Modeling of Life-Limited Spare Units in a Steady-State Scenario

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DEGREE PROJECT, IN OPTIMIZATION AND SYSTEMS THEORY , SECOND LEVEL

STOCKHOLM, SWEDEN 2015

Modeling of Life-Limited Spare Units in a Steady-State Scenario

SARA HALLIN

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Modeling of Life-Limited Spare Units in a Steady-State Scenario

S A R A H A L L I N

Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master Programme in Applied and Computational Mathematics

(120 credits) Royal Institute of Technology year 2015 Supervisor at Systecon AB was Thord Righard Supervisor at KTH was Per Enqvist Examiner was Per Enqvist

TRITA-MAT-E 2015:32 ISRN-KTH/MAT/E--15/32--SE

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Abstract

This thesis studies the problem of modeling life-limited spare units in a steady-state scenario. This means that units that have a predefined lifespan are to be modeled in a scenario where all conditions are kept constant and all transients have faded out.

OPUS10 is a spare parts optimization software developed by Systecon AB. There is no way to explicitly model the life-limited units in OPUS10, although there are different approximate models that are built on adjustments of the failure rate and repair fraction or the definition of preventive maintenance.

The objective of this thesis is to analyze the usage of life-limited items in real life and to investigate what approximated models different OPUS10 users will utilize in their modeling of life-limited units. Furthermore, the objective is to analyze the consequences of the approximated models and to investigate the possibility of an improved model.

The results show that the main interest when choosing which approximated model to use is the type of life limit. There are three different types of operating time life limits investigated. Either the unit is discarded immediately after the life limit is reached, or it is instead discarded at the next failure. There is also the possibility of resetting of the life limit timer at each maintenance. In all three cases, it is shown that if choosing the most fitting approximate model, the results are very accurate. If the life limit is instead measured in calendar time, even the best approximation will give an under-estimation of the expected number of backorders. It is also shown that most of the OPUS10 users model life-limited units as preventive maintenance with discard, which is not the best approximation in any of the types of life limits.

Keywords: Life-limited items, Spare parts optimization, Inventory systems, OPUS10.

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Acknowledgments

I would like to thank Tomas Eriksson at Systecon AB for giving me the opportunity to write this thesis. I would also like to thank my supervisor at Systecon AB, Thord Righard, for great guidance and support. Also, thanks to my supervisor at KTH, Per Enqvist, for the feedback. Finally, thanks to my family and friends for their love an support.

Stockholm, May 2015 Sara Hallin

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Introduction

An airplane, a windmill or a train. These are all examples of complex technical systems on which society relies in order to function properly. In order for a complex technical system to be available when needed, effective maintenance and logistic support is important. Therefore, the support system is vital in order for the system as a whole to meet its objective. In this thesis, when referred to a system, it is considered to consist of two components; the technical system and the support system.

The support system can be designed and analyzed using mathematical modeling. By using different optimization techniques, an optimal design of the support system together with an optimal assortment and allocation of spare parts can be found.

Systecon AB has since the 1970’s developed a spare parts optimization software called OPUS10. It uses analytic methods in order to find the optimal design of the support system and an optimal spare parts allocation given different prerequisites.

1.1 Problem Description

One limitation of the methods used in OPUS10 is the modeling of life-limited items, i.e., spare parts that have a predefined lifespan. It could for example be an engine in an aircraft that must be discarded after a certain amount of flight hours due to safety regula- tions.

OPUS10 is built on the assumption of steady-state, which means that all demand rates and other conditions are kept constant. Since a life-limited item will cause an extra demand for spare parts as the age limit is reached, this needs to be modeled in steady-state.

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1.2. Outline of the Thesis Chapter 1. Introduction

In the model in OPUS10, there is no way of explicitly modeling the life-limited items. There are approximations that are built on adjustment of the failure rate, the repair fraction and/or the definition of preventive maintenance.

This thesis has three main objectives;

• To analyze the use of life-limited spare parts for users of the OPUS10 software.

Furthermore, to identify what approximate models they use in order to include the life-limited items in their OPUS10 models.

• To analyze the consequences of the approximations used when modeling life-limited items in OPUS10.

• To investigate the possibility of an improved way of modeling life-limited spare parts in OPUS10. This will also include a thorough literature search for existing models of life-limited items.

1.2 Outline of the Thesis

This master’s thesis starts with a background chapter, which gives theoretical information about spare parts optimization. Furthermore, it introduces terminology that is used in the thesis, such as different types of repairable units, discardable units, partially repairable units and life-limited units. It also describes the support organization and different types of maintenance. Some different measures of efficiency that can be used when performing the spare parts optimization are proposed. In order to determine the demand for spare parts, two different models called the METRIC model and the Vari-METRIC model are explained. There is also a section about aircraft engine reliability. Finally, there is a short description of the OPUS10 software.

Followed by the background is Chapter 3, where the methods used in order to meet the objective of the thesis are explained. These methods include interviews that are used in order to identify the use of life-limited items and the approximate models that different users of OPUS10 utilize. The methods also consist of a generic test case, which is used both in simulations of the true scenario and in order to analyze the approximate models in OPUS10.

Next, Chapter 5 gives an overview of the conducted interviews. The users describe the approximate models that they use for modeling life-limited items in OPUS10. These models are further developed and discussed in the following chapter, Chapter 6. Here, the mathematical foundation of the models is explained.

The results and the analysis of the results are presented in Chapter 7. The results of the approximated models are compared to the outcome of the simulations of the true scenario

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1.3. Literature Review Chapter 1. Introduction

for four different types of life limits. The results are followed by the conclusions of the thesis. Lastly, some future extensions of the scope are suggested.

1.3 Literature Review

In 1968, Sherbrooke presented the METRIC model, [15], which handles multi-echelon inventory systems when assuming independent and identically distributed lead-times. In 1973, this model was extended by Muckstadt to include more than one indenture [11].

The general idea behind the Vari-METRIC model was proposed by Graves in 1985, [7], and the idea was that the Poisson distribution can be generalized to a negative binomial distribution, which has two parameters. This makes it possible to relax the assumption of independent and identically distributed lead-times. Sherbrooke later extended the idea in 1986 [16]. Sherbrooke’s work also laid the foundation for coming theories and for example his book ”Optimal Inventory Modeling of Systems: Multi-Echelon Techniques”, printed in 1992 [17].

In 1997, Alfredsson presented his PhD thesis ”On the Optimization of Support Systems”, [2]. His work has laid ground for many of the models used in OPUS10. He has also proposed the approximations for life-limited items used in OPUS10 today, see [3].

