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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.1 pp.52-63
DOI : https://doi.org/10.7232/iems.2017.16.1.052

# Optimizing Concurrent Spare Parts Inventory Levels for Warships Under Dynamic Conditions

Seongmin Moon*, Jinho Lee
KSS-III Program Group, Defense Acquisition Program Administration
Corresponding author : m1s1m@hotmail.co.uk
August 13, 2016 December 29, 2016 February 16, 2017

## ABSTRACT

The inventory level of concurrent spare parts (CSP) has a significant impact on the availability of a weapon system. A failure rate function might be of particular importance in deciding the CSP inventory level. We developed a CSP optimization model which provides a compromise between purchase costs and shortage costs on the basis of the Weibull and the exponential failure rate functions, assuming that a failure occurs according to the (non-) homogeneous Poisson process. Computational experiments using the data obtained from the Korean Navy identified that, throughout the initial provisioning period, the optimization model using the exponential failure rate tended to overestimate the optimal CSP level, leading to higher purchase costs than the one using the Weibull failure rate. A Pareto optimality was conducted to find an optimal combination of these two failure rate functions as input parameters to the model, and this provides a practical solution for logistics managers.

## 1.INTRODUCTION

We developed a concurrent spare parts (CSP) optimization model which provides a compromise between purchase costs and shortage costs on the basis of the Weibull and the exponential failure rate functions, and this model provides a practical solution for logistics managers. Maintaining the operational availability of weapon systems with a limited budget is a major concern in military logistics. Keeping an appropriate inventory level of the spare parts at the proper place and time might be a prerequisite to sustaining the availability of weapon systems (Menhert, 1983). Since the study on the multi-echelon technique for recoverable item control (METRIC) by Sherbrooke (1968), a considerable amount of literature has been published on this research area. Concurrent spare parts (CSP) refers to the spare parts purchased at the time of deployment of a new weapon system in order to sustain the required operational availability of the weapon system without resupply of the spare parts for some initial time period due to their conditions of contracts and resupply processes that incur long delivery lead times (Yoon and Sohn, 2007).

With the advent of high technology weapon systems, there has been an increasing interest in the determination of the CSP inventory level (Yoon and Sohn, 2007; Yoon and Lee, 2011). However, the majority of authors have presented spare parts optimization models that rely on analytical studies or simulations, with the exception of some researchers, such as Sherbrooke (2004) and Yoon and Sohn (2007), who have presented empirical evidence to validate their models.

Furthermore, most of the spare parts optimization models have assumed a stationary failure rate derived from the exponential distribution: for example, the optimization models by Sherbrooke (1986) and Yoon and Sohn (2007). As an input parameter, failure rates of parts have a great effect in determining the optimal spare parts inventory level that minimizes purchase costs and shortage costs incurred by backorders. We superposed individual part failure processes for CSP periods to form an overall demand process for the spare part. The superposed process can in many cases be close to a HPP due to Palm's Theorem. However, such a stationary failure rate cannot explain the failure rates in the early or wear-out stages, which confuses logistical decisions. In fact, the failure rate at each stage of the lifetime of many spare parts might follow a bathtub-shaped curve (Xie and Lai, 1996; Slack et al., 2010). The Weibull failure rate function is generally applicable to the failure rate in the early or wear-out stage, owing to its time-variant (Xie and Lai, 1996; Wang et al., 2001; Wessels, 2007). The widely used spare parts optimization models such as VARI-METRIC (Sherbrooke, 1986) did not focus thoroughly on such nonstationary conditions. The dynamic multi-echelon technique for recoverable item control (Dyna-METRIC) proposed by Hillestad (1982) and the aircraft sustainability model (ASM) suggested by Slay et al. (1996) considered a non-stationary failure rate dependent on flying hours. Although they provided the general form of nonstationary failure rate functions, to the best of our knowledge little attempt has been made to incorporate a failure rate function derived from the Weibull distribution into a spare parts optimization model, especially when determining the CSP inventory level.

