Name: Solving Complex Maintenance Planning Optimization Problems Using Stochastic Simulation and Multi-Criteria Fuzzy Decision Making
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Author: Sahar Tahvili
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The main goal of this project is to explore the use of stochastic simulation, genetic algorithms, fuzzy decision making and other tools for solving complex maintenance planning optimization problems. We use two different maintenance activities, corrective maintenance and preventive maintenance. Since the evaluation of specifc candidate maintenance policies can take a long time to execute and the problem of finding the optimal policy is both non-linear and non-convex, we propose the use of genetic algorithms (GA) for the optimization. The main task of the GA is to find the optimal maintenance policy, which involves: (1) the probability of breakdown, (2) calculation of the cost of corrective maintenance, (3) calculation of the cost of preventive maintenance and (4) calculation of ROI (Return On Investment). Another goal of this project is to create a decision making model for multi-criteria systems. To find a near-optimal maintenance policy, we need to have an overview over the health status of the system components. To model the health of a component we should find all the operational criteria that affect it. We also need to analyze alternative maintenance activities in order to make the best maintenance decisions. In order to do that, the TOPSIS method and fuzzy decision making has been used. To evaluate the proposed methodology, internal combustion engine cooling of a typical Scania truck was used as a case study.

Extrait

MASTER THESIS IN MATHEMATICS/APPLIED

MATHEMATICS

Solving Complex Maintenance Planning Optimization

Problems Using Stochastic Simulation and

Multi-Criteria Fuzzy Decision Making

Sahar Tahvili

Masterarbete i Matematik/Till¨

ampad matematik

DIVISION OF APPLIED MATHEMATICS

M ¨

ALARDALEN UNIVERSITY

SE-721 23 V ¨

ASTER°

AS, SWEDEN

Master Thesis In Mathematics/Applied Mathematics

Date:

2014-06-03

Project name:

Solving Complex Maintenance Planning Optimization Problems Using Stochas-

tic Simulation and Multi-Criteria Fuzzy Decision Making

Author:

Sahar Tahvili

Comprising:

30 ECTS credits

Dedication

Dedicated to my parents and Razieh Matini who always encouraged me

Acknowledgements

This thesis was done with the aid, help and great support of my supervisors

Professor Sergei Silvestrov and Jonas ¨

Osterberg.

Special thanks to my supervisor Dr. Jonas Biteus at Scania who responded

to all of my questions and also the IRIS-project team at Scania CV AB who

cared so much about my work.

September 5, 2014

V¨

aster°

as, Sweden

Abstract

The main goal of this project is to explore the use of stochastic simulation,

genetic algorithms, fuzzy decision making and other tools for solving complex

maintenance planning optimization problems. We use two different mainte-

nance activities, corrective maintenance and preventive maintenance. Since

the evaluation of specific candidate maintenance policies can take a long time

to execute and the problem of finding the optimal policy is both non-linear

and non-convex, we propose the use of genetic algorithms (GA) for the op-

timization. The main task of the GA is to find the optimal maintenance

policy, which involves: (1) the probability of breakdown, (2) calculation of

the cost of corrective maintenance, (3) calculation of the cost of preventive

maintenance and (4) calculation of ROI (Return On Investment). Another

goal of this project is to create a decision making model for multi-criteria

systems. To find a near-optimal maintenance policy, we need to have an

overview over the health status of the system components. To model the

health of a component we should find all the operational criteria that af-

fect it. We also need to analyze alternative maintenance activities in order

to make the best maintenance decisions. In order to do that, the TOPSIS

method and fuzzy decision making has been used. To evaluate the proposed

methodology, internal combustion engine cooling of a typical Scania truck

was used as a case study.

Keywords: Genetic algorithm, Corrective maintenance, Preventive mainte-

nance, ROI, Multi-criteria decision making, TOPSIS, Fuzzy decision making,

Discrete event simulation, Intelligent agent

Sammanfattning

Det huvudsakliga m°

alet med det h¨

ar projektet ¨

ar att utforska anv¨

andan-

det av stokastisk simulering, genetiska algoritmer, fuzzy beslutsst¨

od och an-

dra verktyg f¨

or optimering av komplexa underh°

alls-planerings-problem.Vi

anv¨

ander oss av tv°

a olika underh°

allsaktiviteter, korrektivt underh°

all och

preventivt underh°

all. Eftersom utv¨

arderingen av specifika kandidater for

underh°

allspolicys kan ta l°

ang tid att genomf¨

ora och problemet med att hitta

den optimala policyn ¨

ar b°

ade icke-linj¨

art och icke-konvext s°

a f¨

oresl°

ar vi

anv¨

andning av genetiska algoritmer (GA) f¨

or optimeringen. Den viktigaste

uppgiften f¨

or GA ¨

ar att hitta den optimala underh°

allspolicyn, vilket inneb¨

ar:

(1) sannolikheten f¨

or break-down, (2) Ber¨

akningen av kostnaden f¨

or korrek-

tivt underh°

all, (3) ber¨

akning av kostnaden f¨

or preventivt underh°

all och (4)

ber¨

akning av ROI (Return On Investment). Ett annat m°

al med projektet ¨

att skapa en beslutsmodell f¨

or multiobjektiv-system. F¨

or att hitta en n¨

ara-

optimal underh°

allspolicy s°

a m°

aste vi ha en ¨

overblick ¨

over h¨

alsotillst°

andet

hos systemkomponenterna. F¨

or att modellera h¨

alsan hos en komponent s°

vi beh¨

over hitta alla kriterier som p°

averkar den. Vi m°

aste ocks°

a analysera

alternativa underh°

allsaktiviteter f¨

or att kunna fatta de b¨

asta besluten f¨

underh°

allet. F¨

or att g¨

ora det s°

a har TOPSIS-metoden och fuzzy beslutsst¨

anv¨

ants. F¨

or att utv¨

ardera den f¨

oreslagna metoden s°

a valdes kylsystemet i

en typisk Scania lastbil f¨

or en fallstudie.

Keywords: Genetisk algoritm, Korrektivt underh°

all, Preventivt underh°

all,

ROI, Multicriteria decision making, TOPSIS, Fuzzy beslutsst¨

od, Diskret

event simulering, Intelligent agent

Contents

List of Tables

List of Figures

1 Introduction

1.1

Preventive Maintenance

. . . . . . . . . . . . . . . . . . . . . 11

1.1.1

Application and Advantage

. . . . . . . . . . . . . . . 11

1.2

Corrective Maintenance . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1

Advantage and Disadvantage

. . . . . . . . . . . . . . 12

1.3

Return on Investment . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1

Life Cycle Cost . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2

Total Cost Calculation . . . . . . . . . . . . . . . . . . 14

1.4

Replacement Strategies . . . . . . . . . . . . . . . . . . . . . . 16

1.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Component health

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2

Definitions and Functions

. . . . . . . . . . . . . . . . . . . . 19

2.3

Cooling System . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4

The Consequences of PM on the Component Health . . . . . . 25

2.5

Dynamic Reliability . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1

Riccati differential equation . . . . . . . . . . . . . . . 25

2.5.2

Dynamic Reliability Equation . . . . . . . . . . . . . . 27

2.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Multi Criteria Fuzzy Decision Making (MCFD)

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2

Fuzzy Set and Membership Function . . . . . . . . . . . . . . 31

3.3

IFS Generalize Fuzzy Sets . . . . . . . . . . . . . . . . . . . . 31

3.4

Fuzzy Implication Operators . . . . . . . . . . . . . . . . . . . 32

3.5

Inclusion Degree Function of IFS . . . . . . . . . . . . . . . . 32

3.6

TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6.1

The Structure of TOPSIS Method . . . . . . . . . . . . 33

CONTENTS

3.6.2

TOPSIS Method in Multiple Criteria Fuzzy Decision

Making

. . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7

Problem Statement

. . . . . . . . . . . . . . . . . . . . . . . 36

3.8

A Numerical Example

. . . . . . . . . . . . . . . . . . . . . . 38

3.9

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Optimization Methods

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2

Classical and Non-classical Optimization Methods . . . . . . . 42

4.3

Global Optimization . . . . . . . . . . . . . . . . . . . . . . . 43

4.4

Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . 45

4.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Genetic Algorithm

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2

Structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3

Applications and Advantages

. . . . . . . . . . . . . . . . . . 48

5.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Maintenance Optimization Model

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2

Simulation Framework Model . . . . . . . . . . . . . . . . . . 50

6.3

The Simulation Algorithm . . . . . . . . . . . . . . . . . . . . 52

6.4

Example Problem Class

. . . . . . . . . . . . . . . . . . . . . 53

6.5

A Numerical Example

. . . . . . . . . . . . . . . . . . . . . . 55

6.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Summary, Conclusions and Future Work

7.1

Summary

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.2

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.3

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Summary of reflection of objectives in the thesis

8.1

Objective 1 - Knowledge and Understanding . . . . . . . . . . 63

8.2

Objective 2 -Methodological Knowledge . . . . . . . . . . . . . 64

8.3

Objective 3 - Critically and Systematically Integrate Knowledge 64

8.4

Objective 4 - Independently and Creatively Identify and Carry

out Advanced Tasks

. . . . . . . . . . . . . . . . . . . . . . . 64

8.5

Objective 5 - Present and Discuss Conclusions and Knowledge

8.6

Objective 7 - Scientific, Social and Ethical Aspects

. . . . . . 65

A Java code - Event Triggers

CONTENTS

B Java code - Genetic Algorithm

C Java code - Simulation Classes

D Java code - Example Problem Class

List of Tables

1.1

Variable descriptions . . . . . . . . . . . . . . . . . . . . . . . 17

2.1

Idler Roller parts description . . . . . . . . . . . . . . . . . . . 23

2.2

The values in this table has been hidden by the request from

Scania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1

Binary implication . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2

Decision Making by TOPSIS . . . . . . . . . . . . . . . . . . . 35

3.3

The Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4

The alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5

The inclusion degrees of A

in M

, M

and inclusion degrees

of A

in M

, M

. . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6

The inclusion degrees of A

in M

, M

and inclusion degrees

of A

in M

, M

. . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7

The inclusion degrees D

) and D

) . . . . . . . . . . . 40

6.1

Parameters defining random events. . . . . . . . . . . . . . . . 56

6.2

Parameters defining plannable events. . . . . . . . . . . . . . . 56

6.3

Parameters defining efficiency measures.

. . . . . . . . . . . . 56

6.4

Parameters defining the default event.

. . . . . . . . . . . . . 56

List of Figures

1.1

Corrective maintenance function. . . . . . . . . . . . . . . . . 12

1.2

Costs illustration [6]

. . . . . . . . . . . . . . . . . . . . . . . 15

1.3

An illustration of component age as a function of time with

periodic preventive replacement . . . . . . . . . . . . . . . . . 15

2.1

The bathtub curve hazard function [26] . . . . . . . . . . . . . 21

2.2

Cooling system parts for chassis types P-, G-, R-, T-series

. . 22

2.3

Idler roller [28]

. . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1

TOPSIS illustration . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1

Point 'A' shows a global optimization and the other points

indicate a local optimization [12]

. . . . . . . . . . . . . . . . 44

4.2

An illustration of Evolutionary algorithm

. . . . . . . . . . . 46

5.1

Genetic code of the parents and the offspring before and after

the crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.1

The structure of a Decidor . . . . . . . . . . . . . . . . . . . . 51

6.2

Schematic view of our simulation algorithm. . . . . . . . . . . 52

6.3

Illustration of the model . . . . . . . . . . . . . . . . . . . . . 55

6.4

Results from optimization of linear decidors in the example

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.5

Convergence of expected profit. . . . . . . . . . . . . . . . . . 58

CHAPTER

Introduction

One of the most important factors in the operations of many corporations

today is to maximize profit and one important tool to that effect is the

optimization of maintenance activities. The main goals of all maintenance

activities are: 1- wait as long as possible to perform maintenance so that the

amount of useful lifetime of parts that is thrown away is minimized while

avoiding failure, 2- find the optimal maintenance policy consisting of sched-

uled and unscheduled maintenance so that the life cycle cost is minimized

while satisfying system availability and safety requirements [1].

Maintenance activities is at the largest level divided into two major areas,

corrective maintenance activities (CM) and preventive maintenance activities

(PM). In this work, we define inspection and imperfect maintenance

, which

can be used as a maintenance activity in some situations. Each maintenance

policy contains various activities such as `replacement', `minimal' or `main

repair', etc.

A near-optimal maintenance policy can utilize a single activity or a com-

bination of two or more activities that depends on company policies and

procedures. The work reported in this thesis has been conducted at Sca-

nia, a major Swedish automotive industry manufacturer of heavy trucks and

buses. According to Scania's policy, there is no repair strategy for many

components and replacement is an acceptable strategy for both corrective

and preventive maintenance.

The optimization of the maintenance activities is in large affected by their

financial implications for a specific corporation, where given two equivalent

systems (mechanical or otherwise) under similar operations may require two

quite different maintenance policies for two different corporations.

is a standard maintenance which reduces failure intensity but does not leave the system

as good as new [45]

CHAPTER 1. INTRODUCTION

Reducing the total cost of ownership for vehicles, maintenance cost analysis

and the resulting decisions becomes the subjects of scrutiny today [1]. A

Maintenance Decision Support System (MDSS) must satisfy an extensive

cost benefit analysis. With accurate information about vehicle health state

and failure prediction capabilities we can make a more informed decision

process.

In this chapter we define some engineering and economic concepts and also

analyze various strategies.

1.1

Preventive Maintenance

Definition 1. Preventive maintenance corresponds to a type of planned

maintenance that improves remaining useful life for a component by pre-

venting excess depreciation and impairment [2].

The main goal with PM is avoiding or mitigating a breakdown in the sys-

tem. PM includes tests, measurements, adjustments, cleaning, lubrication,

minimal repairs, main repair and part replacements for avoiding component

failure. PM has a flexible structure and is not limited to the above activities

[3].

1.1.1

Application and Advantage

The most important application of using PM is energy optimization. We

summarize some other advantages of Preventive Maintenance as:

· Increasing the efficiency of equipment

· Extending the remaining useful life

· Increasing the system performance

· Increasing customer service because maintenance teams have less un-

planned maintenance and can respond quicker to new problems [2]

Moreover, PM measures increased overall safety levels and reduce insurance

inventories.

CHAPTER 1. INTRODUCTION

1.2

Corrective Maintenance

Definition 2. Corrective maintenance corresponds to a maintenance type

with different subtasks such as identify, isolate, and rectify a failure so that

the failed component can be restored to an operational condition within the

tolerances for in service operations [4].

In Figure 1.1 we summarize the function of CM:

Figure 1.1: Corrective maintenance function.

1.2.1

Advantage and Disadvantage

With corrective maintenance we improve product quality, increase compo-

nent lifetime and increase safety.

Higher investment in diagnostic equipment and training is a disadvantage of

CM [5].

CHAPTER 1. INTRODUCTION

1.3

Return on Investment

Definition 3. Return on investment denotes the benefit derived from having

spent money for a system (or product) development, change and manage-

ment. ROI is also a performance measure used to evaluate the efficiency of

an investment opportunity[1].

ROI is calculated by:

ROI =

Return - Investment

Investment

(1.1)

The 'Return - Investment' in the denominator is the loss or gain realized by

making the investment [1].

For maintenance purposes ROI is defined as:

ROI

- C

(1.2)

where

· C

is the life-cycle cost of the system when managed using unscheduled

maintenance

· C

is the life-cycle cost of the system when managed using a health

management (HM) approach

· I

is the investment in HM

In this work C

is to the total life cycle costs of a system using corrective

maintenance and C

is the total life cycle costs with preventive mainte-

nance.

If we assume I

= 1, then ROI simplifies to:

ROI

= C

- C

(1.3)

Our goal is to maximize ROI. To obtain this we want to make C

as small

as possible, of course C

is fixed for a particular system.

CHAPTER 1. INTRODUCTION

1.3.1

Life Cycle Cost

The life-cycle cost (LCC) can be divided into eight parts,

LCC = C

ini

+ C

ins

+ C

env

+ C

where

· C

ini

: initial cost (purchase price of the component and all items unique

to that component)

· C

ins

: installation cost (shipping cost, rigging, installation and start up

of the component)

· C

: energy cost (predicted energy cost for system operation)

· C

: operation cost (labor cost of normal system supervision)

· C

: maintenance and repair cost (include both routine and predicted

repairs)

· C

: downtime cost (loss of production cost)

· C

env

: environmental cost

· C

is the decommissioning and disposal cost (include disposal of com-

ponent, associated equipment and site restoration)

1.3.2

Total Cost Calculation

We calculate total maintenance cost G at time t by

G(t) = K + C(t) R(t)

(1.4)

where K is the replacement cost, C(t) is the maintenance cost and R(t) is

rescue cost [7].

A preventive replacement applicability is an appropriate decision if and only

if a component has negligible failure rate and if the preventive replacement

cost is cheaper than the corrective maintenance cost.

CHAPTER 1. INTRODUCTION

In Fig. 1.2, we illustrate every maintenance cost per time.

