order to save money, time and manpower, managers are suggested using three indicators simultaneously, including reliability, availability and maintainability, to find the critical components that affects the performance of the entire system mostly (Garg, 2014a,b).With the development of the sensor technology, condition-based maintenance (CBM) has been considered a more efficient strategy (Ahmad, 2012;Jardine, Lin, & Banjevic, 2006), as CBM is based on the actual conditions of a component monitored through sensors.Over the past few years, the research on the CBM decision-making of multi-component systems has developed vastly.In the study of multi-component systems maintenance, components may depend on each other in three different ways including stochastic dependence, economic dependence and structural dependence (Dao & Zuo, 2017).
The economic dependence provides opportunities to maintain several components jointly, and thus reducing high fixed set-up costs (such as sending a maintenance team to the site, which incurred once maintenance action is performed on one component).Several literatures have pointed out that group maintenance actions can reduce the total costs of the system (Bouvard, Artus, Bérenguer, & Cocquempot, 2011;Qian & Wu, 2014;Tian, Jin, Wu, & Ding, 2011;Tian & Liao, 2011;Zhang, Zhou, Sun, & Ma, 2012).When considering economic dependence, it is sensible to take group or opportunistic maintenance policies into account.Opportunistic maintenance can be recognized as dynamic group maintenance, because workers will not maintain non-failed components unless they are in opportunistic zone (Cavalcante & Lopes, 2014).Compared with group maintenance, opportunistic maintenance can reduce the waste of maintenance resources.
Several papers have proposed opportunistic maintenance policy in CBM in terms of economic dependence.Wijnmalen and Hontelez (1997) is the first one to propose a condition-based opportunistic maintenance policy based on deterioration level of components.It points out that if one component is going to repair, group maintenance actions only occur when the deterioration level of another component is over its opportunistic threshold.Barbera, Schneider, and Watson (1999) proposes a condition-based opportunistic maintenance policy based on exponential failures for a two-component series system.And finally, the long-term average cost is minimized by dynamic programming.When comes to a two-component series system, Castanier, Grall, and Berenguer (2005) also formulates a condition-based opportunistic maintenance policy, in which the deterioration level can be obtained by non-periodic inspections.It points out that the degradation process of the system after maintenance has the semi-regenerative properties and this policy finally establishes a minimum cost rate model based on the semi-renewal theory.For a two-component parallel system, Li, Deloux and Dieulle (2016) studies a condition-based maintenance policy considering both stochastic and economic dependence among components.It presents a new decision rule which permits the maintenance grouping in advance or postponed maintenance.Zhu, Peng, and Houtum (2015) presents an optimal CBM policy that minimizes the long-term mean maintenance cost per unit time for a multi-component system with continuous deterioration.They point out that it is more economical to preventively maintain several components simultaneously.Finally, the advantage of the presented policy is analyzed by a three-component series system of wind turbine.In recent years, some scholars intend to optimally plan maintenance activities for multi-component systems based on prognostic information and presents a dynamic predictive maintenance policy for multi-component systems (Bian & Gebraeel, 2014;Nguyen, Do, & Grall, 2014;Shi & Zeng, 2016).Furthermore, Jiang, Duan, Tian, and Wei (2015) and Keizer, Teunter and Veldman (2016) propose a condition-based opportunistic maintenance policy for redundancy systems separately.Jiang et al. (2015) establishes a reliability analysis model of the system with the basis of the remaining useful life (RUL) of components.This real-time sampling can reduce unnecessary preventive maintenance costs and high fixed maintenance costs of components, thereby increasing the efficiency of the whole redundant system.Finally, three numerical examples are used to verify the feasibility and flexibility of the model.Keizer et al. (2016) points out that in redundancy systems, the maintenance of some failed components can be postponed, under the condition that the reliability of the whole system is not reduced.Finally, the optimal maintenance policy can be obtained by dynamic programming.Despite the single-objective model above, some scholars are inclined to multi-objective optimization.Garg (2016) presents a method for obtaining the optimum maintenance interval by considering maximum availability and minimum maintenance cost.For a series system, Garg and Sharma (2013) consider the maximum reliability of system and minimum design cost as the two objectives.Then, they solve the reliability allocation problem of subsystems by fuzzy nonlinear programming.Based on the two objectives, Garg, Rani, Sharma, and Vishwakarma (2014a) allocate the reliability for a series-parallel system.As well known, reliability and cost parameters are under uncertainties normally in design phases.Therefore, under the condition that parameters are imprecise, Garg, Rani, Sharma, and Vishwakarma (2014b) utilize intuitionistic fuzzy programming techniques to solve a multi-objective reliability optimization problem.
In summary, traditional condition-based maintenance strategies merely focuses on reducing maintenance costs.Although these strategies ensure the safety and reliability of the system, they ignore the ecological impacts.Additionally, the fact that the maintenance activities can increase or reduce the ecological impacts resulted from system's normal operation is ignored.Generally, reasonable maintenance activities can reduce the ecological impact caused by the system itself, otherwise the maintenance activities can increase the ecological burden.However, with the increasingly serious problems of energy and environment, several literatures of CBM with ecological awareness have been proposed over the past decade.Hoang, Do, andIung (2016a, 2017) point out that maintenance activities (such as maintenance time, preventive or corrective maintenance, etc) can be influenced when considering some ecological aspects.Mora, Vera, Rocamora, and Abadia (2013) and Xu and Cao (2014) conclude that different maintenance actions on deteriorated components can lead to different effect on ecology.Horenbeek et al. (2014) is the first one to integrate ecological factors in a maintenance optimization model.This model can be considered as an ecological analysis tool which covers many maintenance strategies (such as failure-based, block-based and usebased maintenance).Then, the presented model, which concerns an integrated economy and ecology, can determine the optimum maintenance interval.Chouikhi, Dellagi, and Rezg (2012) propose a CBM model from the respect of probability and statistics and combine environmental problems with maintenance activities.Preventive maintenance takes place when the amount of released refrigerant gas exceeds the alarm threshold and this action helps enterprises avoid a large penalty cost caused by excessive refrigerant gas release.Tlili, Radhoui, and Chelbi (2015) has made a further improvement in 2015.It proposes a preventive maintenance threshold value below the alarm threshold, which can avoid a huge economic penalty more effectively.In addition to the environmental problems (the release of refrigerant gas, etc.), some scholars also consider energy consumption in CBM decisionmaking.Hoang et al. (2016b) proposes a CBM model for a single-component system considering the energy consumption, in which maintenance activities can be performed based on the energy efficiency (EE) of the components.This model aims to minimize the total cost including maintenance cost and energy consumption cost.Nevertheless, literatures, in which CBM model stresses energy consumption during components' normal operation for multi-component systems are absent till now.In view of this situation, we propose a new condition-based opportunistic maintenance model integrated with energy efficiency for a two-component parallel system in this paper.

