Modeling of Life-Limited Spare Units in a Steady-State ...kth.diva-portal.org/smash/get/diva2:826623/FULLTEXT01.pdf · OPUS10 is a spare parts optimization software developed by Systecon

DEGREE PROJECT, IN , SECONDOPTIMIZATION AND SYSTEMS THEORYLEVEL

STOCKHOLM, SWEDEN 2015

Modeling of Life-Limited Spare Unitsin a Steady-State Scenario

SARA HALLIN

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCI SCHOOL OF ENGINEERING SCIENCES

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

Abstract

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

OPUS10 is a spare parts optimization software developed by Systecon AB. There is noway to explicitly model the life-limited units in OPUS10, although there are differentapproximate models that are built on adjustments of the failure rate and repair fractionor the definition of preventive maintenance.

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

The results show that the main interest when choosing which approximated model touse is the type of life limit. There are three different types of operating time life limitsinvestigated. Either the unit is discarded immediately after the life limit is reached, or it isinstead discarded at the next failure. There is also the possibility of resetting of the life limittimer at each maintenance. In all three cases, it is shown that if choosing the most fittingapproximate model, the results are very accurate. If the life limit is instead measured incalendar time, even the best approximation will give an under-estimation of the expectednumber of backorders. It is also shown that most of the OPUS10 users model life-limitedunits as preventive maintenance with discard, which is not the best approximation in anyof the types of life limits.

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

Acknowledgments

I would like to thank Tomas Eriksson at Systecon AB for giving me the opportunity towrite 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, forthe feedback. Finally, thanks to my family and friends for their love an support.

Stockholm, May 2015

Sara Hallin

Contents

1 Introduction 11.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theoretical Background 42.1 Spare Parts Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Measures of Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Probability Distribution of the Outstanding Demand for Spares . . . . . . . 9

2.4.1 The METRIC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 102.4.3 The Vari-METRIC Model . . . . . . . . . . . . . . . . . . . . . . . . 102.4.4 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . 11

2.5 Discardable Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Partially Repairable Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Modeling Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . 14

2.7.1 The Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . . 142.7.2 Combining Corrective Maintenance and Preventive Maintenance . . 14

2.8 OPUS10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Method 173.1 The Usage of Life-Limited Items . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Generic Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Approximated Models of Life-Limited Items in OPUS10 . . . . . . . . . . . 183.4 Simulation of the True Scenario . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Types of Life Limits 214.1 Life-Limited Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Aircraft Engine Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.1 Hard life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Minimum Issue Life and Soft Life . . . . . . . . . . . . . . . . . . . . 23

5 Interviews - The Usage of Life-Limited Items 245.1 Life-Limited Items in Helicopters . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Life-Limited Items in Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Aircraft in Sweden . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2 Aircraft in the United Kingdom . . . . . . . . . . . . . . . . . . . . . 26

5.3 Life-Limited Items in Combat Vehicles and Tracked Vehicles . . . . . . . . . 275.4 Life-Limited Items in Rail Vehicles . . . . . . . . . . . . . . . . . . . . . . . 28

6 Approximated Models of Life-Limited Items in OPUS10 296.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1.1 Replacement Rate and Repair Fraction . . . . . . . . . . . . . . . . 296.1.2 Modeling in OPUS10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.2 Variation of the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2.1 Replacement Rate and Repair Fraction . . . . . . . . . . . . . . . . 306.2.2 Modeling in OPUS10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3 Combined Demand Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3.1 Replacement Rate and Repair Fraction . . . . . . . . . . . . . . . . 326.3.2 Modeling in OPUS10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.4 Preventive Maintenance with Discard . . . . . . . . . . . . . . . . . . . . . 336.4.1 Modeling in OPUS10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.5 Summary of the Approximated Models . . . . . . . . . . . . . . . . . . . . . 34

7 Results and Analysis 357.1 Operational Time Life Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2 Operational Time Life Limit with Resetting . . . . . . . . . . . . . . . . . . 41

7.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.3 Calendar Time Life Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4 Operational Time Life Limit, Variation . . . . . . . . . . . . . . . . . . . . 48

7.4.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 Conclusions 51

9 Future Extensions 53

A Results from Simulations - EBO(s) 54

B Results from Simulations - Percentage Differences 59B.1 Percentage Difference, Operational Time Life Limit . . . . . . . . . . . . . . 59B.2 Percentage Difference, Operational Time Life Limit with Resetting . . . . . 62B.3 Percentage Difference, Calendar Time Life Limit . . . . . . . . . . . . . . . 65B.4 Percentage Difference, Operational Time Life Limit, Variation . . . . . . . . 68

Bibliography 71

Chapter 1

Introduction

An airplane, a windmill or a train. These are all examples of complex technical systemson which society relies in order to function properly. In order for a complex technicalsystem to be available when needed, effective maintenance and logistic support is impor-tant. Therefore, the support system is vital in order for the system as a whole to meetits objective. In this thesis, when referred to a system, it is considered to consist of twocomponents; the technical system and the support system.

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

Systecon AB has since the 1970’s developed a spare parts optimization software calledOPUS10. It uses analytic methods in order to find the optimal design of the supportsystem 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 anaircraft 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 andother conditions are kept constant. Since a life-limited item will cause an extra demandfor spare parts as the age limit is reached, this needs to be modeled in steady-state.

1

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. Thereare approximations that are built on adjustment of the failure rate, the repair fractionand/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 thelife-limited items in their OPUS10 models.

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

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

1.2 Outline of the Thesis

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

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

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

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

2

1.3. Literature Review Chapter 1. Introduction

for four different types of life limits. The results are followed by the conclusions of thethesis. 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 in-ventory systems when assuming independent and identically distributed lead-times. In1973, 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 binomialdistribution, which has two parameters. This makes it possible to relax the assumption ofindependent and identically distributed lead-times. Sherbrooke later extended the idea in1986 [16]. Sherbrooke’s work also laid the foundation for coming theories and for examplehis book ”Optimal Inventory Modeling of Systems: Multi-Echelon Techniques”, printed in1992 [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 alsoproposed 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 commonlyassume unlimited shelf-life. The first work to relax this assumption was that of Schmidtand Nahmias in 1985 [14]. They look at spare parts optimization subject to costs only anddo not include backorders, but instead assume that excess demand is lost. In 2010, Olssonand Tydesjo extended the model to include backorders [12]. Their work considers a single-product and a single-stock location with Poisson demand, fixed replenishment lead-timeand 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 notapplicable 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 studiesand specifies how lifetime limits are used in real life. Ackert’s paper, [1], is specific foraircraft 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.

3

Chapter 2

Theoretical Background

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

2.1 Spare Parts Optimization

The technical system is designed to fulfill some kind of need. Therefore, it is beneficial if thetechnical system is available when required. The measure of availability is consequentlya common criteria that is used in order to evaluate different system designs. Anotherimportant criteria used is the life support cost, LSC, which is the expected investment andoperating 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 thesystem. For example, the failure of an airplane can be due to, for instance, an enginefailure 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 withinthe support system are called stock points.

In order to respond to the demand for spare parts, each stock point takes a spare from thestock, given that there is stock available. If the inventory at the stock point is empty atthe 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 numberof backorders is often used to measure the performance of the system. The probabilitydistribution 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 mereplenished. This is done either by reordering from a supplying stock point or by receiving

4

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 forthe system, and thereby, the time the system is operational. Spare parts optimization isthe exercise of finding optimal stock sizes and assortment of spares in order to maximizethe 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. Onekey assumption of the theory in this thesis is that the support system is designed to aidmany identical technical systems that together make up a fleet. A common example of thisis 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 com-ponents such as an engine, landing gears, auxiliary units, etc, called replaceable items. Allthese 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 amulti-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 correspondto the support organization (stores, workshops and bases) and the triangles correspondto the technical systems in operation.

