TELKOM
NIKA
, Vol. 13, No. 4, Dece
mb
er 201
5, pp. 1486
~1
494
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i4.2910
1486
Re
cei
v
ed Au
gust 25, 20
15
; Revi
sed O
c
t
ober 2
1
, 201
5; Acce
pted
No
vem
ber 1
1
,
2015
Optimal Economic Ordering Policy with Trade Credit
and Discount Cash-Flow
Approach
Hao Jiaqin*
1
,
Mo Jiangtao
2
, Min Jie
3
1
School of Mat
hematics a
nd
Statistics, Suz
hou
U
n
ivers
i
t
y
, Suzho
u
,
Anhui 234
00
0,
Chin
a
2
Colle
ge of Mat
hematics a
nd Informatio
n
Sci
ence, Gu
a
n
g
x
i
Univers
i
t
y
, Na
n
n
in
g, Guang
xi 530
00
4, Chin
a
3
School of Mat
hematics a
nd
Ph
y
s
ics,
Anhui
Ji
anz
hu Un
iver
sit
y
, Hefei,
Anh
u
i 30
60
1, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: h-j-q-2-0-
@1
63.com
A
b
st
r
a
ct
In this paper
, an inv
entory
mode
l for deteri
o
ratin
g
ite
m
s u
nder tw
o level
s
of trade cred
it w
ill be
establ
ishe
d. T
he trad
e cred
it policy
de
pen
ds on th
e
reta
iler
’
s
order
qu
antity
.
W
hen the reta
iler
’
s
or
de
r
qua
ntity is greater than or
equ
al to a predeter
mi
ne
d qua
nt
ity, both of the su
ppli
e
r
and the retail
er are takin
g
trade cre
d
it p
o
licy; otherw
i
se,
the
d
e
lay in p
a
yments
is not
per
mitted
.
S
i
n
c
e the s
a
me c
a
sh a
m
ou
nt h
a
s
different val
u
e
s
at different points
of time, the disco
unt ca
sh-flow
(DCF
)
is used to ana
l
ysis the invent
ory
mo
de
l. T
he
pur
pose
of th
is p
a
per is
to fi
nd
a
n
o
p
ti
ma
l
ord
e
r
i
ng
po
licy to
mi
ni
mi
z
i
n
g
the
pr
esent v
a
lu
e
of
all
future cas
h
-flo
w
s
cost by us
ing
DCF
a
ppr
oach.
T
h
e
me
thod to
det
er
mi
ne t
he
opti
m
a
l
or
deri
n
g
polic
y
efficiently is pr
esente
d
. Some
numerica
l
exa
m
p
l
es ar
e pr
ov
ide
d
to demon
strate the mod
e
l an
d sensitiv
i
t
y
of some import
ant para
m
eters
are
ill
ustrated
the opti
m
a
l
sol
u
tions.
Ke
y
w
ords
: tw
o-lev
e
l trad
e cr
edit, det
erior
a
ti
ng ite
m
s, or
der
qua
ntity de
pe
nde
nt cred
it, di
scount cas
h
-flo
w
,
EOQ
Copy
right
©
2015 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
In the traditional economi
c
orde
r quantit
y m
odels a
s
sumed the pu
rcha
se
r mu
st pay for
the items a
s
soo
n
as th
e items rece
ived. Howeve
r
,
in real m
a
rkets, to sti
m
ulate retailer
’
s
orde
rin
g
qual
ities the supp
lier allows a certai
n fi
xed
permi
ssible d
e
lay in payment to settle
the
amount. Simil
a
rly
,
a retailer may of
fer hi
s/her custo
m
e
r
s a
pe
rmissi
ble delay p
e
ri
od to settle the
outstan
ding b
a
lan
c
e when
he/sh
e re
ceiv
ed a tra
de
cr
edit by the supplie
r
,
whi
c
h is a two-l
e
vel
trade credit. Hua
ng [1] was the first to
explor
e an EOQ model un
der the two-l
e
vel trade credit.
Kreng an
d
T
an [2] and Ouyang
et a
l
[3] propose
d
to determi
ne the optim
al reple
n
ish
m
en
t
deci
s
io
ns if the purch
asers ord
e
r q
uant
ity is gr
eater
than or eq
ual
to a predete
r
mine
d quanti
t
y
.
Teng
et al
[4
] extended th
e co
nsta
nt d
e
mand to
a l
i
near
non
-de
c
re
asi
ng d
e
m
and fun
c
tio
n
of
time and i
n
co
rpo
r
ate
sup
p
li
er offers
a pe
rmissibl
e del
a
y
linked to
order
quantity u
nder two l
e
ve
ls
of trade credi
t.
T
eng et al [5] establishe
d an EOQ wi
th trade credi
t financing fo
r a linea
r no
n-
decrea
s
in
g d
e
mand fu
ncti
on of time. Paulu
s
[6] sh
e
d
light on ho
w se
arch
stra
tegy can b
e
use
d
to gain the
maximum be
nefit of information sea
r
ch activities.