Figure 1.2: Costs illustration [6]

The maintenance cost indicates preventive maintenance which increases with

time, that is, more PM. The breakdown cost relates to corrective maintenance

which decreases with time because of the increased PM. As Figure 1.2 shows

there is an optimum point when preventive maintenance should be performed

when maintenance cost and breakdown cost are equal, which is point A in

this case [7].

Figure 1.3: An illustration of component age as a function of time with periodic preventive replacement

CHAPTER 1. INTRODUCTION

In Figure 1.3, the component gets older with time, the component age is

reset upon each preventive replacement.

1.4

Replacement Strategies

As we mentioned before, there are two maintenance activities: corrective

maintenance and preventive maintenance.

Barlow and Hunter examined optimal use of preventive maintenance in their

model in 1960 [8]. We summarize Barlow and Hunter model as:

h(t

)

- F (t)]dt - F (t

) =

- 1

(1.5)

where

h(t) =

f (t)

-F (t)

f (t) is density function for F (t)

F (t) is error probable function for the component

indicates to corrective maintenance

indicates to preventive maintenance

< C

Although Eq. 1.5 provides a simple solution for optimization of fix interval

replacement times, the assumptions in the construction of the underlying

model are actually quite limited and subsequently fails to capture many real

world problems in sufficient detail.

Wang classified Barlow and Hunter's model in 2002 and introduced three

replacement strategies which are: minimal repair, imperfect reparation and

perfect reparation [9].

In this section we review different strategies for replacement and select one

replacement strategy for our project.

The variables in table 1.1 are used in our set of possible maintenance strate-

gies.

CHAPTER 1. INTRODUCTION

Variables

Description

Preventive maintenance cost

Corrective maintenance cost

Mean failure time

,2t

,3t

,...

Planned time for replacement

Table 1.1: Variable descriptions

· Strategy 1:

Component replacement occurs when a component fails, there is no

preventive maintenance for this strategy. We estimate

failures up

to time T . The maintenance cost here depends solely on corrective

maintenance cost, we formulate the strategy as

Cost

· Strategy 2:

In this strategy, we replace the component at planned time t

regardless

of the component age. We also replace a component whenever it breaks

down. This strategy can be formulated as

Cost

P M

+ C

H(t

)

where H(t

) indicate the failure replacement numbers in the time inter-

val (0, t

). To determine H(t

) we used renewal theory [43]. To obtain

the above equation, Chapman and Hall used the probability density

function (PDF) for the first failure, then they took inverse Laplace

transform of the function.

· Strategy 3:

The replacement occurs in this strategy if and only if the component

age comes to the planned times, we replace also the component if we

have a breakdown as usual, we determine the cost as:

Cost

P M

- P (t

)) + C

P (t

)

- P (t

)) +

t(t)dt

CHAPTER 1. INTRODUCTION

That (t) indicates the density function and P (t

) is a probability of

failure before time t

If we assume the same situation as strategy 2 we are able to calculate down-

time. Assume T

is the downtime that occur during a planned replacement

and T

the downtime for replacement due to failure then:

Downtime

+ T

H(t

)

+ T

With the same assumption as in strategy 3 we calculate a new downtime

as:

Downtime

- P (t

)) + T

P (t

)

+ T

)(1

- P (t

)) +

t(t)dt + T

P (t

)

We accept strategy 2 applied to the component level as a better starting

point in this work. In chapter 6 we determine t

and calculate maintenance

costs and also ROI by using Eq. 1.3 and Eq. 1.4.

1.5

Conclusions

In this chapter we introduced some definitions in the maintenance concept.We

also described some economics terms such as `Return on Investment' and

introduced some formula to calculate it. We are going to select the best

maintenance activity for our problem based on maximizing profit in chapter

In subsection 1.3.2 we illustrated different maintenance cost and suggested

some optimal point for a preventive maintenance plan. In section 1.4 we in-

troduced and analyzed various strategies for replacement and selected strat-

egy 2 for our problem according to Scania's policy. We are going to intro-

duce `component health' as a new engineering concept in the next chapter

and trying to find the consequences of preventive maintenance on the system

performance.

As future work for 'replacement strategy' part of this project we are going

to examine strategy 3 and compare the result in the different cases.

CHAPTER

Component health

2.1

Introduction

Component health status or component health is an interpretation of 'Reli-

ability', R(t), which we use in this project. As we mentioned earlier, with

a preventive maintenance, we are able to maximize ROI and also minimize

potential risks for breakdown. In this chapter we calculate the system relia-

bility.

2.2

Definitions and Functions

· Failure rate () is a frequency which indicates component failures per

time unit, it can be expressed as [22]:

× H × A

(2.1)

where

r: number of failures

D : numbers of components tested

H : test hours per component

is the acceleration factor derived from the Arrhenius equation that

can be calculated by [24]:

= e

(2.2)

where

CHAPTER 2. COMPONENT HEALTH

E: Activation energy of the failure mode

: Boltzmann's Constant=8.617

× 10

-5

J/K

: Use temperature

is the test Temperature

· Mean Time To Failure () or MTTF is a standard industry value

which shows the average time to failure. MTTF calculate by [23]:

M T T F =

(2.3)

· Mean Time Between Failure (MTBF) is an expected time between

failures of a system during operation, which can be calculate by [23]:

M T BF = M T T F + M T T R

(2.4)

where MTTR is the Mean Time to Repair.

· Hazard function h(t) is a calculation of failure rate over time interval

- t

)

· Failure rate function (t) shows the number of failures per unit of

time and it is related to Hazard function that its plot over time has the

same shape [25].

(t) =

f (t)

R(t)

(2.5)

where f (t) indicate time to first failure [27] and R(t) = 1

- f(t) is

reliability function.

· Degraded factor (A) The amount multiplied by mean time between

failures of a component to get the operational MTBF [27].

The "Bathtub curve" Figure 2.1 is a well known curve used in reliability

engineering and illustrates a particular form of the hazard function.

The bathtub curve includes three phases [27]:

1. Failures phase: decreasing failure rate

2. Phase with constant (random) failure rate: constant failure rate

3. Wear-out failures phase: increasing failure rate

CHAPTER 2. COMPONENT HEALTH

Figure 2.1: The bathtub curve hazard function [26]

As The bathtub curve represents Mean Time To Failure is in phase with

constant failure rate that shows the predicted elapsed time between inherent

failures of a system during operation [27].

As we mentioned earlier, internal combustion engine cooling (cooling system)

was used as a case study in this project. We analyzed the performance of

cooling system in various areas.

To get a better understanding about the system health status, we calculate

failure rate, mean time between failure and time to fist failure by using

equations Eq. 2.1, Eq. 2.3 and Eq. 2.4.

CHAPTER 2. COMPONENT HEALTH

2.3

Cooling System

Cooling system is an important system in the trucks engine. The system

task is cooling continuously the engine by circulating coolant liquid

The cooling system consists of various parts such as: coolant, Idle roller,

compressor, belt tensioner, axeltapp, generator and ploy-V belt.

Figure 2.2 was taken of the Scania workshop manual and shows cooling

systems parts in the typical Scania trucks (P-,G-,R- and T-series).

Figure 2.2: Cooling system parts for chassis types P-, G-, R-, T-series

To avoid lengthy calculations and to get a better analysis for the health of

the cooling system we chose the idler roller as a representative subcompo-

nent.

a mixture of an antifreeze and water

CHAPTER 2. COMPONENT HEALTH

Idler roller consists of 10 parts:

Figure 2.3: Idler roller [28]

where:

Item

Description

Quantity

Shell

Bearing Housing

Shaft

Inner Snap Ring

Bearing

Female Labyrinth Seal

Male Labyrinth Seal

Outer Labyrinth Seal

Outer Snap Ring

Cover

Table 2.1: Idler Roller parts description

As our study shows 'Bearing' (part number 5) is the most vulnerable part

for failure.

Table 2.2 in the next page represents failure rate,TTFF, MTBF and time

to first failure for the idler roller, the values calculated by Eq. 2.1, Eq. 2.3

and Eq. 2.4. To compute the Idler Roller health status we used the Scania

workshop manual and warranty statistics (survival analysis).

Notation: The values in Table 2.2 has been hidden according to Scania's

security policy for confidential data.

CHAPTER 2. COMPONENT HEALTH

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CHAPTER 2. COMPONENT HEALTH

2.4

The Consequences of PM on the Compo-

nent Health

In this section we calculate the consequences of PM on a mechanical system

which increases system reliability and efficiently.

We redefine the Hazard rate function as new function of reliability as [29]:

h(t) =

R(t)

dR(t)

(2.6)

where h(t) and R(t) denote hazard rate function and reliability function

respectively. According to `The dynamic reliability models for fatigue crack

growth problem' see [30], we can rewrite hazard function as:

h(t) =

+ A(R

- R(t))

(2.7)

where

and A represent initial failure rate and degraded factor respectively

and R

is the initial reliability.

2.5

Dynamic Reliability

Definition 4. Dynamic reliability method provides a mathematical frame-

work capable of handling interactions among components and process vari-

ables explicitly [31].

In other words, the dynamic reliability method represents a more realistic

image of the system, ability, risk and also safety.

In this section, we combine Eq. 2.6 and Eq. 2.7 to define a new equation for

dynamic reliability in a mechanical system. To obtain this new equation, we

use Riccati differential equation.

2.5.1

Riccati differential equation

Definition 5. The Riccati

equation is a nonlinear first order differential

equation which is not in the group of classical equations [32].

The equation is named after Jacopo Francesco Riccati (16761754)

CHAPTER 2. COMPONENT HEALTH

Riccati equation appears in the different areas of mathematics such as the

theory of conformal mapping [33], algebra and geometry.

A general form for Riccati equation can be written as

= p(x)y

+ q(x)y + r(x)

(2.8)

If r(x) = 0 then Riccati differential equation transfers to Bernoulli's princi-

ple(differential equation).

If r(x) = 0, and if we accept u(x) as a solution for the differential equation

then:

y = u +

(2.9)

differentiating in Eq. 2.9 with respect to x:

(

) =

substitute this into Eq. 2.8

= p(x) u +

+ q(x) u +

+ r(x)

= p(x) u

+ z

+ q(x) u +

+ r(x)

= p(x)u

p(x) +

p(x) + q(x)u + q(x)

+ r(x)

= (p(x)u

+ q(x)u + r(x)) +

p(x) +

q(x)

p(x) +

q(x)

CHAPTER 2. COMPONENT HEALTH

-2uzp(x) - p(x) - zq(x)

(2.10)

then:

- (2up(x)) + q(x)z = -p(x)

(2.11)

as we see Eq. 2.11 is a linear differential equation.

2.5.2

Dynamic Reliability Equation

By using Riccati differential equation, we are able to create a general form

for dynamic reliability based on R(t). To obtain this we find R(t) from

Eq. 2.7:

R(t) =

{(

+ AR

)

- h(t)}

(2.12)

we substitute Eq. 2.12 into Eq. 2.6:

dR(t)

-1

dh(t)

(2.13)

where Eq. 2.16 is a Riccati differential equation based on h(t):

h(t) =

dh(t)

(

+ AR

)

- h(t)

(2.14)

Eq. 2.14

dh(t)

+ h

(t)

- (

+ AR

)h(t) = 0

we define h

(t) as a particular solution for the problem:

(t) = (

+ AR

)

CHAPTER 2. COMPONENT HEALTH

then

h(t) = h

(t) +

u(t)

(2.15)

to solve Eq. 2.15, we need to define an acceptable solution such as u which

satisfies the linear 2nd order ODE:

u(t)

(

+ AR

)

+ AR

) exp((

+ AR

)t)

- 1

h(t) =

+ AR

)

exp((

+ AR

)t)

+ AR

) exp((

+ AR

)t)

- 1

according to Eq. 2.12 we have:

R(t) =

(

+ AR

)

+ AR

)

exp((

+ AR

)t)

+ AR

) exp((

+ AR

)t)

- 1

R(t) =

+ AR

)

+ AR

) exp((

+ AR

)t)

- 1

R(t) =

(

+ AR

)

- cA(

+ AR

) exp((

+ AR

)t)

R(t) =

(

+ AR

)

- cR

(

+ AR

) exp((

+ AR

)t)

we consider:

c =

+ AR

)

then:

R(t) =

(

+ AR

)

exp((

+ AR

)t)

(2.16)

CHAPTER 2. COMPONENT HEALTH

where R

is initial reliability,

is initial failure rate which can be calculated

by Eq. 2.1 and A indicate the degraded factor that can be find by fitting

Eq. 2.16 with the experimental data or simulation result. By using Eq. 2.16

the health promotion of a mechanical system can be calculated [34].

Eq. 2.16 introduced by Yuo-Tern Tsai and Kuo-Shong Wan in April 2004 (see

[35] , page 91) as a dynamic reliability equation that depicts the degraded

behavior of component. The system reliability shows how far we are getting

the particular outcome for the given input with as much less wastage as

possible. As future work we are going to use Eq. 2.16 for calculate the

system reliability after every preventive, corrective maintenance and also

inspection.

By using this approach we are able to determine the consequences of different

maintenance policies on the system health. This approach helps us to develop

our model and gives us also a better visibility of the health of a system.

2.6

Conclusions

In this chapter we introduced some new concepts such as failure rate, time

to first failure, etc. to get better understanding about the health of a compo-

nent. We selected cooling system as an example and calculated failure rate,

time to first failure, mean time to failure and mean time between failures for

the idle roller which is a subcomponent in cooling system.

In section 2.4 we analyzed the consequences of a preventive maintenance on

the component health by using Hazard rate function and reliability function.

In subsection 2.5.1 we introduced Riccati differential equation and used it in

subsection 2.5.2 to find a new formula to calculate the health promotion of

a mechanical system.

As future work for this part of this project we are going to simulate the

system reliability for a Multi-component system.

In the next chapter we are going to find the best maintenance activity for

cooling system.

CHAPTER

Multi Criteria Fuzzy Decision Making

(MCFD)

3.1

Introduction

In this chapter, we provide a decision making model for a mechanical system

based on maintenance activities and return on investment. We use the health

of system as an indicator to measure the systems performance.

The health of a system depends on the health of all the components that

make up the system. Since, there are various criteria that affect the health

on a mechanical system, our decision making process become a multi-criteria

decision making. We consider the health of a system to be between 0 and 1 in

this project. By health equal to 0 we mean that the system fails and a health

equal to 1 indicates a fully healthy system. By this consideration, we are able

to formulate our decision making problem in a fuzzy environment.

There are different multi-criteria decision making techniques such as: AHP

(The Analytical Hierarchy Process),TOPSIS (The Technique for Order of

Preference by Similarity to Ideal Solution), SAW(Simple Additive Weight-

ing), ELECTRE (Elimination er Choice Translation Reality), SMART(The

Simple Multi Attribute Rating Technique) and ANP(The Analytical Network

Process) for the problem solving. In this project we use TOPSIS technique

for decision making model under fuzzy environment.

To reach this goal we need to identify all criteria that affect the health of

the components. Some of these criteria are completely quantifiable, some

partially quantifiable, and others criteria are completely subjective. We need

also to define different maintenance activities, such as preventive, corrective,

inspective and imperfect maintenance.

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

The result from this chapter shows the best maintenance activities to perform

on each component, the policy of when to perform these activities is in fact

the main output from the simulation and optimization outlined in the next

chapter.

3.2

Fuzzy Set and Membership Function

A membership function indicates the degree of truth as an extension of eval-

uation. This concept was introduced by Zadeh in 1965. Fuzzy truth rep-

resents membership in vaguely defined sets. Some basic definitions of fuzzy

sets, membership function and intuitionistic fuzzy sets are reviewed by Yun

Shi [36] , KERRE [37] and Yang[38].

Definition 6. A fuzzy set is a pair (A, m) where A is a set and m : A

[0, 1].

For each x

A, m(x) is called the grade of membership of x in (A, m).

For a finite set A =

, . . . , x

}, the fuzzy set (A, m) is often denoted by

{m(x

)/x

, . . . , m(x

)/x

}. Let x A. Then x is called fully included in

the fuzzy set (A, m) if m(x) = 1 and is called not included if m(x) = 0. The

set

{x A|m(x) > 0} is called the support of (A, m) and the set is called a

kernel. x is a fuzzy member if 0 < m(x) < 1, [38].

Definition 7. For any set X a membership function on X is any function

from X to the real unit interval [0, 1], the membership function which repre-

sents a fuzzy set A is denoted by

. For an element x of X, the value

(x)

is called the membership degree of x in the fuzzy set A, [39].

According to [40] we are able to model unknown information by using an

additional degree and Intuitionistic fuzzy sets (IFS)

3.3

IFS Generalize Fuzzy Sets

Definition 8. An Intuitionistic Fuzzy Set A on a universe U is defined as

an object of the following form:

A =

{(u,

(u),

(u))

| u U}, where the functions u

: U

[0, 1] and

: U

[0, 1] define the degree of membership and the degree of non-

membership of the element u

U in A, respectively, and for every u U we

have 0

(u) +

(u)

1, [41].

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

According to [40] a fuzzy set can be written as:

{(u,

(u), 1

(u))

| u U}

(3.1)

IFS distribute fuzzy sets for every membership function and non-membership

functions where = 1

- .