(S-1,S)-models often make up the base for multi-echelon models, but they most commonly assume unlimited shelf-life. The first work to relax this assumption was that of Schmidt and Nahmias in 1985 [14]. They look at spare parts optimization subject to costs only and do not include backorders, but instead assume that excess demand is lost. In 2010, Olsson and Tydesj¨o extended the model to include backorders [12]. Their work considers a single- product and a single-stock location with Poisson demand, fixed replenishment lead-time and fixed lifetime. However, they only consider the lifetime when the units are in storage, and not when they are installed in a technical system. Therefore, these studies are not applicable to the work in this thesis due to the different nature of the analysis.

Blischke and Murthy’s book, [5], is of a more practical nature and describes case studies and specifies how lifetime limits are used in real life. Ackert’s paper, [1], is specific for aircraft applications.

In addition to the sources mentioned above, Systecon AB’s own material such as ”OPUS10 - Algorithms and Methods”, [18], and ”OPUS10, Getting Started, Part 3 - Spares Calcu- lations”, [19], have been used.

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Chapter 2

Theoretical Background

This chapter will give background information on spare parts optimization and associated terms.

2.1 Spare Parts Optimization

The technical system is designed to fulfill some kind of need. Therefore, it is beneficial if the technical system is available when required. The measure of availability is consequently a common criteria that is used in order to evaluate different system designs. Another important criteria used is the life support cost, LSC, which is the expected investment and operating cost associated with the support system design [2, p. 3].

When a technical system fails, it is due to the failure of some specific item within the system. For example, the failure of an airplane can be due to, for instance, an engine failure or problems with the landing gears. In order to make the system operational again, the faulty item is replaced with a spare one. The locations of the spare part stocks within the support system are called stock points.

In order to respond to the demand for spare parts, each stock point takes a spare from the stock, given that there is stock available. If the inventory at the stock point is empty at the time of the demand, a backorder is generated. This means that the technical system, in this case the airplane, would be grounded due to lack of spares. The expected number of backorders is often used to measure the performance of the system. The probability distribution for the backorders will vary depending on the support system design.

When the demand for a spare part is met at the stock point, the stock level must me replenished. This is done either by reordering from a supplying stock point or by receiving

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2.2. System Description Chapter 2. Theoretical Background

a repaired item from a repair shop.

The assortment of spare parts at the stock points will affect the number of backorders for the system, and thereby, the time the system is operational. Spare parts optimization is the exercise of finding optimal stock sizes and assortment of spares in order to maximize the performance of the technical system to the lowest possible cost.

2.2 System Description

The system consists of two components; the support system and the technical system. One key assumption of the theory in this thesis is that the support system is designed to aid many identical technical systems that together make up a fleet. A common example of this is a fleet of aircrafts, and this example will be used throughout this thesis.

The technical system, here the aircraft, will in turn also consist of several technical components such as an engine, landing gears, auxiliary units, etc, called replaceable items. All these items need to function in order for the aircraft to be operational.

As stated above, the spares are stocked at various places in the support organization.

The support organization is made up of local levels, for example bases, and central levels, for example depots. A system with several levels in the support organization is called a multi-echelon inventory system [17, p. 7], and an example can be seen in Figure 2.1.

Figure 2.1: An example of a multi-echelon inventory system. The squares correspond to the support organization (stores, workshops and bases) and the triangles correspond to the technical systems in operation.

The echelons describe how the supply system is organized. There is also a hierarchy describing the engineering parts, referred to as the indenture structure. The aircraft consists of many subsystems, as described above. If for example the engine fails, it will be replaced

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2.3. Measures of Effectiveness Chapter 2. Theoretical Background

by a spare engine, which makes the aircraft operational again. These first-indenture items are called line replaceable units, LRUs.

When the first-indenture item, here the engine, has been taken to the repair shop, the faulty second-indenture items are replaced. The second-indenture items are called shop- replaceable units, SRUs. In the case of the engine, it could for example be the compressor.

Each SRU can have its own sub-items as well. If there is more than one indenture, it is called a multi-indenture structure [17, p. 8]. An example of an LRU with two SRUs can be seen in Figure 2.2.

Figure 2.2: An example of an LRU with two SRUs.

The demand of spare parts is typically low, the holding cost of having spares in stock is high compared to the replenishment cost, and backorders are expensive. These presump- tions lead to the ordering policy for repairable items. It is assumed that a one-for-one replenishment is applied, which is also called (S-1,S)-policy. This means that each time there is a demand appearance, the stock point immediately issues a resupply order. The (S-1,S)-policy is a well-accepted assumption for spare parts inventory systems for expensive equipment [2, p. 4].

Both the LRUs and SRUs are assumed to be repairable. There are also consumable items that are discarded when faulty. These are typically less expensive items, which makes it more beneficial to discard them when deficient instead of repairing them. A discardable first-indenture item is called discardable unit, DU, while a second-indenture item is called a discardable part, DP. For non-repairable spare parts of low failure rate and high consump- tion, so called (r, Q) inventory policy is usually applied [9]. This is developed further in Section 2.5.

2.3 Measures of Effectiveness

As mentioned in Section 2.1, one objective of the spare parts optimization is to maximize the operational availability, A, of the technical system. Measuring the availability gives a way of evaluating the system, a so called measure of effectiveness.

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The operational availability of a system is defined as the percent of the total fleet that is ready to operate. Let nor (Not Operational Ready) denote the expected number of non- operational aircraft at any given point in time, and let N be the total number of aircraft.

Then the operational availability is given by A = (N − nor)/N .

There are three main properties of the system that will affect the measure nor; the reliability, maintainability and supportability of the system. These three properties will depend both on the technical system and on the support system.

The reliability is defined as the technical system’s capacity to remain in operational status and is built-in in the technical design. The measure of the reliability is the failure rate of the system, λ, which is defined as λ = 1/mtbf, where mtbf is the mean time between failures. The maintainability of the system is the ease of restoring the system to operational status. It is measured in mttr, the mean time to repair. The supportability of the system is associated with the capability of the support system. It could for example be if there are necessary spares, test equipment and personnel available when a maintenance task needs to be performed. The supportability is measured in mwts, the mean waiting time for spares.

The relationship between nor, mttr, mwts and λ can be derived by considering a fleet of N aircraft, which will produce a total failure rate N λ. When an aircraft breaks down, the repair time will be the sum of the actual repair time and the waiting time for spares, mttr and mwts, which gives the expected number of non-operational aircraft as

nor = N λ(mttr + mwts).

Since both λ and mttr to a large extent are a result of the technical design, the focus when designing the support system is mwts [2, p. 8].

