1.INTRODUCTION
Owing to the global pressure on defence budgets, military forces need to sustain operational availability at a required level and have to reform maintenance policies (Ahmadi et al., 2009). Under this circumstance, one of the most significant issues to be considered when deploying a weapon system might be the optimization of lifetime operating costs by determining suitable maintenance intervals. The majority of the defence standards, such as MILHDBK217F, MILSTD1388, and MILSTD2173, utilize mean time between failures (MTBF) for planning maintenance. MTBF cannot be used to model agerelated failure mechanisms owing to the memoryless property of the exponential distribution, which does not include a timevariant in spite of its theoretical ease of use. Maintenance models based on MTBF admit that failure cannot be evaded and lead to corrective maintenance (Long et al., 2009). In military establishments the penalty costs caused by corrective maintenance can greatly outweigh the preventive maintenance costs (Moon et al., 2012). For example, a long preventive maintenance interval for saving $100 preventive maintenance cost could cause corrective maintenance that could lead to a $100 million warship to be nonoperational. This could result in a military defeat that could cause casualties and deaths.
A maintenancefree operating period (MFOP) can be defined as “a period of operation during which an item will be able to carry out all its assigned missions, without the operator being restricted in any way due to system faults or limitations, with the minimum of maintenance” (Kumar et al., 1999, p. 128). MFOP survivability (MFOPS) refers to the probability that the item will survive for the duration of MFOP, as shown in Equation (1) (Kumar et al., 1999).
where, MFOPS_{i} = the probability that component i will survive when it is up at time I_{i}+t, given that it survives at I_{i}; R_{i} = reliability of component i; I_{i} = maintenance interval for component i.
MFOP can be employed to analyze agerelated failure mechanisms and can help improve the reliability and design of the system (Kumar et al., 1999). This decreases the uncertainty present in maintenance planning and allows logistics managers to focus on selective systems (Cini and Griffith, 1999; Wu et al., 2004). MFOP has been used in many projects, such as future offensive aircraft systems, joint strike fighter, ultrareliable aircraft, and BAe Airbus aircraft (Cini and Griffith, 1999; Kumar, 1999; Relf, 1999). Several authors (Kumar et al., 1999; Long et al., 2009) have presented maintenance models based on MFOPS. MFOPS is a useful concept for a hightech weapon system. This is because MFOPS allows logistics manager to trace its failure mechanism and warranties its survivability expressed as a specified probability during the military operations. In spite of the usefulness of MFOPS, however, far too little attention has been paid to the development of a practical optimization model determining lifetime operating costs and maintenance intervals for a multicomponent system based on MFOPS. The objective of this paper is to present a twostep maintenance planning model for multicomponent weapon systems based on MFOPS.
The remainder of this paper is organized as follows. Section 2 reviews the theoretical framework for MFOP and optimization models for lifetime operating costs. Section 3 develops the maintenance planning model. This is followed in Section 4 by the results and analysis using data obtained from the South Korean navy, and insights for practitioners. Finally, Section 5 presents the concluding remarks.
2.LITERATURE REVIEW
This section summarizes the research that has investigated MFOP and maintenance optimization models. MFOP is a reliability measure used by the Ministry of Defence, UK (Kumar, 1999). MFOP requires the manufacturers to analyze the failure mechanism and to improve the reliability and the design of the system. This is because MFOP can trace the failures of the system throughout its lifetime (Kumar et al., 1999). The main idea of MFOP is the elimination of unplanned maintenance for the period of operations (Relf, 1999).
A considerable body of research has proposed a variety of maintenance optimization models. Table 1 compares sixteen major studies that have examined maintenance optimization models. The Weibull failure rate function was the most commonly used failure mechanism as six papers adopted the Weibull failure rate.
Most of the researchers cited in Table 1 provided models minimizing the total costs occurring throughout the lifetime of the systems. Ten studies considered breakdown costs. The majority of the literature employed the preventive maintenance interval (or time) as a decision variable. Eight studies looked at multicomponent systems. Some researchers (Horenbeek and Pintelon, 2013) considered grouping maintenance activities.