3.4

Fuzzy Implication Operators

The following table summarizes the classical binary implication:

Table 3.1: Binary implication

Definition 9. A mapping I : [0, 1]

[0, 1] is a fuzzy implication if it

satisfies the boundary conditions:

I(0, 0) = I(0, 1) = I(1, 1) and I(1, 0) = 0, [36].

A fuzzy implication can be generated by using three different approaches,

R-implications, S-implications and QL-implications. In the present paper we

use R-implications.

3.5

Inclusion Degree Function of IFS

Assume U is a finite universe and R is an implication. I

IF S

is a an inclusion

degree function of IFS if R satisfies the following conditions [36]:

· a, b [0, 1] and a b R(a, b) = 1

· R(a, b) is non-decreasing with respect to b and non-increasing with

respect to a.

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

By using this definition we can write

IF S

(A, B) =

| U |

[R(

(u),

(u)) + (1

- )R(

(u),

(u))],

[0, 1],

(3.2)

where

| U | is the cardinality of U which can be calculated by, [42],

| U |=

1 +

(u)

(3.3)

There are different methods to calculate an R-implication which was intro-

duced by several mathematicians. we use Lukasiewics implication:

(a, b) = min(1

- a + b, 1).

(3.4)

3.6

TOPSIS Method

The Technique for Order of Preference by Similarity to Ideal Solution (TOP-

SIS) is an analysis method that is one of the best methods for multi criteria

decision making. TOPSIS was developed by Hwang and Yoon in 1981 and

also by Yoon in 1987.

TOPSIS method is based on two main solutions: 1- The positive ideal solu-

tion which has the best attribute values 2- The negative ideal solution which

has the worst attribute values.

TOPSIS measures the geometric distance between all alternatives, positive

and negative ideal solutions and selects the best one. The best alternative

is an alternative which has the shortest distance from positive ideal solution

and also the farthest distance from the negative ideal solution [44].

3.6.1

The Structure of TOPSIS Method

TOPSIS method consists of six steps. We assume a decision making prob-

lem that has m alternatives and n criteria. The steps can be performed as

follow:

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

Step 1 :

Obtain m alternatives and n criteria, create the evaluation matrix with m

rows and n columns. Obtain the intersections of alternatives and a criteria,

define it as x

, standardization x

as matrix (x

)

m×n

Step 2 :

Create a set of weight for the criteria, define it as w

, normalize (x

)

m×n

Step 3 :

Identify the positive ideal solution, show it as A

Identify the negative ideal solution show it as A

Step 4 :

Measure the distance between all criteria and A

, define it as: D

Measure the distance between all criteria and A

, define it as: D

Step 5 :

Determine the ranking index ( p

) of each alternative, calculate p

by:

)

) + D

)

Step 6 :

Order the rank of alternatives according to proportion in step 5.

To illustrate the TOPSIS method, we assume a problem with 5 criteria and

3 alternatives:

Decision

Figure 3.1: TOPSIS illustration

As we see, every criterion affects every single alternative. With a set of

alternatives and criteria, the decision maker can make for example three

decisions as follow:

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

Criteria

Alternatives

Decision 1

Decision 2

Decision 3

, M

Table 3.2: Decision Making by TOPSIS

Notation: A best decision at a time can be a single alternative or a combi-

nation of two or more alternatives.

3.6.2

TOPSIS Method in Multiple Criteria Fuzzy De-

cision Making

Since we consider our problem to be a multi criteria decision problem in a

fuzzy environment we define A

as a Fuzzy Positive Ideal Solution and A

as a Fuzzy Negative Ideal Solution.

We now use the TOPSIS method to calculate the distance between A

and

Assume that we have a set of criteria C and a set of alternatives M :

C =

, C

, ..., C

}

M =

, M

, ..., M

}

According to [40] we assume that the alternatives and criteria are represented

(using IF S) as:

{(C

1,1

), (C

1,2

), ..., (C

1,m

)

}

{(C

2,1

), (C

2,2

), ..., (C

2,m

)

}

{(C

n,1

), (C

n,2

), ..., (C

n,m

)

where

i,j

indicates the degree by which the alternative M

satisfies criterion

i,j

indicates the degree by which the alternative M

does not satisfy

criterion C

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

Definition 10. A fuzzy positive ideal solution is defined as

{(C

, Max

{

i,1

}, min{

i,1

}),

, Max

{

i,2

}, min{

2,m

}),

, Max

{

i,m

}, min{

i,m

})}.

Definition 11. A fuzzy negative ideal solution is defined as

{(C

, min

{

i,1

}, Max{

i,1

}),

, min

{

i,2

}, Max{

i,2

}),

, min

{

i,m

}, Max{

i,m

})}.

To calculated the distance between alternatives A

and A

we define two

inclusion degrees as follows:

Definition 12. The inclusion degree D

) of the positively ideal solution

in alternative M

and the inclusion degree D

) of the negatively ideal

solution in alternative M

are respectively defined as

) = Max(I(A

, M

))

(3.5)

) = min(I(M

, A

)),

(3.6)

where I denotes the inclusion degree function, see Equation (2).

Definition 13. The ranking index of alternative M

is defined as

)

) + D

)

(3.7)

where 0

If there exists i

{1, 2, ..., n} such that p

= M ax

, p

, ..., p

} then the

alternative M

is the best alternative, [40].

3.7

Problem Statement

To use MCFD for solving our problem we need to identify all criteria that

effect on the component's health.To perform this process, we analyzed various

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

components in Scania's technical module system with the help of engineers

at Scania R&D. In table 3.3 we classify the 15 most important criteria with

high effectivity on a mechanical system.

Number

Description

Calendar time

Mileage

Chassis load and strength

Material operation

Components health status

Humidity

Temperature

Quality of roads

Road dust

Usage

Fuel quality

Driving styles

Environment

Speed

Transport tasks

Table 3.3: The Criteria

we categorize different maintenance policies in table 3.4, with `No Action' we

mean no maintenance activity should be run at some special time-intervals.

For example for a component in the end of its remaining useful life `No

Action' is an optimal decision.

Number

Description

Corrective Maintenance

Imperfect Maintenance

Preventive Maintenance

Inspection

No Action

Table 3.4: The alternatives

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

3.8

A Numerical Example

To have a better understanding of MCFD, we analyse internal combustion

engine cooling (cooling system for short) as a real world example. In this

section, we try to find the best maintenance activity for the cooling system

in a typical engine.

As we mentioned in the last section, we need to define different alternatives

and also identify all criteria which have a direct effect on the health of a

component.

To define the maintenance activities, we need to study the company policies,

the customer's perspective and requirements, which depend on the company's

task operating systems.

Let us assume that M

, M

are three maintenance alternatives which in-

dicates corrective maintenance, imperfect maintenance and preventive main-

tenance respectively.

To identify the criteria with the highest influence we need knowledge of the

mechanical properties of the component. We use table 3.3 to choose related

criteria with the highest impact on the cooling system's health.

Let C

, C

and C

be the criteria that represent mileage, temperature,

time and humidity.

As a decision maker, we want to find which of the alternatives M

that

best satisfy the criteria C

and C

or just C

, according to the customers

perspective and the company's policies.

Suppose that the relationships between alternatives and criteria are:

{(C

, (0.5, 0.6)), (C

, (0.5, 0.1)), (C

, (0.2, 0.4)), (C

, (0.1, 0.5))

}

{(C

, (0.5, 0.6)), (C

, (0.5, 0)), (C

, (0.3, 0.6)), (C

, (0.5, 0.2))

}

{(C

, (0.6, 0.2)), (C

, (0.4, 0.3)), (C

, (0.2, 0.3)), (C

, (0.4, 0.1))

}

To find the above values, we studied Scania's survival analysis - warranty

data. In this study we compared the cooling system efficiency in the different

regions.

To estimate the exact coefficients for these relationships we need to perform

an accurate data mining with some suitable tool such as RapidMiner.

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

Now we can construct the positive and negative ideal solutions:

{(C

, (0.6, 0.2)), (C

, (0.5, 0))

}

{(C

, (0.3, 0.3))

}

{(C

, (0.5, 0.6)), (C

, (0.4, 0.3))

}

{(C

, (0.2, 0.6))

}

where the fist elements in the A

and A

are maximum values and the second

elements are minimum values. It means that a positive ideal solution is a set

of elements that have maximum values and a negative ideal solution is a set

of least value.

We then calculate the inclusion degree function by using Eq. 3.2, but be-

fore that we need to calculate R

by using Eq. 3.4 (Lukasiewicz implica-

tion):

(

) =

min(1

- 0.6 + 0.5, 1) = 0.9 × 0.5 = 0.45

(

) = min(1

- 0.6 + 0.5, 1)

= 0.6

× (1 - 0.5) = 0.3

(

) =

min(1

- 0.5 + 0.5, 1) = 1 × 0.5 = 0.5

(

) = min(1

- 0.1 + 0, 1)

= 0.9

× (1 - 0.5) = 0.45

Note that is an optimal value between 0 and 1, we determine = 0.5 in

this example and

|U| is the cardinality of U which is |U| = 2.

By using Eq. 3.2 we have:

I(A

, M

) =

× (0.45 + 0.3 + 0.5 + 0.45) = 0.85

then:

I(A

, M

)

0.85

0.825

0.9

I(M

, A

)

0.925

0.9

0.875

Table 3.5: The inclusion degrees of A

in M

, M

and inclusion degrees of A

in M

, M

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

and also:

I(A

, M

)

0.45

0.425

0.475

I(M

, A

)

0.45

0.475

0.425

Table 3.6: The inclusion degrees of A

in M

, M

and inclusion degrees of A

in M

, M

By using Eq. 3.5 and Eq. 3.6 we can calculate the inclusion degrees D

)

and D

)

0.85

0.825

0.9

)

0.45

0.475

0.425

Table 3.7: The inclusion degrees D

) and D

)

The ranking index of alternatives (p

) can be calculated by using Eq. 3.7

as:

0.85

0.85 + 0.45

= 0.65

0.825

0.825 + 0.475

= 0.634

0.9

0.9 + 0.425

= 0.679

As we see p

= 0.679 is the best alternative and indicates preventive main-

tenance in this case.

3.9

Conclusions

In this chapter we define our problem in fuzzy environment and used the

TOPSIS method for the decision making. As we mentioned earlier the most

advantage of using fuzzy logic is flexibility.

CHAPTER 3. MULTI CRITERIA FUZZY DECISION MAKING (MCFD)

We started this chapter with an introduction of fuzzy logic. We defined some

concepts in fuzzy environment in sections 3.2, 3.3, 3.4 and also section 3.5.

We fined the TOPSIS method as a suitable decision making method for our

problem and described the structure of this method in subsection 3.6.1.

By define a positive and negative ideal solution in subsection 3.6.2 we esti-

mated the best and worst case for our problem. In fact we sketched a fuzzy

frame for the problem and our goal is to have a long distance from the worst

case (negative ideal solution) and rise up to catch the best case (positive

ideal solution).

We identified fifteen criteria with direct effect on the cooling system's health

and defined some alternative as maintenance activities in section 3.7.

As future work for the decision making part of this project we are going to

examine some other multiple-criteria decision methods such as: value analysis

(VA) fuzzy, VIKOR method and weighted product model.

Until now we defined different maintenance activities, analyze the system

health and making decision to choose the best alternative. In the next chapter

we analyzed different optimization methods to find a suitable algorithm for

optimization.

CHAPTER

Optimization Methods

4.1

Introduction

In the decision making problems, it is normal to have multiple design objec-

tives and there exists different optimization methods today for solving this

kind of problems.

Most of the optimization algorithms and methods have been developed and

improved from their previous versions. As discussed in [10] the solutions

that generated by most of the optimization algorithms, cannot be used for

problems that have a combination of discrete and continuous variables. Those

algorithms mostly provide local optimization as final solutions.

In this chapter, we will look at different ways to solve multi objective design

optimization problems and also find the best optimization algorithm to solve

our problem that provides a global optimization.

4.2

Classical and Non-classical Optimization

Methods

We classify the optimization problems in consideration of objective functions

and constraints into four groups:

1. Unconstrained Single - Objective Optimization (USOOP)

2. Constrained Single - Objective Optimization (SOOP)

3. Unconstrained Multi - Objective Optimization (UCMOP)

4. Constrained Multi - Objective Optimization (CMOP)

CHAPTER 4. OPTIMIZATION METHODS

There are two approaches to solve this problem: classical optimization tech-

niques and intelligent(Non-classical) optimization techniques.

A classical optimization method is an analytical method that solves differen-

tiable functions. This method is efficient when the underlying assumptions

are fulfilled [11]. The classical optimization techniques don't support non

differentiable optimization problem. This method cannot solve a large scale

problem and it is also sensitive to changes the parameters, which is one po-

tential disadvantage of using classical optimization method. Trust region

method and Interior point method are two well known classical optimization

methods.

The intelligent (Non classical) optimization method has been specifically de-

veloped for those cases where the classical method was not suitable, high

dimensional search or problems with many local optimizations. Since the

intelligent method investigates all possible solutions, the numbers of eval-

uations can be very high and therefore this method is applied in connec-

tion with computer experiments[13]. The intelligent optimization method

is also able to find the optimum solution for a CMOP. Penalty function

method, Resource allocation optimization methods, Multi-objective method

and Co-evolutionary method are some well known intelligent optimization

methods.

4.3

Global Optimization

Global optimization, per definition, indicates to finding the extreme value

of a given non convex function in a certain feasible region [13]. In most

cases, the classical optimization methods are not able to solve the global

optimization problems, because this methods usually entrap in a local op-

timization. Moreover, classical methods could neither generate nor use the

global information that needed to find the global solution.

To get a better understanding of local and global optimization, we analyze

Figure 4.1 in follow:

CHAPTER 4. OPTIMIZATION METHODS

Figure 4.1: Point 'A' shows a global optimization and the other points indicate a local optimization [12]

Point B refers to a local optimization, because it's the highest place in

their area. In other words, the peak of mountain always indicates a local

optimization[13]. As we see in Figure 4.1, there are several mountains; the

highest mountain always is the global optimization, which is point A in this

example.

At find the global optimum point is the challenge in many cases. Metaheuris-

tic algorithms are new solvers which designed to find the global optimum for

optimization problems. Metaheuristics algorithms are able to implement a

stochastic optimization. Compared to other optimization algorithms, meta-

heuristics algorithms do not assure that a globally optimal solution can be

found on the all class of optimization problems. The final solutions that

provided by metaheuristics algorithms, are dependent on the set of random

variables [14].

Metaheuristics algorithms can be classified by different approaches, for exam-

ple population-based searches or categorized by the type of search strategy.

CHAPTER 4. OPTIMIZATION METHODS

In this project, we review Metaheuristics of population-based searches type,

to find a suitable optimization algorithm for solving our problem. Population

based metaheuristics include evolutionary algorithms, particle swarm opti-

mization (PSO), Imperialist Competitive Algorithm (ICA) and Intelligent

Water Drops algorithm (IWD) [15].

4.4

Evolutionary Algorithms

As we mentioned before, evolutionary algorithms (EA) are a subset of population-

based metaheuristic that inspired by biological evolutionary mechanism in

nature, such as reproduction, mutation, recombination, and selection [16].

We summarize EA's functions in two processes (algorithms) that work simul-

taneously:

1. Evaluation

2. Optimization

To get a better understanding of EA's functions, we analyse an example:

min f (x)

s.t.

g(x)

------------ min f(x) + max 0, 1 -

g(x)

EA tries to optimize the objective function and simultaneously tries to find a

feasible set for the solutions. In fact, we are going to find the minimum value

for f (x) and also find the value of . We illustrate the process as:

In fact, for solve this problem; we need to run two algorithms parallel with

different tasks. The first algorithm's task is finding the value of , and the

second algorithm trying to optimizes the objective function f (x).

As we see in Figure 4.2, Algorithm 1 finds some , that can be assumed as

...

Algorithm 2 uses and represents a set of solution such as:

, x

, ...

}

Algorithm 1 recognizes the best and transforms every . After the trans-

formation, algorithm 2 produces new set of x

CHAPTER 4. OPTIMIZATION METHODS

Alg 1

. . .

Alg2

Figure 4.2: An illustration of Evolutionary algorithm

Note: for recognize the best by algorithm 1, an optimization process should

be performed every time.

In fact, algorithm 1 is a 'Meta-Algorithm' (external) that consists of various

members () and every members in algorithm 1 calls algorithm 2.

Genetic algorithm (GA), artificial bee colony algorithm, ant colony opti-

mization algorithms, evolution strategy (ES) and imperialist competitive al-

gorithm are some examples of evolutionary algorithms.

4.5

Conclusions

According to our problem's conditions we need to find a robust and fast

optimization algorithm for problem solving. As we reviewed in this chapter

the classical optimization algorithms are not able to solve the large-scale

optimization problems such as our problem in this work.

We compared various optimization algorithms in both classical and intelligent

optimization methods. According to the problem's conditions, limitation and

our goals the intelligent optimization methods are better suited for solving

our problem.

After an accrue pre-study we find Genetic algorithm as a suitable optimiza-

tion algorithm to solving our problem. In the next chapter we analyze Ge-

netic algorithm.

