IFAC-Pape sOnLine 49-28 (2016) 250–255
ScienceDi ec
A ailable online a www.sciencedi ec .com
2405-8963 © 2016, IFAC (In e na ional Fede a ion o Au oma ic Con ol) Hos ing by Else ie L d. All igh s ese ed.
Pee e iew unde esponsibili y o In e na ional Fede a ion o Au oma ic Con ol.
10.1016/j.i acol.2016.11.043
©
2016, IFAC (In e na ional Fede a ion o Au oma ic Con ol) Hos ing by Else ie L d. All igh s ese ed.
Case S udy based on Inequali y Indices o he Assessmen s o Indus ial Flee s
V. González-P ida*, M. O cha d**, C. Ma ín***, A. Guillén*, J. Shambhu****, S. Sha i *****

* Uni e si y o Se ille, Spain ( icen e.gonzalezp [email protected]; [email protected])
** Uni e si y o San iago de Chile, Chile (mo cha [email protected])
*** Uni e si y o Toulouse, F ance (Ca men.Ma in@eni . )
**** Uni e si y o S a ange , No way (shambuce @gmail.com)
***** Uni e si y o Taibah, Saudi A abia (samsha i @yahoo.com)
A bs ac : This con ibu ion illus a es he ad an ages o measu es and indica o s based on he no ions o
Shannon en opy which is widely implemen ed in he modynamics, in o ma ion heo y, econome ics,
and biology. The p esen ed case s udy applies hese indica o s o a lee o indus ial asse s, in oducing
inno a i ely he en opy concep o he ield o a ailabili y. The e is no eason o deal he e ec o
pe o mance a iables in asse s independen ly wi hou aking in o accoun dispa i ies in e ms o
eliabili y, ope a ion hou s, applied main enance, ope a ing condi ions, and usage p o iles, e c. The e o e,
he cha ac e is ic o he me hod and hei independence om he he e ogenei y o he sample ha
compa es is in ended o be p omising o applica ions in a ailabili y analysis. In o he wo ds, his
esea ch p esen s a de i a ion om well-known concep s such as he Gini, Hoo e and Theil indices
illus a ing hei applica ion by he suppo o an example whe e a ibu es o di e en g oups o asse s
a e compa ed. To illus a e he applica ion we a e using wo ele an and inno a i e scena ios as case
s udy examples.
Keywo ds: Asse and main enance managemen ; Reliabili y, S a is ical App oaches; Diagnos ics,
P ognos ics, Reasoning, Decision Suppo .

1. INTRODUCTION
F om he no ion o Shannon en opy, his con ibu ion
p esen s an applica ion o indices o inequali y measu es
(Gini, Hoo e , and Theil index) o he analysis o he
a ailabili y in lee s o indus ial asse s. This is in o de o
make possible compa isons among lee s ope a ing unde
di e en bounda y condi ions and usage p o iles. The
ob ained indices e e o ha po ion o he o al g oup o
asse s ha has o be edis ibu ed in o de o ob ain a 100%
equal lee , in e ms o a ailabili y and pe o mance. The
gi en app oach and applica ion o he en opy concep o he
ield o a ailabili y is inno a i e. Pa icula ly he
cha ac e is ics o he me hod and hei independence om he
he e ogenei y o he sample ha compa es, is in ended o be
p omising o applica ions in a ailabili y analysis. The
applica ion, p oposed wi h he help o wo scena ios, is also
ele an in his new a ea o s udy and ies o illus a e his
de elopmen .
I is assumed ha componen s ope a e indi idually. Howe e
hei a ailabili y has ce ain dependency wi h he associa ed
componen s since hey belong o he same sys em. This
aspec is cla i ied h oughou his pape . Wi h ha aim, he
pape p esen s how hese indica o s can be implemen ed o
he analysis o he li e cycle o an indus ial asse . The
heo e ical de elopmen o such me hodology (including
Shannon en opy, applica ion o in -dices, and all hese
keywo ds) is de eloped in e e ence o Gonzalez-P ida e al
2016. A summa y o o mulae a e shown he e below:
Shannon En opy:
Gin i Inde x:
Hoo e Index:
Theil Index:
These concep s a e implemen ed in a p ac ical case s udy,
e e ing o he compa ison o wo scena ios (A and B).
Ne e heless, his si ua ion can be used in a ious
appli
ca ions including di e en phases o wind a ms whe e
he se e i y o he en
i onmen is di e en om one phase o
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3 d IFAC Wo kshop on Ad anced Main enance Enginee ing, Se ice and Technology
Oc obe 19-21, 2016. Bia i z, F ance
Copy igh © 2016 IFAC 250
Case S udy based on Inequali y Indices o he Assessmen s o Indus ial Flee s
V. González-P ida*, M. O cha d**, C. Ma ín***, A. Guillén*, J. Shambhu****, S. Sha i *****