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

5

2.3. Measures of Effectiveness Chapter 2. Theoretical Background

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

When the first-indenture item, here the engine, has been taken to the repair shop, thefaulty 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 iscalled a multi-indenture structure [17, p. 8]. An example of an LRU with two SRUs canbe 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 ishigh 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-onereplenishment is applied, which is also called (S-1,S)-policy. This means that each timethere 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 expensiveequipment [2, p. 4].

Both the LRUs and SRUs are assumed to be repairable. There are also consumable itemsthat are discarded when faulty. These are typically less expensive items, which makes itmore beneficial to discard them when deficient instead of repairing them. A discardablefirst-indenture item is called discardable unit, DU, while a second-indenture item is called adiscardable 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 inSection 2.5.

2.3 Measures of Effectiveness

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

6


The operational availability of a system is defined as the percent of the total fleet that isready 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 relia-bility, maintainability and supportability of the system. These three properties will dependboth on the technical system and on the support system.

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

The relationship between nor, mttr, mwts and λ can be derived by considering a fleetof N aircraft, which will produce a total failure rate Nλ. 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 = Nλ(mttr + mwts).

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

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

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

7


operational for a long time, so that the system is at steady state, X is interpreted as thenumber of units in the pipeline at an arbitrary point in time. An example of a single sitewith 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 numberof backorders is X − S. The expected number of backorders, EBO, will therefore be afunction 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 andx+ = max{0, x}.

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

EBO(s) =∞∑k=S

(k − S)P (X = k). (2.2)

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

If the expected number of backorders is known, it is straightforward to find the meanwaiting time for a spare, mwts, since it is equal to the mean duration of a backorder. Thisis done by using Little’s formula [10], which states that λW = B, where λ is the demandrate at the specific stock point, W is the waiting time and B is the number of backorders.This gives that the waiting time is Wak = Bak/λak, where the index a denotes site a andk denotes LRU k. In order to get mwts, the demand-weighted average is taken. mwts isalso a commonly used measure of effectiveness.

8

2.4. Probability Distribution of X Chapter 2. Theoretical Background

2.4 Probability Distribution of the Outstanding Demand forSpares

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

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

There are two widely-spread models of the outstanding demand for spares, called theMETRIC 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 onlyhandled LRUs, i.e., only one indenture. In 1973, this model was extended by Muckstadtto more indentures [11].

There is one key assumption to the METRIC model, which is that the lead-times are as-sumed independent and identically distributed. This means that the if Ln is the time thatunit n spends in the pipeline, then {Ln}∞n=1 are independent and identically distributedrandom variables according to any distribution with mean L. Palm’s theorem [13] statesthat when the demand process is a Poisson process with rate λ and the lead-times areindependent and identically distributed with mean lead-time L, then the stationary prob-ability distribution for the number of items in the pipeline is a Poisson distribution withmean λL.

The expected number of backorders for the LRU at central level is calculated as the sum ofthe expected number of units in transit, units being repaired directly, units being repairedby the replacement of an SRU and units waiting for a spare SRU [2, p. 18]. The inventorylead-time for unit k at the local level is given by the sum of the mean waiting time forLRU k at the central level, plus the time it takes to ship it to the local level. Following theMETRIC assumption, it is given that the number of units in local pipeline a is Po(λakL

ak)

for LRU k. For a given stock level Sak , the expected number of backorders is then calculatedfrom (2.2).

9

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 fromsome probability distribution. When the time between events is given by an exponentialdistribution, it is called a Poisson process. The exponential distribution is a memorylessdistribution, in which the time of the last demand has no influence on the time of the nextdemand.

It can be shown that for a Poisson process, the number of demands in a time period offixed length T is given by the Poisson distribution, X ∈ Po(λT ), with probability densityfunction 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 isPo(λakL

ak). 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 isgiven by E[X] = λT .

2.4.3 The Vari-METRIC Model

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

The general idea behind the Vari-METRIC model was proposed by Graves in 1985 [7]. Theidea 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 toobserved data, which is useful for the reason mentioned above.

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

10

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 theprobability 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− pand 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 = 0is allowed by letting this mean a Poisson approximation.

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

VMR[X] =V ar[X]

E[X]=

1

1− p. (2.6)

When VMR = 1 the distribution will degenerate to become a pure Poisson distribution.Thus, when the mean and the variance are known, VMR can also be calculated. Theparameters in the NegBin-distribution can be fitted by

r =E[X]

(VMR− 1), p =

(VMR− 1)

VMR. (2.7)

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

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 meansthat the number of units in the system will decrease unless they are replenished. Forrepairables, there is only one decision variable per stock point, which is the stock level. For

11

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 positionreaches the reorder point r, a batch of Q units is ordered. The parameter Q is called thereorder size. The inventory position is defined as the stock on hand minus the number ofbackorders plus units in outstanding orders (ordered but yet not received). A graphicalillustration 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. Whenthe 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 beforethe order is received. Due to the lead-time, there is a risk of having shortages orbackorders.

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

P (X = k) =1

Q

Q−1∑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

12

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 ∈Po(λL). The expected number of backorders can then be calculated as a function of thereorder position r as

EBO(r) = E[(X −Q(r)− r)+] =∞∑

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 repairableunits, PRUs. These units can in some cases be repaired, while they in other cases mustbe discarded. The decision between discard and repair is governed by, for example, failuremode and the cost of repairing the item [4, p. 33]. In some instances, a straightforwardrepair action can restore the functionality of the item, while in other cases, the item isbeyond repair. A lot of items fall into this category, since even if an item is repairablethere could be a risk that the item is completely wrecked. However, if this risk is small itis 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 unitscan be repaired. Therefore, for each faulty unit arriving at the workshop, the probabilityof 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 ofunits in resupply for a PRU is of the same type as for discardables. This means that themodel 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, thelead-time must be modified to be pT + (1− p)L, where T is the time to repair a repairableand L is the lead-time for the discardable. This modification will of course change theprobability P (Y = k).

13

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 processand X is described according to the METRIC or Vari-METRIC model. For cyclic demandgenerated from preventive maintenance (PM), the probability function P (X = n) is insteadapproximated 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 twointeger values around the mean. The main reason for choosing the Bernoulli distributionis that it is the distribution that has the smallest possible variance-to-mean ratio, VMR.That fits well with the assumption that cyclic demand is perfectly regular.

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

Pcyclic(X = n) =

1− ph 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 ph are selected given the mean value m as

k = int[m/q]q, ph = (m− k)/q,

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

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 randomand cyclic demands, the probability function P (X = n) is calculated as a convolution ofthe contribution from the random demand, Prandom(X = n), and the contribution fromthe cyclic demand, Pcyclic(X = n) [18, p. 35]. Since Pcyclic(X = n) 6= 0 for two values of nonly, see (2.10), the convolution contains only two terms and can be written as

P (X = n) = (1− ph)Prandom(X = n− k) + phPrandom(X = n− k − q), (2.11)

14

2.8. OPUS10 Chapter 2. Theoretical Background

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

2.8 OPUS10

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

A central element of an OPUS10 optimization is the Cost/Efficiency (C/E) curve. Anexample can be seen in Figure 2.5. Each point on the curve represents an optimal sparesassortment for a given spares investment budget. The overall efficiency resulting from anassortment 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 thestocking policies for each store in the support organization, that considered together areoptimal with respect to the overall efficiency of the support organization for that specificcost.

Figure 2.5: Example of a Cost/Efficiency curve.

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

15

2.8. OPUS10 Chapter 2. Theoretical Background

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

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 isdescribed in [18].

