Preventive Maintenance and Replacement Scheduling: Models and Algorithms

Preventve Mantenance and Replacement Schedulng: Models and Algorthms By Kamran S. Moghaddam B.S. Unversty of Tehran 200 M.S. Tehran Polytechnc 2003 A Dssertaton Proposal Submtted to the Faculty of the Graduate School of the Unversty of Lousvlle n Partal Fulfllment of the Requrements for the Doctor of Phlosophy Canddacy Department of Industral Engneerng Unversty of Lousvlle Lousvlle Kentucky USA November 2008

Copyrght 2008 by Kamran S. Moghaddam All Rghts Reserved

Preventve Mantenance and Replacement Schedulng: Models and Algorthms By Kamran S. Moghaddam B.S. Unversty of Tehran 200 M.S. Tehran Polytechnc 2003 A Dssertaton Proposal Approved on November 2008 By the followng Dssertaton Commttee Professor John S. Usher Commttee Char Professor Gerald W. Evans Professor Gal W. DePuy Professor Sunderesh S. Heragu Professor Al M. Shahhossen

TABELE OF CONTENTS LIST OF TABLES... v LIST OF FIGURES... v. Introducton.. Preventve Mantenance and Replacement Schedulng....2. Research Contrbutons... 2.3. Outlne... 3 2. Lterature Revew 4 2.. Introducton... 4 2.2. Optmzaton Models... 4 2.2.. Exact Algorthms... 4 2.2.2. Heurstcs and Meta-Heurstcs Algorthms... 9 2.2.3. Hybrd Algorthms... 3 2.2.4. Mult-Obectve Algorthms... 5 2.3. Smulaton Models... 7 2.3.. Monte Carlo Smulaton... 7 2.3.2. Dscrete-Event and Contnuous Smulaton... 9 2.4. Age Reducton and Improvement Factor Models... 24 2.5. Applcatons... 29 2.5.. Manufacturng and Producton Systems... 29 v

2.5.2. Servce Systems... 35 2.5.3. Power Systems... 36 2.6. Chapter Summary... 37 3. Optmzaton Models - Exact Algorthms 39 3.. Introducton... 39 3.2. Formulaton... 39 3.2.. Mantenance... 40 3.2.2. Replacement... 4 3.2.3. Do Nothng... 42 3.2.4. Cost of Preventve Mantenance and Replacements... 42 3.2.4.. Falure Cost... 42 3.2.4.2. Mantenance Cost... 44 3.2.4.3. Replacement Cost... 44 3.2.4.4. Fxed Cost... 44 3.2.4.5. Total Cost... 45 3.3. Optmzaton Models... 47 3.3.. Model - Mnmzng total cost subect to a relablty constrant... 47 3.3.2. Model 2 - Maxmzng relablty subect to a budget constrant... 48 3.4. Soluton Procedure... 49 3.5. Computatonal Results... 5 3.6. Chapter Summary... 57 4. Optmzaton Models - Heurstc Algorthms 58 4.. Introducton... 58 4.2. Formulaton... 58 v

4.3. Optmzaton Model... 59 4.4. Mult-Obectve Genetc Algorthms... 60 4.4. Representaton of Solutons... 60 4.4.2 Ftness Functons... 6 4.4.3. Crossover Procedure... 62 4.4.4. Mutaton Procedure... 63 4.4.5. Generatonal GA... 63 4.4.6. Steady State GA... 64 4.5. Computatonal Results... 65 4.5.. Computatonal Results of Ftness Functon... 65 4.5.2. Computatonal Results of Ftness Functon 2... 68 4.5.3. Computatonal Results of Ftness Functon 3... 7 4.6. Chapter Summary... 74 5. Research Plan 75 Bblography 76 v

LIST OF TABLES 3.. Parameters for the Numercal Example... 5 3.2. Model - Mantenance and Replacement Schedule Mnmzes Total Cost... 52 3.3. Model 2 - Mantenance and Replacement Schedule Maxmzes Relablty... 52 3.4. Effectve age of components n Model... 53 3.5. Effectve age of components n Model 2... 53 4.. Parameters of Genetc Algorthms... 65 4.2. Non-nferor solutons resulted from Ftness Functon... 66 4.3. Mantenance and Replacement Schedule Ftness Functon GGA... 66 4.4. Mantenance and Replacement Schedule Ftness Functon SSGA... 66 4.5. Non-nferor solutons resulted from Ftness Functon 2... 69 4.6. Mantenance and Replacement Schedule Ftness Functon 2 GGA... 69 4.7. Mantenance and Replacement Schedule Ftness Functon 2 SSGA... 69 4.8. Non-nferor solutons resulted from Ftness Functon 3... 7 4.9. Mantenance and Replacement Schedule Ftness Functon 3 GGA... 72 4.0. Mantenance and Replacement Schedule Ftness Functon 3 SSGA... 72 v

LIST OF FIGURES 3.. Effect of perod- mantenance on component ROCOF... 4 3.2. Effect of perod- replacement on system ROCOF... 42 3.3. Effectve age of components n Model... 55 3.4. Effectve age of components n Model 2... 56 4.. Pareto Optmal Solutons for Ftness Functon... 67 4.2. Cost Improvement for Ftness Functon... 68 4.3. Relablty Improvement for Ftness Functon... 68 4.4. Pareto Optmal Solutons for Ftness Functon 2... 70 4.5. Cost Improvement for Ftness Functon 2... 70 4.6. Relablty Improvement for Ftness Functon 2... 7 4.7. Pareto Optmal Solutons for Ftness Functon 3... 73 4.8. Cost Improvement for Ftness Functon 3... 73 4.9. Relablty Improvement for Ftness Functon 3... 73 4.0. Pareto Optmal Solutons for all Ftness Functons... 74 v

Chapter Introducton.. Preventve Mantenance and Replacement Schedulng Preventve mantenance s a broad term that encompasses a set of actvtes amed at mprovng the overall relablty and avalablty of a system. All types of systems from conveyors to cars to overhead cranes have prescrbed mantenance schedules set forth by the manufacturer that am to reduce the rsk of system falure. Preventve mantenance actvtes generally consst of nspecton cleanng lubrcaton adustment algnment and/or replacement of sub-components that wear-out. Regardless of the specfc system n queston preventve mantenance actvtes can be categorzed n one of two ways component mantenance or component replacement. An example of component mantenance would be mantanng proper ar pressure n the tres of an automoble. Note that ths actvty changes the agng characterstcs of the tres and f done correctly ultmately decreases ther rate of occurrence of falure. An example of component replacement would be smply replacng one or more of the tres wth new ones. Obvously preventve mantenance nvolves a basc trade-off between the costs of conductng mantenance/replacement actvtes and the cost savngs acheved by reducng the overall rate of occurrence of system falures. Desgners of preventve mantenance schedules must wegh these ndvdual costs n an attempt to mnmze

the overall cost of system operaton. They may also be nterested n maxmzng the system relablty subect to some sort of budget constrant. Other crtera such as avalablty and demand satsfacton mght be consdered as the obectve functons but they wll not be studed n ths dssertaton. The problem s to fnd the best sequence of mantenance actons for each component n the system n each perod over a plannng horzon such that overall costs are mnmzed subect to a constrant on relablty or the relablty of the system s maxmzed subect to a constrant on budget..2. Research Contrbutons In ths dssertaton proposal optmzaton models are developed and solved va exact heurstc and meta-heurstc algorthms. Analytcal and statstcal agereducton and mprovement factor models are developed and can be consdered as the man research contrbuton. In partcular the followng contrbutons are made:. Two optmzaton models wll be constructed based on extensons of prevous work n partcular by Usher et al (998). The optmzaton models are solved by usng a dynamc programmng approach. These models also provde a general framework to acheve optmal preventve mantenance and replacement polces and wth modfcatons can be used as a basc closedform model for any type of system. 2. A mult-obectve optmzaton model s developed based on a set of basc assumptons and engneerng economy consderatons. Ths model s optmzed va mult-obectve generatonal and steady state genetc algorthms as well as 2

by a mult-obectve smulated annealng algorthm whch allows for the comparson of these optmzaton approaches. 3. In order to estmate the parameters of optmzaton models an analytcal model for estmatng age reducton and mprovement factor parameters wll be developed. In addton a procedure wll be developed to estmate the mprovement factor of any general component due to mperfect mantenance actvtes. 4. Fnally a real case study wll be consdered as the applcaton of developed models and preventve mantenance and replacement schedule resulted from optmzaton models wll be compared wth the current mantenance polcy n that case study..3. Outlne The remander of ths dssertaton proposal s organzed as follows: In Chapter 2 a comprehensve lterature revew of models and applcatons of preventve mantenance and replacement schedulng s presented. In Chapter 3 a formulaton of the optmzaton models s presented and ther computatonal results are analyzed. Chapter 4 ncludes the extenson of Chapter 3 optmzaton models by consderng engneerng economy features. These models have been optmzed by mult-obectve generatonal and steady state genetc algorthms and the computatonal results obtaned by mplementaton of these algorthms are demonstrated. Fnally n Chapter 5 the plan and ts schedule for the research s presented. 3

Chapter 2 Lterature Revew 2.. Introducton Ths chapter has four man sectons. The frst secton presents a complete revew on varous optmzaton models and algorthms related to preventve mantenance and replacement schedulng. Secton 2.3 presents a revew of key works that utlze smulaton models. In Secton 2.4 models that ntroduce and develop age reducton and mprovement factor models are presented. Fnally applcatons of preventve mantenance and replacement schedulng n manufacturng and producton systems servce systems and power systems are revewed. 2.2. Optmzaton Models 2.2.. Exact Algorthms Determnstc optmzaton algorthms have been proposed by varous authors. Yao et al (200) present a two-layer herarchcal model that optmzes the preventve mantenance schedulng n semconductor manufacturng operatons. They develop a Markov decson process and optmze ths model va a mxed nteger lnear programmng model. They defne proft of cluster tools producton as the obectve functon to be maxmzed and consder tme wndow for preventve mantenance

actvtes and lmtaton of resources as the constrant whch were nonlnear functons. In order to acheve a global optmum they transfer the nonlnear functons nto lnear ones and use EasyModeler and OSL as the optmzaton software. In addton they utlze AutoSched AP as the smulaton software n order to construct a smulaton model to evaluate the performance of the optmzaton model n a real case study wth preventve mantenance tasks n a one-week plannng horzon and compare the obtaned results wth the actual preventve mantenance plan. Later Yao et al (2004) extend ther prevous model to be more general apply ths extended model n a producton lne of a semconductor manufacturng system and show the applcaton of t va numercal examples. Jayakumar and Asgarpoor (2004) present a lnear programmng model n order to optmze the mantenance polcy for a component wth deteroraton and random falure rate. They determne optmal mean tmes of mnor and maor preventve mantenance actons based on maxmzng the avalablty of the component. They utlze MAPLE and LINGO for solvng the lnear programmng model of Markov decson process. Duarte et al (2006) present a model and algorthm for mantenance optmzaton of a system wth seres components. In ths research they assume that all components have lnearly ncreasng falure rate wth a constant mprovement factor for mperfect mantenance. In addton they consder the total cost as the obectve functon and the total downtme as the man constrant. In terms of mantenance actvtes they defne preventve and correctve mantenance for each component. Fnally ther algorthm optmzes the nterval of tme between mantenance actons for each component over a plannng horzon. Canto (2006) presents an optmzaton model to schedule a preventve mantenance of a real power plant over a plannng horzon. He consders the total 5

cost of varous operatons as the obectve functon and uses Bender s decomposton to solve a mxed-nteger lnear programmng model. Buda et al (2006) present two mxed-nteger lnear programmng models for preventve mantenance schedulng problems. The authors assume the total cost ncludng possesson costs mantenance costs and the penalty costs of early consecutve mantenance actvtes as the obectve functon for both models. They present and prove a theorem about the NP-hard structure of the preventve mantenance schedulng problem and use GAMS to mplement the optmzaton models. They use CPLEX as the optmzaton software to fnd the optmal preventve mantenance schedule. They apply ther model to a case study of ralway mantenance schedulng. In addton they develop four heurstc optmzaton algorthms two for each model and compare the computatonal results obtaned from exact algorthms n CPLEX wth the results acheved from heurstc algorthms and menton the advantages of each soluton methodology. Another excellent study n ths area s by Tam et al (2006) who develop three nonlnear optmzaton models: one that mnmzes total cost subect to satsfyng a requred relablty one that maxmzes relablty at a gven budget and one that mnmzes the expected total cost ncludng expected breakdown outages cost and mantenance cost. They utlze MS-Excel Solver as the optmzaton software that uses a generalzed reduced gradent (GRG) algorthm to solve the nonlnear optmzaton models. Usng these models they determne the optmal mantenance ntervals for a mult-component system but ther models consder only mantenance actons for components and do not consder replacement actons. Alardh et al (2007) present a bnary nteger lnear programmng model n order to fnd the best preventve mantenance schedule n separated and lnked cogeneraton plants. The 6