Degradation Model and Energy Efficiency Indicator
Throughout this paper, we consider a two-component parallel system.The energy consumption of the system is the sum of energy consumption of individual component.The two components have their own different energy consumption process incurred by different degradation process of them.

Notations
Some notations used in this paper are summarized as follows: The energy consumption during one-time unit of component i (from t to t + 1), i = 1,2 EEI i (t ): The energy efficiency indicator during one-time unit of component i, i = 1,2 EEIL i : Corrective maintenance threshold for component i in new proposed maintenance policy, i =1,2 The cumulative useful output of the whole system during the period (0, t ] C t : Cumulative total cost during the period (0, t ] IC t : Cumulative inspection cost of the whole system during the period (0, t ] MC t : Cumulative maintenance cost of the whole system during the period (0, t ]

Degradation Model
The Gamma process has been successfully selected to describe the gradual degradation process of individual component in different industrial systems (Dieulle, Bérenguer, Grall, & Roussignol, 2003;Noortwijk, 2009).The Gamma process has several characteristics as follows: • X i (t ) has independent increments, • For t > τ > 0, the increment of deterioration level for component i X i (t ) -X i (τ) follows a Gamma probability density function with the shape parameter α i (t -τ) and the scale parameter β i : (1) Given α 1 = 1, β 1 = 1 and α 2 = 1, β 2 = 2, then the degradation process of two components are illustrated in Figure 1.

