Research ArticleReliability-Centered Maintenance Analyzing Failure inHarvest Sugarcane Machine Using Some Generalizations ofthe Weibull Distribution

Pedro L Ramos Diego C Nascimento Camila Cocolo Maacutercio J NicolaCarlos Alonso Luiz G Ribeiro Andreacute Ennes and Francisco Louzada

Institute of Mathematical and Computer Sciences Sao Paulo University Sao Carlos SP Brazil

Correspondence should be addressed to Pedro L Ramos pedrolramosuspbr

Received 7 December 2017 Revised 22 January 2018 Accepted 14 February 2018 Published 1 April 2018

Academic Editor Farouk Yalaoui

Copyright copy 2018 Pedro L Ramos et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We considered five generalizations of the standard Weibull distribution to describe the lifetime of two important components ofsugarcane harvesting machines The harvesters considered in the analysis harvest an average of 20 tons of sugarcane per hour andtheir malfunctionmay lead tomajor losses therefore an effective maintenance approach is of main interest for cost savings For theconsidered distributions mathematical background is presented Maximum likelihood is used for parameter estimation Furtherdifferent discrimination procedures were used to obtain the best fit for each component At the end we propose a maintenancescheduling for the components of the harvesters using predictive analysis

1 Introduction

The arrival of the sugarcane culture in Brazil has had asignificant impact on the national economy which led thecountry to become the largest producer in the world [1] Itssubproducts are used in the food and chemical industries aswell as in electricity generation and fuel production Mecha-nized harvesting is one of the most important stages in thesugar and ethanol mills since it provides the raw materialwith quality time and competitive costs for later processingAmong the used machines in the mechanized harvest theharvesters stand out for having a large number of correctivestops given the functionality in such extreme environmentalconditions In addition its operation is in a regime of 24 hourson the workdays having impact on fatigue and wear of theirparts During operation the harvester processes an average of20 tons of sugarcane per hour and its malfunction may leadto major losses therefore an effective maintenance approachis of keen interest [2]

Reliability-centeredmaintenance consists of determiningthe most effective maintenance approach [3 4] This processwas firstly developed in the aviation industry for decidingwhat maintenance work is needed to keep aircraft airborne

driven by the need to improve reliability while reducing thecost of maintenance [5] Reliability analysis can be used toestimate time-related parameters to the next machine stop[6] providing information to manage and control the pre-ventive maintenance of harvesters which could result in in-creased production and has potential for cost savings

In reliability common procedures are usually based onthe assumption that the data follows a Weibull distributionIntroduced by Weibull [7] this distribution has convenientmathematical properties and its physiological failure processarises in many areas (see Manton and Yashin [8]) Addition-ally McCool [9] provided an extensive discussion about itsuse in reliability However this distribution cannot be usedto describe data with nonmonotone hazard function (bath-tub upside-down bathtub to list a few) To overcome thisproblem many generalizations of the standard Weibull dis-tribution have been proposed Murthy et al [10] presentedthe application of some generalized Weibull distributions formodeling complex failure data sets Pham and Lai [11] dis-cussed recent generations of Weibull-related lifetime Fur-ther Lai [12] reviewed more than 25 generalizations of theWeibull distribution and Tahir and Cordeiro [13] cited morethan 30 compounded Weibull models

HindawiModelling and Simulation in EngineeringVolume 2018 Article ID 1241856 12 pageshttpsdoiorg10115520181241856

2 Modelling and Simulation in Engineering

In this paper we consider five important generalizedWeibull distributions with three parameters to describe thelifetime of two important components of the sugarcane har-vesting machines Our main goal here is to correctly predictthe next failure of the components not to present an extensivereview of the generalizations of Weibull distribution In thecited papers the authors only described mathematical prop-erties of the distributions and conducted the fit for differentdata However reliability is more about correctly predictingthe future than describing the past in this sense no predictiveanalysis was presented considering such generalizations

The distributions considered are the gamma-Weibull dis-tribution [14] generalizedWeibull (GW)distribution [15] ex-ponentiated Weibull (EW) distribution [16] Marshall-OlkinWeibull (MOW) distribution [17] and the extended Poisson-Weibull (EPW) distribution [18] While the first three distri-butions are themost common three parameter generalizationof the Weibull the MOW and the EPW arise in the com-petitive and complementary risk scenario (see Louzada [19]for a detailed discussion) In these cases the latent variablesfollow respectively a geometric and a zero-truncated Poissondistribution and each of components in risk came from aWeibull baseline distribution

For each distribution themathematical background is re-viewed and the parameters estimators are presented using themaximum likelihood estimators Further different discrimi-nation procedures are used to obtain the best fit for each com-ponent At the end we propose a maintenance scheduling forthe components of the harvesters using predictive analysis

The remainder of this paper is organized as follows Sec-tion 2presents the literature review related to the survivalmod-els adopted Section 3 exposes the data collection and empir-ical analysis as well as carrying out the predictive analysisbased on the parametric models Finally in Section 4 wepresent some final remarks related to the contribution of thisstudy

2 Theoretical Background

In this section we present the statistical background on theadopted distributions and itsparameter estimation proceduresThe following distributions are considered gamma-WeibullgeneralizedWeibull exponentiatedWeibullMarshall-Olkin-Weibull and Marshall-Olkin-Weibull Their choice is basedon their flexibility to accommodate lifetime dataset withhazard functions with different shapes for instance constantincreasing decreasing bathtub and upside-down bathtub

21 The Gamma-Weibull Distribution Introduced by Stacy[14] the gamma-Weibull distribution with three parametersis a flexible model for reliability data due to its ability toaccommodate various forms of the hazard functionThis dis-tribution is also known as generalized gamma (GG) distribu-tion as it generalizes the two-parameter gamma distributionhereafter we will refer to this model as GG distribution toavoid confusion with the GW distribution A random vari-able has GG distribution if its probability density function(PDF) is given by

119891 (119905 120601 120583 120572) = 120572Γ (120601)120583120572120601119905120572120601minus1119890minus(120583119905)120572 119905 gt 0 (1)

where 120572 gt 0 120601 gt 0 and 120583 gt 0 The mean and variance of GGare given by

119864 (119883) = Γ (120601 + 1120572)120583Γ (120601)

119881 (119883) = 11205832 Γ (120601 + 2120572)Γ (120601) minus (Γ (120601 + 1120572)Γ (120601) )2 (2)

Some relevant distributions are special cases such as theWeibull distribution (when 120601 = 1) the distribution gamma(120572 = 1) log-normal (case limit when 120601 rarr infin) and the gener-alized normal distribution (120572 = 2) For example the gener-alized normal distribution is also a distribution that includesseveral distributions known as half-normal (120601 = 12 120583 =1radic2120590) Rayleigh (120601 = 1 120583 = 1radic2120590) Maxwell-Boltzmann(120601 = 32) and chi (120601 = 1198962 119896 = 1 2 )The cumulative dis-tribution function (CDF) is given by

119865 (119905 120601 120583 120572) = int(120583119905)1205720

1Γ (120601)119908120601minus1119890minus119908119889119908 = 120574 [120601 (120583119905)120572]Γ (120601) (3)

where 120574[119910 119909] = int1199090119908119910minus1119890minus119908119889119908 is the lower incomplete gam-

ma function The survival function is

119878 (119905 120601 120583 120572) = 1 minus 119865 (119905 120601 120583 120572) = Γ [120601 (120583119905)120572]Γ (120601) (4)

where Γ[119910 119909] = intinfin119909119908119910minus1119890minus119908119889119908 is the upper incomplete

gamma functionThe hazard function of the GG distribution is

ℎ (119905 120601 120583 120572) = 119891 (119905 120601 120583 120572)119878 (119905 120601 120583 120572)

= 120572120583120572120601119905120572120601minus1 exp (minus (120583119905)120572)Γ [120601 (120583119905)120572]

(5)

where the hazard function has constant increasing decreas-ing bathtub and upside-down bathtub hazard rate

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim GG(120572 120583 120601) Then the likelihoodfunction related to the PDF (1) is given by

119871 (120601 120583 120572 t)= 120572119899Γ (120601)119899 120583119899120572120601

119899prod119894=1

119905120572120601minus1119894 expminus120583120572 119899sum119894=1

119905120572119894 (6)

The log-likelihood is given by

119897 (120601 120583 120572 t) = 119899 log (120572) minus 119899 log Γ (120601) + 119899120572120601 log (120583)+ (120572120601 minus 1) 119899sum

119894=1

log (119905119894) minus 120583120572 119899sum119894=1

119905120572119894 (7)

Modelling and Simulation in Engineering 3

Setting the partial derivatives (120597120597120572)119897(120601 120583 120572 t) (120597120597120583)119897(120601 120583 120572 t) and (120597120597120601)119897(120601 120583 120572 t) equal to 0 we obtain thefollowing maximum likelihood estimators

120583 = ( 1120572sdot 119899sum119899119894=1 119905119894 log (119905119894) minus ((sum119899119894=1 119905119894 ) 119899)sum119899119894=1 log (119905119894))

1120572

120601 = ( 1120572sum119899119894=1 119905119894sum119899119894=1 119905119894 log (119905119894) minus ((sum119899119894=1 119905119894 ) 119899)sum119899119894=1 log (119905119894))

119899 log (120583) + 119899sum119894=1

log (119905119894) minus 119899120595 (120601) = 0

(8)

where 120595(119896) = (120597120597119896) log Γ(119896) = Γ1015840(119896)Γ(119896) The solutionprovides the maximum likelihood estimates (MLEs) See forinstance Ramos et al [20 21] and Achcar et al [22] for a de-tailed discussion

Under mild conditions the estimators become unbiasedfor large samples and asymptotically efficient Moreoversuch estimators have asymptotically normal joint distributiongiven by

(120601 120583 ) sim 1198733 [(120601 120583 120572) 119868minus1 (120601 120583 120572)] for 119899 997888rarr infin (9)

where 119868(120579) is the Fisher information matrix that is

[[[[[[[[[

1 + 2120595 (120601) + 1206011205951015840 (120601) + 120601120595 (120601)21205722 minus1 + 120601120595 (120601)120583 minus120595 (120601)120572

minus1 + 120601120595 (120601)120583 12060112057221205832 120572120583minus120595 (120601)120572 120572120583 1205951015840 (120601)

]]]]]]]]] (10)

and 1205951015840(119896) = (120597120597119896)120595(119896) is the trigamma function

22 The Generalized Weibull Distribution Introduced byMudholkar et al [15] the generalizedWeibull distributionhasPDF given by

119891 (119905 120582 120573 120572)= (120572120573)minus1 ( 119905120573)

1120572minus1 (1 minus 120582( 119905120573)1120572)1120582minus1 (11)

where 120582 isin R 120573 gt 0 and 120572 gt 0 The CDF and the survivalfunction are respectively given by

119865 (119905 120582 120573 120572) = 1 minus (1 minus 120582( 119905120573)1120572)1120582

119878 (119905 120582 120573 120572) = (1 minus 120582( 119905120573)1120572)1120582

(12)

The hazard function of the GW distribution is

ℎ (119905 120582 120573 120572) = (119905120573)1120572minus1120572120573 (1 minus 120582 (119905120573)1120572) sdot (13)