The objectives of this study are: (i) to develop an optimization model determining CSP inventory levels using the Weibull failure function; (ii) to compare the performance of CSP optimization models using the exponential failure rates function and the Weibull failure rates function fitted to the empirical data obtained from the Korean Navy; and (iii) to identify an optimal mixture of the exponential and Weibull failure rates functions which minimizes the backorder and purchase costs of the parts.

## 2.LITERATURE REVIEW

In this section, we review the research that has investigated the system failure distributions and the multiechelon models for spare parts.

### 2.1.Failure Rate Function

The lifetime failure of a single spare part for the warships in use was difficult to model. This is because of the intermittent failure rate of the spare parts. Therefore, we superposed individual part failure processes to form an overall demand process for the spare part. The exponential distribution is straightforward to understand and employ, and has been claimed to present good estimations of machine failure distributions (Das, 2008; Wessels, 2007).

However, the exponential distribution has a practical limitation in representing dynamic failure behaviour. In practice, for the majority of mechanical parts, the failure mechanism could be represented by a bathtub curve (Xie and Lai, 1996; Wang et al., 2001). The Weibull distribution is one of the distributions which fit each distinct stage of the bathtub failure patterns.

The Weibull distribution has been reported as being more practical and as describing failure data more precisely than the exponential distribution. For instance, Wessels (2007) demonstrated that, with increasing true failure rates for the drive motor of unmanned ground vehicles, the failure rate in the exponential distribution was over 20% higher; it then became at least 10% lower than that of the Weibull distribution with β > 1.0. This error of the exponential failure rate makes logistical support difficult to cover the spare parts life costs (Wessels, 2007). Das (2008) also showed that, in designing a cellular manufacturing system, the Weibull distribution presented better cell configurations than the exponential distribution in terms of machine utilizations and inter-cell material movement with almost the same cost.

### 2.2.Multi-Echelon Inventory Model

Table 1 summarizes the results of fourteen major studies that have been developed and have contributed to multi- echelon inventory theories and practices. While five studies were conducted by military research organizations as technical reports, nine studies were carried out for academic purposes. The empirical failure data might be difficult to model using the Poisson distribution or the negative binomial distribution used by the authors cited in Table 1. This is because the empirical data are intermittent or have highly variable, low-average volumes (Moon et al., 2012; Moon et al., 2013). Hence, it is important to demonstrate some empirical evidence to support the performance of the inventory theory. This is the first research gap.

The assumption of the failure rate function cited in Table 1 has varied. Earlier research adopted only the homogeneous Poisson process. Authors such as Hillestad (1982) and Graves (1985) have employed the negative binomial distribution since the early 1980s. Some authors used the exponential distribution (Kaplan, 1987) and the Erlang distribution (Verrijdt and Kok, 1996). The majority of the researchers considered only stationary failure conditions, with the exception of Hillestad (1982) and Slay et al. (1996), who included non-stationary failure conditions such as wartime. The negative binomial distribution (NB) can reflect changes in mean failure rates, such that the va riance-to-mean ratios are not one (Sherbrooke, 2004). However, using NB it is difficult to analyse the failure mechanism at each stage of the lifetime of many spare parts. In spite of the usefulness of the Weibull failure rate function in the analysis of life data, little attention has been paid to the adoption of the Weibull failure rate function to analyse the failure patterns for a multi-echelon inventory model. This is the second research gap. This paper attempts to fill these research gaps.

## 3.THE DEVELOPMENT OF A SPARE PARTS OPTIMIZATION MODEL

This section presents the optimization model which minimizes the total costs (purchase cost plus shortage cost) subject to an available budget. The objective of this model is to determine the inventory levels at each echelon that minimizes the total costs. The current research considers a homogeneous Poisson process (HPP) with a constant (stationary) failure (demand) rate and a non-homogeneous Poisson process (NPP) with a time-varying (non-stationary) failure rate function. Note that λ (t) = λ, ∀t for an HPP. Adapted from the model suggested by Yoon and Sohn (2007), the optimization model minimizing the total costs to achieve an optimal CSP inventory level was formulated in the following way. We assumed that all failed parts are repaired at the depot.

