International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
438
EFFECT OF SHORTAGES ON LIMITED
STORAGE INVENTORY SYSTEM FOR
DECAYING ITEMS WITH EXPONENTIAL
DEMAND
Dr. Pankaj Agarwal
Information Technology
Inst i tute of Management Studies Noida ,U.P. , India
Abstract - In this p aper a two -warehouse
inventory model fo r de ter io ra t ing i tems wi th
expo nent ia l demand r a te i s deve loped and
ana lysed . In the p resent model shor tages a re
a l lo wed and co mp le te ly backlo gged . I t i s
assumed tha t t he o wned wareho use has a f ixed
capac i ty whe ther the r ented warehouse (RW)
has un l imi ted capac i ty . Ef fec t o f i n f la t ion i s
taken in considera t io n.
Keywords: Demand, Inventory, I n f l a t io n,
Shor t ages E tc .
I . INTRODUCTION
The impor tant p rob lem assoc ia ted wi th the
inventory ma intenance i s to dec ide where to
s tock the goods . This p rob lem does no t have
a t t rac t ed the a t t en t io n of researchers . I n the
exi s t ing l i te ra ture , i t i s fo und tha t c la ss ica l
inventory mode ls genera l ly dea l wi th a s ingle
s to rage fac i l i t y. The bas ic a ssump t io n in these
model s i s t ha t t he management has o wned
s to rage wi th un l imi t ed capac i ty . I n the f ie ld o f
inventory management , the unl imi ted capac i ty
o f s to rage i s no t t rue . When an a t t rac t ive p r ice
d isco unt fo r bulk p urchase i s ava i l ab le o r the
cost o f p rocur ing goods i s higher than the o the r
inventory re la t ed cos ts o r there a re so me
prob lems in f req uent p rocurement o r the
demand o f i tems i s very h igh, management then
dec ides to purchase (o r p roduce) a huge
quant i ty o f i tems a t a t ime. These i t ems can no t
be s to red in the ex is t ing s to rage , viz the o wned
wareho use (OW) wi th l imi ted capac i ty . Then
for s to r ing the excess o f i tems, a wareho use i s
hi red on a renta l bas i s . This rented warehouse
(RW) may be loca ted ne ar the OW or a l i t t l e
away fro m i t . I t i s genera l ly assumed tha t ,
ho ld ing co st i n the RW is grea ter t han the same
as in OW. Hence , t he i tems are s to red f i r s t i n
OW and o nly excess s tock i s s to red in RW.
Fur ther the i tems of RW are t rans fer red to OW
in a con t inuous re lea se pa t te rn to mee t the
demand unt i l t he s tock leve l i n the RW i s
empt ied and then the i t ems o f OW are re l eased .
A two wareho use inventory model was
d iscussed b y Har t e ly [1976] . Sarma [1983]
developed a de termin is t ic inven tory model wi th
in f in i te p roduct ion ra te and two leve l s o f
s to rage . Dave [1988] rec t i f ied the e r rors and
gave a co mple te so lu t io n o f the model given b y
Sarma [1983] . In the above models , t he
ana lys i s i s car r ied out wi tho ut tak ing
shor tages . Gos wami and Chaudhur i [1992]
cons ide red two s to rage mode ls wi th and
wi tho ut shor tages , a l lo wing t ime dependent
demand. Bhunia and Mai t i [1994] developed
the same inventory model s cor rec t ing and
modi fying the assump t ion of Gos wami and
Chaudhur i [1992] . Also , b y co ns ider ing
cons tant demand, S arma [1987] p resented a
model fo r de ter io ra t ing i tems wi th an in f in i te
rep len ishment ra te a l lo wing for shor tages .
Bhunia and Mai t i [ 1997] s tud ied a two
wareho use inven tory model fo r de ter io ra t ing
i tems co nsider ing l inear ly t ime dependent
demand and shor tages . Buzaco t t and Misra
[1975] s imul taneo us ly developed EOQ mode ls
wi th constant demand and a s ing le in f la t io n
ra te fo r a l l a s soc ia t ed cost s . B ierman and
Tho mas [1977] then proposed an in f la t ion
model fo r the EOQ in which the t ime va lue o f
mo ney i s a l so inc orpora ted . Bose e t a l . [1995]
developed an EOQ inventory model unde r
in f l a t io n and t ime d iscount ing. Recen t ly , Yang
e t a l . [2006] genera l ized the inventory mode l
under in f la t ion for var i ab le demand.
In the las t few years , i nventory
prob lems involving t ime var iab le demand
pa t te rn have rece ived a t ten t io n fro m severa l
researcher s . This type of p rob lem was f i r s t
d iscussed b y Stanfe l and Sivaz l ian [1975] .