This model is very flexible to describe lifetime data sinceit has the hazard function with constant increasing decreas-ing bathtub and upside-down bathtub hazard rate Thequantile function of the GWdistribution has closed form andis given by

119876 (119906 120582 120573 120572) =

120573 (minus log (1 minus 119906))120572 if 120582 = 0120573(1 minus (1 minus 119906)120582120582 )120572 if 120582 = 0 (14)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where119879 sim GW(120572 120583 120573)Then the likelihoodfunction related to the PDF (11) is given by

119871 (120582 120573 120572 t)= (120572120573)minus119899 119899prod

119894=1

(119905119894120573)1120572minus1 (1 minus 120582(119905119894120573)

1120572)1120582minus1 (15)

The log-likelihood is given by

119897 (120582 120573 120572 t) = ( 1120582 minus 1) 119899sum119894=1

log(1 minus 120582(119905119894120573)1120572)

minus 119899 log (120572120573) + ( 1120572 minus 1) 119899sum119894=1

log(119905119894120573) (16)

Setting the partial derivatives equal to 0 we obtain themaximum likelihood estimators Here we followMudholkaret al [15] which considers the direct maximization of (16)Undermild conditions the obtained estimators are consistentand efficient with an asymptotically normal joint distributiongiven by

Θ sim 1198733 [Θ 119868minus1 (Θ)] as 119899 997888rarr infin (17)

where 119868(Θ) is the 3 times 3 Fisher information matrix associatedwith the vector of parameters Θ and 119868119894119895(Θ) is the Fisherinformation elements in 119894 and 119895 given by

119868119894119895 (Θ) = 119864 [minus 1205972120597Θ119894120597Θ119895 119897 (ΘD)2] 119894 119895 = 1 2 3 (18)

Since the Fisher informationmatrix does not have closed-form expression for some terms an alternative is to considerthe observed information matrix where the terms are givenby

119867119894119895 (Θ) = minus 1205972120597Θ119894120597Θ119895 119897 (Θ t)2 119894 119895 = 1 2 3 (19)

Hereafter we considered the same approach to obtain theconfidence intervals for the parameters from other distribu-tions

4 Modelling and Simulation in Engineering

23 The Exponentiated Weibull Distribution Introduced byMudholkar et al [16] the exponentiatedWeibull distributionwith PDF is given by

119891 (119905 120590 120601 120572) = 120572120601120590 ( 119905120590)120572minus1

exp(minus( 119905120590)120572)

sdot (1 minus exp (minus( 119905120590)120572))120601minus1

(20)

where 120590 gt 0 120601 gt 0 and 120572 gt 0The exponentiatedWeibull distribution includes theWei-

bull distribution (120601 = 1) and the exponentiated exponentialdistribution (120572 = 1) The survival function is given by

119878 (119905 120590 120601 120572) = 1 minus (1 minus exp(minus( 119905120590)120572))120601 (21)

The hazard function of the GG distribution is

ℎ (119905 120601 120583 120572)= 120572120601 (119905120590)120572minus1 exp (minus (119905120590)120572) (1 minus exp (minus (119905120590)120572))120601minus1

120590 (1 minus (1 minus exp (minus (119905120590)120572))120601) (22)

The shapes of the hazard function are analogous to theGG and GW distribution Additionally the quantile functionof the EW distribution has closed form and is given by

119876 (119906 120590 120601 120572) = 120590 (minus log (1 minus 1199061120601))1120572 (23)

The 119896th moment of the EW distribution is given by

120583119896 = int10119876 (119906 120590 120601 120572)119896 119889119906

= 120579120590119896Γ(119896120572 + 1)(1 + infinsum119894=1

119886119894 [(119894 + 1)119896120572+1]) (24)

where 119896 isin N 119886119894 = (minus1)119894(120579 minus 1)(120579 minus 2) sdot sdot sdot (120579 minus 1 minus 119894 minus 1)(119894)minus1The proof of this equality is presented by Choudhury [23]

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EW(120590 120601 120572)Then the likelihoodfunction related to the PDF (20) is given by

119871 (120590 120601 120572 t) = 120572119899120601119899120590119899119899prod119894=1

(119905119894120590)120572minus1

sdot (1 minus exp(minus(119905119894120590)120572))120601minus1

sdot exp(minus 119899sum119894=1

(119905119894120590)120572)

(25)

The log-likelihood is given by

119897 (120590 120601 120572 t) = 119899 log (120572120601)+ (120601 minus 1) 119899sum

119894=1

log(1 minus exp (minus(119905119894120590)120572))

+ (120572 minus 1) 119899sum119894=1

log (119905119894) minus 119899sum119894=1

(119905119894120590)120572

minus 119899120572 log (120590)

(26)

Setting the partial derivatives (120597120597120590)119897(120590 120601 120572 t) (120597120597120601)119897(120590 120601 120572 t) and (120597120597120572)119897(120590 120601 120572 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120572 + 119899sum119894=1

log (119905119894) + (120601 minus 1)120590120572

119899sum119894=1

119905120572119894 log (119905119894120590)exp ((119905119894120590)120572) minus 1

minus 119899sum119894=1

(119905119894120590)120572

log(119905119894120590) exp (minus(119905119894120590)120572)

minus 119899 log (120590) = 0minus 119899120572120590 minus 120572120590120572+1

119899sum119894=1

119905120572119894 + 120572120590120572119899sum119894=1

(120601 minus 1) 119905120572119894120590 minus 120590 exp ((119905119894120590)120572) = 0120601 = minus 119899

sum119899119894=1 log (1 minus exp (minus (119905119894120590)120572)) sdot

(27)

24 The Marshall-Olkin-Weibull Distribution Marshall andOlkin [17] presented a new procedure for introducing anadditional parameter into a family of distribution In this casethe authors applied such procedure in the Weibull distribu-tion The obtained PDF of the MOW distribution is given by

119891 (119905 120582 120572 120574) = 120572120574120582119905120574minus1119890minus120582119905120574(1 minus (1 minus 120572) 119890minus120582119905120574)2 (28)

where 120582 gt 0 120572 gt 0 and 120574 gt 0 The MOW distributionarises naturally in competing risks scenarios Let 119883 =min(1198791 1198792 119879119872) where119872 is a random variable with geo-metrical distribution and 119879119894 are assumed to be independentand identically distributed according to a Weibull distribu-tion then the119883 has a PDF given by (28) Cordeiro and Lem-onte [24] derived many properties and the parameter esti-mators for the MOW distribution the following results wereobtained from the cited work The survival function is givenby

119878 (119905 120582 120572 120574) = 1 minus 1 minus 119890minus1205821199051205741 minus (1 minus 120572) 119890minus120582119905120574 sdot (29)

The hazard function of the MOW distribution is

ℎ (119905 120582 120572 120574) = 120574120582119905120574minus11 minus (1 minus 120572) 119890minus120582119905120574 (30)

Modelling and Simulation in Engineering 5

where its behavior is constant increasing decreasing bath-tub and unimodal Moreover the quantile function of theMOW distribution has closed form and is given by

119876 (119906 120582 120572 120574) = 120582minus1120574 (log(1 minus (1 minus 120572) 1199061 minus 119906 ))1120574 (31)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim MOW(120582 120572 120574) Then the like-lihood function related to the PDF (28) is given by

119871 (120582 120572 120574 t)= 120572119899120574119899120582119899 119899prod

119894=1

119905120574minus1119894(1 minus (1 minus 120572) 119890minus120582119905120574119894 )2 exp(minus120582

119899sum119894=1

119905120574119894 ) (32)

The log-likelihood is given by

119897 (120582 120572 120574 t) = 119899 log (120572) + 119899 log (120574) + 119899 log (120582)+ (120574 minus 1) 119899sum

119894=1

log (119905119894)

minus 2 119899sum119894=1

log (1 minus (1 minus 120572) 119890minus120582119905120574119894 ) minus 120582 119899sum119894=1

119905120574119894 (33)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120574 t) (120597120597120572)119897(120582 120572 120574 t) and (120597120597120574)119897(120582 120572 120574 t) equal to 0 we obtainthe following maximum likelihood estimators

119899120572 minus 2 119899sum119894=1

119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0

119899120582 minus 119899sum119894=1

119905120574119894 minus 2 (1 minus 120572)119899sum119894=1

119905120574119894 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0119899120574 +119899sum119894=1

log (119905119894) minus 2 (1 minus 120572) 119899sum119894=1

120582119905120574119894 log (119905119894) 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894= 120582 119899sum119894=1

119905120574119894 log (119905119894)

(34)

for more details see Cordeiro and Lemonte [24]

25 The Extended Poisson-Weibull Distribution Ramos et al[18] introduced the extended Poisson-Weibull (EPW) distri-bution as a generalization of Weibull-Poisson distribution(see Hemmati et al [25]) where its PDF is given by

119891 (119905 120582 120572 120601) = 120572120582120601119905120572minus1119890minus120601119905120572minus120582119890minus1206011199051205721 minus 119890minus120582 (35)

where 120582 isin Rlowast 120601 gt 0 and 120572 gt 0 Analogously to the MOWdistribution the EWP model arises naturally in competingrisks scenarios Let 119883 = min(1198791 1198792 119879119872) where 119872 isa random variable with a zero-truncated Poisson distribution

and 119879119894 are assumed to be independent and identically dis-tributed according to aWeibull distribution then the119883 has aPDF given by (35) The survival function is given by

119878 (119905 120582 120572 120601) = 1 minus exp (minus120582119890minus120601119905120572)1 minus 119890minus120582 (36)

The hazard function of the GG distribution is

ℎ (119905 120582 120572 120601) = 120582120601119905120572minus1119890minus120601119905120572minus120582119890minus120601119905120572 (1 minus 119890minus120582119890minus120601119905120572)minus1 (37)

For the EWP distribution the hazard function has differ-ent shapes such as constant increasing decreasing bathtuband upside-down bathtub Furthermore the quantile func-tion of the EPW distribution has closed form and is given by

119876 (119906 120582 120572 120601)= (minus 1120601 log(1 minus log ((119890120582 minus 1) 119901 + 1)

120582 ))1120572

(38)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EPW(120582 120572 120601) Then the like-lihood function related to the PDF (20) is given by

119871 (120582 120572 120601 t)= 120572119899120582119899120601119899(1 minus 119890minus120582)119899

119899prod119894=1

119905120572minus1119894 exp(minus120601 119899sum119894=1

119905120572119894 minus 120582 119899sum119894=1

119890minus120601119905120572119894 ) (39)

The log-likelihood is given by

119897 (120582 120572 120601 t) = 119899 log (120572120582120601) minus 119899 log (1 minus 119890minus120582) minus 120601 119899sum119894=1

119905120572119894+ (120572 minus 1) 119899sum

119894=1

log (119905119894) minus 120582 119899sum119894=1

119890minus120601119905120572119894 (40)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120601 t) (120597120597120572)119897(120582 120572 120601 t) and (120597120597120601)119897(120582 120572 120601 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120582 + 1198991 minus 119890120582 minus119899sum119894=1

