Impacts of weibull parameters estimation on preventive maintenance cost

Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Available online at www.sciencedirect.com Information Control Problems in Manufacturing Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Bergamo, Italy, June 11-13, 2018 Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Manufacturing Information Control in Bergamo, Italy, JuneProblems 11-13, 2018 Bergamo, Italy, Italy, June June 11-13, 11-13, 2018 2018 Bergamo,

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IFAC PapersOnLine 51-11 (2018) 508–513

Impacts of Impacts of weibull weibull parameters parameters estimation estimation on on preventive preventive maintenance maintenance cost cost Impacts Impacts of of weibull weibull parameters parameters estimation estimation on on preventive preventive maintenance maintenance cost cost Impacts of weibull parameters estimation onFlorian*, preventive maintenance cost Fabio Sgarbossa*, Ilenia Zennaro*, Eleonora Alessandro Persona*

Fabio Sgarbossa*, Ilenia Zennaro*, Eleonora Florian*, Alessandro Persona* Fabio Florian*,  Fabio Sgarbossa*, Sgarbossa*, Ilenia Ilenia Zennaro*, Zennaro*, Eleonora Eleonora Florian*, Alessandro Alessandro Persona* Persona*  Fabio*Department Sgarbossa*, of Ilenia Zennaro*, Eleonora Florian*, Alessandro Persona*  Management andEleonora Engineering, University of Padova, Fabio Sgarbossa*, Ilenia Zennaro*, Florian*, Alessandro Persona*  *Department of Management and Engineering, University of Padova,  *Department of Engineering, University of Vicenza, Italy (e-mail: [email protected]; [email protected]; [email protected]; *Department of Management Management and and Engineering, University of Padova, Padova, Vicenza, Italy (e-mail: [email protected]; [email protected]; [email protected]; *Department of Management and Engineering, University of Vicenza, Italy (e-mail: [email protected]; [email protected]; [email protected]; *Department of Management and Engineering, University of Padova, Padova, [email protected]) Vicenza, Italy (e-mail: [email protected]; [email protected]; [email protected]; [email protected]) Vicenza, Italy Italy (e-mail: (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]) Vicenza, [email protected]; [email protected]) [email protected]) [email protected]) Abstract: To maintain a production or logistic system in a state as it could perform its output is the core Abstract: To maintain a production or logistic system in a state as it could perform its output is the core Abstract: maintain aa production or system in as is the maintenance goal. Maintenance represents a cost for industrial realities but perform it is alsoits strategic function Abstract: To To goal. maintain production or logistic logistic system in aa state state realities as it it could could perform itsaa output output is function the core core maintenance Maintenance represents a cost for industrial but it is also strategic Abstract: To To maintain production or logistic logistic system in aa state state realities asPolicy it could could perform itsa output output is function the core core maintenance goal. Maintenance aa cost industrial but it is also strategic Abstract: maintain aa production or system in as it perform its is the necessary to be competitive in therepresents market. The Agefor Replacement is the most diffused maintenance maintenance goal. Maintenance represents cost for industrial realities but it is also a strategic function necessary to be competitive in the market. The Age Policy is diffused maintenance maintenance goal. Maintenance represents areplacement cost forReplacement industrial realities butthe it most is also also a strategic strategic function necessary to be competitive in the market. The Age Replacement Policy is the most diffused maintenance maintenance goal. Maintenance represents a cost for industrial realities but it is a function strategy and it is characterized by periodic of components to avoid failures. Data as MTTF necessaryand to be competitive in the market. The Age Replacement Policy to is the most diffused maintenance strategy it is characterized by periodic replacement of components avoid failures. Data as MTTF necessary be competitive in market. The Age Replacement is most diffused maintenance strategy it is characterized by periodic replacement of components to avoid failures. Data as necessary toestimate be competitive in the the market. The Agerelative Replacement Policy is the the most diffused maintenance are used and toto replacement intervals and the costs ofPolicy maintenance. strategy and it is characterized by periodic replacement of components to avoid failures. Data as MTTF MTTF are used to estimate replacement intervals and the relative costs of maintenance. strategy and it is characterized by periodic replacement of components to avoid failures. Data as MTTF are used to estimate replacement the relative of maintenance. strategy and it is characterized periodicand replacement ofcosts components to avoid failures. Data MTTF Weibull distribution is the mostbyintervals diffused model to fit spare parts lifetime, and so to define theasoptimal are used to estimate replacement intervals and the relative costs of maintenance. Weibull distribution is the most diffused model to fit spare parts lifetime, and so to define the optimal are used to estimate replacement intervals and the relative costs of maintenance. Weibull distribution is the most diffused model to fit spare parts lifetime, and so to define the optimal are used to estimate replacement intervals and the relative costs of maintenance. periodic interval of replacement; in case of censored data, Weibull parameters could be wrong, and be Weibull distribution is the most diffused model to fit data, spareWeibull parts lifetime, and so to define the optimal periodic interval of replacement; in case of censored parameters could be wrong, and be Weibull distribution is the diffused model to fit spare parts lifetime, so to the periodic interval of replacement; in case of censored Weibull parameters could be wrong, and Weibull distribution is % theofmost most diffused model to study fit data, spare parts lifetime, and so to define define the optimal optimal influenced by a certain error. The aim of this is to investigate theand relationship between the be % periodic interval of replacement; in case of censored data, Weibull parameters could be wrong, and be influenced by aa certain % of error. aim this study is to investigate the relationship the % periodic interval ofestimation replacement; inThe case of of censored data, Weibull parameters coulddecision be between wrong, and be influenced by certain % of error. The aim of this study is to the relationship between the % periodic interval of replacement; in case of censored data, Weibull parameters could be wrong, and be error in parameter and the additional costs related toinvestigate this error. Economical graph and influenced by a certain % of error. The aim of this study is to investigate the relationship between the % error in parameter estimation and the additional costs related to this error. Economical decision graph and influenced by a certain % of error. The aim of this study is to investigate the relationship between the % error in parameter estimation and the additional costs related to this error. Economical decision graph and influenced by a certain % of error. The aim of this study is to investigate the relationship between the % maps have been considered toand support the study.costs Finally, considerations about consequences on costsand in error in parameter estimation the additional related to this error. Economical decision graph maps have been considered to support the study. Finally, considerations about consequences on costs in error in parameter estimation and the additional costs related to this error. Economical decision graph and maps have been considered to support the study. Finally, considerations about consequences on costs error in parameter estimation and the additional costs related to this error. Economical decision graph and case ofhave overestimation or underestimation are carried out. considerations about consequences on costs in maps been considered to support the study. Finally, in case of overestimation or underestimation carried out. maps have been considered considered to support support the theare study. Finally, considerations about about consequences consequences on on costs costs in in case of overestimation or underestimation are carried out. maps have been to study. Finally, considerations case of overestimation or underestimation are carried out. Keywords: Preventive Maintenance; MTTF; Censored data; Maintenance Costs Ltd. All rights reserved. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier case of overestimation or underestimation are carried out. Keywords: Preventive Maintenance; MTTF; Censored data; Maintenance Costs case of overestimation or underestimation are carried out. Keywords: Keywords: Preventive Preventive Maintenance; Maintenance; MTTF; MTTF; Censored Censored data; data; Maintenance Maintenance Costs Costs  Keywords: Preventive Preventive Maintenance; Maintenance; MTTF; MTTF; Censored Censored data; Maintenance Maintenance Costs Costs Keywords: data;  1. INTRODUCTION situation as there could be lack in data collection or new  1. INTRODUCTION situation as there could be lack in data collection or new  1. situation there could be data or productionas systems; in literature, works investigate 1. INTRODUCTION INTRODUCTION situation as there could be lack lack in inmany data collection collection or new new production systems; in literature, many works investigate 1. INTRODUCTION INTRODUCTION situation as systems; thereofcould could bemodels lack in in data collection or new production in literature, works investigate 1. situation as there be lack data or new suitability failure inmany thesecollection cases and how to Maintenance of production and logistic systems is a strategic about production systems; in literature, many works investigate about suitability of failure models in these cases and how to Maintenance of production and logistic systems is a strategic production systems; in literature, many works investigate about suitability of failure models in these cases and how to production systems; in literature, many works investigate Maintenance of production and logistic systems is a strategic reduce bias in parameters estimation [Zhang et al., 2006]. function