An Inventory System with Retrial Demands and Working Vacation

8/10/2019 An Inventory System with Retrial Demands and Working Vacation

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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 1

ISSN 2250!15!

An Inventory System with Retrial Demands and

Working Vacation

J. Kathiresan*, . An!a"hagan*and K. Jeganathan**

"De#artment of $athematics, %la&a##a 'ni(ersit), *arai+udi,India""Ramanu-an Institute for %d(anced Stud) in $athematics, 'ni(ersit) of $adras, .he#au+, .hennai, India

Abstract# /his article considers a continuous re(ie retrialin(entor) s)stem at a ser(ice facilit), herein an item demanded

b) a customer is issued after #erformin& ser(ice on the item /he

arri(al time #oints of customers form a Poisson #rocess /he

in(entor) re#lenished accordin& to an , Qs #olic) and thelead times are assumed to follo an e3#onential distribution /he

demands that occur durin& the stoc+ out #eriod or the ser(er bus)

re&ular or or+in& (acation are #ermitted to enter into the orbitof infinite sie hen the in(entor) le(el is ero or no demands

in the s)stem or both, ser(er &oes to a or+in& (acation hich ise3#onentiall) distributed If the ser(er is in or+in& (acation or

the in(entor) le(el is ero, the im#atience occurs in orbitin&customers, that follos an e3#onential distribution /he -oint

#robabilit) distribution of the number of demands in the orbit,

the in(entor) le(el and the ser(er status is obtained in the stead)

state case Some s)stem #erformance measures are deri(ed, thelon&run total e3#ected cost rate is calculated and the results are

illustrated numericall)

Index Terms .ontinuous re(ie in(entor) s)stem, , Qs

Polic), Positi(e leadtime, Retrial demand, or+in& (acation

A$S %lassi&ication' 60705, 80J29

I IN/R:D'./I:N

he conce#t of ser(er (acation in in(entor) ith to ser(ers

as first introduced b) Daniel and Ramanara)anan ;1


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rates are b

and v

F b , hen the s)stem is in re&ular

and or+in& (acation, res#ecti(el), hich are e3#onentiall)

distributed /he ser(er ta+es a or+in& (acation at times henno customers in the s)stem or the in(entor) le(el is ero or both

or+in& (acation durations are e3#onentiall) distributed ith

#arameter %t the com#letion time of the or+in& (acation,

the ser(er sitches a re&ular bus) ie ser(ice rate from v

to

b if there are customers #rimar) or retrial in the s)stem:therise, the ser(er continues the or+in& (acation /heorbitin& customers ma) either retr) or ma) lea(e the orbit /he

lea(in& orbitin& customers are described as im#atientrene&in&

customers If the ser(er is in or+in& (acation or the in(entor)

le(el is ero, the im#atience occurs in orbitin& customers %n

im#atient customer lea(es the orbit inde#endenl) after a random

time hich is distributed e3#onentiall) ith #arameter0E

e assume the constant retrial #olic) for these orbi

demands, that is #robabilit) of a re#eated attem#t of an orbitin&

demand is inde#endent of the number of demands in the orbit

Retrial re=uests from the orbit follo an e3#onential distribution

ith #arameter 0E %s the , Qs re#lenishment #olic)hen the onhand in(entor) le(el dro#s to a #refi3ed le(el, sa)

0Es an order for 1EG + ssSQ units is #laced /he#ositi(e lead time is e3#onentiall) distributed ith #aramete

0E e assume that the interdemand times beteen the

#rimar) demands, the lead times, retrial demand times, ser(er

re&ular #eriods and ser(er or+in& (acation #eriods are mutuall)

inde#endent random (ariables

otations

ijArom the assum#tions made on the in#ut and out#ut #rocesses, it can be shon that the tri#let{ }0,,, ttYtLtX

ith the state s#ace E is a $ar+o( #rocess

/o determine the infinitesimal &enerator

,,,,,G nljmkipP,

Enljmki ,,,,,

of this #rocess e use the folloin& ar&uments

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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 !

ISSN 2250!15!