For each stock point, there are several important parameters. One of these is S, the stock level, which is the number of spares at that are allocated at the specific stock point. If there are units in the pipeline, it means that they are in repair or are ordered but not yet received. Therefore, the spares on hand are equal to the stock level minus the units in the pipeline, provided that the stock level is larger than or equal to the number of units in the pipeline. The units in the pipeline are sometimes referred to as outstanding orders. Each time there is a failure, it causes a demand, which means that the pipeline increases by one unit. Similarly, the pipeline decreases by one unit when a pipeline unit is received. The expected time a unit spends in the pipeline is called the inventory lead-time and is here denoted T [2, s. 9].

The number of items in the pipeline at time t is denoted Xt. Since Xtis a random variable, {X_t | t ≥ 0} is a random process, which is assumed to be asymptotically stationary and tend to the distribution of the random variable X. When the support system have been

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operational for a long time, so that the system is at steady state, X is interpreted as the number of units in the pipeline at an arbitrary point in time. An example of a single site with the corresponding variables can be seen in Figure 2.3.

Demand'rate'(λ) Resupply'2me'(T)

Stock'level'(S) Number'in'resupply'(X)

Figure 2.3: A single site with the corresponding variables.

If X ≤ S, all demand is met and there will be no backorders. If instead X > S, the number of backorders is X − S. The expected number of backorders, EBO, will therefore be a function of S and is given as

EBO(S) = E[(X − S)⁺], (2.1)

where the distribution of X is assumed to be known. In (2.1), E denotes expectation and x⁺= max{0, x}.

Using the definition of expected value, (2.1) can be written as

EBO(s) =

∞

X

k=S

(k − S)P (X = k). (2.2)

Minimizing the number of backorders is equivalent to maximizing the availability. It is therefore a very commonly used measure of effectiveness [17, p. 38].

If the expected number of backorders is known, it is straightforward to find the mean waiting time for a spare, mwts, since it is equal to the mean duration of a backorder. This is done by using Little’s formula [10], which states that λW = B, where λ is the demand rate at the specific stock point, W is the waiting time and B is the number of backorders.

This gives that the waiting time is W_ak = B_ak/λ_ak, where the index a denotes site a and k denotes LRU k. In order to get mwts, the demand-weighted average is taken. mwts is also a commonly used measure of effectiveness.

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2.4. Probability Distribution of X Chapter 2. Theoretical Background

2.4 Probability Distribution of the Outstanding Demand for Spares

In order to determine the availability of the technical system, it is, as stated in the previous section, convenient to use the expected number of backorders. As can be seen in (2.2), the probability distribution of the number of outstanding LRU demands, P (X = k), also called the pipeline distribution, is needed.

Since the technical systems are operating from the local levels, this is where the backorders are generated. The aim is to fit a probability distribution in all intermediate steps of the multi-echelon system in the backorder calculations.

There are two widely-spread models of the outstanding demand for spares, called the METRIC model and the Vari-METRIC model.

2.4.1 The METRIC Model

In 1968, Sherbrooke presented the Multi-Echelon Technique for Recoverable Item Control (METRIC) for the Air Force, which is described in [15]. The original METRIC model only handled LRUs, i.e., only one indenture. In 1973, this model was extended by Muckstadt to more indentures [11].

There is one key assumption to the METRIC model, which is that the lead-times are assumed independent and identically distributed. This means that the if L_n is the time that unit n spends in the pipeline, then {Ln}^∞_n=1 are independent and identically distributed random variables according to any distribution with mean L. Palm’s theorem [13] states that when the demand process is a Poisson process with rate λ and the lead-times are independent and identically distributed with mean lead-time L, then the stationary probability distribution for the number of items in the pipeline is a Poisson distribution with mean λL.

The expected number of backorders for the LRU at central level is calculated as the sum of the expected number of units in transit, units being repaired directly, units being repaired by the replacement of an SRU and units waiting for a spare SRU [2, p. 18]. The inventory lead-time for unit k at the local level is given by the sum of the mean waiting time for LRU k at the central level, plus the time it takes to ship it to the local level. Following the METRIC assumption, it is given that the number of units in local pipeline a is Po(λâ_kLâ_k) for LRU k. For a given stock level S_kâ, the expected number of backorders is then calculated from (2.2).

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2.4. Probability Distribution of X Chapter 2. Theoretical Background

2.4.2 The Poisson Distribution

Let P (X = k) be the probability that a random variable X takes on a specific value k from some probability distribution. When the time between events is given by an exponential distribution, it is called a Poisson process. The exponential distribution is a memoryless distribution, in which the time of the last demand has no influence on the time of the next demand.

It can be shown that for a Poisson process, the number of demands in a time period of fixed length T is given by the Poisson distribution, X ∈ P o(λT ), with probability density function given by

P (X = k) = e^−λT(λT )^k

k! , k = 0, 1, . . . (2.3)

where λT is the expected number of occurrences in the time interval T [17, p. 20].

According to the METRIC model, the number of units of LRU k in local pipeline a is Po(λ^a_kL^a_k). When a Poisson distribution is to be fitted to existing data, as is the case in the METRIC model, it is only the mean value that has to be fitted. The mean value is given by E[X] = λT .

2.4.3 The Vari-METRIC Model

In the METRIC model, all pipelines were assumed to be Poisson distributed, which gives that E[X] = V ar[X], i.e., the variance in the pipeline equals the expected value. Due to backordering at supplying stocks, the assumption of independent and identically distributed lead-times will not be correct and therefore, the variance usually exceeds the mean, often substantially [17, p. 59].

The general idea behind the Vari-METRIC model was proposed by Graves in 1985 [7]. The idea is that the Poisson distribution can be generalized to a negative binomial distribution, which has two parameters. This makes it possible to fit a mean and variance separately to observed data, which is useful for the reason mentioned above.

Sherbrooke later extended the idea in 1986 [16]. He derived formulas for the pipeline variances at all stock points within the inventory system. The formulas also include expected values and variances of backorder levels that influence the pipeline of the stock point in question. When the expected value and the variance of the pipeline are determined, the parameters of the negative binomial distribution are fitted to these values.

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2.5. Discardable Units Chapter 2. Theoretical Background

2.4.4 The Negative Binomial Distribution

The negative binomial distribution, NegBin(p, r), where r > 0 and 0 < p < 1, has the probability function

P (X = n) =r + n − 1 n

pⁿ(1 − p)^r (2.4)

and the mean and variance are given as

E[X] = rp

1 − p and V ar[X] = rp

(1 − p)². (2.5)

The negative binomial distribution has a relationship with the Poisson distribution [17, p.

60]. As p → 0, the distribution approaches a Poisson distribution. This means that p = 0 is allowed by letting this mean a Poisson approximation.

When fitting data to a negative binomial distribution, the variance-to-mean ratio, V M R is essential. It is defined as

V M R[X] = V ar[X]

E[X] = 1

1 − p. (2.6)

When V M R = 1 the distribution will degenerate to become a pure Poisson distribution.