Sets/Indices/Parameters

• iI indexes an echelon 0: depot, 1: base)

• iJ indexes a CSP

• tT indexes a discrete time period (day)

• cj :  unit purchase cost of CSP j

• gi :  unit shortage cost of CSP j

• rj :  probability that CSP j is repairable at base

• λi (t) :  (instantaneous) failure rate at time t of CSP j

• M :  maximum available budget

Decision/Random Variables

• sij ∶  number of CSP j at echelon i

• Dij (t) ∶  random variable representing demand of CSP j at echelon i

• ( nij )  at time t

CSP Optimization Model

$min s ∑ i ∈ I ∑ j ∈ J c j s i j + ∑ i ∈ I ∑ j ∈ J g j E ( D i j ( t ) − s i j ) +$
(1)

$s.t. ∑ i ∈ I ∑ j ∈ J c j s i j ≤ M$
(2)

$s.t. ∑ i ∈ I ∑ j ∈ J c j s i j − ∑ i ∈ I ∑ j ∈ J g j E ( D i j ( t ) − s i j ) + ≥ 0$
(3)

$1 ≤ s i j ≤ u i j and integer , ∀ i ∈ I , ∀ j ∈ J$
(4)

Equation (1) represents the objective function in which the first term is the total purchase costs of CSPs and the second term indicates the total expected shortage costs incurred by backorder quantities, where (x)+ = max (0, x) . Note that t is the end time (1,095th day) for observing stock and backorder levels, and the CSP period was considered as t. This was because the largest backorder was expected at the end time. Constraint (2) only allows the purchase of CSPs within the available budget; and Constraint (3) requires that the total expected shortage costs should not exceed the total purchase costs, as in the model of Yoon and Sohn (2007). Finally, each level of CSP has a proper upper bound, as shown in Constraint (4) (e.g. $u i j = E [ D i j ( t ) ]$); in addition, at least one spare for each part was purchased, as in the model of Yoon and Sohn (2007). This constraint was included in the present model to provide a degree of conservativeness, considering CSP as playing a critical role in maintaining a high level of readiness in weapon systems. Note that we did not constrain the conditions for the availability of the weapon systems. However, this was already considered by minimizing the total expected shortage costs as Slay et al. (1996) mentioned in their study.

The current research has assumed that demand for CSP j coming in to the base at time t follows NPP with the failure rate of λj (t) . Computing demand rate at echelon i for CSP j at time t, denoted by Ωij (t) , requires consideration of the following: base repair probability rj , which is independent of time t as shown by Yoon and Sohn (2007); base repair time brtj ; depot repair time drtj ; and order and ship time (OST) from/to depot when CSPs need to be delivered to be repaired at depot. We assumed that brtj, drtj , and OST are deterministic.

In order to compute failure rate at echelon i, we adopted ASM (Slay et al., 1996) in which a base repair pipeline and an order and ship pipeline were separately considered for obtaining a failure rate. Using their idea, a demand rate at base for CSP j at time t becomes the sum of the parts, which have failed and can be repaired at the base. The parts that cannot be repaired at the base and thus remain in pipeline on shipping to depot for repair. This resulted in Equation (5). Furthermore, a demand rate at depot for CSP j at time t is the sum of all CSPs’ failure rates that arrived at depot and are now being repaired, which led to Equation (6). Note that since we assumed discrete time horizon, we can sum the possible failure rates that occur independently in spite of NPP nature.