researchers defne the avalablty of the power and desaltng equpments as the obectve functon to be maxmzed and consder the mantenance tme wndow mantenance completon duraton logcal operatonal resource lmtaton mantenance crew avalablty effcency measures and demand as the set of constrants. They apply ther model n two cogeneraton plants wth seven unts and 42 peces of equpment n Kuwat over a 52-week plannng horzon and utlze LINGO as the optmzaton software to optmze the model. In addton they perform a senstvty analyss on the model to assess the robustness and analyze the effect of expandng the plannng horzon reducng the resources and ncreasng the demand on the mantenance strateges. Panagotdou and Tagaras (2007) develop an optmzaton model that optmzes the preventve mantenance schedules n a manufacturng process. The authors consder two dfferent states for components n-control or out-of-control before complete falure. They treat the tme to shft and the tme to falure as random varables and express them wth Webull and Gamma dstrbutons. In addton they combne age-based and condton-based concepts nto the optmzaton model wth the mnmzaton of total cost and solve t by applyng Karush-Kahn-Tucker (KKT) condtons of optmalty to obtan the optmal preventve mantenance schedule. Fnally they present several numercal examples to demonstrate the effectveness of ther methodology. Shrmohammad et al (2007) develop an agebased nonlnear optmzaton model to determne the optmal preventve mantenance schedule for a sngle component system. They defne two types of decson varables the tme between preventve replacements and the cut-off age and assume an expected cost of falures mantenance replacement costs and total cycle cost n the cost functon and consder cost per unt tme as the obectve 7

functon. In order to solve the optmzaton model and show the effectveness of the proposed approach they utlze MAPLE and run the program for a numercal example by settng dfferent values for an mprovement factor whch s assumed as a constant n the model. Dynamc programmng has been broadly used as a standard optmzaton technque to acheve the optmal mantenance and replacement actons n engneerng problems. Canfeld (986) studes preventve mantenance optmzaton models va focusng on dfferent aspects of falure functon on systems relablty. He mentons that preventve mantenance actons do not change or affect deteroraton behavor of falure rate so the developed falure functon s constant wth mantenance actons. He consders ncreasng falure rate based on the Webull dstrbuton for hs study and determnes the optmal cost of mantenance polces by defnng the average cost-rate of system operaton and applyng dynamc programmng as the soluton approach. Robeln and Madanat (2006) develop a mantenance optmzaton model for brdge decks va a Markov chan process. In ths paper they classfy optmzaton models nto two categores () physcally based deteroraton models wth lmted number of decson varables and (2) smpler deteroraton models wth more and sophstcated decson varables. They apply Markov chan methodology wth states based on hstory of deteroraton and mantenance actons and utlze dynamc programmng as the soluton approach to solve Markov decson process. As a case study they apply ther approach to optmze the mantenance polcy of brdges. 8

2.2.2. Heurstcs and Meta-Heurstcs Algorthms Genetc algorthm as a maor optmzaton approach has been presented n several research papers. Usher et al (998) present an optmzaton mantenance and replacement model for a sngle-component system. They determned an optmal preventve mantenance schedule for a new system subect to deteroraton by consderng the tme value of money n all future costs ncreasng rate of occurrence of falure over tme and the use of the mprovement factor to provde for the case of mperfect mantenance actons. In addton they provde a comparson of computatonal results among random search genetc algorthm and branch and bound algorthms. One of the most notable studes n the area of relablty and mantenance optmzaton for mult-state mult-component systems s found n Levetn and Lsnansk (2000). They defne a mult-state system as a system n whch all or some of components have dfferent performance levels from proper functonng to complete falure and the relablty of the system as ts ablty of satsfyng the demand levels. They formulate an optmzaton model to determne preventve mantenance actons that affect the effectve age of components. Ther model s based on mnmzaton of cost subect to requred level of relablty. They apply a unversal generatng functon technque and use a genetc algorthm to determne the best mantenance strategy. Levetn and Lsnansk (2000) present addtonal research n whch an optmzaton model was developed n order to determne the optmal replacement schedulng n mult-state seres-parallel systems. They consdered an ncreasng falure rate based on the expected number of falures durng tme ntervals and defned summaton of mantenance actvtes cost along 9

wth cost of unsuppled demand due to falures of components as the obectve functon. Fnally they utlzed unversal generatng functon approach and appled genetc algorthm to fnd the optmal mantenance polcy. Wang and Handschn (2000) develop a new genetc algorthm by modfyng the basc operators crossover and mutaton of a standard genetc algorthm based on the specfc characterstc of preventve mantenance schedulng problem for power systems. They mprove the tme computatonal complexty of genetc algorthm by consderng a code-specfc and constrant-transparent ntegrated codng method to acheve faster convergence and to prevent producton of nfeasble solutons. As the mplementaton methodology an obect orented programmng approach s appled and the effectveness of the new genetc algorthm shown va theoretcal analyss and smulaton results to compare wth a tradtonal genetc algorthm. Tsa et al (200) consder two actvtes mperfect mantenance and replacement n ther preventve mantenance optmzaton model. They model mperfect mantenance actvtes based on the concept of an mprovement factor whch s determned by a quanttatve assessment procedure. They use a genetc algorthm to fnd the optmal preventve mantenance actvtes whle the system unt-cost lfe s consdered as the obectve functon. As a case study they test a mechatronc system to show the effectveness of ther model and algorthm. Cavory et al (200) present an optmzaton model to schedule the best preventve mantenance tasks of all machnes n a sngle product manufacturng producton lne. They assume that each machne should be assgned to each operator and consdered the total throughput of the lne as the obectve functon to be maxmzed. At the frst step they formulate the optmzaton model and analyze t va analytcal approach. Then the researchers used C++ as a programmng 0

envronment and appled genetc algorthm n order to fnd the best combnaton of preventve mantenance tasks. In addton they construct an expermental desgn to set and analyze the parameters of genetc algorthm and utlz the Taguch method and statstcal analyss to valdate the results. Fnally an applcaton of the approach was performed n an actual producton lne of car engnes. Leou (2003) presents an optmzaton model to fnd the optmal preventve mantenance schedule for a mult-component system. He consders total cost of operatons and mantenance actvtes along wth relablty as the crtera of the system and transfer them nto the obectve functon by defnng degree of volaton from requred relablty. In addton he defnes mantenance crew and duraton of mantenance as the system s constrants. He apples hs optmzaton model n a case study wth sx electrc generators and utlzes genetc algorthm as the optmzaton methodology to determne the best preventve mantenance schedule. Han et al (2003) consder the recursve nature of falure rate between preventve mantenance cycles and develop a nonlnear optmzaton model based on repar cost preventve mantenance cost and producton loss cost n a producton system. They apply a genetc algorthm as the optmzaton technque and menton that ther model can be consdered n decson support systems for mantenance and ob shop schedulng. Brs et al (2003) consder cost and avalablty as the systems crtera n ther research. They optmze a model ncludng cost n the obectve functon and avalablty as the constrant by usng a genetc algorthm to fnd the best preventve mantenance schedule. They use a tme-dependent Brnbaum mportance factor to generate the ordered sequence of frst nspecton tmes and utlze MATLAB to calculate the system avalablty va a Monte Carlo smulaton approach.

Lmbourg and Kochs (2006) propose several technques to represent the decson varables n preventve mantenance schedulng models that use heurstcs and meta-heurstcs optmzaton algorthms. They test varous non-standard approaches and compare them to bnary representatons by a heurstc algorthm and the computatonal results show that effectveness of ther approaches. In addton they apply some modfed crossover and mutaton procedures n a genetc algorthm and show the mprovement n performance of ther algorthm n terms of computatonal tme and accuracy. Other research on the applcaton of genetc algorthms to mantenance optmzaton has been recently done by Lapa et al (2006). They consder flexble ntervals between mantenance actons and menton the advantage of ths assumpton over the common methodologes of contnuous fttng of the schedules. They develop a model that ncludes preventve and correctve mantenance actons and the assocated cost wth them outage tmes relablty of the system and probablty of mperfect mantenance. Because ther model s a nonlnear large-scale optmzaton model they utlze a genetc algorthm as the soluton procedure. In addton and as a case study they apply ther model to a hgh-pressure necton system to measure the effectveness of ther methodology. Verma and Ramesh (2007) group systems and sub-systems of a large engneerng plant nto hgher modular assembles (HMA) and apply a mult-obectve preventve mantenance schedulng method. They model ths problem as a constraned nonlnear mult-obectve mathematcal program wth relablty cost and nonconcurrence of mantenance perods and mantenance start tme factor as elements of the obectve functons and use a genetc algorthm to solve the model. Shum and Gong (2007) recently present an applcaton of a genetc algorthm for optmzaton of preventve mantenance schedulng of a producton machne. They consder 2

mantenance and replacement frequency along wth purchasng strategy and the sze of the mantenance workforce as the decson varables and the total cost as the obectve functon. They examne the effect of these costs on the optmal mantenance schedule n numercal example. Other meta-heurstcs have been used as the combnatoral optmzaton technques to solve mantenance schedulng problems. Samrout et al (2005) use an ant colony algorthm to optmze the problem that was prevously optmzed va genetc algorthm. They defne seres of component mantenance and nspecton perods and use MATLAB as the programmng envronment. 2.2.3. Hybrd Algorthms Km et al (994) combne genetc algorthm wth smulated annealng n order to optmze a large-scale and long-term preventve mantenance and replacement schedulng problem. In ther research the acceptance probablty of a smulated annealng method s consdered as a measure for ndvdual survval n the genetc algorthm. By usng ths approach they acheve a near optmal soluton n a short perod of tme compare to the computatonal tme of smple genetc algorthm. As a case study they optmze a long-term mantenance schedulng problem of a thermal system. Tan and Kramer (997) develop a general framework for preventve mantenance optmzaton n chemcal process operatons. They assume a Webull model for falure rate and consder dfferent mantenance actvtes that can be performed. They develop a methodology that combnes Monte Carlo smulaton wth a genetc algorthm to solve opportunstc mantenance problems wth a nondetermnstc obectve functon. They apply ther approach to two case studes to 3

compare the results obtaned from the proposed model wth the results acheved from analytc approach and Monte Carlo smulaton wth a neural network. Fnally they menton the advantages of ther approach over other approaches. Marseguerra et al (2002) develop a condton-based mantenance (CBM) model for mult-component systems and use a Monte Carlo smulaton model to predct the degradaton level n a contnuously montored system. They apply a genetc algorthm to optmze the degradaton level after mantenance actons n a multobectve optmzaton model wth proft and avalablty as the obectve functons. In addton they consder the smulaton model to descrbe the dynamcs of a stressdependent degradaton process n load-sharng components. Based on the computatonal results they menton that the combnaton of a genetc algorthm wth Monte Carlo smulaton s an effectve approach to solve the combnatoral optmzaton problems. Shalaby et al (2004) develop an optmzaton model for preventve mantenance schedulng of mult-component and mult-state systems. They defne sequence of preventve mantenance actvtes as the decson varables and the summaton of preventve mantenance mnmal repar and downtme costs as the obectve functon. In addton they consder system relablty mnmum ntervals between mantenance actons and crew avalablty as the constrants of ther model. Fnally a combnaton of genetc algorthm and smulaton was utlzed to optmze the model. Allaou and Artba (2004) present a combnaton of smulaton and optmzaton models n order to solve the NP-hard hybrd flow shop schedulng problem wth mantenance constrants and multple obectve functons based on flow tme and due date. In addton they consder setup tmes cleanng tmes and transportaton tmes n the model and menton that the performance of the algorthm can be 4