16

Chapter 3

Method

As previously stated, there is no way of explicitly model life-limited items in OPUS10. Theaim of the thesis is to understand the way Systecon’s customers use life-limited spare partsin their applications. Further, it is to analyze the consequences of the approximations usedwhen modeling life-limited items in OPUS10, and investigate the possibility of an improvedway 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 interviewsare used to show the need for modeling of life-limited spare parts and what types ofapproximate models that the different OPUS10 users employs.

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

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

17

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 whensimulating the true scenario in MATLAB.

The generic test case is a one-echelon and one-indenture problem. The reason for choosingthis 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 aone-echelon problem, there is only one base, with its own inventory of spare items and itsown workshop. Furthermore, since it is a one-indenture problem, only one type of LRUis 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 calculatethe age of each engine. Each engine is assumed to have a life limit L after which it mustbe discarded. It is assumed that aircraft with a malfunctioning engine arrive to the basewith 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 theworkshop 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 holdingcost 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 inOPUS10. 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 modeledin OPUS10 using the different approximated models in order to include the life limits.

18

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. Thesimulation is run repeatedly and averaging is performed in order to get accurate resultsand to get rid of deviations. In order to confirm the truthfulness of the model and thesimulations, the results attained using no life limits are compared to analytic results andsimulated results using Systecon’s own simulation software SIMLOX.

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

1. Life Limits Measured in Operational TimeWhen measuring the life limit in operational time, as described in Section 4.1, theitems only age when installed in an operating system. As soon as an item has reachedits 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 ResettingThe lifetimes are measured in the same way as in the previous case. However, if arandom failure occurs and the unit is sent to repair, the age of the unit is reset tozero. As soon as an item has reached its life limit, it is taken out of operation and ademand for a new item is issued.

3. Life Limits Measured in Calendar TimeThe item is assumed to age in the same rate when in storage, in repair, or wheninstalled in the technical system. The lifetime of the item is assumed to be measuredfrom delivery. As soon as an item has reached its life limit, it is taken out of operationor storage and a demand for a new item is issued.

4. Life Limits Measured in Operating Time, VariationThe items only age when installed in an operating system. When reaching the lifelimit, the item is not immediately taken out of operation, but is instead discarded atthe 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 thebackorders from 10 simulations is calculated.

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

19

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 correspondingto 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 threetimes the mean time between failure, i.e., L = 3mtbf. When the life limit is medium, asin case 2, 5 and 8, it is set equal to the mean time between failure, L = mtbf, and whenit is short, as in the remaining cases, it is a third of mtbf, L = mtbf/3.

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

mtbf [h] TR [h] TL [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, . . . .

20

Chapter 4

Types of Life Limits

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

4.1 Life-Limited Units

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

A life-limited item has a given age L after which the item is obligatorily scrapped. Thelimits 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 measuredeither from production of the item or from delivery. In other words, this means thatthe lifetime of the unit is measured in calender time and that the unit will reach thelimit with the same rate either if it is installed in the technical system or if it is instorage.

• Operating Time Limit (OTL)This gives a limit on the amount of time the unit is allowed to operate. In for examplean aircraft, the operating time is measured automatically and there is a nationwidesystem that tracks all flight hours.

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

21

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 lifelimit, 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 operatebefore it must be examined. When the overhaul is done, several units are usuallyrepaired. During the overhaul, items are examined for damages. It can for examplebe to measure the growth of a crack or the thickness of a brake pad. If the measuresare below a certain safety margin, the item must be repaired, and if the wear is toohigh, the item must be discarded.

• Number of operational cyclesThe number of operational cycles could for example be the number of landings andtake-offs for a landing gear or the number of times a winch is used. When the unithas performed a certain amount of operational cycles, it must be discarded.

• Number of maintenancesThis 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 LifeLimit and Operating Time Limit. Sometimes, the number of operational cycles can alsobe 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-limiteditems. The life-limited parts in aircraft engines generally consist of disks, seals, spools andshafts. 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 andscrapped. The hard life can be flight-, cycle-, and calendar-limited. That is, as soon as thecomponent 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 basicreasons for this: These parts are impossible or very difficult to inspect when they are

22

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 wouldfail, 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 componentstill has remaining before it is due for a scheduled replacement when refitted to a moduleor 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 arandom failure occurs, then the unit would be discarded or have to be reconditioned beforebeing used as a spare part. To recondition a unit means that it is thoroughly repaired atdepot level or by the contractor [6]. Thus, a low MISL would result in a smaller number ofparts 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 itssoft life, it would be reconditioned or replaced the next time the engine in which it isinstalled is removed for maintenance. Essentially, it is the same as the MISL, except thatit can apply to any part and not just those with a hard life, and it is the age (fromnew), not the hours remaining to the hard life. Thus, a low soft life would result in morereconditions/replacements of this type of component than a high soft life. The fact that acomponent has exceeded its soft life would not be sufficient reason to ground the aircraftin order to remove the engine, whereas this would be cause for rejection if it had exceededits hard life [6].

The hard lives of the components exist because of safety reasons, and are not subject tomanipulation. However, the minimum issue life and the soft life have nothing to do withsafety, but is subject to economical questions [5, p. 387]. If a component has a life limit of1000 flight hours and breaks after 950 hours, it is often more economical to discard it thanto repair it. This is what is regulated by the soft life and minimum issue life.

23

Chapter 5

Interviews - The Usage ofLife-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 questionsasked include whether or not there exist life-limited items in the projects, and if the answeris yes, how and if these are modeled in OPUS10. There were also questions about typicallife-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,Forsvarets 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 documentationfrom the constructor. The life limits can be measured in different ways, for example flighthours, calender time or cycles (such as the number of landings for a landing gear or thenumber 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, whichwas discussed in Section 4.2.

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

24

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

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

In order to model the life-limited units in OPUS10, preventive maintenance is used. Thetime of the preventive maintenance is specified to equal the life limit, and it is specifiedthat the unit should be discarded at the preventive maintenance. The unit is modeled asa 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. Oneinterview was with Dick Ryman at Saab in Sweden and one was with Phil Bean at SysteconUK in the United Kingdom.

5.2.1 Aircraft in Sweden

Dick Ryman works with modeling of support of aircraft at Saab. They have two maincategories of life-limited items; time based limits and tolerance based limits. For the timebased limits, the time can be measured either in calendar time from production, calendartime 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 theaircraft is used to see which limit that is most likely to be reached first. In the case withtolerance based life limit, the tolerance can be measured in the level of wear, the maximumnumber of maintenance, or the number of operational cycles.

There is a term called ”Beyond Economical Repair”, which is usually on a 70-75 % levelof the life limit of the component. If there is a random failure after this limit is reached, itis not considered economically beneficial to repair the item. This fills the fame purpose asthe 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 couldbe accurately modeled in OPUS10, especially when modeling autonomous missions or whensetting up the initial stock levels.

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

25

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

Sometimes, there is also a problem with transportation time, which consumes a lot of thecalendar time in which the item is allowed to operate. This is solved by buying the itemslocally. This is however not a very big problem in OPUS10. Some units also have a varyinglife limit depending on different circumstances. For example, an o-ring has a life limit whenit is stored in a fridge, but another limit if it is stored in room temperature. Since this isquite 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. Theitems 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 Britainfor many years. He says that the most obvious life-limited items in the air force are theengine components that have ”item hard lives”. The term ”hard life” is described in Section4.2. Some of the items with hard lives also have a so called ”Minimum issue life”, which isdiscussed in Section 4.2 as well.

Working for the Royal Air Force, Phil Bean has experienced items or systems that havelife 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), fatigueindex (g)/stress/strain. Presently, they tend to try and convert most of these into operatinghours based on a mean rate, but this can cause problems, especially if adjusting existingmodels to represent different operating profiles.

Most items that reach their life limits are discarded, but this is not always the case. Someitems may be refurbished and then, if needed, overhauled. Then they can be re-issued witha 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 compo-nents 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 gettingcomponents that will all have similar lives remaining. Often engine modules are held asseparate entities and only assembled into an engine when needed, at which time the age ofeach module would be matched to ensure all modules have approximately the same timeremaining to the next scheduled task.