$Ω 1 j ( t ) = ∑ k = t − b r t j + 1 t r j λ j ( k ) + ∑ k = t − O S T + 1 t ( 1 − r j ) λ j ( k ) .$
(5)

$Ω 0 j ( t ) = ∑ k = t − b r t j − O S T + 1 t − O S T ( 1 − r j ) λ j ( k )$
(6)

Then, Dij (t) is the random variable whose distribution follows the Poisson distribution with mean and variance of Ωij (t) . Taking this into account,(7)

$E ( D i j ( t ) − s i j ) + = ∑ n i j = s i j + 1 ∞ ( n i j − s i j ) P ( D i j ( t ) = n i j ) = Ω i j ( t ) − ∑ n i j = 0 s i j n i j e − Ω i j ( t ) Ω i j ( t ) n i j n i j ! − s i j ( 1 − P ( D i j ( t ) ≤ s i j ) )$
(7)

Since this partial expectation calculation involves the decision variable itself, the model is a nonlinear, nonconvex integer programming formulation, and in general is very difficult to solve optimally within a reasonable time. To overcome this difficulty, it was reformulated as a linear integer program in the following way. Let us define bij (sij) as(8)

$b i j ( s i j ) = Ω i j ( t ) − ∑ n i j = 0 s i j n i j e − Ω i j ( t ) Ω i j ( t ) n i j n i j ! − s i j ( 1 − P ( D i j ( t ) ≤ s i j ) )$
(8)

which is a function of decision variable sij , given a specific time t. For a predetermined t, suppose that we fix sij to be some value satisfying Constraint (4). Then, bij (sij) can be easily obtained because no other decision variable exists. Using this, we denote $b i j ( k ) ≡ b i j k = Ω i j ( t ) − ∑ n i j = 0 k n i j e − Ω i j ( t ) Ω i j ( t ) n i j n i j ! − k ( 1 − P ( D i j ( t ) ≤ k ) )$. An auxiliary binary decision variable, zijk , was introduced such that

$z i j k = { 1 , s i j if k 0 , other$
(9)

Then, the second term of Equation (1) can be reformulated as

$∑ i ∈ I ∑ j ∈ J g j ∑ k = 1 u i j b i j k z i j k$
(10)

From (9), there is a relationship between s and z such that

$s i j = ∑ k = 1 u i j k z i j k$
(11)

The following additional constraint was also needed:

$∑ k = 1 u i j z i j k = 1 , ∀ i ∈ I j ∈ J$
(12)

so that sij selects exactly one integer value in [1, uij ].

Combining Equations (9), (10), (11), and (12) with the initial model, the following linear binary integer programming formulation was finally obtained:

$min z ∑ i ∈ I ∑ j ∈ J c j ∑ k = 1 u i j k z i j k + ∑ i ∈ I ∑ j ∈ J g j ∑ k = 1 u i j b i j k z i j k$
(13)

$s.t. ∑ i ∈ I ∑ j ∈ J c j ∑ k = 1 u i j k z i j k ≤ M$
(14)

$∑ i ∈ I ∑ j ∈ J c j ∑ k = 1 u i j k z i j k − ∑ i ∈ I ∑ j ∈ J g j ∑ k = 1 u i j b i j k z i j k ≥ 0$
(15)

$∑ k = 1 u i j = 1 , ∀ i ∈ I j ∈ J$
(16)

$z i j k ∈ { 0 , 1 } , ∀ i ∈ I j ∈ J$
(17)

The objective function (13) can be simplified as(18)

$min z ∑ i ∈ I ∑ j ∈ J ∑ k = 1 u i j h i j k z i j k ,$
(18)

where hijk = kcj + gjbijk. Equation (14) is similar to a knapsack constraint. Using this with Constraints (16) and (17) provides a preprocessing infeasibility: If $∑ i ∈ I ∑ j ∈ J c j = ∑ j ∈ J | I | c j > M$, then the problem is in feasible. This results in a naive condition for feasibility of the problem such that $∑ j ∈ J | I | c j ≤ M$ which means that when we decide our budget level we have to take this into consideration.