* Uni e si y o Se ille, Spain ( icen e.gonzalezp [email protected]; [email protected])
** Uni e si y o San iago de Chile, Chile (mo cha [email protected])
*** Uni e si y o Toulouse, F ance (Ca men.Ma in@eni . )
**** Uni e si y o S a ange , No way (shambuce @gmail.com)
***** Uni e si y o Taibah, Saudi A abia (samsha i @yahoo.com)
A bs ac : This con ibu ion illus a es he ad an ages o measu es and indica o s based on he no ions o
Shannon en opy which is widely implemen ed in he modynamics, in o ma ion heo y, econome ics,
and biology. The p esen ed case s udy applies hese indica o s o a lee o indus ial asse s, in oducing
inno a i ely he en opy concep o he ield o a ailabili y. The e is no eason o deal he e ec o
pe o mance a iables in asse s independen ly wi hou aking in o accoun dispa i ies in e ms o
eliabili y, ope a ion hou s, applied main enance, ope a ing condi ions, and usage p o iles, e c. The e o e,
he cha ac e is ic o he me hod and hei independence om he he e ogenei y o he sample ha
compa es is in ended o be p omising o applica ions in a ailabili y analysis. In o he wo ds, his
esea ch p esen s a de i a ion om well-known concep s such as he Gini, Hoo e and Theil indices
illus a ing hei applica ion by he suppo o an example whe e a ibu es o di e en g oups o asse s
a e compa ed. To illus a e he applica ion we a e using wo ele an and inno a i e scena ios as case
s udy examples.
Keywo ds: Asse and main enance managemen ; Reliabili y, S a is ical App oaches; Diagnos ics,
P ognos ics, Reasoning, Decision Suppo .

1. INTRODUCTION
F om he no ion o Shannon en opy, his con ibu ion
p esen s an applica ion o indices o inequali y measu es
(Gini, Hoo e , and Theil index) o he analysis o he
a ailabili y in lee s o indus ial asse s. This is in o de o
make possible compa isons among lee s ope a ing unde
di e en bounda y condi ions and usage p o iles. The
ob ained indices e e o ha po ion o he o al g oup o
asse s ha has o be edis ibu ed in o de o ob ain a 100%
equal lee , in e ms o a ailabili y and pe o mance. The
gi en app oach and applica ion o he en opy concep o he
ield o a ailabili y is inno a i e. Pa icula ly he
cha ac e is ics o he me hod and hei independence om he
he e ogenei y o he sample ha compa es, is in ended o be
p omising o applica ions in a ailabili y analysis. The
applica ion, p oposed wi h he help o wo scena ios, is also
ele an in his new a ea o s udy and ies o illus a e his
de elopmen .
I is assumed ha componen s ope a e indi idually. Howe e
hei a ailabili y has ce ain dependency wi h he associa ed
componen s since hey belong o he same sys em. This
aspec is cla i ied h oughou his pape . Wi h ha aim, he
pape p esen s how hese indica o s can be implemen ed o
he analysis o he li e cycle o an indus ial asse . The
heo e ical de elopmen o such me hodology (including
Shannon en opy, applica ion o in -dices, and all hese
keywo ds) is de eloped in e e ence o Gonzalez-P ida e al
2016. A summa y o o mulae a e shown he e below:
Shannon En opy:
Gin i Inde x:
Hoo e Index:
Theil Index:
These concep s a e implemen ed in a p ac ical case s udy,
e e ing o he compa ison o wo scena ios (A and B).
Ne e heless, his si ua ion can be used in a ious
applica ions including di e en phases o wind a ms whe e
he se e i y o he en i onmen is di e en om one phase o

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

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1 1
1
1
1
))])/(/()
) / (
/
ln(())/(
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) / (
[(
)2/1(
3 d IFAC Wo kshop on Ad anced Main enance Enginee ing, Se ice and Technology
Oc obe 19-21, 2016. Bia i z, F ance
Copy igh © 2016 IFAC 250
Case S udy based on Inequali y Indices o he Assessmen s o Indus ial Flee s
V. González-P ida*, M. O cha d**, C. Ma ín***, A. Guillén*, J. Shambhu****, S. Sha i *****

* Uni e si y o Se ille, Spain ( icen e.gonzalezp [email protected]; [email protected])
** Uni e si y o San iago de Chile, Chile (mo cha [email protected])
*** Uni e si y o Toulouse, F ance (Ca men.Ma in@eni . )
**** Uni e si y o S a ange , No way (shambuce @gmail.com)
***** Uni e si y o Taibah, Saudi A abia (samsha i @yahoo.com)
A bs ac : This con ibu ion illus a es he ad an ages o measu es and indica o s based on he no ions o
Shannon en opy which is widely implemen ed in he modynamics, in o ma ion heo y, econome ics,
and biology. The p esen ed case s udy applies hese indica o s o a lee o indus ial asse s, in oducing
inno a i ely he en opy concep o he ield o a ailabili y. The e is no eason o deal he e ec o
pe o mance a iables in asse s independen ly wi hou aking in o accoun dispa i ies in e ms o
eliabili y, ope a ion hou s, applied main enance, ope a ing condi ions, and usage p o iles, e c. The e o e,
he cha ac e is ic o he me hod and hei independence om he he e ogenei y o he sample ha
compa es is in ended o be p omising o applica ions in a ailabili y analysis. In o he wo ds, his
esea ch p esen s a de i a ion om well-known concep s such as he Gini, Hoo e and Theil indices
illus a ing hei applica ion by he suppo o an example whe e a ibu es o di e en g oups o asse s
a e compa ed. To illus a e he applica ion we a e using wo ele an and inno a i e scena ios as case
s udy examples.
Keywo ds: Asse and main enance managemen ; Reliabili y, S a is ical App oaches; Diagnos ics,
P ognos ics, Reasoning, Decision Suppo .