Energy Efficiency Indicator
Hoang et al. ( 2016b) has pointed out that energy consumption of component i varies over time and it depends on the degradation level and the running speed.In order to ensure the stable operation of the whole system, we need to control the running speed of different components (constants in this paper).Therefore, the energy consumption during one-time unit depends only on the deterioration level X(t ): (2) Suppose that S 1 = S 2 = 200, the energy consumption process of the two components are illustrated in Figure 2.
Similar to Figure 1, Figure 2 shows that the energy consumption rate of component 1 is lower than that of component 2. However, the energy consumption rate of each component is growing over time.This means that there would be much more energy consumption in the manufacture process if the components are in bad condition, even if they can still run.Therefore, we propose a new maintenance policy integrated with energy efficiency for a two-component parallel system.The useful output during one-time unit of each component depends on its running speed as follow: (3) Therefore, the useful output during one-time unit of each component is constant.
Energy efficiency indicator (EEI) has been introduced in maintenance decision-making.EEI represents the amount of energy consumption needed to produce one useful output.Its mathematical expression is as follow: (4) Hence, energy efficiency indicator also depends on the deterioration level of components: (5)

Maintenance Policy
In this section, we intend to propose a new condition-based opportunistic maintenance policy integrated with energy efficiency for a two-component parallel system.The policy is based on the following assumptions: 1.A component is not affected by the degradation of another and only the economic dependence between them is considered in this paper.
2. Inspection tasks for each component are only performed with its own fixed interval and their duration can be negligible.
3. An inspection cost of energy consumption is not higher than a deterioration inspection cost .6.Both preventive and corrective maintenance are assumed to be perfect and the maintenance time can be ignored.
In general, both the energy consumption and deterioration level of each component can be monitored.However, Hoang et al. (2016b) point out that measuring the current deterioration level of a component is more complicated than inspecting the current energy consumption, therefore, we also assume in condition-based opportunistic maintenance policy as Hoang et al. (2016b).But we also make the additional sensitivity analysis with various and in 5.2.1 section.
In case an inspection indicates that the energy efficiency indicator of an component exceeds its legislation, the component would be considered in a rather serious state and will generate large energy consumption which would lead to a penalty, and then a corrective maintenance is performed.In order to reduce the chances of being in such serious situation, a preventive maintenance threshold lower than a legislation is considered.It should be noted that both preventive and corrective maintenance are instantaneous as Li et al. (2016).In order to comparatively analyze the minimum long-run expected cost of system per useful output unit for new proposed maintenance policy in this paper and for traditional maintenance policy as in Li et al. (2016), we also assume that the maintenance time can be negligible in this paper.

Long-Run Expected Cost of System Per Useful Output Unit
In order to assess the effectiveness of new proposed maintenance policy, we use the long-run expected cost of system per useful output unit as the objective function: (6) The cumulative useful output of the whole system during period (0, t ] is calculated as: (7) Where O 1 and O 2 is the useful output during one time unit of component 1 and component 2 respectively.
During the period (0, t ], cumulative cost of the whole system includes cumulative inspection cost, maintenance cost and energy cost: (8) Where: An inspection cost of energy consumption is trigged by each inspection operation and thus IC t depends on the number of inspection.The cumulative inspection cost of the whole system includes the inspection costs of both two components: Similarly, MC t is related to the number of both preventive and corrective maintenance of each component.A setup cost and an individual maintenance cost are both generated once each maintenance (preventive or corrective) is performed.Thus, the cumulative maintenance cost of the whole system is written as: (10) EC t is the product of energy price and the total amount of energy consumed by the whole system: (11) Therefore, the cumulative cost of the whole system is the sum of Equations ( 9), ( 10) and ( 11), so it can finally be expressed as: (12) In industry, taking maintenance actions of two components jointly is more economical than repairing them separately.When a component is repaired, the cost of maintenance (both preventive and corrective) includes two parts: a fixed set-up cost c s shared by two components and individual maintenance cost of each component ( or ).Set-up cost refers to the cost resulted from the same maintenance preparation, such as maintenance devices, technology, workers and so on.Therefore, two individual preventive or corrective maintenance incur two set-up costs c s while grouping maintenance of two components only incurs single c s .Thus, a set-up cost can be saved when two components are maintained simultaneously, which can ultimately reduce total maintenance cost and improve the economic benefit for a company.In order to make the best use of economic dependence, it is necessary to find an opportunity to postpone the preventive maintenance of one component or repair the other in advance.Hence, the cumulative cost of the whole system can be reduced by a set-up cost multiplied by the number of grouping maintenance (N GM ) as follows: (13)

