Really ageing systems undergoing a discrete maintenance optimization

Citation: Briš, R.; Jahoda, P. Really Ageing Systems Undergoing a Discrete Maintenance Optimization. Mathematics 2022,10, 2865. https:// doi.org/10.3390/math10162865 Academic Editors: Gurami Tsitsiashvili and Alexander Bochkov Received: 15 July 2022 Accepted: 9 August 2022 Published: 11 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Really Ageing Systems Undergoing a Discrete Maintenance Optimization Radim Briš * and Pavel Jahoda Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB—Technical University of Ostrava, 708 00 Ostrava-Poruba, Czech Republic *Correspondence: [email protected] Abstract: In general, a complex system is composed of different components that are usually subject to a maintenance policy. We take into account systems containing components that are under both preventive and corrective maintenance. Preventive maintenance is considered as a failure-based preventive maintenance model, where full renewal is realized after the occurrence of every nth failure. It offers an imperfect corrective maintenance model, where each repair deteriorates the component or system lifetime, the probability distribution of which gradually changes via increasing failure rates. The reliability mathematics for unavailability quantification is demonstrated in the paper. The renewal process model, involving failure-based preventive maintenance, arises from the new corresponding renewal cycle, which is designated a real ageing process. Imperfect corrective maintenance results in an unwanted rise in the unavailability function, which can be rectified by a properly selected failure-based preventive maintenance policy; i.e., replacement of a properly selected component respecting both cost and unavailability after the occurrence of the nth failure. The number nis considered a decision variable, whereas cost is an objective function in the optimization process. The paper describes a new method for finding an optimal failure-based preventive maintenance policy for a system respecting a given reliability constraint. The decision variable nis optimally selected for each component from a set of possible realistic maintenance modes. We focus on the discrete maintenance model, where each component is realized in one or several maintenance mode(s). The fixed value of the decision variable determines a single maintenance mode, as well as the cost of the mode. The optimization process for a system is demanding in terms of computing time because, if the system contains kcomponents, all having three maintenance modes, we need to evaluate 3 k maintenance configurations. The discrete maintenance optimization is shown with two systems adopted from the literature. Keywords: unavailability; imperfect repair; failure-based replacement; renewal theory; optimization MSC: 60K10; 90B25 1. Introduction Various maintenance strategies have recently been intensively studied and developed to improve the reliability, availability and usability of relevant industrial systems. Unexpected failures may cause dangerous situations for human lives, unplanned production outages, etc. This is why systems must be protected against them. System reliability can be significantly improved by applying the optimal maintenance strategy. One can read in the work of [1] that maintenance is no longer a necessary evil and that production companies should invest in maintenance to maximize revenues. To find the best and most suitable strategy for every unit or subsystem, preferred economic decisions have to be made that make it possible to achieve a profit. Therefore, evaluation of the performances of different strategies is often used ([1,2]). Mathematics 2022,10, 2865. https://doi.org/10.3390/math10162865 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 2865 2 of 17 We can distinguish between the different maintenance strategies that can be used: corrective maintenance (CM) and preventive maintenance (PM) strategies. CM strategies, often known as restoration or repair strategies, are only launched when a failure occurs and the system is broken. The system is then returned to a functioning state by applying a maintenance action. PM strategies aim to prevent the system from undergoing undesired breakdowns. PM is usually carried out on operating systems and it reduces ageing processes, which means that the probability of system failures is decreased. Maintenance modelling is a dynamically developing recent scientific discipline, and we do not aim to present