Thus, when the mean and the variance are known, V M R can also be calculated. The parameters in the NegBin-distribution can be fitted by

r = E[X]

(V M R − 1), p = (V M R − 1)

V M R . (2.7)

This means that it is possible to take observed mean and variance-to-mean ratio (greater than one), determine the parameters r and p, and generate the probability distribution.

2.5 Discardable Units

When the spare units are repairable, the total amount of units in the system is constant.

This is not the case for discardables, since they are discarded when faulty. This means that the number of units in the system will decrease unless they are replenished. For repairables, there is only one decision variable per stock point, which is the stock level. For

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2.5. Discardable Units Chapter 2. Theoretical Background

discardables, however, the replenishment strategy must be decided upon. In OPUS10, a (r, Q)-policy is assumed [18, p. 47]. This means that when the so called inventory position reaches the reorder point r, a batch of Q units is ordered. The parameter Q is called the reorder size. The inventory position is defined as the stock on hand minus the number of backorders plus units in outstanding orders (ordered but yet not received). A graphical illustration of the (r, Q)-policy can be seen in Figure 2.4.

Time Inventory

Q units ordered

Lead time r

Q units recieved

Q units ordered

Shortage Q units recieved

Figure 2.4: Graphical illustration of the (r, Q)-policy for discardable units. When the inventory position drops to r, Q units are ordered, and the inventory position (the dotted line) is restored to r + Q. The lead-time corresponds to the time before the order is received. Due to the lead-time, there is a risk of having shortages or backorders.

Hadley & Whitin showed in [8] that the probability function for the number of outstanding demands X at a stock-based reorder position in steady-state can be written as

P (X = k) = 1 Q

Q−1

X

j=0

P (Y = k − j), (2.8)

where P (Y = k) is the probability of k demands during the lead-time. For a Poisson

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2.6. Partially Repairable Units Chapter 2. Theoretical Background

process, this corresponds to (2.3), where the average lead-time demand is λL, i.e., Y ∈ P o(λL). The expected number of backorders can then be calculated as a function of the reorder position r as

EBO(r) = E[(X − Q(r) − r)⁺] =

∞

X

k=r+Q(r)

(k − Q(r) − r)P (X = k), (2.9)

where the reorder size Q(r) is a function of the reorder position r.

2.6 Partially Repairable Units

In addition to the repairable and discardable units, there are so called partially repairable units, PRUs. These units can in some cases be repaired, while they in other cases must be discarded. The decision between discard and repair is governed by, for example, failure mode and the cost of repairing the item [4, p. 33]. In some instances, a straightforward repair action can restore the functionality of the item, while in other cases, the item is beyond repair. A lot of items fall into this category, since even if an item is repairable there could be a risk that the item is completely wrecked. However, if this risk is small it is a reasonable approximation to treat it as 100 percent repairable [4, p. 33].

In OPUS10 it is assumed, for a partially repairable unit, that a fraction p of all faulty units can be repaired. Therefore, for each faulty unit arriving at the workshop, the probability of repair is p. This holds independent of what has happened to previously arrived units.

The model does not handle an individual maximum number of repairs per item. However, if each individual unit can sustain 3 repairs, and is then discarded at the fourth failure, this is well approximated by using p = 3/4.

Using generating functions, it is shown in [4, p. 34] that the distribution for the number of units in resupply for a PRU is of the same type as for discardables. This means that the model described in Section 2.5 for discardables is also used for partially repairable units.

In (2.8), P (Y = k) is the probability of k demands during the lead-time. For PRUs, the lead-time must be modified to be pT + (1 − p)L, where T is the time to repair a repairable and L is the lead-time for the discardable. This modification will of course change the probability P (Y = k).

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2.7. Modeling Preventive Maintenance Chapter 2. Theoretical Background

2.7 Modeling Preventive Maintenance

When there is corrective maintenance, the failures are assumed to follow a Poisson process and X is described according to the METRIC or Vari-METRIC model. For cyclic demand generated from preventive maintenance (PM), the probability function P (X = n) is instead approximated by a Bernoulli distribution [18, p. 35].

2.7.1 The Bernoulli Distribution

The Bernoulli Distribution is a two-point distribution where P (X = n) > 0 only for two integer values around the mean. The main reason for choosing the Bernoulli distribution is that it is the distribution that has the smallest possible variance-to-mean ratio, V M R.

That fits well with the assumption that cyclic demand is perfectly regular.

The cyclic demand process is based on three parameters, p_h, k and q and is defined as

P_cyclic(X = n) =







1 − p_h if n = k ph if n = k + q

0 otherwise

(2.10)

where q is the cyclic demand batch quantity, and is assumed to be a fixed integer value.

The parameters k and p_h are selected given the mean value m as

k = int[m/q]q, p_h= (m − k)/q,

where int[m/q] denotes the integer part of the fraction. From (2.10), one can see that the number of outstanding cyclic demands can be either k with probability 1 − ph or k + q with probability p_h. This gives the mean value m = k + p_hq.

2.7.2 Combining Corrective Maintenance and Preventive Maintenance In the general case, when the total demand at a stock position is given by a mix of random and cyclic demands, the probability function P (X = n) is calculated as a convolution of the contribution from the random demand, Prandom(X = n), and the contribution from the cyclic demand, P_cyclic(X = n) [18, p. 35]. Since P_cyclic(X = n) 6= 0 for two values of n only, see (2.10), the convolution contains only two terms and can be written as

P (X = n) = (1 − p_h)P_random(X = n − k) + p_hP_random(X = n − k − q), (2.11)

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2.8. OPUS10 Chapter 2. Theoretical Background

where it is assumed that P_random(X = n) = P_random(X = 0) when n < 0 [18, p. 35].

2.8 OPUS10

OPUS10 is a software tool for optimizing logistics support solutions and spare part supply for complex technical systems. The program is owned and developed by Systecon AB, a Swedish employee owned consultancy business that was founded around 1970.

A central element of an OPUS10 optimization is the Cost/Efficiency (C/E) curve. An example can be seen in Figure 2.5. Each point on the curve represents an optimal spares assortment for a given spares investment budget. The overall efficiency resulting from an assortment is shown on the y-axis and the associated spares investment cost on the x-axis.

Each point is thus related to both an efficiency and a cost. Each point also contains the stocking policies for each store in the support organization, that considered together are optimal with respect to the overall efficiency of the support organization for that specific cost.

Figure 2.5: Example of a Cost/Efficiency curve.

The OPUS10 model assumes that the demand for spares is represented by a Poisson process, i.e., a stochastic process where events occur continuously and independently of each another. It also assumes is that the model is stationary. This means that the demand

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2.8. OPUS10 Chapter 2. Theoretical Background

processes and other conditions do not change over time. OPUS10 uses the Vari-METRIC model described briefly in Section 2.4.3, but also gives the choice to use the METRIC model.