119890minus120601119905120572119894 = 0119899120572 + 119899sum119894=1

log (119905119894) minus 120601 119899sum119894=1

119905120572119894 log (119905119894) + 120601120582 119899sum119894=1

119905120572119894 log (119905119894) 119890minus120601119905120572119894= 0

119899120601 minus 119899sum119894=1

119905120572119894 + 120582 119899sum119894=1

119905120572119894 119890minus120601119905120572119894 = 0

(41)

26 Goodness of Fit Firstly in order to verify the behaviorof the empirical data the Total Time on Test plot (TTT-plot)was considered (Barlow and Campo [26]) The TTT-plot isobtained through the plot of [119903119899 119866(119903119899)] where

119866( 119903119899) =(sum119903119894=1 119905119894 + (119899 minus 119903) 119905(119903))sum119899119894=1 119905119894 (42)

6 Modelling and Simulation in Engineering

Table 1 Maintenance distribution preventive (P) and corrective (C)stops per crop

Crop 1 Crop 2 Crop 3P C P C P C

Machine A 39 232 32 255 23 127Machine B 37 199 32 182 23 166

119903 = 1 119899 119894 = 1 119899 and 119905(119894) is the ordered data For datawith concave (convex) curve the hazard function has increas-ing (decreasing) shape If the behavior starts convex andthen becomes concave (concave and then convex) the hazardfunction has bathtub (inverse bathtub) shape

The goodness of fit is checked considering the Kolmog-orov-Smirnov (KS) test This procedure is based on the KSstatistic 119863119899 = sup |119865119899(119905) minus 119865(119905 120579)| where sup 119905 is the supre-mumof the set of distances119865119899(119905) is the empirical distributionfunction and 119865(119905 120579) is CDF A hypothesis test is conductedat the 5 level of significance to test whether or not the datacomes from119865(119905 120579) In this case the null hypothesis is rejectedif the returned 119901 value is smaller than 005

The following discrimination criterion methods wereadopted Akaike information criteria (AIC) and the correctedAIC (AICc) computed respectively by AIC = minus2119897( t) + 2119896and AICc = AIC + 2119896(119896 + 1)(119899 minus 119896 minus 1)minus1 where 119896 is thenumber of parameters to be fitted and isMLEs of 120579 For a setof candidatemodels for t the best one provides theminimumvalues

3 Data Collection and Empirical Analysis

The dataset came from two sources a manual stop systemwhich brings the history of revisions and corrective stopsof two sugarcane harvesters and data from the onboardcomputers of the harvesters which provide information onthe operation of the machine The data were collected fromJanuary 2015 to August 2017 a period corresponding to 25harvests (crops) that is a period of thirty months of activity

31 Empirical Analysis Firstly considering all the stops andtheir reasons records of the performance of the predictivemaintenance are required to be observed In total 1347 stopswere observed of which 186 were preventive and 1161 correc-tive stops Thus it is possible to observe the superior amountof unplanned stops thus questioning the effectiveness ofpreventive maintenance Table 1 shows the failure among theharvests considering both machines analysis

The Pricker and transmission from each machine wereselected given their complexity in the maintenance Figure 1describes the number of failures per year divided by harvestconsidering their temporal sparsity by which items analyzedin this report correspond to 18 of the stops

It is possible to notice a difference in the machinesrsquobehavior both machines appear to be equally affected by theproblems of transmission and Pricker but the machine B is

Table 2 Dataset related to the sugarcane harvesterrsquos Pricker

1 1 1 1 1 1 1 1 2 2 2 22 3 3 3 3 3 4 4 4 5 5 56 6 7 8 9 11 11 12 14 16 18 1818 22 22 23 29 32 34 38 41 46 53 53

Table 3 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker A

Criteria GG GW EW MOW EPWAIC 341013 343696 340435 341317 342750AICc 335559 338241 334981 335862 343296KS 06735 06046 07447 07457 05751

Table 4 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0379 0044 (03181 04969)120579 6446 0703 (47070 76915)120590 0727 0290 (03897 14691)119910lowast 3093 0582 (17374 40236)

more affected by problems with the Pricker Further relia-bility models were individually adjusted and thereby com-pared as described in the next section

32 Preventive Maintenance In this section we discuss aparametric approach in order to perform a predictive analysisfor the lifetime of the components

321 Pricker from Machine A Table 2 presents a high defectrate after a short repair time as well compromising the cost ofthe production The experiment considered a total period of30months as said beforeThe operating equipment had threeoff-seasons these periods were not included in the datasetThe equipmentwas only observed during the time of its activeoperation

Figure 2 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

From the TTT-plot we observed that the proposed datahas unimodal hazard rate which implies that all the proposedmodels may be used to describe the proposed datasetAdditionally the survival function adjusted by the differentdistributions shows that the proposedmodels provide a goodfit for the proposed data In order to discriminate the best fitwe considered the results of AIC and AICc (see Table 3)

Among the proposed models the exponentiated Weibulldistribution has superior goodness of fit since the AIC andAICc returned smaller values Therefore using the expo-nentiated Weibull distribution we computed the maximumlikelihood estimates and the predictive value for 25 (seeTable 4) Hereafter as we considered the quantile functionto obtain the predictive value the confidence intervals (CI)related to this estimate were obtained from bootstrap tech-nique [27]

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Modelling and Simulation in Engineering 3

Setting the partial derivatives (120597120597120572)119897(120601 120583 120572 t) (120597120597120583)119897(120601 120583 120572 t) and (120597120597120601)119897(120601 120583 120572 t) equal to 0 we obtain thefollowing maximum likelihood estimators

120583 = ( 1120572sdot 119899sum119899119894=1 119905119894 log (119905119894) minus ((sum119899119894=1 119905119894 ) 119899)sum119899119894=1 log (119905119894))

1120572

120601 = ( 1120572sum119899119894=1 119905119894sum119899119894=1 119905119894 log (119905119894) minus ((sum119899119894=1 119905119894 ) 119899)sum119899119894=1 log (119905119894))

119899 log (120583) + 119899sum119894=1

log (119905119894) minus 119899120595 (120601) = 0

(8)

where 120595(119896) = (120597120597119896) log Γ(119896) = Γ1015840(119896)Γ(119896) The solutionprovides the maximum likelihood estimates (MLEs) See forinstance Ramos et al [20 21] and Achcar et al [22] for a de-tailed discussion

Under mild conditions the estimators become unbiasedfor large samples and asymptotically efficient Moreoversuch estimators have asymptotically normal joint distributiongiven by

(120601 120583 ) sim 1198733 [(120601 120583 120572) 119868minus1 (120601 120583 120572)] for 119899 997888rarr infin (9)

where 119868(120579) is the Fisher information matrix that is

[[[[[[[[[

1 + 2120595 (120601) + 1206011205951015840 (120601) + 120601120595 (120601)21205722 minus1 + 120601120595 (120601)120583 minus120595 (120601)120572

minus1 + 120601120595 (120601)120583 12060112057221205832 120572120583minus120595 (120601)120572 120572120583 1205951015840 (120601)

]]]]]]]]] (10)

and 1205951015840(119896) = (120597120597119896)120595(119896) is the trigamma function

22 The Generalized Weibull Distribution Introduced byMudholkar et al [15] the generalizedWeibull distributionhasPDF given by

119891 (119905 120582 120573 120572)= (120572120573)minus1 ( 119905120573)

1120572minus1 (1 minus 120582( 119905120573)1120572)1120582minus1 (11)

where 120582 isin R 120573 gt 0 and 120572 gt 0 The CDF and the survivalfunction are respectively given by

119865 (119905 120582 120573 120572) = 1 minus (1 minus 120582( 119905120573)1120572)1120582

119878 (119905 120582 120573 120572) = (1 minus 120582( 119905120573)1120572)1120582

(12)

The hazard function of the GW distribution is

ℎ (119905 120582 120573 120572) = (119905120573)1120572minus1120572120573 (1 minus 120582 (119905120573)1120572) sdot (13)

This model is very flexible to describe lifetime data sinceit has the hazard function with constant increasing decreas-ing bathtub and upside-down bathtub hazard rate Thequantile function of the GWdistribution has closed form andis given by

119876 (119906 120582 120573 120572) =

120573 (minus log (1 minus 119906))120572 if 120582 = 0120573(1 minus (1 minus 119906)120582120582 )120572 if 120582 = 0 (14)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where119879 sim GW(120572 120583 120573)Then the likelihoodfunction related to the PDF (11) is given by

119871 (120582 120573 120572 t)= (120572120573)minus119899 119899prod

119894=1

(119905119894120573)1120572minus1 (1 minus 120582(119905119894120573)

1120572)1120582minus1 (15)

The log-likelihood is given by

119897 (120582 120573 120572 t) = ( 1120582 minus 1) 119899sum119894=1

log(1 minus 120582(119905119894120573)1120572)

minus 119899 log (120572120573) + ( 1120572 minus 1) 119899sum119894=1

log(119905119894120573) (16)

Setting the partial derivatives equal to 0 we obtain themaximum likelihood estimators Here we followMudholkaret al [15] which considers the direct maximization of (16)Undermild conditions the obtained estimators are consistentand efficient with an asymptotically normal joint distributiongiven by

Θ sim 1198733 [Θ 119868minus1 (Θ)] as 119899 997888rarr infin (17)

where 119868(Θ) is the 3 times 3 Fisher information matrix associatedwith the vector of parameters Θ and 119868119894119895(Θ) is the Fisherinformation elements in 119894 and 119895 given by

119868119894119895 (Θ) = 119864 [minus 1205972120597Θ119894120597Θ119895 119897 (ΘD)2] 119894 119895 = 1 2 3 (18)

Since the Fisher informationmatrix does not have closed-form expression for some terms an alternative is to considerthe observed information matrix where the terms are givenby

119867119894119895 (Θ) = minus 1205972120597Θ119894120597Θ119895 119897 (Θ t)2 119894 119895 = 1 2 3 (19)

Hereafter we considered the same approach to obtain theconfidence intervals for the parameters from other distribu-tions

4 Modelling and Simulation in Engineering

23 The Exponentiated Weibull Distribution Introduced byMudholkar et al [16] the exponentiatedWeibull distributionwith PDF is given by

119891 (119905 120590 120601 120572) = 120572120601120590 ( 119905120590)120572minus1

exp(minus( 119905120590)120572)

sdot (1 minus exp (minus( 119905120590)120572))120601minus1

(20)

where 120590 gt 0 120601 gt 0 and 120572 gt 0The exponentiatedWeibull distribution includes theWei-

bull distribution (120601 = 1) and the exponentiated exponentialdistribution (120572 = 1) The survival function is given by

119878 (119905 120590 120601 120572) = 1 minus (1 minus exp(minus( 119905120590)120572))120601 (21)

The hazard function of the GG distribution is

ℎ (119905 120601 120583 120572)= 120572120601 (119905120590)120572minus1 exp (minus (119905120590)120572) (1 minus exp (minus (119905120590)120572))120601minus1

120590 (1 minus (1 minus exp (minus (119905120590)120572))120601) (22)

The shapes of the hazard function are analogous to theGG and GW distribution Additionally the quantile functionof the EW distribution has closed form and is given by

119876 (119906 120590 120601 120572) = 120590 (minus log (1 minus 1199061120601))1120572 (23)

