We consider production systems which generate damage to environment as they get older and degrade. The system is submitted to inspections to assess the generated environmental damage. The inspections can be periodic or nonperiodic. In case an inspection reveals that the environmental degradation level has exceeded the critical level
Because of the degradation of the environment in the world and the growing public pressure, governments imposed several constraining and penalizing measures to companies whose production processes potentially generate any form of environmental damage. These measures have been taken globally (Kyoto Protocol in 1997) and nationally. Moreover, the companies’ greenhouse gas emissions (primarily CO_{2}) are severely limited and penalized financially. These constraints led decisionmakers to establish and implement effective environmental sustainability policies considering environmental and cost criteria.
Numerous efforts have emerged to address the environment issues related to environmental impact assessment within the company. A set of qualitative tools such as FMEA [
The excess of environmental damages generated by production systems is, in numerous situations, caused indirectly by the deterioration of those systems (e.g., degradation and/or corrosion yielding leakages, wear causing excess of energy consumption, etc.). For instance, in a nuclear power plant, the degradation of the mechanical shaft seal of the refrigeration compressor induces toxic refrigerant leakages [
Therefore, when it is technically possible, conditionbased monitoring, repair, and maintenance are the most appropriate activities to be adopted. It allows monitoring the degradation level and the resulting environmental damage in order to take the appropriate preventive actions and limit the risk of penalties and sometimes catastrophes. In the literature, two approaches are proposed to monitor the degradation: continuous monitoring [
Golmakani and Fattahipour [
Recently, Chouikhi et al. [
To make conditionbased maintenance more effective, both the inspection sequence and the alarm threshold level of degradation need to be optimized. In this context, Grall et al. [
In the same context of conditionbased policies optimizing alarm threshold levels and inspection schedules, we focus in this paper on production systems whose degradation generates directly (or indirectly) environmental damage. A conditionbased maintenance strategy is proposed for such systems considering two threshold levels related to the amount of environmental damage: a critical one,
The remainder of this paper is organized as follows. The main notations used in the mathematical model are presented in Notations. Other notations will be introduced throughout the following sections. The next section is devoted to the modeling of the environmental degradation process. The detailed problem description and the definition of the assumptions are given in Section
Stochastic processes are appropriate to model the degradation process involving independent increments. Many stochastic processes have been studied in the literature (for more details see [
The Wiener process (also called Brownian motion process) had firstly been used to model the irregular motion of the pollen particles floating in water. The mathematical theory of Brownian motion was discussed in detail in Wiener’s dissertation in 1918 and in papers that followed. The theory was completed by Lévy, Ito, McKean, and others [
The Wiener process as a degradation model is based on the consideration that the degradation increment in an infinitesimal time interval might be viewed as an additive superposition of a large number of small external effects [
Suppose that the degradation process,
For any time sequence
The paths of
The failure time is defined as the first hitting time (FHT) which is the time from present time to the instant
We define the remaining useful life (RUL) of a deteriorating system associated with the degradation process given that the degradation level
Based on property (b) of the Wiener process, the degradation increments are independent; hence, if
We consider a production system assimilated to a single unit that causes a random amount of damage to environment as it gets older and degrades. It is assumed that the environmental degradation process is modeled by Wiener process. The value of the environmental degradation level (damage) can be known (measured) only by inspection. Hence, the system is submitted to inspections to assess the generated environmental damage. In case an inspection reveals that the environmental degradation level has exceeded a critical level
The conditionbased maintenance strategy.
The following working assumptions are considered.
The degradation of the system induces the degradation of the environment.
Inspections are perfect and their duration is negligible.
The durations of PM and CM actions are also negligible.
After each inspection, only one of the three following events is possible: do nothing, perform a PM action, or perform a CM action.
Both PM and CM actions renew or bring back the system to a state as good as new.
All costs related to maintenance, inspection, and environmental penalty are considered as average costs. They are known and constant.
The resources necessary for the achievement of maintenance actions are always available.
In this section, we assume that periodic inspections are performed at times
By using classical renewal arguments, the total average cost per time unit can be expressed over a renewal cycle
Hence, the expression of the longrun average cost per unit of time is given by
The following analysis will lead to the expression of the average longrun cost per time unit.
The expected total cost during a cycle
The analytical expressions of these different components of the total expected cost are developed below.
Example of cycle that ends by CM action.
Example of cycle that ends by PM action.
Performing
The duration of generation of excess amount of damage.
The cycle is considered to be the interval between consecutive maintenance activities either PM or CM. Therefore, the expected cycle length is given as follows:
Hence, the average longrun cost rate function
It is expressed below as a function of the decision variables which are the inspection period
Consider a production system subject to continuous CO_{2} gas emissions (it could be any other source of environmental damage). We suppose that the amount of environmental damage generated
The following input parameters of the problem have been arbitrarily chosen. The parameters of the process’s pdf are
Input data.