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

26

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 thatthey 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 mightbe observed and a measure of its impact. Removal of one item may result in a percentagechance of removal of other sub-items/items.

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

Lars Sjodin works at BAE Systems Hagglunds. They produce combat vehicles and trackedvehicles and have around 10 − 20 life-limited items per model in OPUS10. Examples ofunits with a life limit are all the tracks and wheels in the vehicle, and also filters, batteriesand barrels.

The life limit for a track is measured in the wear. When a certain limit is reached, thetrack is discarded. The wear can approximately be translated into kilometers, which inturn usually can be translated into operating hours. For a filter, the life limit is measuredin transmission hours, i.e., operating hours for the filter. For a barrel, the life limit ismeasured in operating cycles, i.e., the number of shots. For batteries, it is measured incalendar time. The batteries are modeled as life-limited items, but they are not discardedafter 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 quitea large variance on the limits.

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

Lars Sjodin does not think the results of the model correspond very well to reality. If therefor example are 40 band wheels on one vehicle, they will be worn out with different ratesdepending on placement. Therefore, it is difficult to translate the wear into kilometersor time. The total amount of items is usually correct, but the maintenance resources areallocated poorly since not everything is replaced at the same time. Another problem inOPUS10 is when there is a high flow of articles. For example, there are 168 belt-plates inone steel chain. On each belt-plate, there is a rubber pad. The life limit is measured inwear. If there is a fleet of 100 − 150 vehicles, the flows of rubber pads will be extremelylarge. 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 inOPUS10. 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.

27

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 transportin Stockholm. They both work on the rail division of Systecon.

For trains and subways, most life limits are measured in wear. The most common exampleis the wheels, which are worn down after a certain amount of kilometers. The time untilpreventive maintenance has to be carried out is approximated considering the utilization.When the preventive maintenance is carried out, the wheels are sharpened in order tomaintain the correct shape. There is usually room for 2− 3 grindings before the wheel hasto be scrapped. There is also a combination of normal wear and corrective maintenancedue to random failures. If a random failure leads to a damage in the wheel and it has tobe extra grind down, the lifetime is shortened and there may only be room for one or twomaintenance occasions.

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

28

Chapter 6

Approximated Models ofLife-Limited Items in OPUS10

In OPUS10, there are different approximate models that can estimate the effect of havinglife-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, atwhich it will be obligatorily scrapped. Before the time L, it is assumed that the item canbe 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 repairsover the installed life is λL = L/mtbf, where λ denotes the failure rate and mtbf is theMean Time Between Failure. This gives the repair rate RR, which is the expected numberof 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 asimilar way and is given by DR = 1/L. Then the stationary probability that an item is

29

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 apartially 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 rateDRT treated above in (6.2). This means that the failure rate is adjusted for the forceddiscard 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 nextfailure 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 operationallife of the item is then also the time between discard, which gives the variation discardrate, DRV , as

DRV =1

L+ mtbf.

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

30

6.3. Combined Demand Rate Chapter 6. Approximated Models

RRV = DRT V −DRV =1

mtbf− 1

L+ mtbf=

L

mtbf (L+ mtbf).

Using the result in the previous equation, the variation repair fraction ρV is given by

ρV =RRV

DRT V=

L

L+ mtbf, (6.3)

which is the same as in the basic model. The replacement rate will however not be thesame. 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 theVari-Basic Model.


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

6.3 Combined Demand Rate

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

Using results from probability theory, the demand rate for a combination of random failuresand demand caused by life limits can be found. The theoretical results can be used inOPUS10 as an approximate way to model life-limited units.

31

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 duringtime 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 theitem. 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 tonext 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 theexponential 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 , iscalculated using the law of total expectation, which is a well-known law in probabilitytheory. 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 ] =

∫ L

0λxe−λxdx+ Lp,

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

E[T ] =1− e−λL

λ=

1− pλ

,

32

6.4. Preventive Maintenance with Discard Chapter 6. Approximated Models

where the results from (6.4) have been used.

The total event rate, i.e., the total demand rate for spares, is then given by the inverse ofthe previous result, i.e.,

DRT =λ

1− pif λ > 0. (6.8)

In reality, the demand will not be strict Poisson, but instead a combination of randomdemand and cyclic demand.


When using the approximation in OPUS10, the failure rate is adjusted following the resultsin (6.8). The engine will be modeled as a partially repairable unit, PRU. The repair fractionis given by (6.5).

6.4 Preventive Maintenance with Discard

The interviews have shown that Preventive Maintenance (PM) with Discard is the mostcommon ways to approximately model life-limited units. The mathematical methodsOPUS10 uses to model preventive maintenance are described briefly in Section 2.7.


The unit is specified as a partially repairable unit, PRU, which is described in Section2.6. The life limit of the item is seen as the time between preventive maintenance and isspecified in OPUS10. It is also specified that the item should be discarded when there ispreventive maintenance.

Since the (S-1,S)-model is assumed in the generic test case, see Section 3.2, OPUS10 isforced to only order one item at a time by setting the maximum reorder size to 1.

33

6.5. Summary of the Approximated Models Chapter 6. Approximated Models

6.5 Summary of the Approximated Models

In order to get a clear overall picture of the approximated models that are used in OPUS10,a summary is presented in Table 6.1.

Model Presumption

Basic Unit discarded immediately after the life limit being reached

Vari-Basic Unit discarded at the next failure after the life limit being reached

Combined Demand Rate Failure will reset the service life limit timer or the unit

PM with Discard The interval between discards caused by life limits is perfectly regular

Model Size of Demand Rate (DRT)

Basic Large: Combination of L and mtbf, straight addition

Vari-Basic Small: The same as in the unlimited life case

Combined Demand Rate Medium: Combination of L and mtbf, calculated from assumption

PM with Discard Large: Combination of L and mtbf, calculated automatically in OPUS10

Model Periodicity in DRT

Basic Medium: A combination of random demands and life limits

Vari-Basic Small: Discards from life limits at the random failures only

Combined Demand Rate Medium: A combination of random demands and life limits

PM with Discard Large: Preventive maintenance is completely periodical

Table 6.1: A summary of the approximated models.

34

Chapter 7

Results and Analysis

This chapter will describe the results when comparing the outcome of the simulations tothe approximate models. Each of the approximations described in Chapter 6 are modeledin OPUS10. The true scenario when life limits are measured in different ways, explained inSection 3.4, is simulated in MATLAB for the different parameters described in Section 3.5.The simulations are numbered in the same way as in Section 3.4. This means that thereare results from simulation 1−4 for each of the nine cases, giving resulting cases numbered1.1− 1.4, 2.1− 2.4, . . . , 9.1− 9.4, culminating in a total of 36 different cases.

For the nine cases described in Table 3.1, the resulting expected number of backordersfor different stock solutions, EBO(s), can be seen in Appendix A. The results from thedifferent simulation models are shown in the same figures. The key results will be presentedand discussed in this chapter.

When the scenario is simulated, the demands that are caused by either a discard or a failureare stored in a vector. The duration of each demand is either TL or TR depending on thecause of the demand. Figure 7.1 shows the typical shape of the generated backorders fora randomly chosen scenario. The reason why the backorders do not only take on integervalues is because the values are averaged over multiple simulations.

Figure 7.2 shows another random scenario. Here, the backorders show a periodical patternwith period length equal to the life limit L. The reason for this is a poor choice of pa-rameters, where the demand rate for the whole fleet is extremely low compared to the lifelimit. At the start of the simulation, all items are given a start age uniformly distributedbetween 0 and L. Within the first L time units, all items will reach their life limit, causinga demand for new spares. However, since there is a lead time TL, the oldest new unitarrives at earliest at time TL. Therefore, during the period between L and L+TL, no unitswill reach their life limits. This is what causes the periodicity. When choosing the generic

35

Chapter 7. Results and Analysis

test case, this was taken into consideration when choosing the parameters.