Feng et al [7
] investigated
the
retaile
r’s o
p
timal cycl
e time and o
p
tima
l payment
time und
er the
suppli
e
r’
s ca
sh di
scount a
n
d
trade
credit policy withi
n
the EPQ fra
m
ewo
r
k. Wa
ng et al [8] establi
s
h
ed a
n
economi
c
orde
r
quantity mod
e
l for dete
r
i
o
rating ite
m
s with maxim
u
m lifetime and
cre
d
it p
e
riod i
n
crea
sing
deman
d an
d
default ri
sk.
Liao [9] deve
l
oped a
n
inv
entory mo
del
by con
s
ide
r
i
ng two level
s
of
trade
credit, limited sto
r
ag
e
cap
a
city. Wu
et al
[10] di
scu
s
sed a
n
e
c
onomi
c
o
r
de
r quantity mod
e
l
unde
r two lev
e
ls trad
e cred
it, and assum
ed deteri
o
rating items h
a
ve their expiration date
s
.
Enda
et al [11] presente
d
a gen
eric
solution t
o
the s
ensitiv
e issu
e of PCI Complian
c
e
.
Teng et al
[12]
prop
osed a
n
EPQ model f
r
om the
selle
r's
pro
s
p
e
ct
iv
e to determi
n
e
his/he
r opti
m
al trade
cre
d
it
perio
d, and in
his pap
er p
r
o
ductio
n
co
st
decli
ned an
d
obeyed a le
arning curve ph
enome
non.
Ho
weve
r
,
the above invent
ory model
s di
d not co
nsi
d
e
r
the ef
fect
s of the time value of
money
. In fact, as the
value of money chang
es with
time, it is necessary to take the ef
fect of th
e
time value of money on the
inventory pol
icy into con
s
i
deratio
n. Cha
ng
et al
[13] investigate
d
the
DCF
ap
pro
a
ch to e
s
tablish
an invento
r
y
model fo
r det
erio
rating ite
m
s
with tra
d
e
credit ba
se
d
on
the ord
e
r
qua
ntity
.
Chung
a
nd Lia
o
[14] a
dopted th
e
DCF ap
proa
ch
to discu
ss th
e
ef
fect of trad
e
cre
d
it depen
d
i
ng on the orderin
g quantit
y
.
Liao and
Huang [15] extende
d the inventory model
to
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 148
6 – 1494
1487
con
s
id
er the factors of two
levels of trad
e
cre
d
it, deterioration a
nd time discou
nting.
In this pap
er
,
we devel
op
an invento
r
y system
fo
r d
e
terio
r
ating it
ems. Firstly
,
the items
start d
e
terio
r
ating from th
e mome
nt they are
put
int
o
inventory.
Secon
d
ly, if the ret
a
iler
’
s
orde
r
quantity is greater than o
r
equal to a pre
determi
ned q
uantity, both
of the supplie
r and the reta
iler
are takin
g
trade credit po
licy; otherwi
se,
the delay
in payments is not permitted
.
Thirdly, the
pre
s
ent valu
e of all
future cash-flo
ws cost
instea
d
of the aver
age co
st. The theorem
s are
develop
ed to ef
ficiently determin
e
the optimal cycle ti
me and the p
r
esent value
of the total cost
for the retail
er
. Finally
, n
u
meri
cal exa
m
ples
and
sensitive an
al
ysis of majo
r param
eters
are
given to illustrate the theoretical re
sult o
b
tain som
e
m
anag
erial in
si
ght.
2. Nota
tions
and As
sump
tion
2.1. Nota
tion
s
The followi
ng
notations a
r
e
used th
roug
h
out this pap
er
.
A
the orde
rin
g
cost one o
r
de
r;
c
unit purcha
s
i
ng co
st per it
em;
p
unit selling pri
c
e per item
p
c
;
h
holdin
g
cost
per unit
time excludi
ng
int
e
re
st
cha
r
g
e
s;
D
deman
d rate
per yea
r
;
r
the contin
uou
s rate of di
scount;
W
quantity at whi
c
h the d
e
lay in
payments i
s
permitted;
d
T
the time interval that
W
units are
d
eplete
d
to z
e
ro;
T
the cycle time
;
Q
the retailer
’
order qu
antity per cy
cle;
()
It
the inventory level at the time of
t
;
()
P
VT
the present
value of all f
u
ture
ca
sh
-flow
co
st
.
2.2. Assump
tions
The assu
mpti
ons in thi
s
pa
per a
r
e a
s
follows:
(1)
T
i
me h
o
ri
zon
is infinite, a
nd the le
ad t
i
me is n
egligi
b
le; repl
eni
sh
ment are in
stantane
ou
s,
and shorta
ge
is not allo
wed
;
(2) A
con
s
tant
(
01
) fraction of the on-hand i
n
ventory det
erio
rate
s per unit of time
and
there is n
o
re
pair o
r
repl
acement
of the deterio
rate
d inventory;
(3)
If
QW
, both the fixed trade
cre
d
it period
M
offe
red by the su
pplier a
nd th
e trade
credit
N
offered by the retaile
r are
permitted. Ot
herwise
, the delay in payments is n
o
t permitted.