to achieve industrial goals and gain success; it also about suitability of failure models in these cases and how to Maintenance of production andgoals logistic systems is a strategic reduce bias in parameters estimation [Zhang et al., 2006]. function to achieve industrial and gain success; it also about suitability of failure models in these cases and how to reduce bias in parameters estimation [Zhang et al., 2006]. Maintenance of production and logistic systems is a strategic about suitability of failure models in these cases and how to function to achieve industrial goals and gain success; it also Anyway, consequences of errors of parameters estimation Maintenance ofrelevant production logistic systems is represents aachieve costand and toand establish thea strategic optimal reduce bias in parameters estimation [Zhang et al., 2006]. function to industrial goals gain success; it also Anyway, consequences of errors of parameters estimation represents aaachieve relevant cost and to establish the optimal reduce bias in parameters estimation [Zhang et al., 2006]. Anyway, consequences of errors of parameters estimation function to achieve industrial goals and gain success; it also reduce bias in parameters estimation [Zhang et al., 2006]. represents relevant cost and to establish the optimal derived from censored data on maintenance costs are not function to industrial goals and gain success; it also maintenancea interval is aand critical to be the competitive consequences of errors of parameters estimation represents relevanttime cost to point establish optimal Anyway, derived from censored data on maintenance costs are not maintenance time is aaand critical point to be competitive Anyway, consequences of of estimation derived censored data on maintenance are represents relevant cost and to establish the optimal Anyway, consequences of errors errors ofanparameters parameters estimation maintenance interval time is critical point to be competitive In from particular, censored data has impact costs on preventive represents aa interval relevant cost to establish the optimal in the market. To define the ideal maintenance policies and clear. derived from censored data on maintenance costs are not not maintenance interval time is a critical point to be competitive clear. In particular, censored data has an impact on preventive in the market. To define the ideal maintenance policies and derived from censored data on maintenance costs are clear. In particular, censored data impact on preventive maintenance interval time is aaideal critical pointisto toabe bevery competitive derived from censored data onsohas maintenance are not not in the market. To the maintenance policies and maintenance interval time and onan costs. Tocosts quantify the maintenance interval is critical point competitive strategies mix of define a time production system critical clear. In particular, censored data has an impact on preventive in the market. To define the ideal maintenance policies and maintenance interval time and and sohas onan costs. Toon quantify the strategies mix of aa production system is aa very critical clear. In censored data impact preventive maintenance interval time on costs. To quantify the in the market. To define the ideal maintenance policies and clear. In particular, particular, censored dataso has an impact on preventive strategies mix of production system is very critical additional cost of incorrect estimation needs to be investigate. in the market. To define the ideal maintenance policies and challenge of industrial realities as it has direct consequences maintenance interval time and so on costs. To quantify the strategies mix of a production system is a consequences very critical additional cost of incorrect estimation needs to be investigate. challenge of industrial realities as it has direct maintenance interval time and so on costs. To quantify the additional cost of incorrect estimation needs to be investigate. strategies mix of a production system is a very critical maintenance interval time and so on costs. To quantify the challenge of industrial realities as it has direct consequences strategies mix of a production system is a very critical on production efficiency and economic performance additional cost of incorrect estimation needs to be investigate. In this study we focused on consequences of wrong challenge of industrial realitiesand as it economic has direct consequences on production efficiency performance additional cost of incorrect estimation needs to be investigate. In this study we focused on consequences of wrong challenge of industrial realities as it has direct consequences additional cost of incorrect estimation needs to be investigate. on production efficiency and performance In this we on wrong challenge ofetindustrial realities asnumber it economic has direct consequences [Regattieri al.(2010)]. A great of models has been parameters estimation in terms of additional of of on production efficiency and economic performance In this study study we focused focused on consequences consequences ofcosts wrong [Regattieri et al.(2010)]. A great number of models has been parameters estimation in terms of additional costs of on production efficiency and economic performance In this study we focused on consequences of wrong [Regattieri et al.(2010)]. A great number of models has been parameters estimation in terms of additional costs of on production efficiency and economic performance In this study we focused on consequences of wrong carried out from RAMS Analysis (Reliability, Availability, preventive maintenance (TYPE I) in case of censored data. It [Regattieri etfrom al.(2010)]. AAnalysis great number of models has been parameters estimation (TYPE in terms of additional costs of carried out RAMS (Reliability, Availability, preventive maintenance I) in case of censored data. It [Regattieri et al.(2010)]. A great number of models has been parameters estimation in terms of additional costs of carried out from RAMS Analysis (Reliability, Availability, preventive maintenance (TYPE I) in case of censored data. It [Regattieri etfrom al.(2010)]. AAnalysis great number of models been parameters estimation in terms of additional costs of Maintainability and Safety) to define optimal mixAvailability, ofhas policies is investigated the additional % I) cost when MTTF estimation, carried out RAMS (Reliability, preventive maintenance (TYPE in case of censored data. It Maintainability and Safety) to define optimal mix of policies is investigated the additional % cost when MTTF estimation, carried out from RAMS Analysis (Reliability, Availability, preventive maintenance (TYPE I) in case of censored data. It Maintainability and Safety) to define optimal mix of policies is investigated the additional % cost when MTTF estimation, carried out from RAMS Analysis (Reliability, Availability, preventive maintenance (TYPE I) in case of censored data. It and strategies to improve performance andmix efficiency of is or investigated the shape parameter (β) one, has a when relative % error. Results Maintainability and Safety) to define optimal of policies the additional % cost MTTF estimation, and strategies to improve performance and efficiency of or the shape parameter (β) one, has aa when relative % error. Results Maintainability and Safety) to define optimal mix of policies is investigated the additional % cost when MTTF estimation, and strategies to improve performance and efficiency of or the shape parameter (β) one, has relative % error. Results Maintainability and Safety) to define optimal mix of policies is investigated the additional % cost MTTF estimation, production systems. arethecarried out from(β)parameters analysis takenResults from and strategies to improve performance and efficiency of or shape parameter one, has a relative % error. production systems. out from analysis taken from and strategies to improve improve performance performance and and efficiency efficiency of of are or thecarried shape parameter (β)parameters one, has aa relative relative % error. error. Results production systems. are carried out from parameters analysis taken from and strategies to the shape parameter (β) one, has % Results analytical standard models. Economical decision graph and production systems. Age replacement policy (preventive maintenance) is the most or are carried out from parameters analysis taken from analytical standard models. Economical decision graph and production systems. Age replacement policy (preventive maintenance) is the most are carried outconsidered from parameters analysis taken from analytical standard models. Economical decision graph and production systems. are carried out from parameters analysis taken from Age replacement policy (preventive maintenance) is the most maps have been to support the study. diffused industrialpolicy policy to achieve failures decrease and analytical standard models. Economical decision graph and Age replacement (preventive maintenance) is the most maps have been considered to support the study. diffused industrial policy to achieve failures decrease and analytical standard models. Economical decision graph and maps have been considered to support the study. Age replacement policy (preventive maintenance) is the most analytical standard models. Economical decision graph and diffused industrial policy to achieve failures decrease and Age replacement policy (preventive maintenance) is the most production performance improvement through periodic maps have been considered to support the study. diffused industrial policy to achieve failures decrease and The paper is divided into four sections: as first it is presented production performance improvement through periodic maps have been considered to support the study. diffused industrial policy to achieve failures decrease and maps have been considered to support the study. production performance improvement through periodic diffused industrial policy to achieve failures decrease and The paper is divided into four sections: as first it is presented replacementsperformance defined by fixed interval ofthrough time or periodic worked The paper four sections: as presented production improvement the literature review into about failure replacements defined by fixed interval of time or periodic worked The paper is is divided divided into fourWeibull sections:distribution as first first it it is isfor presented production performance improvement through periodic replacements defined by fixed