1. INTRODUCTION
F om he no ion o Shannon en opy, his con ibu ion
p esen s an applica ion o indices o inequali y measu es
(Gini, Hoo e , and Theil index) o he analysis o he
a ailabili y in lee s o indus ial asse s. This is in o de o
make possible compa isons among lee s ope a ing unde
di e en bounda y condi ions and usage p o iles. The
ob ained indices e e o ha po ion o he o al g oup o
asse s ha has o be edis ibu ed in o de o ob ain a 100%
equal lee , in e ms o a ailabili y and pe o mance. The
gi en app oach and applica ion o he en opy concep o he
ield o a ailabili y is inno a i e. Pa icula ly he
cha ac e is ics o he me hod and hei independence om he
he e ogenei y o he sample ha compa es, is in ended o be
p omising o applica ions in a ailabili y analysis. The
applica ion, p oposed wi h he help o wo scena ios, is also
ele an in his new a ea o s udy and ies o illus a e his
de elopmen .
I is assumed ha componen s ope a e indi idually. Howe e
hei a ailabili y has ce ain dependency wi h he associa ed
componen s since hey belong o he same sys em. This
aspec is cla i ied h oughou his pape . Wi h ha aim, he
pape p esen s how hese indica o s can be implemen ed o
he analysis o he li e cycle o an indus ial asse . The
heo e ical de elopmen o such me hodology (including
Shannon en opy, applica ion o in -dices, and all hese
keywo ds) is de eloped in e e ence o Gonzalez-P ida e al
2016. A summa y o o mulae a e shown he e below:
Shannon En opy:
Gin i Inde x:
Hoo e Index:
Theil Index:
These concep s a e implemen ed in a p ac ical case s udy,
e e ing o he compa ison o wo scena ios (A and B).
Ne e heless, his si ua ion can be used in a ious
applica ions including di e en phases o wind a ms whe e
he se e i y o he en i onmen is di e en om one phase o









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


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1
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[(
)2/1(
3 d IFAC Wo kshop on Ad anced Main enance Enginee ing, Se ice and Technology
Oc obe 19-21, 2016. Bia i z, F ance
Copy igh © 2016 IFAC 250
Case S udy based on Inequali y Indices o he Assessmen s o Indus ial Flee s
V. González-P ida*, M. O cha d**, C. Ma ín***, A. Guillén*, J. Shambhu****, S. Sha i *****

* Uni e si y o Se ille, Spain ( icen e.gonzalezp [email protected]; [email protected])
** Uni e si y o San iago de Chile, Chile (mo cha [email protected])
*** Uni e si y o Toulouse, F ance (Ca men.Ma in@eni . )
**** Uni e si y o S a ange , No way (shambuce @gmail.com)
***** Uni e si y o Taibah, Saudi A abia (samsha i @yahoo.com)
A bs ac : This con ibu ion illus a es he ad an ages o measu es and indica o s based on he no ions o
Shannon en opy which is widely implemen ed in he modynamics, in o ma ion heo y, econome ics,
and biology. The p esen ed case s udy applies hese indica o s o a lee o indus ial asse s, in oducing
inno a i ely he en opy concep o he ield o a ailabili y. The e is no eason o deal he e ec o
pe o mance a iables in asse s independen ly wi hou aking in o accoun dispa i ies in e ms o
eliabili y, ope a ion hou s, applied main enance, ope a ing condi ions, and usage p o iles, e c. The e o e,
he cha ac e is ic o he me hod and hei independence om he he e ogenei y o he sample ha
compa es is in ended o be p omising o applica ions in a ailabili y analysis. In o he wo ds, his
esea ch p esen s a de i a ion om well-known concep s such as he Gini, Hoo e and Theil indices
illus a ing hei applica ion by he suppo o an example whe e a ibu es o di e en g oups o asse s
a e compa ed. To illus a e he applica ion we a e using wo ele an and inno a i e scena ios as case
s udy examples.
Keywo ds: Asse and main enance managemen ; Reliabili y, S a is ical App oaches; Diagnos ics,
P ognos ics, Reasoning, Decision Suppo .

1. INTRODUCTION
F om he no ion o Shannon en opy, his con ibu ion
p esen s an applica ion o indices o inequali y measu es
(Gini, Hoo e , and Theil index) o he analysis o he
a ailabili y in lee s o indus ial asse s. This is in o de o
make possible compa isons among lee s ope a ing unde
di e en bounda y condi ions and usage p o iles. The
ob ained indices e e o ha po ion o he o al g oup o
asse s ha has o be edis ibu ed in o de o ob ain a 100%
equal lee , in e ms o a ailabili y and pe o mance. The
gi en app oach and applica ion o he en opy concep o he
ield o a ailabili y is inno a i e. Pa icula ly he
cha ac e is ics o he me hod and hei independence om he
he e ogenei y o he sample ha compa es, is in ended o be
p omising o applica ions in a ailabili y analysis. The
applica ion, p oposed wi h he help o wo scena ios, is also
ele an in his new a ea o s udy and ies o illus a e his
de elopmen .
I is assumed ha componen s ope a e indi idually. Howe e
hei a ailabili y has ce ain dependency wi h he associa ed
componen s since hey belong o he same sys em. This
aspec is cla i ied h oughou his pape . Wi h ha aim, he
pape p esen s how hese indica o s can be implemen ed o
he analysis o he li e cycle o an indus ial asse . The
heo e ical de elopmen o such me hodology (including
Shannon en opy, applica ion o in -dices, and all hese
keywo ds) is de eloped in e e ence o Gonzalez-P ida e al
2016. A summa y o o mulae a e shown he e below:
Shannon En opy:
Gin i Inde x:
Hoo e Index:
Theil Index:
These concep s a e implemen ed in a p ac ical case s udy,
e e ing o he compa ison o wo scena ios (A and B).
Ne e heless, his si ua ion can be used in a ious
applica ions including di e en phases o wind a ms whe e
he se e i y o he en i onmen is di e en om one phase o