Th
e
retaile
r ca
n
accumul
a
te revenue a
nd
earn i
n
tere
st
after his/h
e
r cu
stome
r
p
a
ys for the
amount of pu
rch
a
si
ng cost
until the end of the
trade cre
d
it perio
d of
fered by the sup
p
lier
.
That is to say
,
the retailer can accumul
a
te revenue an
d earn intere
st during the perio
d
N
to
M
with rate
e
I
under the
con
d
ition of trad
e cre
d
it; Whe
n
TM
, the acco
u
n
t is settled
at
TM
and the retail
er wo
uld pay for the intere
st
cha
r
ge
s on
items in stock with rate
p
I
ov
e
r
the interval
,
M
T
; when
TM
, the accou
n
t is al
so
settled
at
TM
a
nd the
retaile
r do
e
s
not need to p
a
y any intere
st cha
r
g
e
of items in
stock
durin
g the wh
ole cy
cl
e;
Th
e fixed cre
d
it
perio
d of
fere
d by the supp
lier to the reta
ile
r is no le
ss to his/her
cu
stomers, i.e.
0
NM
.
3. Mathema
t
i
cal Model
Based
on ab
ove assum
p
tions, de
pletio
n due to d
e
m
and a
nd d
e
terio
r
ation
will occu
r
simultan
eou
sl
y
.
The inventory level of th
e syst
em can
be describ
ed
by the following dif
f
erentia
l
equatio
n
'
()
()
I
tI
t
D
,
0
tT
,
()
0
IT
.
The sol
u
tion to the above e
quation i
s
1
Tt
It
D
e
,
0
tT
.
So the retaile
r
’
s orde
r si
ze
per cy
cle is
(0
)
1
T
QI
D
e
,
0
tT
.
If
QW
, we get
d
T
:
1
ln
1
d
W
T
D
.
The present value of all future cash-flo
w cost
PV
T
con
s
i
s
ts
of the followin
g
element
s:
(1)
T
he prese
n
t value of order cost:
1
rT
O
VA
e
;
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Optim
a
l Econ
om
ic Orde
rin
g
Policy
with Trad
e Credit and Di
scou
nt Ca
sh-Flo
w …
(Hao Ji
aqin
)
1488
(2) T
he prese
n
t value of holding cost excluding inte
re
st charg
e
s:
1
1
Tr
T
r
T
H
rT
hD
e
e
e
V
rr
e
(3) T
he prese
n
t value of purch
asi
ng cost
:
whe
n
QW
(
d
TT
),
1
1
T
C
rT
cD
e
V
e
;
whe
n
QW
(
d
TT
),
1
1
rM
T
C
rT
cD
e
Ve
e
;
(4)
T
he prese
n
t values of intere
st cha
r
g
ed and e
a
rn
e
d
are ad
dressed as follo
ws:
whe
n
0
d
TT
QW
, there is no inte
re
st
earn
ed, that i
s
0
IE
V
.
The pre
s
e
n
t value of
int
e
re
st
cha
r
g
ed is
1
1
Tr
T
r
T
p
IP
rT
cI
D
ee
e
V
rr
e
whe
n
0
TN
and
d
TT
WQ
, there i
s
no inte
rest ch
arged, that is
0
IP
V
. The
prese
n
t
value of interest earned i
s
1
rN
r
M
e
IE
rT
pI
D
T
Ve
e
re
.
whe
n
NT
M
and
d
TT
WQ
, th
ere i
s
no int
e
re
st ch
arg
e
d
, that is
0
IP
V
, and the
pre
s
ent value
of interest ea
rned i
s
2
1
1
rN
rM
rT
e
IE
rT
pI
D
Vr
N
e
r
T
e
e
re
.
whe
n
M
T
and
d
TT
WQ
, the pre
s
ent val
ue of intere
st cha
r
ge
d is gi
ven by
;
1
Tr
M
r
T
rT
rM
p
IP
rT
cI
D
ee
e
e
V
rr
e
The present value of intere
st earn
ed is
2
11
1
rN
rM
e
IE
rT
pI
D
Vr
N
e
r
M
e
re
.
Therefore, th
e pre
s
e
n
t value of all future ca
sh
-flow
cost,
()
P
VT
,
can be expre
s
sed
a
s
OH
C
I
P
I
E
PV
T
V
V
V
V
V
.
Con
s
e
quently
, based
on th
e value
s
of
d
T
,
N
,
M
,thr
ee poss
ible cases
:
(
1
)
0
d
TN
, (2)
d
NT
M
, and (3)
d
M
T
will be occur.