Simulated time (hours) �1050 0.5 1 1.5 2

Num

ber o

f bac

kord

ers

0

0.5

1

1.5

2

2.5

3

3.5

4Number of backorders when s = 4, 6 = 0.0001, L = 30000, TL = 300λ

Figure 7.1: Number of backorders for a random scenario.

Simulated time (hours) �1040 1 2 3 4 5

Num

ber o

f out

stan

ding

dem

and

0

10

20

30

40

50

60Number of backorders when s = 20, 6 = 0.008, L = 1500, TL = 300λ

Figure 7.2: Number of backorders for a scenario with poorly chosen parameters.

36

7.1. Operational Time Life Limit Chapter 7. Results and Analysis

7.1 Operational Time Life Limit

In order to display the accuracy of each model, the results are shown as percentage of thesimulated value for EBO(s), i.e., 100(EBOmodel/EBOsimulation). The results of Case 1− 3are very similar to each other. The results of Case 3 will be discussed here, and can seenin Figure 7.3. The results of Case 1 and 2 are found in Appendix B, Section B.1. SinceCase 3 corresponds to a shorter life limit than Case 1 and 2, it will give more clear results.However, the discussion and analysis for Case 3 can approximately be applied to Case 1and 2 as well.

Number of spares s0 2 4 6 8 10 12 14

EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100

120Case 3.1, Percent of EBO(s), MTBF =10000, T

R = 300, T

L = 300, L = 3333.3333

VariBasicBasicPMCombined DRT

Figure 7.3: Difference between simulation and approximate models for Case 3.1

As can be seen in Figure 7.3, the Basic model is very accurate for all different stocksolutions, while the PM with Discard-model is good for small stocks but not when thereare more units in stock. Using the Combined Demand Rate-model gives poor results andusing the Vari-Basic Model gives extremely poor results. These will give a large under-estimation of the expected number of backorders, which will cause the user to buy toosmall quantities of spares.

When comparing Case 4 and Case 5, their results are similar to each other. The percentagedifference between the simulated results and the models in Case 5 can be seen in Figure7.4, while the results for Case 4 can be found in Appendix B. The results of Case 6 canbe seen in Figure 7.5.

37


Number of spares s0 1 2 3 4 5 6

EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 100, L = 10000



Number of spares s0 1 2 3 4 5 6 7 8 9

EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 100, L = 3333.3333



38


In Figure 7.4, it can be seen that the Basic model is the best in Case 5, when TL = TR/3and L = mtbf. The PM with Discard-model is accurate when s = 0 or s = 1, but forlarges stocks it gives an increasing under-estimation of EBO(s). Similarly as for Case 5,the Basic model is most accurate in Case 6. Here, it is more evident that the PM withDiscard-model gives a poor approximation for large stocks of spares.

In Case 7 − 9, the failure rate is decreased. The results for Case 7 and 8 can be seen inAppendix B, Section B.1, while the percentage difference between the simulated resultsand the models in Case 9 can be seen in Figure 7.6.


EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 10000



Again, the Basic model is the most accurate. However, it gives a slight under-estimationof EBO(s) when s is increasingly large. It can also be seen in Figure 7.6 that in this case,the PM with Discard-model is second best when there are less than 3 spares, while if thereare more than 3 spares, the Combined Demand Rate-model is better.

7.1.1 Analysis

Since the Combined Demand Rate-model assumes that the life limit timer is reset to zeroat each corrective maintenance, the number of discarded items in this model will be lowerthan in reality. Therefore, it could be expected that this model would under-estimate thenumber of backorders in all the cases.

39


In the Vari-Basic model, the failure rate is the same as when there are no life-limited items,i.e., DRT = λ = 1/mtbf, and the only thing that is changed is the repair fraction, which iscalculated from (6.1). Since TR = TL in Case 1−3 and 7−9, the duration of each backorderis the same for the life limit discarded items as for the random failures. This means thatthe repair fraction will have no affect on EBO(s) and the result of the Vari-Basic modelwill be the same as if not including the life-limited items at all in the OPUS10 model. InCase 4 − 6, TL = TR/3, i.e., the delivery time of a new item is a third of the repair time.The demands occur at the same rate as if there were no life-limits, but the mean durationof each demand will be shorter since the delivery time is shorter than the repair time. Therepair fraction in Case 6 is calculated from (6.1) to ρ = 0.25, which specifies that thereare three times as many discards as there are repairs. This gives a lower value of EBO(s)in this case than in Case 3 for all the approximated models that use the repair fraction,which is all except the PM with Discard-model. The results of EBO(s) can be seen in thegraphs for Case 3 and 6 in Appendix A, Figure A.3 and Figure A.6.

Since the Basic model does not assume resetting of life limits, it could be expected thatit would be the best fit for the simulation when life limits are measured in operating timewithout resetting of the life limit counter. In Case 1 − 3, the failure rate is averagelylarge and the repair time and the delivery time are equal. The Basic model combines thetotal demand from random failures and the demand caused by life limits. However, sinceTR = TL, the backorder will last the same amount of time either if it is caused by failureor by a life limit discard. Because of this, the repair fraction has no affect on the numberof backorders. In Case 4− 6, the repair fraction does affect the number of backorders, butFigure 7.5 shows that the Basic model is nevertheless the best approximation. In Case 9,the Basic model gives a slight under-estimation of EBO(s) when s is increasingly large.However, the value of EBO(s) is very small (see Appendix A, Figure A.9) which meansthat very small deviations on the value of EBO(s) between the simulation and the modelgive large differences when measured in percent. In fact, the approximation is still close tothe simulated values in absolute values.

In the PM with Discard-model, the demand rate is adjusted automatically in OPUS10according to the theory in Section 2.7 in order to account for the preventive maintenance.In Case 3, if the value of s is small, this approximation is very good, which can be seenin Figure 7.3. The reason for this is that when s is small, there are almost constantbackorders, especially in the cases when the life limit L is short, such as in Case 3. Havingconstant backorders means that all the the units are either in the workshop or installed inan operating system. This means that the PM with Discard-model will be very accurate,since it gives a constant interval on when the units are to be discarded. However, as sgrows, so does the error. When there are no longer constant backorders, the units will alsospend time in storage. This means that the perfect regularity that the PM with Discard-model assumes is no longer true, and this will cause a large under-estimation of the numberof backorders. The shapes of the PM approximation are similar in Case 6 and 9 as well.

40

7.2. Operational Time Life Limit with Resetting Chapter 7. Results and Analysis

The reason for this is the same as in Case 3, i.e., short life limits (which means that thebackorders are dominated by the life limit discards rather than repairs) and large stocksmeans that the perfect regularity of the PM model is lost.

Both Case 3, 6 and 9 have a life limit constraint that is a third of the mean time betweenfailure, L = mtbf/3. If comparing the results of these three cases, they are very similarwhen considering the shapes of the curves. Since the total number of backorders is higherin Case 3 and 6 than in Case 9, the curves are shifted. This shows that the Basic modelis always to prefer when modeling life-limited items when the life limit is measured inoperating hours and there is no reset of the life limit timer.

7.2 Operational Time Life Limit with Resetting

Similarly to the previous section, the results of the approximated models are shown aspercentage of the simulated value for EBO(s) in order to display the accuracy of eachmodel.

The results of Case 1− 3 are very similar. The results of Case 3 are seen in Figure 7.7 andthe other two are found in Appendix B, Section B.2.


EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200

250


R = 300, T

L = 300, L = 3333.3333



It can be seen in Figure 7.7 that for Case 3, the Combined Demand Rate-model is very

41


accurate. The Basic model gives an over-estimation of the number of backorders for alls, and the over-estimation increases as s gets larger. The Vari-Basic model gives a verypoor approximation of EBO(s) when comparing to the simulated result. For the PM withDiscard-model, EBO(s) is over-estimated except for very large values of s.

Case 5 and Case 6 both assume that TL = TR/3, i.e., the delivery time of a new unit is athird of the repair time. The results can be seen in Figure 7.8 and Figure 7.9.


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100

120

140

160


R = 300, T

L = 100, L = 10000



For Case 5, when L = mtbf, and Case 6, when L = mtbf/3, the results for the CombinedDemand Rate-model, Basic model and Vari-Basic Model are very similar. The PM withDiscard-model, however, differs between the two cases. In Figure 7.8, it can be seen that itgives an over-estimation of EBO(s) for all s, while it in Figure 7.9 is evident that EBO(s)in the PM with Discard-model decreases for large values of s when the life limit is moredominant compared to the random failures. The best approximation in both Case 5 andCase 6 is the Combined Demand Rate-model.

The results for Case 7 and Case 8 can be found in Appendix B, Section B.2. The percentagedifference between the simulated EBO(s) and the models in Case 9 can be seen in Figure7.10.

The result is similar to the previous cases; the Combined Demand Rate-model gives themost accurate result, the Basic model gives an over-estimation of EBO(s) and the PM withDiscard-model gives an over-estimation of EBO(s) for small s and an under-estimation for

42



EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100

120

140


R = 300, T

L = 100, L = 3333.3333




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100

120

140

160


R = 300, T

L = 300, L = 10000



43


larger s. The Vari-Basic model always gives a very large under-estimation of EBO(s).

7.2.1 Analysis

Since the Combined Demand Rate-model assumes that the life limit timer is set to zeroat the time of maintenance and the steady-state total demand rate is calculated from thatassumption, it could be expected that this approximate model would give the best result.In Case 3, if calculating p from (6.4), it gives that p = 0.7165. This corresponds to theprobability of no failure during the life time L, i.e., it gives the probability of the itemreaching its life limit during L. For a longer life limit, i.e., larger L, this probability willdecrease if the failure rate is kept constant.

In Case 1 − 3 and 7 − 9, since TL = TR, the repair fraction will not affect the resultsince a backorder will last the same amount of time either if it is caused by an itembeing repaired or an item being discarded. In the Basic model, the demand rate is notcalculated assuming life limit resetting, but instead as the sum of the demand causedby life limits being reached and the failure rate. In Case 3, this gives DRT = 4 · 10−4

failures/hour, compared to the demand rate of the Combined Demand Rate-model, whichis DRT = 3.527 · 10−4 failures/hour. Since the demand rate in the Basic model is largerthan for the Combined Demand Rate-model, it could be expected that it would give anover-estimation of EBO(s). For larger values of s, the difference is more evident.

The Vari-Basic model gives a very poor result in all cases for this type of life limit. Thereason is the same as discussed in the previous section. In the cases where TL = TR, therepair fraction ρ does not affect the value of EBO(s). Since the demand rate is not changedin the Vari-Basic model, the model will in Case 1− 3 and 7− 9 give the same result as ifnot including life limits in the OPUS10 model at all. In Case 4 − 6, the model will givefewer backorders than if not including the life limits at all. Obviously, this is not whatwill happen in reality since the demand is caused both by random failures and items beingdiscarded because of life limits.

In Case 3, using the PM with Discard-model, the value of EBO(s) is over-estimated exceptfor very large number of s. The cause for this is that the PM with Discard-model does notassume that the ages of the units are set to zero at the time of maintenance. Furthermore,when s is large, the system will lose the perfect regularity that the PM-model assumes. InCase 5 and Case 6, the PM with Discard-model differs between the two cases. In Figure7.8, it can be seen that it gives an over-estimation of EBO(s) for all s, while it in Figure7.9 is evident that EBO(s) in the PM with Discard-model decreases for large values of swhen the life limit is more dominant compared to the random failures. This is because inCase 5, where L = mtbf, there are fewer backorders than in Case 6, when L = mtbf/3. Iflooking at the x-axis in Figure 7.8 and Figure 7.9, it can be seen that the maximum value

44

7.3. Calendar Time Life Limit Chapter 7. Results and Analysis

of s is s = 6 in Case 5 and s = 9 in Case 9. For smaller values of s, the regularity that thePM-approximation assumes is still more intact than for larger s.

It is interesting to compare the results of Case 3 and Case 9, which corresponds to Figure7.7 and Figure 7.10, since the relationship between the failure rate and life limit is keptconstant, but the failure rate is lower in Case 9. The overall shapes of the different approxi-mations are similar, but the break-point at which the PM with Discard-model stops to givean over-estimation and instead gives an under-estimation of EBO(s) is shifted. Overall, thePM with Discard-model works better when the failure rate is higher. In both Case 3 andCase 9, the Basic model gives an over-estimation and the Combined Demand Rate-modelis by far the most accurate.

Looking at all the cases, it is clear that the Combined Demand Rate-model is always thebest approximation when the life limits are measured in operational time and the life limittimer is reset at each maintenance.

7.3 Calendar Time Life Limit

In the same way as in the previous sections, the results of the approximated models inOPUS10 are shown as percentage of the simulated value of EBO(s). The results of Case1 − 3 are similar in the overall performance of the different models. Case 3 is shown inFigure 7.11, while the results of Case 1 and Case 2 can be seen in Appendix B, SectionB.3.

In Case 3, Figure 7.11 shows that the Basic model gives the most exact result whencompared to the simulation. However, all the models give an under-estimation of EBO(s).It can also be seen that although the Basic model is most accurate, it gives an increasingunder-estimation of EBO(s) as s grows larger. When s = 13, which corresponds to asimulated value of EBO(13) ≈ 1.3, no approximated model will give more than 80% of thesimulated value.

The Combined Demand Rate-model gives a poor approximation of EBO(s). It can also beseen in Figure 7.11 that the PM with Discard-model gives approximately the same resultsas the Basic model when there are few items in stock, s = 0, 1, ..., 6, but gives poor resultsfor large stocks. The Vari-Basic model gives extremely poor results.

The results of Case 4− 6 are similar to each other. The percentage difference between thesimulated results and the approximate models for Case 4 and 5 can be seen in AppendixB, Section B.3, while Case 6 can be seen in Figure 7.12.

For Case 6, it can be seen in Figure 7.12 that the Basic model is the closest approximation,although all the models give an under-estimation of the number of backorders. The Vari-

45



EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 3333.3333




EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 100, L = 3333.3333



46


Basic model is worst. For small values of s, the PM with Discard-model is second best,but for larger s, the Combined Demand Rate-model gives a better approximation.

The results of Case 7 and 8 can be seen in Appendix B, Section B.3. The percentagedifference between simulation and model for Case 9 is displayed in Figure 7.13.


EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 10000



It can be seen that as in the other cases, the Basic model gives the best approximation.Similarly to Case 3 and 6, all approximate models give an under-estimation of EBO(s).When the life limit is measured in calendar time, this will also hold for all the other casesexcept in Case 1, 5 and 7, when all models except the Basic model give under-estimations.Case 1 and 7 correspond to longer life limits, which is the reason why the Basic model inthese cases does not give an under-estimation but instead is almost exact. This also holdsfor case 4, where the Basic model is very accurate.

7.3.1 Analysis

In Case 3, all of the approximated models give an under-estimation of the number ofbackorders. Since both the Basic model and the Combined Demand Rate-model are builton life limits being measured in operational time, this could be expected. The units willbecome old faster when they age in storage as well as when installed in the technical system,which causes more backorders than in the two previous simulations.