900  500  100  10000  10 
Hence, the pdf of the FHT for the critical level
The obtained optimal solution for periodic inspection policy.



CM cost ($) 

PM cost ($) 

Inspection 

Environmental 





17.76  160  82.24  412  1.08  108  0.02  200  7.1  123.94 
Numerical procedure for the periodic strategy.
From the results presented in Table
In what follows, while keeping the original combination of input parameters, the unitary cost of environmental penalty
Effect of




CM cost ($) 

PM cost ($) 

Inspection cost ($) 

Environmental 






61.43  553  38.57  193  1.44  144  7.1  0  10.06  88.47 



17.76  160  82.24  412  1.08  108  0.02  200  7.1  123.94 



0.43  3.87  99.57  498  1.00  100  7 10^{−3}  700  5  260.37 
Effect of




CM cost ($) 

PM cost ($) 

Inspection cost ($) 

Environmental 






0.21  2  99.79  499  6.65  0  8 10^{−6}  8 10^{−2}  6.65  75.35 



17.76  160  82.24  412  1.08  108  0.02  200  7.1  123.94 



95.25  857  4.75  24  1.00  1000  1.13  11300  9  1464.56 
Effect of




CM cost ($) 

PM cost ($) 

Inspection 

Environmental 






67.47  405  32.53  163  1.00  100  0.0262  262  8  116.25 



17.76  160  82.24  412  1.08  108  0.02  200  7.1  123.94 



0.43  3.87  99.57  498  1.00  100  7 10^{−3}  70  5  134.37 
From Table
From Table
In case of costly inspections, interinspection interval is high (less frequent inspections), the PM threshold level is reduced, and the cycle is most likely to finish with a corrective action and an important penalty.
Finally, Table
In this section, it is assumed that the system is submitted to nonperiodic inspections to assess the generated environmental damage. Let
The expression of the expected total cost per time unit is developed as follows.
The probability that the cycle ends with a corrective maintenance action is
The probability that the cycle ends with a preventive maintenance action is
The expected number of inspections during a cycle is
The average time of generation of excess amount of environmental damage during a cycle is
The expected renewal cycle length is
By collecting (
The developed procedure is based on the NelderMead algorithm. It is presented in Figure
Numerical procedure for the nonperiodic strategy.
It consists in an iterative procedure comparing the values of the objective function at the (
Sort the simplex vertices according to the function value at that point:
Compute the centroid
Compute the reflection point
Compute new
The convergence of the algorithm is achieved when the standard deviation of the objective function at the (
The standard values of NelderMead parameters are chosen to be
In the numerical procedure described in Figure
We consider the same example of Section
Given the above input parameters, we applied the procedure of Figure
The obtained nearly optimal solution for the nonperiodic inspection policy.