An important feature is that OPUS10 is purely analytical. This gives it the great com- petitive advantages of speed and application handiness. The mathematical foundation is described in [18].

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Chapter 3

Method

As previously stated, there is no way of explicitly model life-limited items in OPUS10. The aim of the thesis is to understand the way Systecon’s customers use life-limited spare parts in their applications. Further, it is to analyze the consequences of the approximations used when modeling life-limited items in OPUS10, and investigate the possibility of an improved way of modeling life-limited spare parts.

In this chapter, the methods used to attain the results are described.

3.1 The Usage of Life-Limited Items

In order to investigate the usage of life-limited items in different areas of applications, customers and users of the software OPUS10 are interviewed. The results of the interviews are used to show the need for modeling of life-limited spare parts and what types of approximate models that the different OPUS10 users employs.

Users from different types of industries are interviewed in order to get a complete picture of the problem. The industries include helicopters, aircraft, trains, subways and combat vehicles.

3.2 Generic Test Case

A generic test case is constructed in order to investigate how the approximations of life- limited items used in OPUS10 will differ from the true scenario. The generic test case will

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3.3. Approximated Models of Life-Limited Items in OPUS10 Chapter 3. Method

be used in OPUS10 in order to evaluate the approximations, and it will also be used when simulating the true scenario in MATLAB.

The generic test case is a one-echelon and one-indenture problem. The reason for choosing this as the scenario is that it is the core problem that exists within all larger problems.

In the generic test case, it is assumed that the demand rate is constant. Since it is a one-echelon problem, there is only one base, with its own inventory of spare items and its own workshop. Furthermore, since it is a one-indenture problem, only one type of LRU is considered. The generic test case is assumed to consist of a fleet of technical systems, which will here be assumed to be aircraft, where the LRU considered is the engine.

The population of aircraft is assumed to be finite, which makes it possible to calculate the age of each engine. Each engine is assumed to have a life limit L after which it must be discarded. It is assumed that aircraft with a malfunctioning engine arrive to the base with a rate that is modeled by a Poisson process with intensity λ engines per time unit.

Following the METRIC model described in Section 2.4.1, the number of engines in the workshop at a randomly chosen time is a Poisson random variable with expected value λT , where T is the repair time.

The life-limited units in the generic test case are assumed to follow the (S-1,S)-model, i.e., one-for-one replenishment. This is because the demand is typically low and the holding cost of having spares in stock is high compared to the replenishment cost.

3.3 Approximated Models of Life-Limited Items in OPUS10

There are several different approximations used when modeling life-limited spare parts in OPUS10. The approximations considered in this thesis are the following support strate- gies:

• Basic Model

• Variation to the Basic Model

• Using the Combined Demand Rate

• Preventive Maintenance with Discard

These will be further presented and discussed in Chapter 6.

In order to investigate the accuracy of the approximations, the generic test case is modeled in OPUS10 using the different approximated models in order to include the life limits.

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3.4. Simulation of the True Scenario Chapter 3. Method

3.4 Simulation of the True Scenario

A model of the generic test case is built in MATLAB and life limits are included. The simulation is run repeatedly and averaging is performed in order to get accurate results and to get rid of deviations. In order to confirm the truthfulness of the model and the simulations, the results attained using no life limits are compared to analytic results and simulated results using Systecon’s own simulation software SIMLOX.

Four different simulation models are built, which are described below.

1. Life Limits Measured in Operational Time

When measuring the life limit in operational time, as described in Section 4.1, the items only age when installed in an operating system. As soon as an item has reached its life limit, it is taken out of operation and a demand for a new item is issued.

2. Life Limits Measured in Operational Time with Resetting

The lifetimes are measured in the same way as in the previous case. However, if a random failure occurs and the unit is sent to repair, the age of the unit is reset to zero. As soon as an item has reached its life limit, it is taken out of operation and a demand for a new item is issued.

3. Life Limits Measured in Calendar Time

The item is assumed to age in the same rate when in storage, in repair, or when installed in the technical system. The lifetime of the item is assumed to be measured from delivery. As soon as an item has reached its life limit, it is taken out of operation or storage and a demand for a new item is issued.

4. Life Limits Measured in Operating Time, Variation

The items only age when installed in an operating system. When reaching the life limit, the item is not immediately taken out of operation, but is instead discarded at the next failure.

The simulations of the generic test case are used to get the number of outstanding LRU- demands X. This is in turn used in order to calculate the expected number of backorders.

The mission time is set to 200 000 hours. For each stock solution, a mean value of the backorders from 10 simulations is calculated.

The ages of the engines at the start of the simulation are set to be uniformly distributed between 0 and L, the life limit. This is done both to the engines in storage and the engines installed in the operating systems. The reason for doing this is to more quickly reach steady-state.

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3.5. Choice of Parameters Chapter 3. Method

3.5 Choice of Parameters

The generic test case described in Section 3.2 is run for 9 different cases, each corresponding to a specific scenario. These are labeled according to Table 3.1.

Case 1. Repair time and delivery time equal. Long life limit.

Case 2. Repair time and delivery time equal. Medium life limit.

Case 3. Repair time and delivery time equal. Short life limit.

Case 4. Shorter delivery time than repair time. Long life limit.

Case 5. Shorter delivery time than repair time. Medium life limit.

Case 6. Shorter delivery time than repair time. Short life limit.

Case 7. Repair time and delivery time equal. Lower failure rate. Long life limit.

Case 8. Repair time and delivery time equal. Lower failure rate. Medium life limit.

Case 9. Repair time and delivery time equal. Lower failure rate. Short life limit.

Table 3.1: The nine different cases.

In case 1, 4 and 7, when the life limits are considered long, the life limit is set to three times the mean time between failure, i.e., L = 3mtbf. When the life limit is medium, as in case 2, 5 and 8, it is set equal to the mean time between failure, L = mtbf, and when it is short, as in the remaining cases, it is a third of mtbf, L = mtbf/3.

When the delivery time T_L is shorter than the repair time T_R, as in case 4 − 6, it is set to a third of the repair time, T_L= T_R/3. The lower failure rate used in case 7 − 9 is set to a third of the previous value used in case 1 − 6, which corresponds to multiplying the mean time between failure, mtbf, by three. The corresponding parameter values are presented in Table 3.2.

mtbf [h] T_R [h] T_L [h] L [h]

Case 1 10 000 300 300 30 000

Case 2 10 000 300 300 10 000

Case 3 10 000 300 300 3333.33

Case 4 10 000 300 100 30 000

Case 5 10 000 300 100 10 000

Case 6 10 000 300 100 3333.33

Case 7 30 000 300 300 90 000

Case 8 30 000 300 300 30 000

Case 9 30 000 300 300 10 000

Table 3.2: The input parameters of the different cases.