In terms of solving this model, Constraint (16) may play an important role in forcing z to take one in any value of k other than simply k = 1 for each pair of (i, j), because without these constraints z would always take zero for all (i, j, k) in order to achieve a minimum cost to the objective function, and this makes the budget constraint (14) non-tight and thus meaningless. However, involving Constraint (16) in the model prevents such a situation from occurring. Instead, combining with Constraint (15) may require zijk = 1 for some high index of k. In particular, when shortage cost gj is high and/or the corresponding value of bijk is large, the model would be likely to purchase more spare parts to satisfy Constraint (15). In this sense, the model seems to be well-defined, although it minimizes the total costs under a knapsack-type constraint. Using this model, the following section discusses our computational experiments and shows the corresponding results based on the empirical data from the Korean Navy.

## 4.CASE STUDY

This section analyses the features of the spare parts used for this research and presents failure rate functions fitted to the failure data. We mainly focus on obtaining an optimal CSP inventory level for the warships’ weapon systems in the Korean Navy.

### 4.1.The Characteristics of the System Failure Data

The Korean Navy purchases CSP for the initial three years of logistical support at the time of the deployment of a new weapon system for about 5% of the system’s purchase budget (Kim and Kim, 2010).

Table 2 presents the characteristics of the spare parts for the Gas Turbine (GT) and Missile Launcher (ML) in a specific type of warship. The number of failures for each part as shown in Table 2 was generally low. The number of failures for GT was double the number for ML. The mean unit price of the spare parts for GT was lower than the price for ML. “Repair time” consists of the time it takes to repair a specific part in a designated place, plus lead time. All of the parts for GT require 730-day repair processes, and this leads to much longer repair times than for ML. Most of the parts were repaired in the depot except for Parts 10 and 11 of ML. Parts 5 and 6 of GT are consumable, and so their repair time and place are left blank.

### 4.2.Fitting Failure Rate Functions to the Data

The failure records of eight parts of GT and seventeen parts of ML for the initial three years were collected. In order to aggregate failure data from several warships, the initial deploying date of each warship was adjusted to the identical initial time, 0, and the failure records for parts in each warship were adjusted accordingly.

The exponential model (as a traditional stationary failure rate) and the Weibull model were used as the alternative failure rate functions for this research.

Mean time between failures (MTBF) of a part was calculated as the operating period (i.e. 1,095 days) divided by the number of failures, λ. The Weibull distribution with two parameters, β and η, was hypothesised to fit the data. The parameters can be estimated using the Weibull probability paper (Abernethy, 2001): (i) ranking failure data (i.e. rearranging so the earliest failure is listed first and the oldest failure is last); (ii) plotting the data on the Weibull probability paper; and (iii) interpreting the plot.

In order to eliminate the drudgery of hand calculations, we used the reliability software Relex 7.5 (Relex Scandinavia, 2014) employing four methods to fit the Weibull distributions to the data. The four methods employed are the mean rank, the binomial method, Hazen’s method, and Benard’s method (see Abernethy (2001) for more detail). Among them, the best fit method for each part was used. For example, for Part 5 of GT, the values of β, η, and R2 fit by the mean rank were 1.416, 689.791, and 0.912, Benard’s method were 1.471, 685.756, and 0.918; by Hazen’s method were 1.518, 682.34, and 0.922; the binomial method were 1.474, 685.548, and 0.919. Thus, we employed the result from Hazen’s method.

Table 3 presents the results of fitting the exponential failure rate λ or the Weibull failure rate λ(t) to the data. The longer MTBF for ML than for GT in Table 3 corresponds to the smaller number of failures for ML than for GT shown in Table 2. More than half of the R2 values indicating goodness of fit were higher than 0.9. However, the R2 of Part 6 of GT was 0.69. This might be owing to the widely varying demand size of Part 6, ranging from 1 to 8. In an early life stage, the failure rate of a machine usually decreases, as illustrated by Wang et al. (2001). Although the initial three years seem to be an early stage of the weapon system, the value of β, 1.03-1.518, which is close to 1.0, appears to indicate a constant failure rate independent of time. As Rausand and Hoyland (2004) noted, in a useful-life stage, the failure rate of many mechanical items increases slightly. Hence, the CSP period, 1,095 days, might span a useful-life or early wear-out stage for the spare parts with slightly increasing failure rates. Figure 1 illustrates λ(t) and λ for Part 1 of GT: λ was higher than λ(t) with β = 1.267 for the first 534 days. This was followed by a value of λ which was lower than λ(t).