47

7.4. Operational Time Life Limit, Variation Chapter 7. Results and Analysis

The reason why the Vari-Basic model give poor result is the same as discussed in theprevious sections, i.e., in Case 1 − 3 and 7 − 9 it will not take the life limits into accountat all. Similarly, the analysis about why the PM with Discard-model behaves as it does isthe same as in the previous simulations.

When comparing the overall shape of EBOsimulation/EBOmodel in Case 6, Figure 7.12to those in Case 3, Figure 7.11, it can be seen that they are similar. Some differencesare that the Combined Demand Rate-model gives a better approximation in Case 6 whenTL = TR/3, which is the only difference between Case 3 and Case 6. When the deliverytime of a unit is shorter, the total number of backorders will be smaller, which can alsobe seen in the results of EBO(s) in Appendix A. For the Combined Demand Rate-model,the units are assumed to have a constant failure rate λ. The model assumes that when afailure occurs and the unit is maintained, the life limit timer is set to zero. When the unitis kept in storage, the probability of a failure is zero. However, the age of the items willcontinue to increase in storage since the life limit is measured in calendar time. This meansthat the combined demand rate calculated in Section 6.3 will no longer be correct. Theprobability of the unit reaching its life limit will be increased, since there are no failureswhen kept in storage. This means that this model will give a larger under-estimation ofEBO(s) than in the case with operational time life limits, especially for large values of s,since the larger s is, the more time will the units spend in storage. This is also the reasonwhy the Combined Demand Rate-model works better in Case 6 than in Case 3, i.e., thevalues of s are smaller.

The reason why the simulated value of EBO(s) when s is zero, i.e. EBO(0), is not thesame when the life limits are measured in operational time or in calendar time is that,although there are no units in storage, there will be units in the workshop during repair.The calendar time model assumes that the units will continue to age in the workshop aswell as when in operation. Therefore, the value for EBO(0) is larger in the simulationswith life limits measured in calendar time than in the simulations with life limits measuredin operational time. This can also be seen in the figures in Appendix A.

In total, the Basic model is the best in all cases when the life limit is measured in calendartime. The more dominant the life limit is compared to the random failure, the larger theunder-estimation of EBO(s) will be.

7.4 Operational Time Life Limit, Variation

The results of the fourth simulation are very different from the results of the other simu-lations, since the total number of backorders is much lower. This because the items thathave reached the life limit are discarded after a failure, instead of immediately after thelife limit is reached. The results can be seen in the plots of EBO(s) in Appendix A.

48


By looking at the graphs of EBO(s), it is obvious that the Vari-Basic model gives the bestresult in all different cases with a very large margin to the second best fitting model. Allthe other models give a very large over-estimation of the expected number of backorders,EBO(s). Therefore, only the result for the Vari-Basic model in OPUS10 is shown as per-centage of the simulated value for EBO(s), i.e., 100(EBOV ari−Basic/EBOsimulation).

The results of Case 6 is shown in Figure 7.14, while the other cases can be found inAppendix B, Section B.4. The reason for choosing to display only this result is because inCase 1− 3 and 7− 9, the Vari-Basic model will be the same as not including life limits atall in OPUS10. The reason for this has been explained in Section 7.1.1. Case 6 is one ofthe three cases that have the shortest life limits, i.e., the life limits will be most dominantcompared to the random failures in these cases.


EB

Om

odel

/EB

Osi

mul

atio

n

0

500

1000

1500

2000

2500

3000Case 6.4, Percent of EBO(s), MTBF = 10000, T

R = 300, T

L = 100, L = 3333.3333

VariBasicOPUS10 with no approximation


The Vari-Basic model will give values between 95% and 100.7% of the simulated values.It can also be seen that if not including life limits at all when TL 6= TR can have largeconsequences, since the model will give a value approximately 3000% of the simulated valuein the worst case. However, since the values of EBO(s) are extremely small, the differencewhen measured in absolute values is in fact not very large.

49


7.4.1 Analysis

The results in Figure 7.14 shows that for s = 0, ..., 3, the Vari-Basic model is very accurate.The reason why EBOV ari−Basic/EBOsimulation is zero for larger values of s is that theapproximated model in OPUS10 does not handle as many decimals as the simulation does.Since the value of EBO(s) is very small, it will be rounded off to zero in OPUS10. Therefore,the points when s is large can be ignored since the value from the model are in fact veryclose to the simulated values, which can be seen in the plots for EBO(s) in AppendixA.

Overall, the Vari-Basic model is by far the best in all the cases for this type of life limit.However, in Case 1 − 3 and 7 − 9, the Vari-Basic model will give exactly the same resultas if not including life limits at all in OPUS10. The reason for this has been explained inSection 7.1.1. Therefore, if TL = TR, there is no need to use an approximated model inOPUS10 at all. If TL 6= TR, it can however give a very large difference of EBO(s), as couldbe seen in Figure 7.14.

50

Chapter 8

Conclusions

The first objective of this thesis was to investigate the usage of life-limited items and toidentify what approximate models that different OPUS10 users utilize in their modeling.This was done by conducting interviews. The results show that the most common way toapproximate life-limited items in OPUS10 is to model the life limit as preventive mainte-nance with discard.

In order to analyze the consequences of the approximations used when modeling life-limiteditems in OPUS10, a generic test case was built and used both for the different models inOPUS10 and for running simulations in MATLAB. The results show that the choice ofapproximated model depends entirely on the type of life limit for the unit.

When the life limit restriction is measured in operational time, the Basic model gives avery accurate result for all different input parameters. The worst case is when the failurerate is low compared to the life limits and the lead times and the maximum number ofstock s is large, for which the Basic model gives an under-estimation of EBO(s). However,there is no point in any of the cases where the model is outside the range 90− 105% of thesimulated values.

If the life limit is measured in operating time and the life limit timer is reset at eachmaintenance, the Combined Demand Rate-model is the most accurate in all different cases.The outcome of the model is in all cases 90−110% of the results from the simulations.

When the life limit is measured in calendar time, the Basic model is again the most exact.However, it will give an under-estimation of EBO(s), especially for large values of s. Therange for the outcome of the Basic model is in this case 75 − 101% of the simulatedvalues.

The last type of life limits investigated in this thesis is when the limit is measured in

51

Chapter 8. Conclusions

operating time but the unit is not discarded immediately, but instead at the next failure.In this case, the Vari-Basic model gives by far the best result. However, if TL ≈ TR,there is no need to use an approximated model in OPUS10 at all. In the cases whereTL 6= TR, the outcome of the Vari-Basic model is within 95− 106% of the results from thesimulations.

The last objective of the thesis was to investigate the possibility of an improved way ofmodeling life-limited spare parts in OPUS10. Since the most common way to to modelthe life limits is shown to be the PM with Discard-model, it would give improved results ifinstead using the best suited model for the type of life limit. If choosing the most fittingmodel, the approximations are very accurate in all the cases except for when the life limitsare measured in calendar time.

When the life limit is measured in calendar time, a possibility to improve the modelingwould be to adjust the failure rate in order to take the aging in storage into account. Thiscould possibly be done by first calculating the probability of each unit being in storage andthe replacement rate in storage, which then only corresponds to the life limit discards. Sec-ondly, the probability of the unit being in operation and the replacement rate in operation,which will then be a combination of the failure rate and the life limit discards, should becalculated. Finally, the expected time to next replacement can be calculated after whichit is possible to determine the total demand rate.

52

Chapter 9

Future Extensions

This thesis has only investigated life limits measured in calendar time and different typesof operational time. There are, however, other types of life limits, which was explained inSection 4.1. It would be interesting to analyze the approximated models in the cases forother life limits as well. One very common measure of life limits in aircraft is the numberof cycles, which could for example be the number of landings for a landing gear. To havea way of modeling cycles in steady-state would be of great benefit when modeling aircraftsupport systems.