The 119896th moment of the EW distribution is given by

120583119896 = int10119876 (119906 120590 120601 120572)119896 119889119906

= 120579120590119896Γ(119896120572 + 1)(1 + infinsum119894=1

119886119894 [(119894 + 1)119896120572+1]) (24)

where 119896 isin N 119886119894 = (minus1)119894(120579 minus 1)(120579 minus 2) sdot sdot sdot (120579 minus 1 minus 119894 minus 1)(119894)minus1The proof of this equality is presented by Choudhury [23]

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EW(120590 120601 120572)Then the likelihoodfunction related to the PDF (20) is given by

119871 (120590 120601 120572 t) = 120572119899120601119899120590119899119899prod119894=1

(119905119894120590)120572minus1

sdot (1 minus exp(minus(119905119894120590)120572))120601minus1

sdot exp(minus 119899sum119894=1

(119905119894120590)120572)

(25)

The log-likelihood is given by

119897 (120590 120601 120572 t) = 119899 log (120572120601)+ (120601 minus 1) 119899sum

119894=1

log(1 minus exp (minus(119905119894120590)120572))

+ (120572 minus 1) 119899sum119894=1

log (119905119894) minus 119899sum119894=1

(119905119894120590)120572

minus 119899120572 log (120590)

(26)

Setting the partial derivatives (120597120597120590)119897(120590 120601 120572 t) (120597120597120601)119897(120590 120601 120572 t) and (120597120597120572)119897(120590 120601 120572 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120572 + 119899sum119894=1

log (119905119894) + (120601 minus 1)120590120572

119899sum119894=1

119905120572119894 log (119905119894120590)exp ((119905119894120590)120572) minus 1

minus 119899sum119894=1

(119905119894120590)120572

log(119905119894120590) exp (minus(119905119894120590)120572)

minus 119899 log (120590) = 0minus 119899120572120590 minus 120572120590120572+1

119899sum119894=1

119905120572119894 + 120572120590120572119899sum119894=1

(120601 minus 1) 119905120572119894120590 minus 120590 exp ((119905119894120590)120572) = 0120601 = minus 119899

sum119899119894=1 log (1 minus exp (minus (119905119894120590)120572)) sdot

(27)

24 The Marshall-Olkin-Weibull Distribution Marshall andOlkin [17] presented a new procedure for introducing anadditional parameter into a family of distribution In this casethe authors applied such procedure in the Weibull distribu-tion The obtained PDF of the MOW distribution is given by

119891 (119905 120582 120572 120574) = 120572120574120582119905120574minus1119890minus120582119905120574(1 minus (1 minus 120572) 119890minus120582119905120574)2 (28)

where 120582 gt 0 120572 gt 0 and 120574 gt 0 The MOW distributionarises naturally in competing risks scenarios Let 119883 =min(1198791 1198792 119879119872) where119872 is a random variable with geo-metrical distribution and 119879119894 are assumed to be independentand identically distributed according to a Weibull distribu-tion then the119883 has a PDF given by (28) Cordeiro and Lem-onte [24] derived many properties and the parameter esti-mators for the MOW distribution the following results wereobtained from the cited work The survival function is givenby

119878 (119905 120582 120572 120574) = 1 minus 1 minus 119890minus1205821199051205741 minus (1 minus 120572) 119890minus120582119905120574 sdot (29)

The hazard function of the MOW distribution is

ℎ (119905 120582 120572 120574) = 120574120582119905120574minus11 minus (1 minus 120572) 119890minus120582119905120574 (30)

Modelling and Simulation in Engineering 5

where its behavior is constant increasing decreasing bath-tub and unimodal Moreover the quantile function of theMOW distribution has closed form and is given by

119876 (119906 120582 120572 120574) = 120582minus1120574 (log(1 minus (1 minus 120572) 1199061 minus 119906 ))1120574 (31)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim MOW(120582 120572 120574) Then the like-lihood function related to the PDF (28) is given by

119871 (120582 120572 120574 t)= 120572119899120574119899120582119899 119899prod

119894=1

119905120574minus1119894(1 minus (1 minus 120572) 119890minus120582119905120574119894 )2 exp(minus120582

119899sum119894=1

119905120574119894 ) (32)

The log-likelihood is given by

119897 (120582 120572 120574 t) = 119899 log (120572) + 119899 log (120574) + 119899 log (120582)+ (120574 minus 1) 119899sum

119894=1

log (119905119894)

minus 2 119899sum119894=1

log (1 minus (1 minus 120572) 119890minus120582119905120574119894 ) minus 120582 119899sum119894=1

119905120574119894 (33)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120574 t) (120597120597120572)119897(120582 120572 120574 t) and (120597120597120574)119897(120582 120572 120574 t) equal to 0 we obtainthe following maximum likelihood estimators

119899120572 minus 2 119899sum119894=1

119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0

119899120582 minus 119899sum119894=1

119905120574119894 minus 2 (1 minus 120572)119899sum119894=1

119905120574119894 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0119899120574 +119899sum119894=1

log (119905119894) minus 2 (1 minus 120572) 119899sum119894=1

120582119905120574119894 log (119905119894) 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894= 120582 119899sum119894=1

119905120574119894 log (119905119894)

(34)

for more details see Cordeiro and Lemonte [24]

25 The Extended Poisson-Weibull Distribution Ramos et al[18] introduced the extended Poisson-Weibull (EPW) distri-bution as a generalization of Weibull-Poisson distribution(see Hemmati et al [25]) where its PDF is given by

119891 (119905 120582 120572 120601) = 120572120582120601119905120572minus1119890minus120601119905120572minus120582119890minus1206011199051205721 minus 119890minus120582 (35)

where 120582 isin Rlowast 120601 gt 0 and 120572 gt 0 Analogously to the MOWdistribution the EWP model arises naturally in competingrisks scenarios Let 119883 = min(1198791 1198792 119879119872) where 119872 isa random variable with a zero-truncated Poisson distribution

and 119879119894 are assumed to be independent and identically dis-tributed according to aWeibull distribution then the119883 has aPDF given by (35) The survival function is given by

119878 (119905 120582 120572 120601) = 1 minus exp (minus120582119890minus120601119905120572)1 minus 119890minus120582 (36)

The hazard function of the GG distribution is

ℎ (119905 120582 120572 120601) = 120582120601119905120572minus1119890minus120601119905120572minus120582119890minus120601119905120572 (1 minus 119890minus120582119890minus120601119905120572)minus1 (37)

For the EWP distribution the hazard function has differ-ent shapes such as constant increasing decreasing bathtuband upside-down bathtub Furthermore the quantile func-tion of the EPW distribution has closed form and is given by

119876 (119906 120582 120572 120601)= (minus 1120601 log(1 minus log ((119890120582 minus 1) 119901 + 1)

120582 ))1120572

(38)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EPW(120582 120572 120601) Then the like-lihood function related to the PDF (20) is given by

119871 (120582 120572 120601 t)= 120572119899120582119899120601119899(1 minus 119890minus120582)119899

119899prod119894=1

119905120572minus1119894 exp(minus120601 119899sum119894=1

119905120572119894 minus 120582 119899sum119894=1

119890minus120601119905120572119894 ) (39)

The log-likelihood is given by

119897 (120582 120572 120601 t) = 119899 log (120572120582120601) minus 119899 log (1 minus 119890minus120582) minus 120601 119899sum119894=1

119905120572119894+ (120572 minus 1) 119899sum

119894=1

log (119905119894) minus 120582 119899sum119894=1

119890minus120601119905120572119894 (40)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120601 t) (120597120597120572)119897(120582 120572 120601 t) and (120597120597120601)119897(120582 120572 120601 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120582 + 1198991 minus 119890120582 minus119899sum119894=1

119890minus120601119905120572119894 = 0119899120572 + 119899sum119894=1

log (119905119894) minus 120601 119899sum119894=1

119905120572119894 log (119905119894) + 120601120582 119899sum119894=1

119905120572119894 log (119905119894) 119890minus120601119905120572119894= 0

119899120601 minus 119899sum119894=1

119905120572119894 + 120582 119899sum119894=1

119905120572119894 119890minus120601119905120572119894 = 0

(41)

26 Goodness of Fit Firstly in order to verify the behaviorof the empirical data the Total Time on Test plot (TTT-plot)was considered (Barlow and Campo [26]) The TTT-plot isobtained through the plot of [119903119899 119866(119903119899)] where

119866( 119903119899) =(sum119903119894=1 119905119894 + (119899 minus 119903) 119905(119903))sum119899119894=1 119905119894 (42)

6 Modelling and Simulation in Engineering

Table 1 Maintenance distribution preventive (P) and corrective (C)stops per crop

Crop 1 Crop 2 Crop 3P C P C P C

Machine A 39 232 32 255 23 127Machine B 37 199 32 182 23 166

119903 = 1 119899 119894 = 1 119899 and 119905(119894) is the ordered data For datawith concave (convex) curve the hazard function has increas-ing (decreasing) shape If the behavior starts convex andthen becomes concave (concave and then convex) the hazardfunction has bathtub (inverse bathtub) shape

The goodness of fit is checked considering the Kolmog-orov-Smirnov (KS) test This procedure is based on the KSstatistic 119863119899 = sup |119865119899(119905) minus 119865(119905 120579)| where sup 119905 is the supre-mumof the set of distances119865119899(119905) is the empirical distributionfunction and 119865(119905 120579) is CDF A hypothesis test is conductedat the 5 level of significance to test whether or not the datacomes from119865(119905 120579) In this case the null hypothesis is rejectedif the returned 119901 value is smaller than 005

The following discrimination criterion methods wereadopted Akaike information criteria (AIC) and the correctedAIC (AICc) computed respectively by AIC = minus2119897( t) + 2119896and AICc = AIC + 2119896(119896 + 1)(119899 minus 119896 minus 1)minus1 where 119896 is thenumber of parameters to be fitted and isMLEs of 120579 For a setof candidatemodels for t the best one provides theminimumvalues

3 Data Collection and Empirical Analysis

The dataset came from two sources a manual stop systemwhich brings the history of revisions and corrective stopsof two sugarcane harvesters and data from the onboardcomputers of the harvesters which provide information onthe operation of the machine The data were collected fromJanuary 2015 to August 2017 a period corresponding to 25harvests (crops) that is a period of thirty months of activity

31 Empirical Analysis Firstly considering all the stops andtheir reasons records of the performance of the predictivemaintenance are required to be observed In total 1347 stopswere observed of which 186 were preventive and 1161 correc-tive stops Thus it is possible to observe the superior amountof unplanned stops thus questioning the effectiveness ofpreventive maintenance Table 1 shows the failure among theharvests considering both machines analysis

The Pricker and transmission from each machine wereselected given their complexity in the maintenance Figure 1describes the number of failures per year divided by harvestconsidering their temporal sparsity by which items analyzedin this report correspond to 18 of the stops

It is possible to notice a difference in the machinesrsquobehavior both machines appear to be equally affected by theproblems of transmission and Pricker but the machine B is

Table 2 Dataset related to the sugarcane harvesterrsquos Pricker

1 1 1 1 1 1 1 1 2 2 2 22 3 3 3 3 3 4 4 4 5 5 56 6 7 8 9 11 11 12 14 16 18 1818 22 22 23 29 32 34 38 41 46 53 53