## 5.COMPUTATIONAL EXPERIMENTS AND ANALYSIS

The previous section discussed the results of fitting a constant failure rate, λ , and a more practical failure rate function, λ (t) , considering time-dynamics, to the data. With the failure rates obtained in Section 4, the optimization model formulated in Section 3 was used for numerical, empirical tests. A two-echelon model (i.e. depot–base) having one single base in which an initial demand occurs was considered, and in fact this model is similar to the Korean Navy’s repair process for the type of warship used in this paper. A single-depot–multi-base model can be extended in a straightforward way by simply putting more bases into index i, although we used i = 0 to indicate depot and i = 1 for base. Table 2 provides cj , brtj , drtj , and rj for each CSP of Part. Shortage cost gj was set to be 3 cj . This was because in military establishments the shortage cost can outweigh the purchase cost, as noted by Moon et al. (2012). Further research could examine various ratios between the shortage cost and purchase cost. Using the GAMS optimization software which solves an integer programming model via branch-and-bound algorithm (GAMS Development Corporation, 2014), the optimization model was solved for each system: that is, the models for GT and ML were separately solved, in turn. When the model for each system was being solved, both a stationary failure rate and a non-stationary one were considered. In other words, failure rate function λj (t) becomes λ under a stationary failure rate (i.e. HPP) and the Weibull failure rate function is used with parameters β and η , all of which have been obtained from Table 3.Then, Ωij (t) was specified by plugging the values of λj (t) into Equations (5) and (6). As shown in Table 2, many CSPs were not able to be repaired at the base (i.e. rj = 0), and there is no reason to have spare parts at the base for such CSPs. Therefore, Constraint (16) was released for these CSPs. The same was done for CSPs 10 and 11 of ML, which cannot be repaired at the depot. CSPs 5 and 6 of GT are not repairable but consumable. To reflect this in the model, drtj was set to be the same as the whole time period (i.e. 1,095 days), so that once the part breaks down it cannot be repaired by the end of the time period.

The optimization model for the two systems, GT and ML, has been solved on two separate occasions, once with an HPP and once with an NPP. As shown in Table 4, an HPP (stationary failure rate) incurred more costs in both purchases and shortages than an NPP for all systems. Note that the purchase and shortage costs were computed using the first term and the second term of Equation (1) respectively. This could be interpreted that, under an NPP including time dynamics, a more precise and accurate failure rate might be predicted.

This result also corresponds to the results in Tables 5 and 6: that is, optimal purchase quantity and the resulting expected backorders are shown to be smaller when NPP is considered as a failure rate. Here, the expected backorder quantity for each CSP j at echelon i was computed via $E ( D i j ( t ) − s i j * ) = E ( D i j ( t ) ) − s i j * = Ω i j ( t ) − s i j *$, where $s i j *$ is an optimal solution that can be obtained from Equation (11).

For both GT and ML, the optimal inventory level was higher under HPP than under NPP, causing higher purchase costs. While the backorder quantity for GT had more surpluses with an HPP (almost twice as many as with an NPP), the one for ML gave a smaller level of surplus with HPP despite its larger purchase quantities. This was because HPP for ML estimated larger failure rates than NPP, and this induced HPP to have a smaller level of surplus. The overestimation of failure rates by HPP was remarkable for Parts 5 and 6 for GT, as shown in Table 5. Parts 5 and 6 for GT had higher surplus inventories with HPP than with NPP. The large number of failures, low unit price, and consumable features for these parts (as shown in Table 2) might have a more sensitive effect on the CSP inventory level determined with HPP. A positive relationship between the repair time and the backorder quantity was also identified. In Table 6, Parts 10 and 11 for ML had rather higher surplus inventories than other parts with both HPP and NPP. This might be caused by the short repair times of those spare parts (as shown in Table 2). A short repair process required small spare parts inventories in the base or depot, and this led to small backorders.