Ca
se 1
0
d
TN
1
2
3
4
,0
,
,,
,,
,,
d
d
PV
T
T
T
PV
T
T
T
N
PV
T
PV
T
N
T
M
PV
T
M
T
whe
r
e
1
1
11
;
1
Tr
T
p
T
rT
hc
I
D
cD
re
e
PV
T
A
e
er
r
2
1
11
;
1
Tr
T
rM
T
r
N
r
M
e
rT
Dh
D
r
e
e
D
PV
T
A
c
e
e
p
I
T
e
e
er
r
r
3
2
1
11
1
;
1
Tr
T
rM
T
rN
rM
rT
e
rT
ch
D
r
e
e
D
PV
T
A
De
e
p
I
r
N
e
rT
e
e
er
r
r
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 148
6 – 1494
1489
4
2
1
11
1
11
.
Tr
M
Tr
T
r
T
rM
T
rM
p
rT
rN
rM
e
ch
D
r
e
e
D
r
e
e
PV
T
A
De
e
c
I
e
er
r
r
r
pI
D
rN
e
r
M
e
r
Ca
se 2
d
NT
M
.
1
3
4
,0
,
,,
,.
d
d
PV
T
T
T
PV
T
P
V
T
T
T
M
PV
T
M
T
Ca
se 3
d
M
T
1
4
,0
,
,.
d
d
PV
T
T
T
PV
T
PV
T
T
T
4. Theore
t
ic
al Results
The obje
c
tive
in this pape
r is to find the repleni
sh
me
nt time
*
T
to minimize the p
r
e
s
ent
value of all future c
a
s
h
-flow c
o
s
t
of the retailer
.
T
o
sim
p
ly the proof p
r
o
c
e
s
s of this model
, the following
lemma is giv
en.
Lemma 1
. Let
*
x
denote
s
th
e minimum
p
o
int of the fu
nction
of
F
x
on i
n
terval
,
ab
. Suppose
f
x
is continu
o
u
s
function and
increa
sin
g
on
,
ab
, and
2
'1
rx
rx
Fx
f
x
e
e
. W
e
have the
following results.
(a) if
0
fa
, then
*
=
x
a
; (b) if
0
f
af
b
, then
*
0
=
x
x
, where
0
x
is the
uniqu
e sol
u
tion of
0
fx
on
,
ab
; (c) if
0
fb
, the
n
*
=
x
b
.
4.1 w
h
en
Cas
e
1
T
a
k
i
ng derivative of
1
PV
T
with res
p
ec
t to
T
, we obtain
2
'
11
1.
rT
r
T
PV
T
f
T
e
e
whe
r
e
11
(1
)
1
rT
rT
T
r
T
T
p
fT
r
e
P
V
T
e
D
h
c
I
e
e
r
c
e
and
1
0
f
rA
.
Thus, we
kn
ow that
'
1
10
.
Tr
T
p
fT
D
e
h
c
I
r
c
e
From the a
b
o
v
e analysi
s
a
nd lemma 1
(
1
-
2), we have the followi
ng result
s.
Lemma 2.
Let
*
1
T
is the minim
u
m point of
1
PV
T
on
0,
d
T
.
If
1
0
d
fT
, then
*
1
d
TT
; els
e
,
*0
11
TT
, where
0
1
T
is the
unique
soluti
on of
1
0
fT
on
0,
d
T
.
Cas
e
2
T
a
ki
ng de
rivative of
2
PV
T
with res
p
ec
t to
T
, we obtain
2
'
22
1,
rT
rT
PV
T
f
T
e
e
whe
r
e
22
(1
)
1
.
rT
rT
T
r
T
T
rM
rN
rM
e
hp
I
fT
r
e
P
V
T
e
D
e
e
c
e
e
e
rr
Si
m
ilarly
,
tak
i
ng derivative
of
with res
p
ec
t to
, we k
n
ow that
'
2
1,
rT
f
Te
D
g
T
whe
r
e
.
T
T
rM
rN
rM
e
gT
h
e
r
c
e
p
I
e
e
Lemma 3.
Let
*
2
T
is the minim
u
m point of
2
PV
T
on
,
d
TN
.
(1) whe
n
0
d
gT
(a) if
2
0
d
fT
, then
*
2
d
TT
; (b) if
22
0
d
f
Tf
N
,
*0
22
TT
, w
h
er
e
0
2
T
is the u
n
iqu
e
sol
u
tion of
2
0
fT
on
,
d
TN
; (c) if
2
0
fN
, then
*
2
TN
.
0
d
TN
0
d
TT
d
TT
N
2
f
T
T
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
1693-6
930
Optim
a
l Econ
om
ic Orde
rin
g
Policy
with Trad
e Credit and Di
scou
nt Ca
sh-Flo
w …
(Hao Ji
aqin
)
1490
(2) whe
n
0
d
g
Tg
N
, where
#
2
T
is uniqu
e solutio
n
of
0
gT
on
,
d
TN
.