of time worked production improvement through the literature review about Weibull distribution for failure hours (TYPEperformance I); the key point isinterval the interval time or definition, The paper is divided into four sections: as first it is presented the literature review about Weibull distribution for replacements defined by fixed interval of time or worked The paper is divided into four sections: as first it is presented events and statistical tools Weibull to estimate MTTF and βfailure with hours (TYPE I); the key point is the interval time definition, the literature review about distribution for failure replacements defined by fixed interval of time or worked hours (TYPE I); the key point is the interval time definition, replacements defined by fixed interval of time or worked events and statistical tools to estimate MTTF and β with directly related to failures distribution and to maintenance the literature review about Weibull distribution for failure events and statistical tools to estimate MTTF with hours (TYPE I); to thefailures key point is the interval time definition, the literature review about for failure censored data. It emerges the Weibull lack of a distribution study aboutand the β impact directly related distribution and to maintenance events and statistical tools to estimate MTTF and β with hours (TYPE I); the key point is the interval time definition, directly related to failures distribution and to maintenance hours (TYPE I); the key point is the interval time definition, censored data. It emerges the lack of a study about the impact costs. events and statistical tools to estimate MTTF and β with censored data. It emerges the lack of a study about the impact directly related to failures distribution and to maintenance events and statistical tools to estimate MTTF and β with on costs of a wrong estimation. In the second part it is costs. censored data. It emerges the lack of a study about the impact directly related to failures distribution and to maintenance costs. directly related to failures distribution and to maintenance on costs of a wrong estimation. In the second part it is Weibull distribution is one of the most diffused probability censored data. It emerges the lack of a study about the impact on costs of a wrong estimation. In the second part it is costs. censored data. It emerges the lack of a study about the impact presented the problem and the mathematical models used Weibull distribution is one of the most diffused probability on costs of aproblem wrong and estimation. In the second part it to is costs. Weibull distribution is one of the most diffused probability costs. presented the the mathematical models used to function used to describe systems lifetime and failure events on costs of a wrong estimation. In the second part it is presented the problem and the mathematical models used to Weibull distribution is one of the lifetime most diffused probability on costs of a wrong estimation. In the second part it is define maintenance parameters. Then, in the third section, it function used to describe systems and failure events presented the problem and the mathematical models used to Weibull distribution is one of the most diffused probability function used to describe systems lifetime and failure events Weibull is one of the lifetime most diffused probability define maintenance parameters. Then, in the third section, it [Lawless,distribution 1982; Hallinan, 1993]; various methodologies to is presented the problem and the mathematical models used to define maintenance parameters. Then, in the third section, it function used to describe systems and failure events presented thethe problem and cost the mathematical models used to calculated additional related to a wrong estimation [Lawless, 1982; Hallinan, 1993]; various methodologies to define maintenance parameters. Then, in the third section, it function used to describe describe systems lifetime and failure events [Lawless, 1982; Hallinan, 1993]; various methodologies to function used to systems failure is calculated the additional cost related to aa wrong estimation estimate Weibull parameters fromlifetime failuresand data haveevents been define maintenance parameters. Then, in the third section, it is calculated the additional cost related to wrong estimation [Lawless, 1982; Hallinan, 1993]; various methodologies to define maintenance parameters. Then, in the third section, it and its trend varying maintenance model parameters (cost estimate Weibull parameters from failures data have been is calculated the additional cost related to a wrong estimation [Lawless, 1982; Hallinan, 1993]; various methodologies to and estimate Weibull parameters from failures data have been [Lawless, 1982; Hallinan, 1993]; various methodologies to its trend varying maintenance model parameters (cost carried out, like MLE (Maximum Likelihood Estimation) and is calculated the additional cost related to a wrong estimation and its trend varying maintenance model parameters (cost estimate Weibull parameters from failures data have been is calculated the additional cost related to a wrong estimation Finally,maintenance considerations aboutparameters the value added carried out, like MLE (Maximum Likelihood Estimation) and and reliability). its trend varying model (cost estimate Weibull parameters from failures data haveallows been carried out, like MLE (Maximum Likelihood Estimation) and estimate Weibull parameters from failures data have been reliability). Finally, considerations about the value added LSE (Least Square Estimation). Weibull distribution its trend trend varying maintenance model parameters (cost reliability). Finally, considerations about the value added carried out, like MLE (Maximum Likelihood Estimation) and and and its varying maintenance model parameters (cost of results and the future research are presented. LSE (Least Square Estimation). Weibull distribution allows and reliability). Finally, considerations about the value added carried out,when like MLE (Maximum Likelihood Estimation) and LSE (Least Square Estimation). Weibull distribution allows carried out, like MLE (Maximum Likelihood and and the future research are presented. to define preventive maintenance is Estimation) suitable and to of andresults reliability). Finally, considerations about the the value value added added of results and the future research are presented. LSE (Least Square Estimation). Weibull distribution allows and reliability). Finally, considerations about to define when preventive maintenance is suitable and to of results and the future research are presented. LSE (Least Square Estimation). Weibull allows to when preventive maintenance is suitable and to LSE (Least Square Estimation). Weibull distribution allows 2. LITERATURE REVIEW estimate the optimal maintenance interval,distribution so its relative research to define define when preventive maintenance isand suitable and to of of results results and and the the future future research are are presented. presented. 2. LITERATURE REVIEW estimate the optimal maintenance interval, and so its relative to define when preventive maintenance is suitable and to 2. LITERATURE REVIEW estimate the optimal maintenance interval, and so its relative to define when preventive maintenance is suitable and to costs. 2. LITERATURE REVIEW estimate the optimal maintenance interval, and so its relative costs. 2. LITERATURE REVIEW estimate the optimal maintenance interval, and so its relative 2. LITERATURE REVIEW costs. estimate the optimal maintenance interval, and so its relative Weibull distribution is one of the most diffused model used Availability of correct and complete data is a common costs. Weibull distribution is one of the most diffused model used Availability of correct and complete data is a common costs. Weibull distribution is one of the most diffused model used Availability of correct and complete data is a common costs. to describe failure time in ofreliability analysis complex assumption inofmaintenance models analysis butis often it does Weibull distribution is one the most diffused of model used Availability correct and complete data a common to describe failure time in reliability analysis of complex assumption in maintenance models analysis but often it does Weibull distribution is one one ofreliability the mostdescribes diffused the model used Availability of correct and complete data is a common to describe failure time in analysis of complex Weibull distribution is of the most diffused model used assumption in maintenance models analysis but often it does Availability of correct and complete data is a common systems [Hossain et al., 2003]. It well different not reflect in reality [Ross, models 1996; Montanari etoften al., it1997]. to describe failure time in reliability analysis of complex assumption maintenance analysis but does systems [Hossain et al., 2003]. It well describes the different not reflect reality [Ross, 1996; Montanari et al., 1997]. to describe failure time in reliability analysis of complex assumption in maintenance models analysis but often it does systems [Hossain et al., 2003]. It well describes the different to describe failure time in reliability analysis of complex not reflect reality [Ross, 1996; Montanari et al., 1997]. assumption in maintenance models analysis but often it does phases of lifetime of components, characterized by the Censored interval data 1996; and incomplete [Hossain et al., 2003]. It well describes the different not reflectdata, reality [Ross, Montanaridata et are al.,diffused 1997]. systems of lifetime of components, characterized by the Censored interval data and incomplete systems [Hossain et al., 2003]. It well describes the different not reflectdata, reality [Ross, 1996; Montanaridata et are al.,diffused 1997]. phases phases of lifetime of components, characterized by systems [Hossain et al., 2003]. It well describes the different Censored data, interval data and incomplete data are diffused not reflect reality [Ross, 1996; Montanari et al., 1997]. of lifetime of components, characterized by the the Censored data, interval data and incomplete data are diffused phases phases of lifetime of components, characterized by Censored data, interval data and incomplete data are diffused phases of lifetime of components, characterized by the the Censored©data, data and incomplete are diffused 2405-8963 2018,interval IFAC (International Federation data of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