Cet 0Gi , Sk ,1,2,G

L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,0, ki to ,1, ki ith the intensit)

L an) arri(in& #rimar) demand ta+es the state of the #rocess from,, mki

to ,1, mki + ith theintensit) , 1,!Gm

L /he ser(er chan&es from or+in& (acation to re&ular ta+es the state of the #rocess from,1, ki

to

,!, ki ith the intensit)

L /he com#letion of ser(ice from #rimar) demand ma+es a transition from,1, ki

to1,0, ki ith the

intensit) v

L /he com#letion of ser(ice from #rimar) demand ma+es a transition from ,!, ki to 1,0, ki ith the

intensit) b

Cet 0i , 0Gk

L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,0, ki to ,01, ki + ith the intensit)

Cet 1i , Sk ,1,2,G

L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,, mki to 1,, +mki ith the intensit) , mG0, 2

L an) arri(in& #rimar) demand ta+es the state of the #rocess from,, mki

to ,1, mki + ith theintensit) , mG1, !

L /he ser(er chan&es from or+in& (acation to re&ular ta+es the state of the #rocess from ,, mki to2,, +mki ith the intensit) , mG0, 1

L a retrial re=uests ta+es the state of the #rocess from,, mki

to1,1, + mki ith the intensit) ,

mG0, 2

L an im#atient customer ta+es the state of the #rocess from,, mki

to,1, mki

ith the intensit)

,

mG0, 1

L an im#atient customer ta+es the state of the #rocess from,0,0i

to1,0,0 i

ith the intensit)

L /he com#letion of ser(ice in the s)stem ma+es a transition from ,1, ki to 1,0, ki ith the

intensit) v

Cet 1i , Sk ,2,!,G

L /he com#letion of ser(ice in the s)stem ma+es a transition from,!, ki

to1,2, ki

ith the

intensit) b

L /he com#letion of ser(ice in the s)stem ma+es a transition from ,1,!i to ,0,0i ith the intensit)

b

Cet 0i , sk 1,2,G

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L a transition from ,, mki to ,, mQki + for 0,1,2,!Gm ta+es #lace ith the intensit) hen are#lenishment occurs

L a transition from ,0,0i to ,0, Qi ta+es #lace ith the intensit) hen a re#lenishment occurs

e obser(e that no transition other than the abo(e is #ossible

>inall), the (alue of ,,,,, mkimkip is obtained b)