When deciding which model to use, this thesis has used the expected number of backorders,EBO(s), as the measure of effectiveness. However, the cost of the items has not beenconsidered. In Section 4.2, terms such as soft life and minimum issue life are explained.It would be interesting to use optimization in order to determine a life limit for when itis more profitable to buy a new item instead of repairing a broken one. Additionally, itwould be interesting to see if any of the approximate models can be modified in order toaccount for minimum issue life or soft life as well.

Another extension of this thesis would be to investigate the possibility of finding an im-proved approximated model for the life limits measured in calendar time. As explained inthe conclusion chapter, a possibility to improve the modeling would be to adjust the failurerate. This could be done by calculating the probability of the item being in storage or not,and then calculate the replacement rate in storage and in operation, respectively. Thatwould make it possible to calculate the expected time to next replacement, which makes itpossible to determine the total demand rate.

53

Appendix A

Results from Simulations -EBO(s)


EB

O(s

)

0

0.5

1

1.5

2

2.5

3

3.5

4Case 1, EBO(s) when MTBF =10000, T

R = 300, T

L = 300, L = 30000

Simulation, Op TimeSimulation, Op Time with ResetSimulation, Calendar TimeSimulation, Op Time, VariationVariBasicBasicPMCombined DRT

Figure A.1: EBO(s) for Case 1.

54

Appendix A. Results from Simulations - EBO(s)

Number of spares s0 1 2 3 4 5 6 7 8

EB

O(s

)

0

1

2

3

4

5

6


R = 300, T

L = 300, L = 10000




EB

O(s

)

0

2

4

6

8

10

12


R = 300, T

L = 300, L = 3333.3333



55



EB

O(s

)

0

0.5

1

1.5

2

2.5

3

3.5Case 4, EBO(s) when MTBF =10000, T

R = 300, T

L = 100, L = 30000




EB

O(s

)

0

0.5

1

1.5

2

2.5

3

3.5

4


R = 300, T

L = 100, L = 10000



56



EB

O(s

)

0

1

2

3

4

5

6


R = 300, T

L = 100, L = 3333.3333



Number of spares s0 0.5 1 1.5 2 2.5 3

EB

O(s

)

0

0.2

0.4

0.6

0.8

1

1.2


R = 300, T

L = 300, L = 90000



57


Number of spares s0 0.5 1 1.5 2 2.5 3 3.5 4

EB

O(s

)

0

0.5

1

1.5

2


R = 300, T

L = 300, L = 30000




EB

O(s

)

0

0.5

1

1.5

2

2.5

3

3.5

4


R = 300, T

L = 300, L = 10000



58

Appendix B

Results from Simulations -Percentage Differences

B.1 Percentage Difference, Operational Time Life Limit


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 30000


Figure B.1: Difference between simulation and approximate models for Case 1.1

59

B.1. Operational Time Appendix B. Percentage Difference


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 10000




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 100, L = 30000



60

B.1. Operational Time Appendix B. Percentage Difference


EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 90000




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 30000



61

B.2. Operational Time, Resetting Appendix B. Percentage Difference

B.2 Percentage Difference, Operational Time Life Limit withResetting


EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200

250


R = 300, T

L = 300, L = 30000



62



EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200

250

300

350


R = 300, T

L = 300, L = 10000




EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100


R = 300, T

L = 100, L = 30000



63



EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200


R = 300, T

L = 300, L = 90000




EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200

250


R = 300, T

L = 300, L = 30000



64

B.3. Calendar Time Appendix B. Percentage Difference

B.3 Percentage Difference, Calendar Time Life Limit


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 30000




EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 10000



65



EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 100, L = 30000




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 100, L = 10000



66



EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 90000




EB

Om

odel

/EB

Osi

mul

atio

n

0

10

20

30

40

50

60

70

80

90


R = 300, T

L = 300, L = 30000



67

B.4. Operational Time, Variation Appendix B. Percentage Difference

B.4 Percentage Difference, Operational Time Life Limit, Vari-ation


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 30000



68



EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 10000




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 3333.3333



69



EB

Om

odel

/EB

Osi

mul

atio

n

0

50

100

150

200


R = 300, T

L = 100, L = 30000




EB

Om

odel

/EB

Osi

mul

atio

n

0

100

200

300

400

500


R = 300, T

L = 100, L = 10000



70

Appendix B. Percentage Difference


EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 90000




EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 30000



71

Appendix B. Percentage Difference

Number of spares s0 1 2 3 4 5

EB

Om

odel

/EB

Osi

mul

atio

n

0

20

40

60

80

100


R = 300, T

L = 300, L = 10000



72

Bibliography

[1] S. Ackert. Basics of Aircraft Maintenance Reserve Development and Management.Written for Aircraft Monitor, Version 1.0, 2012.

[2] P. Alfredsson. On the Optimization of Support Systems. PhD thesis, Royal Instituteof Technology, Stockholm, Sweden, 1997.

[3] P. Alfredsson. Lifed Items Revisited. Research paper for Systecon AB, 2000.

[4] P. Alfredsson. Lecture Notes in Spare Parts Optimization. Written for Systecon AB,2002.

[5] W. R. Blischke and D. N. P. Murthy. Case Studies in Reliability and Maintenance.Wiley Series in Probability and Statistics. John Wiley & Sons, Hoboken, New Jersey,2003.

[6] J. Crocker and U.D. Kumar. Age-related maintenance versus reliability centred main-tenance: a case study on aero-engines. Reliability Engineering and System Safety,(67):113–118, 2000.

[7] S. C. Graves. A multi-echelon inventory model for a repairable item with one-for-onereplenishment. Management Science, (31):1247–1256, 1985.

[8] G. Hadley and T.M. Whitin. Analysis of Inventory Systems. Prentice-Hall Quantita-tive Methods Series. Prentice-Hall, 1963.

[9] F. S. Hillier and G. J. Lieberman. Introduction to Operations Research. McGraw-Hill,1989.

[10] J.D.C. Little. A proof of the queueing formula: L = λW . Operations Research,(9):383–387, 1961.

[11] J. Muckstadt. A model for a multi-item, multi-echelon, multi-indenture inventorysystem. Management Science, (20):472–481, 1973.

73

Bibliography

[12] F. Olsson and P. Tydesjo. Inventory problems with perishable items: Fixed lifetimesand backlogging. European Journal of Operational Research, (1):131–137, 2010.

[13] C. Palm. Analysis of the Erlang traffic formula for busy-signal arrangements. EricssonTechnics, (5):39–58, 1938.

[14] C.P. Schmidt and S. Nahmias. (S-1,S) Policies for Perishable Inventory. ManagementScience, (6):719–728, 1985.

[15] C. C. Sherbrooke. METRIC: A Multi-Echelon Technique for Recoverable Item Control.Operations Research, (16):122–141, 1968.

[16] C. C. Sherbrooke. Vari-METRIC: Improved approximation for multi-indenture, multi-echelon availability models. Operations Research, (34):311–319, 1986.

[17] C. C. Sherbrooke. Optimal Inventory Modeling of Systems: Multi-Echelon Techniques.John Wiley & Sons, Bethesda, Maryland, 1992.

[18] Systecon AB. OPUS10 - Algorithms and Methods. Version 10.0, 2014.

[19] Systecon AB. OPUS10, Getting Started, Part 3 - Spares Calculations. Version 10.0,2014.

74

TRITA -MAT-E 2015:32

ISRN -KTH/MAT/E--15/32--SE

www.kth.se

Documents

Modeling of Life-Limited Spare Units in a Steady-State ...kth.diva-portal.org/smash/get/diva2:826623/FULLTEXT01.pdf · OPUS10 is a spare parts optimization software developed by Systecon