CHAPTER

Genetic Algorithm

5.1

Introduction

As we mentioned in the last chapter, genetic algorithm is a type of evolu-

tionary algorithms that was introduced by John Holland. As the other evo-

lutionary algorithm, GA uses the biological processes of reproduction and

natural selection to solve for the `fittest' solutions [17].

Genetic algorithm solves both constrained and unconstrained multi objective

optimization problems and it is also able to solve problems with discontinu-

ous, stochastic, highly nonlinear or ill-defined objective function [18]. As we

explained in the last chapter, classical optimization methods are not able to

solve a wide range of redundancy allocation problem. A recent study that

published in 'Reliability Engineering and System Safety' shows that genetic

algorithm is an efficient meta-heuristic method to solving combinatorial op-

timization problems [19].

In this chapter, we introduce, illustrate and discuss genetic algorithm as

suitable algorithm for solving our problem.

5.2

Structure

Before we begin describing the structure of GA, we need to define the so-

lution encoding which called `chromosomes' in GA's concept. In fact, the

chromosomes are a set of parameters which propose a solution to the initial

problem and they are represented as a simple string[19]. The design of the

chromosome and the parameters depends on the initial problem. We describe

our design for both chromosomes and parameters in chapter 5.

CHAPTER 5. GENETIC ALGORITHM

We summarize the structure of genetic algorithm as:

1. fitness function

2. initial population of chromosomes

3. selection of parents for generation of new population

4. crossover to produce next generation of chromosomes

5. random mutation

Figure 5.1 represents the structure of GA with a numerical example:

010111010110 100010101010

101101101010 011010101010

After crossover

Parent genetic code

010111010110 011010101010

101101101010 100010101010

Offspring genetic code

Randomly chosen crossover point

Figure 5.1: Genetic code of the parents and the offspring before and after the crossover

5.3

Applications and Advantages

The most important GAs application is optimization. GA is suitable method

to solving both synthetic and numerical problems such as graph coloring,

routing, partitioning and also TSP. Machine learning is also the second most

important GAs usage which can be categorized as

· Prediction and classification

· Prediction of weather

· Designing neural networks

Population genetics, Ecology (model symbiosis), Immune systems, Auto-

matic programming and filter design are also another important applications

of genetic algorithms [20].

CHAPTER 5. GENETIC ALGORITHM

By using GA, we avoid the local optimization and have a chance to finding a

global optimization solution for the problem and the solution becomes better

and better with time.

By using GA, the fitness functions are able to be changed from iteration

to iteration that allows incorporating new data in the model if it becomes

available [21].

GA supports the multi objective optimization which is a most important

benefit with GA. The modular genetic algorithm (MGA) separate from ap-

plication; building blocks are able to use in hybrid applications

which is

another GA's advantage.

We use Genetic algorithm as a suitable optimization algorithm in the next

chapter.

5.4

Conclusions

The goal of this chapter was introduction to genetic algorithm. We fined

genetic algorithm as a robust, fast and suitable optimization algorithm to

optimize our problem. We analyzed the structure of genetic algorithm and

discussed some advantage of using genetic algorithm in this chapter. In the

next chapter we implement genetic algorithm in our maintenance optimiza-

tion model.

As future work for optimization part of this project we are going to ex-

amine another optimization algorithms such as: ant colony optimization

algorithm, artificial bee colony algorithm and natural evolution strategies

algorithm.

Is a combination elements of web application and native application

CHAPTER

Maintenance Optimization Model

6.1

Introduction

As we mentioned before, maintenance activities is at the largest level divided

into two major areas, corrective maintenance activities (CM) and preven-

tive maintenance activities (PM). Corrective maintenance is, per definition,

performed as a response to a system failure while preventive maintenance is

performed when the system is operational and to avoid future system failure.

When optimizing maintenance activities, by a maintenance plan or policy,

we seek to find the best activities to perform at each point in time, be it

PM or CM. The optimization of these activities is in large affected by their

financial implications for a specific corporation, where given two equivalent

systems (mechanical or otherwise) under similar operations may require two

quite different maintenance policies for two different corporations. A con-

cise review and analysis of different maintenance optimization models can be

found in [1]. In the article the authors describe several models for analytical

optimization of PM policies and mention computer simulation as a good tool

whenever simplifications of systems, to make them analytically tractable,

would lead to unrealistic results. In light of this we have focused our efforts

towards a simulation approach to maintenance optimization with the benefit

of a capability to optimize more complex systems.

6.2

Simulation Framework Model

In this section we introduce a framework model for simulation of a stochastic

system, the reader may think of it in terms of a mechanical system operating

under some cooperate environment and subject to corrective and preventive

maintenance activities. Consider a discretization of time into time-steps t

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

and a description of a system in which all events fill up a whole number of

such time-steps. By making t sufficiently small, such a model can describe

the system with arbitrary precision. We consider a time horizon of T such

discrete time units. At each point in time the system is described in all

important aspects by a state vector S

, the current state includes the

system time (multiple of t) and any variables describing the components

of the system.

Furthermore, consider three types of events, random events which happen

stochastically depending on the evolution of the system state, plannable

events that may happen by choice depending on the current system state

and a default event that happen whenever neither a random nor a plannable

event occur. Let P =

} be the set of plannable events, R = {R

} be

the set of random events and D be the default event. All events are con-

sidered as functions that does nothing else than change the current state,

, R

, D :

. Let r

be functions r

[0, 1] correspond-

ing to each random event R

such that r

) is the probability that the

event R

was triggered before time t. Let p

be functions (called decidors)

{T rue, F alse} corresponding to each plannable event P

such

that if p

) = T rue then the plannable event P

is triggered at time t and

if p

) = F alse the event is not triggered at time t.

Decidor

Decision Methods

Decision Data

Figure 6.1: The structure of a Decidor

For the rationale of this simulation system we also assume that if several

plannable and random events compete to run at a specific time, the order in

which they are executed has limited effect on the results of the simulation.

We also assume that the plannable events, when executed, has no significant

increasing effect on the probability of the random events to be triggered in

the next time-step.

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

6.3

The Simulation Algorithm

Consider Figure 6.2 describing the major structure of our simulation algo-

rithm.

t = 0

¯r

Figure 6.2: Schematic view of our simulation algorithm.

We begin by setting the current state to the initial state and in each it-

eration of the algorithm we first execute all plannable events P

for which

(S) = T rue, we then randomize the level ¯

at which the random events

will trigger in the future, execute the default event D, execute each random

event R

for which the current level exceeds the trigger level (r

¯r

), and

we iterate this until our termination-time T is reached. A more detailed de-

scription can be found in the following algorithm.

initial

while t

T do

for all i where p

(S) = T rue do

(S)

end for

for all i do

sample from uniform distribution on [r

(S), 1]

end for

D(S)

10:

for all i where r

(S)

¯r

11:

(S)

12:

end for

13:

end while

Note that the state S may change for subsequent i in the for-loops at lines

3 and 10 and our implementation iterates through i in increasing order, of

course the values taken of i depends on the number of plannable and random

events respectively.

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

Normally, since we have randomly triggered events, we would need to run

this simulation a number of times, perhaps 10 to 1000 repetitions, and collect

statistics for the state variables in S as they develop over time. In our

implementation we gather the first two moments of all state variables for each

point in time, that is, their mean value and their mean square value.

6.4

Example Problem Class

The framework model and simulation algorithm in the previous sections al-

lows for easy modeling of many real-world systems. In a specific model one of

the most important decisions to make is the time discretization t. Making

it too small will increase the run-time of any simulation and making it too

large will introduce greater error in the simulation output and subsequent

decisions based on the output. Another key factor is choosing specific ways

in which to encode the decidors p

. One of the simplest decidors, which we

have used in this paper and specifically in our example problem class, is the

linear decidor.

A linear decidor p

in our framework model is a real vector p

such

that the outputted decision value, True or False, is equal to the truth value

of the statement p

· S > 1. This allows the decision to perform a plannable

event to depend to different degrees on different elements in the state vector

S. In many cases then, when modeling a maintenance system, all aspects

of the system is constructed to describe the real world system in sufficient

detail, while the decidors are the desired output to be chosen to give optimal

performance for the maintenance policy.

Since there is only one type of decidors (one decision method) in our example

problem class and data for all decidors are real vectors of the same size, we

may describe the decision process in a more compact form. Let M be the

matrix defined by

M =

· · · p

. ..

· · · p

N M

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

where N is the number of plannable events in the system. Furthermore,

consider the current system state S to be a column vector. Then the elements

in the vector

D = M S,

will take a value d

1 for each plannable event P

which is decided to be

performed.

Our simulation algorithm however, will not in the current version execute all

such decided plannable events, but only in effect iterate i from 1 to N and if

1 perform the event and in anticipation of the next value for i recompute

D. A variation to this method could be to calculate D, then execute the

event with greatest d

, then recalculate D and again pick the event with the

greatest d

, until all d

< 1, in which case the simulation would continue to

execute the default event as usual. This method will add overhead since the

matrix multiplication may need recalculation several times, but might add

extra decision intelligence by letting the decidors have different degrees of

preference for the execution of different plannable events.

In this section we provide a method for constructing members from a class

of problems, using linear decidors, such that these problems in many aspects

could be considered to describe real world maintenance activities on systems

with one or more components. Specifically, the single most important out-

putted value of the simulations is the expected profit at the end of the time

horizon T . We also allow for the components to have evolving efficiency

measures that affect the profit development over time.

Let N

be the number of components in or problem system, this value is a

free parameter in the class and can for example be chosen randomly. Each

such component is granted a number of independent state variables s

, a

part of the total state vector S.

For each component we allow for any number of randomly triggered events

. Each such event has a probability distribution r

which is a Weibull

distribution with a randomly chosen scale () and shape (k). To compute

the probability levels as functions of the state r(S) we introduce a linear de-

pendency on the independent state variables associated with the component

· S). That is, r(S) = r(a

· S).

For each component we also allow for a number of dependent state variable

in a similar fashion, e

. These state variables are computed from the current

independent state variables associated with the component by a function

f (S) = f (v

·S), where f is a randomly chosen Weibull distributions and v

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

suitable randomly chosen dependency vector. We interpret these dependent

variables as a measure of efficiency for some function of the component which

soon will be seen to affect simulated profit.

What remains in our example class is to define, actually parameterize, the

actual event functions P

, R

, D.

We generate the plannable and the random events P

, R

in the same fashion:

MS + w, where M is a matrix of suitable dimensions and w a vector,

both chosen randomly such that the new values for the dependent state

values associated with the corresponding component are free to change, but

only dependent on the old values for these state variables. The only other

state variables allowed to change is the time t and the current profit, time

must be advanced by an integer amount and the profit must be increased

by a constant term. The default event D is somewhat different. It may add

a constant term to any dependent state for any component s, the current

profit must increase by addition of a term p and may also decrease linearly

by a factor c

multiplied by any dependent state variable (efficiency measure)

and the time must increase by 1 (t).

Simulation

Decidor

Fitness

Figure 6.3: Illustration of the model

6.5

A Numerical Example

We selected the following example system, randomized from our stated prob-

lem class. The number of components is N

= 2. Horizon time is T = 504.

The first component has two random events (failures) and one independent

state variable. The second component has only one random event and also

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

one independent state variable. Both components have two efficiency mea-

sures.

Component

Event

M w t

43.385

1.6374

0.33321

-7.8147

66.813

1.5975

0.38519

-8.8816

66.487

1.1860

0.30687

-9.8577

Table 6.1: Parameters defining random events.

Component

Event

M w t p

-1

Table 6.2: Parameters defining plannable events.

Component

Efficiency

504

0.55770

0.36798

504

1.2823

0.70517

504

0.85805

0.071498

504

0.51121

0.34842

Table 6.3: Parameters defining efficiency measures.

(

-0.0096735, -0.69229, -0.77131, -0.71269) (0.21124, 0.78309)

Table 6.4: Parameters defining the default event.

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

Figure 6.4: Results from optimization of linear decidors in the example problem.

In Figure 6.4 the expected value for the relevant state variables from 2000

iterations of a simulation of the example problem is shown.

The linear decidors of the system (plannable events) was optimized using a

genetic algorithm with the expected profit at the time horizon (T ) as fitness

function. All 6 graphs are rescaled to lie between 0 and 1 to allow placement

in the same figure.

The two green curves shows the two independent state variables (one for

each component). The red spikes represent a high probability of having

a plannable event at those points in time.

Where there is an increased

probability of the plannable events to occur there is also a visible drop in the

corresponding state variable because M and w are both 0 for all plannable

events, efficiently resetting the states (component age) to zero.

The fitness can be seen to increase approximately linearly, which is a common

feature of maintenance planning simulations. As can be seen, the red spikes,

denoting a high probability for plannable events, decrease in height at later

times. This is likely a result from the increased number of random events that

has occurred on average at later times, causing a decreased determinability

of the system.

Figure 6.5 in the next page shows the convergence of the expected profit

at the end of horizon T for the genetic algorithm used. We begin with an

expected profit of 377 and after 272 generations of the optimization we have

reached a fairly long plateau beginning at a profit of 421 and we terminate

the algorithm after 512 generations with a profit of 423.

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

Figure 6.5 shows convergence of expected profit at the end of horizon T for

the genetic algorithm used to optimize the linear decidors in the example

problem.

Figure 6.5: Convergence of expected profit.

We summarize our algorithm as a flowchart in the next page. This approach

allows a better understanding of the process.

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

Start

Initialize system

Time

M ax

Stop

For each plannable event:

Perform event it the

events decidor returns true

For each random event:

Generate a random level

for the cumulative failure

distribution at which point

the component will fail

Perform default event

YES

For each random event:

Perform the event if

cumulative probabil-

ity

random gener-

ated cumulative level

CHAPTER 6. MAINTENANCE OPTIMIZATION MODEL

6.6

Conclusions

In this chapter we introduced our maintenance optimization model.

defined three different events: random, plannable and default events with

different tasks. We also defined the decidor for each plannable event and

described the structure and function of a decidor. In section 6.3 we illustrated

our simulation algorithm with details.

In section 6.4 we explained a specific model and introduced the linear decider

as a type of decidors and formulated the decision process in a compact form.

We finished this chapter with a numerical example by using genetic algorithm

and analyzed the result.

The main goal of this chapter was creating a simulation approach to mainte-

nance optimization with the benefit of a capability to optimize more complex

systems.

CHAPTER

Summary, Conclusions and Future

Work

7.1

Summary

As was discussed in the abstract, this work is split into two separate parts, one

dealing with decision making (chapter 3) and one dealing with maintenance

optimization model (chapter 6). The aim has been to create an optimization

model for maintenance problem. The component health has been examined

in chapter 2. Several criteria have been identified for DSS in chapter 3 and

a maintenance decision has been found. We analyzed different optimization

methods and algorithm in chapter 4 and Genetic algorithm had been chosen

as a suitable optimization algorithm (see chapter 5). The maintenance plan,

algorithm, simulation and optimization are summarized in chapter 6 and

then in section 6.5 the results of the numerical example is discussed.

7.2

Conclusions

We have provided some introduction to maintenance planning optimization.

Simulation and suitable optimization algorithms provides freedom in de-

scribing complex systems otherwise intractable by analytical methods. The

method of finding optimal decidors defined by some parameterization of the

decision logic allows for prompt establishment of a fair approximation to the

optimal use of resources. Although some additional layers are useful, per-

haps necessary, to establish a clear decision support for human consideration.

The decidors make their decision only as a function of the current state and

so provides only indirect decision patterns for planning of future activities.

CHAPTER 7. SUMMARY, CONCLUSIONS AND FUTURE WORK

One good way to transition from the decidors, and the statistics collected for

their decisions on a given system, is the realization that decisions closer in

time are more readily understandable and predictable. As a system evolves,

the inherent stochastic nature makes for example precise timing of mainte-

nance activities more and more difficult. In light of this we may consider

optimizing a plan for the close future in which activities are decided and

fixed, and will be performed unless unexpected events occur, in which case

a new good close future plan must be found. For a one-component system

such a short-time plan may be based on `time to first intervention', where we

for all possible times to first the plannable event consider enforcing the plan,

and in the simulations, if unexpected behavior occur, fall back on the deci-

dors to evaluate a near optimal `just in time' decision policy for the future.

This provides the decision maker with a nice single-value (profit) expected

result for planning the first intervention on different times, and may apply

additional non-modeled preferences on the decision based on these additional

simulations.

7.3

Future Work

There are three main lines of this research which should be pursued. Firstly,

the work on the criteria that effect on the component health. It is our inten-

tion to investigate thoroughly the various criteria, with a view to measuring

how different criteria affect the different components. To obtain this goal

we need to have an accurate data mining with some tools such as Rapid

Miner. A second line, which follows from chapter 2, is to investigate and cal-

culate the system reliability by the equations that we introduced and proved

in chapter 2. As a last line, we are going to examine the result by differ-

ent optimization algorithm such as artificial bee colony algorithm, Memetic

algorithms etc.

CHAPTER

Summary of reflection of objectives in

the thesis

We summarize our views of the objectives achieved in this thesis in this

chapter.

8.1

Objective 1 - Knowledge and Understand-

ing

In writing this thesis I had to research many journal articles in mechanical,

mathematical modeling, simulation and optimization.