Next , S i lver and Meal [1973] es tab l i shed an
approximate so lut io n techniq ue o f a
de terminis t ic inventory mod e l wi th t ime
dependent demand. Donald son [197 7]
developed an op t imal a lgor i thm for so lving the
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
439
c lass ica l no -shor tage inventory mode l
ana lyt ica l ly wi th l i near t rend in demand over a
f ixed t ime hor izon. Ho wever , t he so lu t io n
procedure requi r es a lo t o f co mp u ta t io nal work
and can no t be ea s i ly emp lo yed to de termine
the va lues o f the op t imal dec is ion var iab le . To
remo ve the co mp uta t ional and conceptua l
co mplexi ty o f Do naldson’s [1977] op t ima l
analyt ic approach, severa l researchers
emplo yed search me thods fo r so lving the
p rob lem. Amo ng them, Si lver [1979] , Henry
[1979] , Phelps [1980] , Buchanan [1980] , Mi t ra
e t a l . [1984] , Ri tchie [1984] and o thers a r e
wor th ment io ning, bu t no ne o f them co nsidered
shor tages . Co ns ider ing shor tages , Dave [1989] ,
Deb and Chaud har i [1 987] , Goyal e t . a l .
[1992] , Dut ta and Pal [1992] , and Hor iga
[1993 , 1994] developed a so lut ion fo l lo wing
Dona ldso n’s o r a l te rna t ive approaches . In these
papers , t he so lut ion procedure was
co mp uta t ional ly co mpl i ca ted excep t Dut ta and
Pal where i t was very s imple and easy to
ca lcula t e the va lues o f dec is ion var iab les .
Other paper s o f r e la t ed top ic were wr i t t en b y
Chung and T ing [1993] , Go yal e t a l . [1996] ,
Bhunia and Mai t i [1 999] , Chakravar t i and
Chaudhar i [1997] and o ther s .
I I . ASSUMPTIONS
The ma themat i ca l mo d el o f the t wo -warehouse
inventory prob lems i s based on the fo l lo wing
assumpt ions:
1 . Shor t ages a re a l lo wed and co mple te ly
backlo gged .
2 . Lead t ime i s zero and the ini t ia l
inventory leve l i s zero .
3 . The o wned warehouse (OW) has a f ixed
capac i ty o f W uni t s .
4 . The rented wareho use (RW) has
unl imi ted capac i ty .
5 . The inventory co st s ( inc lud ing ho ld ing
cost and de ter io ra t io n
cos t ) i n RW are h igher than tho se in OW.
In add i t io n the fo l lo wi ng no ta t io ns a re
used througho ut thi s paper .
W The capac i t y o f o wned wareho use (OW).
aeb t
The demand r a te a t t ime t , we a ssume
tha t aeb t
i s de termini s t ic expo nent ia l
ra te o f D un i t s per un i t t ime.
The de ter io ra t io n ra te in OW, where
10 .
The de ter io ra t ion ra te in RW, where
10 and .
r The in f l a t io n ra t e
pc The purchas ing cost per uni t .
0c The rep len ishment cost per o rder .
1ch The ho ld ing cost per uni t per un i t t ime
in OW.
2ch The ho ld ing cost per uni t per un i t t ime
in RW.
sc The shor tage cos t per uni t t ime.
iTc The present va lue o f the to ta l re levant
cos t per uni t t ime fo r model 2,1i,i .
)t(I0 The inventory leve l i n OW at t ime t .
)t(I r The inventory leve l i n RW at t ime t .
)t(S The shor tage leve l a t t ime t .
0t The t ime a t which the inventory l eve l
reaches to zero in OW.
rt The t ime a t which the inventory leve l
reaches to zero in RW.
st The t ime a t which the shor tages leve l
reaches to the lo wes t po int i n the
rep len ishment cyc le .
M athemat ica l For mula t ion and Ana lysi s
There i s on ly o ne s to rage model ( i . e . t he
t rad i t io na l model ) as soc ia ted wi th thi s two
wareho uses inven tory prob lem fo und in the
p revio us l i te ra ture . The t rad i t io na l mode l i s
dep ic ted grap hica l ly in f ig 1 . I t s ta r t s wi th an
ins tant rep len ishment and end s wi th shor tages .
In the o ther model t he demand wi l l be met a t
the end of each rep leni shment cyc le , t ha t i s ,
the inven tory leve l s t a r t wi th shor tages and
ends wi thout shor tages . The proposed inven tory
model i s dep ic ted graphica l ly in f ig 2 .