11,,21,,,22G11,,21,,,22

1,,21,,2

nljmkipmkimkipnlj

nljmki

Mence e ha(e,,,,,, nljmkip

G

+

+

+

+

0,2G,,1,2,G1,

1G,G,G

0G0,G0,

G,G1,G

1,!G,,1,2,G0,G,G1,G

0G,,1,2,G0,G

1G,G,G,

mSki

mnklij

or

mki

mnklij

or

mSkimnklij

or

mSki

mnklij

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==

==

=

+

+

+

00

,

,1

,1

0,1G

,,1,2,G1,

G,G1,G,

0,2G,,1,2,G1,1G,G1,G,

0,1G,,1,2,G1,

2G,G,G

1G,,1,2,G0,G

2G,G,G,

!G1,G1,

0G1,G,G

!G,,2,G1,

1G1,G,G

!G,,1,2,G0,G

0G1,G,G,

1G,,1,2,G0,

0G1,G,G,

m

mn

k

kl

i

ij

or

m

Ski

mnklij

mSki

mnklij

mSki

mnklij

or

mSki

mnklij

mki

nklij

or

mSki

mnklij

or

mSki

nklij

mSki

nklij

b

v

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=++=+++

++

+++

+

++

+

++++

++

+

+

0

G

,,,2,1

,G

,1

,G,

2G,,1,2,G0,G

G,G,G,

!G,,1,2,G0,

G,G,G,

1G,,1,2,G0,G

G,G,G,

0G,,0,1,2,G0,G

G,G,G,

!G,,2,1,G0,

G,G,G,

1G,,2,1,G0,G

G,G,G,

0G,,2,1,G0,G

G,G,G,

0G0,G0,

G,G,G

0,1,2,!G,,1,2,G0,

G,G,G,

m

mn

Sssk

kl

i

ij

mski

mnklij

mski

mnklij

mski

mnklij

mski

mnklij

mSsski

mnklij

mSsski

mnklij

mSsski

mnklij

mki

mnQklij

or

mski

mnQklij

b

v

b

v

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++

++

++++

++++

+++

+++++

otherise0,

0G0,G1,G,G,G,

2G,,1,2,G1,

G,G,G,

1G,,1,2,G1,

G,G,G,

0G,,1,2,G1,

G,G,G,

2G,,2,1,G1,

G,G,G,

1G,,2,1,G1,

G,G,G,

mkimnklij

mski

mnklij

mski

mnklij

mski

mnklij

mSsski

mnklij

mSsski

mnklij

v

v

Denotin&,!,,2,,,1,,,0,,,,1,!,,1,2,,1,1,,,1,0,0,0,G SqSqSqSqqqqqqq fo

0,1,Gq 7) orderin& states le3ico&ra#hicall), the infinitesimal &enerator P can be con(enientl) e3#ressed in a bloc+#artitioned matri3 ith entries

+

otherise0,

2,1,G1,G,

2,1,0,G1,G,

2,1,G,G,

0G,G,

G


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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 @

ISSN 2250!15!

LAM

LAM

LAM

LA

P00

00

00

000

G

1

here

+

+

==

+

++

otherise0,

0G,G,

,,21,G,G,

11

,,2G1,G,

0G,G,

,,2,1G,G,

,2,1,G,G,

G


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klJ


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ith

klF


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kl!


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ijQ


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ISSN 2250!15!

kl"


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.learl) 4!10 ,,, """"

and C are s=uare matrices of order 4 /he matri3 4" is of sie 4 1 and the matri3 1C isof sie 1 4 /he matri3 2" is of sie 1 1

Cet + be the stead)state #robabilit) (ector of Q /hat is, + satisfies

1G0,G +e+Q

/he (ector + can be re#resented b)

+ ,,, 10 S

+++

here0

+ G 0,0

i+ G ,,,

,!,2,1,0 iiii , Si ,1,2,G

-heorem #he ste%&'(st%te prob%bilit' ve)tor + )orespon&in* to the *ener%tor Q is *iven b'

1,G0,G21

4

i"" ii

+

1,,!,2,G0,G11

!

++ si"" ii

,,,!2,G0,G01

!

Qssi""

ii+++

1G0,G11

0

1

!

+++ QiC"" Qiii

,,,!2,G0,G101

!

SQQiC"" Qiii ++++

0G

0

C"

sS +

roo&'e ha(e0GQ+ , 1G+e

%fter lon& sim#lications, the abo(e e=uations, e3ce#t the last one, )ields

++

+

+

+

+

++

+

SQQi"

C

"

"

"

"

"

"

"

"

"

"

"

"

"

"

Qi"

C

"

"

"

"

"

"

Qssi"

"

"

"

"

"

si"

"

"

"

QikQikQi

k

Qisis

i

sis

i

sis

i

i

i

i

,!,2,G,

1

1G,1

,!,2,G,1

1,2,1,G,1

G

!

0

1

!

0

1

!

1

4

21

1G

1

!

0

1

!

0

!

1

4

20

!

1

1

!

0

!

1

4

20

1

!

0

!

1

4

20

1

!

1

4

20

here0 can be obtained b) sol(in&,

0G

0

C"

sS +and

1G2

0G

ei

S

i

,

that is

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0G1

1

1

!

1

4

2

0

!

1

!

0

1

!

1

4

21

1G

1

!

0

1

!

0

!

1

4

2

+

+

+

s

s

QSkQSkQS

k

QS

S

sSs

C

"

"

"

"

""

C

"

"

"

"

"

"

"

"

"

"

"

"

"

"/(0+

and

+

+

+

+

+

1

!

0

!

1

4

2

2G

1

!

1

4

21

1G

11

sis

iQ

si

i

is

i "

"

"

"

"

"

"

"

"

"I

/(0+

+

+

+

+

+ 1

!

0

!

1

4

2

2G!

1

!

0

!