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Fabio Sgarbossa et al. / IFAC PapersOnLine 51-11 (2018) 508–513

bathtub curve (initial, random and wear-out failure modes) [Hisada at al., 2002]; in particular the shape parameter β indicates when the component is in the setting phase (β <1), in a random failures phase (β =1) or in the degradation phase (β >1). Many authors describe suitability of Weibull distribution for reliability models and discuss about the optimal methods to estimate its parameters. Murthy et al. presented a study of various models derived from two-parameter Weibull distribution, which aims to describe complex failure data set using WPP (Weibull Probability Plot). Pham et al. described industrial cases about different failure models based on this distribution. Methods to estimate scape and scale parameters like MLE and LSE focusing on the suitability of final models are deeply discussed in literature. In their work Genschel et al. discussed about different methods of estimating weibull distribution parameters like maximum likelihood and rank regression; Yavuz compared various regression methods for estimating Weibull shape parameter using a performance estimator based on bias and mean square error criteria in Monte-Carlo simulations. Xie et al., used a graphical estimation technique based on Weibull plot to evaluate parameters of a modified weibull distribution. In particular knowing that weibull distribution has been widely used for modeling different phases of lifetime, it is investigated an additive model as the failure rate function is expressed as the sum of two failure rate functions of weibull form. Ling et al., presented a model based on Kolmogorov-Smirnov distance for parameter evaluation. Complete and correct data are often not available as systems could be new installation or there could be lack in data collection. Regattieri et al. presented a framework about robustness of reliability estimation considering the critical role of censored data. The uncertainty and lack of information is a common situation that needs to be considered; Red-Horse et al. used polynomial chaos expansion to investigate the correlation between failure model output and the limited data as input. Coit et al. also studied suitability of Weibull distribution in case of censored data; they develop a framework to test the suitability of the exponential distribution for grouped data with censoring. Zhang, instead, focused the analysis on LSE method, proposing a bias correction method linked to the sample size and the censoring level; he demonstrated that bias correction is more affected by the censoring level then the sample size. Economou et al., developed a model for reliability performance study of underground water pipes considering data with right censoring. In their study, Montanari et al., compared various methods for estimating the parameters of the two-parameters weibull distribution from uncensored data. Finally Yang at al. proposed an analysis of biasness of MLE of Weibull parameters; it is demonstrated that bias increases as the degree of censorship increases and as more population is involved; a new method called MMLE is proposed. Many application have been discussed about failure models; for example, Guo et al, introduced censored data in the analysis applying two parameter weibull model to a fleet wind turbines.