There a re two poss ib le shor tages model s
under the assump t io n descr ibed above . These
two model s a re dep ic ted graphica l ly in f igs 1
and 2 . The t rad i t io na l model ( i . e . mode l 1 ) , i t
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
440
s ta r t s wi th an ins tant o rder and ends wi th
shor tages .
For the t r ad i t io na l model , a t t ime t = 0 ,
a lo t s ize o f cer ta in uni t s enters the sys tem,
f ro m which a por t ion i s backlo gged to ward s
p revio us shor tages . W uni t s a r e kep t in OW and
the re s t i s s to red in RW. The goods o f OW a re
consumed o nly a f t e r consuming the goods kep t
in RW. Dur ing the int e rva l )t,0( r , the
inventory in RW grad ual ly decreases due to
demand and de ter io ra t ion and i t van ishes a t
rtt . In OW, the inventory W decreases
dur ing )t,0( r due to de ter io ra t ion o nly, but
dur ing )t,t( 0r the inventory i s dep le ted due to
bo th demand and de ter io ra t ion. B y the t ime 0t ,
bo th wareho uses beco me empty a nd therea f t e r
the shor tages a re a l lo wed to occur . The
shor tage quant i ty i s suppl ied to custo mers a t
the beg inn ing o f the next cyc le . B y the t ime st ,
the rep len ishment cyc le res ta r t s . The ob jec t ive
o f the t rad i t iona l mod el i s to de termin e the
t imings o f rt and 0s tt so tha t the to ta l
re levan t cos t ( inc lud ing ho ld ing, de ter io ra t io n,
shor tage and order ing cost s) per uni t t ime o f
the inven tory sys tem i s minimu m.
For model I d ur ing the int e rva l )t,0( r , the
inventory leve l a t t ime t i n RW and OW i s
go verned b y the fo l lo wing d i f fe rent ia l
equat ion :
)t(Iaedt
)t(dIr
btr ,
rtt0 … (1)
wi th the bo undary co ndi t ion 0)t(I rr and
)t(Idt
)t(dI0
0 ,
rtt0 … (2)
wi th the ini t ia l co ndi t ion W)0(I0 ,
r espec t ive ly. Whi le d ur ing the inte rva l )t,t( 0r
, t he inventory leve l a t OW, )t(I0 , i s go verned
b y the fo l lo wing d i f fe rent i a l eq uat ion
)t(Iaedt
)t(dI0
bt0 ,
0r ttt … (3)
wi th the bo undary cond i t io n 0)t(I 00
. S imi lar ly , dur ing )t,t( s0 , the shor tage leve l
a t t ime t , S( t ) i s governed b y the fo l lo wing
d i f fe rent ia l eq ua t io n :
btae
dt
)t(dS ,
s0 ttt … (4)
With the boundary co ndi t io n 0)t(S 0 .
The so lut io n o f (1 ) i s as fo l lo ws
Cdtaee)t(I rt)b(t
rrrr
0)t(I rr
rt
t
u)b(tr duaee)t(I
… (5)
The so lut io n o f (2 ) i s as fo l lo ws
t
0 We)t(I ,
rtt0 … (6)
The so lut io n o f (3 ) i s as fo l lo ws
dtaedtaee)t(I t)b(
0t)b(t
00
0t
t
u)b(t0 duaee)t(I ,
0r ttt … (7 )
The so lut io n o f (4 ) i s as fo l lo ws
0btbt dtaedtae)t(S 0
t
t
bu
0
duae)t(S ,
s0 ttt … (8)
Using the co ndi t ion for )t(I0 a t rtt , we have fro m (6) and (7) , tha t
t
0 We)t(I
and
0t
t
u)b(t0 duaee)t(I
o r
0
r
rr
t
t
u)b(ttr0 duaeeWe)t(I
… (9)
which impl ie s tha t
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
441
0
r
t
t
t)b( dtaeW
… (10)
r0 t)b(t)b( ee)b(
aW
1eea
W)b(r0r t)b(t)b(t)b(
rt(b )0 r
(b )t t ln 1 We (b )
a
… (11)
Note tha t f ro m (11) t 0 i s a funct io n o f t r ,
there fore t 0 i s no t a dec is io n var iab le in mode l
I . Thus , t he cumula t ive inventor i es in RW
dur ing )t,0( r and OW dur ing )t,0( 0 a re
r rr t
0
t
t
u)b(t
t
0
r dtduaeedt)t(I
… (12)
and 0
r
r0 t
t
0
t
0
0
t
0
0 dt)t(Idt)t(Idt)t(I
0
r
0r0 t
t
t
t
u)b(t
t
0
t
t
0
0 dtdueedtWedt)t(I
… (13)
r espec t ive ly and the cumula t ive shor t ages
dur ing )t,t( s0 i s
s
0 0
t
t
t
t
bu dtduae .