A New Opportunistic Maintenance Policy Integrated with Energy Efficiency (Policy 0)
Traditional condition-based opportunistic maintenance policies do not consider energy consumption during the operation of components.In response to the enhanced awareness of energy conservation over the whole world, this paper proposes a new condition-based opportunistic maintenance decision rule by using energy efficiency indicator (EEI ).
Since the actual energy consumption of components can be achieved by inspection (then EEI is calculated), we can consider the inspection time as the decision moment.Suppose that the current time is nΔT 1 (the moment of nth inspection for component 1).If the energy efficiency indicator of component 1 EEI 1 (nΔT 1 ) exceeds its preventive maintenance threshold, a maintenance action (preventive or corrective) can be performed for component 1 according to CBM decision rule.On the one hand, once the energy efficiency indicator of component 2 EEI 2 (nΔT 1 ) also reaches its preventive maintenance threshold at the same time, then we can take the preventive maintenance for component 2 in advance.On the other hand, the time of next inspection (m + 1)ΔT 2 (the moment of (m + 1)th inspection for component 2) is very close to nΔT 1 (see Figure 3).Pp 2 = P(EEI 2 (nΔT 1 ) ≥ EEIM 2 | EEI 2 (mΔT 2 ) = a) represents the probability that the energy efficiency indicator of component 2 reaches its preventive threshold at current time nΔT 1 , under the condition that the energy efficiency indicator of component 2 does not reach its preventive maintenance threshold at its last inspection time mΔT 2 (a < EEIM 2 .According to Equation ( 5), the value of Pp 2 can be obtained by firstly transforming energy efficiency indicator of component 2 into the corresponding deterioration level, as X 2 (t ) = f -1 (t(EEI 2 (t )).Pp 2 is related to the last energy efficiency indicator of component 2 and it is calculated as follows: (14) Where X 2 EEI2(nΔT1) -X 2 a follows Gamma distribution with parameter (nΔT 1 -mΔT 2 )α 2 and β 2 .Then according to Equation ( 1), Pp 2 can be calculated as follows: (15) is the probability that the energy efficiency indicator of component 1 will exceed its legislation at the next inspection time (m + 1)ΔT 2 and the current EEI of component 1 is denoted by b.Similar to Pp 2 , the value of Pc 1 is computed as follows: (16) Li et al. (2016) points out that PP and PC, the threshold of Pp 2 and Pc 1 , are difficult to compute.Therefore, PP and PC will be set in advance and they are optimized through simulation.
From Figure 4, we conclude the following six maintenance activities: • No opportunity of grouping maintenance: 1.If EEI 1 (nΔT 1 ) < EEIM 1 , no maintenance activity will be carried out and we need to wait for the next inspection.This means that component 1 still operates well and does not cause a great amount of energy consumption.
Pp 2 represents the probability that component 2 needs a preventive maintenance at current time.
Obviously, taking a maintenance action of component 2 too early will increase maintenance cost (Pp 2 is too small).Therefore, no maintenance action will take place when Pp 2 ≤ PP.It's worthy to take group maintenance only when Pp 2 is large enough (Pp 2 > PP).If Pp 2 → 1, component 2 will be repaired immediately at the next inspection time (m + 1)ΔT 2 .In this case, a preventive maintenance for component 2 is carried out in advance, in order to save one set-up cost c s and avoid the energy efficiency indicator of component 2 beyond its legislation before next inspection time.
• Opportunity of grouping maintenance is identified: 4. If EEI 1 (nΔT 1 ) ≥ EEIL 1 and Pp 2 > PP, a corrective maintenance is performed for component 1 and the preventive maintenance is preempted for component 2.
It's inadvisable to postpone maintenance for component 1 when the energy efficiency indicator of component 1 exceeds its legislation.This indicates that the current energy consumption of component 1 is too large, thus component 1 must be maintained immediately.Therefore, we need to take a corrective maintenance for component 1 and preempt the preventive maintenance for component 2 at current time.
Pc 1 is the probability that the energy efficiency indicator of component 1 will exceed its legislation at the next inspection time (m + 1)ΔT 2 .An unreasonable postponed maintenance (when Pc 1 is large) maybe take a risk that the energy efficiency indicator of component 1 reaches its legislation before repaired at the next inspection time.If Pc 1 → 0, then the energy efficiency indicator of component 1 will almost be lower than its legislation before the next inspection time.Therefore, the maintenance of component 1 can be postponed only when Pc 1 is small enough (Pc 1 < PC).
Specifically, if Pp 2 is small enough, it is useless to postpone maintenance for component 1, because component 2 would hardly be repaired at the next inspection time (m + 1)ΔT 2 .In such case, grouping maintenance will rarely take place at the next inspection time even if the maintenance of component 1 is postponed.Therefore, we should check the condition Pp 2 > PP at first.