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It is evident that Weibull distribution models for failures events and bias correction of its parameter estimation have been deeply studied and investigated; anyway, there is a lack about effects of parameters estimation errors (like MTTF or β) on preventive maintenance costs. These costs are defined by spare parts and resources. Preventive Maintenance policy aims to reduce failures through periodic spare parts replacements related to failure rate function; errors of the interval time evaluation have an impact on maintenance costs that needs to be investigated. The purpose of this work is to analyze the additional maintenance cost related to different levels of % error in parameter estimation; graph, plots and economical maps are carried out to support strategical decisions. 3. AGE REPLACEMENT POLICY TYPE I (ARP-I) AND FAILURE DATA 3.1 ARP-I based on Weibull parameters The first formulation of the ARP-I problem was developed by Barlow and Hunter (1960). They introduced a simple mathematical model based on the reliability function 𝑅𝑅(𝑡𝑡) of the component and the average cost of preventive action, called 𝐶𝐶𝑝𝑝 , and the average cost of intervention after a failure, called 𝐶𝐶𝑓𝑓 . Under this policy (ARP-I), the component is replaced at a certain time 𝑡𝑡𝑝𝑝 or after a failure, whichever occurs first. The replacement is made with a new component, so under the assumption “as good as new”. Thus, the Unit Expected Cost varying the time to replacement 𝑡𝑡𝑝𝑝 is expressed by:

𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝 ) =

𝐶𝐶𝑝𝑝 𝑅𝑅(𝑡𝑡𝑝𝑝 )+𝐶𝐶𝑓𝑓 [1−𝑅𝑅(𝑡𝑡𝑝𝑝 )] 𝑡𝑡𝑝𝑝

∫0 𝑅𝑅(𝑠𝑠)𝑑𝑑𝑑𝑑

(1)

where 𝑅𝑅(𝑡𝑡𝑝𝑝 ) follows the Weibull distribution, with 𝛽𝛽 as the shape parameter and 𝜃𝜃 as the scale one, as described by: 𝛽𝛽 𝑡𝑡 𝑅𝑅(𝑡𝑡𝑝𝑝 ) = 𝑒𝑒𝑒𝑒𝑒𝑒 [− ( 𝑝𝑝⁄𝜃𝜃) ]

(2)

The optimal value of 𝑡𝑡𝑝𝑝 , which minimize the 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝 ), indicated with 𝑡𝑡𝑝𝑝∗ , cannot be found with a closed form equation. For this reason, the authors in a previous paper (Faccio et al., 2014), developed an easy-to-use abacus to calculate the correct value of 𝑡𝑡𝑝𝑝∗ and the related 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ). As described in this paper, the maintenance engineers start with the estimation of the two cost parameters 𝐶𝐶𝑓𝑓 and 𝐶𝐶𝑝𝑝 using typical accounting methodology. Then, from the time to failure available related to the component under analysis, the Weibull parameters are assessed using one of the method already developed and discussed in the literature review section. Once all these parameters are estimated, the practitioners can use the figures 1 and 2, adapted from Faccio et al. (2014), in order to estimate the ratio 𝑡𝑡𝑝𝑝∗ ⁄𝜃𝜃 and the [𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) ∙ 𝜃𝜃]⁄𝐶𝐶𝑓𝑓 based on the shape parameter 𝛽𝛽 and cost ratio 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 . 516

IFAC INCOM 2018 510 Bergamo, Italy, June 11-13, 2018

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Fig. 1. Curves to assess the optimal time to replace.

Fig. 3. Hazard function 𝜆𝜆(𝑡𝑡), fixing 𝜃𝜃 = 1000 unit time.

Fig. 2. Abacus for the estimation of the 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ).

Fig. 4. 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ⁄𝜃𝜃 as function of Weibull shape parameter.

3.2 ARP-I when ttf set is not available

̂ ⁄𝜃𝜃̂ . This graph has been developed by the authors 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 based on the following equation:

Since the correct time to replace and minimal cost depend on the reliability parameters and often the data necessary to estimate them are not available or accurate, it is mandatory to have an alternative method to assess the Weibull parameters.

𝛽𝛽 ∞ ∞ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = ∫0 𝑅𝑅(𝑠𝑠)𝑑𝑑𝑑𝑑 = ∫0 𝑒𝑒𝑒𝑒𝑒𝑒 [−(𝑠𝑠⁄𝜃𝜃) ] 𝑑𝑑𝑑𝑑

In fact, as shown in previous section, the shape factor 𝛽𝛽 has significant impact on the optimal ARP-I solution, while the parameter 𝜃𝜃 is just a scale factor of the problem.

(3)

̂ and the Based on this set of reliability factors 𝛽𝛽̂ , 𝜃𝜃̂, 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 cost parameters 𝐶𝐶𝑓𝑓 and 𝐶𝐶𝑝𝑝 , the optimal time to replacement 𝑡𝑡̂𝑝𝑝∗ is estimated using the previous graphs (Fig. 1 and 2).

Noted the assumption that the ARP-I is applicable only if there is wearing condition, 𝛽𝛽 has to be higher than 1. Its value can be estimate by analyzing the hazard function 𝜆𝜆(𝑡𝑡) which expressed the failure rate of the component. Figure 3 shows different 𝜆𝜆(𝑡𝑡) varying the value of 𝛽𝛽. In this case, the maintenance engineers can estimate the shape parameter assuming the best curve for the component, based on their own experiences. We call it 𝛽𝛽̂ in order to make it different from real value of 𝛽𝛽.