As a r esul t t he p re sent va lue o f the
inventory ho ld ing cos t i n RW and OW are
rt
0
rrt
2 dt)t(Iech
r rt
0
t
t
u)b(t)r(2 dtduaeech
rt
0
u
0
t)r(u)b(2 dudteaech
rt
0
u)r(u)b(2 due1ae)r(
ch
rt
0
t)rb(t)b(2 dteea)r(
ch
… (14)
and
0t
0
0rt
1 dt)t(Iech
0
r
r t
t
0rt
t
0
0rt
1 dt)t(Iedt)t(Iech
0
r
0r t
t
t
t
u)b(t)r(
t
0
t)r(1 dtduaeedtWech
0
r
0r 0
r
t
t
t
t
u)b(t)r(
t
0
t
t
u)b(t)r(1 dtduaeedtduaeech
0
r r
0
r
r t
t
u
t
t)r(u)b(
t
t
t
0
t)r(u)b(1 dudteaedudteaech
0
r
r
t
t
t)r(u)b(1 due1e
r
ach
0
r
r
t
t
u)r(t)r(u)b( dueeer
a
0
r
t
t
u)rb(u)b(1 dueear
ch
0
r
t
t
t)rb(t)b(1 dteear
ch
… (15)
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
442
Respect ive ly and the p resent va lue of the
shor tage cos t i s
s
0 0
t
t
t
t
burts dtduaeec
s
0
st
t
t
u
rtbus dudteaec
s
0
s
t
t
rtrtbts dteeaer
c
s
0
s
t
t
rtbtt)rb(s dteear
c
… (16)
In add i t ion, the amount s o f de ter io ra t ed
i tems in bo th RW and OW d ur ing )t,0( 0 a re
dt)t(Irt
0
r and dt)t(I0t
0
0 . Therefore , t he
p resent va lue o f the cost fo r the de ter io ra ted
i tems i s
0r t
0
0rt
t
0
rrt
p dt)t(Iedt)t(Iec
r
r
r r t t
t
ubtr
t t
t
ubtr
p dtduaeedtduaeec0
)()(
0
)()(0
0
r
0t
t
t
t
u)b(t)r( dtduaee
0
r
r
r t
t
t)r(u)b(
t
0
u)r(u)b(p due1ae
rdue1ae
rc
0
r
r
t
t
u)r(t)r(u)b( dueeae
0
r
r t
t
u)r(u)b(
t
0
u)rb(u)b(p due1ae
rdueea
rc
0
r
r t
t
t)rb(t)b(
t
0
t)rb(t)b(p dteea
rdteea
rc
… (17)
Conseq uent ly, t he p resent va lue of the
to ta l re levan t cos t per uni t t ime for mode l I
dur ing the cyc le )t,0( s i s
0
r
r t
t
t)rb(t)b(1
t
0
t)rb(t)b(201 dteea
r
chdteea
r
chcTc
s
0
s
t
t
rtbtt)rb(s dteear
c
rt
0
t)rb(t)b(p dteea
rc
s
t
t
t)rb(t)b( tdteear
0
r
ae1
br
11e
b
1
r
cchc
rt
0
t)rb(t)b(p2
0
aerb
1e
b
1
r
cch 0
r
t
t
t)rb(t)b(p1
s
t
t
rtbtt)rb(s ta
b
e
rb
e
r
cs
0
s
ae1
br
11e
b
1
r
cchc rr t)rb(t)b(p2
0
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
443
abr
e
br
e
b
e
b
e
r
cch r0r0 t)rb(t)rb(t)b(t)b(p1
s
rtbtt)rb(t)rb(t)rb(s ta
b
e
b
e
br
e
br
e
r
c s0s0s
ae1
br
11e
b
1
r
cchc rr t)rb(t)b(p2
0
r00rr rtrtbtbtrtp1eee
br
aW
r
cch
s
btbtrtrtbtbtrts t
b
e
b
e
br
e
br
ee
r
c 0ss00s
s
ae1
br
11e
b
1
r
cchc rr t)rb(t)b(p2
0
)tt(rbtbtrtp1 r00rr eeebr
aW
r
cch
sbtbtbt)tt(rbtrts tee
b
1ee
br
1e
r
c0sss00s
ae1
br
11e
b
1
r
cchc rr t)rb(t)b(p2
0
10rr rbtbtrtp1eee
br
aW
r
cch
s2021r btrbt)t(rs eebr
1e
r
c
)t(btbtb
121r0s
ae1
br
11e
b
1
r
cchc rr t)rb(t)b(p2
0
1r )rb(t)br(p1e1e
br
aW
r
cch
)t(aeebr
1e
r
c21r2
btrbt)t(rs s2021r
… (18)
where
)b(ea
W)b(1Intt rt)b(
r01
… (19)
and 0s2 tt
To d is t ingu ish eas i ly , t he no ta t io ns 0t ,
rt , st a re rep laced b y 0t , rt , st in model 2 . fo r