Numerical Experiments
This section will compare the proposed Policy 0 with the classical maintenance policy (Policy 1) in order to testify the superiority of Policy 0. We use Monte-Carlo method to imitate the operation process and obtain the optimum decision variables.According to Li et al. (2016), the maintenance decision rule of Policy 1 is similar to that of Policy 0. But the nature of Policy 1 is that all maintenance activities in this policy are determined by deterioration level of components, whereas Policy 0 is based on energy efficiency.Finally, the long-run expected cost of system per useful output unit of Policy 0 and that of Policy 1 are both estimated by Equation ( 6).However, it is significant to notice that the inspection cost of Policy 1 depends on deterioration inspection operation ( ), instead of energy consumption inspection cost ( ) in Equation ( 13).Detailed description of Policy 1 is shown in Appendix A.

Simulation Analysis
All maintenance activities in Policy 0 are determined by the energy efficiency of components.According to Equation (4), the energy efficiency indicator of component i is a function of the energy consumption and the useful output during one time unit.The energy consumption can be estimated by the non-linear fitting of some historical data from different operating status and it is expressed as (Hoang et al., 2016b): (17) Without loss of generality, we define S i as a constant in the range of 400 to 1200.
Meanwhile, the useful output during one time unit of component i is provided as (Hoang et al., 2016b): (18) Then the energy efficiency indicator of component i can be obtained based on Equations ( 4), ( 17) and ( 18).
In Policy 0, the decision variables are ΔT 1 , ΔT 2 , EEIM 1 , EEIM 2 , PP, PC.In Policy 1, the decision variables are ΔT 1 , ΔT 2 , M 1 , M 2 , PP', PC'.Total time should be large enough to ensure the number of inspection and acquire more reliable results.The parameters of two policies are shown in Table 1.
Table 2 shows costs of inspection and activities, all the costs are unit cost.
For the record of computing long-run expected cost of system per useful output unit, large amount of simulation is preceded.Different combinations of decision variables can lead to different results.The optimal long-run expected cost of system per useful output unit (CO * ) and decision variables of two policies are found and specifically shown in Table 3.
Table 3 shows that the optimal long-run expected cost of system per useful output unit of Policy 0 is 8.8046($/product ).And it also presents that the optimal long-run expected cost of system per useful output unit of Policy 1 is 9.4622($/product ), which is higher than that of Policy 0. Therefore, Policy 0 is superior to Policy 1 in terms of the long-run expected cost of system per useful output unit.Additional, the optimal decision variables of Policy 0 are ΔT 1 * = 9, ΔT 2 * = 8.5, PP * = 0.9, PC * = 0.05, EEIM 1 * = 2800(Wh/product), EEIM 2 * = 2750(Wh/product).The optimal decision variables of Policy 1 are ΔT 1 * = 10.5, ΔT 2 * = 9.5, PP' * = 0.88, PC' * = 0.06, M 1 * = 10.5, M 2 * = 9.Table 3 shows that PP * and PP' * are both close to 1.It also reveals that PC * and PC' * are both close to 0. In general, the value of PP * , PP' * should be large enough and the ideal value of them are 1.But the value of PC * , PC' * ought to be small enough and the ideal value of them are 0. Thus these results are consistent with actual situation.

