Published Online July 2019 in IJEAST ( ...Tesma403,IJEAST.pdf · without shortages, allowing time...

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



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



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)


t

0 We)t(I ,

rtt0 … (6)


dtaedtaee)t(I t)b(

0t)b(t

00

0t

t

u)b(t0 duaee)t(I ,

0r ttt … (7 )


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



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)



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



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



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



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



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 ) ,



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)



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(



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



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


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(



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)


r

1e

br

1ec

rr t)rb(t)b(p2ee

r

cch

1rt)rb( )r(p1

e1er

cch


r

1e

br

1ec

rr t)rb(t)b(p2ee

r

r

r

cch

1

rt)rb( )r(

p2

p1e1e

cch

cch


r

1e

br

1ec



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


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


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


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



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 .

V. REFERENCES:

Buzacot t , J .A. (1975) :Econo mic order

quant i t ie s wi th in f la t io n.

Opera t io nal Research Quar te r ly, 26 , 3 , 553 -

558 .

Bierman, H. and Thomas, J .

(1977) : Inventory dec is io ns under

in f l a t io nary condi t io ns . Decis io n Sc iences ,

8 , 151 -155 .

Bose , S . , Go swami , A. and Chaud hur i , K.S.

(1995) :An EOQ mode l fo r de ter io ra t ing

i tems wi th l i near t ime dependent demand

ra te and shor tages und er in f la t io n and t ime

d isco unt ing. Journa l o f the Opera t io na l

Research Soc ie ty , 46 , 6 , 771 -782 .

Teng, J .T . and Yang, H.L. (2004) :

Determini s t ic eco no mic order quan t i ty

model s wi th par t ia l back logging when

demand and cos t a re f luc tua t ing wi th t ime .

Journa l o f the Op era t io nal Research

Socie ty , 55(5) , 495 -503 .

Pal , A.K. , Bhunia , A.K. and Mukher j e e ,

R.N. (2006) : Op t imal lo t s ize mode l fo r

de ter io ra t ing i tems wi th demand ra te

dependent on d i sp layed s tock leve l (DSL)

and par t ia l backorder ing. European Journa l

o f Opera t io nal Research , 175(2) , 977 -991 .

Singh, C. and Singh, S .R. (2010) ‘Supply

chain mode l wi th s tochast i c lead t ime under

imprec ise par t ia l ly backlo gging and fuzzy

ramp - t ype demand for exp i r ing i tems’ , In t .

J . Opera t iona l Resea rch, Vol . 8 , No . 4 ,

pp .511–522 .

Sko ur i , K. , Konstantar as , I . , Papachr i s tos ,

S . , & Teng, J .T . (2011) . Supply cha in

model s fo r de ter io ra t ing prod uct s wi th ramp

type demand ra te under permiss ib le de lay in

payment s . Exper t Systems wi th

Appl ica t ions , 38 , 14861 -14869 .

Singh, S .R. and Saxena , N. (2012) ‘An

opt ima l re turned po l icy fo r a reverse

logi s t ics inven tory mod el w i t h backorders ’ ,

Ad vances in Deci s io n Sciences , Ar t ic le ID

386598 , 21pp .

Zang J, Bai Z, Tang W. Optimal pricing policy for

deteriorating items with preservation technology

investment. J Ind Manag Optim. 2014;10(4):2014.

H. Nagar and P. Surana (2015) Fuzzy Inventory Model

for Deteriorating Items with Fluctuating Demand and

Using Inventory Parameters as Pentagonal Fuzzy

Numbers, Journal of Computer and Mathematical

Sciences, Vol. 6(2), 55-66

Banu A, Mondal SK. Analysis of credit linked demand

in an inventory model with varying ordering

cost. Springerplus. 2016;5:926. doi: 10.1186/s40064-

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