Previous research (Kumar et al., 1999, Long et al., 2009) also has suggested the concept of MFOPS which might be suitable for hightech weapon systems in view of their requirement of high reliability. Kumar et al. (1999) proposed a mathematical model to predict MFOPS for a multicomponent system. Recently, the concept of MFOPS has been developed by Long et al. (2009) as a maintenance cost calibration model for a single unit system. However, to the best of our knowledge, no attempt has been made to optimize maintenance interval and lifetime operating costs based on MFOPS for a multi component system. The present work is designed to be the first to develop the maintenance optimization model based on MFOPS for a multicomponent system. This paper also demonstrates some empirical evidence to support the performance of the maintenance optimization model.
3.THE DEVELOPMENT OF A TWOSTEP MAINTENANCE PLANNING MODEL
This section describes the twostep maintenance planning model for hightech weapon systems such as a missile launcher in a warship. We assumed that the failures of the component in the missile launcher are independent of each other, and multiple failures between maintenance activities can be occurred. The missile launcher is required to launch missiles at any time before its major overhaul made in the Naval repair shop. In order to be kept in operational condition before the major overhauls, the missile launcher needs to be maintained periodically. Adapted from the model of Tam et al. (2006), the first step minimizing the lifetime operating costs can be formulated as an integer program, as shown in Equation (2).
Index and Set
Decision Variable
Parameters

L: the lifetime period of the system before a major overhaul;

C_{D}: downtime cost per unit period (i.e. day);

MFOPS_{s} : the mean MFOPS of the system over the lifetime period before a major overhaul;

C_{M,i} : the unit preventive maintenance cost for component i.
Lifetime Operating Costs Optimization Model
According to the definition of MFOP presented in Section 1, we did not consider the corrective maintenance. The preventive maintenance indicates the replacement of line replaceable unit (LRU) made on board. We assumed the LRU become as good as new after the replacement, and the weapon system become as good as new after a major overhaul. The replacement or repair of shop replaceable unit (SRU) made in the Naval repair shop was beyond the scope of this study. A field survey identified that the replacement times for LRU for the missile launcher were less than 30 minutes. 30 minutes might be a short time period compared to operating period (e.g. 14 days). Hence, we assumed that the replacement time does not affect the survivability of the weapon system. MFOPS_{i} seems to be a function of I_{i} and t as shown in Equation (2). In order to describe the maintenance model with periodic preventive maintenance, we here define MFOPS_{i} as the probability that component i will survive when it is up at time I_{i} + I_{i}, given that it survives at I_{i}.
MFOPS decreases by the end of a maintenance interval and was assumed to become as good as new after the replacement, as shown in Figure 1. In order to compute MFOPS over the lifetime period, we employed the mean MFOPS.
Under the assumption of a multicomponent system in series with the Weibull failure rate function such that ${\text{MFOPS}}_{i}=\text{exp}\left[\left({I}_{i}^{{\beta}_{i}}{\left({I}_{i}+{I}_{i}\right)}^{{\beta}_{i}}\right)/{\eta}_{i}^{{\beta}_{i}}\right]$, where we considered the MFOPS of each component i by the very next maintenance interval from the current one, MFOPS_{s} can be expressed as the product of all MFOPS_{t}, and furthermore, it can be approximated as shown in Equation (3). The total downtime is a period of the lifetime during which the system will not be able to carry out its assigned missions before a major overhaul. The total downtime can be estimated as L • (1−MFOPS_{s}) .