4. ECONOMIC IMPACT OF ESTIMATION OF WEIBULL PARAMETERS The main objective of this paper is to evaluate the impact of the wrong estimation of Weibull parameters on the Unit Expected Cost. Thus, the following equation permits to calculate 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡), when the time to replacement is found ̂ , but the component has a real reliability using 𝛽𝛽̂ , 𝜃𝜃̂, 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 function described by 𝛽𝛽, 𝜃𝜃 and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:

The same can be done for the estimation of 𝜃𝜃 parameter and the Mean Time To Failure (𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀). Once the 𝛽𝛽̂ is defined, it can be use the figure 4 to assess the ratio 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ⁄𝜃𝜃 , called

𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡̂𝑝𝑝∗ ) =

517

∗ ̂∗ 𝐶𝐶𝑝𝑝 𝑅𝑅(𝑡𝑡̂ 𝑝𝑝 )+𝐶𝐶𝑓𝑓 [1−𝑅𝑅(𝑡𝑡𝑝𝑝 )] ∗ 𝑡𝑡̂ 𝑝𝑝

∫0 𝑅𝑅(𝑠𝑠)𝑑𝑑𝑑𝑑

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IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018

10

𝛽𝛽̂

1.4 3% 0% 0% 1% 1% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

1.6 8% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 1% 1% 1% 1% 1% 2% 2% 2%

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 14% 19% 23% 26% 29% 30% 32% 32% 33% 33% 33% 32% 31% 30% 29% 28% 27% 1% 2% 2% 2% 1% 1% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 6% 8% 10% 11% 13% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 6% 7% 9% 11% 13% 15% 17% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 6% 8% 9% 11% 13% 15% 17% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 7% 8% 10% 12% 14% 16% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 6% 7% 9% 10% 12% 14% 0% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 6% 7% 8% 10% 12% 0% 0% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 7% 8% 10% 0% 1% 1% 0% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 6% 8% 1% 1% 1% 1% 1% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 6% 1% 1% 1% 1% 1% 1% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 1% 2% 2% 2% 1% 1% 1% 0% 0% 0% 0% 0% 1% 1% 2% 3% 3% 2% 2% 2% 2% 2% 1% 1% 1% 0% 0% 0% 0% 0% 1% 1% 2% 2% 2% 3% 3% 3% 2% 2% 1% 1% 1% 0% 0% 0% 0% 0% 1% 1% 2% 2% 3% 4% 3% 3% 3% 2% 2% 1% 1% 0% 0% 0% 0% 0% 1% 1% 3% 4% 4% 4% 4% 3% 3% 2% 2% 1% 1% 0% 0% 0% 0% 0% 1% 3% 4% 5% 5% 4% 4% 3% 3% 2% 1% 1% 1% 0% 0% 0% 0% 0% 4% 5% 5% 5% 5% 5% 4% 3% 3% 2% 1% 1% 1% 0% 0% 0% 0% 4% 5% 6% 6% 6% 5% 5% 4% 3% 3% 2% 1% 1% 0% 0% 0% 0%

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

1.2 0% 1% 2% 1% 1% 1% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

1.4 2% 0% 0% 0% 0% 0% 0% 0% 1% 1% 1% 2% 2% 3% 3% 3% 4% 4% 5% 5%

1.6 4% 0% 0% 0% 0% 0% 1% 1% 2% 2% 3% 4% 5% 6% 6% 7% 8% 9% 9% 10%

1.8 6% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13%

2 6% 0% 0% 0% 0% 0% 1% 1% 2% 3% 4% 5% 7% 8% 9% 10% 12% 13% 14% 15%

2.2 5% 0% 1% 0% 0% 0% 0% 1% 1% 2% 3% 5% 6% 7% 9% 10% 12% 13% 15% 16%

2.4 4% 1% 2% 1% 1% 0% 0% 0% 1% 1% 2% 4% 5% 6% 8% 9% 11% 12% 14% 16%

2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 3% 2% 1% 0% 0% 0% 0% 1% 2% 3% 5% 6% 2% 3% 5% 8% 11% 14% 18% 21% 25% 29% 33% 37% 3% 6% 8% 12% 15% 19% 24% 28% 33% 37% 42% 46% 3% 5% 8% 11% 14% 18% 22% 26% 31% 35% 40% 44% 2% 3% 6% 8% 11% 15% 19% 23% 27% 31% 35% 39% 1% 2% 4% 6% 8% 11% 15% 18% 22% 25% 29% 33% 0% 1% 2% 4% 6% 8% 11% 14% 17% 20% 23% 27% 0% 0% 1% 2% 3% 5% 8% 10% 13% 16% 19% 22% 0% 0% 0% 1% 2% 3% 5% 7% 9% 12% 14% 17% 1% 0% 0% 0% 1% 2% 3% 5% 7% 9% 11% 13% 2% 1% 0% 0% 0% 1% 2% 3% 4% 6% 8% 10% 3% 1% 1% 0% 0% 0% 1% 2% 3% 4% 5% 7% 4% 3% 1% 1% 0% 0% 0% 1% 1% 2% 4% 5% 5% 4% 2% 1% 1% 0% 0% 0% 1% 1% 2% 3% 7% 5% 4% 2% 1% 1% 0% 0% 0% 1% 1% 2% 8% 6% 5% 3% 2% 1% 1% 0% 0% 0% 0% 1% 10% 8% 6% 5% 3% 2% 1% 1% 0% 0% 0% 0% 11% 10% 8% 6% 5% 3% 2% 1% 0% 0% 0% 0% 13% 11% 9% 8% 6% 4% 3% 2% 1% 0% 0% 0% 14% 13% 11% 9% 7% 6% 4% 3% 2% 1% 0% 0%

5 8% 41% 51% 48% 43% 37% 30% 25% 20% 16% 12% 9% 7% 5% 3% 2% 1% 0% 0% 0%

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

1.2 0% 1% 1% 0% 0% 0% 0% 0% 0% 1% 1% 1% 2% 2% 2% 3% 3% 3% 3% 4%

1.4 1% 0% 0% 0% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 11% 12% 13% 14%