In addition to the computation of the expected backorder quantity on the basis of a failure rate (function), a simulated backorder level was also computed by tracking the empirical failure records from the Korean Navy.

Figure 2 shows the change in backorder quantities over time, up to the end of the time period (by the 1,095th day). With only 66% of the purchase cost, NPP for ML presented comparable simulated backorders to HPP throughout the period of the experiment. However, in spite of the purchase cost of GT, which was much lower with NPP than HPP, the simulated backorders induced by NPP grew enormously as time passed. A possible explanation for this result may be the high failure rate, low unit price, and long repair time of parts for GT, as shown in Table 2. These characteristics of parts for GT require and cause the inventory system to hold very large CSP inventories. In fact, this is an advantageous condition to HPP, which overestimates the failure rate. However, as stated above, NPP is less sensitive to these characteristics than HPP in determining the CSP inventory level. A logistics manager should be careful of using the Weibull failure rate function within the optimization model when dealing with those spare parts that have a high failure rate, a low unit price, and a long repair time.

The optimal purchase quantity especially for the NPP may be underestimated: In the case of GT Part 5 under the NPP, 28 failures occurred on average, although the optimal purchase quantity was 3. This might be due to the intermittent failures from empirical data and furthermore unavoidable inaccuracy through fitting to the Weibull failure rate function. To overcome this drawback, we attempted to combine the two failure rate functions, the exponential and Weibull ones.

Although dealing with an expected failure rate provides a low level of cost and backorders with NPP, applying that solution to real demand needs particular care. In this sense, considering only NPP as a failure rate, obtained by fitting the data to the Weibull failure rate function, might not be the best choice for GT, although NPP results in a lower purchase cost than HPP. For this reason, a mixed model of failure rates was considered in the following way:(19)

$μ ( t ) = ω 1 λ + ω 2 λ ( t )$
(19)

where μ (t) is a mixed failure rate, and 0 ≤ ω1 ≤ 1 and 0 ≤ ω2 ≤ 1 such that ω1 + ω2 = 1. We attempted to obtain an efficient frontier, a.k.a. Pareto optimality: see Feldman and Serrano (1980), by changing the values of ω1 and ω2 . Starting with ω1 = 0 ( ω2 =1 ), a mixed model increases the value of ω1by 0.1 up to 1.0 (thus, ω2 decreases from 1.0 to 0.0). For both GT and ML, Figure 3 plots the corresponding results that report the relationships between purchase cost and simulated backorder quantity in time, depending on the weight of an HPP and NPP. The top left point of each line represents the case where ω1 : ω2 = 1 : 0, and the bottom right point is the case where ω1 : ω2 = 0 : 1. The intermediate eight points between these two are the corresponding cases where ω1 :ω2 = 0.9 : 0.1 , ω1 : ω2 = 0.8 : 0.2 , and so on. As shown in Figure 3, GT has a more rapidly increasing backorder quantity as the fraction of an HPP decreases, while ML seems not to be quite as sensitive in both simulated backorders and purchase cost. As time goes by, a high weight of an NPP provides a great amount of simulated backorders for GT. However, backorders for ML remain at a low level even when we change the weights.

The mixed model of failure rates could provide a logistics manager with a more flexible CSP level choice: a manager with a strictly limited budget is recommended to use a higher ω2 because of the purchase cost issue; a manager who deals with a critical system such as a weapon system is recommended to employ a higher ω1 in order to maintain a more conservative inventory level. In the case of ML, using a high ω2 is reasonable, because the simulated backorder quantity is not sensitive to the purchase cost. For GT, the manager should take care in selecting the weight, because the simulated backorder quantities are very sensitive to the purchase cost. This high level of sensitivity of GT might also be caused by the features of the parts of GT, i.e. having a high failure rate, a low unit price, and a long repair time, as shown in Table 2.