a general overview of references on maintenance here. However, several papers can be discussed, some of them dealing with CM, others with PM. For example, authors of [ 3 ] mention that there are several different types of PM and CM actions depending on the degree of restoration, including perfect maintenance action, which restores the system to an as-good-as-new function state; and imperfect maintenance action, which can have several restoration levels between perfect maintenance and minimal maintenance, with the latter restoring the system to the same state it was in just before the failure occurred, with the same failure rate that it had just before the minimal maintenance action. A repair action can gradually restore a system to its initial level, with the system being returned step-by-step to the operating state, resulting in perfect CM. This approach is in detail introduced for example in [ 4 ] and the authors call it a gradual CM repair strategy. In general, we can say that imperfect maintenance is a kind of intervention where the system is returned to someplace between being as good as new and as bad as it used to be. However, for any maintenance intervention, it is necessary to establish the level of imperfect maintenance. Authors in [ 5 ] introduced such imperfect restoration when they assumed that the system age is affected by the kind of maintenance intervention. Apart from the system age, the failure rate may also become worse due to maintenance actions ([ 6 ]). Both age shortening and failure rate adaptation are assumed in the so-called hybrid model, which seems to be a more realistic model (see more in [ 7 , 8 ]). Whereas the former authors addressed the optimal maintenance decision for binary systems to maximize the reliability of the next mission under imperfect maintenance, the latter introduced the imperfect maintenance model, where fixed maintenance action corrects a system functionality to any state between minimal repair and full renewal; i.e., perfect repair. For extensive discussions regarding imperfect maintenance models, readers may refer to authors in [9,10]. In this article, we use the imperfect CM process, which degrades the system lifetime at any CM intervention due to the growing failure rate. Gradual changes in the failure rate result in corresponding changes in the lifetime probability distribution. The basic aim of this article is to describe a realistic failure-based PM model that undergoes a process of discrete optimization. PM is considered a failure-based preventive maintenance model (FBM), where full renewal is realized at the occurrence of every nth failure. This model is related to the imperfect CM process, an overview of which is provided in [ 11 ]. For instance, a comparable strategy, where a unit is replaced at the nth failure and (n − 1) previous failures are repaired with minimal repair, was proposed in the cost-focused study [ 12 ]. However, the author supposed that the failure rate would not be violated after carrying out minimal repair. Stochastic models that describe the failure pattern of repairable units subject to minimal maintenance are discussed in [ 13 ]. The unit can be replaced at time T or at the nth failure, whichever occurs first, and ncan be minimized in the context with both repair and replacement costs, as is developed in [ 14 ]. We use the imperfect CM model, where each repair action deteriorates the component or system lifetime, the probability distribution of which is gradually changed via the increasing failure rate. This model is of particular interest for experts analyzing components of power distribution networks ([ 15 ]), the reliability of which will be studied by the authors of this article in the near future. Innovative reliability mathematics for the unavailability quantification of components is presented in this article. The new renewal process model, involving the FBM, is designated as a real ageing process. The imperfect CM results in an unwanted rise in the unavailability function, which can be rectified with a properly selected FBM process. This Mathematics 2022,10, 2865 3 of 17 means that the renewal of a component starts with the occurrence of the nth failure. The number nis considered a decision variable, whereas cost is an objective function in the optimization process. This article describes a new method that can be used to find an optimal FBM strategy to solve a particular optimization problem while respecting a given reliability constraint. The above-mentioned decision variable determining