If we let ${f}_{i}\left({I}_{i}\right)=({\displaystyle {\sum}_{k=1}^{{I}_{i}}\text{exp}\left[{\gamma}_{i}{k}^{{\beta}_{i}}\right]})/{I}_{i}$ where ${\gamma}_{i}=\left(1{2}^{{\beta}_{i}}\right)/{\eta}_{i}{}^{{\beta}_{i}}$ which is a constant, Equation (2) is (approximately) equivalent to minimizing ${\text{C}}_{\text{D}}{\displaystyle {\prod}_{i=1}^{n}{f}_{i}\left({I}_{i}\right)}+{\displaystyle {\sum}_{i=1}^{n}{\text{C}}_{\text{M,i}}/{I}_{i}}$. Note that ${\gamma}_{i}\le 0$ for ${\beta}_{i}\ge 0$ and thus f_{i} is monotonically decreasing in I_{i} since ${f}_{i}\left({k}_{1}\right)\ge {f}_{i}\left({k}_{2}\right)$ for any $0\le {k}_{1}\le {k}_{2}$ where k_{1} and k_{2} are integers. Because of this, the first term $({\text{C}}_{\text{D}}{\displaystyle {\prod}_{i=1}^{n}{f}_{i}({I}_{i})})$ decreases, whereas the second term ${\sum}_{i=1}^{n}\left({\text{C}}_{\text{M,}\hspace{0.17em}\text{i}}{\text{/I}}_{\text{i}}\right)$ does so as I_{i} increases. Therefore, the two terms of the objective function conflict with each other, and we may take an optimal interval achieved at some point which is suboptimal for both but optimal for their combination. As stated in Section 1, the breakdown outages cost can greatly outweigh the preventive maintenance cost. This implies that C_{D} ≫ C_{M,i} for all i , and would make some bias on an optimal interval more closely to the consideration of the breakdown outages cost. Before dealing with a multicomponent system, we first discuss an optimal interval for a single component. When considering only a single component, the objective becomes:
The solution strategy for this objective is identical to the one that Tam et al. (2006) used: to plot the two values of preventive maintenance cost and breakdown outages cost independently on a grid and take the sum of these two values to obtain an optimal interval that presents the minimum value of the objective. For each component i, taking its optimal interval, ${I}_{i}^{*}$, that achieves objective (4) might not ensure an optimality for objective (2) under a multicomponent system. A heuristic approach, similar to the marginal analysis used by Sherbrooke (2004), can be applied to this problem as shown in objective (5). In order to find the marginal or incremental value for each component, the lifetime operating cost of the item is divided by its unit preventive maintenance cost. However, as a practical approach, we solved objective (4) for each single component and used that solution for the multicomponent model. The interdependency of the maintenance intervals for the components in the system was considered in the second step.
The second step, adopting the formula of Horenbeek and Pintelon (2013), is to group maintenance activities to save setup costs. The lifetime operating costs, taking into account preventive maintenance costs including the component dependent maintenance cost and the systemdependent maintenance cost (i.e. setup cost), can be written as shown in Equation (6):
where

${\widehat{I}}_{i}$ : a modified interval by grouping maintenance activities;

a_{j} : the number of maintenance activities for group j during the lifetime period, L, before a major overhaul;

S_{j} : the setup cost for group j.
Without grouping components, the lifetime operating costs can be described as shown in Equation (7). Since each component i was included in only one set of G_{j}, SC_{i} was welldefined. The costs according to Equation (6) were hypothesised to be lower than the costs according to Equation (7), owing to the suboptimal interval for each component achieved without grouping maintenance activities. The next section provides the results from some computational experiments based on a multicomponent weapon system of the Korean navy.
where SC_{i} = S_{j}, for i∈G_{j} (set of group j).
4.REALLIFE EXAMPLE AND ANALYSIS
This section presents the failure rates fitted to the failure data of the components for a weapon system, discusses some computational studies using the fitting results, and provides insights for practitioners. The failure history for the seventeen components for the missile launcher of a specific type of warship in the Korean navy before a major overhaul (1,095 days), used by Moon and Lee (2017), was analyzed. The Weibull failure rate function $\left(\lambda \left(t\right)=\frac{\beta}{\eta}{(\frac{t}{\eta})}^{\beta 1},\hspace{0.17em}t\ge 0\right)$ was assumed to fit the failure data for this research. Table 2 presents the results of fitting the Weibull to the failure records of the components and their material costs. The values of β as shown in Table 2 indicate that 1095 days might span a usefullife or early wearout stage for the components with slightly increasing failure rates (Abernethy, 2001; Rausand and Hoyland, 2004).