Each case is also tested for different stock solutions, i.e., s = 0, 1, 2, . . . .

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Chapter 4

Types of Life Limits

This chapter aims to explain the type of life limits that exist and some terms associated with aircraft engine reliability.

4.1 Life-Limited Units

Spare parts that have a predefined lifetime are sometimes referred to as lifed items. There can be different reasons for an item to have lifetime restrictions, which will be discussed further in Section 4.2.

A life-limited item has a given age L after which the item is obligatorily scrapped. The limits for the age L can be measured in different ways:

• Service Life Limit (SLL)

The unit’s service life is the expected lifetime of the unit. The time can be measured either from production of the item or from delivery. In other words, this means that the lifetime of the unit is measured in calender time and that the unit will reach the limit with the same rate either if it is installed in the technical system or if it is in storage.

• Operating Time Limit (OTL)

This gives a limit on the amount of time the unit is allowed to operate. In for example an aircraft, the operating time is measured automatically and there is a nationwide system that tracks all flight hours.

There are also combination cases when items have both an SLL and an OT L.

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4.2. Aircraft Engine Reliability Chapter 4. Types of Life Limits

• On Occurrence (OO)

When there has been for example bad weather or a problematic landing, i.e., a non- predicted incident, an item can be discarded. This is a randomly distributed life limit, as opposed to the previously described deterministic life limits.

• Time Between Overhaul (TBO)

The time between overhaul is a measure on how long an item is allowed to operate before it must be examined. When the overhaul is done, several units are usually repaired. During the overhaul, items are examined for damages. It can for example be to measure the growth of a crack or the thickness of a brake pad. If the measures are below a certain safety margin, the item must be repaired, and if the wear is too high, the item must be discarded.

• Number of operational cycles

The number of operational cycles could for example be the number of landings and take-offs for a landing gear or the number of times a winch is used. When the unit has performed a certain amount of operational cycles, it must be discarded.

• Number of maintenances

This is a measure that is more common for smaller items. For example, some screw- nuts are discarded after 10 disassembles.

In this thesis, the focus has been on the first two types of life limits, i.e., Service Life Limit and Operating Time Limit. Sometimes, the number of operational cycles can also be approximated by a certain amount of operational time.

4.2 Aircraft Engine Reliability

In aircraft engines, there are some specific terms and measures when considering life-limited items. The life-limited parts in aircraft engines generally consist of disks, seals, spools and shafts. A complete set of life-limited parts will generally represent a high proportion, greater than 20%, of the overall cost of the engine [1, p. 16].

4.2.1 Hard life

The hard life of the item is a limit after which the component needs to be replaced and scrapped. The hard life can be flight-, cycle-, and calendar-limited. That is, as soon as the component age reaches its hard time, it is replaced with a new component.

For example, most of the rotating engine units are hard-timed. There are three basic reasons for this: These parts are impossible or very difficult to inspect when they are

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4.2. Aircraft Engine Reliability Chapter 4. Types of Life Limits

installed in the engine, their times to failure are strongly age-related, and if they would fail, the risk of catastrophic consequences is unacceptably high [5, p. 380].

4.2.2 Minimum Issue Life and Soft Life

The term called issue life is the number of hours or cycles that a life-limited component still has remaining before it is due for a scheduled replacement when refitted to a module or engine. Hence, the issue life is the hard life minus the current age [5, p. 387].

The minimum issue life (MISL) is an age limit. If a unit has exceeded this limit and a random failure occurs, then the unit would be discarded or have to be reconditioned before being used as a spare part. To recondition a unit means that it is thoroughly repaired at depot level or by the contractor [6]. Thus, a low MISL would result in a smaller number of parts being replaced before their hard life than would be the case for a higher MISL.

The soft life of an item is not related to its hard life. If a component has exceeded its soft life, it would be reconditioned or replaced the next time the engine in which it is installed is removed for maintenance. Essentially, it is the same as the MISL, except that it can apply to any part and not just those with a hard life, and it is the age (from new), not the hours remaining to the hard life. Thus, a low soft life would result in more reconditions/replacements of this type of component than a high soft life. The fact that a component has exceeded its soft life would not be sufficient reason to ground the aircraft in order to remove the engine, whereas this would be cause for rejection if it had exceeded its hard life [6].

The hard lives of the components exist because of safety reasons, and are not subject to manipulation. However, the minimum issue life and the soft life have nothing to do with safety, but is subject to economical questions [5, p. 387]. If a component has a life limit of 1000 flight hours and breaks after 950 hours, it is often more economical to discard it than to repair it. This is what is regulated by the soft life and minimum issue life.

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Chapter 5

Interviews - The Usage of Life-Limited Items

This chapter will analyze what types of life-limited items that exist and how they are used.

It will also investigate how they are modeled in OPUS10.

People working in different projects using OPUS10 have been interviewed. The questions asked include whether or not there exist life-limited items in the projects, and if the answer is yes, how and if these are modeled in OPUS10. There were also questions about typical life-limited items, tolerances and the need for more accurate modeling in OPUS10.

5.1 Life-Limited Items in Helicopters

Jan Karlsson works as a consultant from Systecon AB in a helicopter project for FMV, F¨orsvarets materielverk, where he does spare parts optimization using OPUS10 and Sys- tecon’s simulation tool SIMLOX for Helicopter NH90.

The helicopter has many life-limited items, all specified in the technical documentation from the constructor. The life limits can be measured in different ways, for example flight hours, calender time or cycles (such as the number of landings for a landing gear or the number of engine starts). The different types of life limits are described in Section 4.1.

When the limits are measured in operating time, there is a ”hard life” of the item, which was discussed in Section 4.2.

When modeling in OPUS10, life limits of 10 years or less are included. Longer life limits are not included because the lifespan of a helicopter is usually between 20 and 30 years

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5.2. Life-Limited Items in Aircraft Chapter 5. Interviews

with planned flight-time of approximately 200 flight hours per year. The life lengths are mixed in relation to mtbf.

In order to model the life-limited units in OPUS10, preventive maintenance is used. The time of the preventive maintenance is specified to equal the life limit, and it is specified that the unit should be discarded at the preventive maintenance. The unit is modeled as a PRU, partially repairable unit, which can be repaired up until discard.

5.2 Life-Limited Items in Aircraft

Two interviews were performed with people with experience in the aircraft industry. One interview was with Dick Ryman at Saab in Sweden and one was with Phil Bean at Systecon UK in the United Kingdom.