model 2 , t he p resent va lue o f o rder ing co st i s
str0eC
. B y us ing a s imi lar a rgument as in
model 1 , t he p re sent va lue of the to ta l re levant
cos t per uni t t ime for model 2 i s ob ta ined as
fo l lo ws :
s
ss
t
0
bt)tt()rb(s0
tr2 dteea
r
cceTc
r
s
ss
t
t
)tt()rb()tt()b(p2dteea
r
cch
)t(dteear
cch43s
t
t
)tt)(rb()tt)(b(p10
r
ss
International Journal of Engineering Applied Sciences and Technology, 2019
Vol. 4, Issue 3, ISSN No. 2455-2143, Pages 438-453 Published Online July 2019 in IJEAST (http://www.ijeast.com)
444
ss
s
t
0
bt)tt()rb(s
0tr
b
e
rb
e
r
acce
r
s
sst
t
)tt()rb()tt()b(p2
rb
e
b
e
r
ccha
)t()rb(
e
)b(
e
r
ccha 43s
t
t
)tt)(rb()tt()b(p1
0
r
ss
b
1
rb
e
b
e
rb
1
r
acce
ss
s
t)rb(tbs
0tr
rb
1
b
1
rb
e
b
e
r
ccha
)tt()rb()tt()b(p2
srsr
b
e
rb
e
b
e
r
ccha
)tt()b()tt)(rb()tt()b(p1
srs0s0
)t(rb
e43s
)tt)(rb( sr
a1e
b
11e
br
1
r
cce sss tbt)br(s
0tr
ae1br
11e
b
1
r
cch33 )br()b(p2
)tt()b()tt()b(p1srs0 ee
)b(
a
r
cch
)t(eeebr
a43s
)tt()tt()rb( s0sr
a1e
b
11e
br
1
r
cce sss tbt)br(s
0tr
ae1br
11e
b
1
r
cch33 )br()b(p2
r
cch p1
)t(e1ebr
aW 43s
)rb()rb( 43
… (20)
where
0
r
s
t
t
tt)b(dtaeW , sr3 tt
and
)b(Wea
)b(1Intt 3)b(
r04
… (21)
The ob jec t ive here i s to de termine the
op t ima l va lues o f st and 4 in o rder to
minimize the to ta l re levant co s t per uni t t ime.
I I I . SOLUTIONS OF THE MODEL
No w, the op t imal so lu t ions fo r bo th model I
and I I wi l l be e s tab l i shed . Model I , f ro m (19)
adt
d)rb(ee
br
1e1e
r
1)rb(t)br()rb(t)br( 1r1r
)b(ea
W)b(1Intt rt)b(
r01
a
1
ea
W)b(1
We)b(
)b(
1
dt
d
r
r
t)b(
t)b(2
r
1
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445
a
e)b(Waa
We)b(
r
r
t)b(
t)b(
r
1
dt
d
r
r
t)b(
t)b(
e)b(Wa
We)b(
Also we have
rt)b(1 e
a
W)b(1In
)b(
1
a
We)b(aIn)b(
rt)b(
1
a
We)b(ae
r
1
t)b()b(
1We)b(a
a1e
r
1
t)b(
)b(
r
r
1
t)b(
t)b()b(
We)b(a
We)b(1e
r
1
dt
d
1ee)b(Wa
We)b(1
r
r)b(
t)b(
t)b(
… (22)
The necessary co ndi t io ns fo r Tc 1 i n (18) to b e
minimu m are
r
1
t
Tc
aeer
cchrr t)rb(t)b(p2
r
cch p1
aeebr
1
dt
d1)r(e
r
c2
btrbt
r
1)t(rs s2021r
0)t(dt
d1Tc 21r
r
11
r
1
t
Tc
rr t)rb(t)b(p2
eer
cch
ae1er
cch1r )r(t)br(p1
r
12
btrbt)t(rs
dt
d1ee
br
1aec s2021r
0)t(dt
d1Tc 21r
r
11
… (23)
and
2
btrbt)t(rs
2
1 s2021r eebr
1)r(e
r
acTc
0)t(Tc1ebr
re
r
c21r1
rbt)t(rs 2021r
2021r rbt)t(rs
2
1 ebr
1eac
Tc
0)t(Tcr
1e
br
1e
br
121r1
rbt2
bt 20s
2
1Tc
0)t(Tcr
1e
br
1eac 21r12
bt)t(rs
s21r
… (24)
Fro m eq ua t io n (23) and (24) , t he
fo l lo wing can be eas i ly ob ta ined
Fro m (23)
)ee(
r
cch
dt
d1Tc rr t)rb(t)b(p2
r
11
ae1er
cch1r )r(t)rb(p1
)t(rs
21raec