(i) If
#
22
0
fT
, then
*
2
d
TT
.
(ii) when
#
22
0
fT
, (a) if
2
0
fN
, then
*
22
2
2
mi
n
,
d
PV
T
P
V
T
PV
N
and
*
22
2
ar
g
m
i
n
,
d
TP
V
T
P
V
N
; (b) if
2
0
fN
, when
2
0
d
fT
,
*0
22
TT
where
0
2
T
is the
uniqu
e so
lution of
2
0
fT
on
,
d
TN
; els
e
,
*2
22
2
2
2
mi
n
,
d
PV
T
P
V
T
PV
T
and
*2
22
2
2
arg
m
i
n
,
d
TP
V
T
P
V
T
, where
2
2
T
is the
large
s
t sol
u
tion of
2
0
fT
on
,
d
TN
;
(3) whe
n
0
gN
, the
n
*
22
2
2
mi
n
,
d
PV
T
P
V
T
PV
N
and
*
22
2
ar
g
m
i
n
,
d
TP
V
T
P
V
N
.
Proof:
Sinc
e
'
0
Tr
M
gT
e
h
r
c
e
, we k
n
ow
g
T
is increa
sing o
n
,
d
TN
.
(1) whe
n
0
d
gT
, the
n
0
gT
, that is
2
0
fT
. From lemma 1, the re
sults a
r
e
proofed.
(2) W
hen
0
d
g
Tg
N
, the equation of
0
gT
ha
s a uniqu
e root (say
#
2
T
). In this situation,
0
gT
for
#
2
,
d
TT
,
0
gT
for
#
2
,
TN
. Thus
,
2
f
T
is decrea
s
ing on
#
2
,
d
TT
and in
cre
a
si
ng on
#
2
,
TN
.
As follows, we will di
scuss the property of
2
f
T
on
#
2
,
d
TT
and
#
2
,
TN
.
(A). In this ca
se we discu
s
s the prope
rty of
2
f
T
on
#
2
,
d
TT
.
From the ab
ove analysi
s
, we kno
w
th
at
2
f
T
is decrea
s
ing on
#
2
,
d
TT
, and have the
following results.
If
#
22
0
fT
, then we have
2
0
fT
for
#
2
,
d
TT
; If
#
22
2
0
d
f
Tf
T
, then the e
q
u
a
tion
2
0
fT
has a
uni
que root
(say
1
2
T
) on
#
2
,
d
TT
, and
2
0
fT
for
1
2
,
d
TT
T
,
2
0
fT
for
1#
22
,
TT
T
; If
2
0
d
fT
, then
2
0
fT
on
#
2
,
d
TT
.
(B) In this case we di
scuss
the prop
erty of
2
f
T
on
#
2
,
TN
.
From the a
b
o
v
e analysi
s
,
2
f
T
is increasing on
#
2
,
TN
, and have the followin
g
results.
If
#
22
0
fT
, then we
have
2
0
fT
for
#
2
,
TN
; If
#
22
2
0
d
f
Tf
T
, then the e
q
u
a
tion
2
0
fT
has a uniq
u
e
root (say
2
2
T
) on
#
2
,
TN
, and
2
0
fT
for
2
2
,
d
TT
,
2
0
fT
for
2
2
,
TN
; I
f
2
0
fN
, then
2
0
fT
on
#
2
,
TN
.
From (A
) and
(B), we have
the following
results
(i) if
#
22
0
fT
, then
2
0
fT
on
,
d
TN
. Thus
,
2
PV
T
is increasing on
,
d
TN
. Thus
,
*
2
d
TT
;
(ii) if
#
22
0
fT
, (a) In this
c
a
s
e
,
we
s
u
ppos
e
2
0
fN
.If
2
0
d
fT
, then
2
0
fT
; els
e
,
2
0
fT
for
1
2
,
d
TT
and
2
0
fT
for
1
2
,
TN
. therefor,
we obt
ain that if
2
0
d
fT
, then
2
PV
T
is decre
asi
ng
on
,
d
TN
; els
e
,
2
PV
T
is incre
a
si
ng on
1
2
,
d
TT
and de
creasi
ng on
1
2
,
TN
. Hence, wh
en
2
0
fN
,
*
22
2
2
mi
n
,
d
PV
T
P
V
T
PV
N
and
*
22
2
ar
g
m
i
n
,
d
TP
V
T
P
V
N
. (b) I
n
this
situation, we sup
p
o
s
e
2
0
fN
. If
2
0
d
fT
, then
2
0
fT
for
2
2
,
d
TT
and
2
0
fT
for
2
2
,
TN
.
For
the conveni
e
n
ce of that problem
, we de
note
0
2
T
is the un
ique sol
u
tion
of
2
0
fT
on
,
d
TN
. In
this
ca
se,
02
22
=
TT
. Thus,
we obtain t
hat
2
PV
T
is de
cre
a
s
ing o
n
0
2
,
d
TT
and increa
sing
o
n
0
2
,
TN
.