In the process of writing chapter 4, I used a broad range of optimization books

as source material. Originally I compared a lot of optimization methods and

algorithms in details, and the material grew to many pages and I decided

under recommendation of my examiner and supervisor to present the main

result of the chapter in a more concise format.

At the research level, I recognized that the initial problem is a multi- objec-

tive optimization problem, and therefore I transfered the problem to a fuzzy

environment, I find 15 criteria which affect the component health. I also ana-

lyze the structure of the component and also studied a wide range of articles

in different areas of multi-criteria decision making, then I find TOPSIS as a

suitable method for DSS.

In section 2.5.1, I find Riccati differential equation as a suitable equation to

express the system reliability.

CHAPTER 8. SUMMARY OF REFLECTION OF OBJECTIVES IN

THE THESIS

8.2

Objective 2 -Methodological Knowledge

In writing this thesis I find two different approaches for decision making.

The first approach is a combination of fuzzy decision making and TOP-

SIS method. As second approach, I used genetic algorithm for optimiza-

tion.

For the reader get a better understanding about the background to the chap-

ters I began every chapter with a short introduction and definitions.

8.3

Objective 3 - Critically and Systemati-

cally Integrate Knowledge

The sources that I used in the thesis was obtained by searches at M¨

alardalen

University computer system and also journal articles which I found in the

University library and through Google search.

I divided the thesis in three different phases: pre-study, execution and im-

plementation. In the pre-study phase I read a broad range of articles about

maintenance policies and optimization. I identified the problem and indica-

tors in these phases.

The execution phase contains design, make and test the initial algorithm

with Matlab. The implementation phase, is a collaboration between me and

Jonas ¨

Osterberg, the initial algorithm was implemented in JavaScript.

8.4

Objective 4 - Independently and Creatively

Identify and Carry out Advanced Tasks

In section 2.5.1 I was used Riccati differential equation to finding Eq. 2.16

which is a combination of two equations that was introduced by Yuo-Tern

Tsai and Kuo-Shong.

The numerical example in section 3.8, is defined by the author. I identified 15

criteria which affect on health of component. To calculate the relationship

between criteria and alternatives, I used my knowledge in solid mechanics

and strength of materials and I also discussed about the results with senior

mechanical engineers in the maintenance department at Scania.

CHAPTER 8. SUMMARY OF REFLECTION OF OBJECTIVES IN

THE THESIS

8.5

Objective 5 - Present and Discuss Con-

clusions and Knowledge

I believe that this thesis is understandable for readers who have a moderate

familiarity with English and moderate knowledge in optimization topics. The

reader need to have a good knowledge in DSS to understand the different

methods that we used in this project for decision making (such as TOPSIS

and fuzzy decision making).

I assigned chapter 1to definitions and terms and also tried to define all con-

cepts which are required to understand the topic. In this project, I also bring

some numerical example to get a better understanding.

I explained the future work in the conclusions section, which is to explore

the use of data mining with Rapid Miner to get a more accurate model for

the health of components.

8.6

Objective 7 - Scientific, Social and Ethi-

cal Aspects

This thesis was a part of the IRIS-project, (in Swedish: Integrerat dynamiskt

prognostiserande underh°

allsst¨

od) which is a five-year project in Scania's Re-

search and Development department in Sweden. The approval process for

this project at Scania involves review and approval at each phase of project

development.

To experience the results, we had scheduled meeting with the IRIS-project

team every week and also with my supervisor Dr. Jonas Biteus who is project

leader for the IRIS-project.

I also presented this project in the ICNPAA 2014 Congress [47] in Norway

(Mathematical problems in engineering, aerospace and sciences). The at this

point unpublished congress article has the title: ` Solving complex mainte-

nance planning optimization problems using stochastic simulation and multi-

criteria fuzzy decision making ` and was written under supervision of profes-

sor Sergei Silvesterov, Jonas ¨

Osterberg and Dr. Jonas Biteus.

APPENDIX

Java code - Event Triggers

In this appendix we list the three types of event triggers used in our simula-

tions.

Trigger: The superclass describing a decision to perform an event at a

specific time. The prepare method is called at the beginning of every event

dispatch iteration and allows the implementors of this class to adjust their

behavior depending on a change in system state. The decide method is called

whenever this trigger is asked if its payload event should be executed. The

mutate and mate methods are used by the genetic algorithm for constructing

new triggers from one or two ancestors, leaving the specifics of the mutations

up to the implementing classes.

AlwaysTrigger: Represents the decision to perform the event whenever it is

asked to decide. This is suitable for "default" events, representing continuous

normal system operation.

StochasticTrigger: This trigger offers stochastic behavior due to a change

in the value method from the prepare phase to the decide phase.

LinearTrigger: This trigger returns a True decision if the linear combina-

tion of the state variables exceeds 1. The coefficients for the linear combina-

tion are stored in the val array, and these values will change on a mutate or

mate method call.

APPENDIX A. JAVA CODE - EVENT TRIGGERS

package com . vuebe . vhen ;

i m p o r t

java . util . R a n d o m ;

p u b l i c a b s t r a c t

class

T r i g g e r {

p u b l i c s t a t i c R a n d o m r a n d o m = new R a n d o m () ;

public Event event ;

p u b l i c a b s t r a c t void prepare ( State state ) ;

p u b l i c a b s t r a c t

b o o l e a n

decide ( State state ) ;

p u b l i c T r i g g e r m u t a t e ( d o u b l e p ) {

r e t u r n

this

;

}

p u b l i c T r i g g e r mate ( Trigger b , d o u b l e i n t e r p o l ) {

r e t u r n

this

;

}

p u b l i c T r i g g e r r a n d o m () {

r e t u r n

this

;

}

package com . vuebe . vhen ;

p u b l i c

class

A l w a y s T r i g g e r e x t e n d s T r i g g e r {

public A l w a y s T r i g g e r ( Event event ) {

this

. event = event ;

}

@ O v e r r i d e

public void prepare ( State state ) {

}

@ O v e r r i d e

p u b l i c

b o o l e a n

decide ( State state ) {

return true ;

}

package com . vuebe . vhen ;

APPENDIX A. JAVA CODE - EVENT TRIGGERS

p u b l i c a b s t r a c t

class

S t o c h a s t i c T r i g g e r e x t e n d s T r i g g e r {

p r i v a t e d o u b l e v a l u e ;

p r i v a t e d o u b l e t r i g g e r v a l u e ;

p u b l i c S t o c h a s t i c T r i g g e r ( Event event ) {

this

. event = event ;

}

p u b l i c a b s t r a c t d o u b l e v a l u e ( St a t e s t a t e ) ;

@ O v e r r i d e

public void prepare ( State state ) {

value = value ( state ) ;

if ( value < 0)

value = 0;

if ( value > 1)

value = 1;

t r i g g e r v a l u e = value + r a n d o m . n e x t D o u b l e () * (1 - value ) ;

}

@ O v e r r i d e

p u b l i c

b o o l e a n

decide ( State state ) {

value = value ( state ) ;

if ( value < 0)

value = 0;

if ( value > 1)

value = 1;

r e t u r n value >= t r i g g e r v a l u e ;

}

package com . vuebe . vhen ;

i m p o r t

java . util . R a n d o m ;

i m p o r t

com . vuebe . util . Util ;

p u b l i c

class

L i n e a r T r i g g e r e x t e n d s T r i g g e r {

p u b l i c d o u b l e [] val ;

public double [] mask ;

public L i n e a r T r i g g e r ( Event event , d o u b l e [] val , d o u b l e []

mask ) {

this

. event = event ;

this

. val = val ;

this

. mask = mask ;

}

APPENDIX A. JAVA CODE - EVENT TRIGGERS

p ub l ic L i n e a r T r i g g e r ( Ev e nt event , d o u b l e [] val ) {

100

this

. event = event ;

101

this

. val = val ;

102

this

. mask = new double [ val . length ];

103

s e t M a s k () ;

104

}

105

106

p ub l i c L i n e a r T r i g g e r ( Ev e nt event , State S ) {

107

this

. event = event ;

108

this

. val = new d o u b l e [ S . val . l e n g t h ];

109

this

. mask = new double [ val . length ];

110

s e t M a s k () ;

111

}

112

113

public void setM a s k () {

114

for ( int i = 0; i < mask . length ; i ++)

115

mask [ i ] = 1;

116

}

117

118

public void setM a s k ( int i ) {

119

mask [ i ] = 1;

120

}

121

122

public void c l e a r M a s k () {

123

for ( int i = 0; i < mask . length ; i ++)

124

mask [ i ] = 0;

125

}

126

127

public void c l e a r M a s k ( int i ) {

128

mask [ i ] = 0;

129

}

130

131

@ O v e r r i d e

132

p u b l i c L i n e a r T r i g g e r m u t a t e ( d o u b l e p ) {

133

d o u b l e [] mal = new d o u b l e [ val . l e n g t h ];

134

for ( int i = 0; i < mal . l e n g t h ; i ++) {

135

if ( r a n d o m . n e x t D o u b l e () < p ) {

136

d o u b l e v = val [ i ];

137

mal [ i ] = v + ( random . n e x t D o u b l e () * 2 - 1) * mask [ i ];

138

if ( mal [ i ] < 0)

139

mal [ i ] = 0;

140

} else {

141

mal [ i ] = val [ i ];

142

}

143

}

144

r e t u r n new L i n e a r T r i g g e r ( event , mal , mask ) ;

145

}

146

APPENDIX A. JAVA CODE - EVENT TRIGGERS

147

@ O v e r r i d e

148

p u b l i c T r i g g e r mate ( Trigger b , d o u b l e i n t e r p o l ) {

149

if ( b i n s t a n c e o f L i n e a r T r i g g e r ) {

150

L i n e a r T r i g g e r B = ( L i n e a r T r i g g e r ) b ;

151

d o u b l e [] va = val ;

152

d o u b l e [] vb = B . val ;

153

d o u b l e [] vm = new d o u b l e [ va . l e n g t h ];

154

for ( int i = 0; i < vm . l e n g t h ; i ++) {

155

d o u b l e v1 = va [ i ];

156

d o u b l e v2 = vb [ i ];

157

vm [ i ] = v1 + ( v2 - v1 ) * i n t e r p o l ;

158

}

159

r e t u r n new L i n e a r T r i g g e r ( event , vm , mask ) ;

160

161

} else

162

throw

new R u n t i m e E x c e p t i o n (

" I n v a l i d T r i g g e r type

c o m b i n a t i o n . "

) ;

163

}

164

165

p u b l i c T r i g g e r r a n d o m () {

166

d o u b l e [] vm = new d o u b l e [ val . l e n g t h ];

167

for ( int i = 0; i < vm . l e n g t h ; i ++) {

168

vm [ i ] = mask [ i ] * random . n e x t D o u b l e () * 0.001;

169

}

170

r e t u r n new L i n e a r T r i g g e r ( event , vm , mask ) ;

171

}

172

173

@ O v e r r i d e

174

public void prepare ( State state ) {

175

}

176

177

@ O v e r r i d e

178

p u b l i c

b o o l e a n

decide ( State state ) {

179

d o u b l e sum = 0;

180

for ( int i = 0; i < val . l e n g t h ; i ++) {

181

sum += val [ i ] * state . val [ i ];

182

}

183

r e t u r n sum >= 1;

184

}

185

186

p u b l i c S t r i n g t o S t r i n g () {

187

r e t u r n

" val "

+ Util . t o S t r i n g ( val ) +

" mask "

+ Util .

t o S t r i n g ( mask ) ;

188

}

189

}

APPENDIX

Java code - Genetic Algorithm

In this appendix we list the Java code for the genetic algorithm used in our

simulations.

Individual: The genetic algorithm implementation will consider a popula-

tion of individuals, but the specifics of these individuals are subject to cus-

tomization by implementing this class. Specifically, the evaluate method

will compute the fitness of this individual and the getValue method will

return the last evaluated fitness. The fitness function may be stochastic and

so the evaluate method may have to be called more than once.

GA: This is the main genetic algorithm implementation providing possibility

to optimize the fitness function for an individual and its mutated offspring.

The initiatePopulation method provides the algorithm with a seed indi-

vidual and from it constructs a population. This population is then evolved,

ideally to a better fitness, by the evolve method. The evolve method may

be run any number of times, and the currently best individual can then be

retrieved by the getBestIndividual method.

APPENDIX B. JAVA CODE - GENETIC ALGORITHM

p a c k a g e com . v u e b e . ga ;

p u b l i c i n t e r f a c e I n d i v i d u a l e x t e n d s C o m p a rable < Individual > {

/* * Compute the value of this i n d i v i d u a l . */

p u b l i c d o u b l e e v a l u a t e () ;

/* * Return the c o m p u t e d value of this i n d i v i d u a l . */

p u b l i c d o u b l e g e t V a l u e () ;

p u b l i c I n d i v i d u a l m u t a t e ( d o u b l e p ) ;

public I n d i v i d u a l [] mate ( I n d i v i d u a l mate ) ;

p u b l i c I n d i v i d u a l r a n d o m () ;

}

p a c k a g e com . v u e b e . ga ;

i m p o r t

java . util . C o l l e c t i o n s ;

i m p o r t

com . vuebe . vhen .*;

i m p o r t

java . util . R a n d o m ;

i m p o r t

java . util . V e c t o r ;

i m p o r t

com . vuebe . util . Util ;

p u b l i c

class

GA {

p u b l i c s t a t i c final int T H R E A D C O U N T = 8;

p u b l i c s t a t i c f i n a l int T O U R N A M E N T = 0;

p u b l i c s t a t i c final int R O U L E T T E _ W H E E L = 1;

p r i v a t e int Ne = 2;

// Elite i n d i v i d u a l s

p r i v a t e int Ns = 50;

// S e l e c t i o n i n d i v i d u a l s

p r i v a t e int N = Ne + Ns ;

// P o p u l a t i o n size N = Ne + Ns

private double Pmut = 0.2;

// M u t a t e p r o b a b i l i t y

p r i v a t e int s e l e c t i o n m e t h o d = T O U R N A M E N T ;

// S e l e c t i o n

m e t h o d

p r i v a t e int Nt = 6;

// T o u r n a m e n t size

p r i v a t e d o u b l e Pt = 0.5;

// T o u r n a m e n t p r o b a b i l i t y

p r i v a t e R a n d o m r a n d o m = new R a n d o m () ;

p r i v a t e Vector < Individual > p o p u l a t i o n ;

p r i v a t e int g e n e r a t i o n ;

p r i v a t e int e v a l u a t i o n s ;

APPENDIX B. JAVA CODE - GENETIC ALGORITHM

p u b l i c I n d i v i d u a l g e t B e s t I n d i v i d u a l () {

I n d i v i d u a l ind = p o p u l a t i o n . get (0) ;

double best = ind . g e t V a l u e () ;

for ( int i = 1; i < p o p u l a t i o n . size () ; i ++) {

I n d i v i d u a l ind2 = p o p u l a t i o n . get ( i ) ;

double value = ind2 . g e t V a l u e () ;

if ( value > best ) {

best = value ;

ind = ind2 ;

}

r e t u r n ind ;

}

public void s e t S e l e c t i o n S i z e ( int ns ) {

if ( ns < 0 || ( ns & 1) != 0)

throw

new R u n t i m e E x c e p t i o n () ;

this

. Ns = ns ;

this

. N = Ns + Ne ;

}

public void s e t E l i t e S i z e ( int ne ) {

if ( ne < 0)

throw

new R u n t i m e E x c e p t i o n () ;

this

. Ne = ne ;

this

. N = Ns + Ne ;

}

p u b l i c int g e t G e n e r a t i o n () {

r e t u r n g e n e r a t i o n ;

}

p u b l i c int g e t E v a l u a t i o n s () {

r e t u r n e v a l u a t i o n s ;

}

public void i n i t i a t e P o p u l a t i o n ( I n d i v i d u a l seed ) {

p o p u l a t i o n = new Vector < Individual >() ;

e v a l u a t i o n s = 0;

for ( int i = 0; i < N ; i ++) {

I n d i v i d u a l i n d i v i d u a l = seed . random () ;

i n d i v i d u a l . e v a l u a t e () ;

e v a l u a t i o n s ++;

p o p u l a t i o n . add ( i n d i v i d u a l ) ;

}

public void sort () {

APPENDIX B. JAVA CODE - GENETIC ALGORITHM

C o l l e c t i o n s . sort ( p o p u l a t i o n ) ;

}

100

101

public void evolve () {

102

// Util . dump (" E v o l v e ") ;

103

sort () ;

104

105

S t r i n g s =

" "

;

106

for ( int i = 0; i < p o p u l a t i o n . size () ; i ++) {

107

s += p o p u l a t i o n . get ( i ) . g e t V a l u e () +

" "

;

108

}

109

// Util . dump (" Values : " + s ) ;

110

111

Vector < Individual > n e x t p o p u l a t i o n = new Vector < Individual

>() ;

112

113

// S e l e c t i o n

114

if ( s e l e c t i o n m e t h o d == T O U R N A M E N T ) {

115

for ( int tourn = 0; tourn < Ns / 2; tourn ++) {

116

Vector < Individual > tour = new Vector < Individual >() ;

117

for ( int i = 0; i < Nt ; i ++) {

118

tour . add ( p o p u l a t i o n . get ( r a n d o m . n e x t I n t ( p o p u l a t i o n .

size () ) ) ) ;

119

}

120

C o l l e c t i o n s . sort ( tour ) ;

121

122

I n d i v i d u a l first = null ;

123

I n d i v i d u a l second = null ;

124

int f i r s t i n d e x = -1;

125

combine : while ( true ) {

126

d o u b l e P = Pt ;

127

for ( int i = Nt - 1; i >= 0; i - -) {

128

if ( i != f i r s t i n d e x && r a n d o m . n e x t D o u b l e () < P ) {

129

if ( first == null ) {

130

first = tour . get ( i ) ;

131

f i r s t i n d e x = i ;

132

} else {

133

second = tour . get ( i ) ;

134

I n d i v i d u a l [] c h i l d r e n = first . mate ( second ) ;

135

136

c h i l d r e n [0] = c h i l d r e n [0]. mutate ( Pmut ) ;

137

c h i l d r e n [1] = c h i l d r e n [1]. mutate ( Pmut ) ;

138

139

n e x t p o p u l a t i o n . add ( c h i l d r e n [0]) ;

140

n e x t p o p u l a t i o n . add ( c h i l d r e n [1]) ;

141

142

c h i l d r e n [0]. e v a l u a t e () ;

143

c h i l d r e n [1]. e v a l u a t e () ;

144

APPENDIX B. JAVA CODE - GENETIC ALGORITHM

145

e v a l u a t i o n s += 2;

146

147

b r e a k c o m b i n e ;

148

}

149

}

150

P = P * (1 - Pt ) ;

151

152

}

153

}

154

155

}

156

} else

157

throw

new R u n t i m e E x c e p t i o n (

" U n i m p l e m e n t e d s e l e c t i o n

m e t h o d . "

) ;

158

// Elite

159

for ( int i = 0; i < Ne ; i ++) {

160

I n d i v i d u a l ind = p o p u l a t i o n . get ( p o p u l a t i o n . size () - 1 -

i ) ;

161

ind = (( com . vuebe . vhen . System ) ind ) . clone () ;

162

ind . e v a l u a t e () ;

163

n e x t p o p u l a t i o n . add ( ind ) ;

164

}

165

166

g e n e r a t i o n ++;

167

p o p u l a t i o n = n e x t p o p u l a t i o n ;

168

}

169

}

APPENDIX

Java code - Simulation Classes

In this appendix we list the Java code for simulation classes used in our

simulations.