## 6.CONCLUSIONS

The lifetime distribution of many spare parts is not purely random and generally follows the typical bathtub curve. Logistics managers might have to consider the change of failure rates throughout the lifetime and are recommended to use a non-stationary failure rate for an inventory decision.

In order to determine the spare parts inventory level, a multi-echelon optimization using a non-stationary failure rate was proposed. Hence, it could be contended that the research gaps mentioned in Subsection 2.2 have been filled. This paper used empirical data obtained from the South Korean Navy. In addition to the traditional stationary failure rate (derived from the exponential function), this study employed a specific nonstationary failure rate (i.e. the Weibull function) to fit to the empirical data (research gap i). Although each stage of the lifetime of many spare parts usually follows the Weibull model, little attention has been paid to the adoption of the Weibull failure rate function to analyse the failure patterns for a multi-echelon inventory model. This paper has developed a multi-echelon optimization model to determine concurrent spare parts (CSP) inventory levels using both stationary and non-stationary failure rates with a budgetary limitation. A homogeneous Poisson process (HPP) as a stationary failure rate and a non-homogeneous Poisson process (NPP) based on a Weibull function as a non-stationary failure rate were employed (research gap ii).

The models using the alternative failure rates were compared in terms of backorders and purchase and shortage costs. HPP tended to overestimate the actual failure rate and incur a higher CSP inventory level; NPP, using the Weibull failure rate function, had a tendency to request a lower CSP inventory level and to reduce purchase costs. For Gas Turbine (GT) parts, the purchase cost by NPP was only 33% of that by HPP. For Missile Launcher (ML) parts, NPP presented comparable simulated backorders, with only 66% of the purchase cost of ML using HPP. However, GT with NPP produced simulated backorders that were too large, despite having much lower purchase costs. This might be caused by a high failure rate, a low unit price, and a long repair time of parts for GT, as shown in Table 2. Therefore, a Pareto optimality to find an optimal mixture between HPP and NPP was constructed. The mixed model presents a practical solution for logistics managers.

The findings of this study have an important implication for future practice. Although the CSP optimization model using NPP can reduce purchase costs with backorder quantities comparable to those in the model using HPP, and the mixed model can provide a practical solution for managers, determining a spare parts inventory level for a system with parts characterized as having a high failure rate, a low unit price, and a long repair time requires more attention. These features of parts require and cause the inventory system to hold very large inventories, are an advantageous condition to HPP which overestimates a failure rate, and can finally lead NPP to overlarge backorders compared to HPP.

One may think that our proposed model might be less useful than the typical one using HPP, because the Weibull distribution can be modelled from hindsight. However, when a database including many spare parts that fit to Weibull distributions is built, the proposed model can provide a strong solution to logistic managers. In practice, the Korean military forces are developing such databases for the acquisition of spare parts (Paik et al., 2012).

One source of weakness of this study is that the optimization model adopting the Weibull failure rate did not dominate the model using the stationary failure rate in terms of backorders as shown in Section 5. Possible reasons for this are the potential data distortion caused by a small data set from a small fleet of the specific warships. It is suggested that a larger data set from a larger fleet would help us to identify the more precise performance of the model with the Weibull failure rate function.

## Figure

Failure rate distributions for Part 1 of Gas Turbine.

Simulated backorder quantity in time.

Pareto optimality for a mixed failure rate model.

## Table

A review of multi-echelon inventory models

T = type of reference; AR = academic report; TR = technical report; M = method of research; A = analytical study; S = simulation; E = empirical study; F = failure rate function; HP = homogeneous Poisson; NB = negative binomial; Ex= exponential; Er = Erlang.

Characteristics of the spare parts for the two systems

Fitting failure rates to data: Results

Optimal purchase and shortage costs for GT and ML with HPP and NPP

Optimal purchase and backorder quantities for GT with HPP and NPP

The minus sign in backorder quantity means that there is surplus stock.

Optimal purchase and backorder quantities for ML with HPP and NPP

The minus sign in backorder quantity means that there is surplus stock.

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