Sensitivity Analysis
In order to discuss the performance of our new proposed condition-based opportunistic maintenance policy integrated with energy efficiency (Policy 0), various sensitivity analyses are carried out in the following.

Sensitivity Analysis to Inspection Costs
It is significant to conduct a sensitivity analysis with various and (i = 1,2).All simulation results for two policies are given in Table 4

Sensitivity Analysis to Running Speed
Concerning the simulation analysis and the sensitivity analysis on inspection costs above, we have concluded that Policy 0 performs better than Policy 1 in terms of the long-run expected cost of system per useful output unit.Since both the energy consumption and useful output are associated with the running speed, this section makes a sensitivity analysis of the running speeds for two policies.All the optimal long-run expected costs of system per useful output unit of two policies are given in Table 5.
As shown in Table 5, when S i changes, CO * of Policy 0 fluctuates between about 6.5 and 10.5 with , while corresponding CO * varies between about 7.0 and 11 with .However, CO * of Policy 1 fluctuates in the range of about 7.5 and 14.5 as S i changes.Thus it can be concluded that CO * of Policy 0 is relatively more stable because CO * of Policy 0 varies less than that of Policy 1.Therefore, we can conclude that the maintenance decision making by using EEI can lead to a better performance.
In addition, it is very clear that CO * of Policy 0 with is higher than that of Policy 0 with regardless of the values of running speed.
It is significant to notice that CO * of Policy 0 is always lower than that of Policy 1 when (i = 1,2).It's also true even if .In summary, Policy 0 is superior to Policy 1 because Policy 0 leads to the lower long-run expected cost of system per useful output unit.

Sensitivity Analysis to Deterioration Parameter
In the above sections, Policy 0 has been justified to perform better than Policy 1 in terms of the long-run expected cost of system per useful output unit.This section studies the impact of variation of degradation process denoted by β i on CO * in two policies.Table 6 provides all the simulation results.
According to Table 6, as β i (i = 1,2) rise, CO * of Policy 0 increases with various (i = 1,2), as well as CO * of Policy 1.When , , we find that Policy 0 always provides lower CO * than Policy 1 with varying β i (i = 1,2).It is significant to notice that the superiority of Policy 0 still exists even if = 10 and = 15.
In addition, an interesting outcome about the cost saving between Policy 0 and Policy 1 has been found.It is shown in the following Figure 5 and Figure 6.   Figure 5 shows that when β 2 is fixed, the cost saving increases as β 1 increases.Similarly, when β 1 is fixed, the larger β 2 the more cost saving is.Moreover, the finding is also suitable for the cost saving when = 10 and = 15, seeing Figure 6.
Figure 6 reveals clearly that when β 2 is fixed, the cost saving increases gradually as β 1 rises.And it also presents that when β 1 is fixed, the cost saving also increases as β 2 rises.Above all, larger variation of degradation process (high value of β i ) leads to a higher cost saving even if each energy consumption inspection cost equals to each deterioration inspection cost.To sum up, Policy 0 is superior to Policy 1 and the superiority of Policy 0 rises as β i increases.