 n
i
n
i
iii
iii
n
i
iii
iii
NpUA
NpUA
NpUA
NpUA
H
1
11
]
)) / ((
) / (
ln[
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 
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1
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1 1
1
1
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))])/(/()
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/
ln(())/(
)) / ((
) / (
[(
)2/1(
3 d IFAC Wo kshop on Ad anced Main enance Enginee ing, Se ice and Technology
Oc obe 19-21, 2016. Bia i z, F ance
Copy igh © 2016 IFAC 250
Case S udy based on Inequali y Indices o he Assessmen s o Indus ial Flee s
V. González-P ida*, M. O cha d**, C. Ma ín***, A. Guillén*, J. Shambhu****, S. Sha i *****

* Uni e si y o Se ille, Spain ( icen e.gonzalezp [email protected]; [email protected])
** Uni e si y o San iago de Chile, Chile (mo cha [email protected])
*** Uni e si y o Toulouse, F ance (Ca men.Ma in@eni . )
**** Uni e si y o S a ange , No way (shambuce @gmail.com)
***** Uni e si y o Taibah, Saudi A abia (samsha i @yahoo.com)
A bs ac : This con ibu ion illus a es he ad an ages o measu es and indica o s based on he no ions o
Shannon en opy which is widely implemen ed in he modynamics, in o ma ion heo y, econome ics,
and biology. The p esen ed case s udy applies hese indica o s o a lee o indus ial asse s, in oducing
inno a i ely he en opy concep o he ield o a ailabili y. The e is no eason o deal he e ec o
pe o mance a iables in asse s independen ly wi hou aking in o accoun dispa i ies in e ms o
eliabili y, ope a ion hou s, applied main enance, ope a ing condi ions, and usage p o iles, e c. The e o e,
he cha ac e is ic o he me hod and hei independence om he he e ogenei y o he sample ha
compa es is in ended o be p omising o applica ions in a ailabili y analysis. In o he wo ds, his
esea ch p esen s a de i a ion om well-known concep s such as he Gini, Hoo e and Theil indices
illus a ing hei applica ion by he suppo o an example whe e a ibu es o di e en g oups o asse s
a e compa ed. To illus a e he applica ion we a e using wo ele an and inno a i e scena ios as case
s udy examples.
Keywo ds: Asse and main enance managemen ; Reliabili y, S a is ical App oaches; Diagnos ics,
P ognos ics, Reasoning, Decision Suppo .

1. INTRODUCTION
F om he no ion o Shannon en opy, his con ibu ion
p esen s an applica ion o indices o inequali y measu es
(Gini, Hoo e , and Theil index) o he analysis o he
a ailabili y in lee s o indus ial asse s. This is in o de o
make possible compa isons among lee s ope a ing unde
di e en bounda y condi ions and usage p o iles. The
ob ained indices e e o ha po ion o he o al g oup o
asse s ha has o be edis ibu ed in o de o ob ain a 100%
equal lee , in e ms o a ailabili y and pe o mance. The
gi en app oach and applica ion o he en opy concep o he
ield o a ailabili y is inno a i e. Pa icula ly he
cha ac e is ics o he me hod and hei independence om he
he e ogenei y o he sample ha compa es, is in ended o be
p omising o applica ions in a ailabili y analysis. The
applica ion, p oposed wi h he help o wo scena ios, is also
ele an in his new a ea o s udy and ies o illus a e his
de elopmen .
I is assumed ha componen s ope a e indi idually. Howe e
hei a ailabili y has ce ain dependency wi h he associa ed
componen s since hey belong o he same sys em. This
aspec is cla i ied h oughou his pape . Wi h ha aim, he
pape p esen s how hese indica o s can be implemen ed o
he analysis o he li e cycle o an indus ial asse . The
heo e ical de elopmen o such me hodology (including
Shannon en opy, applica ion o in -dices, and all hese
keywo ds) is de eloped in e e ence o Gonzalez-P ida e al
2016. A summa y o o mulae a e shown he e below:
Shannon En opy:
Gin i Inde x:
Hoo e Index:
Theil Index:
These concep s a e implemen ed in a p ac ical case s udy,
e e ing o he compa ison o wo scena ios (A and B).
Ne e heless, his si ua ion can be used in a ious
applica ions including di e en phases o wind a ms whe e
he se e i y o he en i onmen is di e en om one phase o















 n
i
n
i
iii
iii
n
i
iii
iii
NpUA
NpUA
NpUA
NpUA
H
1
11
]
)) / ((
) / (
ln[
)) / ((
) / (

 

 

n
i
ii
n
i
n
i
ii
pUA
NG
G
1
1 1
) / (
)/()(
1
 

 

 n
i
n
i
ii
n
i
iii
iii NN
NpUA
NpUA
H
1 1
1
|)/(
)) / ((
) / (
|)2/1(
 



 