affected by the number of the breakdown tmes. Fnally they prove that the effectveness of the smulated annealng algorthm s better than other heurstc algorthms wth the same condtons. Suresh and Kumarappan (2006) develop an optmzaton model and use a combnaton of genetc algorthm wth smulated annealng. The authors apply ther method to determne the preventve mantenance schedule n a power system. They menton that the method could produce better solutons f some changes and modfcaton are made to the soluton procedure. As a case study they test the method on 62-unt state electrcal system of Vctora. Samrout et al (2006) present another paper about the combnaton of an ant colony algorthm and genetc algorthm to optmze a large-scale preventve mantenance problem. They dvde the obectve functon of ther problem nto two sectons and then utlze each algorthm to mprove the sectons separately. They menton that usng hybrd algorthm n a large-scale problem s more effcent than the smple algorthm. 2.2.4. Mult-Obectve Algorthms Mult-obectve preventve mantenance optmzaton models have been presented n several papers. Kral and Petrovc (995) present a novel approach n preventve mantenance schedulng of thermal generatng systems. The authors develop a large-scale mult-obectve combnatoral optmzaton model wth three obectve functons and a set of the constrants. They consder mnmzaton of total fuel costs maxmzaton of relablty n term of expected unserved energy and mnmzaton of technologcal concerns as the obectve functons. In addton they defne mantenance duraton mantenance contnuty mantenance season mantenance 5

sequence of thermal unts of the same class lmtaton on smultaneous mantenance of thermal unts and lmtaton on total capacty on mantenance due to labor and resources as the constrants. They develop a mult-obectve preventve mantenance schedulng software based on a mult-obectve branch and bound algorthm mplemented n FORTRAN. Fnally the researchers apply ther methodology to a real system of 8 power plants wth 2 thermal unts wth mantenance classes over 3 weeks as the plannng horzon. Chareonsuk et al (997) develop a mult-crtera preventve mantenance optmzaton model to fnd the optmal preventve mantenance ntervals of components n a producton system. In ths study the authors consder an age-based falure rate for components by fttng a Webull dstrbuton to the data and defne expected total cost per unt tme and the relablty of the producton system as the man crtera. In followng they utlze a preference rankng organzaton method for enrchment evaluatons (PROMETHEE) as the soluton approach and defne the alternatve decsons as the preventve mantenance ntervals. By usng ths approach they can aggregate preferences of alternatves by combnng the weghted values of the preference functons of the complete set of crtera. As a case study they apply ther methodology n a paper factory and used PROMCALC as the optmzaton software. Fnally they menton the advantage of ther approach n whch decson makers and managers can nput varous crtera nto the model and do senstvty analyss on the optmal solutons. Konak et al (2006) present a comprehensve study on mult-obectve genetc algorthms and ther applcatons n relablty optmzaton problems. They revew 55 research papers and demonstrate the recent technques and methodologes. Quan et al (2007) develop a novel mult-obectve genetc algorthm n order to optmze 6

preventve mantenance schedule problems. They defne the problem as a multobectve optmzaton problem by consderng the mnmzaton of workforce dle tme and the mnmzaton of mantenance tme and menton that there s a tradeoff between the obectve functons. As the soluton procedure they use utlty theory nstead of domnance-based Pareto search to determne the non-nferor solutons and show the advantage of ths method va numercal example. Taboada et al (2008) present a recent study n ths area. They develop a mult-obectve genetc algorthm n order to solve mult-state relablty desgn problems. The authors utlze the unversal moment generatng functon to measure the relablty and avalablty crtera n the system. They appled ther approach nto two examples; the frst one s a system of fve unts connected n seres n whch each component has two states functonng properly or falure and the second one s a system of three unts connected n seres. In ths system each component has mult states wth dfferent levels of performance whch range from maxmum capacty to total falure. They utlzed MATLAB as the programmng envronment and shown the effectveness of ther approach n terms of computatonal tmes and obtaned non-nferor solutons. 2.3. Smulaton Models 2.3.. Monte Carlo Smulaton Bottaz et al (992) present the results of a systematc collecton of actual falure tmes and preventve and correctve mantenance actvtes of 900 buses over a perod of fve years. They create an updatable database to estmate the falure dstrbutons and to evaluate the nfluence of systematc preventve and correctve mantenance actons. They consder the total cost and avalablty as the obectve 7

functons apply Monte Carlo smulaton approach to evaluate and compare dfferent mantenance polces and present the computatonal results. Bllnton and Pan (2000) develop a model whch s based on the use of Monte Carlo smulaton to determne the total falure frequency and the optmum mantenance nterval for a parallel-redundant system. The authors present a modfed dstrbuton functon assumng an exponental dstrbuton for component useful lfe perod and the Webull dstrbuton for the wear out perod. The procedure ncludes constructon of a mathematcal model and defnton of the stoppng rule n smulaton for a parallel-redundant system. They state that f the shape parameter β of the Webull dstrbuton ncreases the optmum mantenance nterval decreases. Fnally they show that a two-component parallel-redundant system s a bass structure n mnmal cut set analyss that s used n evaluaton of power systems relablty. Zhou et al (2005) present an approach for sequental preventve mantenance schedulng based on the concept of age reducton due to mperfect mantenance actons. They consder an assumpton for the tme of mperfect mantenance actons based on requred relablty of the system. They utlze a hybrd recursve method based on an assumed mprovement factor and ncreasng falure rate and develop an optmzaton model wth a mantenance cost rate n the lfe cycle of the system as the obectve functon. Fnally they apply Monte Carlo smulaton and descrbe how ther computatonal results can be used n decson support systems for mantenance schedulng. Marquez et al (2006) develop a smulaton model to fnd the best preventve mantenance strategy n semconductor manufacturng plants. The authors model the age of equpment avalablty of equpment mantenance actvty backlog and preventve mantenance polces and consder dfferent wafer 8

producton scenaros n a Monte Carlo contnuous tme smulaton model. They analyze and compare the dfferent mantenance strateges on the status of manufacturng equpments and operatng condtons of the wafer producton flow. Furthermore they descrbe how the combnaton of age and avalablty-based models ncreases the throughput and provdes better results than the smple agebased models. 2.3.2. Dscrete-Event and Contnuous Smulaton Goel et al (973) present a smulaton model and develop a statstcal analyss that consders three dfferent types of preventve mantenance actvtes for components by defnng stochastc and determnstc decson varables as well as unavalablty and cost as the obectves. In addton they make a 2-level sequental fractonal factoral desgn n order to facltate ther smulaton. By desgnng the smulaton model based on expermental desgn approach ther model produces the preventve mantenance schedule for ground electroncs systems. Burton et al (989) develop a smulaton model to evaluate the performance of a ob shop. In ths research the effectveness of the preventve mantenance schedulng under dfferent condtons such as shop load ob sequencng rule mantenance capacty and strategy s determned and presented. Krshnan (992) develops a smulaton model to determne the mantenance schedule for an automated producton lne n a steel rollng mll plant. He consders three dfferent mantenance polces as opportunstc falure and block wth the percent of avalablty as the obectve functon. He shows that the exstng mantenance polcy only ncludes the falure and block mantenance actons. By 9

usng the hstorcal data of mantenance actvtes n the smulaton model the optmal preventve mantenance schedule s obtaned n the form of checklst. Mathew and Raendran (993) present a smulaton model n order to determne the frequency of the shutdown for perodc system overhaul preventve and correctve mantenance and nspectons n a sugar manufacturng plant. They utlze a tmedependent smulaton model to mnmze the total cost ncludng mantenance costs and downtme losses. Paz et al (994) develop a two-stage knowledge base for a mantenance supervsor assstant system. Ths knowledge base nteracts wth the mantenance manger on a perodc bass to select the proper preventve mantenance plan for the next perod. The frst stage deals wth an obect-orented computer smulaton model to montor dfferent preventve mantenance schedules that nclude preventve mantenance polces staffng polces downtme costs smultaneous downtme practces travel tme mpacts and blockng stuatons as the systems specfcatons. In addton they consder overall machne avalablty crtcal machne avalablty worker utlzaton cost of the mantenance actvtes and work order completon tme as the systems crtera. At the second stage they make a knowledge engneerng envronment to use the computatonal results obtaned from a smulaton model and send feedback to the frst stage. Joe (997) develops a smulaton model n order to evaluate dfferent preventve mantenance strateges for a fleet of vehcles of the St. Lous metropoltan polce department. He utlzes GPSS as the smulaton software analyzes several polces to mprove the effectveness and effcency of operatons and presents the best polcy. Savar (997) develops a smulaton model n order to nvestgate effect of dfferent preventve mantenance strateges n a ust-n-tme producton system. He 20

constructs a smulaton model on a 5-staton producton system and consders throughput rate average equpment utlzatons and total work-n-process as the performance measures of the producton system. After runnng the smulaton model and analyzng the computatonal results he mentons that preventve mantenance and correctve mantenance polces have a hgh mpact on the performance measures of ust-n-tme producton systems and by combnng the mantenance actvtes and ust-n-tme operatons one can mprove the effectveness of the ths knd of systems. Mohamed-Salah et al (999) develop a smulaton model n order to acheve opportunstc mantenance strateges n a mult-component producton lne. The authors consder two dfferent strateges and defne total cost as the functon of preventve and correctve mantenance actvtes as well as fxed cost due to any stop or falure n producton lne. The frst strategy assumes that the mantenance actvtes are allowed on all non-faled components f the dfference between the expected preventve tme of non-faled components and the falure nstant of faled components s less than certan value. The second one consders that the mantenance actvtes are allowed on all non-faled components f the dfference between the expected preventve tme of non-faled components and the preventve tme or correctve nstant of faled components s less than certan value. They utlze PROMODEL and descrbe that the total cost functon has a unque optmum. Fnally they express that the optmal nterval of mantenance for the strateges s 5.5 and 3.5 days respectvely. Greasley (2000) presents a smulaton model to fnd the optmal mantenance plannng n tran mantenance depot for an underground transportaton faclty n UK. He develops a smulaton based on two dfferent stuatons. The frst stuaton assumes there s no random arrval and the second one consders random arrvals 2

and nvestgates the effect of the arrval on servce level performance measures. He utlzes ARENA as the smulaton software and shows the effectveness of the mantenance polces obtaned by the smulaton model. Chan (200) presents a smulaton model to analyze the effects of preventve mantenance polces on buffer sze nventory sortng rules and process nterruptons n a flow lne of a push producton system. He presents the performance of the producton system under dfferent operatonal condtons and preventve mantenance polces. Duffuaa et al (200) present a generc conceptual smulaton model for mantenance systems. They defne ths smulaton model by constructng seven modules ncludng an nput module mantenance load module plannng and schedulng module materals and spares module tools and equpment module qualty module and fnally a performance measure module. The authors menton that ths model could be used to develop a dscrete event smulaton models n one of the commercal smulaton softwares. In addton they suggest that by usng ths model one can evaluate the need for contract mantenance and effect of avalablty of spare parts on performance measures n the system. Devulapall et al (2002) develop a smulaton model n order to determne the best preventve mantenance polces for brdge management systems (BMS). They utlze STROBOSCOPE as the smulaton software and examne the condtons of brdges under dfferent strateges. They apply ther model to a set of brdges n Vrgna and argue that the model can be used to provde varous mantenance polces for a brdge management system. Alfares (2002) presents a smulaton model to obtan the preventve mantenance schedule for components of a detergent-packng lne and consders two dfferent stuatons n hs model. The frst one assumes a constant tme nterval that s not 22

affected by mantenance actons or unexpected falures. In the second stuaton the tme nterval s affected and restarted by mantenance actons or unexpected falures. In order to mnmze the total cost he develops a smulaton model to optmze the mantenance schedule of components for each stuaton. Houshyar et al (2003) present a smulaton model to measure the mpact of preventve mantenance schedulng on the producton rate of a machne. They utlze PROMODEL to make the smulaton model and consder two dfferent scenaros for the smulaton run. They use statstcal analyss on the smulaton outputs n order to determne the mpact of recommended yearly preventve mantenance on the producton throughput of the machne. Fnally they menton that the preventve mantenance polcy does not affect the producton rate but can reduce yearly mantenance costs of the system. Han et al (2004) develop a fnte tme horzon model to acheve preventve mantenance schedulng of manufacturng equpment based on setback based resdual factors and use smulaton to solve the model. They menton the consstency of computatonal results and shown that smulaton s a useful and effectve method to solve such models. Jn et al (2006) develop a preventve mantenance optmzaton model for a mult-component producton process. They defne a combnaton of mechancal servce repar and replacement actvtes for each component and use Markov decson process to present the transton functon of probablty for mantenance actvtes. In addton they consder requred relablty of the system as the constrant and total preventve mantenance cost as the obectve functon of the model. A smulaton approach was utlzed to fnd the optmal schedule as the soluton procedure. The authors descrbe that consderng the combnaton of 23