5.2.1 Aircraft in Sweden

Dick Ryman works with modeling of support of aircraft at Saab. They have two main categories of life-limited items; time based limits and tolerance based limits. For the time based limits, the time can be measured either in calendar time from production, calendar time from delivery, operating time or a combination of these. For the combination case, the limit that occurs first is limiting. When modeling this, the predicted utilization of the aircraft is used to see which limit that is most likely to be reached first. In the case with tolerance based life limit, the tolerance can be measured in the level of wear, the maximum number of maintenance, or the number of operational cycles.

There is a term called ”Beyond Economical Repair”, which is usually on a 70-75 % level of the life limit of the component. If there is a random failure after this limit is reached, it is not considered economically beneficial to repair the item. This fills the fame purpose as the term ”minimum issue life” which was described in Section 4.2.

Dick Ryman claims that it would be of great importance to Saab if life-limited items could be accurately modeled in OPUS10, especially when modeling autonomous missions or when setting up the initial stock levels.

If there are items with limited calendar time, there is a problem with the minimum order quantity and batch handling for these items. In order to avoid life limits being reached in storage, they either put an interest rate on the item, or designate a higher storage cost than there actually is. This forces OPUS10 to buy smaller batches more often, which decreases the time in storage for each unit before discard. There is no specific method to choose the extra storage cost, and they have used an extra 2-5% of the values in some models.

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5.2. Life-Limited Items in Aircraft Chapter 5. Interviews

Sometimes, there is also a problem with transportation time, which consumes a lot of the calendar time in which the item is allowed to operate. This is solved by buying the items locally. This is however not a very big problem in OPUS10. Some units also have a varying life limit depending on different circumstances. For example, an o-ring has a life limit when it is stored in a fridge, but another limit if it is stored in room temperature. Since this is quite uncommon, there is not a big need to be able to model this in OPUS10.

When modeling in OPUS10, some life limits are included in the model, but not all. The items that are included are modeled as PRUs with preventive maintenance and discard, the same as in the helicopter case.

5.2.2 Aircraft in the United Kingdom

Phil Bean works at Systecon UK and has worked with the Royal Air Force in Great Britain for many years. He says that the most obvious life-limited items in the air force are the engine components that have ”item hard lives”. The term ”hard life” is described in Section 4.2. Some of the items with hard lives also have a so called ”Minimum issue life”, which is discussed in Section 4.2 as well.

Working for the Royal Air Force, Phil Bean has experienced items or systems that have life limits assigned to them based on operating hours, calendar hours, number of missions, cycles, shots fired, landings, distance traveled, starts, firings (e.g. missile launchers), fatigue index (g)/stress/strain. Presently, they tend to try and convert most of these into operating hours based on a mean rate, but this can cause problems, especially if adjusting existing models to represent different operating profiles.

Most items that reach their life limits are discarded, but this is not always the case. Some items may be refurbished and then, if needed, overhauled. Then they can be re-issued with a lesser life assigned, and perhaps also a more strict inspection regime.

According to Phil, a major issue with most engine components is the matching of components and modules when engines are reassembled. This is more of a management issue, but it can require a significant effort. Whilst matching components, the focus is on getting components that will all have similar lives remaining. Often engine modules are held as separate entities and only assembled into an engine when needed, at which time the age of each module would be matched to ensure all modules have approximately the same time remaining to the next scheduled task.

When modeling in OPUS10, it is not always straight forward. Usually, they would specify the life for the component and then remove the item and discard/repair depending on the item requirement. Most life-limited items would result in a discard. Other issues also come into play, for example, items with a Minimum issue life might be removed early, and on

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5.3. Life-Limited Items in Combat Vehicles Chapter 5. Interviews

other occasions they might reach, or even exceed, their intended life. Phil Bean means that they often suggest modeling the worst case, e.g. (a) removal with life remaining and (b) removal at life, since this will at least give an indication of the range of demand that might be observed and a measure of its impact. Removal of one item may result in a percentage chance of removal of other sub-items/items.

5.3 Life-Limited Items in Combat Vehicles and Tracked Ve- hicles

Lars Sj¨odin works at BAE Systems H¨agglunds. They produce combat vehicles and tracked vehicles and have around 10 − 20 life-limited items per model in OPUS10. Examples of units with a life limit are all the tracks and wheels in the vehicle, and also filters, batteries and barrels.

The life limit for a track is measured in the wear. When a certain limit is reached, the track is discarded. The wear can approximately be translated into kilometers, which in turn usually can be translated into operating hours. For a filter, the life limit is measured in transmission hours, i.e., operating hours for the filter. For a barrel, the life limit is measured in operating cycles, i.e., the number of shots. For batteries, it is measured in calendar time. The batteries are modeled as life-limited items, but they are not discarded after a certain amount of time but instead when they ”run out” (which is not deterministic).

They also have a usual failure rate. Since the life limits are not deterministic, there is quite a large variance on the limits.

In OPUS10, the life-limited items are modeled as PRUs. For the PRUs, preventive maintenance with discard is scheduled at the interval of the life limit of the item.

Lars Sj¨odin does not think the results of the model correspond very well to reality. If there for example are 40 band wheels on one vehicle, they will be worn out with different rates depending on placement. Therefore, it is difficult to translate the wear into kilometers or time. The total amount of items is usually correct, but the maintenance resources are allocated poorly since not everything is replaced at the same time. Another problem in OPUS10 is when there is a high flow of articles. For example, there are 168 belt-plates in one steel chain. On each belt-plate, there is a rubber pad. The life limit is measured in wear. If there is a fleet of 100 − 150 vehicles, the flows of rubber pads will be extremely large. In OPUS10, only one article per transport is modeled which makes the result poor.

Therefore, they see a large improvement potential when modeling life-limited items in OPUS10. They think that to model as preventive maintenance on single items works well, but not when there is preventive maintenance on many units at the same time.

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5.4. Life-Limited Items in Rail Vehicles Chapter 5. Interviews

5.4 Life-Limited Items in Rail Vehicles

Both Kristina Abelin and Jon Haugsbak are consultants from Systecon who work at SL, Stockholms Lokaltrafik, which is the company that is responsible for the public transport in Stockholm. They both work on the rail division of Systecon.

For trains and subways, most life limits are measured in wear. The most common example is the wheels, which are worn down after a certain amount of kilometers. The time until preventive maintenance has to be carried out is approximated considering the utilization.

When the preventive maintenance is carried out, the wheels are sharpened in order to maintain the correct shape. There is usually room for 2 − 3 grindings before the wheel has to be scrapped. There is also a combination of normal wear and corrective maintenance due to random failures. If a random failure leads to a damage in the wheel and it has to be extra grind down, the lifetime is shortened and there may only be room for one or two maintenance occasions.

Kristina and Jon both agree that it is difficult to model life-limited units in OPUS10. It is quite common that the life-limits are not included at all. Jon Haugsbak says that one method they use it is to model the wheels with corrective maintenance only, but to let the failure rate be a combination of the life limit and the random failure rate.