1

4

21 11

sis

iS

Qi

sQs

Q

"

"

"

"

"

"

"

C

"

"

"

"

"

"

1G!

1

!

0

1

!

1

4

21

1G

1

!

0 e"

C

"

"

"

"

"

"

"

"QikQik

Qi

k

Qi

+

Ne3t, e deri(e the condition under hich the s)stem is stable

1emma #he st%bilit' )on&ition of the s'stem +n&er st+&' is *iven b'

ee E 11

0FFF

1

roo&'>rom the ell +non result of Neuts9 on the #ositi(e recurrence of P e ha(e

e+e+ LM E

and b) the e3#loitin& the structure of matrices M and L and + the stated result follos

).2 Steady state analysis

It can be seen from the structure of the rate matri3 P and from the lemma 1 that the mar+o( #rocess

1121,21,2I2 tYtLtX,

0Jt ith the state s#ace E is re&ular Mence the limitin& #robabilit) distribution

0J0,0,NG,G,GIlimG,,

YLXmtYktLitXprtmki

, e3ists and is inde#endent of the initial state

/hat is, G ,, 10

satisfies

0GP3 , 1G3e

e #artition the (ectori , for 0,1,2,Gi as follos

i G ,,, ,,1,0 Siii

hich is #artitioned as follos, for Sk1,0i G

,0,0i

, ki G ,,, ,!,,2,,1,,0, kikikiki

-heorem 2 ,hen the st%bilit' )on&ition -./ hol&s *oo&0 the ste%&' st%te prob%bilit' ve)tor 3 is *iven b'

0,1,2,G,G i1 ii /(0

3 2

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here the matri3 1 satisfies the matri3 =uadratic e=uation

0G2

L1AM1 ++ !

and the (ector/(0

3 is obtained b) sol(in&

0G 1 1MA+/(03sub-ect to normaliin& condition

1G 1e1I /(03

roo&'/he theorem follos from the ell +non result on matri3&eometric methods Neuts9

).) %om45tation o& R matri6

In this subsection e #resent an efficient al&orithm for com#utin& the rate matri3 1 hich is the main in&redient for discussin&

=ualitati(e beha(ior of the model under stud) /he 1 matri3 is of sie 4S O 1 can be com#uted b) usin& lo&arithmic reductional&orithm

1ogarithmic red5ction algorithm

Co&arithmic reduction al&orithm is de(elo#ed b) Catouche and Ramasami ;11< hich has e3tremel) fast =uadratic con(er&ence

Mere e discuss onl) the im#ortant ste#s in(ol(ed in this al&orithm e refer the reader to Catouche and Ramasami ;11< for more

details about this al&orithm

Ste4 (' LA" 1

, MAF

1 , F!G , and "#G

Ste4 '

F""F2 +G2

G"E

E2I" 1

2FE

E2IF 1 #L!! +

#"#

.ontinue Ste4 'until FNNNN !ee

Ste4 2'1G + L!AL1

IV SKS/B$PBR>:R$%N.B$B%S'RBS

In this section some #erformance measures of the s)stem under consideration in the stead) state are deri(ed7. 864ected inventory level

Cet i denote the a(era&e in(entor) #osition in the stead) state /hen[ ],,

!

0G1G0G

G kji

k

S

ji

i j

47.2 864ected reorder rate

Cet r

denote the e3#ected reorder rate in the stead) state /hen

[ ]1,!,1,1,0G

G ++

+ sibsi

vi

r 5

7.) 864ected n5m!er o& demands in the or!it

Cet o

denote the a(era&e in(entor) #osition in the stead) state /hen

[ ] [ ],0,01G

,,!

0G1G1G

G i

i

kji

k

S

ji

o ii

+8

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7.7 8&&ective reneging rate &or an or!iting c5stomer

Cet ro

denote the effecti(e rene&in& rate for an orbitin& customer in the stead) state /hen

[ ],,,0,01

0G1G1G

G kjii

k

S

ji

ro +

97.9 :verall rate o& retrials

Cet or

denote o(erall rate of retrials in the stead) state /hen

[ ] [ ],0,01G

,,!