n
i
n
i
ii
n
i
ii
ii
n
i
ii
n
i
iii
iii
sym
NN
pUA
pUA
NN
NpUA
NpUA
T
1 1
1
1
1
))])/(/()
) / (
/
ln(())/(
)) / ((
) / (
[(
)2/1(
3 d IFAC Wo kshop on Ad anced Main enance Enginee ing, Se ice and Technology
Oc obe 19-21, 2016. Bia i z, F ance
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ano he ; expo a ions o p oduc s om eme ging coun ies o
de eloped coun ies compe ing wi h he local manu ac u ing,
e c. A e he case s udy, he pape concludes wi h some
in e es ing u u e esea ch possibili ies.
2. CASE STUDY
Nobel P ize winne Ake lo G. indica ed ha , in a second-
hand ma ke , cus ome s do no eally know exac ly i he
ela ionship be ween quali y and p ice is ai when hey a e
buying a used ca (Ake lo G, 1970). Unce ain y due o his
asymme ic in o ma ion, causes he selle s o high quali y
ca s o be posi ioned ou o he ma ke , since owne s o ca s
wi h lowe quali y ha e he same oppo uni ies o sell hei
ehicles a a high enough p ice. The p oposed app oach is
in ended o com-pa e di e en lee s o indus ial asse s,
whe e scena ios ha e di e en a ibu es (e.g., p oduc i i y
and una ailabili y). Ne e heless, i could be use ul o
compa e second-hand ca ma ke s. Compa ing inequali y
indices enables o de ec wha scena io p esen mo e equali y
in he i ems’ beha iou o i s popula ion. The highe is he
equali y, easie he o eseeing o how will be he emaining
use ul li e o he physical asse s in ope a ion and, as a
consequence, he wo hwhileness o one lee o e he o he
one. In o de o p o ide a p ac ical applica ion o he abo e
men ioned heo e ical subjec s, we p oceed now o syn hesize
an example ha shows he u ili y o he desc ibed inequali y
indices, wi h a case s udy ela ed o a lee o complex
indus ial asse s wi h di e en beha iou and deg ada ion
along hei li e ime.
2.1 S udy Scena io
Le us conside wo di e en scena ios (A and B). These
scena ios can ela e o, o example, speci ic geog aphic a eas
o di e en lee s om he same company. Each scena io will
include a lee o ehicles wi h speci ic se ings and
con igu a ion, which a e assumed o be equal o all o hem.
Howe e , he age o he ehicles will be anging om 1 o 10
yea s old. Vehicles om a speci ic age (1 o 10) and scena io
(A o B), will p esen : di e en popula ion (Ni); di e en
p oduc i i y (pi); di e en una ailabili y (UAi). As
commen ed, bo h pi and UAi, a e a iables ha ep esen
cha ac e is ics o a popula ion Ni o ehicles o he same age
(i).
Table 1. Amoun , una ailabili y and p oduc i i y pe
yea and scena io
i
(age)
Ni
(A)
Ni (B) UAi (A) UAi (B) pi
(A)
pi
(B)
1
800
3.500
0000000480
0000000400
24.0
20.0
2
850
4.250
0000006010
0000005893
17.0
16.8
3
975
4.750
0000015988
0000014677
14.6
11.5
4
1.000
5.195
0000025971
0000023356
13.4
10.
0
5
1.033
6.000
0000034030
0000031018
12.6
9.0
6
1.050
6.100
0000041912
0000037601
12.0
8.0
7
1.075
7.000
0000047880
0000043243
11.6
7.5
i
(age)
Ni
(A)
Ni (B) UAi (A) UAi (B) pi
(A)
pi
(B)
8
1.100
8.000
0000053999
0000048107
11.3
7.0
9
1.125
8.500
0000058487
0000052335
11.2
6.6
10
1.125
9.000
0000062424
0000056044
11.1
6.3
2.2 Condi ions and objec i e
Table 1 ga he s da
a om he wo scena ios (A and B).
Scena io A, in compa ison o Scena io B, assumes ehicles
wi h p oduc i i y ha emains highe as hey age, al hough
hey become less a ailable.
I is assumed ha he deg ada ion
end o each ehicle, o same age and scena io, is simila
yea a e yea . The ques ion he e is how o choose he g oup
o asse s wi h he bes a ailabili y-p oduc i i y ela ionship.
In pa icula , we can obse e ha Scena io A has a lowe
uck popula ion han Scena io B (Fig. 1).
Fig 1. Amoun o asse s in ope a ion pe yea o bo h
scena ios
Ne e heless, due o he p ope ies o Shannon en opy, he
model is also applicable e en when he scale changes. In
o he wo ds, i is applicable e en wi h di e en sizes o
lee s. The e o e,
he indices we a e going o ob ain a e
compa able. In igu e 1, he abscissa e e s o he yea and he
o dina e e e s o he amoun o ucks in use. In addi ion o
his, we obse e ha he asse ope a ion is di e en om one
scena io o ano he ; since he una ailabili y o llo ws di e en
ends (Fig. 2).
0,0000000%
0,0000100%
0,0000200%
0,0000300%
0,0000400%
0,0000500%
0,0000600%
0,0000700%
1 2 3 4 5 6 7 8 9 10
UAi (A)
UAi (B)
Fig. 2. Una ailabili y pe age and scena io (%)
These di e ences p o ide us he no ion o di e se usage
p o ile o main enance policies in bo h scena ios. In o he
wo ds, o e ime, i he usage
p o ile is simila in bo h
con ex s, hen Scena io A applies a wo se main enance policy
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han Scena io B. Finally, compa ing p oduc i i y, such