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Chapter 6

Approximated Models of

Life-Limited Items in OPUS10

In OPUS10, there are different approximate models that can estimate the effect of having life-limited items in the system. The first two, described in Section 6.1 and Section 6.2, are defined in [3].

6.1 Basic Model

It is assumed that the life-limited item has a given age L,measured in operating hours, at which it will be obligatorily scrapped. Before the time L, it is assumed that the item can be repaired. It is also assumed that the age of the item is preserved during repair.

6.1.1 Replacement Rate and Repair Fraction

The mean operational or installed life of the item is L. The expected number of repairs over the installed life is λL = L/mtbf, where λ denotes the failure rate and mtbf is the Mean Time Between Failure. This gives the repair rate RR, which is the expected number of repairs per operational time unit, as

RR = L/mtbf

L = 1

mtbf.

Since there is only one discard per life cycle, the discard rate, DR, can be calculated in a similar way and is given by DR = 1/L. Then the stationary probability that an item is

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6.2. Variation of the Basic Model Chapter 6. Approximated Models

repaired, the repair fraction ρ, is given by

ρ = RR

RR + DR = 1/mtbf

1/mtbf + 1/L = L

L + mtbf. (6.1)

The total demand rate, denoted DRT , is given as the sum of repair and discard rate, as

DRT = RR + DR = 1 mtbf + 1

L. (6.2)

6.1.2 Modeling in OPUS10

When modeling the Basic model in OPUS10, the life-limited item is approximated as a partially repairable item, PRU, which was described in Section 2.6.

The failure rate of the item in OPUS10 should be interpreted as the total replacement rate DRT treated above in (6.2). This means that the failure rate is adjusted for the forced discard every L time units.

In addition to this, the repair fraction must be specified according to (6.1). In Section 2.6, this corresponds to the factor p.

6.2 Variation of the Basic Model

If the item is not scrapped at time L, the repair life of the item, but instead at the next failure after L, there is a slight variation in the model.

The mean operational life of the item is in this case L+mtbf instead of L. The operational life of the item is then also the time between discard, which gives the variation discard rate, DR^V, as

DR^V = 1

L + mtbf.

The variation replacement rate is given by DRT^V = 1/mtbf, since there is only replacement when the item has failed. This gives the repair rate as

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6.3. Combined Demand Rate Chapter 6. Approximated Models

RR^V = DRT^V − DR^V = 1

mtbf − 1

L + mtbf = L

mtbf (L + mtbf). Using the result in the previous equation, the variation repair fraction ρ^V is given by

ρ^V = RR^V

DRT^V = L

L + mtbf, (6.3)

which is the same as in the basic model. The replacement rate will however not be the same. By comparing DRT^V with the replacement rate of the basic model given in (6.2), the following relation is obtained:

DRT = 1

mtbf+ 1

L = 1

mtbf

1 +mtbf L

= 1

mtbf

L + mtbf

L = DRT^V L + mtbf L

.

When referring to the variation of the basic model in the results, it will be called the Vari-Basic Model.

6.2.2 Modeling in OPUS10

The same modeling approximations that were made for the Basic model are also applied for the Vari-Basic model. This means that the life-limited item is in OPUS10 approximated as a partially repairable unit, PRU. The repair fraction is also specified in the same way as for the Basic model according to (6.3). However, in the Vari-Basic model, the failure rate of the item is unchanged compared to unlimited life and therefore, it does not have to be adjusted in OPUS10.

6.3 Combined Demand Rate

In this model, it is assumed that the operating time limit L is given for an item. When the limit is reached, the item is obligatory discarded. The failure rate λ is known. A key assumption is that a failure will reset the service life limit timer.

Using results from probability theory, the demand rate for a combination of random failures and demand caused by life limits can be found. The theoretical results can be used in OPUS10 as an approximate way to model life-limited units.

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6.3. Combined Demand Rate Chapter 6. Approximated Models

The random failures will, as explained in Section 2.4, arrive according to a Poisson process.

Let p be the probability that life limit is reached, i.e., the probability of no failure during time period L. Then by letting k = 0 in (2.3), p is given by

p = P (X = 0) = e^−λL. (6.4)

The probability that the life limit is reached corresponds to the discard fraction of the item. Therefore, the repair fraction is given by

ρ = 1 − p = 1 − e^−λL. (6.5)

The combined total event rate is calculated by using the expected value of the time to next replacement. The time between repairs when there is random failure only is exponen- tially distributed, as explained in Section 2.4.2. The probability density function for the exponential distribution is given by

f (x) =

(λe^−λx if x ≥ 0

0 if x < 0, (6.6)

where λ is the demand rate. The expected value of the time to next replacement, T , is calculated using the law of total expectation, which is a well-known law in probability theory. This gives

E[T ] = E[T | failure during L]P (failure during L)+E[T | life limit reached]P (life limit reached), (6.7) where P (life limit reached) = p as described in (6.4). Using the probability density function (6.6) in (6.7) and the definition of expected value gives

E[T ] = Z L

0

λxe^−λxdx + Lp,

where it has been used that the time to next replacement given that there are no failures is L. The integral is solved using partial integration, which yields

E[T ] = 1 − e^−λL

λ = 1 − p λ ,

Modeling of Life-Limited Spare Units in a Steady-State Scenario

Modeling of Life-Limited Spare Units in a Steady-State Scenario

Modeling of Life-Limited Spare Units in a Steady-State Scenario

Contents

Chapter 1

Introduction

1.1 Problem Description

1.2 Outline of the Thesis

1.3 Literature Review

Chapter 2

Theoretical Background

2.1 Spare Parts Optimization

2.2 System Description

2.3 Measures of Effectiveness

2.4 Probability Distribution of the Outstanding Demand for Spares

2.5 Discardable Units

2.6 Partially Repairable Units

2.7 Modeling Preventive Maintenance

2.8 OPUS10

Chapter 3

Method

3.1 The Usage of Life-Limited Items

3.2 Generic Test Case

3.3 Approximated Models of Life-Limited Items in OPUS10

3.4 Simulation of the True Scenario

3.5 Choice of Parameters

Chapter 4

Types of Life Limits

4.1 Life-Limited Units

4.2 Aircraft Engine Reliability

Chapter 5

Interviews - The Usage of Life-Limited Items

5.1 Life-Limited Items in Helicopters

5.2 Life-Limited Items in Aircraft

5.3 Life-Limited Items in Combat Vehicles and Tracked Ve- hicles

5.4 Life-Limited Items in Rail Vehicles

Chapter 6

Approximated Models of

Life-Limited Items in OPUS10

6.1 Basic Model

6.2 Variation of the Basic Model

6.3 Combined Demand Rate