0G1G1G

G i

i

kji

k

S

jior

+@

7.; -he s5ccess&5l retrial rate

Cet sr

denote successful retrial rate in the stead) state /hen

[ ],2,,0,1G1G

G jijiS

ji

sr +

67.< -he &raction o& s5ccess&5l rate o& retrial

Cet sr

denote successful retrial rate in the stead) state /hen

or

srfr

G

10

V /:/%CBPB./BD.:S/R%/B

/o com#ute the total e3#ected cost #er unit time, e consider the folloin& costs

s) Setu# cost #er order

h) /he in(entor) carr)in& cost #er unit item #er unit time

w) aitin& cost of a customer in the orbit #er unit time

r) rene&in& cost #er customer #er unit time

/he lon& run total e3#ected cost rate is &i(en b)

rorowrhis

))))Ss#C +++G,>rom e=uations 4,5,8 and 9, e obtain

[ ] +

++

++ 1,!,1,1,0G

,,!

0G1G0G

G, sibsi

v

i

h

kji

k

S

ji

s )j)Ss#C

[ ] ,,,0,01

0G1G1G

,0,0

1G

,,!

0G1G1G

++

+

kjii

k

S

ji

r

i

i

kji

k

S

ji

w )ii)

VI N'$BRI.%CICC'S/R%/I:NS

In this section, e discuss some numerical e3am#les indicates the function, Ss#C

to be con(e3>i&ure 1refers the chan&es

of s and S are ho to affect the total e3#ected cost rate /he table 1 #resents the total e3#ected cost rate for (arious combinations ofs and Sb) fi3in& the other #arameters and costs as 4,0G1@,G,5G 02G16,G199,G2,G bv and

1G04@,G@,!G006,G rwhs )))) /he sim#le numarical search #rocedure are used to obtain the o#timal (alues o

, Ss#Csa) ,

""" Ss#C /he o#timal (alue of the total e3#ected cost rate is20,55#C

G641942 /he o#timal cost foreach S is shon in underline and the o#timal cost for each s is bold Some of the results are #resented in /ables 2 throu&h 1! here

the loer entr) in each cell &i(es the o#timal total e3#ected cost rate and the u##er entries the corres#ondin&"S and

"s

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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 1@

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-a!le

'

-otal

e64ected cost rate as a &5nction o& S and s

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s

S

19 1@ 16 20 21 22 2!

52 64!448 642984 642990 64!400 64481@ 64841! 64@9605! =.7))>9 64249! 64221! 642540 64!419 644@26 648999

54 64!594 =.7279( 641648 641668 642582 64!828 6451@5

55 64!660 64289! =.7=79 =.7


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4,0G1@,G,5G 02G16,G199,G2,G bv 1G04@,G@,!G006,G rwhs ))))

?ig. '% three dimensional #lot of the cost function, Ss#C

86am4le In this e3am#le e stud) the im#act of arri(al rate , ser(ice rates v

and b

, the lead time #arameter , the

retrial rate , and the rena&in& rate , on the o#timal (alues , ""Ss and the corres#ondin& total e3#ected cost rate

"#C 7)

fi3in& #arameter

02Gand the costs (alues as

1G048,G@,!G006,G rwhs ))))

e obser(e the folloin& from table 2to 9

1 /he total e3#ected cost rate increases hen increases and the total e3#ected cost rate decreases hen ,,

bv ,, increase

2 If , v

increase, then"S monotonicall) increases If ,, b

,increase, then

"S monotonicall) decrease

! If increases, then"s monotonicall) increases If ,, bv

,,increase, then

"s monotonicall) decreases

-a!le 2' 8&&ect o& arrival rate and sevice rate v

on the o4timal val5es

4,0G1@,G 16G2,G b

v

195 198 199 19@ 196

466 55 20 55 20 55 20 58 20 58 20

@6@@82 @69811 @68!91 @651!9 66!@6@

500 55 20 55 20 55 20 55 20 58 20

62002! 61@944 619495 618219 614656

501 58 21 58 21 58 21 58 21 58 21

64!021 641926 64044@ 6!6198 6!9615

502 58 21 58 21 58 21 58 21 58 21

68@191 688@50 6855!6 6842!6 68264@

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50! 59 22 59 22 58 21 58 21 58 21