Thus,
*0
22
22
PV
T
P
V
T
and
*0
22
TT
. If
2
0
d
fT
, then
2
0
fT
for
1
2
,
d
TT
and
2
2
,
TN
,
2
0
fT
for
12
22
,
TT
. Thus
,
2
PV
T
is in
crea
sing o
n
1
2
,
d
TT
and
2
2
,
TN
, and de
creasi
ng on
12
22
,
TT
. Hence,
*2
22
2
2
2
mi
n
,
d
PV
T
P
V
T
PV
T
and
*2
22
2
2
arg
m
i
n
,
d
TP
V
T
P
V
T
, where
2
2
T
is the larg
est
solution
of
2
0
fT
on
,
d
TN
.
(3) whe
n
0
gN
, we have
0
gT
. Thus,
2
f
T
is decrea
s
ing o
n
,
d
TN
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 148
6 – 1494
1491
(a) if
2
0
fN
, then
2
0
fT
. T
herefo
r
,
2
PV
T
is in
crea
sing o
n
,
d
TN
. T
hus
,
*
2
d
TT
;
(b) if
22
0
d
f
Nf
T
, there e
x
ists a u
n
iqu
e
sol
u
tion
3
2
T
on
3
22
0
fT
, and
2
0
fT
for
3
2
,
d
TT
,
2
0
fT
for
3
2
,
TN
.
2
PV
T
is
inc
r
ea
s
i
ng
on
3
2
,
d
TT
and de
crea
sing
o
n
3
2
,
TN
. Hence,
*
22
2
2
mi
n
,
d
PV
T
P
V
T
PV
N
and
*
22
2
ar
g
m
i
n
,
d
TP
V
T
P
V
N
;
(c
) if
2
0
d
fT
, then
2
0
fT
.
2
PV
T
is decrea
s
in
g on
,
d
TN
. Henc
e,
*
2
TN
. From the a
b
o
ve
analysi
s
, we kn
ow that when
0
gN
,
*
22
2
2
mi
n
,
d
PV
T
P
V
T
PV
N
and
*
22
2
ar
g
m
i
n
,
d
TP
V
T
P
V
N
.
Cas
e
3
Tak
i
ng der
i
vative of
3
PV
T
with res
p
ec
t to
T
, we obtain
2
'
33
1.
rT
rT
PV
T
f
T
e
e
whe
r
e
33
1
(1
)
1
.
rT
rT
T
r
T
T
rM
rT
r
M
e
h
fT
r
e
P
V
T
e
D
e
e
c
e
p
I
e
e
rr
Thus,
we kno
w
that
'
3
10
.
rT
T
T
rM
rM
e
fT
e
D
h
e
r
c
e
p
I
e
From the a
b
o
v
e analysi
s
a
nd lemma 1,
we obtai
n the
lemma 4.
Lemma 4.
Let
*
3
T
is the mini
mum point of
3
PV
T
on
,
NM
.
(a)
if
3
0
fN
, then
*
3
TN
; (b) if
33
0
f
Nf
M
,then
*0
33
TT
, where
0
3
T
is the unique solutio
n
of
3
0
fT
on
,
NM
; (c) if
3
0
fM
, then
*
3
TM
.
Cas
e
4
Tak
i
ng der
i
vative of
4
PV
T
with respec
t to
T
, we obtain
2
'
44
1,
rT
rT
PV
T
f
T
e
e
whe
r
e
44
(1
)
1
,
Tr
M
p
rT
rT
T
r
T
T
rM
r
T
cI
h
fT
r
e
P
V
T
e
D
e
e
c
e
e
e
rr
and
4
li
m
T
fT
.
Therefor
, we kno
w
that
'
4
10
.
rM
Tr
T
r
M
p
fT
e
D
e
h
r
c
e
c
I
e
From the a
b
o
v
e analysi
s
a
nd lemma 1
(
1
-
2), we obtain
the lemma 5.
Lemma 5.
Let
*
4
T
is the minim
u
m point of
4
PV
T
on
,
M
if
4
0
fM
, then
*
4
TM
; els
e
,
*0
44
TT
, where
0
4
T
is the uni
que
so
lution of
4
0
fT
on
,
M
.
From lem
m
as 2-5, the following the
o
re
m
is obtaine
d.
Theorem 1.
The optimal
cycle time
*
T
and
the pre
s
ent value of all future cash-flo
w cost
*
()
PV
T
will be dete
r
mined by the followin
g
equ
ation
**
*
*
*
1
1
22
3
3
44
mi
n
,
,
,
P
V
T
P
V
T
PV
T
P
V
T
PV
T
.
4.2 w
h
en
d
NT
M
Cas
e
5
, we obtain the follo
wing lem
m
a.
Lemma 6.