preventve mantenance actvtes can reduce more cost n comparson wth the stuaton that dfferent actvtes are consdered separately. One of the most recent studes on applcaton of smulaton n preventve mantenance schedulng s presented by Hagmark and Vrtanen (2007). They develop a smulaton model to determne the level of relablty avalablty and correctve and preventve mantenance at the early stage of desgn. Ther method consders repar tme delays and effect of preventve mantenance on the system s falure observed by condton montorng and dagnostc resources. Yn et al (2007) recently propose a smulaton model n order to analyze the dynamc structure of mantenance systems. The researchers consder varous subsystems such as preventve mantenance subsystem defects subsystem condton-based subsystem falure subsystem correctve mantenance subsystem and performance subsystem and utlzed SIMULINK to buld up the model. They analyze the structure of components and the relaton of ther constrants n a mantenance system and present the advantages of the model over classcal stochastc process methods n a numercal example. In addton they menton that obtaned smulaton results express the dynamc nature of mantenance systems. 2.4. Age Reducton and Improvement Factor Models Nakagawa (988) presents a basc and notable approach for models that utlze mprovement factor. The work has been referenced by many researchers. He develops two analytcal models n order to fnd the optmal preventve mantenance schedule based on an assumpton of ncreasng falure rate over tme. The frst model called a preventve mantenance hazard rate model calculates the average 24

falure cost of mnmal repars along wth costs of preventve mantenance and replacement under the assumpton that preventve mantenance actons reduce the next effectve age to zero the falure rate s assumed to ncrease wth the ncreasng the frequency of preventve mantenance actons. Furthermore ths model assumes that mantenance actvtes take place at fxed ntervals between each predetermned replacement. The second model called an age reducton preventve mantenance model consders the average falure cost of mnmal repars as well as costs of preventve mantenance and replacement by assumng the age reducton after each mnmal repar. In order to fnd the optmal schedule both models are optmzed by calculus methods. He apples the models n a numercal example and descrbes that based on obtaned computatonal results the second model s more practcal than the frst model. Jayabalan and Chaudhur (992) propose another referenced work on age reducton and mprovement factors models. They develop an optmzaton model and a branchng algorthm that mnmzes the total cost of preventve mantenance and replacement actvtes. They assume a constant mprovement factor and defne a requred falure rate. In addton they assume a zero falure cost and do not consder tme value of money for future costs. Ther algorthm determnes the optmal schedule of mantenance actons before each replacement acton n order to mnmze the total cost n a plannng horzon. They utlze FORTRAN to mplement the algorthm and prove the effectveness of the algorthm va several numercal examples. Dedopoulos and Smeers (998) develop a nonlnear optmzaton model to fnd the best preventve mantenance schedule by consderng the degree of age reducton as the varable n the model. The researchers defne mprovement factor tme and 25

duraton of preventve mantenance actvtes as the decson varables consder fxed cost and varable cost for mantenance actons and defne the varable cost as a functon of the degree of age reducton the duraton of the acton and the effectve age of the component. Moreover they present the falure rate n each perod as a recursve functon of age reducton from a prevous perod and consder the net proft as the obectve functon of the model. They mplement the model n GAMS and use GAMS/MINOS optmzaton software. Fnally the effectveness of the model s shown va three numercal examples. Martorell et al (999) present an agedependent preventve mantenance model based on the survellance parameters mprovement factor and envronmental and operatonal condtons of the equpment n a nuclear power plant. They consder rsk and cost as the crtera of the model based on the age of the system and made the senstvty analyss to show the effect of the parameters on the preventve mantenance polces. They express that the results obtaned from ther model are dfferent from those resulted from the models that do not consder the mprovement factor and workng condtons. Ln et al (200) combne the models were developed by Nakagawa (988) and present hybrd models n whch effects of each preventve mantenance acton are consdered by two aspects; one for ts mmedate effects and the other one for the lastng effects when the equpment s put to use agan. The authors construct two models that reflect the concept of mantanable and non-mantanable falure modes. In the frst model they assume that preventve mantenance and replacement tme are ndependent decson varables and consder the mean cost rate as the obectve functon to be mnmzed. In the second model they assume that preventve mantenance actvtes are performed whenever the falure rate of the system exceeds the certan level and lke the frst model the mean cost rate s consdered as 26

the obectve functon. Fnally they present numercal examples to show the applcaton of the developed models and menton that for a system wth a Webull lfe dstrbuton optmal schedules can be acheved analytcally but for the general case t cannot be solved by analytc methods. X et al (2005) develop a sequental preventve mantenance optmzaton model over a fnte plannng horzon. They defne a recursve hybrd falure rate based on the mprovement factor and ncreasng falure rate n order to estmate the systems relablty n each perod of plannng horzon. In addton they consder the total cost of preventve mantenance actvtes and assume that mean cost n each perod s a functon of requred relablty and the mprovement factor. Fnally they utlze a smulaton approach to optmze the model and menton that the computatonal results can be used n a mantenance decson support system of ob shop schedulng. Jaturonnatee et al (2006) develop an analytcal model n order to fnd the optmal preventve mantenance schedule of leased equpment by mnmzng a total cost functon. They defne mantenance actons as preventve and correctve each wth assocated costs and then consder the concept of reducton n falure ntensty functon along wth penalty costs due to volaton of leased contact ssues. They present a numercal example for a system wth Webull falure rate solve the model analytcally and examne the effect of penalty terms on the optmal preventve mantenance polces. Bartholomew-Bggs et al (2006) present several preventve mantenance schedulng models that consder the effect of mperfect mantenance on effectve age of component. The researchers develop optmzaton models that mnmze the total cost of preventve mantenance and replacement actvtes. In ths study they assume a known falure rate to express the expected falures as a functon of age and consder age reducton n the effectve age based on the concept 27

of an mprovement factor. They develop a new mathematcal programmng formulaton to acheve the optmal mantenance schedule and utlze automatc dfferentaton as the numercal approach nstead of analytcal approach to compute the gradents and hessans n the optmzaton procedure whch s the global mnmzaton of non-smooth performance functon. Fnally the effectveness of the presented model and algorthm s shown n several numercal examples. El-Ferk and Ben-Daya (2006) present an age-based hybrd model for mperfect preventve mantenance. The authors revew dfferent polces and the models developed by other researchers and propose a new sequental age-based analytcal model. They assume that the mperfect preventve mantenance actvtes reduce the effectve age of the system but ncrease the falure rate and presented mathematcal formulatons to determne the adustment factors for both falure rate and age reducton coeffcent. They construct an optmzaton model based on ther analytcal models consder the mnmzaton of the total cost as the obectve functon and solve the optmzaton model va a new heurstc algorthm for a numercal example. One of the recent works on methods for estmatng age reducton factor s by Che- Hua (2007). In ths research he consders an optmal preventve mantenance for a deteroratng one-component system va mnmzng the expected cost over a fnte plannng horzon. He develops a model for estmatng mprovement factor to measure the restoraton of component under the mnmal repar. The proposed mprovement factor s a functon of effectve age of component the number of preventve mantenance actons and the cost rato of each mantenance acton to the replacement acton. Fnally the researcher could obtan the optmal preventve mantenance schedule for a case study wth the Webull hazard functon by applyng a partcle swarm optmzaton method. 28

Cheng et al (2007) present a paper about models for estmatng the degradaton rate of the age reducton factor. They present two optmzaton models whch mnmze the cost subect to requred relablty. The frst model has a perodc preventve mantenance tme nterval for every replacement and the second one contans the mantenance schedule where the tme nterval between the fnal mantenance and replacement s not constant. Lm and Park (2007) present three analytcal preventve mantenance models that consder the expected cost rate per unt tme as the obectve functon. In ths research they assume that each preventve mantenance actvty reduces the startng effectve age but does not change the falure rate and consder the mprovement factor as the functon of number of preventve mantenance actvtes. They also assume that the falure functon s based on a Webull dstrbuton and develop mathematcal formulaton for three dfferent stuatons; preventve mantenance perod s known number of preventve mantenance s known and number and perod of preventve mantenance s unknown. They derve the optmal preventve mantenance and replacement schedules by takng an analytcal approach and apply them to a numercal example to show an applcaton of ther models. 2.5. Applcatons 2.5.. Manufacturng and Producton Systems The applcaton of preventve mantenance schedulng has been wdely used n manufacturng and producton systems. For example Hsu (99) develops an optmzaton model n order to determne the optmal preventve mantenance schedules for a seral mult-staton manufacturng system. He mentons that most of 29

models use smulaton at that tme but hs model s focused on mathematcal programmng approach. The computatonal results of hs study show that operatng features of the statons are nterrelated and one must nvestgate the effect of preventve mantenance actvtes on all statons at the same tme. Cassady et al (999) develop an ntegrated control chart and preventve mantenance schedulng to reduce the total operatng cost of manufacturng systems. The researchers formulate an economc model that ncludes the product nspecton costs process downtme costs and poor qualty costs and analyze t va a smulaton model. In addton they construct a smulaton-optmzaton model n order to evaluate and optmze the parameters of control chart and preventve mantenance strategy. They demonstrate ther approach n a numercal example and shown the feasblty and effectveness of ther methodology. Westman and Hanson (2000) develop a model to determne the mean tme to falure (MTTF) as a functon of the uptme for a workstaton n a mult-stage manufacturng system. The authors assume that the uptme of the workstaton has an ncreasng rate and s reduced f preventve mantenance actons are performed. They menton that ths methodology captures the flexblty and mult-stage propertes of manufacturng systems and can generate the preventve mantenance polces. Westman et al (200) formulate a mathematcal model to fnd the optmal producton schedulng va lnear quadratc Gaussan Posson functon wth state dependent Posson process. They consder the total cost of producton and mantenance polces as the obectve functon and demonstrate the applcaton of the model by a numercal example. Charles et al (2003) present a preventve mantenance optmzaton model n order to mnmze the total mantenance costs n a producton system. In ths paper 30

they consder the total productve mantenance correctve mantenance and preventve mantenance actons along wth producton operatons as well as the related assocated costs. They assume a Webull lfe dstrbuton and utlze MELISSA C++ as dscrete-event producton-orented smulaton software to evaluate dfferent scenaros. As a case study they analyze a prototype semconductor manufacturng workshop to demonstrate the approach and mentoned that ths model has general structure that can be appled for other knd of manufacturng systems. Han et al (2004) develop a nonlnear optmzaton model to mnmze the cost of mantenance and replacement actons under the relablty constrants for producton machne n a producton system. Ther model consders Webull dstrbuton as the falure functon of the machne and can be used as a decson support system for ob shop schedulng. Sawhney et al (2004) present a smulaton model to determne mantenance strateges of a manufacturng system. Ther model s constructed for ntegratng reactve and proactve mantenance schedulng n order to ncrease productvty of operatons n the lean manufacturng structure. Preventve mantenance optmzaton s also used n semconductor manufacturng. L and Qan (2005) present a real tme preventve mantenance optmzaton model for cluster tools n a semconductor manufacturng system. They consder the standpont of the system and used genetc algorthm as the soluton procedure. In the area of applcaton of preventve mantenance n manufacturng and producton systems many researchers are nterested n ntegratng preventve mantenance and producton schedulng. Adzakpa et al (2004) present an applcaton of combnaton of mantenance schedulng and ob assgnment n dstrbuton systems. They develop an optmzaton model that consders the total cost of 3

mantenance actons as the obectve functon and avalablty n a gven tmewndow and precedence among consecutve standby obs and ther emergency as the constrants of the model. They show that ther problem s NP-hard to solve and because of that they use a heurstc optmzaton algorthm to solve the problem. Yng et al (2005) develop an ntegrated model that smultaneously consders preventve mantenance and producton schedulng decson varables. Ther model mnmzes the total tardness of obs and makes a 30% reducton n expected total tardness of obs. Rezg et al (2004) present an ntegrated preventve mantenance and nventory control smulaton model for a producton lne wth mult-component. The authors defne preventve and correctve mantenance actvtes along wth nventory control varables and parameters to develop approxmate analytcal models for the sngle machne under dfferent scenaros. In addton they utlze PROMODEL to construct an age-based smulaton model and apply genetc algorthm to optmze the parameters of the smulaton model and evaluate dfferent producton scenaros. Fnally they test ther methodology on three numercal examples of a producton lne and compare the computatonal results wth results obtaned from analytcal formulas. They menton that applyng combnaton of mantenance strateges producton plannng polces leads to a sgnfcant reducton of the total cost. Rezg et al (2005) present another paper n ths area. He and hs colleagues develop an ntegrated age-based preventve mantenance and nventory control smulaton model n a manufacturng system wth ust-n-tme confguraton. They present two approaches; the frst one s a mathematcal model to determne the average cost per unt tme and the second one s the combnaton of smulaton and expermental desgn. They use MAPLE for solvng the analytcal model utlze PROMODEL for 32