665956 6644!0 66!060 661956 6604!6

-a!le )' 8&&ect o& arrival rate and sevice rate b


4,0G1@,G 919G2,G v

b

1@@ 1@6 160 161 162

466 58 21 55 20 55 20 58 20 55 16

68!!94 629!95 @68!91 @86495 @45612

500 58 21 58 21 55 20 55 20 58 20

661251 651501 619495 @@@091 @82489

501 59 22 58 21 58 21 55 20 58 20

102166@ 699624 640449 60@2!2 @@0211

502 59 22 59 22 58 21 58 21 58 20

1058016 100900! 6855!6 6!0142 @66502

50! 59 22 59 22 58 21 58 21 58 21

1081!6@ 10!6092 66!060 654012 620521

-a!le 7' 8&&ect o& arrival rate and retrial rate on the o4timal val5es4,0G1@,G 16G9,19G vv

166 200 201 202 20!

466 55 20 55 20 55 20 55 16 55 16

61@425 @68!91 @98!08 @59648 @4100@

500 58 21 55 20 55 20 55 20 55 16

641545 619495 @6582! @95926 @5952!

501 58 21 58 21 55 20 55 20 55 20688@04 640449 618812 @6465! @95221

502 58 21 58 21 58 21 55 20 55 20

664546 6855!6 6!644! 615@!2 @64!58

50! 59 22 58 21 58 21 58 21 55 20

10250@5 66!060 684!96 6!@5!1 6151!5

-a!le 9' 8&&ect o& lead time and sevice rate v


4,0G5,G 16G2,G b

v

195 198 199 19@ 196

196 58 21 58 21 55 20 55 20 58 20

6200@8 61@@20 619558 618268 6150!!

1@0 55 20 55 20 55 20 55 20 58 20

62002! 61@944 619495 618219 614656

1@1 55 20 55 20 55 20 55 20 58 20

616644 61@888 619400 618144 614@@6

1@2 55 20 55 20 55 20 55 20 58 20

616@86 61@564 619!!0 618098 614@24

1@! 55 20 55 20 55 20 55 20 58 20

616@00 61@529 619284 618012 614984

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ISSN 2250!15!

-a!le ;' 8&&ect o& lead time and sevice rate b


4,0G5,G 919G2,G v

b

1@@ 1@6 160 161 162

196 58 21 58 21 55 20 55 20 58 16

661!!5 651586 619558 @@@1!9 @82516

1@0 58 21 58 21 55 20 55 20 58 20

661251 651501 619495 @@@091 @82489

1@1 58 21 58 21 55 20 55 20 55 16

661192 6514!9 619400 @@@006 @82409

1@2 58 21 55 20 55 20 55 20 55 16

66106@ 651!91 619!!0 @@965! @82!!@

1@! 58 21 55 20 55 20 55 20 55 16

66102@ 651261 619284 @@9601 @82294

-a!le


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ISSN 2250!15!

60!909 60946@ 611200 614602 61@524

00@@ 55 20 55 20 58 20 58 20 59 20

60595@ 606588 61!!01 61900! 620895

00@6 55 20 55 20 58 20 58 20 58 20

609@06 611819 615402 616104 622@08

0060 55 20 55 20 55 20 58 20 58 20

606@80 61!88@ 619695 621205 624609

0061 54 20 55 20 55 20 58 20 58 20

611@9! 615916 616528 62!!08 62900@

-a!le =' Sensitivity o& s)

and w)


1G@,!G rh ))

w)

s)

04! 044 045 048 049

00@9 58 20 58 20 58 20 58 20 58 20

@94@28 @@8650 @66095 611200 62!!25

00@@ 58 20 58 20 58 20 58 20 58 20

@98629 @@6051 601198 61!!01 625428

00@6 55 20 58 20 58 20 58 20 58 20

@960244 @6115! 60!299 615402 6295290060 55 20 55 20 55 20 55 20 55 20

@@1095 @6!206 605!42 619495 626806

0061 55 20 55 20 55 20 55 20 55 20

@@!128 @65280 609!6! 616528 6!1880

-a!le (' Sensitivity o& s)

and r)


048G@,!G wh ))

r)

s)

066 100 101 102 10!