Let
*
5
T
is the minim
u
m point of
3
PV
T
on
,
d
TM
.
(a) if
3
0
d
fT
, then
*
5
d
TT
; (b) if
33
0
d
f
Tf
M
,then
*0
55
TT
, where
0
5
T
is the uniqu
e solutio
n
of
3
0
fT
on
,
d
TM
; (c) if
3
0
fM
, then
*
5
TM
.
From the lem
m
as of 2, 3an
d 6, we have
the follow the
o
rem.
NT
M
M
T
d
TT
M
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Optim
a
l Econ
om
ic Orde
rin
g
Policy
with Trad
e Credit and Di
scou
nt Ca
sh-Flo
w …
(Hao Ji
aqin
)
1492
Theorem 2
whe
n
d
NT
M
, the optimal cycle ti
me
*
T
and the
prese
n
t value o
f
all future ca
sh-
f
l
ow co
st
*
()
PV
T
will
be determined by the following
**
*
*
11
3
5
4
4
mi
n
,
,
PV
T
P
V
T
P
V
T
P
V
T
.
4.3 w
h
en
d
M
T
Cas
e
6
, we obtain the follo
wing lem
m
a.
Lemma 7.
Let
*
6
T
is the minimum point of
4
PV
T
on
,
d
T
.
If
4
0
d
fT
, then
*
6
d
TT
; els
e
,
*0
66
TT
, where
0
6
T
is the uniq
ue sol
u
tion of
6
0
fT
on
,
d
T
.
From the lem
m
a of 2 and 7
,
we have the
theorem 3.
Theorem 3
whe
n
d
M
T
,
the optimal cycle time
*
T
and the prese
n
t value of all
future ca
sh-flo
w
co
st
*
()
PV
T
will be d
e
t
ermine
d by the followi
ng e
quation
**
*
11
4
6
mi
n
,
.
PV
T
P
V
T
PV
T
5. Numerical
examples
T
o
illu
strate
the re
sults
o
b
tained in th
is pap
er
,
we
provide th
e
following
nu
meri
cal
example
s
.
Example 1.
Le
t
5
c
,
7
p
,
0.15
p
I
,
0.
1
e
I
,
0.
1
h
,
2500
D
,
0.
2
r
,
0.
08
,
0.
3
M
,
0.
1
N
.
Re
sults a
r
e summari
ze
d in
Table
1
.
T
abl
e 1. The impact of cha
nge of
A
and
W
on
*
()
PV
T
a
nd
*
T
A
W
d
T
*
PV
T
*
T
10 150
0.0599
58683
*0
22
0.0666
TT
450
0.1787
59637
*
5
0.1787
d
TT
950
0.3743
62488
*
6
0.374
3
d
TT
100 150
0.0599
62196
*0
33
0.1855
TT
450
0.1787
62196
*0
55
0.1855
TT
950
0.3743
63736
*
6
0.374
3
d
TT
350 150
0.0599
67131
*0
44
0.3421
TT
450
0.1787
67131
*0
44
0.3421
TT
950
0.3743
67202
*
6
0.374
3
d
TT
Example 2
.
Let
10
0
A
,
5
c
,
7
p
,
0.15
p
I
,
0.
1
e
I
,
0.
1
h
,
2500
D
,
0.
2
r
,
0.
08
,
450
W
.
Re
sults a
r
e summari
ze
d in
Table
2
.
The followi
ng
inferen
c
e
s
can be mad
e
b
a
se
d on table
1 and table2:
(1)
For fixed othe
r paramete
r
s,
the large
r
the value of
A
, the large
r
the value
s
of
*
PV
T
and
*
T
.
(2)
For fixed other pa
ramete
rs, whe
n
0
d
TN
, the values of
*
PV
T
an
d
*
T
are not ch
a
nged
whateve
r
the value of
W
; when
d
NT
M
and
d
M
T
, if
*
d
TT
, then larger th
e value of
W
, the larger th
e value of
*
PV
T
but the value of
*
T
is not cha
n
ged; if
*
d
TT
, the large
r
the
value of
W
, the
large
r
the val
ues of
*
PV
T
and
*
T
.
(3)
For fixed other paramete
r
s, when
QW
, the large
r
the value of
M
, the smaller the valu
es of
d
TT
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93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 148
6 – 1494
1493
*
PV
T
; the larger th
e value of
N
, the large
r
the v
a
lue
s
of
*
PV
T
; wh
en
*
d
TT
, the lager
the value
s
of
M
and
N
, the la
rg
er the
value
of
*
T
; wh
en
*
d
TT
, the value
of
*
T
is
k
e
ep
ing
a con
s
tant when the value
s
of
M
and
N
.