the different maintenance modes of a system component is optimally selected from a set of possible realistic maintenance modes. Thus, the discrete maintenance model is considered, where each component can work in one or several maintenance modes. The fixed value of the decision variable ndetermines one maintenance mode of the component, which predetermines both the unavailability course and cost. Different maintenance modes of system components result in different system configurations, each having a specific unavailability course, as well as cost. The optimization process is demanding in terms of computing time because a complex system can have many maintenance configurations. The discrete maintenance optimization is demonstrated with two systems adopted from the literature. 2. Optimization Problem Each optimization problem works on the presumption that an objective function f( x ) that varies in a given range must be optimized; i.e., either maximized or minimized, constrained by several restrictions imposed on the decision variables. The optimization problem in this article can be formulated using the following objective function f( x ), which represents the minimum cost: f(x) = min CS(1) subject to the constraint US(x)≤U0(2) where x = (x 1 , . . . , x k ) ∈Rk is a decision variable, and kis the number of system components each having the decision variable x i =n i , which can be optimized if needed. Each component undergoes a real ageing process, including imperfect CM, until the occurrence of the ni-th failure, which starts its restoration (renewal). In most cases, both f( x ) and U S ( x ) are complicated, either linear or nonlinear, functions of the decision variable vector: x= (x1, . . . , xk)=(n1, . . . nk) that constitute parameters for which optimal values must be found. 3. Discrete Maintenance Model Maintenance optimization can be classified in different ways. A recent thorough classification can be found in [ 16 ]. Concerning the optimization outcome and decision variables, the discrete maintenance model can be classified as a process that finds optimized parameter values defining a single maintenance strategy selected a priori; e.g., in this paper, the type of action performed (repair, replacement). There are different optimization approaches that take into account the previously selected decision variables. The methodology used in this paper can be included among the mathematical approaches in which the optimization problem is formulated utilizing mathematical equations, which are then solved using differential calculus to identify optimal parameters for the maintenance strategy. We introduced a discrete maintenance model for real multi-component systems with non-identical components for the first time in [ 17 ], in which systems with repairable components and latent failures were optimized using changeable periods of inspection as a decision variable. Real complex systems are composed of a finite number of repairable and maintained components. Each component can be operated in different discrete maintenance modes. A maintenance mode of the i-th component is determined by a prescribed value of the decision variable (x i =n i ) that influences the maintenance cost of the mode. Given that a system is composed of kcomponents and each component has four maintenance modes in general, we have to investigate 4 k maintenance configurations of the system. Any system Mathematics 2022,10, 2865 4 of 17 configuration can be described by a maximal system unavailability U S ( x ) and a total cost C S , which is usually obtained as a sum of the costs of all component modes forming the configuration. The optimal system configuration is detected under requirements (1) and (2). This maintenance model is defined in this article as a discrete maintenance model. Similar models of maintenance optimization have been used in other publications. For example, the work of authors in [ 18 – 20 ] addresses the optimization problem under maintenance policies including two decision variables. One of them is the maximum number of failures before the system undergoes a perfect restoration and the second is the inspection interval used to detect hidden failures. In [ 21 ] authors consider only one component system subject to two types of age-dependent failure. Catastrophic failures are detected through periodic inspections, whereas minor failures are followed by minor repairs. The system is preventively replaced either at an optimal multiple of the inspection time or after the n-th minor