smulaton and use STATGRAPHICS to analyze the data for expermental desgn and regresson analyss. The authors menton that both approaches could gve approxmately same results and exstng dfference due to approxmaton assumptons consdered n the analytcal model that was elmnated n the smulaton model. Sortrakul et al (2005) present an optmzaton model of ntegrated preventve mantenance plannng and producton schedulng for a sngle machne. The authors menton that these problems have been tackled separately n several papers but they have not been consdered together n real manufacturng systems. They consder the total weghted expected ob completon tme as the obectve functon and optmze the combnatoral optmzaton model va genetc algorthm. As the result they express the advantages and effectveness of ther approach that can be used to solve real manufacturng problems. Cassady and Kutanoglu (2005) develop and present an ntegrated preventve mantenance and producton schedulng mathematcal model for a sngle-machne. They consder the total weghted expected completon tme as the obectve functon to be mnmzed. Ther model allows multple mantenance actvtes and explctly captures the rsk of not performng mantenance. They use a heurstc approach to solve the model and compare the obtaned computatonal results of ntegrated model wth the results acheved from the solvng preventve mantenance and ob schedulng problems ndependently. Leng et al (2006) present an ntegrated preventve mantenance schedulng and producton plannng mult-obectve optmzaton model for a sngle machne. They use chaotc partcle swarm optmzaton algorthm to solve the model and show the applcaton and effectveness va numercal examples. L and Zuo (2007) recently develop a smulaton model that determnes that mpact of preventve and correctve 33

mantenance actvtes on the total cost of nventores n a producton system. They apply smulaton as the soluton methodology to fnd the optmal number of falures and the optmal level of safety stock smultaneously and menton that combnng the preventve and correctve mantenance wth producton schedulng can reduce the large amount of total operatng cost n system. Kou and Chang (2007) develop an ntegrated producton and mantenance optmzaton model for a sngle machne based on cumulatve damage process and the effect of preventve mantenance polcy on the producton schedulng n order to mnmzaton of the total tardness. The authors express that n the optmal strategy f obs have certan process tme wth dfferent due dates the optmal producton schedule sorts the obs by earlest due date and f obs have certan due dates wth dfferent process tme t sorts them by shortest process tme. In addton they menton that the optmal mantenance polcy s a constrant on the producton schedule when machne shuts down due to cumulatve damage falure process. The computatonal results show that by ncreasng the number of obs the effect of obs due dates on the optmal mantenance polcy control lmt s decreased. Zhou et al (2007) demonstrate an age based preventve mantenance schedulng combned wth producton plannng optmzaton model n order to maxmze the avalablty of a producton machne. The authors use a heurstc algorthm to obtan the optmal schedule that mnmzes the make span. They also apply a smulaton approach to valdate the heurstc algorthm and to show ts effectveness to solve the flow shop schedulng problems of ntegrated producton and preventve mantenance. Ruz et al (2007) present comprehensve research n area of ntegratng preventve mantenance and producton schedulng. They defne three dfferent polces for preventve mantenance schedulng; preventve mantenance at fxed 34

predefned tme ntervals preventve mantenance for maxmzng the equpment avalablty and mantanng a mnmum relablty threshold over the plannng horzon. The mnmzaton of the total manufacturng tme of the sequence s consdered as the man crteron. The authors apply sx dfferent adaptatons of heurstc and meta-heurstc algorthms to evaluate the last two polces for two sets of problems and menton that ant colony and genetc algorthm solve these problems effectvely. Fnally they conclude that ntegrated preventve mantenance and producton schedulng optmzaton problems along wth meta-heurstc algorthms can be successfully appled n flowshop problems. In addton they suggest that one can defne more crtera and consder the problem as a mult-obectve optmzaton model. 2.5.2. Servce Systems Jayabalan and Chaudhur (992) present two dfferent preventve mantenance models for mantanng bus engnes n a publc transt network based on mnmzaton of the total cost over a fnte plannng horzon. They construct the models based on the concept of mean tme to falure (MTTF) of the engnes and assume the upper bound for the falure rates. The frst model s based on dfferent Webull falure functons between preventve mantenance actvtes and the second assumes that the each preventve mantenance acton reduces the effectve age of the system. The authors present the obtaned computatonal results and show the effectveness of the models n a real case study. Pongpech et al (2006) present an optmzaton model that mnmzes the total mantenance costs and penalty costs for used equpment under lease. They assume 35

Webull dstrbuton as the falure functon for equpment develop a 4-parameter model and apply a 4-stage algorthm to solve t. They apply ther model to several numercal examples wth dfferent contract assumpton and analyze the optmal polcy n each stuaton. Martn (988) presents a preventve mantenance optmzaton model whch has been developed and mplemented by Columba Hosptal n Mlwaukee based on plant technology and safety management standards. The hosptal desgned ths program n order to use the optmal preventve mantenance plan for electrcal dstrbuton equpment wth consderng safety servceablty relablty and the total cost. Fard and Nukala (2004) study and revew the applcaton of dfferent stochastc process such as homogenous Posson process (HPP) non-homogenous Posson process (NHPP) branchng Posson process (BPP) and supermposed renewal process (SRP) n preventve mantenance schedulng. They present current methods based on non-homogenous processes for modelng and optmzaton of sngle and mult-component systems. They assume that mantenance actons do not affect the falure rate of system; hence the NHPP can be appled to present and model reparable servce systems. 2.5.3. Power Systems Applcatons of preventve mantenance schedulng are not restrcted to manufacturng or servce systems. Power plants use preventve mantenance strateges to ncrease the relablty and avalablty of equpments. McClymonds and Wnge (987) present methods to acheve optmal preventve mantenance 36

schedulng for nuclear power plants though they have not been appled successfully. They consder the plant avalablty and relablty as the obectve functons and develop models based on assgnng resources to preventve and correctve mantenance actvtes. Zhao et al (2005) present an age-based preventve mantenance optmzaton model for a gas turbne power plant. They develop a model wth proft nstead of cost as the obectve functon and consdered power plant performance relablty and the market dynamcs effects n the model. In order to determne the effects of economcs on mantenance costs and frequences they utlze a sequental approach and show ts effectveness by usng real data of based load combned cycle power plant wth a gas turbne unt. 2.6. Chapter Summary In ths chapter recent work pertanng to methods and applcatons of preventve mantenance and replacement schedulng were revewed. They were categorzed as optmzaton models smulaton models age reducton and mprovement factor models and applcatons n manufacturng servce and power systems. We fnd that most studes focus on sngle-component systems or smple and specfc systems whch s not always applcable for real and general systems. In addton not much work has been done n the area of age reducton and mprovement factor models. Hence we propose preventve mantenance and replacement schedulng models that deal wth mult-component system and can be appled to a wde varety of systems. Because we use the concept of age reducton and mprovement factor n these models we also develop mathematcal and statstcal models to estmate the 37

mprovement factor for mperfect mantenance actvtes. These are our research contrbutons and they are appled to a real system. 38

Chapter 3 Optmzaton Models - Exact Algorthms 3.. Introducton Ths chapter wll present a new modelng approach to fnd optmal preventve mantenance and replacement schedules for mult-component systems. We construct new closed-form optmzaton models based on the cost and relablty characterstcs and solve them usng a standard optmzaton procedure. These models provde a general framework that can be appled and used n a wde varety of systems. Computatonal results show the feasblty of the proposed approach. 3.2. Formulaton Consder a new reparable seres system of components each subect to deteroraton. Each component s assumed to have an ncreasng rate of occurrence of falure (ROCOF) v (t) where t denotes actual tme ( t > 0 ). In ths paper we assume that component falures follow the well-known Non-Homogeneous Posson Process (NHPP) wth ROCOF gven as: v ( t) λ βt β for... (3.) where λ and β are the characterstc lfe (scale) and the shape parameters of component respectvely. The NHPP s smlar to the Homogeneous Posson Process

(HPP) wth the excepton that the falure rate s a functon of tme. For more on ths well-known stochastc process see Ascher and Fengold (984). We seek to establsh a schedule of future mantenance and replacement actons for each component over the perod [0 T]. The nterval [0 T] s segmented nto J dscrete ntervals each of length T/J. At the end of perod the system s ether mantaned replaced or no acton s taken. We assume that mantenance or replacement actvtes n perod reduce the effectve age of the system and thus t s ROCOF. For smplcty we also assume that these actvtes are nstantaneous.e. the tme requred to replace or mantan s neglgble relatve to the sze of the nterval and thus s assumed to be zero however we do mpose a cost assocated wth the repar or mantenance acton. To account for the nstantaneous changes n system age and system falure rate we ntroduce the followng notaton. Let X denote the effectve age of component at the start of perod and X denotes the age of component at the end of perod. It s clear that: T X X + for... ;... T (3.2) J 3.2.. Mantenance Consder the case where component s mantaned n perod. For smplcty we assume that the mantenance actvty occurs at the end of the perod. The mantenance acton effectvely reduces the age of component for the start of the next perod. That s: X + α X for... ;... T and (0 α ) (3.3) 40

The term α s an mprovement factor smlar to that proposed by Malk (979) and Jayabalan and Chaudhur (992). Ths factor allows for a varable effect of mantenance on the agng of a system. Whenα 0 the effect of mantenance s to return the system to a state of good-as-new. When α mantenance has no effect and the system remans n a state of bad-as-old. Note that the mantenance acton at the end of perod results n an nstantaneous drop n the ROCOF of component as shown n Fgure 3.. Thus at the end of perod the ROCOF for component s v ). At the start of perod + we fnd that the ROCOF drops to v ). ( X ( X Fgure 3.. Effect of perod- mantenance on component ROCOF 3.2.2. Replacement If component s replaced at the end of perod we fnd that: X + 0 for... ;... T (3.4).e. the system s returned to a state of good-as-new. The ROCOF of component ' nstantaneously drops from v ( ) to v (0) as shown n Fgure 3.2. X 4

Fgure 3.2. Effect of perod- replacement on system ROCOF 3.2.3. Do Nothng If no acton s performed n perod we see no effect on the ROCOF of component and we fnd that: T X X + for... ;... T J (3.5) X X + for... ;... T (3.6) v ( X ) v ( X + ) for... ;... T (3.7) 3.2.4. Cost of Preventve Mantenance and Replacements For a new system we seek to fnd the cost assocated wth a gven schedule of future mantenance and replacement actvtes. The cost assocated wth all componentlevel mantenance and replacement actons n perod wll be a functon of the all the actons taken durng that perod. 3.2.4.. Falure Cost When we vew the future perods of operaton for the system we must account for the nevtable costs due to unplanned component falures. From our vantage pont 42

at the start of perod however we cannot know when such falures wll occure. However we know that f the system carres a hgh ROCOF through a perod then we are at rsk of experencng hgh number and hence hgh cost of falures. Conversely a low ROCOF n perod should yeld a low cost of falure. To account for ths we propose the computaton of the expected number of falures n each perod for each component n the system. (We depart here from the approach found n Usher et al (998) where an average falure rate concept was used wth a cost constant.) Here we compute the expected number of falures of component n perod as: E X [ ] v ( t) dt for... ; T... X (3.8) Under the NHPP assumpton we fnd the expected number of component falures n perod to be: E β β [ ] ( X ) λ ( X ) for... ; T λ (3.9)... We assume that the cost of each falure s F (n unts of $/falure event) whch n turn allows us to compute F the cost of falures attrbutable to component n perod as: [ ] for... ; T F F E... (3.0) Hence regardless of any mantenance or replacement actons (whch are assumed to occur at the end of the perod) n perod there s stll a cost assocated wth the possble falures that can occur durng the perod. 43