Table 3 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker A

Criteria GG GW EW MOW EPWAIC 341013 343696 340435 341317 342750AICc 335559 338241 334981 335862 343296KS 06735 06046 07447 07457 05751

Table 4 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0379 0044 (03181 04969)120579 6446 0703 (47070 76915)120590 0727 0290 (03897 14691)119910lowast 3093 0582 (17374 40236)

more affected by problems with the Pricker Further relia-bility models were individually adjusted and thereby com-pared as described in the next section

32 Preventive Maintenance In this section we discuss aparametric approach in order to perform a predictive analysisfor the lifetime of the components

321 Pricker from Machine A Table 2 presents a high defectrate after a short repair time as well compromising the cost ofthe production The experiment considered a total period of30months as said beforeThe operating equipment had threeoff-seasons these periods were not included in the datasetThe equipmentwas only observed during the time of its activeoperation

Figure 2 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

From the TTT-plot we observed that the proposed datahas unimodal hazard rate which implies that all the proposedmodels may be used to describe the proposed datasetAdditionally the survival function adjusted by the differentdistributions shows that the proposedmodels provide a goodfit for the proposed data In order to discriminate the best fitwe considered the results of AIC and AICc (see Table 3)

Among the proposed models the exponentiated Weibulldistribution has superior goodness of fit since the AIC andAICc returned smaller values Therefore using the expo-nentiated Weibull distribution we computed the maximumlikelihood estimates and the predictive value for 25 (seeTable 4) Hereafter as we considered the quantile functionto obtain the predictive value the confidence intervals (CI)related to this estimate were obtained from bootstrap tech-nique [27]

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

4 Modelling and Simulation in Engineering

23 The Exponentiated Weibull Distribution Introduced byMudholkar et al [16] the exponentiatedWeibull distributionwith PDF is given by

119891 (119905 120590 120601 120572) = 120572120601120590 ( 119905120590)120572minus1

exp(minus( 119905120590)120572)

sdot (1 minus exp (minus( 119905120590)120572))120601minus1

(20)

where 120590 gt 0 120601 gt 0 and 120572 gt 0The exponentiatedWeibull distribution includes theWei-

bull distribution (120601 = 1) and the exponentiated exponentialdistribution (120572 = 1) The survival function is given by

119878 (119905 120590 120601 120572) = 1 minus (1 minus exp(minus( 119905120590)120572))120601 (21)

The hazard function of the GG distribution is

ℎ (119905 120601 120583 120572)= 120572120601 (119905120590)120572minus1 exp (minus (119905120590)120572) (1 minus exp (minus (119905120590)120572))120601minus1

120590 (1 minus (1 minus exp (minus (119905120590)120572))120601) (22)

The shapes of the hazard function are analogous to theGG and GW distribution Additionally the quantile functionof the EW distribution has closed form and is given by

119876 (119906 120590 120601 120572) = 120590 (minus log (1 minus 1199061120601))1120572 (23)

The 119896th moment of the EW distribution is given by

120583119896 = int10119876 (119906 120590 120601 120572)119896 119889119906

= 120579120590119896Γ(119896120572 + 1)(1 + infinsum119894=1

119886119894 [(119894 + 1)119896120572+1]) (24)

where 119896 isin N 119886119894 = (minus1)119894(120579 minus 1)(120579 minus 2) sdot sdot sdot (120579 minus 1 minus 119894 minus 1)(119894)minus1The proof of this equality is presented by Choudhury [23]

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EW(120590 120601 120572)Then the likelihoodfunction related to the PDF (20) is given by

119871 (120590 120601 120572 t) = 120572119899120601119899120590119899119899prod119894=1

(119905119894120590)120572minus1

sdot (1 minus exp(minus(119905119894120590)120572))120601minus1

sdot exp(minus 119899sum119894=1

(119905119894120590)120572)

(25)

The log-likelihood is given by

119897 (120590 120601 120572 t) = 119899 log (120572120601)+ (120601 minus 1) 119899sum

119894=1

log(1 minus exp (minus(119905119894120590)120572))

+ (120572 minus 1) 119899sum119894=1

log (119905119894) minus 119899sum119894=1

(119905119894120590)120572

minus 119899120572 log (120590)

(26)

Setting the partial derivatives (120597120597120590)119897(120590 120601 120572 t) (120597120597120601)119897(120590 120601 120572 t) and (120597120597120572)119897(120590 120601 120572 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120572 + 119899sum119894=1

log (119905119894) + (120601 minus 1)120590120572

119899sum119894=1

119905120572119894 log (119905119894120590)exp ((119905119894120590)120572) minus 1

minus 119899sum119894=1

(119905119894120590)120572

log(119905119894120590) exp (minus(119905119894120590)120572)

minus 119899 log (120590) = 0minus 119899120572120590 minus 120572120590120572+1

119899sum119894=1

119905120572119894 + 120572120590120572119899sum119894=1

(120601 minus 1) 119905120572119894120590 minus 120590 exp ((119905119894120590)120572) = 0120601 = minus 119899

sum119899119894=1 log (1 minus exp (minus (119905119894120590)120572)) sdot

(27)

24 The Marshall-Olkin-Weibull Distribution Marshall andOlkin [17] presented a new procedure for introducing anadditional parameter into a family of distribution In this casethe authors applied such procedure in the Weibull distribu-tion The obtained PDF of the MOW distribution is given by

119891 (119905 120582 120572 120574) = 120572120574120582119905120574minus1119890minus120582119905120574(1 minus (1 minus 120572) 119890minus120582119905120574)2 (28)

where 120582 gt 0 120572 gt 0 and 120574 gt 0 The MOW distributionarises naturally in competing risks scenarios Let 119883 =min(1198791 1198792 119879119872) where119872 is a random variable with geo-metrical distribution and 119879119894 are assumed to be independentand identically distributed according to a Weibull distribu-tion then the119883 has a PDF given by (28) Cordeiro and Lem-onte [24] derived many properties and the parameter esti-mators for the MOW distribution the following results wereobtained from the cited work The survival function is givenby

119878 (119905 120582 120572 120574) = 1 minus 1 minus 119890minus1205821199051205741 minus (1 minus 120572) 119890minus120582119905120574 sdot (29)

The hazard function of the MOW distribution is

ℎ (119905 120582 120572 120574) = 120574120582119905120574minus11 minus (1 minus 120572) 119890minus120582119905120574 (30)

Modelling and Simulation in Engineering 5

where its behavior is constant increasing decreasing bath-tub and unimodal Moreover the quantile function of theMOW distribution has closed form and is given by

119876 (119906 120582 120572 120574) = 120582minus1120574 (log(1 minus (1 minus 120572) 1199061 minus 119906 ))1120574 (31)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim MOW(120582 120572 120574) Then the like-lihood function related to the PDF (28) is given by

119871 (120582 120572 120574 t)= 120572119899120574119899120582119899 119899prod

119894=1

119905120574minus1119894(1 minus (1 minus 120572) 119890minus120582119905120574119894 )2 exp(minus120582

119899sum119894=1

119905120574119894 ) (32)

The log-likelihood is given by

119897 (120582 120572 120574 t) = 119899 log (120572) + 119899 log (120574) + 119899 log (120582)+ (120574 minus 1) 119899sum

119894=1

log (119905119894)

minus 2 119899sum119894=1

log (1 minus (1 minus 120572) 119890minus120582119905120574119894 ) minus 120582 119899sum119894=1

119905120574119894 (33)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120574 t) (120597120597120572)119897(120582 120572 120574 t) and (120597120597120574)119897(120582 120572 120574 t) equal to 0 we obtainthe following maximum likelihood estimators

119899120572 minus 2 119899sum119894=1

119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0

119899120582 minus 119899sum119894=1

119905120574119894 minus 2 (1 minus 120572)119899sum119894=1

119905120574119894 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0119899120574 +119899sum119894=1

log (119905119894) minus 2 (1 minus 120572) 119899sum119894=1

120582119905120574119894 log (119905119894) 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894= 120582 119899sum119894=1

119905120574119894 log (119905119894)

(34)

for more details see Cordeiro and Lemonte [24]

25 The Extended Poisson-Weibull Distribution Ramos et al[18] introduced the extended Poisson-Weibull (EPW) distri-bution as a generalization of Weibull-Poisson distribution(see Hemmati et al [25]) where its PDF is given by

119891 (119905 120582 120572 120601) = 120572120582120601119905120572minus1119890minus120601119905120572minus120582119890minus1206011199051205721 minus 119890minus120582 (35)

where 120582 isin Rlowast 120601 gt 0 and 120572 gt 0 Analogously to the MOWdistribution the EWP model arises naturally in competingrisks scenarios Let 119883 = min(1198791 1198792 119879119872) where 119872 isa random variable with a zero-truncated Poisson distribution

and 119879119894 are assumed to be independent and identically dis-tributed according to aWeibull distribution then the119883 has aPDF given by (35) The survival function is given by

119878 (119905 120582 120572 120601) = 1 minus exp (minus120582119890minus120601119905120572)1 minus 119890minus120582 (36)

The hazard function of the GG distribution is

ℎ (119905 120582 120572 120601) = 120582120601119905120572minus1119890minus120601119905120572minus120582119890minus120601119905120572 (1 minus 119890minus120582119890minus120601119905120572)minus1 (37)

For the EWP distribution the hazard function has differ-ent shapes such as constant increasing decreasing bathtuband upside-down bathtub Furthermore the quantile func-tion of the EPW distribution has closed form and is given by

119876 (119906 120582 120572 120601)= (minus 1120601 log(1 minus log ((119890120582 minus 1) 119901 + 1)

120582 ))1120572

(38)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EPW(120582 120572 120601) Then the like-lihood function related to the PDF (20) is given by

119871 (120582 120572 120601 t)= 120572119899120582119899120601119899(1 minus 119890minus120582)119899

119899prod119894=1

119905120572minus1119894 exp(minus120601 119899sum119894=1

119905120572119894 minus 120582 119899sum119894=1

119890minus120601119905120572119894 ) (39)

The log-likelihood is given by

119897 (120582 120572 120601 t) = 119899 log (120572120582120601) minus 119899 log (1 minus 119890minus120582) minus 120601 119899sum119894=1

119905120572119894+ (120572 minus 1) 119899sum

119894=1

log (119905119894) minus 120582 119899sum119894=1

119890minus120601119905120572119894 (40)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120601 t) (120597120597120572)119897(120582 120572 120601 t) and (120597120597120601)119897(120582 120572 120601 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120582 + 1198991 minus 119890120582 minus119899sum119894=1

119890minus120601119905120572119894 = 0119899120572 + 119899sum119894=1

log (119905119894) minus 120601 119899sum119894=1

119905120572119894 log (119905119894) + 120601120582 119899sum119894=1

119905120572119894 log (119905119894) 119890minus120601119905120572119894= 0

119899120601 minus 119899sum119894=1

119905120572119894 + 120582 119899sum119894=1

119905120572119894 119890minus120601119905120572119894 = 0

(41)

26 Goodness of Fit Firstly in order to verify the behaviorof the empirical data the Total Time on Test plot (TTT-plot)was considered (Barlow and Campo [26]) The TTT-plot isobtained through the plot of [119903119899 119866(119903119899)] where