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446
2
bt)rbt()ee(
br
1s20
r
1
dt
d1
1)b(
1eTc
rr t)rb(t)b(p2ee
r
cch
a)e1(er
cch1r )r(t)rb(p1
)t(rs
21raec
1s20 )b(2
btrbte)ee(
br
1
and fro m (24)
r
1e
br
1aecTc 2
bt)t(rs1
s21r
1s21r1 )b(2
bt)t(rs
)b(1 e
r
1e
br
1eaceTc
1s21r )b(2
bt)t(rs e
r
1e
br
1eac
rrr t)rb(p1t)rb(t)b(p2e
r
cchee
r
cch
ae1 1)r(
)t(rs
21raec
1s20 )b(2
btrbteee
br
1
1
s
20s21r )b(2
btrbt
2bt)t(r
s ebr
ee
br
1
r
1e
br
1ec
1rrr )r(t)rb(p1t)rb(t)b(p2e1e
r
cchee
r
cch
12021r )b(rbt)t(rs e
r
1e
br
1ec
1rrr )r(t)rb(p1t)rb(t)b(p2e1e
r
cchee
r
cch
… (25)
This impl ies tha t
1
20
1r )b(rbt
)t(rs e
r
e
br
eec
1rr )r(p1t)r(p2t)rb(e1
r
cch1e
r
cche
r11r
20t)rb()b()t(r
rbt
s er
e
br
ec
1r )r(p1t)r(p2
e1r
cch1e
r
cch
r111
20btbr
rbt
s er
e
br
ec
1r )r(p1t)r(p2
e1r
cch1e
r
cch
r1
20bt)br(
rbt
s er
e
br
ec
r 12 p 1 p( r)t ( r)ch c ch c
e 1 1 er r
… (26)
Thus 2 i s a l so a funct ion o f rt
consequent ly , i f rt i s kno wn, then 1 and 2
can be uniq uely de termined b y (19) and (26 ) ,
International Journal of Engineering Applied Sciences and Technology, 2019
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447
respec t ive ly. The so lu t ion tha t sa t i s f ies (23)
and (24) min imizes 1Tc . S imi l ar ly , fo r mode l
I I , f ro m (21)
)b(Wea
)b(1In 3)b(
4
3
3
)b(
)b(2
3
4
We)b(aa)b(
Wae)b(
d
d
3
3
)b(
)b(
We)b(a
We)b(
Also ,
)b(a
We)b(1In
3)b(
4
3
3
4
)b(
)b()b(
We)b(a
We)b(1e
3
4
d
d
1eWe)b(a
We)b(4
3
3)b(
)b(
)b(
… (27)
The necessary co ndi t io ns fo r Tc 2 i n (20) to b e
minimu m are
s
2
t
Tc
a1e
b
11e
br
1
r
c
te sss tbt)br(s
s
tr
ae1
br
11e
b
1
r
cchcer 33s )br()b(p2
0tr
r
cch p1
)t(e1ebr
aW 43s
)rb()rb( 43
0)t(Tc 43s2
1e
)b(
1
r
cchcratce 3s )b(p2
0sstr
ae1)br(
13)br(
r
cch p1
)t(e1e)br(
aW 43s
)rb()rb( 43
0)t(Tc 43s2 …
(28)
and
r
cchaee
r
cche
Tc p1)br()b(p2tr
3
2 33s
)t(d
deaee1ae 43s
3
4)rb()rb()rb()rb( 4343
0)t(d
d1Tc 43s
3
42
333s )rb(p1)br()b(p2tre
r
cchee
r
cchae
)t(e143s
)r( 4
0)t(d
d1Tc 43s
3
42
… (29)
Aga in fro m (28)
1e
)b(
1
r
cchcratceTc 3s )b(p2
0sstr
2
r
cchae1
)br(
1 p1)br( 3
43 )rb()rb(e1e
)br(
aW
And fro m (29)
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448
33s )br()b(p2tr
3
42 ee
r
cchae
d
d1Tc
43 )r()rb(p1e1e
r
cch
333s )br()b(p2)b(tr2 ee
r
cchWe)b(aeTc
43 )r()rb(p1e1e
r
cch
Fro m (28) and (29)
1e)b(
1
r
cchcratc 3)b(p2
0ss
ae1)br(
13)br(
r
cch p1
43 )rb()rb(e1e
)br(
aW
333 )br()b(p2)b(ee
r
cchWe)b(a
43 )r()rb(p1e1e
r
cch
33 )br()b(p2
0ss e1br
ar1e
b
ar
r
cchrcatc
33333 )br()b()b()br()b(eeWe)b(eaae
r
cch p1
43 )rb()rb(ee
br
arrW
43 )r()rb(e1e
. . . (30)
Which impl ies tha t st i s a func t io n of 3 .