State: The Represents a system state consisting of a set of double values

(real numbers) in the val field. The state variables have names by the

corresponding array values in the name field. The dep array provides the

possibility to have some state variables dynamically computed as a function

of the other state variables.

Event: The superclass of all events that may occur in the simulation, the

events only result is to change the values of the state variables in the run

method.

StateFunction: Used in the State class to provide the possibility for dy-

namically computed state variables.

System: This is the main simulation class realizing the simulation algorithm.

The system is defined by the initialstate field and the set of triggers (and

the event they trigger) in the triggers field. The toMatlab method creates

a string in Matlab syntax constructing the experimental data constructed

from a number of simulations performed on this system from the simulate

method. The class also provides mutation and mating methods since the class

can be used as an individual in the genetic optimization algorithm. The main

simulation algorithm is contained in the simulateOne method, performing a

simulation of the system over the entire time horizong horizon. This method

is a bit more generally written than the algorithm described previously in the

report for experimentation purposes and easability for adaptation to more

complex scenarios. The algorithm, given a correctly ordered list of event

triggers, works equivalently except for a short time step time difference.

APPENDIX C. JAVA CODE - SIMULATION CLASSES

package com . vuebe . vhen ;

i m p o r t

java . util . H a s h t a b l e ;

p u b l i c

class

State {

public Hashtable < String , Integer > ind = new Hashtable <

String , Integer >() ;

public String [] name ;

p u b l i c d o u b l e [] val ;

p u b l i c S t a t e F u n c t i o n [] dep ;

public State ( State s ) {

ind = ( Hashtable < String , Integer >) s . ind . clone () ;

val = s . val . clone () ;

dep = s . dep . clone () ;

name = s . name ;

}

public State ( String ... vars ) {

for ( int i = 0; i < vars . length ; i ++) {

if ( ind . put ( vars [ i ] , i ) != null ) {

throw

new R u n t i m e E x c e p t i o n (

" D u p l i c a t e v a r i a b l e "

) ;

}

val = new double [ vars . length ];

this

. name = vars . clone () ;

}

p u b l i c S t a t e ( S t r i n g [] vars , S t a t e F u n c t i o n [] dep ) {

this

( vars ) ;

this

. dep = dep . clone () ;

if ( dep . length != vars . length )

throw

new R u n t i m e E x c e p t i o n (

" I n v a l i d d e p e n d e n c y f u n c t i o n

count . "

) ;

}

p u b l i c S t r i n g t o S t r i n g () {

S t r i n g s =

" "

;

for ( int i = 0; i < name . length ; i ++)

s += name [ i ] +

" = "

+ val [ i ] +

" "

;

r e t u r n s ;

}

public void c o m p u t e D e p e n d e n t () {

if ( dep == null )

r e t u r n ;

for ( int i = 0; i < dep . l e n g t h ; i ++) {

if ( dep [ i ] != null )

val [ i ] = dep [ i ]. value (

this

) ;

APPENDIX C. JAVA CODE - SIMULATION CLASSES

}

p u b l i c int in d e x ( S t r i n g var ) {

r e t u r n ind . get ( var ) ;

}

package com . vuebe . vhen ;

p u b l i c a b s t r a c t

class

Event {

public a b s t r a c t void run ( State state ) ;

}

package com . vuebe . vhen ;

p u b l i c a b s t r a c t

class

V a l u e F u n c t i o n {

p u b l i c a b s t r a c t d o u b l e v a l u e ( S y s t e m s y s t e m ) ;

}

package com . vuebe . vhen ;

p u b l i c a b s t r a c t

class

S t a t e F u n c t i o n {

p u b l i c a b s t r a c t d o u b l e v a l u e ( St a t e s t a t e ) ;

}

package com . vuebe . vhen ;

i m p o r t

java . util . R a n d o m ;

i m p o r t

java . util . V e c t o r ;

i m p o r t

com . vuebe . ga . I n d i v i d u a l ;

i m p o r t

com . vuebe . util . Util ;

p u b l i c

class

S y s t e m

i m p l e m e n t s

I n d i v i d u a l {

p u b l i c s t a t i c R a n d o m r a n d o m = new R a n d o m () ;

p r i v a t e State i n i t i a l s t a t e ;

p r i v a t e int h o r i z o n ;

p r i v a t e V a l u e F u n c t i o n v a l u e f u n c t i o n ;

APPENDIX C. JAVA CODE - SIMULATION CLASSES

p u b l i c Vector < Trigger > t r i g g e r s = new Vector < Trigger >() ;

p u b l i c Vector < String > t r i g g e r n a m e = new Vector < String >() ;

100

// d e r i v e d or s i m u l a t e d

101

int itime ;

102

103

p u b l i c int N ;

104

public double [][] M ;

// State Mean

105

public double [][] V ;

// State V a r i a n c e

106

public double [][] C ;

// T r i g g e r c o u n t s

107

108

p u b l i c d o u b l e v a l u e ;

109

p r i v a t e int s i m u l a t i o n c o u n t ;

110

111

p u b l i c S t r i n g t o M a t l a b () {

112

S t r i n g B u i l d e r sb = new S t r i n g B u i l d e r () ;

113

114

sb . a p p e n d (

" i t e r a t i o n s = "

+ N ) ;

115

sb . a p p e n d (

" ;\ n "

) ;

116

117

sb . a p p e n d (

" s t a t e v a r s = "

+ Util . t o M a t l a b ( i n i t i a l s t a t e . name )

) ;

118

sb . a p p e n d (

" ;\ n "

) ;

119

120

for ( int i = 0; i < M . l e n g t h ; i ++) {

121

122

sb . a p p e n d (

" M "

+ i +

" = "

123

+ Util . t o M a t l a b ( Util . m u l t i p l y N e w ( M [ i ] , 1 d / N ) ) ) ;

124

sb . a p p e n d (

" ;\ n "

) ;

125

126

sb . a p p e n d (

" V "

+ i +

" = "

127

+ Util . t o M a t l a b ( Util . m u l t i p l y N e w ( V [ i ] , 1 d / N ) ) ) ;

128

sb . a p p e n d (

" ;\ n "

) ;

129

}

130

131

sb . a p p e n d (

" M ={ "

) ;

132

for ( int i = 0; i < M . l e n g t h ; i ++) {

133

sb . a p p e n d (

" M "

+ i ) ;

134

if ( i != M . l e n g t h - 1)

135

sb . a p p e n d (

" ,"

) ;

136

}

137

sb . a p p e n d (

" }\ n "

) ;

138

139

sb . a p p e n d (

" V ={ "

) ;

140

for ( int i = 0; i < M . l e n g t h ; i ++) {

141

sb . a p p e n d (

" V "

+ i ) ;

142

if ( i != M . l e n g t h - 1)

143

sb . a p p e n d (

" ,"

) ;

144

}

APPENDIX C. JAVA CODE - SIMULATION CLASSES

145

sb . a p p e n d (

" }\ n "

) ;

146

147

for ( int i = 0; i < C . l e n g t h ; i ++) {

148

sb . a p p e n d (

" C "

+ i +

" = "

149

+ Util . t o M a t l a b ( Util . m u l t i p l y N e w ( C [ i ] , 1 d / N ) ) ) ;

150

sb . a p p e n d (

" ;\ n "

) ;

151

}

152

153

sb . a p p e n d (

" C ={ "

) ;

154

for ( int i = 0; i < C . l e n g t h ; i ++) {

155

sb . a p p e n d (

" C "

+ i ) ;

156

if ( i != C . l e n g t h - 1)

157

sb . a p p e n d (

" ,"

) ;

158

}

159

sb . a p p e n d (

" }\ n "

) ;

160

161

sb . a p p e n d (

" t r i g g e r n a m e ={ "

) ;

162

for ( int i = 0; i < C . l e n g t h ; i ++) {

163

sb . a p p e n d (

" '"

+ t r i g g e r n a m e . get ( i ) +

" '"

) ;

164

if ( i != C . l e n g t h - 1)

165

sb . a p p e n d (

" ,"

) ;

166

}

167

sb . a p p e n d (

" }\ n "

) ;

168

169

r e t u r n sb . t o S t r i n g () ;

170

}

171

172

public State g e t I n i t i a l S t a t e () {

173

r e t u r n i n i t i a l s t a t e ;

174

}

175

176

p u b l i c T r i g g e r g e t T r i g g e r ( int i n d e x ) {

177

r e t u r n t r i g g e r s . get ( i n d e x ) ;

178

}

179

180

public void s e t I n i t i a l S t a t e ( State S ) {

181

i n i t i a l s t a t e = S ;

182

itime = i n i t i a l s t a t e . index (

" time "

) ;

183

}

184

185

public void s e t V a l u e F u n c t i o n ( V a l u e F u n c t i o n vf ) {

186

this

. v a l u e f u n c t i o n = vf ;

187

}

188

189

public void s e t H o r i z o n ( int h o r i z o n ) {

190

this

. h o r i z o n = h o r i z o n ;

191

}

192

193

public void s e t S i m u l a t i o n C o u n t ( int s i m u l a t i o n c o u n t ) {

APPENDIX C. JAVA CODE - SIMULATION CLASSES

194

this

. s i m u l a t i o n c o u n t = s i m u l a t i o n c o u n t ;

195

}

196

197

p u b l i c int g e t H o r i z o n () {

198

r e t u r n h o r i z o n ;

199

}

200

201

p u b l i c int a d d T r i g g e r ( T r i g ger trigger , String name ) {

202

t r i g g e r s . add ( t r i g g e r ) ;

203

t r i g g e r n a m e . add ( name ) ;

204

return t r i g g e r s . size () - 1;

205

}

206

207

public void dump () {

208

Util . dump (

" I n i t i a l : "

+ Util . t o S t r i n g ( i n i t i a l s t a t e . val ) ) ;

209

Util . dump (

" H o r i z o n : "

+ h o r i z o n ) ;

210

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

211

Util . dump (

" T r i g g e r "

+ i +

" : "

+ t r i g g e r s . get ( i ) .

t o S t r i n g () ) ;

212

}

213

}

214

215

private void s i m u l a t e O n e () {

216

State S = new State ( i n i t i a l s t a t e ) ;

217

S . c o m p u t e D e p e n d e n t () ;

218

Random rand = new Random () ;

219

S . val [ itime ] = 0;

220

w h i l e ( S . val [ it i m e ] < h o r i z o n ) {

221

// p r e p a r e t r i g g e r s

222

for ( int i = 0; i < t r i g g e r s . size () ; i ++)

223

t r i g g e r s . get ( i ) . p r e p a r e ( S ) ;

224

// d e c i d e t r i g g e r s

225

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

226

if ( t r i g g e r s . get ( i ) . d e c i d e ( S ) ) {

227

int t0 = ( int ) Math . round ( S . val [ itime ]) ;

228

if ( t0 < h o r i z o n )

229

C [ i ][ t0 ]++;

230

231

t r i g g e r s . get ( i ) . e ve nt . run ( S ) ;

232

S . c o m p u t e D e p e n d e n t () ;

233

234

int t1 = ( int ) Math . round ( S . val [ itime ]) ;

235

for ( int t = t0 ; t < t1 && t < h o r i z o n ; t ++) {

236

for ( int p = 0; p < S . val . l e n g t h ; p ++) {

237

d o u b l e v = S . val [ p ];

238

M [ p ][ t ] += v ;

239

V [ p ][ t ] += v * v ;

240

}

241

}

APPENDIX C. JAVA CODE - SIMULATION CLASSES

242

}

243

}

244

}

245

}

246

247

public void s i m u l a t e () {

248

// Util . dump (" Sim ") ;

249

// dump () ;

250

M = new d o u b l e [ i n i t i a l s t a t e . val . l e n g t h ][];

251

V = new d o u b l e [ i n i t i a l s t a t e . val . l e n g t h ][];

252

C = new d o u b l e [ t r i g g e r s . size () ][];

253

for ( int i = 0; i < M . l e n g t h ; i ++) {

254

M [ i ] = new d o u b l e [ h o r i z o n ];

255

}

256

for ( int i = 0; i < V . l e n g t h ; i ++) {

257

V [ i ] = new d o u b l e [ h o r i z o n ];

258

}

259

for ( int i = 0; i < C . l e n g t h ; i ++) {

260

C [ i ] = new d o u b l e [ h o r i z o n ];

261

}

262

263

for ( N = 0; N < s i m u l a t i o n c o u n t ; N ++) {

264

s i m u l a t e O n e () ;

265

}

266

267

if ( v a l u e f u n c t i o n != null ) {

268

value = v a l u e f u n c t i o n . value (

this

) ;

269

}

270

}

271

272

p r i v a t e S y s t e m c l o n e N o T r i g g e r s () {

273

S y s t e m s = new S y s t e m () ;

274

s . s e t I n i t i a l S t a t e ( i n i t i a l s t a t e ) ;

275

s . s e t H o r i z o n ( h o r i z o n ) ;

276

s . s e t V a l u e F u n c t i o n ( v a l u e f u n c t i o n ) ;

277

s . itime = itime ;

278

s . s i m u l a t i o n c o u n t = s i m u l a t i o n c o u n t ;

279

s . t r i g g e r n a m e = t r i g g e r n a m e ;

280

r e t u r n s ;

281

}

282

283

p u b l i c S y s t e m c l o n e M u t a t e ( d o u b l e p ) {

284

S y s t e m s = c l o n e N o T r i g g e r s () ;

285

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

286

s . t r i g g e r s . add ( t r i g g e r s . get ( i ) . m u t a t e ( p ) ) ;

287

}

288

r e t u r n s ;

289

}

290

APPENDIX C. JAVA CODE - SIMULATION CLASSES

291

p u b l i c S y s t e m c l o n e () {

292

S y s t e m s = c l o n e N o T r i g g e r s () ;

293

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

294

s . t r i g g e r s . add ( t r i g g e r s . get ( i ) ) ;

295

}

296

r e t u r n s ;

297

}

298

299

public System [] c l o n e M a t e ( System mate ) {

300

double i n t e r p o l = random . n e x t D o u b l e () * 1.4 - 0.2;

301

S y s t e m s1 = c l o n e N o T r i g g e r s () ;

302

S y s t e m s2 = c l o n e N o T r i g g e r s () ;

303

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

304

s1 . t r i g g e r s . add ( t r i g g e r s . get ( i )

305

. mate ( mate . t r i g g e r s . get ( i ) , i n t e r p o l ) ) ;

306

s2 . t r i g g e r s . add ( t r i g g e r s . get ( i ) . mate ( mate . t r i g g e r s . get (

i ) ,

307

0.5 - i n t e r p o l ) ) ;

308

}

309

r e t u r n new S y s t e m [] { s1 , s2 };

310

}

311

312

p u b l i c S y s t e m c l o n e R a n d o m () {

313

S y s t e m s = c l o n e N o T r i g g e r s () ;

314

for ( int i = 0; i < t r i g g e r s . size () ; i ++) {

315

s . t r i g g e r s . add ( t r i g g e r s . get ( i ) . r a n d o m () ) ;

316

}

317

r e t u r n s ;

318

}

319

320

@ O v e r r i d e

321

p u b l i c int c o m p a r e T o ( I n d i v i d u a l ind ) {

322

return ( int ) Math . signum ( g e t V a l u e () - ind . g e t V a l u e () ) ;

323

}

324

325

@ O v e r r i d e

326

p u b l i c d o u b l e e v a l u a t e () {

327

s i m u l a t e () ;

328

return value ;

329

}

330

331

@ O v e r r i d e

332

p u b l i c d o u b l e g e t V a l u e () {

333

return value ;

334

}

335

336

@ O v e r r i d e

337

p u b l i c I n d i v i d u a l m u t a t e ( d o u b l e p ) {

338

r e t u r n c l o n e M u t a t e ( p ) ;

APPENDIX C. JAVA CODE - SIMULATION CLASSES

339

}

340

341

@ O v e r r i d e

342

public I n d i v i d u a l [] mate ( I n d i v i d u a l mate ) {

343

return c l o n e M a t e (( System ) mate ) ;

344

}

345

346

@ O v e r r i d e

347

p u b l i c I n d i v i d u a l r a n d o m () {

348

r e t u r n c l o n e R a n d o m () ;

349

}

350

}

APPENDIX

Java code - Example Problem Class

In this appendix we list the Java code for the construction and analyzis of

instanses of the example problem class, Section 6.4.