failure, whichever comes first. Both parameters are decision variables. The cost per unit of time is an objective function. The objective is to obtain a cost-minimizing policy. Authors in [ 22 ] also present a maintenance model for a system subject to two types of unrevealed failures: minor and catastrophic. The system is replaced at the occurrence of the n-th minor failure, a catastrophic failure, or due to working age, etc. 4. Unavailability Analysis and Cost Model 4.1. The Method for Unavailability Calculation for a Complex System The basic methodology, including algorithms, for unavailability calculation for a complex system with maintenance was developed in [ 23 , 24 ]. The system structure and its functionality are described through the use of a directed acyclic graph (AG). An AG contains nodes and edges. The system function or non-function state constitutes the highest node, which is at the top of the AG. Subsystems (components) are described through internal (terminal) nodes, all of which are interconnected by edges. An AG cannot contain feedback loops because it is acyclic. Terminal nodes characterize the stochastic behavior of the input components of a system, which means that each terminal node must be provided with information about the probability distribution of its lifetime, as well as maintenance characteristics and parameters. Stemming from this information, the unavailability function of each terminal node is further computed through the renewal process model described in the following section. The final system unavailability function is obtained through the unavailability functions of all terminal nodes. Other details concerning the computing algorithm that calculates the system unavailability function from component (i.e., terminal node) unavailability functions are described in our previous research work in ([ 17 ], pp. 86–87). 4.2. Unavailability Analysis of Terminal Nodes The reliability mathematics used in this section results partially from the work of authors in [ 25 ] and partially from other articles presented in the references ([ 26 – 30 ]). The mathematics used in these sources was developed to a large extent to consider the new renewal cycle of a terminal node, as introduced below. A complex maintained system consists of particular components that, in the context of the AG system structure, are denoted terminal nodes. In this section, the unavailability function of a terminal node is investigated. The renewal cycle of a terminal node starts at time t= 0. The first failure occurs at the time X1 and CM of the node starts immediately. This CM action takes the time Y1 . The next failure occurs after the time X2 after the end of the repair of the first failure elapses. In this way, the renewal cycle continues until the nth failure occurs. This is followed by the FBM replacement of the node. When this is completed, the renewal cycle ends, as is demonstrated in Figure 1, where the length of the renewal cycle is T. We will call the progress over one renewal cycle a real ageing process. This evolution is characterized by gradually changing probability distributions of random variables X1,X2, . . . Xndue to different degradation processes caused by CM. Mathematics 2022,10, 2865 5 of 17 Mathematics 2022, 10, x FOR PEER REVIEW 5 of 18 the repair of the first failure elapses. In this way, the renewal cycle continues until the nth failure occurs. This is followed by the FBM replacement of the node. When this is completed, the renewal cycle ends, as is demonstrated in Figure 1, where the length of the renewal cycle is T. We will call the progress over one renewal cycle a real ageing process. This evolution is characterized by gradually changing probability distributions of random variables X1, X2, … Xn due to different degradation processes caused by CM. First, the unavailability function of the terminal node at a given time 𝑡 in the first renewal cycle is determined. Figure 1. The first renewal cycle. The time from the beginning of the renewal cycle to the occurrence of the second failure is a random variable 𝑋 ∗=𝑋+𝑌 +𝑋, etc. The renewal cycle T is terminated by the n-th failure and followed by the replacement. The replacement time is the random variable 𝑌. The random variables 𝑋,…,𝑋,𝑌,…,𝑌 are supposed to be independent. The terminal node is out of service at time 𝑡 only if it is in the process of being repaired (or replaced) after the i-th failure. This happens only when 0≤𝑋∗≤𝑡≤𝑋∗+𝑌  for some 𝑖 ∈ {1,…,𝑛}. It is well-known that 𝑃(𝑋∗+𝑌<𝑡)=𝐹∗(𝑡)= 𝐹(𝑡−𝑥)d   