119866( 119903119899) =(sum119903119894=1 119905119894 + (119899 minus 119903) 119905(119903))sum119899119894=1 119905119894 (42)

6 Modelling and Simulation in Engineering

Table 1 Maintenance distribution preventive (P) and corrective (C)stops per crop

Crop 1 Crop 2 Crop 3P C P C P C

Machine A 39 232 32 255 23 127Machine B 37 199 32 182 23 166

119903 = 1 119899 119894 = 1 119899 and 119905(119894) is the ordered data For datawith concave (convex) curve the hazard function has increas-ing (decreasing) shape If the behavior starts convex andthen becomes concave (concave and then convex) the hazardfunction has bathtub (inverse bathtub) shape

The goodness of fit is checked considering the Kolmog-orov-Smirnov (KS) test This procedure is based on the KSstatistic 119863119899 = sup |119865119899(119905) minus 119865(119905 120579)| where sup 119905 is the supre-mumof the set of distances119865119899(119905) is the empirical distributionfunction and 119865(119905 120579) is CDF A hypothesis test is conductedat the 5 level of significance to test whether or not the datacomes from119865(119905 120579) In this case the null hypothesis is rejectedif the returned 119901 value is smaller than 005

The following discrimination criterion methods wereadopted Akaike information criteria (AIC) and the correctedAIC (AICc) computed respectively by AIC = minus2119897( t) + 2119896and AICc = AIC + 2119896(119896 + 1)(119899 minus 119896 minus 1)minus1 where 119896 is thenumber of parameters to be fitted and isMLEs of 120579 For a setof candidatemodels for t the best one provides theminimumvalues

3 Data Collection and Empirical Analysis

The dataset came from two sources a manual stop systemwhich brings the history of revisions and corrective stopsof two sugarcane harvesters and data from the onboardcomputers of the harvesters which provide information onthe operation of the machine The data were collected fromJanuary 2015 to August 2017 a period corresponding to 25harvests (crops) that is a period of thirty months of activity

31 Empirical Analysis Firstly considering all the stops andtheir reasons records of the performance of the predictivemaintenance are required to be observed In total 1347 stopswere observed of which 186 were preventive and 1161 correc-tive stops Thus it is possible to observe the superior amountof unplanned stops thus questioning the effectiveness ofpreventive maintenance Table 1 shows the failure among theharvests considering both machines analysis

The Pricker and transmission from each machine wereselected given their complexity in the maintenance Figure 1describes the number of failures per year divided by harvestconsidering their temporal sparsity by which items analyzedin this report correspond to 18 of the stops

It is possible to notice a difference in the machinesrsquobehavior both machines appear to be equally affected by theproblems of transmission and Pricker but the machine B is

Table 2 Dataset related to the sugarcane harvesterrsquos Pricker

1 1 1 1 1 1 1 1 2 2 2 22 3 3 3 3 3 4 4 4 5 5 56 6 7 8 9 11 11 12 14 16 18 1818 22 22 23 29 32 34 38 41 46 53 53

Table 3 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker A

Criteria GG GW EW MOW EPWAIC 341013 343696 340435 341317 342750AICc 335559 338241 334981 335862 343296KS 06735 06046 07447 07457 05751

Table 4 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0379 0044 (03181 04969)120579 6446 0703 (47070 76915)120590 0727 0290 (03897 14691)119910lowast 3093 0582 (17374 40236)

more affected by problems with the Pricker Further relia-bility models were individually adjusted and thereby com-pared as described in the next section

32 Preventive Maintenance In this section we discuss aparametric approach in order to perform a predictive analysisfor the lifetime of the components

321 Pricker from Machine A Table 2 presents a high defectrate after a short repair time as well compromising the cost ofthe production The experiment considered a total period of30months as said beforeThe operating equipment had threeoff-seasons these periods were not included in the datasetThe equipmentwas only observed during the time of its activeoperation

Figure 2 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

From the TTT-plot we observed that the proposed datahas unimodal hazard rate which implies that all the proposedmodels may be used to describe the proposed datasetAdditionally the survival function adjusted by the differentdistributions shows that the proposedmodels provide a goodfit for the proposed data In order to discriminate the best fitwe considered the results of AIC and AICc (see Table 3)

Among the proposed models the exponentiated Weibulldistribution has superior goodness of fit since the AIC andAICc returned smaller values Therefore using the expo-nentiated Weibull distribution we computed the maximumlikelihood estimates and the predictive value for 25 (seeTable 4) Hereafter as we considered the quantile functionto obtain the predictive value the confidence intervals (CI)related to this estimate were obtained from bootstrap tech-nique [27]

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Modelling and Simulation in Engineering 5

where its behavior is constant increasing decreasing bath-tub and unimodal Moreover the quantile function of theMOW distribution has closed form and is given by

119876 (119906 120582 120572 120574) = 120582minus1120574 (log(1 minus (1 minus 120572) 1199061 minus 119906 ))1120574 (31)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim MOW(120582 120572 120574) Then the like-lihood function related to the PDF (28) is given by

119871 (120582 120572 120574 t)= 120572119899120574119899120582119899 119899prod

119894=1

119905120574minus1119894(1 minus (1 minus 120572) 119890minus120582119905120574119894 )2 exp(minus120582

119899sum119894=1

119905120574119894 ) (32)

The log-likelihood is given by

119897 (120582 120572 120574 t) = 119899 log (120572) + 119899 log (120574) + 119899 log (120582)+ (120574 minus 1) 119899sum

119894=1

log (119905119894)

minus 2 119899sum119894=1

log (1 minus (1 minus 120572) 119890minus120582119905120574119894 ) minus 120582 119899sum119894=1

119905120574119894 (33)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120574 t) (120597120597120572)119897(120582 120572 120574 t) and (120597120597120574)119897(120582 120572 120574 t) equal to 0 we obtainthe following maximum likelihood estimators

119899120572 minus 2 119899sum119894=1

119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0

119899120582 minus 119899sum119894=1

119905120574119894 minus 2 (1 minus 120572)119899sum119894=1

119905120574119894 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894 = 0119899120574 +119899sum119894=1

log (119905119894) minus 2 (1 minus 120572) 119899sum119894=1

120582119905120574119894 log (119905119894) 119890minus1205821199051205741198941 minus (1 minus 120572) 119890minus120582119905120574119894= 120582 119899sum119894=1

119905120574119894 log (119905119894)

(34)

for more details see Cordeiro and Lemonte [24]

25 The Extended Poisson-Weibull Distribution Ramos et al[18] introduced the extended Poisson-Weibull (EPW) distri-bution as a generalization of Weibull-Poisson distribution(see Hemmati et al [25]) where its PDF is given by

119891 (119905 120582 120572 120601) = 120572120582120601119905120572minus1119890minus120601119905120572minus120582119890minus1206011199051205721 minus 119890minus120582 (35)

where 120582 isin Rlowast 120601 gt 0 and 120572 gt 0 Analogously to the MOWdistribution the EWP model arises naturally in competingrisks scenarios Let 119883 = min(1198791 1198792 119879119872) where 119872 isa random variable with a zero-truncated Poisson distribution

and 119879119894 are assumed to be independent and identically dis-tributed according to aWeibull distribution then the119883 has aPDF given by (35) The survival function is given by

119878 (119905 120582 120572 120601) = 1 minus exp (minus120582119890minus120601119905120572)1 minus 119890minus120582 (36)

The hazard function of the GG distribution is

ℎ (119905 120582 120572 120601) = 120582120601119905120572minus1119890minus120601119905120572minus120582119890minus120601119905120572 (1 minus 119890minus120582119890minus120601119905120572)minus1 (37)

For the EWP distribution the hazard function has differ-ent shapes such as constant increasing decreasing bathtuband upside-down bathtub Furthermore the quantile func-tion of the EPW distribution has closed form and is given by

119876 (119906 120582 120572 120601)= (minus 1120601 log(1 minus log ((119890120582 minus 1) 119901 + 1)

120582 ))1120572

(38)

For parameter estimation let 1198791 119879119899 be a randomsample of size 119899 where 119879 sim EPW(120582 120572 120601) Then the like-lihood function related to the PDF (20) is given by

119871 (120582 120572 120601 t)= 120572119899120582119899120601119899(1 minus 119890minus120582)119899

119899prod119894=1

119905120572minus1119894 exp(minus120601 119899sum119894=1

119905120572119894 minus 120582 119899sum119894=1

119890minus120601119905120572119894 ) (39)

The log-likelihood is given by

119897 (120582 120572 120601 t) = 119899 log (120572120582120601) minus 119899 log (1 minus 119890minus120582) minus 120601 119899sum119894=1

119905120572119894+ (120572 minus 1) 119899sum

119894=1

log (119905119894) minus 120582 119899sum119894=1

119890minus120601119905120572119894 (40)

Setting the partial derivatives (120597120597120582)119897(120582 120572 120601 t) (120597120597120572)119897(120582 120572 120601 t) and (120597120597120601)119897(120582 120572 120601 t) equal to 0 we obtain thefollowing maximum likelihood estimators

119899120582 + 1198991 minus 119890120582 minus119899sum119894=1

119890minus120601119905120572119894 = 0119899120572 + 119899sum119894=1

log (119905119894) minus 120601 119899sum119894=1

119905120572119894 log (119905119894) + 120601120582 119899sum119894=1

119905120572119894 log (119905119894) 119890minus120601119905120572119894= 0

119899120601 minus 119899sum119894=1

119905120572119894 + 120582 119899sum119894=1

119905120572119894 119890minus120601119905120572119894 = 0

(41)

26 Goodness of Fit Firstly in order to verify the behaviorof the empirical data the Total Time on Test plot (TTT-plot)was considered (Barlow and Campo [26]) The TTT-plot isobtained through the plot of [119903119899 119866(119903119899)] where

119866( 119903119899) =(sum119903119894=1 119905119894 + (119899 minus 119903) 119905(119903))sum119899119894=1 119905119894 (42)

6 Modelling and Simulation in Engineering

Table 1 Maintenance distribution preventive (P) and corrective (C)stops per crop

Crop 1 Crop 2 Crop 3P C P C P C

Machine A 39 232 32 255 23 127Machine B 37 199 32 182 23 166

119903 = 1 119899 119894 = 1 119899 and 119905(119894) is the ordered data For datawith concave (convex) curve the hazard function has increas-ing (decreasing) shape If the behavior starts convex andthen becomes concave (concave and then convex) the hazardfunction has bathtub (inverse bathtub) shape

The goodness of fit is checked considering the Kolmog-orov-Smirnov (KS) test This procedure is based on the KSstatistic 119863119899 = sup |119865119899(119905) minus 119865(119905 120579)| where sup 119905 is the supre-mumof the set of distances119865119899(119905) is the empirical distributionfunction and 119865(119905 120579) is CDF A hypothesis test is conductedat the 5 level of significance to test whether or not the datacomes from119865(119905 120579) In this case the null hypothesis is rejectedif the returned 119901 value is smaller than 005