SystemFactory: The class that constructs systems dynamically and ran-

domly. Several parameters for the constructed system can be specified, such

as the number of "components" NC, the number of state variables per com-

ponent sv0 to sv1. After specifying the desired parameters the system can

be created by a call to the createSystem method.

TestFactory: The class is a main program entry point through the main

method and subsequently the run method.

A SystemFactory object is

first constructed, some parameters specifyed for the desired system and the

system is then created by the createSystem method call. This created

system is then fed as the seed Individual to a genetic algorithm by the

ga.initiatePopulation(sys); statement. The genetic algorithm is then

run 10000 times and the best individual after the optimization is saved to

file in Matlab format for analyzis and plotting in Matlab.

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

package com . vuebe . vhen ;

i m p o r t

java . util . R a n d o m ;

i m p o r t

java . util . V e c t o r ;

i m p o r t

com . vuebe . math . W e i b u l l D i s t r i b u t i o n ;

i m p o r t

com . vuebe . util . Util ;

p u b l i c

class

S y s t e m F a c t o r y {

p u b l i c R a n d o m r a n d o m = new R a n d o m (0) ;

// T = T1 * T2 * T3 , hours days weeks

p u b l i c int T ;

p u b l i c int T1 = 24;

p u b l i c int T2 = 7;

p u b l i c int T3 = 3;

// N u m b e r of c o m p o n e n t s

p u b l i c int NC ;

// n u m b e r of s t a t e v a r i a b l e s per c o m p o n e n t

p u b l i c int sv0 ;

p u b l i c int sv1 ;

// F a i l u r e Age co u n t

p u b l i c int FA0 ;

p u b l i c int FA1 ;

// w e i b u l l p r o b a b i l i t y d i s t r i b u t i o i n p a r a m e t e r s

p u b l i c d o u b l e F A s c a l e m i n ;

p u b l i c d o u b l e F A s c a l e m a x ;

p u b l i c d o u b l e F A s h a p e m i n ;

p u b l i c d o u b l e F A s h a p e m a x ;

// d e s c r i p t i o n of age d e p e n d e n c y v e c t o r s

p u b l i c d o u b l e F A d e p e n d e n c y m a x ;

// E f f i c i e n c y age count

p u b l i c int EA0 ;

p u b l i c int EA1 ;

// w e i b u l l p r o b a b i l i t y d i s t r i b u t i o i n p a r a m e t e r s

p u b l i c d o u b l e E A s c a l e m i n ;

p u b l i c d o u b l e E A s c a l e m a x ;

p u b l i c d o u b l e E A s h a p e m i n ;

p u b l i c d o u b l e E A s h a p e m a x ;

// d e s c r i p t i o n of age d e p e n d e n c y v e c t o r s

p u b l i c d o u b l e E A d e p e n d e n c y m a x ;

p u b l i c d o u b l e N E p r o f i t m i n ;

p u b l i c d o u b l e N E p r o f i t m a x ;

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

p u b l i c d o u b l e N E p r o f i t e f f i c i e n c y p e n a l t y m a x ;

p u b l i c d o u b l e N E S V a d d m a x ;

p u b l i c

b o o l e a n

useFE ;

p u b l i c d o u b l e F E m u l 0 ;

p u b l i c d o u b l e F E m u l 1 ;

p u b l i c d o u b l e F E a d d 0 ;

p u b l i c d o u b l e F E a d d 1 ;

p u b l i c int F E t i m e 0 ;

p u b l i c int F E t i m e 1 ;

p u b l i c d o u b l e F E f i t n e s s 0 ;

p u b l i c d o u b l e F E f i t n e s s 1 ;

p u b l i c

b o o l e a n

useDE ;

p u b l i c d o u b l e D E m u l 0 ;

p u b l i c d o u b l e D E m u l 1 ;

p u b l i c d o u b l e D E a d d 0 ;

p u b l i c d o u b l e D E a d d 1 ;

p u b l i c int D E t i m e 0 ;

p u b l i c int D E t i m e 1 ;

p u b l i c d o u b l e D E f i t n e s s 0 ;

p u b l i c d o u b l e D E f i t n e s s 1 ;

// c r e a t i o n s

p u b l i c C o m p o n e n t [] c ;

p u b l i c int t o t a l S V i n d e x ;

p u b l i c int t o t a l S V c o u n t ;

p u b l i c int t o t a l F A i n d e x ;

p u b l i c int t o t a l F A c o u n t ;

p u b l i c int t o t a l E A i n d e x ;

p u b l i c int t o t a l E A c o u n t ;

public int itime ;

p u b l i c int i f i t n e s s ;

p u b l i c State i n i t i a l s t a t e ;

p u b l i c d o u b l e N E f i t n e s s ;

p u b l i c d o u b l e [] N E e f f i c i e n c y f i t n e s s ;

p u b l i c d o u b l e [] N E S V a d d ;

public void dump () {

Util . dump ( i n i t i a l s t a t e . t o S t r i n g () ) ;

Util . dump (

" T = "

+ T ) ;

Util . dump (

" NC = "

+ NC ) ;

for ( int i = 0; i < c . l e n g t h ; i ++) {

Util . dump (

" C O M P O N E N T "

+ i ) ;

c [ i ]. dump () ;

}

// Normal Event

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

Util . dump (

" - - - - - - "

) ;

100

Util . dump (

" N E f i t n e s s = "

+ N E f i t n e s s ) ;

101

Util . dump (

" N E e f f i c i e n c y f i t n e s s = "

+ Util . t o S t r i n g (

N E e f f i c i e n c y f i t n e s s ) ) ;

102

Util . dump (

" N E S v a d d = "

+ Util . t o S t r i n g ( N E S V a d d ) ) ;

103

}

104

105

p u b l i c S y s t e m c r e a t e S y s t e m () {

106

S y s t e m sys = new S y s t e m () ;

107

T = T1 * T2 * T3 ;

108

109

sys . s e t H o r i z o n ( T ) ;

110

c = new C o m p o n e n t [ NC ];

111

for ( int i = 0; i < NC ; i ++) {

112

c [ i ] = new C o m p o n e n t () ;

113

}

114

115

Vector < String > s t a t e v a r s = new Vector < String >() ;

116

Vector < S t a t e F u n c t i o n > s t a t e f u n c = new Vector <

S t a t e F u n c t i o n >() ;

117

118

t o t a l S V i n d e x = s t a t e v a r s . size () ;

119

t o t a l S V c o u n t = 0;

120

for ( int i = 0; i < NC ; i ++) {

121

C o m p o n e n t com = c [ i ];

122

com . svindex = s t a t e v a r s . size () ;

123

for ( int j = 0; j < com . s v c o u n t ; j ++) {

124

s t a t e v a r s . add (

" s "

+ i +

" _ "

+ j ) ;

125

s t a t e f u n c . add ( null ) ;

126

}

127

t o t a l S V c o u n t += com . s v c o u n t ;

128

}

129

130

t o t a l F A c o u n t = 0;

131

t o t a l F A i n d e x = s t a t e v a r s . size () ;

132

for ( int i = 0; i < NC ; i ++) {

133

C o m p o n e n t com = c [ i ];

134

com . FAindex = s t a t e v a r s . size () ;

135

for ( int j = 0; j < com . F A c o u n t ; j ++) {

136

s t a t e v a r s . add (

" f "

+ i +

" _ "

+ j ) ;

137

s t a t e f u n c . add ( com . c r e a t e F A F u n c ( j ) ) ;

138

}

139

t o t a l F A c o u n t += com . F A c o u n t ;

140

}

141

142

t o t a l E A c o u n t = 0;

143

t o t a l E A i n d e x = s t a t e v a r s . size () ;

144

for ( int i = 0; i < NC ; i ++) {

145

C o m p o n e n t com = c [ i ];

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

146

com . EAindex = s t a t e v a r s . size () ;

147

for ( int j = 0; j < com . E A c o u n t ; j ++) {

148

s t a t e v a r s . add (

" e "

+ i +

" _ "

+ j ) ;

149

s t a t e f u n c . add ( com . c r e a t e E A F u n c ( j ) ) ;

150

}

151

t o t a l E A c o u n t += com . E A c o u n t ;

152

}

153

154

s t a t e v a r s . add (

" time "

) ;

155

s t a t e f u n c . add ( null ) ;

156

157

s t a t e v a r s . add (

" f i t n e s s "

) ;

158

s t a t e f u n c . add ( null ) ;

159

160

i n i t i a l s t a t e = new State ( s t a t e v a r s . t o A r r a y ( new S t r i n g [0])

161

s t a t e f u n c . t o A r r a y ( new S t a t e F u n c t i o n [0]) ) ;

162

itime = i n i t i a l s t a t e . index (

" time "

) ;

163

i f i t n e s s = i n i t i a l s t a t e . index (

" f i t n e s s "

) ;

164

sys . s e t I n i t i a l S t a t e ( i n i t i a l s t a t e ) ;

165

166

// Normal Event

167

sys . a d d T r i g g e r ( c r e a t e N E () ,

" n "

) ;

168

169

// F a i l u r e E v e n t s

170

if ( useFE ) {

171

for ( int i = 0; i < NC ; i ++) {

172

C o m p o n e n t com = c [ i ];

173

com . F E i n d e x = sys . t r i g g e r s . size () ;

174

for ( int j = 0; j < com . F A c o u n t ; j ++) {

175

sys . a d d T r i g g e r ( com . F E t r i g g e r [ j ] ,

" f "

+ i +

" _ "

+ j )

;

176

}

177

}

178

}

179

180

// D e c i d a b l e E v e n t s

181

if ( useDE ) {

182

for ( int i = 0; i < NC ; i ++) {

183

C o m p o n e n t com = c [ i ];

184

com . D E i n d e x = sys . t r i g g e r s . size () ;

185

for ( int j = 0; j < com . F A c o u n t ; j ++) {

186

sys . a d d T r i g g e r ( com . c r e a t e D E T r i g g e r ( j ) ,

" d "

+ i +

" _

+ j ) ;

187

}

188

}

189

}

190

191

// V al ue F u n c t i o n

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

192

sys . s e t V a l u e F u n c t i o n ( new V a l u e F u n c t i o n () {

193

@ O v e r r i d e

194

p u b l i c d o u b l e v a l u e ( S y s t e m s y s t e m ) {

195

d o u b l e [] p = s y s t e m . M [ i n i t i a l s t a t e . index (

" f i t n e s s "

) ];

196

r e t u r n p [ p . l e n g t h - 1] / s y s t e m . N ;

197

}

198

}) ;

199

200

r e t u r n sys ;

201

}

202

203

p u b l i c T r i g g e r c r e a t e N E () {

204

N E f i t n e s s = i n D o u b l e ( NEprofitmin , N E p r o f i t m a x ) ;

205

N E e f f i c i e n c y f i t n e s s = new d o u b l e [ t o t a l E A c o u n t ];

206

N E S V a d d = new d o u b l e [ t o t a l S V c o u n t ];

207

208

for ( int i = 0; i < t o t a l E A c o u n t ; i ++) {

209

N E e f f i c i e n c y f i t n e s s [ i ] = - i n D o u b l e (0 ,

N E p r o f i t e f f i c i e n c y p e n a l t y m a x ) ;

210

}

211

for ( int i = 0; i < t o t a l S V c o u n t ; i ++) {

212

N E S V a d d [ i ] = i n D o u b l e (0 , N E S V a d d m a x ) ;

213

// Util . dump (" t o t a l S V c o u n t "+ t o t a l S V c o u n t +" : "+ N E S V a d d

[ i ]) ;

214

}

215

216

return new A l w a y s T r i g g e r ( new Event () {

217

@ O v e r r i d e

218

public void run ( State state ) {

219

state . val [ itime ]++;

220

state . val [ i f i t n e s s ] += N E f i t n e s s ;

221

for ( int i = 0; i < t o t a l E A c o u n t ; i ++) {

222

state . val [ i f i t n e s s ] += N E e f f i c i e n c y f i t n e s s [ i ]

223

* state . val [ t o t a l E A i n d e x + i ];

224

}

225

for ( int i = 0; i < t o t a l S V c o u n t ; i ++) {

226

d o u b l e old = state . val [ t o t a l S V i n d e x + i ];

227

state . val [ t o t a l S V i n d e x + i ] += N E S V a d d [ i ];

228

}

229

}

230

}) ;

231

}

232

233

pub l i c int in In t ( int a , int b ) {

234

r e t u r n a + r a n d o m . n e x t I n t ( b - a + 1) ;

235

}

236

237

p u b l i c d o u b l e i n D o u b l e ( d o u b l e a , d o u b l e b ) {

238

r e t u r n a + r a n d o m . n e x t D o u b l e () * ( b - a ) ;

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

239

}

240

241

p u b l i c

class

C o m p o n e n t {

242

p u b l i c int s v i n d e x ;

// index in state

243

p u b l i c int s v c o u n t ;

// state - v a r i a b l e s count

244

245

p u b l i c int F A i n d e x ;

246

p u b l i c int F A c o u n t ;

// c o m p o n e n t age count

247

p u b l i c d o u b l e [] F A s c a l e ;

248

p u b l i c d o u b l e [] F A s h a p e ;

249

p u b l i c W e i b u l l D i s t r i b u t i o n [] F A f u n c ;

250

public double [][] F A d e p e n d e n c y ;

251

252

p u b l i c int F E i n d e x ;

253

public double [][] FEmul ;

254

public double [][] FEadd ;

255

p u b l i c int [] F E t i m e ;

256

p u b l i c d o u b l e [] F E f i t n e s s ;

257

p u b l i c T r i g g e r [] F E t r i g g e r ;

258

259

p u b l i c int D E i n d e x ;

260

public double [][] DEmul ;

261

public double [][] DEadd ;

262

p u b l i c int [] D E t i m e ;

263

p u b l i c d o u b l e [] D E f i t n e s s ;

264

p u b l i c L i n e a r T r i g g e r [] D E t r i g g e r ;

265

266

p u b l i c int E A i n d e x ;

267

p u b l i c int E A c o u n t ;

// e f f i c i e n c y age count

268

p u b l i c d o u b l e [] E A s c a l e ;

269

p u b l i c d o u b l e [] E A s h a p e ;

270

p u b l i c W e i b u l l D i s t r i b u t i o n [] E A f u n c ;

271

public double [][] E A d e p e n d e n c y ;

272

273

p u b l i c d o u b l e N E f i t n e s s ;

274

p u b l i c d o u b l e [] N E e f f i c i e n c y f i t n e s s ;

275

276

public void dump () {

277

Util . dump (

" F A s c a l e = "

+ Util . t o S t r i n g ( F A s c a l e ) ) ;

278

Util . dump (

" F A s h a p e = "

+ Util . t o S t r i n g ( F A s h a p e ) ) ;

279

for ( int i = 0; i < F A d e p e n d e n c y . l e n g t h ; i ++) {

280

Util . dump (

" F A d e p e n d e n c y "

+ i +

" = "

281

+ Util . t o S t r i n g ( F A d e p e n d e n c y [ i ]) ) ;

282

}

283

for ( int i = 0; i < FE m u l . l e n g t h ; i ++) {

284

Util . dump (

" FEmul "

+ i +

" = "

+ Util . t o S t r i n g ( FEmul [ i ])

) ;

285

}

286

for ( int i = 0; i < FE a d d . l en g t h ; i ++) {

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

287

Util . dump (

" FEadd "

+ i +

" = "

+ Util . t o S t r i n g ( FEadd [ i ])

) ;

288

}

289

Util . dump (

" F E t i m e = "

+ Util . t o S t r i n g ( FEtime ) ) ;

290

Util . dump (

" F E f i t n e s s = "

+ Util . t o S t r i n g ( F E f i t n e s s ) ) ;

291

Util . dump (

" - - - - - - "

) ;

292

for ( int i = 0; i < DE m u l . l e ng t h ; i ++) {

293

Util . dump (

" DEmul "

+ i +

" = "

+ Util . t o S t r i n g ( DEmul [ i ])

) ;

294

}

295

for ( int i = 0; i < DE ad d . l e n g t h ; i ++) {

296

Util . dump (

" DEadd "

+ i +

" = "

+ Util . t o S t r i n g ( DEadd [ i ])

) ;

297

}

298

Util . dump (

" D E t i m e = "

+ Util . t o S t r i n g ( DEtime ) ) ;

299

Util . dump (

" D E f i t n e s s = "

+ Util . t o S t r i n g ( D E f i t n e s s ) ) ;

300

Util . dump (

" - - - - - - "

) ;

301

Util . dump (

" E A s c a l e = "

+ Util . t o S t r i n g ( E A s c a l e ) ) ;

302

Util . dump (

" E A s h a p e = "

+ Util . t o S t r i n g ( E A s h a p e ) ) ;

303

for ( int i = 0; i < E A d e p e n d e n c y . l e n g t h ; i ++) {

304

Util . dump (

" E A d e p e n d e n c y "

+ i +

" = "

305

+ Util . t o S t r i n g ( E A d e p e n d e n c y [ i ]) ) ;

306

}

307

308

}

309

310

p u b l i c C o m p o n e n t () {

311

s v c o u n t = in I n t ( sv0 , sv1 ) ;

312

313

// F a i l u r e age

314

F A c o u n t = in I n t ( FA0 , FA1 ) ;

315

F A s c a l e = new d o u b l e [ F A c o u n t ];

316

F A s h a p e = new d o u b l e [ F A c o u n t ];

317

F A f u n c = new W e i b u l l D i s t r i b u t i o n [ F A c o u n t ];

318

for ( int i = 0; i < F A c o u n t ; i ++) {

319

F A s c a l e [ i ] = i n D o u b l e ( F A s c a l e m i n * T , F A s c a l e m a x * T )

;

320

F A s h a p e [ i ] = i n D o u b l e ( FA s h ap e m i n , F A s h a p e m a x ) ;

321

F A f u n c [ i ] = new W e i b u l l D i s t r i b u t i o n ( F A s c a l e [ i ] ,

F A s h a p e [ i ]) ;

322

}

323

F A d e p e n d e n c y = new d o u b l e [ F A c o u n t ][];

324

for ( int i = 0; i < F A c o u n t ; i ++) {

325

F A d e p e n d e n c y [ i ] = new d o u b l e [ s v c o u n t ];

326

for ( int j = 0; j < s v c o u n t ; j ++)

327

F A d e p e n d e n c y [ i ][ j ] = i n D o u b l e (0 , F A d e p e n d e n c y m a x ) ;

328

}

329

330

// E f f i c i e n c y age

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

331

E A c o u n t = in I n t ( EA0 , EA1 ) ;

332

E A s c a l e = new d o u b l e [ E A c o u n t ];

333

E A s h a p e = new d o u b l e [ E A c o u n t ];

334

E A f u n c = new W e i b u l l D i s t r i b u t i o n [ E A c o u n t ];

335

for ( int i = 0; i < E A c o u n t ; i ++) {

336

E A s c a l e [ i ] = i n D o u b l e ( E A s c a l e m i n * T , E A s c a l e m a x * T )

;

337

E A s h a p e [ i ] = i n D o u b l e ( EA s h ap e m i n , E A s h a p e m a x ) ;

338

E A f u n c [ i ] = new W e i b u l l D i s t r i b u t i o n ( E A s c a l e [ i ] ,

E A s h a p e [ i ]) ;

339

}

340

E A d e p e n d e n c y = new d o u b l e [ E A c o u n t ][];

341

for ( int i = 0; i < E A c o u n t ; i ++) {

342

E A d e p e n d e n c y [ i ] = new d o u b l e [ s v c o u n t ];

343

for ( int j = 0; j < s v c o u n t ; j ++)

344

E A d e p e n d e n c y [ i ][ j ] = i n D o u b l e (0 , E A d e p e n d e n c y m a x ) ;

345

}

346

347

// F a i l u r e E v e n t

348

FEmul = new d o u b l e [ F A c o u n t ][];

349

FEadd = new d o u b l e [ F A c o u n t ][];

350

F E t i m e = new int [ F A c o u n t ];

351

F E f i t n e s s = new d o u b l e [ F A c o u n t ];

352

F E t r i g g e r = new T r i g g e r [ F A c o u n t ];

353

for ( int i = 0; i < F A c o u n t ; i ++) {

354

final int index = i ;

355

F E m u l [ i ] = new d o u b l e [ s v c o u n t ];

356

F E a d d [ i ] = new d o u b l e [ s v c o u n t ];

357

for ( int j = 0; j < s v c o u n t ; j ++) {

358

FEmul [ i ][ j ] = i n D o u b l e ( FEmul0 , F E m u l 1 ) ;

359

FEadd [ i ][ j ] = i n D o u b l e ( FEadd0 , F E a d d 1 ) ;

360

}

361

F E f i t n e s s [ i ] = i n D o u b l e ( FEfitness0 , F E f i t n e s s 1 ) ;

362

F E t i m e [ i ] = in I n t ( FEtime0 , F E t i m e 1 ) ;

363

F E t r i g g e r [ i ] = new S t o c h a s t i c T r i g g e r ( new Event () {

364

@ O v e r r i d e

365

public void run ( State state ) {

366

for ( int i = 0; i < s v c o u n t ; i ++) {

367

s t a t e . val [ s v i n d e x + i ] = s t a t e . val [ s v i n d e x + i ]

368

* FEmul [ index ][ i ] + FEadd [ index ][ i ];

369

}

370

state . val [ itime ] += FEtime [ index ];

371

state . val [ i f i t n e s s ] += F E f i t n e s s [ index ];

372

}

373

}) {

374

375

@ O v e r r i d e

376

public double value ( State state ) {

377

d o u b l e sum = 0;

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

378

for ( int j = 0; j < s v c o u n t ; j ++)

379

sum += F A d e p e n d e n c y [ index ][ j ]

380

* st a t e . val [ s v i n d e x + j ];

381

// Util . dump (" sum "+ sum ) ;

382

r et u r n FA f u n c [ in d e x ]. va l u e ( sum ) ;

383

}

384

};

385

}

386

387

// D e c i d a b l e Event

388

DEmul = new d o u b l e [ F A c o u n t ][];

389

DEadd = new d o u b l e [ F A c o u n t ][];

390

D E t i m e = new int [ F A c o u n t ];

391

D E f i t n e s s = new d o u b l e [ F A c o u n t ];

392

D E t r i g g e r = new L i n e a r T r i g g e r [ F A c o u n t ];

393

for ( int i = 0; i < F A c o u n t ; i ++) {

394

final int index = i ;

395

D E m u l [ i ] = new d o u b l e [ s v c o u n t ];

396

D E a d d [ i ] = new d o u b l e [ s v c o u n t ];

397

for ( int j = 0; j < s v c o u n t ; j ++) {

398

DEmul [ i ][ j ] = i n D o u b l e ( DEmul0 , D E m u l 1 ) ;

399

DEadd [ i ][ j ] = i n D o u b l e ( DEadd0 , D E a d d 1 ) ;

400

}

401

D E f i t n e s s [ i ] = i n D o u b l e ( DEfitness0 , D E f i t n e s s 1 ) ;

402

D E t i m e [ i ] = in I n t ( DEtime0 , D E t i m e 1 ) ;

403

}

404

405

}

406

407

public L i n e a r T r i g g e r c r e a t e D E T r i g g e r ( final int index ) {

408

D E t r i g g e r [ index ] = new L i n e a r T r i g g e r ( new Event () {

409

@ O v e r r i d e

410

public void run ( State state ) {

411

for ( int i = 0; i < s v c o u n t ; i ++) {

412

s t a t e . val [ s v i n d e x + i ] = s t a t e . val [ s v i n d e x + i ]

413

* DEmul [ index ][ i ] + DEadd [ index ][ i ];

414

}

415

state . val [ itime ] += DEtime [ index ];

416

state . val [ i f i t n e s s ] += D E f i t n e s s [ index ];

417

}

418

} , i n i t i a l s t a t e ) ;

419

D E t r i g g e r [ index ]. c l e a r M a s k () ;

420

for ( int j = 0; j < s v c o u n t ; j ++) {

421

D E t r i g g e r [ i n d e x ]. s e t M a s k ( s v i n d e x + j ) ;

422

}

423

r e t u r n D E t r i g g e r [ in d e x ];

424

}

425

426

p u b l i c S t a t e F u n c t i o n c r e a t e F A F u n c ( int i ) {

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

427

r e t u r n c r e a t e F u n c ( F A d e p e n d e n c y [ i ] , F A f u n c [ i ]) ;

428

}

429

430

p u b l i c S t a t e F u n c t i o n c r e a t e E A F u n c ( int i ) {

431

r e t u r n c r e a t e F u n c ( E A d e p e n d e n c y [ i ] , E A f u n c [ i ]) ;

432

}

433

434

public S t a t e F u n c t i o n c r e a t e F u n c ( final double [] dependency

435

final W e i b u l l D i s t r i b u t i o n func ) {

436

r e t u r n new S t a t e F u n c t i o n () {

437

@ O v e r r i d e

438

public double value ( State state ) {

439

d o u b l e age = 0;

440

for ( int j = 0; j < s v c o u n t ; j ++) {

441

age += d e p e n d e n c y [ j ] * st a t e . val [ s v i n d e x + j ];

442

}

443

return func . value ( age ) ;

444

}

445

446

};

447

}

448

}

449

}

450

451

452

453

454

package com . vuebe . vhen ;

455

456

i m p o r t

com . vuebe . ga . GA ;

457

i m p o r t

com . vuebe . math . W e i b u l l D i s t r i b u t i o n ;

458

i m p o r t

com . vuebe . util . Util ;

459

460

p u b l i c

class

T e s t F a c t o r y {

461

public System b e s t s y s t e m = null ;

462

463

public void run () {

464

Util . dump (

" Test Factory BEGIN "

) ;

465

final S y s t e m F a c t o r y sf = new S y s t e m F a c t o r y () ;

466

sf . NC = 2;

467

468

sf . sv0 = 1;

469

sf . sv1 = 1;

470

471

sf . N E p r o f i t m i n = 1;

472

sf . N E p r o f i t m a x = 1;

473

sf . N E p r o f i t e f f i c i e n c y p e n a l t y m a x = 1;

474

sf . N E S V a d d m a x = 1;

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

475

476

sf . EA0 = 1;

477

sf . EA1 = 2;

478

sf . E A s c a l e m i n = 1;

479

sf . E A s c a l e m a x = 1;

480

sf . E A s h a p e m i n = 0.5;

481

sf . E A s h a p e m a x = 3;

482

sf . E A d e p e n d e n c y m a x = 1;

// p e n a l t y for i n e f f i c i e n c y

c o e f f i c i e n t max

483

484

sf . FA0 = 1;

// F a i l u r e e v e n t s min

485

sf . FA1 = 2;

486

sf . F A s c a l e m i n = 0.05;

487

sf . F A s c a l e m a x = 0.2;

488

sf . F A s h a p e m i n = 1;

489

sf . F A s h a p e m a x = 2;

490

sf . F A d e p e n d e n c y m a x = 1;

491

492

sf . useFE = true ;

493

sf . F E m u l 0 = 0;

494

sf . F E m u l 1 = 0;

495

sf . F E a d d 0 = 0;

496

sf . F E a d d 1 = 0;

497

sf . F E t i m e 0 = 2;

498

sf . F E t i m e 1 = 4;

499

sf . F E f i t n e s s 0 = -5;

500

sf . F E f i t n e s s 1 = -10;

501

502

sf . useDE = true ;

503

sf . D E m u l 0 = 0;

504

sf . D E m u l 1 = 0;

505

sf . D E a d d 0 = 0;

506

sf . D E a d d 1 = 0;

507

sf . D E t i m e 0 = 1;

508

sf . D E t i m e 1 = 1;

509

sf . D E f i t n e s s 0 = -1;

510

sf . D E f i t n e s s 1 = -1;

511

512

S y s t e m sys = sf . c r e a t e S y s t e m () ;

513

514

GA ga = new GA () ;

515

sys . s e t S i m u l a t i o n C o u n t ( 2 0 0 0 ) ;

516

ga . s e t S e l e c t i o n S i z e (10) ;

517

ga . s e t E l i t e S i z e (2) ;

518

ga . i n i t i a t e P o p u l a t i o n ( sys ) ;

519

for ( int i = 0; i < 1 0 0 0 0 ; i ++) {

520

ga . e v o l v e () ;

521

b e s t s y s t e m = ( S y s t e m ) ga . g e t B e s t I n d i v i d u a l () ;

522

Util . dump (

" N : "

+ ga . g e t E v a l u a t i o n s () +

" Value : "

APPENDIX D. JAVA CODE - EXAMPLE PROBLEM CLASS

523

+ b e s t s y s t e m . g e t V a l u e () ) ;

524

}

525

Util . s a v e S t r i n g ( b e s t s y s t e m . t o M a t l a b () ,

526

" C :/ U s e r s / jo g 0 4 / D r o p b o x / s c a n i a / f i l e d u m p "

" . m "

) ;

527

528

Util . dump (

" Test Factory END "

) ;

529

}

530

531

public static final void main ( String [] args ) {

532

new T e s t F a c t o r y () . run () ;

533

}

534

}

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101

Frequently asked questions

What is the main topic of this document?

This document is a Master Thesis in Mathematics/Applied Mathematics, focusing on solving complex maintenance planning optimization problems using stochastic simulation and multi-criteria fuzzy decision making.

Who is the author of this thesis?

The author of the thesis is Sahar Tahvili.

What are some keywords associated with this research?

Keywords include Genetic algorithm, Corrective maintenance, Preventive maintenance, ROI, Multi-criteria decision making, TOPSIS, Fuzzy decision making, Discrete event simulation, and Intelligent agent.

What are the two main maintenance activities discussed?

The thesis focuses on two main maintenance activities: Corrective Maintenance (CM) and Preventive Maintenance (PM).

What are the methods employed to solve the maintenance planning problem?

The research employs stochastic simulation, genetic algorithms, fuzzy decision making, and the TOPSIS method.

What are the key components considered in the life cycle cost (LCC) analysis?

The LCC is divided into: initial cost, installation cost, energy cost, operation cost, maintenance and repair cost, downtime cost, environmental cost, and decommissioning and disposal cost.

What is the formula used to calculate Return on Investment (ROI) in this context?

The ROI is calculated by ROI = (Return - Investment) / Investment, or, more specifically for maintenance, ROIm = (CH - CHM) / IHM, where CH is the life-cycle cost of unscheduled maintenance, CHM is the life-cycle cost of health management, and IHM is the investment in health management.

What are the three replacement strategies mentioned?

The three replacement strategies mentioned are minimal repair, imperfect reparation, and perfect reparation.

What is the "bathtub curve" hazard function mentioned in Component Health section?

The bathtub curve illustrates the hazard function over time, typically including three phases: failures phase (decreasing failure rate), phase with constant (random) failure rate, and wear-out failures phase (increasing failure rate).

What is the purpose of using fuzzy logic and TOPSIS in the decision-making process?

Fuzzy logic and TOPSIS are used to model uncertainty and prioritize maintenance alternatives based on multiple criteria, considering both the "ideal" and "worst-case" scenarios.

What was used as a case study for evaluating the proposed methodology?

The internal combustion engine cooling of a typical Scania truck was used as a case study to evaluate the proposed methodology.

What is a "decidor" and how it's used in the simulation?

A "decidor" is a component in the simulation framework that determines whether a plannable event should be triggered, based on the current system state and a predefined decision logic.

What is Riccati differential equation in this context?

The Riccati differential equation is used in Chapter 2 to model dynamic reliability equation.

What are some of the variables that the study takes into consideration?

Variable are Genetic algorithm, Corrective maintenance, Preventive maintenance, ROI, Multi-criteria decision making, TOPSIS, Fuzzy decision making, Discrete event simulation, and Intelligent agent.

Fin de l'extrait de 103 pages - haut de page

Résumé des informations

Titre: Solving Complex Maintenance Planning Optimization Problems Using Stochastic Simulation and Multi-Criteria Fuzzy Decision Making
Université: Mälardalen University
Note: A+
Auteur: Sahar Tahvili (Auteur)
Année de publication: 2014
Pages: 103
N° de catalogue: V281688
ISBN (ebook): 9783656756170
ISBN (Livre): 9783656756187
Langue: anglais
mots-clé: solving complex maintenance planning optimization problems using stochastic simulation multi-criteria fuzzy decision making
Sécurité des produits: GRIN Publishing GmbH

Citation du texte: Sahar Tahvili (Auteur), 2014, Solving Complex Maintenance Planning Optimization Problems Using Stochastic Simulation and Multi-Criteria Fuzzy Decision Making, Munich, GRIN Verlag, https://www.grin.com/document/281688