T
abl
e 2. The impact of cha
nge of
M
and
N
on
*
()
PV
T
a
nd
*
T
d
T
M
N
*
PV
T
*
T
0
d
TN
0.0200
0.2
0.05
64083
0.1785
0.1
64264
0.1840
0.3
0.05
62016
0.1799
0.1
62196
0.1855
d
NT
M
0.1787
0.2
0.05
64083
0.1787
0.1
64264
0.1840
0.3
0.05
62016
0.1799
0.1
62196
0.1855
d
M
T
0.3743
0.2
0.05
65785
0.3743
0.1
65875
0.3743
0.3
0.05
63646
0.3743
0.1
63736
0.3743
Example 3.
Let
10
0
A
,
5
c
,
7
p
,
0.15
p
I
,
0.
1
e
I
,
0.
1
h
,
2500
D
,
0.
2
r
,
0.
08
,
0.3
M
,
0.
1
N
,
450
W
. Figures
1 and 2 sho
w
the cha
nge
of the values of
*
()
PV
T
and
*
T
wh
en
the
para
m
eter of
the discou
nt rate
r
is chan
ge
d from (0,1
).
F
i
gure 1. T
he impact of chan
ge of
r
on
*
()
PV
T
F
i
gure 2. T
he impact of chan
ge of
r
on
*
T
The followi
ng
inferen
c
e
s
can be mad
e
b
a
se
d on figue
r1-2:
(1)
For fixed othe
r paramete
r
s,
the large
r
the value of
r
, the smalle
r the
value of
*
PV
T
.
(2)
For fix
ed oth
e
r pa
ram
e
ter
s
, whe
n
*
d
TT
, the
large
r
the val
ue of
r
, the sm
aller the valu
e of
*
T
; when
*
d
TT
, the value of
*
T
is not
impact to the
value of
r
.
6. Conclusi
ons
In this pape
r
,
we develop
an inventory system
for de
teriorating items und
er pe
rmissi
ble
delay in pay
ments. Th
e p
r
imary differe
nce of thi
s
pa
per a
s
comp
a
r
ed to p
r
evio
us
studie
s
is
that
we int
r
odu
ce
a gen
eralized invento
r
y model
by relaxing the t
r
adition
al EO
Q mod
e
l in
the
followin
g
thre
e ways: (1
) the items dete
r
iorate co
nt
in
uou
sly; (2) if the retailer
’
s
orde
r qu
antity is
0
0.
2
0.
4
0.
6
0.
8
1
10
4
10
5
10
6
t
h
e v
a
l
u
e
of
di
s
c
o
unt
r
a
t
e
r
0
0.
2
0.
4
0.
6
0.
8
1
0.
17
0.
18
0.
19
0.
2
0.
21
0.
22
0.
23
t
he
v
a
l
ue
of
d
i
s
c
oun
t
rat
e
r
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TELKOM
NIKA
ISSN:
1693-6
930
Optim
a
l Econ
om
ic Orde
rin
g
Policy
with Trad
e Credit and Di
scou
nt Ca
sh-Flo
w …
(Hao Ji
aqin
)
1494
greate
r
than
or eq
ual to a
pred
etermi
ne
d quantity,
then both of th
e sup
p
lie
r an
d the retaile
r
are
taking trade
cre
d
it poli
c
e;
otherwise, the del
ay
in payments i
s
not permitted
;
(3) the
pre
s
ent
value of all future
cash-fl
o
ws co
st inst
ead of
the a
v
erage
co
st.
The propo
se
d of the pap
er is
minimizi
ng the pre
s
ent val
ue of all future ca
sh-flo
w cost of the retailer
.
In additi
on, the optimal
solutio
n
s to
the mo
del h
a
v
e been
di
scussed i
n
d
e
ta
il unde
r all
p
o
ssible
situati
ons. T
h
ree e
a
sy-
to-use theo
re
ms a
r
e d
e
vel
oped to fin
d
the optimal
orderin
g poli
c
ie
s for th
e con
s
idere
d
p
r
oble
m
,
and the
s
e the
o
retical re
sult
s are illu
strated by some n
u
meri
cal exa
m
ples.
In rega
rd
s to future research, on
e coul
d co
nsi
d
er in
corporat
ing more re
alistic
assumptio
n
s into the model, su
ch
a
s
the
dem
a
nd dep
end
e
n
ts the selli
ng pri
c
e, qu
antity
discou
nts, su
pply chai
n co
ordin
a
tion, etc.
Referen
ces
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ng
YF
. Optimal retai
l
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’
s
o
r
derin
g pol
ici
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e credit financ
i
n
g
. Journ
a
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of the Operatio
nal R
e
searc
h
Society
.
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1–
101
5
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[2]
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a
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he optimal rep
l
en
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m
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decisi
o
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w
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credit poli
c
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din
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e
r qu
anti
t
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l
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r t
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ic
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an
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atic
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eng JT
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eng HR, Li J
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hao D. Retailer
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a
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r
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atical M
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eng JT
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i
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a
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ang L.
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Q model un
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n
the orderi
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i
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leve
ls of tra
d
e
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i
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g
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Acta appl
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