The following discrimination criterion methods wereadopted Akaike information criteria (AIC) and the correctedAIC (AICc) computed respectively by AIC = minus2119897( t) + 2119896and AICc = AIC + 2119896(119896 + 1)(119899 minus 119896 minus 1)minus1 where 119896 is thenumber of parameters to be fitted and isMLEs of 120579 For a setof candidatemodels for t the best one provides theminimumvalues

3 Data Collection and Empirical Analysis

The dataset came from two sources a manual stop systemwhich brings the history of revisions and corrective stopsof two sugarcane harvesters and data from the onboardcomputers of the harvesters which provide information onthe operation of the machine The data were collected fromJanuary 2015 to August 2017 a period corresponding to 25harvests (crops) that is a period of thirty months of activity

31 Empirical Analysis Firstly considering all the stops andtheir reasons records of the performance of the predictivemaintenance are required to be observed In total 1347 stopswere observed of which 186 were preventive and 1161 correc-tive stops Thus it is possible to observe the superior amountof unplanned stops thus questioning the effectiveness ofpreventive maintenance Table 1 shows the failure among theharvests considering both machines analysis

The Pricker and transmission from each machine wereselected given their complexity in the maintenance Figure 1describes the number of failures per year divided by harvestconsidering their temporal sparsity by which items analyzedin this report correspond to 18 of the stops

It is possible to notice a difference in the machinesrsquobehavior both machines appear to be equally affected by theproblems of transmission and Pricker but the machine B is

Table 2 Dataset related to the sugarcane harvesterrsquos Pricker

1 1 1 1 1 1 1 1 2 2 2 22 3 3 3 3 3 4 4 4 5 5 56 6 7 8 9 11 11 12 14 16 18 1818 22 22 23 29 32 34 38 41 46 53 53

Table 3 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker A

Criteria GG GW EW MOW EPWAIC 341013 343696 340435 341317 342750AICc 335559 338241 334981 335862 343296KS 06735 06046 07447 07457 05751

Table 4 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0379 0044 (03181 04969)120579 6446 0703 (47070 76915)120590 0727 0290 (03897 14691)119910lowast 3093 0582 (17374 40236)

more affected by problems with the Pricker Further relia-bility models were individually adjusted and thereby com-pared as described in the next section

32 Preventive Maintenance In this section we discuss aparametric approach in order to perform a predictive analysisfor the lifetime of the components

321 Pricker from Machine A Table 2 presents a high defectrate after a short repair time as well compromising the cost ofthe production The experiment considered a total period of30months as said beforeThe operating equipment had threeoff-seasons these periods were not included in the datasetThe equipmentwas only observed during the time of its activeoperation

Figure 2 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

From the TTT-plot we observed that the proposed datahas unimodal hazard rate which implies that all the proposedmodels may be used to describe the proposed datasetAdditionally the survival function adjusted by the differentdistributions shows that the proposedmodels provide a goodfit for the proposed data In order to discriminate the best fitwe considered the results of AIC and AICc (see Table 3)

Among the proposed models the exponentiated Weibulldistribution has superior goodness of fit since the AIC andAICc returned smaller values Therefore using the expo-nentiated Weibull distribution we computed the maximumlikelihood estimates and the predictive value for 25 (seeTable 4) Hereafter as we considered the quantile functionto obtain the predictive value the confidence intervals (CI)related to this estimate were obtained from bootstrap tech-nique [27]

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Modelling and Simulation in Engineering

Table 1 Maintenance distribution preventive (P) and corrective (C)stops per crop

Crop 1 Crop 2 Crop 3P C P C P C

Machine A 39 232 32 255 23 127Machine B 37 199 32 182 23 166

119903 = 1 119899 119894 = 1 119899 and 119905(119894) is the ordered data For datawith concave (convex) curve the hazard function has increas-ing (decreasing) shape If the behavior starts convex andthen becomes concave (concave and then convex) the hazardfunction has bathtub (inverse bathtub) shape

The goodness of fit is checked considering the Kolmog-orov-Smirnov (KS) test This procedure is based on the KSstatistic 119863119899 = sup |119865119899(119905) minus 119865(119905 120579)| where sup 119905 is the supre-mumof the set of distances119865119899(119905) is the empirical distributionfunction and 119865(119905 120579) is CDF A hypothesis test is conductedat the 5 level of significance to test whether or not the datacomes from119865(119905 120579) In this case the null hypothesis is rejectedif the returned 119901 value is smaller than 005

The following discrimination criterion methods wereadopted Akaike information criteria (AIC) and the correctedAIC (AICc) computed respectively by AIC = minus2119897( t) + 2119896and AICc = AIC + 2119896(119896 + 1)(119899 minus 119896 minus 1)minus1 where 119896 is thenumber of parameters to be fitted and isMLEs of 120579 For a setof candidatemodels for t the best one provides theminimumvalues

3 Data Collection and Empirical Analysis

The dataset came from two sources a manual stop systemwhich brings the history of revisions and corrective stopsof two sugarcane harvesters and data from the onboardcomputers of the harvesters which provide information onthe operation of the machine The data were collected fromJanuary 2015 to August 2017 a period corresponding to 25harvests (crops) that is a period of thirty months of activity

31 Empirical Analysis Firstly considering all the stops andtheir reasons records of the performance of the predictivemaintenance are required to be observed In total 1347 stopswere observed of which 186 were preventive and 1161 correc-tive stops Thus it is possible to observe the superior amountof unplanned stops thus questioning the effectiveness ofpreventive maintenance Table 1 shows the failure among theharvests considering both machines analysis

The Pricker and transmission from each machine wereselected given their complexity in the maintenance Figure 1describes the number of failures per year divided by harvestconsidering their temporal sparsity by which items analyzedin this report correspond to 18 of the stops

It is possible to notice a difference in the machinesrsquobehavior both machines appear to be equally affected by theproblems of transmission and Pricker but the machine B is

Table 2 Dataset related to the sugarcane harvesterrsquos Pricker

1 1 1 1 1 1 1 1 2 2 2 22 3 3 3 3 3 4 4 4 5 5 56 6 7 8 9 11 11 12 14 16 18 1818 22 22 23 29 32 34 38 41 46 53 53

Table 3 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker A

Criteria GG GW EW MOW EPWAIC 341013 343696 340435 341317 342750AICc 335559 338241 334981 335862 343296KS 06735 06046 07447 07457 05751

Table 4 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0379 0044 (03181 04969)120579 6446 0703 (47070 76915)120590 0727 0290 (03897 14691)119910lowast 3093 0582 (17374 40236)

more affected by problems with the Pricker Further relia-bility models were individually adjusted and thereby com-pared as described in the next section

32 Preventive Maintenance In this section we discuss aparametric approach in order to perform a predictive analysisfor the lifetime of the components

321 Pricker from Machine A Table 2 presents a high defectrate after a short repair time as well compromising the cost ofthe production The experiment considered a total period of30months as said beforeThe operating equipment had threeoff-seasons these periods were not included in the datasetThe equipmentwas only observed during the time of its activeoperation

Figure 2 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

From the TTT-plot we observed that the proposed datahas unimodal hazard rate which implies that all the proposedmodels may be used to describe the proposed datasetAdditionally the survival function adjusted by the differentdistributions shows that the proposedmodels provide a goodfit for the proposed data In order to discriminate the best fitwe considered the results of AIC and AICc (see Table 3)

Among the proposed models the exponentiated Weibulldistribution has superior goodness of fit since the AIC andAICc returned smaller values Therefore using the expo-nentiated Weibull distribution we computed the maximumlikelihood estimates and the predictive value for 25 (seeTable 4) Hereafter as we considered the quantile functionto obtain the predictive value the confidence intervals (CI)related to this estimate were obtained from bootstrap tech-nique [27]

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Modelling and Simulation in Engineering 7

8

99 11

116

18 16

25 26

106

2015 2016 20172015 2016 2017

Machine A Machine B

14 11

46 5

12

10

24

7

18

8

17

Transmission Transmission

Pricker Pricker

Transmission Transmission

Pricker Pricker

Figure 1 Maintenance distribution in each harvester

00 02 04 06 08 10

00

02

04

06

08

10

rn

G(rn)

(a)

0 10 20 30 40 50

00

02

04

06

08

10

Time

S(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 2 Pricker A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

FromTable 4 we observe that the predictivemaintenanceshould be done in approximately 3 days after the last failurewith confidence interval between 2 and 4 days

322 Pricker fromMachine B A similar behavior is observedfor the Pricker in the machine B shown in Table 5 presentinga high defect rate as well The approach was maintainedconsidering only the time during its active operation

Figure 3 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionsimilar to the previous machine

TheTTT-plot shows that the proposed data has unimodalhazard rate which implies that all the proposed models maybe used to describe the dataset Analogously to the previous

Table 5 Dataset related to the sugarcane harvesterrsquos Pricker B

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 3 3 4 4 55 5 5 5 5 5 6 7 7 8 8 88 8 9 9 11 11 11 11 11 11 12 1314 16 16 21 23 24 27 28 38 43 44

case the survival function adjusted by the different distribu-tions shows that the proposed models provide a good fit forthe proposed data Therefore to discriminate the best fit weconsidered the results of AIC and AICc (see Table 6)

From the obtained results we observe that the EW distri-bution also provided the best fit among the proposed model

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

8 Modelling and Simulation in Engineering

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 3 Pricker B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function and thehazard function adjusted by distribution

Table 6 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EPWAIC 382063 384102 381790 382641 383772AICc 376500 378538 376226 377077 384209KS 03055 03628 02737 03900 04443

Table 7 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EW distribution

120579 MLE SD CI95(120579)120572 0457 0050 (03955 05835)120579 5434 0763 (35493 68379)120590 1083 0327 (06879 19727)119910lowast 2497 0459 (18212 35760)

Furthermore the maximum likelihood estimates for the EWdistribution were computed as well as the predictive valuefor 25 Table 7 presents the MLEs standard deviations and95 confidence intervals for 120572 120579 120590 and 119910lowast related to the EWdistribution

Table 7 results suggest that predictivemaintenance shouldbe done in approximately 3 days considering a point estima-tion or given a 95 confidence interval it would be between2 to 4 days approximately Thereby Pricker among machinesshowed no difference in performance ever

Table 8 Dataset related to the sugarcane harvesterrsquos transmissionA

1 1 1 1 1 1 1 1 1 1 1 12 2 2 3 3 3 3 4 5 6 6 66 7 7 8 8 8 11 11 12 13 13 1314 15 16 17 18 18 19 19 21 22 23 2931 32 34 44 52

323 Transmission fromMachine A Table 8 shows that morethan 50 of the defect rate appears until 8 days right after itsrepair for the transmission for the machine A

Figure 4 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distribution

As can be seen in the TTT-plot we observed that theproposed data has also fulfilled the hazard rate shape pre-supposition However from the survival function there isan indication that the generalizedWeibull distribution is not agood candidate to describe the propose data Table 9 presentsthe results of AIC and AICc in order to discriminate the bestfit

From Table 9 we can see that the GW distribution has the119901 value of the KS test smaller than 005 therefore it is nota possible candidate to fit the data Overall the GG distribu-tion has a better fit since it has the smaller AIC and AICcTherefore we computed the maximum likelihood estimatesand the predictive value for 25 using the GG distributionTable 10 presents the MLEs standard deviations and 95