Conseq uent ly, i f 3 i s kno wn, then s't and 4can be uniq uely de termined b y (30) and (21 ) ,
respec t ive ly. The so lu t ion tha t sa t i s f ies (28)
and (29) minimizes 2Tc .
Taking the seco nd par t i a l der iva t ives o f
1Tc wi th respec t to rt and 2 r espec t ive ly,
b y (23) & (25) , we have
ae)rb(e)b(
r
cch
t
Tcrr t)rb(t)b(p2
2
r
12
r
cch p1
ae1e)rb(dt
d)r(ee 1r1r )r(t)rb(
r
1)r(t)rb(
2
btrbt)t(rs
s2021r eebr
1aec
2
r
1
dt
d1r
)t(r
s21raec
2
btrbt s20 eebr
12r
12
dt
d
r
1
r
1
2r
12
1t
Tc
td
d1
dt
dTc
)t(dt
d1
t
Tc21r
r
1
r
1
ae)rb(e)b(r
cch
t
Tc rt)rb(rt)b(p2
2r
12
adt
de)r(e1)rb(e
r
ccha
r
1)r()r(p1 11rt)rb(
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449
2
btrbt)t(rs ree
br
raec s2021r
2
r
1
dt
d1
s2021r btrbt)t(rs ee
br
1aec
r
1)b(
dt
de)b( 1
)t(r
s21raec
r
1)b(2
dt
de)b( 1
r
1e
br
1aec 2
bt)t(rs
s21r
)t(dt
de)b( 21r
r
1)b( 1
rt)rb(r e)rb(e)b(
r
ccha
t
Tc t)b(p2
2r
12
r
1)r()r(p1
dt
de)r(e1)rb(e
r
cch11
rt)rb(
2
btrbt)t(rs ree
br
rec s2021r
2
r
1
dt
d1
s2021r btrbt)t(rs eeec
r
1)b(
dt
de
br
b1
121r )b()t(rs e)b(ec
)t(r
1e
br
1
dt
d21r
bt
r
1 s
rt)rb(r e)rb(e)b(
r
ccha
t
Tc t)b(p2
2r
12
r
1)r(p1
dt
de)r(e1)rb(e
r
cch 1)r(1
rt)rb(
2
btrbt)t(rs ree
br
rec s2021r
s2021r btrbt)t(rs
2
r
1 eeecdt
d1
121r )b()t(rs
r
1
r
1 e)b(ecdt
d
dt
d1
br
b
)t(r
1e
br
1
dt
d21r
bt
r
1 s
rt)rb(r e)rb(e)b(
r
ccha
t
Tc t)b(p2
2r
12
1r )r(t)rb(p1e1e)rb(
r
cch
1ee)cch( 11 )b()r(p1
1s2021r )b(2btrbt)t(rs eee
br
rec
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450
s2021r121r btrbt)t(rs
)b(22
)t(rs eeecerec
121r11 )b()t(rs
)b()b(e)b(ece1e
br
b
)t(r
1e
br
11e 21r
bt)b( s1
rt)rb(r e)rb(e)b(
r
ccha
t
Tc t)b(p2
2r
12
12021r )b(rbt)t(rs e
r
1e
br
1)rb(ec
rr t)rb(t)b(p2ee)rb(
r
cch
1eecch 11 )b()r(p1
1s2021r )b(2btrbt)t(rs eee
br
rec
s2021r btrbt)t(rs eeec
br
b
121r11 )b(22
)t(rs
)b()b(erecee1
1ee)b(ec 1121r )b()b()t(rs
)t(r
1e
br
121r
bts
11r )b()r(p1
t)b(p22
r
12
e1ecchecchat
Tc
2020121r rbtrbt)b()t(rs e
br
r
r
rbeeec
1s1 )b(bt)b(ee
br
re
br
b
1e)b(eee1 1s201 )b(btrbt)b(
)t(erecr
1e
br
121r
)b(22
)t(rs
bt 121rs
11r )r()r(p1
t)b(p22
r
12
e1ecchecchat
Tc
2020121r rbtrbt)b()t(rs e
br
r
r
rbeeec
1s1 )b(bt)b(ee
br
re
20 rbte
br
b
201s rbt)b(btee
br
be
br
b
s1s1 bt)b(bt)b(ee
br
bee
br
b
r
be
br
)b(
r
e)b(s
1bt
)b(
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451
)t(erec 21r)b(2
2)t(r
s121r
11)r(
r )r(p1
t)b(p2 e1eccheccha
20121r rbt)b()t(rs e
r
reec
s20 btrbtee
br
r
201 rbt)b(e
br
be
11 )b()b(e
r
be1
)t(erec 21r)b(2
2)t(r
s121r
… (31)
1e
br
reac
t
Tc2021r rbt)t(r
s
r2
12
)t(r2