𝐹∗(𝑥) Hence 𝑃(𝑡≤𝑋∗+𝑌 )=𝐹∗(𝑡) = 1− 𝐹 (𝑡−𝑥)d   𝐹∗(𝑥) =  󰇡1−𝐹(𝑡−𝑥)󰇢d   𝐹∗(𝑥) =  𝐹(𝑡−𝑥)d   𝐹∗(𝑥) Let us denote the probability that the terminal node is unavailable at time 𝑡 due to its CM or FBM after the 𝑖-th failure 𝑃(𝑡). It follows that 𝑃(𝑡)=𝑃(0≤𝑋∗≤𝑡≤𝑋∗+𝑌)=𝐹(𝑡−𝑥)d  𝐹∗(𝑥) The probability that the node is out of order at time 𝑡 in the first renewal cycle (let us denote it 𝑢(𝑡)) fulfils the following equation: 𝑢(𝑡)=𝑃(𝑡)   =𝐹(𝑡−𝑥)d  𝐹∗(𝑥)   (3) The length of the 𝑘-th renewal cycle is the random variable 𝑇 and the 𝑘-th renewal cycle is the time interval (𝑠,𝑠⟩ where 𝑠=0 and 𝑠 is the value of the random variable 𝑆=∑𝑇   , as is shown in Figure 2. Figure 1. The first renewal cycle. The time from the beginning of the renewal cycle to the occurrence of the second failure is a random variable X∗ 2=X1+Y1+X2 , etc. The renewal cycle Tis terminated by the n-th failure and followed by the replacement. The replacement time is the random variable Yn . First, the unavailability function of the terminal node at a given time t in the first renewal cycle is determined. The random variables X1, . . . , Xn,Y1, . . . , Ynare supposed to be independent. The terminal node is out of service at time t only if it is in the process of being repaired (or replaced) after the i-th failure. This happens only when 0 ≤X∗ i≤t≤X∗ i+Yi for some i∈{1, . . . , n}. It is well-known that P(X∗ i+Yi<t)=FX∗ i+Yi(t) = Z∞ −∞FYi(t−x)dFX∗ i(x) Hence P(t≤X∗ i+Yi)=FX∗ i+Yi(t) =1−Z∞ −∞FYi(t−x)dFX∗ i(x) =Z∞ −∞1−FYi(t−x)dFX∗ i(x) =Z∞ −∞FYi(t−x)dFX∗ i(x) Let us denote the probability that the terminal node is unavailable at time t due to its CM or FBM after the i-th failure Pi(t). It follows that Pi(t) = P(0≤X∗ i≤t≤X∗ i+Yi)=Zt 0FYi(t−x)dFX∗ i(x) The probability that the node is out of order at time t in the first renewal cycle (let us denote it u1(t)) fulfils the following equation: u1(t) = n ∑ i=1 Pi(t) = n ∑ i=1Zt 0FYi(t−x)dFX∗ i(x)(3) The length of the k -th renewal cycle is the random variable Tk and the k -th renewal cycle is the time interval (sk−1,ski where s0= 0 and sk is the value of the random variable Sk=∑k i=1Ti, as is shown in Figure 2. Mathematics 2022, 10, x FOR PEER REVIEW 6 of 18 Figure 2. Assuming that the FBM restores the system to an as-good-as-new state, it holds that all renewal cycles are equivalent; i.e., 𝑇=𝑇 =𝑇 for all 𝑖,𝑗∈N. The unavailability function of a terminal node describes the probability that the node is out of order at a given time t. It can be determined as follows. Let us denote this probability 𝑢(𝑡). It is obvious that 𝑢(𝑡)=𝑢(𝑡)   , (4) where 𝑢(𝑡) is the probability that the node is out of order at time 𝑡, which belongs to the 𝑘-th renewal cycle (𝑠,𝑠⟩. The value 𝑢(𝑡) is given by Formula (3). The values 𝑢(𝑡) for 𝑘 = 2,3,… can be determined in the following way, assuming that the FBM restores the terminal node to an as-good-as-new state. It holds that 𝑢(𝑡)=𝑢(𝑡−𝑠), 𝑢(𝑡)=𝑢(𝑡−𝑠), … and, in general, for 𝑘 ≥ 2 (see Figure 3): Figure 3. The value of the unavailability function 𝑢(𝑡) in the 𝑘-th renewal cycle for 𝑡∈(𝑠,𝑠⟩ is the same as the value of the unavailability function 𝑢 at the time 𝑡−𝑠. 𝑢(𝑡)=𝑢(𝑡−𝑠)=𝑢(𝑡−𝑠)d  𝐹(𝑠) (5) where 𝐹 is the cumulative distribution function of the random variable 𝑆 = ∑𝑇   . Under the assumption that the FBM restores the node to an as-good-as-new state, it also holds that 𝑇=𝑇 =𝑇 for all 𝑖,𝑗∈𝑁. This means that 𝐹 is the (𝑘−1)-fold convolution of the cumulative distribution function 𝐹, where the random variable 𝑇= ∑𝑋   +∑𝑌   is the length of the renewal cycle. Hence 𝐸𝑇=𝐸𝑋   +𝐸𝑌   and 𝐹 is the convolution of cumulative distribution functions 𝐹, …, 𝐹, 𝐹, … and 𝐹. Equations (4) and (5) then imply: 𝑢(𝑡)=𝑢(𝑡)+𝑢(𝑡−𝑠)d     𝐹(𝑠) =𝑢(𝑡)+𝑢(𝑡−𝑠)d     𝐹(𝑠) =𝑢(𝑡)+𝑢(𝑡−𝑠)d  𝐺(𝑠), (6) Figure 2. Assuming that the FBM restores the system to an as-good-as-new state, it holds that all renewal cycles are equivalent; i.e., Ti=Tj=Tfor all i,j∈N. Mathematics 2022,10, 2865 6 of 17 The unavailability function of a terminal node describes the probability that the node is out of order at a given time t. It can be determined as follows. Let us denote this probability u(t). It is obvious that u(t) = ∞ ∑ k=1 uk(t), (4) where uk(t) is the probability that the node is out of order at time t , which belongs to the k-th renewal cycle (sk−1,ski. The value u1(t) is given by Formula (3). The values uk(t) for k= 2, 3, . . . can be determined in the following way, assuming that the FBM restores the terminal node to an as-good-as-new state. It holds that u2(t) = u1(t−s1) , u3(t) = u1(t−s2) , . . . and, in general, for k≥2 (see Figure 3): uk(t) = u1(t−sk−1)=Zt 0u1(t−s)dFSk−1(s)(5) where FSk−1 is the cumulative distribution function of the random variable Sk−1=∑k−1 i=1Ti . Under the assumption that the FBM restores the node to an as-good-as-new state, it also holds that Ti=Tj=T for all i , j∈N . This means that FSk−1 is the (k−1) - fold convolution of the cumulative distribution function FT , where the random variable T=∑n i=1Xi+∑n i=1Yiis the length of the renewal cycle. Hence ET = n ∑ i=1 EXi+ n ∑ i=1 EYi and FT is the convolution of cumulative distribution functions FX1 , . . . , FXn , FY1 , . . . and FYn . Mathematics 2022, 10, x FOR PEER REVIEW 6 of 18 Figure 2. Assuming that the FBM restores the system to an as-good-as-new state, it holds that all renewal cycles are equivalent; i.e., 𝑇=𝑇 =𝑇 for all 𝑖,𝑗∈N. The unavailability function of a terminal node describes the probability that the node is out of order at a given time t. It can be determined as follows. Let us denote this probability 𝑢(𝑡). It is obvious that 𝑢(𝑡)=𝑢(𝑡)   , (4) where 𝑢(𝑡) is the probability that the node is out of order at time 𝑡, which belongs to the 𝑘-th renewal cycle (𝑠,𝑠⟩. The value 𝑢(𝑡) is given by Formula (3). The values 𝑢(𝑡) for 𝑘 = 2,3,… can be determined in the following way, assuming that the FBM restores the terminal node to an as-good-as-new state. It holds that 𝑢(𝑡)=𝑢(𝑡−𝑠), 𝑢(𝑡)=𝑢(𝑡−𝑠), … and, in general, for 𝑘 ≥ 2 (see Figure 3): Figure 3. The value of the unavailability function 𝑢(𝑡) in the 𝑘-th renewal cycle for 𝑡∈(𝑠,𝑠⟩ is the same as the value of the unavailability function 𝑢 at the time 𝑡−𝑠. 𝑢(𝑡)=𝑢(𝑡−𝑠)=𝑢(𝑡−𝑠)d  𝐹(𝑠) (5) where 𝐹 is the cumulative distribution function of the random variable 𝑆 = ∑𝑇   . Under the assumption that the FBM restores the node to an as-good-as-new state, it also holds that 𝑇=𝑇 =𝑇 for all 𝑖,𝑗∈𝑁. This means that 𝐹 is the (𝑘−1)-fold convolution of the cumulative distribution function 𝐹, where the random variable 𝑇= ∑𝑋   +∑𝑌   is the length of the renewal cycle. Hence 𝐸𝑇=𝐸𝑋   +𝐸𝑌   and 𝐹 is the convolution of cumulative distribution functions 𝐹, …, 𝐹, 𝐹, … and 𝐹. Equations (4) and (5) then imply: 𝑢(𝑡)=𝑢(𝑡)+𝑢(𝑡−𝑠)d     𝐹(𝑠) =𝑢(𝑡)+𝑢(𝑡−𝑠)d     𝐹(𝑠) =𝑢(𝑡)+𝑢(𝑡−𝑠)d  𝐺(𝑠), (6) Figure 3. The value of the unavailability function uk(t)in the k-th renewal cycle for t∈(sk−1,skiis the same as the value of the unavailability function u1at the time t−sk−1. Equations (4) and (5) then imply: u(t) = u1(t) + ∞ ∑ k=2Zt 0u1(t−s)dFSk−1(s) =u1(t) + ∞ ∑ k=1Zt 0u1(t−s)dFSk(s) =u1(t) + Zt 0u1(t−s)dG(s), (6) where G(s) = ∑∞ k=1FSk(s). Since S1=T, it holds that G(s) = FT(s) + ∞ ∑ k=2 P(Sk≤s) =FT(s) + ∞ ∑ k=1 P(Sk+T≤s) Mathematics 2022,10, 2865 7 of 17 =FT(s) + ∞ ∑ k=1 FSk(s−T) =FT(s) + Zs 0 ∞ ∑ k=1 FSk(s−T)dFT(t) =FT(s) + Zs 0G(s−T)dFT(t). Hence, the function G(s)is the solution of the integral equation G(s) = FT(s) + Zs 0G(s−T)dFT(t). Thus, knowing G(s) , the required unavailability function u(t) can be determined as a solution of the integral Equation (6). The starting point for the development of this methodology lies in our previous work introduced in [ 31 ], where dormant systems under inspection were investigated. The main difference is in the renewal cycle. A dormant system has a different renewal cycle because failures are not detected immediately but only at special inspection times. 4.3. Cost Model The cost model of a system configuration can be derived by adding up all the contributions arising from both the repair and replacement processes of a mode over all of the system components. A maintenance mode of a component has two main cost contributions: the cost of FBM, given by the replacements of the component during a mission time T M that depends on the decision variable of the component; and the cost of the imperfect repair process, which further depends on the mean number of failures during the mission time T M and the CM parameters. In practical situations, the cost contributions result from a database for the year and give an average yearly cost for the system configurations in a monitored period. In the remainder of this article, the cost will be computed in non-identified cost units based on the summation principle. To obtain the cost of one system configuration, we simply add up the costs of all maintenance modes of all system components. The mean cost of one maintenance mode of the j-th component CTM(j)can be computed as follows: CTM(j) = "nR(j) nj#.CR(j) + nR(j)−"nR(j) nj#!.CCM(j)(7) where nR(j) = TM MTTF(j) + MTTR(j) nR(j)is the mean number of failures of the j-th component per mission