1.6 2% 0% 0% 0% 1% 2% 3% 4% 6% 7% 9% 11% 12% 14% 16% 17% 18% 20% 21% 23%

1.8 1% 0% 0% 0% 0% 1% 2% 4% 6% 8% 10% 12% 14% 16% 19% 21% 23% 25% 26% 28%

2 0% 1% 1% 0% 0% 0% 1% 3% 5% 7% 9% 11% 14% 16% 19% 22% 24% 26% 29% 31%

2.2 0% 3% 3% 2% 0% 0% 0% 1% 3% 5% 7% 10% 12% 15% 18% 20% 23% 26% 28% 31%

2.4 0% 7% 7% 4% 2% 0% 0% 0% 1% 3% 5% 7% 10% 13% 15% 18% 21% 24% 27% 30%

2.6 2% 11% 11% 8% 4% 2% 0% 0% 0% 1% 3% 5% 7% 10% 12% 15% 18% 21% 24% 27%

5 62% 104% 104% 91% 76% 62% 51% 40% 32% 25% 19% 14% 10% 7% 5% 3% 2% 1% 0% 0%

𝛽𝛽

2.8 4% 17% 17% 12% 8% 4% 2% 0% 0% 0% 1% 3% 5% 7% 9% 12% 15% 18% 20% 24%

3 8% 24% 24% 18% 12% 8% 4% 2% 0% 0% 0% 1% 3% 4% 7% 9% 11% 14% 17% 20%

𝛽𝛽

3.2 12% 31% 31% 24% 18% 12% 7% 4% 2% 0% 0% 0% 1% 3% 4% 6% 8% 11% 13% 16%

10

̂ 𝛽𝛽

𝑡𝑡̂∗ 𝑅𝑅(𝑡𝑡̂𝑝𝑝∗ ) = 𝑒𝑒𝑒𝑒𝑒𝑒 [− ( 𝑝𝑝⁄̂) ] 𝜃𝜃 ∗ 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡̂ 𝑝𝑝 )

∗) 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝

-25%

20% 9% 0% 0% 0% 1% 2% 3% 3% 4% 5% 5% 6%

11% 0% 0% 3% 1% 0% 0% 0% 1% 1% 1% 1% 1% 2% 2% 2% 2% 2%

-50%

-25%

3% 1%

-10% 3% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

-5% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

-10% 2% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 1% 1% 1% 1% 1% 1% 1% 1%

-5% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

-10% 2% 0% 1% 1% 0% 0% 0% 0% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1%

-5% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

5% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

10% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

25% 6% 1% 1% 1% 0% 0% 0% 0% 1% 1% 1% 1% 2% 2% 2% 2% 2% 2% 2%

50% 14% 2% 3% 4% 0% 0% 1% 2% 4% 5% 5% 5% 5% 6% 6% 6% 6% 6% 6%

75% 21% 1% 7% 8% 0% 2% 4% 6% 9% 9% 10% 10% 11% 11% 11% 11% 11% 11% 11%

100% 26% 1% 12% 13% 1% 4% 8% 10% 14% 15% 16% 16% 16% 16% 16% 16% 16% 16% 16%

25% 3% 0% 2% 2% 0% 1% 1% 2% 2% 3% 3% 3% 3% 3% 3% 3% 3% 3% 3%

50% 6% 0% 8% 9% 2% 4% 6% 7% 9% 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

75% 6% 1% 17% 18% 6% 10% 13% 15% 18% 19% 19% 19% 18% 18% 18% 18% 17% 17% 17%

100% 4% 3% 27% 28% 12% 18% 23% 25% 28% 28% 28% 27% 27% 26% 26% 25% 25% 24% 24%

25% 1% 0% 4% 4% 1% 2% 3% 4% 5% 5% 5% 5% 5% 5% 5% 5% 5% 5% 5%

50% 1% 2% 15% 15% 7% 10% 12% 14% 16% 16% 16% 15% 15% 15% 15% 15% 14% 14% 14%

75% 0% 8% 30% 30% 17% 23% 27% 28% 30% 30% 29% 28% 27% 27% 26% 26% 25% 24% 24%

100% 0% 17% 45% 45% 31% 39% 43% 45% 44% 43% 42% 40% 39% 38% 37% 36% 35% 34% 33%

Percentage error of the estimation 20 1.2 1.4 2.4 2.6 1.6 1.8 2 2.2 2.8 3 3.2 3.6 3.8 4 4.2 4.4 4.6 4.8 5

𝛽𝛽̂

16% 4% 1% 2% 3% 6% 8% 9% 10% 12% 13% 14% 15%

9% 1% 1% 2% 0% 0% 0% 2% 2% 2% 3% 3% 4% 4% 4% 4% 4% 4%

-50%

-25%

1% 0%

0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

5% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

10% 1% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 1% 1% 1% 1% 1% 1% 1% 1%

Percentage error of the estimation 3.4 16% 39% 39% 31% 23% 16% 11% 6% 3% 1% 0% 0% 0% 1% 2% 4% 6% 8% 10% 13%

3.6 21% 47% 47% 39% 30% 21% 15% 10% 6% 3% 1% 0% 0% 0% 1% 2% 4% 5% 7% 10%

3.8 27% 55% 55% 46% 36% 27% 20% 13% 9% 5% 3% 1% 0% 0% 0% 1% 2% 3% 5% 7%

4 33% 64% 64% 54% 43% 33% 24% 17% 12% 8% 5% 3% 1% 0% 0% 0% 1% 2% 3% 5%

4.2 39% 72% 72% 61% 50% 39% 30% 22% 16% 11% 7% 4% 2% 1% 0% 0% 0% 1% 2% 3%

4.4 45% 80% 80% 69% 56% 45% 35% 26% 20% 14% 10% 6% 4% 2% 1% 0% 0% 0% 1% 2%

4.6 51% 88% 88% 76% 63% 51% 40% 31% 24% 18% 13% 9% 6% 4% 2% 1% 0% 0% 0% 1%

4.8 56% 96% 96% 83% 70% 56% 45% 36% 28% 21% 16% 11% 8% 5% 3% 2% 1% 0% 0% 0%

40

𝛽𝛽̂

1.2 1.4 2.4 2.6 1.6 1.8 2 2.2 2.8 3 3.2 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0% 1%

12% 1% 4% 6% 9% 14% 17% 19% 21% 23% 25% 27% 29%

6% 2% 3% 1% 0% 1% 1% 4% 4% 5% 6% 6% 7% 7% 7% 7% 7% 8%

0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

5% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

10% 1% 0% 1% 1% 0% 0% 0% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1%

Fig. 6. Coloured maps of additional cost varying the estimation error. Black cells are for values higher than 100%.

In this first part of analysis, we calculate ∆𝑈𝑈𝑈𝑈𝑈𝑈 for different values of 𝛽𝛽̂ and 𝛽𝛽 (from 1.2 to 5, every 0.2 unit) and varying the cost ratio 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 . (10, 20 and 40).