Maintenance costs are composed of various cost elements, for example material costs plus maintenance manhours multiplied by labour rate (Wu et al., 2004). However, for simplicity, this paper considered that material cost is identical to preventive maintenance cost, C_{M,i}. This was because a field survey identified that material cost constituted the largest element of maintenance costs. The material costs are provided as shown in Table 2. A breakdown outages cost was calculated as ${\text{C}}_{\text{D}}=c({\displaystyle {\sum}_{i=1}^{n}{\text{C}}_{\text{M,i}}})$ where c is a constant of our choice, to consider the cost of breakdown outages compared to the mean preventive cost, $\sum}_{i=1}^{n}{\text{C}}_{\text{M,i}}/\text{n$.
As the first step, for each component i, we examined an optimal interval that provided a minimum value of the lifetime operating costs by plotting those two values on the twodimensional grid. For instance, an optimal interval of Component 1 (${I}_{1}^{*}$) with C_{M,1} = $9,598 for the case where c = 5 (i.e. C_{D} = $36,951 ) was calculated as 16 days. Similarly, optimal intervals for all components were obtained as shown in Table 3. As mentioned in Section 3, for a fixed value of C_{D}, as the interval grew larger, the preventive maintenance cost decreased, whereas the breakdown outages cost increased.
A high breakdown outages cost reduced the interval, because more frequent preventive maintenance made a component stay in the “up” state and reduced the chance of a component being in the “down” state, which was relatively costly, owing to the breakdown outages cost. We also compared the model with MFOPS to one with traditional system reliability (Tam et al., 2006), as shown in Table 3. Traditional system reliability, R(t), can be defined as $R\left(t\right)=P\left[T>t\right]=\text{exp}\left[{\left(t/\eta \right)}^{\beta}\right]$ based on the Weibull function. In general, optimal intervals obtained via MFOPS were shorter than those obtained via traditional system reliability owing to the consideration of conditional probability involved in MFOPS. This pattern was distinct when maintenance costs were very low (e.g. #6, #7, and #15). The shorter optimal intervals from MFOPS might imply a higher probability of survival and a higher requirement for maintenance (resulting in higher lifetime operating costs) compared to those obtained via traditional system reliability (Croker, 1997). For example, for c = 5, the lifetime operating cost using the MFOPS ($15.04 million) was higher than that using traditional system reliability ($13.84 million).
We further examined how sensitively an optimal interval decreases as c increases. Taking preventive maintenance costs into consideration, Components 6, 1, and 12 were compared. This was because, while their Weibull parameters showed similar characteristics, their preventive maintenance costs were distinguishable (i.e. C_{M,6} = $154, C_{M,1} = $9,598, and C_{M,12} = $21,232). Figure 2 illustrates the influence of preventive maintenance costs on determining optimal intervals. With a lower C_{M,i} (e.g. Component 6), the corresponding optimal interval decreased slowly. However, a higher preventive maintenance cost (e.g. Component 12) resulted in a relatively rapid decrease. The relative breakdown outages cost had a more sensitive effect on the optimal maintenance interval of a component having a higher preventive maintenance cost than on that of a component having a lower preventive maintenance cost, as shown in Figure 2. Managers might have to concentrate more on a component having a high preventive maintenance cost and should be more careful to determine the relative costs of breakdown outages and preventive maintenance for the component.