cha ac e is ic in Scena io A is highe han in Scena io B and,
in bo h cases, i apidly dec eases (Fig. 3).
Fig. 3. P oduc i i y pe asse s age and scena io (€)
2.3 Di e si y indices applied o he p ac ical case ns
Applying he pa ame e s de ined in Sec ion 1 o he da a om
igu es 1 o 3, i is possible o ob ain he ollowing alues o
each scena io (see Table 2 and Table 3):
Table 2. Rela i e and accumula e d al ues o pa ame e
o scena io A
#
Ni
( ela i e
)
Ni
(accum
ula ed)
UAi / pi
UAi / pi
( ela i e
)
UAi / pi
(accumu
la ed)
(UAi /
pi) Ni
( ela i
e)
(UAi / pi)
Ni
(accumul
a ed)
1 7.89% 7.89% 1.6E-
10 0.05% 0.0 5% 0.07% 0.0 7%
2 8.39% 16. 28% 3.0 052
E-09 0.95% 1.00% 1.2 0% 1.27%
3 9.62% 25. 91% 1.0 677
E-08 3.37% 4.37% 3.7 3% 5.00%
4 9.87% 35. 77% 1.9 381
E-08 6.12% 10. 49% 6.6 0% 11. 61%
5 10. 19% 45. 97% 2.7899
3E-08 8.81% 19. 30% 9.2 0% 20. 81%
6 10. 36% 56. 33% 3.6673
4E-08 11. 58% 30. 87% 11. 90% 32. 71%
7 10. 61% 66. 94% 4.4 371
2E-08 14. 01% 44. 88% 14. 06% 46. 77%
8 10. 86% 77. 80% 5.2565
5E-08 16. 60% 61. 48% 16. 28% 63. 05%
9 11. 10% 88. 90% 5.8748
1E-08 18. 55% 80. 03% 17. 79% 80. 84%
10 11. 10% 100.0% 6.3267
3E-08 19. 97% 100 .00% 19. 16% 100.00%
Table 3. Rela i e and accumula ed alues o pa ame e
o scena io B.
A-
ge
Ni
( ela i
e)
Ni
(accumu
la ed)
UAi /
pi
UAi / pi
( ela i
e)
UAi / pi
(accum
ula ed)
(UAi / pi)
Ni
( ela i e)
(UAi / pi)
Ni
(accumul
a ed)
1 5.62% 5.62% 7E-10 0.0 2% 0.02% 0.05% 0.0 5%
2 6.82% 12. 44% 1.4 906
8E-08 0.48% 0.50% 0.8 4% 0.89%
3 7.63% 20. 07% 6.0 623
6E-08 1.94% 2.44% 3.0 7% 3.96%
4 8.34% 28. 41% 1.2 133
5E-07 3.89% 6.33% 5.6 2% 9.58%
5 9.63% 38. 04% 2.0 678
8E-07 6.63% 12. 96% 8.2 9% 17. 86%
6 9.79% 47. 83% 2.8 670
5E-07 9.19% 22. 16% 11. 30% 29. 16%
7 11. 24% 59. 07% 4.0359
7E-07 12. 94% 35. 10% 13. 86% 43. 02%
8 12. 84% 71. 91% 5.4979
E-07 17. 63% 52. 72% 16. 52% 59. 55%
9 13. 64% 85. 55% 6.7401
4E-07 21. 61% 74. 33% 19. 06% 78. 61%
10 14. 45% 100.00 % 8.0062
7E-07 25. 67% 100 .0% 21. 39% 100 .00%
Hence, we can ob ain he di e en indices applying he
co esponding o mula, al eady exp essed in Sec ion 1.
Pa icula ly, indices o scena io A a e he e below (see Table
4):
Table 4. Di e si y indices o scena io A.
Age Gini Hoo e Theil (sym) T Tl Sha nnon
H
1 1.6000E-10 7.84% 39. 63% -0.0026 0.3988 0.38%
2 3.3452E-09 7.44% 16. 21% -0.0207 0.1828 4.42%
3 1.7961E-08 6.25% 6.56% -0.0354 0.1009 11. 43%
4 4.8754E-08 3.75% 1.79% -0.0292 0.0472 17. 09%
5 9.8282E-08 1.39% 0.20% -0.0129 0.0149 21. 40%
6 1.6493E-07 1.22% 0.13% 0.0 128 -0.0115 24. 96%
7 2.5078E-07 3.40% 0.94% 0.0 389 -0.0295 27. 53%
8 3.5458E-07 5.74% 2.44% 0.0 704 -0.0461 29. 81%
9 4.7514E-07 7.44% 3.82% 0.0 952 -0.0570 31. 25%
10 5.9716E-07 8.8 7% 5.21% 0.1173 -0.0652 32. 17%
Simila ly o Scena io B (see Table 5):
Table 5. Di e si y indices o scena io B.
Age Gini Hoo e Theil
( sym) T Tl Shann on
H
1 7.0000E-10 5.60% 30. 91% -0.0012 0.3103 0.19%
2 1.6607E-08 6.34% 16. 58% -0.0127 0.1814 2.55%
3 9.5845E-08 5.68% 6.46% -0.0266 0.1042 7.66%
4 2.9246E-07 4.45% 1.22% -0.0297 0.0636 12. 63%
5 6.8471E-07 3.00% -0.89% -0.0248 0.0360 17. 99%
6 1.1931E-06 0.60% -0.49% -0.0058 0.0062 21. 94%
7 2.1017E-06 1.70% 1.94% 0.0 183 -0.0159 26. 46%
8 3.4130E-06 4.78% 6.76% 0.0 558 -0.0407 30. 60%
9 4.8844E-06 7.96% 13. 50% 0.0994 -0.06 27 33. 11%
10 6.6861E-06 11. 22% 21. 71% 0.1 475 -0.0830 34. 91%
Rega ding he Gini index, he alue o his indica o is
be ween ze o and one, inc easing inequali y as he index
app oaches he uppe bound. This measu e also allows a
geome ic in e p e a ion in e ms o he Lo enz concen a ion
cu e (is wice he a ea be ween he concen a ion cu e and
he line o equal dis ibu ion) (Gas wi h J.L. 1972).
Rega ding he Hoo e index, i s alue e e s o he p opo ion
o a pa icula cha ac e is ic (UAi / pi), which would ha e o
be edis ibu e
d aken om he iches pa o he popula ion
o he poo es pa , in o de o ensu e uni o mi y in he lee .
Rega ding he Theil index, i he e is a g ea e dispa i y
be ween he conside ed measu e (UAi / pi), he Theil index
alue inc eases, bu in a
ny case appea s limi ed by he
co a iance be ween ela i e measu e and i s loga i hm. The
in oduc ion o a compensa ing loga i hm unc ion p o ides a
be e iew o he p oblem in in ui i e e ms.
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Fig. 4. Shannon En opy o Scena ios A and B.
Rega ding he Shannon en opy, i measu es he speci ic
di e si y (o inequali y). This en opy index (H) is exp essed
as a posi i e numbe , so ha highe he alue o H, highe he