3.2.4.2. Mantenance Cost If mantenance s performed on component n perod a mantenance cost constant M s ncurred at the end of the perod. 3.2.4.3. Replacement Cost If component s replaced n perod we assume that the replacement cost s the ntal purchase prce of the component denoted R. 3.2.4.4. Fxed Cost For a mult-component system and the cost structure defned above the problem can be shown to reduce to a smple problem of fndng the optmal sequence of mantenance replacement or do-nothng actons for each component ndependent of all other components. That s one could smply fnd the best sequence of actons for component regardless of the actons taken to component 2 and so on. Ths would result n N ndependent optmzaton problems. Such a model seems unrealstc as there should be some overall system cost penalty when an acton s taken on any component n the system. It would seem that there should be some logcal advantage to combnng mantenance and replacement actons e.g. whle the system s shut down to replace one component t may make sense to go ahead and perform mantenance/replacement of some other component even f t s not at ts ndvdual optmum pont where mantenance or replacement would ordnarly be performed. Under ths scenaro the optmal tme to perform mantenance/replacement actons on ndvdual components s dependent upon the decson made for other components. As such we propose that a fxed cost of downtme Z be charged n 44

perod f any component (one or more) s mantaned or replaced n that perod. Consderaton of ths fxed cost makes the problem much more nterestng and more dffcult to solve as the optmal sequence of actons must be determned smultaneously for all components. 3.2.4.5. Total Cost From our vantage pont at the start of perod 0 we wsh to determne the set of actvtes.e. mantenance replacement or do nothng for each component n each perod such that total cost s mnmzed. In order to have X age of component at the end of perod by usng equaton (3.2) frst we defne m and r as bnary varables of mantenance and replacement actons for component n perod as: f component at perod s mantaned (3.) 0 otherwse m f component at perod s replaced (3.2) 0 otherwse r Then we construct the followng recursve functon of X X m r α wth a constrant: X X ( m )( r ) X + m ( α. X ) ' X T + J (3.3) m r (3.4) + In addton we assumed the ntal age for each component s equal to zero: X 0 for... (3.5) 45

If component replacement occurs n the prevous perod then r m 0 so X 0. If a component s mantaned n the prevous perod then r 0 m so X. α X and fnally f we do nothng r 0 m 0 and X X whch corresponds to our basc assumptons gven n Secton 3.. From our defntons of each type of cost we can wrte the total cost functon as: Total Cost T T β β F (. λ X ) ( X ) ) + M. m + R. r + Z ( ( m + r ) (3.6) Ths obectve functon computes the total cost as a smple sum of component costs n each perod based on any mantenance or repar cost the system downtme cost and the cost of the expected number of falures. It s certanly possble to compute a more accurate economc measure of these costs such as Net Present Value (NPV) usng a sutable nterest rate. One could also nclude the effects of nflaton by addng an nflaton rate n the calculaton of future costs. Whle these may make the model more accurate we have avoded those mnor refnements for the sake of notatonal smplcty. See Usher et al (998) for more on these ssues. 46

3.3. Optmzaton Models 3.3.. Model - Mnmzng total cost subect to a relablty constrant In ths model we attempt to mnmze the total cost subect to the constrant that some mnmum level of system relablty over the plannng horzon s acheved and assume that components are arranged n seres. It s mportant to note that other system confguratons (parallel k-out-of-n complex etc) can be handled ust by modfyng and adaptng the fxed cost secton and the relablty functon based on the confguraton of the parallel k-out-of-n or other complex systems but for the sake of smplcty we consder only seres systems n ths paper. To consder the relablty constrant n ths model we defne the relablty functon for component n the perod as follow: E[ ] R e (3.7) By ths defnton we fnd the seres system relablty to be: R T e E [ ] (3.8) Then we formulate the followng non-lnear mxed nteger-programmng model that mnmzes the total cost for a gven relablty n the system: 47

48 ( ) ( ) ( ) ( ) ( ) ( ) ( ) T X X T r m R e T r m T J T X X T X m X r m X X s t r m Z r R m M X X F Cost Total Mn seres T X X T T... and... 0... and... 0 or... and...... and... 2... and... ). ( ) )( (... 0.:.... + + + + + + + β β λ β β α λ (3.9) 3.3.2. Model 2 - Maxmzng relablty subect to a budget constrant Here we modfy the formulaton and ntroduce a budget constrant B. The obectve of ths model s to maxmze the system relablty through our choce of mantenance and replace decsons such that we do not exceed the budgeted total cost. Ths model can be formulated as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) T X X T r m B r m Z r R m M X X F T r m T J T X X T X m X r m X X s t e Relablty Max T T T X X... and... 0... and... 0 or...... and...... and... 2... and... ). ( ) )( (... 0.:. + + + + + + + β β λ λ α β β (3.20)

3.4. Soluton Procedure Based on sequental nature of the preventve mantenance and replacement schedulng and nonlnear mxed-nteger programmng structure of the models presented n Secton 3.2 we apply a Dynamc Programmng (DP) approach to solve the models. Dynamc programmng was developed and ntroduced by Bellman (957) as a computatonal methodology for solvng certan types of optmzaton problems. In ths approach an optmzaton problem nvolvng n decson varables s reduced and transformed to a set of n sngle-varable optmzaton problem. Dynamc programmng provdes an excellent procedure to obtan an optmal soluton when a problem nvolves one constrant at most. However the computatonal complexty of the approach ncreases exponentally wth the number of constrants. We defned state vectors x and x to be the effectve ages at the start and end of perods n spaces X and X called the state spaces and decson vectors m and r bnary varables of mantenance and replacement actons n spaces M and R called the polcy spaces. Note that each polcy n these spaces s a concatenaton of elementary decsons and based on equatons (3) (4) and (5) t can be presented as transton functons as follow: x x g g ( x 2 ( x ) m r ) for for... ;... T... ;... T (3.2) Suppose that the am s to fnd the sequence of decsons that makes extreme value for a return functon. We defne C as a separable functon of falure mantenance and replacement costs for component n perod C 2 as a separable 49

50 functon of fxed cost n perod and R as separable functon of relablty for component n perod. ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + T e x x R r m Z r m C r R m M x x F r m x x C X X... ;... for ) ( ) (... ) ( 2 β β λ β β λ (3.22) We can transfer the nonlnear mxed-nteger optmzaton models to dynamc programmng models by applyng the separable functons presented n equaton (3.22) consderng the frst set of constrants as the ntal condtons needed for recurson and usng the second and thrd sets as the transton functons presented n equaton (3.2). Now we defne ) (w f k and ) (w f k as recursve sequence of functonal equatons at stage k for the total cost and the relablty n whch w and w are the parameters that dscretze the rght hand sde values of the man constrant on a certan grd. Fnally the dynamc programmng formulaton of optmzaton models can be presented as follows: seres k k k r m r m k R w T k r m C r m x r m x g g C w x r m x g g R f Mn w f 0.8... 0.9 ;... for ) ( ) ) ) ( ( ( ) )) ) ) ( ( ( (( ) ( 2 2 2 ; + + + (3.23) B w T k x r m x g g R r m C r m x r m x g g C w f Max w f k k k r m r m k 50000000... 0 ;... for )) ) ) ( ( ( ))) ( ) ) ) ( ( ( ( ( ) ( 2 2 2 ; + + (3.24)

3.5. Computatonal Results In order to llustrate the two models and the proposed soluton procedure we develop the representatve data set shown n Table 3.. In addton we assume Z $800 as the fxed cost R 50% as the requred relablty for Model and B $25000 as the gven budget for Model 2. We utlzed both Mcrosoft Excel Solver and LINGO software (see www.lndo.com) to solve the DPs for each model. Excel Solver s able to solve smaller problems. It s useful to menton that nonlnear mxednteger optmzaton models presented n secton 3.3 have 420 varables 720 of whch are bnary and 062 constrants 352 of whch are nonlnear. Table 3.. Parameters for the numercal example Component λ β α Falure Mantenance Replacement Cost ($) Cost ($) Cost ($) 0.00022 2.20 0.62 250 35 200 2 0.00035 2.00 0.58 240 32 20 3 0.00038 2.05 0.55 270 65 245 4 0.00034.90 0.50 20 42 80 5 0.00032.75 0.48 220 50 205 6 0.00028 2.0 0.65 280 38 235 7 0.0005 2.25 0.75 200 45 75 8 0.0002.80 0.68 225 30 25 9 0.00025.85 0.52 25 48 20 0 0.00020 2.5 0.67 255 55 250 For example a test problem wth 5 components and 20 perods took only 7 mnutes on a desktop computer (Intel/Pentum 4 3.4 GHz and GB RAM). However the example problem descrbed above wth 0 components and 36 perods could not be solved n reasonable tme. The value of obectve functon for the optmum soluton n the Model s $8867.3 and the system relablty under ths model s 50% (bndng constrant). For the second model the budget s equal to $25000 and the system relablty s maxmzed and found to be 60.87%. The results for these two models are presented n the Table 3.2 and 3.3 respectvely. 5

Table 3.2. Model - Mantenance and Replacement Schedule that Mnmzes Total Cost (Relablty50% and Cost$8867.3) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - - M - - - R - - - - R - - M - M - - R - - - M - - R - M - - - - - - 2 - - - M - - - M M - M - M - - M - M - R - - - - R - - - - R - - - - - - 3 - - - M - - - R - - - - R - - - - M - - R - - - M - - - - R - - - - - - 4 - - - M - - - R - - M - M - - M - M - M - - - R - - - - - R - - - - - - 5 - - - - - - - M - - M - M - - M - - - M - - - M M - - M - - - - - - - - 6 - - - M - - - - R - - - M - - M - R - - M - - M M - - - R - - - - - - - 7 - - - - - - - R - - - - - - - R - - - - R - - - - - - - - R - - - - - - 8 - - - - - - - - M - M - M - - M - M - - M - - M M - - - - - - - - - - - 9 - - - - - - - M M - - - M - - M - M - M - - - M M - - - M - - - - - - - 0 - - - - - - - - R - - - - - - R - - - - - - - R - - - - - R - - - - - - Table 3.3. Model 2 - Mantenance and Replacement Schedule that Maxmzes Relablty (Budget$25000 and Relablty60.87%) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component M - - - - - - R - - R - - - M - R M - - M - R - - M - R - M - R - - - - 2 R - - - - - - R - - R - - - M M R M - - M - R - - M - M M M - R - - - - 3 M - - - - - - R - - R - - - R - M M - - R - M - - M - R M M - R - - - - 4 M - - - - - - R - - M - - - M - R - - - R - M - - R - M M - - R - - - - 5 - - - - - - - R - - M - - - M - R - - - M - M - - R - - M - M M - - - - 6 M - - - - - - R - - R - - - M M M M - - R - M - - R - M M M - R - - - - 7 - - - - - - - R - - R - - - - - R - - - R - - - - R - - R - - R - - - - 8 - - - - - - - M - - M - - - - - R - - - M - M - - M - M M - - M - - - - 9 M - - - - - - R - - M - - - M - R - - - M - M - - M - M - - R - - - - - 0 - - - - - - - R - - R - - - - R - - - - R - - - - R - - R - - R - - - - Note that n both models most of mantenance and replacement actons tend to occur n the same perod whch reflects the effect of the fxed cost Z. It s also nterestng to note that once a mantenance or repar acton occurs t s often followed by a perod of nactvty. Such observatons can perhaps lead to the development of smple heurstc soluton procedures n follow on work. Another nterestng aspect of ths type of modelng s that one can analyze the effectve age of each component. Mantenance managers could use the model to track the effectve age of the components and then utlze the nformaton to ntate addtonal montorng actvtes. For example after a component reaches a set effectve age addtonal montorng tests or nspectons mght be warranted to assst n the detecton of mmnent falure. 52

The mnmum maxmum and average effectve age of each component are shown n Tables 3.4 and 3.5. Notce that the mnmum effectve age of each component s equal to zero at the begnnng of plannng horzon. Hence mnmum effectve ages of components are shown from the second month on. Note that every component was replaced at some tme durng the plannng horzon. The effectve age for the components ranges from roughly 0-5 months wth an average age of about 4 months. For model 2 snce we consdered more budget the effectve age of components s less than effectve age of components n model and ranges from 0-2 month wth an average age about 2.9 months. Component Table 3.4. Effectve age of components n Model Mnmum Effectve Age (month) Maxmum Effectve Age (month) Average Effectve Age (month) 0.0 7.2 3.0 2 0.0 6.3 3.2 3 0.0 7.2 3. 4 0.0 6. 3.0 5.0 0.5 4.9 6 0.0 7.6 3.5 7 0.0 9.0 3.8 8.0 5.6 7.2 9.0 0.3 4.8 0 0.0 9.0 3.7 Component Table 3.5. Effectve age of components n Model 2 Mnmum Effectve Age (month) Maxmum Effectve Age (month) Average Effectve Age (month) 0.0 7.6 2.4 2 0.0 7.0 2.3 3 0.0 7.6 2.3 4 0.0 7.5 2.5 5 0.0 8.0 2.9 6 0.0 7.7 2.5 7 0.0 8.0 2.6 8 0.0.7 5.3 9 0.0 7.5 3. 0 0.0 8.0 2.5 53