time TM MTTF(j) . . . mean time to failure of the j-th component: MTTF(j) = ∑nj k=1MTTFk(j) nj (8) MTTFk(j). . . mean time to the k-th failure of the j-th component MTTR(j) . . . mean repair time of the j-th component nj. . . decision variable of the j-th component determining the FBM strategy [x] . . . integral part of the real number x(i.e., f(x)=[x] is the floor function) hnR(j) nji. . . number of FBM replacements of the j-th component per mission time TM nR(j)−hnR(j) nji . . . mean number of repairs (CM) of the j-th component CR(j). . . replacement cost = cost of one FBM action for the j-th component in cost units Mathematics 2022,10, 2865 8 of 17 CCM(j). . . CM cost = cost of one repair action for the j-th component CTM(j). . . mean cost of one maintenance mode of the j-th component It is worth noting that the mean maintenance cost of the j-th component CTM(j) depends on the decision variable n j , not only directly, as follows from Formula (7), but also indirectly via nR(j) (see Formula (8) for MTTF(j)). The total cost of one system configuration C S is obtained by summing up these contributions, described by Formula (7), over all of the system components k: CS= k ∑ j=1 CTM(j)(9) A similar principle for the computation of the total maintenance cost for the whole of a system was used in [32]. 5. Unavailability Optimization of Selected Systems and Discussion 5.1. One Maintained Component System Adopted from the Literature First, we take into account one component system that is under both PM and CM. PM is considered as FBM, where full renewal is realized at the occurrence of every nth failure. The imperfect CM model causes a real ageing process, where each CM intervention deteriorates the system lifetime at an increasing failure rate. Proceeding from the previously derived reliability mathematics, the system unavailability function can be determined from Formula (6). The imperfect CM results in an unwanted rise in the unavailability function that can be reduced by the properly selected FBM process, meaning that renewal of the system starts with the occurrence of the n-th failure. After the renewal, the system is restarted to an as-good-as-new state. The number nis considered here as a changing decision variable permitting optimization. The exact specification of the decision variable determines a system configuration that is connected to a specific cost computed according to Formula (9), where k= 1. The discrete maintenance optimization method is shown for a system adopted from [ 25 ] where a stochastic alternating renewal process model is derived but a different renewal cycle is considered. The distribution function of the first failure time X 1 is a Weibull distribution with the shape parameter β = 2 and scale parameter α = 600 days. Imperfect CM is characterized by a random repair time with rectangular distribution in an interval of <12, 16> days, so that the mean time for CM is 14 days. We further suppose that the replacement time in the FBM model is deterministic—the renewal duration is 7 days; i.e., it is shorter than the CM time since it may be scheduled beforehand, which is given by the fact that the repair team has at its disposal information about (n−1)-th failure. The real ageing process can be realized in the following way. If a failure occurs, it is followed by a standard CM action, which recovers the health of the system to some extent, but its failure rate is always worse when compared with the system health before failure. We presume that, following the first system failure and the CM action, the growth of the failure rate can be estimated by the quotient q a , which ranges from 1 to 1.5 and by which the failure rate is multiplied. Worsening of the failure rate will continue after the second failure and will be followed by a repair time, etc., until the time of the FBM intervention; i.e., the time of the n-th failure. Provided that the first failure time X 1 follows a Weibull distribution, the failure rate can be expressed in the following way: λ(t) = β α2.t(10) X2also has a Weibull distribution failure rate, which is worsened as follows: λ(t) = β α2.qa.t(11) Mathematics 2022,10, 2865 9 of 17 and the worsening continues with each subsequent failure in the same way. In the time Xn−1, the failure rate can be expressed as: λ(t) = β α2.qan−2.t(12) and the FBM is started at the time X n , which launches a renewal (replacement). Thus, the system progressively deteriorates, and its lifetime distribution is modified after each CM action until an FBM time