di e si y (o inequali y). In abo e g aph (Fig. 4), he cu e
ha has highe en opy index p esen s a highe inequali y
deg ee o asse s a ha speci ic age. The e o e, he
ela ionship a ailabili y-p oduc i i y is gene ally wo se
espec ing he o he lee o asse s.
2.4 Resul s discussion
In he example, when compa ing wo lee s, du ing he
speci ic pe iod o ime in which he calcula ion is made, he
lee A has a highe Hoo e index han lee B (26.67%>
25.67%). This means, ha he applica ion o s anda d
main enance policies in o mo e homogeneous lee s (i.e.,
lowe Hoo e index) will be mo e e icien han when hey
a e applied in o mo e di e se lee s (highe Hoo e index).
The a io indica es which scena io is mo e homogeneous in
gene al e ms h oughou he whole pe iod. On he o he
hand, illus a ing he Shannon En opy esul s in a g aph, wo
cu es a e ob ained as i is obse ed in Figu e 4. The igu e
shows a change poin o he end o Shannon En opy
compa ing he wo lee s. I may help iden i ying an
imp o emen in he main enance policy on one lee
compa ed o he o he one. Fu he s udies may de ine
main enance s a egies o bo h lee (i.e. sys ema ic
p e en i e main enance s. a condi ional o p edic i e
main enance o ins ance). The bes choice conside ing
a ailabili y and p oduc i i y pe ehicle and age is p ecisely
in he lowes poin o he cu es. In o he wo ds, when
compa ing wo lee s acco ding o hei Shannon En opy,
scena io A is mo e di e se (has mo e inequali ies) han
scena io B (which is mo e homogeneous) un il he se en h
yea when he si ua ion is e e sed. Tha means, ha he
applica ion o s anda d main enance policies in o mo e
homogeneous lee s (i.e., wi h lowe Shannon En opy) will
be mo e e icien han when applied in mos di e se lee s
(highe Shannon En opy). The a io indica es which scena io
is mo e homogeneous a a gi en ime. O cou se, in a eal
case, he deg ada ion deg ee can be e y di e en conside ing
a ious usage p o ile h oughou he same lee . The e o e,
he e may be g oups o asse s wi h accele a ed deg ada ion
compa ed o he es o he lee . O he ac o s apa om he
aging (e.g., he men ioned usage p o ile) shall be aken in o
ac-coun in u he esea ch as addi ional easons o ehicle
deg ada ion and as an addi ional indica o o es ima e he
e maining li e o he physical asse .
3. A DVA NTA GES AND DISADVA NTA GES
The applied se o indica o s allows desc ibing he e olu ion
o some asse a ibu es, om he poin o iew o g ouping
managemen . Fo analysis o a ailabili y, he e a e a ious
indices/
measu es o he han Shannon en opy, like as pe
SMRP and also by Eu opean Main enance Assessmen
Commi ee KPIs. Ne e heless, in his case, he assessmen
by inequali y indices enables he compa ison o g oups
acco ding o es ablished c i e ia as a ailab
ili y and
p oduc i i y, bu also o he possible speci ic ea u es as
eliabili y, cos s, pe o mance e c. (Goepel K.D. 2013). This
aspec makes Shannon en opy a be e han o he es ablished
KPIs ha do no conside his g ouped poin o iew. The use
o
hese kind o indices ela ed o lee s educes subjec i i y
and p o ide easonable alues, easy o be manipula ed. The
me hodology allows conside ing di e en ac o s o each
asse g oup, acco ding o he cha ac e is ics o he lee and
i s ope a ion.
I is also e y use ul o ep esen hese esul s in
a g aphical way, wi h he measu e
men o each g oup and i s
e olu ion o e ime. This simple p ocess may help o obse e
he g ad
ual change along di e en s ages o an i em/ lee li e
cycle. Fo his
pu pose, he Theil index p o ides an
in e es ing ad an age (compa ed o Gini index) as i can
e xpla in e la i e d i e ences be- ween g oups o he same
lee . As i is de eloped in e e ences like Fields G.S. 1980 o
Holsinge & Jacob 2010, one o he ad an ages o he Theil
index is he ac ha i is a weigh ed a e age o inequali y
wi hin lee s, plus inequali y among hese lee s. In ou case
s udy, i is possible o y wi h o he pa ame e s o examine
whe he simila conclusions can be ob ained. Wi h app oach
o ou case s udy, one can de elop a hyb id measu e o
di ec ly combine he p oduc i i y wi h he una ailabili y
wi hou he need o use in o ma ion indices. The e o e, a
p oblem-based ( a he han me hod-based) app oach would
be p e e ed o eac
h speci ic case s udy o sol e. Acco ding
o his, i is impo an o emphasize ha any indica o could
be p io i ized as a andom a iable and, he e o e, he concep
o en opy could be applied a a b oad sense in many o he
si ua ions ( o examp le e go
dici y cases). In o he wo ds, one
o he heo e ical p oblems he e is ha en opy seeks o
cha ac e ize he in o ma ion o a andom a iable depending