Fgures 3.3. through 3.3.0 and Fgures 3.4. through 3.4.0 show the effectve age of each component. As we can see when a component s mantaned the effectve age of that component drops based on the value of mprovement factor α presented n Table 3.. For example based on the effectve age presented n Fgure 3.3. component does not receve any mantenance acton at the frst 3 perods but t s mantaned at the 4 th perod and replaced at 8 th perod. Ths causes the effectve age drops to zero and the component works as a new one at the begnnng of the next perod. Another mportant feature presented n Fgures 3.3 and 3.4 s the effect of falure rate on the number and frequency of mantenance and replacement actons of components over a plannng horzon. For example compare the varatons n the effectve age of components 7 and 8. It can be seen n Fgures 3.3.7 3.3.8 3.4.7 and 3.4.8 that component 7 s ust replaced and there s no mantenance acton s performed n ths component. On the other hand component 8 s ust mantaned and t s not replaced n Fgure 3.4.8 t s replaced ust once. Ths s related to values of λ and β for each component. In Table 3. component 7 has 0.0005 and 2.25 and component 8 has 0.0002 and.8 for parameters λ andβ whch means that component 7 has the hgher falure rate and the greater probablty to fal than component 8. Therefore t s necessary that component 7 receves more replacement actons n order to satsfy the requred relablty or to maxmze the of the system s relablty. 54

Age (month) Fgure 3.3.. Effectve age of component Age (month) Fgure 3.3.2. Effectve age of component 2 8.0 8.0 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.3.3. Effectve age of component 3 8.0 8.0 Age (month) Fgure 3.3.4. Effectve age of component 4 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.3.5. Effectve age of component 5 2.0 0.0 8.0 6.0 4.0 2.0 Age (month) 0.0 8.0 6.0 4.0 2.0 Fgure 3.3.6. Effectve age of component 6 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.3.7. Effectve age of component 7 Age (month) Fgure 3.3.8. Effectve age of component 8 2.0 0.0 8.0 6.0 4.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 6.0 4.0 2.0 0.0 8.0 6.0 4.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.3.9. Effectve age of component 9 2.0 0.0 Age (month) 2.0 0.0 Fgure 3.3.0. Effectve age of component 0 8.0 6.0 4.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 8.0 6.0 4.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Fgure 3.3. Effectve age of components n Model 55

Age (month) Fgure 3.4.. Effectve age of component Age (month) Fgure 3.4.2. Effectve age of component 2 0.0 0.0 8.0 8.0 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.4.3. Effectve age of component 3 Age (month) Fgure 3.4.4. Effectve age of component 4 0.0 0.0 8.0 8.0 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.4.5. Effectve age of component 5 Age (month) Fgure 3.4.6. Effectve age of component 6 0.0 0.0 8.0 8.0 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.4.7. Effectve age of component 7 Age (month) Fgure 3.4.8. Effectve age of component 8 0.0 2.0 8.0 0.0 6.0 4.0 8.0 6.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Age (month) Fgure 3.4.9. Effectve age of component 9 Age (month) Fgure 3.4.0. Effectve age of component 0 0.0 0.0 8.0 8.0 6.0 6.0 4.0 4.0 2.0 2.0 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) 0.0 0 4 8 2 6 20 24 28 32 36 Effectve age (month) Fgure 3.4. Effectve age of components n Model 2 56

3.6. Chapter Summary Ths chapter presented basc assumptons for the formulaton of preventve mantenance and replacement strateges n order to fnd the best sequence of actons for each component n the system over a plannng horzon such that overall costs are mnmzed or the relablty of the system s maxmzed were presented. Two nonlnear mxed nteger programmng models were proposed and optmzed usng a dynamc programmng approach. The applcaton of the optmzaton models was presented va numercal example and the computatonal results of both models were analyzed. 57

Chapter 4 Optmzaton Models - Heurstc Algorthms 4.. Introducton Chapter 3 presented two nonlnear mxed nteger programmng models that were optmzed usng dynamc programmng. Because of the computatonal complexty of dynamc programmng to solve real large-scale problems and ts weakness to solve such problems n a reasonable tme we apply heurstc methods to tackle the problem. Ths chapter presents a mult-obectve optmzaton model to fnd the optmal preventve mantenance and replacement schedule of mult-component systems whch s an extenson of proposed models n Chapter 3. Two heurstc algorthms are adapted and modfed to solve the mult-obectve optmzaton model. Computatonal results show the feasblty and effectveness of the proposed approaches. 4.2. Formulaton Based on the equatons (3.9) and (3.0) we assume that the general effect of nflaton ncreases the cost of falures over tme at a rate off nffalure percent per perod. Thus we fnd F the cost of falures attrbutable to component n perod as:

' β β ( X ) ( X ) )( + nffalure) for... ;... T F F. λ (4.) In addton we assume a separate nflaton factor nfm nfr and nfz for mantenance replacement and fxed costs ncreases over tme and fnd that the assocated costs of mantenance actvtes of component n perod as follows: ( + nfm) for... ; T M M... (4.2) ( + nfr) for... ; T R R... (4.3) ( + nfz) ( ( m + r ) for... ; T Z Z... (4.4) Note that m and r are bnary varables of mantenance and replacement actons for component n perod and they cannot be equal to one smultaneously. The last term of the above equaton mentons that f a component s mantaned or replaced n each perod the whole system encounters wth the defned fxed cost. 4.3. Optmzaton Model From our defntons of each type of cost and by usng standard tme value of money concepts and an nterest rate nt we can fnd the total net present worth (NPW) of the cost of falure mantenance replacement and fxed over the T perods and we can rewrte the obectve functon of total cost that should be mnmzed. Fnally the mult-obectve optmzaton model corresponds to the cost and relablty functons can be expressed as: 59

60 ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T X X T r m T r m T J T X X T X m X r m X X s t e Relablty Max nt r m nfz Z r nfr R m nfm M nffalure X X F Cost Mn Total T X X T... and... 0... and... 0 or... and...... and... 2... and... ). ( ) )( (... 0.:.... ' ' + + + + + + + + + + + + α λ β β λ β β (4.5) 4.4. Mult-Obectve Genetc Algorthms Genetc Algorthms (GA s) were developed and ntroduced by John Holland (975). They have been desgned as general search strateges and optmzaton methods workng on populatons of feasble solutons. Based on populaton search approach GA s are able to solve mult-obectve optmzaton problems. A generc sngleobectve GA can be easly modfed to search a new set of multple non-domnated solutons. The ablty of GA to smultaneously search dfferent regons of a soluton space makes t possble to fnd a dverse set of solutons for dffcult problems wth non-convex dscontnuous and mult-modal solutons spaces. 4.4. Representaton of Solutons In order to represent the soluton of the proposed preventve mantenance and replacement schedulng problem wth do nothng mantenance and replacement actons; we defne an array wth length of T for N components and T perods

where each cell n that array contans 0 or 2 corresponds to three dfferent actons. 4.4.2 Ftness Functons Snce the optmzaton model presented n (4.5) s a mult-obectve optmzaton problem we consder three dfferent ftness functons n order to represent the problem as a sngle obectve optmzaton problem. ( Total Cost Cost ) + w ( Relablty) Ftness w / max 2 (4.6) Ftness Ftness ( Relablty) + Total Cost Gven budget (4.7) 2 ( Total Cost) + Cost Relablty Requred Relablty (4.8) 3 max Note that the above ftness functons are all subect to mnmzaton. Ftness s based on the weghted summaton of the normalzed cost functon and the relablty functon wth the condton w + w see Cohon (978). In order to normalze the 2 cost functon we defned Cost max as the denomnator. Ths coeffcent s the maxmum amount of cost that the system could ncur whch s when all components are replaced n each perod. The second ftness functon Ftness 2 consders maxmzng the relablty and mnmzng a penalty term of total cost. The penalty term s based on volated values of the total cost of mantenance actvtes and the gven budget for the system. Snce the volated values have larger amount n comparson wth relablty values mnmzaton of the penalty term has a prorty to maxmze of relablty. The thrd ftness functon Ftness 3 mnmzes the total cost and absolute values of subtracton of relablty and requred relablty of the system. We consdered 6

Cost max as the coeffcent of the penalty term n order to make a same magntude for both parts. 4.4.3. Crossover Procedure The crossover procedures create a new soluton as the offsprng of a par of exstng ones (parent solutons). The offsprng nhert some useful propertes of both parents n order to facltate ts propagaton throughout the populaton. We consder several crossover procedures as follows: a) One-Pont Crossover: In ths type of crossover frst we generate a random number between and T then create an offsprng from selected parents n whch all elements located to the left of that random poston are coped from the frst parent and the rest of the elements are coped from the second parent. b) Two-Pont Crossover: In ths type crossover frst we generate two random numbers between and T then create an offsprng from selected parents n whch all elements outsde the poston of those random numbers are coped from the frst parent and nsde elements are coped from the second parents. c) Two-Pont Inverse Crossover: In ths type crossover frst we generate two random numbers between and T then create an offsprng from selected parents n whch all elements outsde the poston of those random numbers are coped from the frst parent but n an nverse order and nsde elements are coped from the second parents. If the chosen parents are dentcal ths type of crossover makes a dfferent offsprng whch s not the same to ts parents. 62

d) N-Pont Crossover: we defne a new procedure for crossover n whch a new offsprng s made based on N components of ts parents. In ths type of crossover the even components are coped from the frst parent and the odd components are coped from the second parent. e) NT-Pont Crossover: In ths type crossover the even genes are coped from the frst parent and odd genes are coped from the second one. 4.4.4. Mutaton Procedure The mutaton procedure s appled to the offsprng soluton. It makes changes nto the soluton encodng strng by modfyng some of the strng elements. Based on the specal structure of the preventve mantenance and replacement schedulng problem n whch f even one mantenance or replacement acton takes place n a perod the whole system encounters a fxed cost we defne a new type of mutaton procedure. In ths type of mutaton frst we generate a random number between and T then change the correspondng gene to or 2 f t s equal to 0 or change t to 0 f t s equal to or 2 and do same procedure n the same perod for other components. Ths helps to create solutons n whch mantenance and replacement actvtes tend to occur n the same perods for all components. 4.4.5. Generatonal GA In the generatonal GA the entre populaton s replaced each generaton. The generatonal GA uses two populatons at the reproducton stage. One populaton contans the parents to be selected and the second one s generated to hold ther progeny. The generatonal GA algorthm s as follows see Goldberg (989) and Lsnansk and Levetn (2003): 63

Begn Generatonal GA g0 Produce ntal populaton P(g) Determne the ftness values of members n P(g) Whle GA termnaton condton s not satsfed do gg+ Select solutons from P(g-) for P(g) based on ther ftness value wth the probablty of p selecton Make an offsprng from selected parents from P(g-) wth the probablty of p crossover Mutate solutons from P(g-) wth the probablty of Determne the ftness values of members n P(g) End whle End Generatonal GA p mutaton 4.4.6. Steady State GA The steady state GA uses the same populaton for both parents and ther progeny. When the generc operaton on the parents s completed the new offsprng takes the place of the members of the prevous generaton wthn that populaton. The steady state GA algorthm s as follows see Whtley (989) and Lsnansk and Levetn (2003): Begn Steady State GA Produce ntal populaton P Determne the ftness values of members n P Whle GA termnaton condton s not satsfed do Whle genetc cycle termnaton condton s not satsfed do Make an offsprng from selected parents Mutate the produced offsprng wth the probablty of p mutaton Determne the ftness values of the new produced soluton Replace the new produced soluton wth the worst soluton n P f ts ftness value s better than the ftness value of the worst soluton Dscard dentcal solutons n P End whle Update P wth new produced solutons End whle End Steady State GA 64