Conclusion
In the past, some companies would not maintain the components when they could still work, even if these components were gradually in poor state which would increase energy consumption.With the enhanced awareness of energy conservation, it will be an inevitable trend to consider energy consumption in maintenance decision-making.Compared with classical maintenance policy (Policy 1), the new proposed maintenance policy (Policy 0) in this paper can save total cost including energy cost.The simulation results and sensitivity analyses show that Policy 0 always results a lower long-run expected cost of system per useful output unit than Policy 1, even if each energy consumption inspection cost equals to each deterioration inspection cost.In addition, Policy 0 always performs better than Policy 1 regardless of the value of the running speeds.Furthermore, larger variation of deterioration process leads to a higher cost saving of the whole system provided by Policy 0 compared to Policy 1.
In summary, the new proposed maintenance policy in this paper performs better than the existing maintenance policy, because the new policy can save total cost including both economic and ecological costs.The new proposed policy integrated with energy efficiency in this paper helps enterprises achieve sustainable development.Therefore, in order to establish green images for companies, managers need to consider both economic cost and energy consumption when they make maintenance plans.In addition, they can integrate energy efficiency into maintenance decision-making if possible.
Although the new proposed maintenance policy performs well, there still exists some limitations.With regard to two-component parallel system, this paper only considers the economic dependence.However, stochastic dependence exists in two-component parallel systems.Furthermore, only energy consumption is considered in this paper, which is one of the ecological impacts.
On the basis of this paper, both stochastic and economic dependence will be studied in the future.Our research will also focus on applying the proposed maintenance policy into a more complex system, such as series-parallel system.Other ecological impacts, such as carbon dioxide emissions, will be studied in the future.In addition, indicators that energy efficiency and any other ecological impacts will be taken into maintenance decision-making.This would be a rather interesting research field in the future.
EC t : Cumulative energy cost of the whole system during the period (0, t ] CO: Long-run expected cost of system per useful output unit : Each deterioration inspection cost of component i, i = 1,2 : Each energy consumption inspection cost of component i : A preventive maintenance cost for component i, i = 1,2 : A corrective maintenance cost for component i, i = 1,2 c s : A fixed set-up cost E p : The energy price : The number of preventive maintenance for component i, i = 1,2 : The number of corrective maintenance for component i : The number of inspection for component i : The number of grouping maintenance : The amount of energy consumption for component i during the period (0, t ] Pp 2 : The probability that the energy efficiency indicator of component 2 exceeds its preventive maintenance threshold at current inspection time Pc 1 : The probability that the energy efficiency indicator of component 1 exceeds its legislation at the next inspection time.PP: The opportunity maintenance threshold in new proposed maintenance policy and it is the threshold of Pp 2 PC: The postponed maintenance threshold in new proposed maintenance policy and it is the threshold of Pc 1 PP': The opportunity maintenance threshold in classical maintenance policy PC': The postponed maintenance threshold in classical maintenance policy CS: The cost saving, it is the difference between CO * of Policy 0 and that of Policy 1

Figure 1
Figure 1 reveals clearly that the deterioration level of each component increases gradually as the time flies.However, the degradation rate of component 1 is lower than that of component 2, because the scale parameter α i β i represents the degradation speed of component i and α 1 β 1 < α 2 β 2 .Generally, different couples of parameters(α and β) can generate different deterioration behaviors.

Figure 1 .
Figure 1.Degradation process of two components

Figure 2 .
Figure 2. Energy consumption process of components 4. A preventive maintenance action takes place for each component when the energy efficiency indicator exceeds its preventive maintenance threshold.The action has a specific maintenance cost for the component and a set-up cost c s . 5. If the energy efficiency indicator exceeds a legislation for component i, a corrective maintenance action is then performed.This action has a specific maintenance cost for the component and a set-up cost c s .

Figure 3 .
Figure 3. Sketch of inspection time

Figure 4 .
Figure 4. Decision rule of maintenance when component 1 is inspected at time nΔT 1

Table 1 .
Parameters for two policies

Table 2 .
Data of various cost parameters

Table 3 .
CO * and optimal decision variables for two policies

Table 4 .
. CO * and optimal decision variables of two policies with different inspection costs

Table 5 .
CO * of two policies with different S 1 and S 2

Table 6 .
CO * of two policies with different β 1 and β 2