on i s dis ibu ion, bu i canno p o e ha he e a e
ealiza ions o he same dis ibu ion in e e y
case. This is he
disad an age o applying i o a s a is ical p oblem, a me hod
designed o wo k in a p obabilis ic en i onmen . In sho , i
mus be unde s ood ha he p esen ed case assumes ha one
can ge samples o UAi/pi indica o , which comes om an
independen , iden ically dis ibu ed p ocess, when ac ually
his is a non- es able ac . Fo a gi en age, i is a c ucial issue
how o speci y UAi. This issue needs pa icula a en ion as
a as an app op ia e quali y measu e and he a ailabili y o
i
s in o ma ion may be he key o he p oblem (Jiang R.
2013).UAi/pi would be meaning ul o use in his cases. In
addi ion o his and acco ding o he applied indices, a iance
is simila o he en opy, bu ac ually, a iance as a simple
measu e may be be e i i can p o ide a easonable and
in ui i e in e p e a ion (Jiang R. & Mu hy DNP. 2013).
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4. FUTURE RESEARCH
This case s udy is jus an app oach wi h e y ew pa ame e s,
because we a e conside ing an ex-ample whe e all asse s end
o deg ade o e ime in he same way. In o he wo ds, ucks
ha e simila use-p o ile and un in simila con ex s o e ime.
Howe e , di e en e sions o he same p oblem can be
analysed, jus changing h ee condi ions in e e ence o he
age: he amoun o uni s, he una ailabili y, and he
p oduc i i y. I could be possible o conside di e en
ope a ing scena ios o di e en models o asse s h oughou
hei li e cycle. Ano he applica ion can be conside ing
complex indus ial asse s as wind u bines (ins ead o
ehicles), connec ed in a ms o ene gy p oduc ion, whe e
he di e en a ms (as ou lee s in he case s udy) wo k
unde di e en bounda y condi ions wi h di e en
deg ada ions o e ime. In addi ion o his, o he di e si y
indices can be applied in main ainabili y assessmen o in
P ognos ics and Heal h Managemen (PHM). The e a e o he
indices wi h hei own s eng hs and weaknesses. The e o e,
scien is s om o he a eas (biology, physics o econome ics)
o en use a combina ion o se e al indices o ake ad an ages
o he s eng hs o each one o hem and de elop a mo e
comple e unde s anding o a communi y s uc u e. In he
con ex o a case based easoning, i is possible o conside
o he c oss- ela ed a eas like he main ainabili y assessmen
in he scope ( o examp le) o CRM o AHP. I is impo an
he e o add ha PHM is a ield wi h plen y o possibili ies on
u he ela ed esea ch lines (Scan E. e Al. 2007; Iz aeli,
O. 2003; Roeme M.J. e Al. 2001), as a as indica o s and
a ibu es assessmen de eloped o main enance (Mo eu e .
Al 2012) can be applied in a simila way as an aid ool o
p edic ing he RUL o c i ical indus ial asse s. Apa om
he abo e u u e esea ches, o he con ibu ions can add a
deepe desc ip ion o he well-known indices (Gini, Hoo e ,
Theil), speci ically when hey a e ocused o o he a eas,
illus a ing he concep behind he o mulas and explaining
how hey can be applied in di e en a eas han he usual
ones.
5. CONCLUSIONS
We ha e b ie ly depic ed he Shannon en opy, as well as
inequali y indices such as he Gini, Hoo e , and Theil index.
They a e success ully applied in o he subjec a eas as
he modynamics, s a is ics, biology o econome ics. This
pape has in ended jus o app oach such concep s o he
physical as-se s managemen , in o de o ind p ac ical and
use ul applica ions. Pa icula ly, one o he possible
applica ions has been he compa ison o wo lee s o
complex asse s wi h di e en beha iou despi e he lee s
being cons i u ed by he same asse con igu a ion. The
una ailabili y-p oduc i i y measu e adop ed in his case
s udy has been (UAi / pi). This measu e dec eases i he
una ailabili y is lowe and he p oduc i i y is highe , and
ice- e sa. Tha means ha he ope a ion and main enance o
one speci ic g oup o asse s is mo e p o i able and e icien .
This compa ison allows us o de e mine:
• A be e main enance policy o he es o he emaining
use ul li e o he i ems,
•
A be e main enance policy o hose componen s ha
a e
o eseen o ope a e unde he same en i onmen
and/o usage p o ile.
• A mo e e icien use o asse s om one lee in
compa ison o ano he , e c.
These indices a e ecommended o be used oge he wi h
o he s al eady known in PHM as a complemen o a be e
judgemen . As u u e e-
sea ch, i is p oposed o de elop
mo e elabo a e in-
dices ha simpli y he choice o
p e e ences om indi idual o lee a ibu es. These new
indices can be ansla ed om o he ields o by he
applica ion o a conjoin analysis.
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