comes. This is in accordance with real practice because each failure followed by a CM action exposes the system to shocks that accumulate, and the system becomes increasingly worse. For example, the level of deterioration after the CM of circuit breakers, components of a power distribution network, has been intensively and personally discussed with experts and authors ([ 15 ]), who stated that each CM action makes the initial technical health of the component worse by approximately 10–25%. That is why we selected the following computing experiments with an appropriate value for the quotient q a = 1.25, which means the limit worsening of the failure rate by 25% after each CM intervention. The difference between real and theoretical ageing is demonstrated in Figure 4, which illustrates the time-dependent unavailability function of the system u(t) , computed according to Formula (6), with a mission time of 4000 days for two system modes: 1. A real ageing mode without FBM; i.e., n= ∞ and ageing quotient q a = 1.25, so that each failure is followed by CM and the subsequent system lifetime has an increasingly worse failure rate, in accordance with Formulas (10)–(12); 2. A traditional ageing mode, where ageing is due to an increasing Weibull failure rate. This ageing process can be denoted as theoretical ageing, where each failure is followed by a standard CM intervention that makes the system as good as new, such that the system is replaced (n= 1). Mathematics 2022, 10, x FOR PEER REVIEW 10 of 18 Figure 4. Theoretical versus real (q a = 1.25) ageing of one-component system discussed. In the first mode, we can see a rapid rise in the unavailability function by the end of mission time, whereas, in the second mode, the unavailability is stabilized shortly after 500 days, which is close to the mean lifetime of the system (531.7 days). These two curves are limiting curves for optimization. The unavailability growth of the real ageing mode can be reduced using a properly selected FBM; i.e., with a properly selected value for the decision variable n respecting a selected unavailability restriction. Let us take into account that the maximal permissible value of U S (x) is U 0 = 0.04. It is necessary to find an optimal system mode for the objective function given by Equation (1); i.e., we look for an optimal value for the decision variable n. To solve this optimization problem, we must provide data related to the maintenance cost: the cost of one FBM action is 𝐶=12 and the cost of one CM action is 𝐶 =6, which is in good agreement with practical experience because FBM entails the replacement of the old system with a new one, which is more expensive than repairing it. The results of our optimization process are shown in Table 1, which gives the clear conclusion that the optimal value for the decision variable n = 5 and the corresponding minimal cost based on Formula (9) is 59.97 units. Figure 5 shows the dependence of unavailability on time in the optimal mode respecting the given restriction, as well as its comparison with all of the computed system modes. Figure 4. Theoretical versus real (qa= 1.25) ageing of one-component system discussed. In the first mode, we can see a rapid rise in the unavailability function by the end of mission time, whereas, in the second mode, the unavailability is stabilized shortly after 500 days, which is close to the mean lifetime of the system (531.7 days). These two curves Mathematics 2022,10, 2865 16 of 17 f(x) = min CSobjective function U(x,t) instantaneous time-dependent unavailability function US(x) maximal system unavailability within a mission time TM U0a specified limitation of US(maximal permissible value) x= (x1, . . . , xk)∈Rkdecision variable ninumber of failures of the i-th component, which starts its renewal knumber of system components X1 the time from the beginning of the renewal cycle to the first failure Xithe time from the occurrence of the i-th failure to the end of its CM repair (for i∈{1, . . . , n−1}) Yithe time from the occurrence of the i-th failure to the end of its CM repair (for i∈{1, . . . , n−1}) Yn the time from the occurrence of the n -th failure to the end of the FBM replacement X∗ i=X1+Y1+· · · + the time from the beginning of the renewal cycle to the occurrence X(i−1)+Y(i−1)+Xiof the i-th failure, X∗ 1=X1,X∗ 2=X1+Y1+X2, etc. FX(t) = P(X≤t)the cumulative distribution function of the random variable X FX(t) = 1−FX(t)the reliability function of the random variable X References 1. 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