In Figure 5, it can be seen that the impact on costs when 𝛽𝛽 is overestimate (𝛽𝛽̂ >𝛽𝛽) is lower than in the case of underestimation. In fact, in case of overestimation, the maximum error varying from a max of +6% in the case of 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 = 10 and +31% in the case 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 = 40. On the other side, in case of underestimation (𝛽𝛽̂ <𝛽𝛽), the ∆𝑈𝑈𝑈𝑈𝑈𝑈 varying from a maximum of +33% in the case of 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 = 10 and + 72% in the case 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 = 40. The overestimation case has a lower impact on costs (with a range between the 6%-31%), while the overestimation cost effect has a range of 33%-72%, also because costs are more affected by failures than preventive activities (as 𝐶𝐶𝑓𝑓 ≫ 𝐶𝐶𝑝𝑝 ). Another interesting analysis is reported in the Figure 6. We calculated ∆𝑈𝑈𝑈𝑈𝑈𝑈, varying the percentage error in 𝛽𝛽̂ (from 100% to + 100%). It useful to understand which is the impact for different estimation errors, fixed the 𝛽𝛽̂ .

(5)

It can be introduced the additional cost function ∆𝑈𝑈𝑈𝑈𝑈𝑈 as follows: ∆𝑈𝑈𝑈𝑈𝑈𝑈 = 1 −

-50% 1.2 1.4 2.4 2.6 1.6 1.8 2 2.2 2.8 3 3.2 3.6 3.8 4 4.2 4.4 4.6 4.8 5

𝛽𝛽̂

Fig. 5. Coloured maps of additional cost for different couple of scale factors. Black cells are for values higher than 100%. where

511

Percentage error of the estimation

1.2 0% 2% 3% 4% 4% 4% 3% 3% 3% 3% 2% 2% 2% 2% 1% 1% 1% 1% 1% 1%

40

𝛽𝛽̂

𝛽𝛽

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

20

𝛽𝛽̂

Fabio Sgarbossa et al. / IFAC PapersOnLine 51-11 (2018) 508–513

(6)

This equation allows to evaluate the impact of the estimation error on the Weibull parameters, where the 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) will be the optimal minimum cost, calculated if the real data are available. 4.1 What if 𝛽𝛽̂ is wrong?

It is known that the optimal time to replacement and the ̂ are only a minimal cost is function of 𝛽𝛽̂ , while 𝜃𝜃̂ and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 scale factors without any impact on the optimal solution but with effects on absolute economic value.

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In Figure 6 are reported the various % of impact on costs varying the % error of estimation. Also in these tables is possible to highlight that the % of impact on ∆𝑈𝑈𝑈𝑈𝑈𝑈 is lower in the overestimation case. Moreover it is shown the increase of % of costs as the underestimation increase. For example, in the case of 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 = 40 and % error = +50%, when 𝛽𝛽̂ is 1,4 the additional cost is +2%, while when 𝛽𝛽̂ is 3,2 the additional cost is +16%.

It indicates that the estimation of the scale and shape factors has limited effect on the economical savings obtained using the preventive maintenance policy ARP-I. Obviously this result is related to one single component, while if the number of different items subjected to this kind of maintenance policy is relevant, also the final annual additional cost could be significantly high. 5. CONCLUSIONS

4.2 Absolute economic impact of estimation

The additional cost function ∆𝑈𝑈𝑈𝑈𝑈𝑈, introduced before, is expressed as percentage of the optimal Unit Expected Cost. Thus, it is important to estimate also the annual additional cost ∆𝐶𝐶 [€/year] in order to evaluate how is the impact of the error in the estimation of Weibull parameters.

Preventive maintenance aims to define the optimal replacement interval time as the strategic point to carry out maintenance activities and avoid failures. This work aims to analyze the impact on costs of a wrong estimation of this interval. Literature highlights as weibull distribution well represents reliability of production and logistic systems; through data as MTTF is possible to recover the scale and shape parameters and then to establish the optimal replacement interval time and related costs. In this study an analysis about consequences of wrong parameters estimation on costs is proposed, focusing on cases with censored data. It is investigated the additional % cost when MTTF estimation, or the shape parameter (β) one, has a relative % error. Tables and diagram about the impact on ∆𝑈𝑈𝑈𝑈𝑈𝑈 are carried out. Results highlight that overestimation of the shape parameter β as a lower additional cost than the underestimation case. Moreover, as the 𝐶𝐶𝑓𝑓 ⁄𝐶𝐶𝑝𝑝 increase, the impact on costs is greater, since failure costs are more relevant. Further researches can be carry out about other maintenance policies, as predictive one or based on condition.

It can be simply calculated by the following: ∆𝐶𝐶 = ∆𝑈𝑈𝑈𝑈𝑈𝑈 ∙ 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) ∙ 𝐻𝐻

(7)

where 𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) is calculated in €/days, as the optimal minimum cost if real data are available, and 𝐻𝐻 is the total working days per year.

As a consequence, its value depends on the percentage error in 𝛽𝛽̂ , as described before, and also on value of 𝐶𝐶𝑓𝑓 , 𝐶𝐶𝑝𝑝 , 𝜃𝜃 and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀.

Some examples of the annual additional cost ∆𝐶𝐶 are reported in table 1.

Although the analysis of additional cost ∆𝑈𝑈𝐸𝐸𝐶𝐶 has highlighted some relevant differences in percentage between optimal cost value and cost of the solution based on estimated Weibull parameters, the table 1 shows how this value are not very significant in terms of annual cost. Table 1. Impact of estimation error on the annual additional cost 𝛽𝛽̂ 2 2 2 2 2

𝜃𝜃̂ 271 271 271 406 542

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 240 240 240 360 480

𝛽𝛽 4 4 4 4 4

𝜃𝜃 265 265 265 397 530

𝐶𝐶𝑓𝑓 10,000 20,000 40,000 20,000 20,000

𝐶𝐶𝑝𝑝 1,000 1,000 1,000 1,000 1,000

𝐶𝐶𝑝𝑝 ⁄𝐶𝐶𝑓𝑓 10 20 40 20 20

𝑡𝑡̂𝑝𝑝∗ 91 62 43 94 125

𝑡𝑡𝑝𝑝∗ 116 96 81 145 193

𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) 11.51 13.85 16.57 9.24 6.93

𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡̂𝑝𝑝∗ ) 12.39 17.06 23.89 11.28 8.48

∆𝑈𝑈𝑈𝑈𝑈𝑈 7.7% 23.1% 44.2% 22.1% 22.4%

𝑈𝑈𝑈𝑈𝑈𝑈(𝑡𝑡𝑝𝑝∗ ) ∙ 𝐻𝐻 2,763 3,325 3,977 2,217 1,663

∆𝑪𝑪 212 769 1,756 490 372

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