As the second step, the components were grouped according to their maintenance intervals determined at the first step. Two distinct groups of components, G_{j}, j = 1, 2 (semimonthly and monthly intervals), were considered. With the breakdown outages cost ${\text{C}}_{\text{D}}=c((1/\text{n}){\displaystyle {\sum}_{i=1}^{n}{\text{C}}_{\text{M,}\hspace{0.17em}\text{i}}})$ where c = 3, the optimal intervals based on MFOPS were considered for groupings in such a way that an optimal interval for each component can be gathered to the closest period between 15 days and 30 days, that is, G_{1} = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 16, 17}, and G_{1} = {9, 12, 14}. Then, the maintenance interval of each component was modified to fit into one of the two groups, that is, ${\widehat{I}}_{i}$ =15 days,$\forall i\in {G}_{1}$, and ${\widehat{I}}_{i}$ = 30 days,$\forall i\in {G}_{2}$. With the setup costs set at S_{1} = S_{2} = $10,000, grouping at the second step outperformed nongrouping intervals, as shown in Table 4. This might be because, although a higher breakdown outages cost was incurred under grouping policy owing to the lower MFOPS, merging maintenance activities into a group reduced the corresponding maintenance costs by more than the additionally incurred breakdown outages cost.
Grouping maintenance activities can be claimed to reduce the lifetime operating costs before major overhaul as shown in Table 4. However, as the grouping increased the breakdown outages, the grouping policy might not be suitable for a system requiring a lower chance of breakdown outages (i.e. higher MFOPS) than that determined by the grouping. In that case, managers should shorten the grouping maintenance intervals in order to sustain the required MFOPS. The grouping policy might be regarded as being useful when it satisfies the required MFOPS for users.
5.CONCLUSIONS
Maintenancefree operating period survivability (MFOPS) can analyze agerelated failure mechanisms and can help improve the reliability and design of the system. This paper suggests a practical twostep maintenance planning model based on MFOPS. At the first step, the lifetime operating costs incurred by breakdown and preventive maintenance for a multicomponent weapon system assumed to be in series are minimized. This provides an optimal maintenance interval for each component of the system. The second step integrates individual preventive maintenance activities for components of a weapon system into temporal groups in order to reduce setup costs. Therefore, the research gap, that is, the lack of a practical maintenance optimization model for a multicomponent system based on MFOPS, might be claimed to be filled.
This study has found that MFOPS has a tendency to require a shorter optimal preventive maintenance interval than traditional system reliability. This implies a higher probability of survival for the system and a higher requirement for maintenance (resulting in higher lifetime operating costs) compared to system reliability. The second major finding was that the relative breakdown outages cost had a greater effect on the optimal maintenance interval of a component having a higher preventive maintenance cost than on that of a component having a lower preventive maintenance cost. This suggests that managers need to focus more on a component with a high preventive maintenance cost and should pay particular attention to deciding the relative cost of breakdown outages and preventive maintenance for the component. The third important finding was that grouping maintenance activities could reduce the lifetime operating costs. However, the advantage of the grouping policy can be offset by the increasing breakdown outages, resulting in a lower MFOPS. This suggests that managers might have to extend the intervals for grouped maintenance activities in order to sustain the required MFOPS.
This research has several theoretical and practical contributions. Firstly, this is the first time that MFOP, which is suitable for hightech weapon systems in view of their requirement of high reliability, has been used to optimize maintenance interval and lifetime operating costs for a multicomponent system. Secondly, the present study provides empirical evidence with respect to the performance of the maintenance optimization model. Thirdly, as stated above, this research provides managers with some practical guidance for the use of the lifetime operating costs optimization model.
A limitation of this study relates to the fact that the results of fitting the Weibull function to the failure history of the components used for this study seemed not to show an obvious agerelated failure mechanism. The β values close to 1 as shown in Table 2 imply an almost constant failure rate. There might not be a significant advantage in adopting MFOP rather than MTBF in this case. Components having values of β which are far from 1 would gain more benefit from adopting MFOP. Further work needs to be done to identify the advantages of MFOP compared to MTBF and this will be done with data presenting an obvious agerelated failure mechanism.