CM cost 

PM cost 

Inspection cost 

Environmental 





6.27  56.41  93.73  468.67  1.09  109  0.0039  39  6.7  100.46 
Note that the first inspection instant is 6.6 weeks. We give only the first three values in this particular example because the cycle average duration is 6.7 weeks and the average number of inspections per cycle is 1.09. Therefore, having more than three inspections per cycle is nearly impossible.
PM action should be performed whenever an inspection reveals that the environmental degradation level has exceeded 2 tons of CO_{2} gas emissions. By adopting this strategy, it would cost in average a total of 100.46 $/week.
In comparison with the periodic inspection policy, one can notice that, in the case of nonperiodic inspections, the probability that the cycle ends with a CM action and the average period of excess CO_{2} gas emissions are lower. This indicates that with the nonperiodic inspection policy PM actions are more likely to be performed in order to avoid the exceeding of the critical level and therefore the reduction of the emission of excess damage to environment.
Moreover, the nonperiodic inspection policy has the lowest average cost rate due to the reduction of the expected CM cost as well as the environmental penalty cost. Hence, even with a nearly optimal solution, the nonperiodic inspection policy is more economical than the periodic inspection policy. This can be explained by the fact that, in the case of the nonperiodic strategy, sequential inspections are scheduled in accordance with the evolution of environmental damage generation.
In this paper, we have considered a conditionbased maintenance policy for production systems which degrade as they get older and generate environmental damage. We have proposed a new modeling approach based on the fact that the degradation process is modeled by the Wiener process. Thus, the first hitting time and the remaining useful life of the system conditional to its environmental degradation level are considered. Moreover, two types of inspection policies, periodic and nonperiodic, can be used to reveal the level of environmental damage and act consequently. According to the observed amount of environmental degradation at each inspection, one decides to undertake or not maintenance actions (PM or CM) on the system.
CM action is undertaken following inspections that reveal the exceeding of a known critical level of environmental degradation. In such situation, an environmental penalty is incurred due to the excess amount of environmental degradation generated. To prevent such event, a lower threshold level has to be considered to trigger a PM action to renew the system at a lower cost and without paying the penalty. This lower threshold level and the inspection schedule were considered as the decision variables.
For the two proposed inspection policies, the total expected cost per time unit has been mathematically modeled and the optimal inspection schedules and PM threshold levels were derived using two numerical procedures.
The developed conditionbased preventive maintenance models permit highlighting the role of preventive maintenance in reducing environmental damage and its consequences that could be caused by the degradation of production systems. The proposed models can be relatively easily used by decisionmakers in the perspective of implementing an effective green maintenance.
This work can be improved in several ways. First, it would be of interest to consider situations in which maintenance actions durations are not negligible and where resources are not always immediately available to perform maintenance actions. Moreover, in many real situations, production systems may not generate only a single damage to environment, but several kinds of damages at the same time at different rates with different impacts. It would be interesting to investigate these issues taking the present model as a start.
Critical level of environmental degradation
PM threshold level of environmental degradation
Preventive maintenance (PM) action cost
Corrective maintenance (CM) action cost
Penalty cost per time unit related to excess environmental damage incurred once the critical level
Inspection cost
The time at which the environmental damage of the system exceeds the critical level
The time at which the environmental damage of the system exceeds the PM threshold level
The system renewal cycle length within the periodic (nonperiodic) inspection policy; a cycle is the time between consecutive maintenance actions (either preventive or corrective)
Probability density function (pdf) and cumulative distribution function (cdf) associated with
pdf and cdf associated with
Discrete random variable associated with the total number of inspections during a cycle within the periodic (nonperiodic) inspection policy
The th inspection instant
Inspection instants sequence
Probability that the cycle ends with a PM action within the periodic (nonperiodic) inspection policy
Probability that the cycle ends with a CM action within the periodic (nonperiodic) inspection policy
The expected number of inspections during a cycle in the case of the periodic (nonperiodic) inspection policy
Period during which excess of damage is generated between the instant when the amount of environment degradation exceeds the critical level
The longrun average cost per unit of time corresponding to the periodic inspection policy
The longrun average cost per unit of time corresponding to the nonperiodic inspection policy.
The authors declare that there is no conflict of interests regarding the publication of this paper.