btrbt 21rs20 reeebr
1
)t(t
Tc
dt
d1 21r
r
1
r
1
2
bt)t(rs
r2
12
rebr
r1aec
t
Tcs21r
)t(dt
d1 21r
r
1
…
(32)
)t(r
r
1s
2r
12
21redt
d1rac
t
Tc
)t(r
1
br
e21r2
bts
s21r bt2
)t(rs
2r
12
ebr
rr1aec
t
Tc
)t(dt
d1 21r
r
1
…
(33)
And
r
1e
br
1)r(eac
Tc2
bt)t(rs2
2
12
s21r
)t(Tc2
e 21r
2
1)t(r 21r
)t(2rebr
race
Tc21r2
bts
)t(r
22
12
s21r
… (34)
Fro m (25)
12021r )b(rbt)t(rs e
r
1e
br
1ec
rr t)rb(t)b(p2ee
r
cch
1rt)rb( )r(p1
e1er
cch
12021r )b(rbt)t(rs e
r
1e
br
1ec
rr t)rb(t)b(p2ee
r
r
r
cch
1
rt)rb( )r(
p2
p1e1e
cch
cch
12021r )b(rbt)t(rs e
r
1e
br
1ec
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452
1rrrr )r(t)rb(t)rb(t)rb(t)b(p2eeeee
r
cch
1rr )r(t)rb(t)b(p2eee
r
cch
… (35)
1
r
rby and
1
cch
cch
p2
p1
Thus fro m (31) i t i s ob v ious tha t
11r )r()r(p1
t)b(p22
r
12
e1e)cch(e)cch(at
Tc
20121r rbt)b()t(rs e
r
reec
1s20 )b(btrbteee
br
r
120 )b(rbte1e
br
b
1)b(e
r
b
0)t(erec 21r)b(2
2)t(r
s121r
Therefore ,
2r
12
r2
12
22
12
2r
12
t
Tc.
t
TcTc.
t
Tc
11r )r()r(p1
t)b(p2
2 e1e)cch(e)cch(a
20121r rbt)b()t(rs e
r
reec
1201s20 )b(rbt)b(btrbte1e
br
beee
br
r
121r1 )b(22
)t(rs
)b(erece
r
b
)t(2rebr
rec 21r2
bt)t(rs
s21r
s21r bt2
)t(rs e
br
rr1ec
2
21r
r
1 )t(dt
d1
1r21r )r(p1
t)b(p2
)t(rs
2 e)cch(e)cch(eca
1)r(e1
20121r rbt)b()t(rs e
r
reec
1201s20 )b(rbt)b(btrbte1e
br
beee
br
r
1)b(e
r
b
121r )b(22
)t(rs erec
s21r bt2
)t(rs2 e
br
rr1ec2r
br
r
0)t(e 221r
)b(2 1
.
As a resu l t , the so lu t ion tha t sa t i s f ie s
(23) and (24) minimizes 1Tc in (18) .
IV. CONCLUSION
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453
In thi s paper , we propose a model fo r
de ter io ra t ing i tems the op t imal rep leni shment
cyc le fo r the two -wareho use inventory prob lem
under in f la t io n, i n which the invento ry
de ter io ra te s a t a co nstant ra te over t ime and
shor tages a re a l lo wed wi th exponent ia l r a te o f
demand. The two -wareho use inven tory mode l
fo r de ter io ra t ing i tems wi th exponent ia l
demand ra t e and shor t ages under in f la t io n i s
cons idered for thi s mod el , t he op t ima l so lu t io n
ob ta ined . The proposed model can be
extended in numerous ways . For examp le , we
may extend the expo nent i a l demand to a more
genera l ized demand pa t te rn tha t f luc tua tes wi th
t ime. Al so , we may co ns ider the uni t purchase
cost , t he inventory ho ld ing cos t , and o ther s a re
a lso f luc tua t ing wi th t ime. Fina l ly , we may
genera l ize the mod el to a l lo w for q uant i ty
d isco unt .
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