00@9 58 20 58 20 58 20 58 20 58 20610600 611200 611500 611@00 612100

00@@ 58 20 55 20 58 20 58 20 58 20

61!001 61!!01 61!801 61!601 614201

00@6 58 20 58 20 58 20 58 20 55 20

615102 615402 615902 618002 618!02

0060 55 20 55 20 55 20 55 20 55 20

619195 619495 619998 61@098 61@!99

0061 55 20 55 20 55 20 55 20 55 20

616228 616528 616@29 62012@ 6,2042@

-a!le ' Sensitivity o& h)

and w)


1G06,0G rs ))

w)

h)

04! 044 045 048 049

!8 54 20 54 20 55 20 55 20 55 20

@9!445 @@55@@ @69929 606@80 62166!

!9 55 20 55 20 55 20 55 20 55 20

@9928@ @@6401 6015!4 61!88@ 625@01

!@ 55 20 55 20 55 20 55 20 55 20

@@1095 @6!206 605!42 619495 626806

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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 2!

ISSN 2250!15!

!6 58 20 58 20 58 20 58 20 58 20

@@4@!1 @68655 6060@0 621205 6!!!!0

40 58 20 58 20 58 20 58 20 58 20

@@@5!! 600859 6129@2 624609 6!90!2

-a!le 2' Sensitivity o& h)

and r)


480G06,0G ws ))

r)

h)

066 100 101 102 10!

!8 55 20 55 20 55 20 55 20 55 20

606556 606@80 610181 610481 610982

!9 55 20 55 20 55 20 55 20 55 20

61!!89 61!88@ 61!68@ 614286 614586

!@ 55 20 55 20 55 20 55 20 55 20

619195 619495 619998 61@098 61@!99

!6 58 20 58 20 58 20 58 20 58 20

620605 621205 621505 621@05 622105

40 58 20 58 20 58 20 58 20 58 20

624809 624609 625221 625509 625@09

-a!le )' Sensitivity o& w)

and r)


@!G006,G hs ))

r)

w)

066 100 101 102 10!

04! 55 20 55 20 55 20 55 20 55 20

@@0995 @@1095 @@1!98 @@1899 @@1699

044 55 20 55 20 55 20 55 20 55 20

@6260@ @6!206 @6!506 @6!@10 @64111045 55 20 55 20 55 20 55 20 55 20

605041 605!42 60584! 60564! 608244

048 55 20 55 20 55 20 55 20 55 20

619195 619495 619998 61@098 61@!99

049 55 20 55 20 55 20 55 20 55 20

626!0@ 626806 626606 6!0210 6!0510

VII .:N.C'SI:N

e anal)sed an , Qs in(entor) s)stem ith retrial

customers and or+in& (acation Primar) inter arri(al times,retrial times, ser(ice times and or+in& (acation times areinde#endent e3#onentiall) distributed random (ariables e ha(e

deri(ed the stead) state distribution of the s)stem usin& $atri3

anal)tic methods and se(eral #erformance measures ha(e also

been calculated Some numerical solutions are #resented toillustrate the =ualitati(e beha(ior of the s)stem

%.*N:CBDQ$BN/

N %nabha&an?s Research as su##orted b) the Nationa

7oard for Mi&her $athematics D%B, Qo(ernment of India

throu&h research #ro-ect 2A4@11A2011ARD IIA1141

RB>BRBN.BS

;1< Daniel, J * and Ramanara)anan, R %n in(entor) s)stem ith to ser(erand rest #eriods .ahiers du .BR:, 'ni(ersite Cibre De 7ru3elles16@9, 26, 65100

;2< Daniel, J * and Ramanara)anan, R%n in(entor) s)stem ith rest #eriodsto the ser(er Na(al Research Co&istics, John ile) and Sons 16@@, !511612!

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An Inventory System with Retrial Demands and Working Vacation

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