4.5. Computatonal Results In order to llustrate the model numercally and the proposed soluton procedure we used data set presented n Table 3. and assume Z $800 as the fxed cost and a 36-perod plannng horzon. In addton we set the GA parameters for both generatonal and steady state GA as presented n Table 4.. Fnally we consder the nflaton rate of falure mantenance replacement and fxed cost equal to %.5% 2% and % respectvely and 3% as the nterest rate for engneerng economy parameters. We utlzed MATLAB R2007a (see www.mathworks.com) programmng envronment to develop the generatonal and steady state GA as well as to defne the ftness functons. Table 4.. Parameters of Genetc Algorthms Generatonal GA Steady State GA Number of Generatons 500 Number of Generatons Populaton Sze 00 Genetc Cycle 500 Probablty of Selecton 20% Number of Iteratons 00 Probablty of Crossover 40% Populaton Sze 00 Probablty of Mutaton 40% Probablty of Mutaton 00% 4.5.. Computatonal Results of Ftness Functon We run both generaton and steady state GA wth ftness functon for the set of weghts for both obectves functons and acheved non-nferor solutons shown n Table 4.2. By usng the new crossover and mutaton procedures we were able to solve the model n less than 2 mnutes for both algorthms. Tables 4.3 and 4.4 show the optmal preventve and replacement schedule of ftness functon for 0.8 and 0.2 as the weghts for cost and relablty obectve functons. Wth these weghts the values of obectve functons are $6240.55 and 62.93% va generatonal GA and are $6979.54 and 65.36% va steady state GA. It should be mentoned that all of 65

replacement actons tend to occur n the same perod whch reflects the effect of the fxed cost Z. It s also nterestng to note that once replacement acton occurs t s always followed by a perod of nactvty. Table 4.2. Non-nferor solutons resulted from Ftness Functon Weghts Generatonal GA Steady State GA W W2 Cost Relablty Cost Relablty 0.0.0 $ 37334.28 9.03% $ 37334.28 9.03% 0. 0.9 $ 37334.28 9.03% $ 37229.57 90.98% 0.2 0.8 $ 33585.74 89.89% $ 32586.72 90.08% 0.3 0.7 $ 28004.50 88.63% $ 27426.80 88.32% 0.4 0.6 $ 2027.67 84.43% $ 244.99 85.48% 0.5 0.5 $ 4602.70 80.23% $ 6697.2 8.97% 0.6 0.4 $ 0599.07 74.85% $ 2694.47 77.29% 0.7 0.3 $ 9080.44 7.7% $ 9638.40 72.86% 0.8 0.2 $ 6240.55 62.93% $ 6979.54 65.36% 0.9 0. $ 358.6 48.79% $ 2602.64 39.80%.0 0.0 $ 454.85 2.22% $ 454.85 2.22% Table 4.3. Mantenance and Replacement Schedule Ftness Functon Generatonal GA (w80% and w220%) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 2 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 3 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 4 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 5 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 6 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 7 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 8 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 9 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - 0 - - - R - - - R - - R - - - - R - - R - - - - R - - - - R - - - - - - - Table 4.4. Mantenance and Replacement Schedule Ftness Functon Steady State GA (w80% and w220%) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 2 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 3 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 4 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 5 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 6 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 7 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 8 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 9 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 0 - R - - R - - - R - - - R - - - R - - - R - - R - - - - R - - - - - - - 66

Fgure 4. represents Pareto optmal solutons (trade off curves) obtaned by generatonal and steady state GA for ftness functon. As t can be seen both Pareto solutons closely correspond and the curvature of both shows that the best part of the curves s somewhere between 50% and 80% of relablty ($5000 to $5000). Fgures 4.2 and 4.3 show the cost and relablty mprovement n terms of number of generatons and genetc cycles n generatonal and steady state GAs. As we can see the convergence of the steady state GA s somewhat faster than the convergence of generatonal GA but the qualty of fnal soluton resulted n generatonal GA s slghtly better than steady state GA. Fgure 4.. Pareto Optmal Solutons for Ftness Functon 67

Fgure 4.2. Cost Improvement for Ftness Functon Fgure 4.3. Relablty Improvement for Ftness Functon 4.5.2. Computatonal Results of Ftness Functon 2 We optmzed the model (4.5) wth ftness functon 2 and by consderng dfferent budget levels for the system and obtaned non-nferor solutons presented n Table 4.5. Tables 4.6 and 4.7 show the optmal preventve and replacement schedule of ftness functon 2 for the gven budget equal to $5000. Wth ths budget the relablty of the system resulted by generatonal and steady state GA respectvely s 54.07% and 5.88% respectvely. As we can see that n ths stuaton all of mantenance and replacement actons take place n the same perod and once 68

mantenance or replacement acton occurs t s often followed by a perod of nactvty. Table 4.5. Non-nferor solutons resulted from Ftness Functon 2 Gven Budget Generatonal GA Steady State GA Cost Relablty Cost Relablty $ 2000.00 $ 2000.6 4.94% $ 2000.2 8.88% $ 4000.00 $ 4000.23 42.4% $ 4000.07 35.6% $ 6000.00 $ 6000.3 58.00% $ 6000.03 56.95% $ 8000.00 $ 7999.97 64.98% $ 7999.87 62.38% $ 0000.00 $ 9999.96 69.07% $ 9999.98 66.39% $ 2000.00 $ 998.88 75.24% $ 999.70 72.3% $ 4000.00 $ 4000.02 77.98% $ 3999.0 75.42% $ 6000.00 $ 5999.56 80.23% $ 6000.65 78.92% $ 8000.00 $ 7999.98 83.56% $ 7999.33 8.25% $ 20000.00 $ 20000.40 85.2% $ 9999.93 83.% Table 4.6. Mantenance and Replacement Schedule Ftness Functon 2 Generatonal GA (Budget$5000 and Relablty54.07%) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - - - R - - R - - - R - - - - R - - - - M - - - R - - - - M - - - - - 2 - - - - R - - - - - - R - - - - R - - - - M - - - R - - - - M - - - - - 3 - - - - R - - R - - - R - - - - R - - - - M - - - R - - - - M - - - - - 4 - - - - R - - - - - - R - - - - R - - - - M - - - R - - - - M - - - - - 5 - - - - R - - - - - - R - - - - R - - - - M - - - R - - - - M - - - - - 6 - - - - R - - R - - - R - - - - R - - - - M - - - R - - - - M - - - - - 7 - - - - R - - R - - - R - - - - R - - - - M - - - R - - - - M - - - - - 8 - - - - R - - M - - - R - - - - R - - - - M - - - R - - - - M - - - - - 9 - - - - R - - - - - - R - - - - R - - - - M - - - R - - - - M - - - - - 0 - - - - R - - R - - - R - - - - R - - - - M - - - R - - - - M - - - - - Table 4.7. Mantenance and Replacement Schedule Ftness Functon 2 Steady State GA (Budget$5000 and Relablty5.88%) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - - - R - - R R - - R - M R - - - - R - - - - R - - - - - - - - - - - 2 - - - - R - - - R - - - - M R - - - - R - - - - R - - - - - - - - - - - 3 - - - - R - - R R - - M - M - - - - - R - - - - R - - - - - - - - - - - 4 - - - - R - - - R - - R - M R - - - - R - - - - R - - - - - - - - - - - 5 - - - - R - - R R - - R - M M - - - - R - - - - R - - - - - - - - - - - 6 - - - - R - - M R - - - - M R - - - - R - - - - R - - - - - - - - - - - 7 - - - - R - - R R - - M - M M - - - - R - - - - R - - - - - - - - - - - 8 - - - - R - - M R - - - - M M - - - - R - - - - R - - - - - - - - - - - 9 - - - - R - - R R - - M - M M - - - - R - - - - R - - - - - - - - - - - 0 - - - - R - - - R - - R - M R - - - - R - - - - R - - - - - - - - - - - 69

Fgure 4.4 shows Pareto optmal solutons (trade off curves) obtaned by generatonal and steady state GA for ftness functon 2. As t can be seen both Pareto solutons are relatvely smlar to each other. The cost and relablty mprovement n terms of number of generatons and genetc cycles n generatonal and steady state GA are shown n Fgures 4.5 and 4.6. As we can see the convergence of the steady state GA s lttle bt faster than the convergence of generatonal GA at the begnnng teratons. Fgure 4.4. Pareto Optmal Solutons for Ftness Functon 2 Fgure 4.5. Cost Improvement for Ftness Functon 2 70

Fgure 4.6. Relablty Improvement for Ftness Functon 2 4.5.3. Computatonal Results of Ftness Functon 3 Fnally Table 4.8 presents the non-nferor solutons of the model wth ftness functon 3 for dfferent requred relablty. Tables 4.9 and 4.0 show the optmal preventve and replacement schedule of ftness functon 3 wth 50% as the desred relablty. Wth ths level of requred relablty the total cost of the system s $409.02 and $525.48 resulted by generatonal and steady state GA respectvely. As t can be seen the structure of both schedules s same as the structure found usng prevous ftness functons. Table 4.8. Non-nferor solutons resulted from Ftness Functon 3 Generatonal GA Steady State GA Requred Relablty Cost Relablty Cost Relablty 0% $ 454.85 2.22% $ 454.85 2.22% 0% $ 908.70 9.82% $ 253.96 0.00% 20% $ 544.45 20.3% $ 843.4 9.88% 30% $ 97.9 30.02% $ 3470.56 29.95% 40% $ 334.55 39.94% $ 4407.27 39.98% 50% $ 409.02 50.00% $ 525.48 49.99% 60% $ 638.03 59.95% $ 7754.48 59.94% 70% $ 8956.37 70.04% $ 8903.02 70.02% 80% $ 4262.8 79.8% $ 4455.02 79.57% 90% $ 4286.09 80.25% $ 500.48 80.40% 00% $ 6076.4 8.53% $ 503.8 80.67% 7

Table 4.9. Mantenance and Replacement Schedule Ftness Functon 3 Generatonal GA (Relablty50% and Cost$409.02) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 2 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 3 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 4 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 5 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 6 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 7 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 8 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 9 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - 0 - - - - R - - - - R - - - - R - - - M - - M - - - R - - - - - - - - - - Table 4.0. Mantenance and Replacement Schedule Ftness Functon 3 Steady State GA (Relablty50% and Cost$525.48) Perod 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 33 34 35 36 Component - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 2 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 3 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 4 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 5 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 6 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 7 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 8 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 9 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - 0 - - R - - R - M - - M - M - R - - - - R - - M M - - - M - - - - - - - - Fgure 4.7 shows Pareto optmal solutons (trade off curves) obtaned by generatonal and steady state GA for ftness functon 3. In ths case the Pareto solutons do not exactly correspond to each other. Fgures 4.8 and 4.9 show the cost and relablty mprovement n both GAs. In ths case the convergence of both algorthms s same but generatonal GA reduces cost better than steady state GA. A comparson between trade off curves from the three ftness functons usng the two algorthms s presented n Fgure 4.0. We can conclude that ftness functon and ftness functon 3 wth generatonal GA gve better non-nferor solutons when compared wth ftness functon 2 and ftness functon 3 wth the steady state GA. 72

Fgure 4.7. Pareto Optmal Solutons for Ftness Functon 3 Fgure 4.8. Cost Improvement for Ftness Functon 3 Fgure 4.9. Relablty Improvement for Ftness Functon 3 73

Fgure 4.0. Pareto Optmal Solutons for all Ftness Functons 4.6. Chapter Summary In ths chapter the extenson of optmzaton models formulated n Chapter 3 was presented by consderng engneerng economy aspects. The new model was formulated as a mult-obectve optmzaton model. Generatonal and steady state genetc algorthms were used to optmze the model and several crossover and mutaton procedures were developed based on the specal structure of the model. In addton three dfferent ftness functons were defned and utlzed to acheve the best non-nferor solutons (Pareto optmal solutons). By analyzng the computatonal results of each algorthm wth each ftness functon we could show the effcency and effectveness of algorthms and ftness functons. Fnally the convergence of algorthms n terms of cost and relablty mprovement was demonstrated and examned. 74

Chapter 5 Research Plan As presented n Chapter the remander of the Ph.D. dssertaton wll progress as follows. Implementaton of mult-obectve smulated annealng n order to solve the mult-obectve optmzaton model and compare the computatonal results wth genetc algorthms and dynamc programmng results s the next step. Ths step completes Chapter 4 optmzaton models-heurstc algorthms of ths dssertaton proposal and t wll be fnshed on December 2008. Analytcal and statstcal models wll be constructed n order to estmate the age reducton and mprovement factor parameters of the optmzaton models. Ths step wll be Chapter 5 of the dssertaton and t wll be fnshed at the end of the summer 2009. The fnal stage ncludes examnng the applcaton of the developed models n a real case study. A comparson between current polces and optmal preventve mantenance and replacement schedules wll be presented and feasblty and effectveness of the models wll be nvestgated. If the related data of that case study s avalable n a proper tme the dssertaton can be completed at the end of 2009.

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