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Modelling and Simulation in Engineering 9

00 02 04 06 08 10rn

G(rn)

00

02

04

06

08

10

(a)

0 10 20 30 40 50Time

S(t)

00

02

04

06

08

10

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 4 Transmission A empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 9 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the Pricker B

Criteria GG GW EW MOW EWPAIC 368074 381036 368271 368375 368385AICc 362563 375526 362761 362864 368875KS 02532 00035 02672 03945 03738

Table 10 MLE standard deviation and 95 confidence intervalsfor 120601 120583 120572 and 119910lowast related to the GG distribution

120579 MLE SD CI95(120579)120601 3011 0543 (17396 39936)120583 1086 0525 (02389 22682)120572 0495 0075 (04214 07124)119910lowast 2807 0635 (18487 43526)

confidence intervals for 120601 120583 120572 and 119910lowast related to the GG dis-tribution

Table 10 results suggest that predictive maintenanceshould be done in approximately 3 days considering a pointestimation or given a 95confidence interval it would be be-tween 2 to 4 days approximately

324 Transmission fromMachine B Comparing to the otherequipment the transmission from the machine B presentedsmaller number of occurrence Table 11 shows the sparsity of

Table 11 Dataset related to the sugarcane harvesterrsquos transmissionB

1 2 3 3 4 5 6 6 7 9 1112 12 18 19 21 23 28 31 31 35 3739 46 61

Table 12 Results of AIC and AICc criteria and the 119901 value from theKS test for all fitted distributions considering the transmission B

Criteria GG GW EW MOW EWPAIC 202220 203201 202833 202368 201997AICc 197363 198344 197975 197511 203140KS 09382 07657 08732 07710 09622

the dataset related to the sugarcane harvesterrsquos transmissionB

Figure 5 presents the TTT-plot and the survival functionfitted by different generalizations of the Weibull distributionconsidering the transmission from machine B

From the TTT-plot we observed that the proposed datahas bathtub shape Moreover the adjusted survival functionsshow that all models are candidates to describe the lifetime ofthe transmission from the machine B

Table 12 presents the results of AIC and AICc in order todiscriminate the best fit

As shown in Table 12 the EWP distribution has theminimum AIC and AICc Therefore we computed its max-imum likelihood estimates and predictive value for 25

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Modelling and Simulation in Engineering

00

02

04

06

08

10

rn

G(rn)

00 02 04 06 08 10

(a)

0 10 20 30 40 50 60

00

02

04

06

08

10

TimeS(t)

EmpiricalGamma-WeibullGen Weibull

Exp WeibullMO-WeibullEP-Weibull

(b)

Figure 5 Transmission B empirical (a) TTT-plot and (b) contains the fitted survival superimposed to the empirical survival function andthe hazard function adjusted by distribution

Table 13 MLEs standard deviations and 95 confidence intervalsfor 120572 120579 120590 and 119910lowast related to the EPW distribution

120579 MLE SD CI95(120579)120601 0022 0034 (00137 01350)120582 minus0572 0541 (minus12492 11886)120572 1206 0137 (07579 13705)119910lowast 6748 1387 (38716 95585)

respectively Table 13 presents theMLEs standard deviationsand 95 confidence intervals for 120572 120579 120590 and 119910lowast related to theEWP distribution

Table 13 results suggest that predictive maintenanceshould be done in approximately 7 days considering a pointestimation or given a 95confidence interval it would be be-tween 4 to 10 days approximately

4 Final Remarks

In this study we considered different distributions to describethe lifetime of sugarcane harvesting machine componentsThe harvesters stand out for having a large number of correc-tive stops given the functionality in such extreme environ-mental conditions However these harvesters do not havean effective preventive maintenance policy which affects itsworking time schedule To overcome this problem we pre-sented a predictive analysis using probability models based

on its percentiles aiming to incorporate intelligence intomaintenance planning

The Weibull distribution is a popular model that can beused to describe a wide range of problems however it cannotbe used to describe data with nonmonotone hazard rateThus many generalizations of the Weibull distribution havebeen proposed to overcome this problem Since the proposeddatasets have nonmonotone hazard rate we considered someflexible generalizations such as the Gamma-Weibull the gen-eralized Weibull the exponentiated Weibull Marshall-OlkinWeibull and the extended Poisson-Weibull distribution Forthe proposed distributions some mathematical functionswere discussed as well as the parameter estimators under themaximum likelihood approach

The proposed distributions were used to fit the datasetsusing maximum likelihood estimators The exponentialWeibull presented a superior fit for both machines consider-ing the Pricker component in these cases we concluded thata predictive maintenance should be done in approximately 3days On the other hand for the transmission component thedistributions that presented better fit were respectivelythe Gamma-Weibull distribution and the extended Poisson-Weibull for machines A and B where a predictive mainte-nance should be done respectively in 3 and 7 days after thelast failure

Further work should be considered beyond the adjustedmodels by including many other generalizations of theWeibull distribution Also a structure of recurrent event datacould be included and its forecast accuracy was analyzed

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Modelling and Simulation in Engineering 11

Finally this approach should be implemented as an applica-tive helping the maintenance section in their individualizedscheduling distributions

Abbreviations

Acronyms

GW Generalized WeibullEW Exponentiated WeibullMOW Marshall-Olkin WeibullEPW Extended Poisson-WeibullGG Generalized gammaMLE Maximum likelihood estimatorPDF Probability density functionCDF Cumulative distribution functionTTT Total Time on TestKS Kolmogorov-SmirnovAIC Akaike Information CriterionAICC Corrected Akaike Information CriterionSD Standard deviationCI Confidence interval

Notations

119891(sdot) Probability density function119864(sdot) Mean functionVar(sdot) Variance function120574(119910 119909) Lower incomplete gamma functionΓ(119910 119909) Upper incomplete gamma function119865(sdot) Cumulative distribution function119878(sdot) Survival functionℎ(sdot) Hazard rate function119871(sdot) Likelihood function119897(sdot) Log-likelihood function120595(sdot) Digamma function1205951015840(sdot) Trigamma function119868(sdot) Expected Fisher information matrix120583 Positive parameter120601 Positive parameter120572 Positive parameter120573 Real parameter120582 Positive parameter119876(sdot) Quantile functionΘ Vector of parameters119867(sdot) Observed Fisher information matrix120583119896 119896th moment119866(sdot) TTT-plot119863119899 Kolmogorov-Smirnov statistic119910lowast Predictive value

Conflicts of Interest

Nopotential conflicts of interestwere reported by the authors

Acknowledgments

The research was partially supported by CNPq FAPESP andCAPES of Brazil

References

[1] B N S Company ldquoAccompanying brazilian safra Sugarcanerdquoin Harvest 201718 vol 4 pp 1ndash57 2017

[2] E Network ldquoFrommanual to mechanical harvesting Reducingenvironmental impacts and increasing cogeneration potential2012rdquo

[3] J Moubray Reliability-Centered Maintenance Industrial PressInc 1997

[4] S Supsomboon and K Hongthanapach ldquoA simulation modelfor machine efficiency improvement using reliability centeredmaintenance Case study of semiconductor factoryrdquo Modellingand Simulation in Engineering vol 2014 Article ID 956182 pp1ndash9 2014

[5] C Sriram and A Haghani ldquoAn optimization model for aircraftmaintenance scheduling and re-assignmentrdquo TransportationResearch Part A Policy and Practice vol 37 no 1 pp 29ndash482003

[6] J F Lawless Statistical Models and Methods for Lifetime Datavol 362 John Wiley amp Sons New York NY USA 1982

[7] W Weibull ldquoWide applicabilityrdquo Journal of Applied Mechanicsvol 103 no 730 pp 293ndash297 1951

[8] K G Manton and A I Yashin Inequalities of Life StatisticalAnalysis And Modeling Perspectives vol 145 Human ClocksThe Bio-cultural Meanings of Age 5 2006

[9] J IMcCoolUsing theWeibull Distribution ReliabilityModelingAnd Inference vol 950 John Wiley and Sons 950 2012

[10] D N PMurthyM Bulmer and J A Eccleston ldquoWeibull modelselection for reliability modellingrdquo Reliability Engineering ampSystem Safety vol 86 no 3 pp 257ndash267 2004

[11] H Pham and C-D Lai ldquoOn recent generalizations of theWeibull distributionrdquo IEEE Transactions on Reliability vol 56no 3 pp 454ndash458 2007

[12] C-D Lai ldquoGeneralized weibull distributionsrdquo in GeneralizedWeibull Distributions pp 23ndash75 Springer Heidelberg Ger-many 2014

[13] M H Tahir and G M Cordeiro ldquoCompounding of distribu-tions a survey andnew generalized classesrdquo Journal of StatisticalDistributions and Applications vol 3 no 1 2016

[14] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[15] G S Mudholkar D K Srivastava and G D Kollia ldquoA generali-zation of the Weibull distribution with application to the anal-ysis of survival datardquo Journal of the American Statistical Associ-ation vol 91 no 436 pp 1575ndash1583 1996

[16] G S Mudholkar D K Srivastava and M Friemer ldquoThe expo-nentiated weibull family a reanalysis of the bus-motor-failuredatardquo Technometrics vol 37 no 4 pp 436ndash445 1995

[17] A W Marshall and I Olkin ldquoA new method for adding a para-meter to a family of distributions with application to the expo-nential andWeibull familiesrdquo Biometrika vol 84 no 3 pp 641ndash652 1997

[18] P L Ramos D K A Dey F Louzada and V H Lachos ldquoAnextended poisson family of life distribution A unified approachin competitive and complementary risksrdquo Tech RepUniversityof Connecticut 2017

[19] F Louzada ldquoPolyhazard models for lifetime datardquo Biometricsvol 55 no 4 pp 1281ndash1285 1999

[20] P L Ramos J A Achcar and E Ramos ldquoMetodo eficientepara calcular os estimadores de maxima verossimilhanca dadistribuicao gama generalizadardquo Revista Brasileira de Biologiavol 32 no 2 pp 267ndash281 2014

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Modelling and Simulation in Engineering

[21] P L Ramos J A Achcar F AMoala E Ramos and F LouzadaldquoBayesian analysis of the generalized gamma distribution usingnon-informative priorsrdquo Statistics vol 51 no 4 pp 824ndash8432017

[22] J A Achcar P L Ramos and E Z Martinez ldquoSome computa-tional aspects to find accurate estimates for the parameters ofthe generalized gamma distributionrdquo Pesquisa Operacional vol37 no 2 pp 365ndash385 2017

[23] A Choudhury ldquoA simple derivation of moments of the Expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[24] G M Cordeiro and A J Lemonte ldquoOn the marshallndasholkin ex-tended weibull distributionrdquo Statistical Papers vol 54 no 2 pp1ndash12 2013

[25] F Hemmati E Khorram and S Rezakhah ldquoA new three-para-meter ageing distributionrdquo Journal of Statistical Planning andInference vol 141 no 7 pp 2266ndash2275 2011

[26] R E Barlow and R A Campo ldquoTotal time on test processes andapplications to failure data analysisrdquo DTIC Document 1975

[27] B Efron and R J Tibshirani An Introduction to the BootstrapCRC Press 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom