![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 28
2. Model Development
2.1 Basis of the Model:
The classical economic order quantity (EOQ) or lot sizing model chooses a batch size that minimises
the total cost calculated as the sum of two conflicting cost functions, the order/setup cost and the
inventory holding costs. The entropic order quantity (EnOQ) is derived by determining a batch size
that minimizes the sum of the above two cost and entropic cost.
Notations
T : the inventory cycle time, which is a decision variable;
C,A : the purchase cost and ordering cost respectively
h : the unit holding cost per year excluding interest change;
D : the demand rate per unit time.
D(T) : the demand rate per unit time where cycle length is T.
r : Discount rate (opportunity cost) per time unit.
Q : Procurement quantity;
M : the credit period;
W : quantity at which the delay payment is permitted;
σ(t) : total entropy generated by time t.
S : rate of change of entropy generated at time t.
E(t) : Entropy cost per cycle;
PV1(T) : Present value of cash-out-flows for the basic EnOQ model;
PV2(T) : present value of cash-out-flows for credit only on units in stock when MT ≤ .
PV3(T) : Present value of cash-out-flows for credit only on units in stock when MT ≥ .
))),( 0 tPtP : unit price and market equilibrium price at time t respectively.
(T)PV∞ : the present value of all future cash-flows.
*
T : the optimal cycle time of
(T)PV∞ when T>0.
ASSUMPTIONS
(1) The demand is constant.
(2) The ordering lead time is zero.
(3) Shortages are not allowed.
(4) Time period is infinite.
(5) If Q<W, the delay in payment is not permitted, otherwise, certain fixed trade credit period M is
permitted. That is, Q<W holds if and only if T<W/D.
(6) During the credit period, the firm makes payment to the supplier immediately after use of the
materials. On the last day of the credit period, the firms pays the remaining balance.
2.2 Commodity flow and the entropy cost:
The commodity flow or demand/unit time is of the form
D= -k(P(t)-P0(t))(1)
The concept represented by equation (1) is analogous to energy flow (heat or work) between
a thermodynamics system and its environment where k (analogous to a thermal capacity) represents
the change in the flow for the change in the price of a commodity and is measured in additional units
sold per year per change in unit price e.g. units/year/$.
Let P(t) be the unit price at time t and P0(t) the market equilibrium price at time t, where
P(t)<P0(t) for every ],0[ Tt ∈ . At constant demand rate P(t)=P and P0(t)=P0 noting that when P<P0
, the direction of the commodity flow is from the system to the surroundings. The entropy generation
rate must satisfy
−+== 2
)( 0
0 P
P
P
P
k
dt
td
S
σ
To illustrate, assume that the price of a commodity decreases according to the following relationship](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-2-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 30
+−++−+ +−+−−−
∫∫ ...)()( )(
0
)(
0
TMr
M
tTrMr
M
rt
eMTDdtDeeMTDdtDeC
( )
( ) ( )[ ]kPkPrTc
eTr
e
rT
rM
02
)0(
1
1
−
−
−
= −
−
( )
( )[ ]TPatTPeMTrkccaMe
erT
rMrM
rT 0)0()(
1
1
−−−+
−
− −−
−
.
Therefore the present value of all future cash-flows in this case is
( )
( )
( )
−
−
+
−
+
+
−−+
−
= −−
−
−
)0(
11
12
)0(
1
)( 02023 PP
r
kT
e
hc
er
ehcka
rT
a
PP
r
hck
e
A
TPV rTrT
rT
rT
( )
( ) ( )[ ]cakkPkPrTc
eTr
e
rT
rM
−−
−
−
+ −
−
02
)0(
1
1
( )
( )[ ]TPatTPeMTrkccaMe
erT
rMrM
rT 0)0()(
1
1
−−−+
−
− −−
−
TP
Pp
aT
a
TP
0
0
2
0
)0(
1ln −
−
−+
(3)
Now our main aim is to minimize the present value of all future cash-flow cost )(TPV∞ . That
is
Minimize
)(TPV∞
subject to T>0.
We will discuss the situations of the two cases,
(A) Suppose M>W/D
In this case we have
≤
<≤
<<
=∞
.)(
/)(
/0)(
)(
3
2
1
TMifTPV
MTDWifTPV
DWTifTPV
TPV
It was found that
0)()( 21 >− TPVTPV and
0)()( 31 >− TPVTPV
for T>0 and MT ≥ respectively.
which implies
)()( 21 TPVTPV > and
)()( 31 TPVTPV >
for T>0 and MT ≥ respectively.
Now we shall determine the optimal replenishment cycle time that minimizes present value of cash-
out-flows. The first order necessary condition for )(1 TPV in (1) to be minimized is expressed as
0
)(1
=
∂
∂
T
TPV
which implies
( )
( )
( )
−+
−
−−
+
−
−
−
−−
−
−
)0(
21
12
1
02
2
2
kPkP
ak
e
CTreeCT
e
Are
rT
rTrT
rT
rT
( ) ( )
( )
−
+−−−
+
−
−−−−
22
1
11
rT
rTrTrTrT
e
reeere
r
hcka](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-4-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 32
( )
( )
( )
( ) ( )
−
+
−+
−
+
−
−
−−
+
−
−
−
−−
−
−
2203
222
2
1
2
)0(
21
1
1
412
rT
rT
rT
rTrT
rT
rT
e
e
r
hcka
kPkP
a
e
eeTcr
e
rcTec
( ) ( )
[ ]
( )3
0
2
2
0
332
2
1
.))0((2
2)1(
1
))0((4
1
2
rT
rT
rT
rTrT
rT
e
eTPPhckr
rrTe
e
PPhck
Tr
hcka
er
hckae
−
−
−
−−
−
−
−
+−+
−
−
++
−
+
We obtain from the above expression
0
)(
2
1
2
>
∂
∂
T
TPV
, which implies
0
)(1
>
∂
∂
T
TPV
is strictly
increasing function of T in the interval (0,W/D).
Also we know that,
0
)(
0
lim 1
<−=
∂
∂
→
rA
T
TPV
T and
1
1 )(
/
lim
∆=
∂
∂
→ T
TPV
DWT
Therefore, if
0
)(
/
lim
1
1
≥∆=
∂
∂
→ T
TPV
DWT , then by applying the intermediate value theorem,
there exists a unique value )/,0(1 DWT ∈ such that
0
)(1
=
∂
∂
T
TPV
and
2
1
2
)(
T
TPV
∂
∂
at the point T1 is
greater than zero.
Thus
)/,0(1 DWT ∈ is the unique minimum solution to
)(1 TPV .
However, if
0
)(
/
lim
1
1
<∆=
∂
∂
→ T
TPV
DWT for all
( )DWT /,0∈ and also we find
0
)(
2
1
2
<
∂
∂
T
TPV
for all
( )DWT /,0∈ . Thus
)(1 TPV is a strictly decreasing function of T in the
interval
( )DW /,0 . Therefore, we can not find a value of T in the open interval
( )DW /,0 that
minimizes
)(1 TPV . This completes the proof.
Lemma-2
(a) If 12 0 ∆≤≤∆ , then the total present value of PV2(T) has the unique minimum value at the
point T=T2 where
),/(2 MDWT ∈ and satisfies
02
=
∂
∂
T
PV
.
(b) If
02 >∆ , then the present value PV2(T) has a minimum value at the lower boundary point
DWT /= .
(c) If 02 <∆ , then the present value PV2(T) has a minimum value at the upper boundary point
T=M.
Proof:
( ) ( )
( )4
222
2
2
2
1
121)(
rT
rTrTrTrT
e
eereer
T
TPV
−
−−−−
−
−+−
=
∂
∂](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-6-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 33
( )
( ) ( ) ( )
( ){ }rrTe
e
PPhck
er
hckae
e
eckare rT
rTrT
rT
rT
rTrT
21
1
))0((
11
1
2
0
4
2
3
−+
+
−
+
+
+
−
+
− −
−−
−
−
−−
( ) 33323
0
2
42
1
))0(((2
Tr
hcka
Tr
cka
e
TePPckhr
rT
rT
−+
−
−
+
−
−
which is >0 where
[ ]MDWT ,/∈ .
Which implies T
PV
∂
∂ 2
is strictly increasing function of T in the interval ),/( MDW .
Also we have
2
/
2 )(
∆=
∂
∂
= DWTT
TPV
.
If 02 ≤∆ , then by applying the intermediate value theorem there exists a unique value
[ ]MDWT ,/2 ∈ so that
0
)(
2
2
=
∂
∂
=TTT
TPV
. Moreover, by taking the second derivative of PV2(T)
with respect to T at the point T2 we have
02
2
2
>
∂
∂
T
PV
.
Thus
[ ]MDWT ,/2 ∈ is the unique solution to
)(2 TPV .
Now if
0
)(
2
/
2
>∆=
∂
∂
= DWTT
TPV
which implies
02
>
∂
∂
T
PV
for all
( )MDWT ,/2 ∈ .
Therefore,
)(2 TPV is strictly increasing function of T in the interval
( )MDW ,/ . Therefore
)(2 TPV
has a minimum value at the lower boundary point DWT /= and similarly we can prove
)(2 TPV
has a minimum value at upper boundary point T=M.
Lemma-3
(a) If
03 ≤∆ , then the total present value of
)(3 TPV has the unique minimum value at the point
T=T3 on
),(31 ∞∈ MT and satisfies
03
=
∂
∂
T
PV
.
(b) If
03 >∆ , then the present value of
)(3 TPV has a minimum value at the boundary T=M.
Proof:
The proof is same to the lemma-2.
Proposition 1
(i) 022 >−− rTert
(ii) 0312
>−+ rTrT
ee
Proof:
(i)
( ) rTerTe rTrT
−−=−−−
1222
( ) rT
rT
rt −
++++= 1.....
!2
12
2
( ) ( )
+++= .....
!3!2
2
32
rTrT
rt
which is always +ve as value of r and T are always positive.
(ii)
rTrT
ee 312
−+](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-7-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 34
( ) ( ) ( ) ( )
++++−+++++= ...
!3!2
131.....
!3
3
!2
2
21
3232
rTrT
rT
rTrT
rT
( ) ( ) ....
6
6
2
1
32
+++−−=
rTrT
rT
( ) ( ) ( )43
2
5.10
2
1 rTrT
rT
rT +++−−=
which is also give a positive value a r and T
are always positive.
By this two position it is easy to say that 21 ∆>∆ .
Then the equations (7) – (9) yield
01 <∆ iff 0)/(/
1 <DWPV iff DWT /*
1 > (10)
02 <∆ iff 0)/(/
2 <DWPV iff DWT /*
2 > (11)
03 <∆
iff
0)(/
2 <MPV iff
MT >*
2 (12)
03 <∆ iff
0)(/
3 <MPV iff
MT >*
3 (13)
From the above equations we have the following results.
Theorem-1
(1) If
0,0 21 ≥∆>∆ and
03 >∆ , then
{ })/(),(min)( *
1
*
DWPVTPVTPV ∞∞∞ = . Hence
*
T is
*
1T or W/D associated with the least cost.
(2) If
0,0 21 <∆>∆ and
03 >∆ , then )()( *
2
*
TPVTPV ∞∞ = . Hence
*
T is
*
2T .
(3) If 0,0 21 <∆>∆ and
03 ≤∆ , then
)()( *
3
*
TPVTPV ∞∞ = . Hence
*
T is
*
3T .
(4) If 0,0 21 <∆≤∆ and
03 >∆ , then )()( *
2
*
TPVTPV ∞∞ = . Hence
*
T is
*
2T .
(5) If 0,0 21 <∆≤∆ and
03 ≤∆ , then
)()( *
3
*
TPVTPV ∞∞ = . Hence
*
T is
*
3T .
Proof:
(1) If 0,0 21 ≥∆>∆ and
03 >∆ , which imply that 0)/(/
1 >DWPV , 0)/(/
2 ≥DWPV ,
0)(/
2 >MPV and
0)(/
3 >MPV . From the above lemma we implies that
(i)
)(3 TPV is increasing on ),[ ∞M
(ii) )(2 TPV is increasing on ),/[ MDW
(iii)
)(1 TPV is increasing on )/,[ *
1 DWT and decreasing on ],0( *
1T
Combining above three, we conclude that )(TPV∞ has the minimum value at
*
1TT = on )/,0( DW
and )(TPV∞ has the minimum value at DWT /= . Hence,
{ })/(),(min)( *
1
*
DWPVTPVTPV ∞∞∞ = . Consequently,
*
T is
*
1T or W/D associated with the least
cost.
(2) If
0,0 21 <∆>∆ and
03 >∆
, which imply that
0)/(/
1 >DWPV ,
0)/(/
2 <DWPV ,
0)(/
2 >MPV and
0)(/
3 >MPV which implies that DWT /*
1 ≤ , DWT /*
2 > , MT <*
2 and
MT <*
3 respectively. Furthermore from the lemma
(i)
)(3 TPV is increasing on ),[ ∞M](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-8-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 35
(ii)
)(2 TPV is decreasing on ],/[ *
2TDW and increasing on ],( *
2 MT
(iii)
)(1 TPV is decreasing on
],0( *
1T and increasing on
]/,( *
1 DWT .
From the above we conclude that
)(TPV∞ has the minimum value at
*
1TT = on )/,0( DW and
)(TPV∞ has the minimum value at
*
2TT = on ),/[ ∞DW . Since
)(1 TPV >
)(2 TPV and T>0. Then
)()( *
2
*
TPVTPV ∞∞ = and
*
T
is
*
2T
(3) If
0,0 21 <∆>∆ and
03 ≤∆ , which implies that 0)/(/
1 >DWPV , 0)/(/
2 <DWPV ,
0)(/
2 ≤MPV and
0)(/
3 ≤MPV which imply that DWT /*
1 < , DWT /*
2 > , MT ≥*
2 and
MT >*
3 respectively. Furthermore, from the lemma it implies that
(i)
)(3 TPV is decreasing on
),( *
3TM and increasing on
),( *
3 ∞T
(ii)
)(2 TPV is decreasing on ),/( MDW
(iii)
)(1 TPV is decreasing on
),0( *
1T and increasing on
]/,( *
1 DWT .
Combining all above, we conclude that
)(TPV∞ has the minimum value at
*
1TT = on
)/,0( DW and
)(TPV∞ has the minimum value at
*
3TT = on ),/[ ∞DW . Since,
)(2 TPV is
decreasing on ),0( *
2T , DWT /*
1 < and DWMT /*
2 >≥ we have )()( *
22
*
11 TPVTPV > ,
)()( 2
*
12 MPVTPV > and
)()( *
333 TPVMPV > . Hence we conclude that )(TPV∞ has the minimum
value at
*
3TT = on ),0( ∞ . Consequently,
*
T
is
*
3T .
(4) If
0,0 21 <∆≤∆ and
03 >∆ , which implies that 0)/(/
1 ≤DWPV , 0)/(/
1 ≤DWPV ,
0)(/
2 >MPV and
0)(/
3 >MPV which imply that
DWT /*
1 ≥ ,
DWT /*
2 > ,
MT <*
2 and
MT <*
3 . Furthermore, we have
(i)
)(3 TPV is increasing on ),[ ∞M
(ii) )(2 TPV is decreasing on ],/[ *
2TDW and increasing on ],( *
2 MT
(iii) )(1 TPV is decreasing on )/,0( DW .
Since
)/()/( 21 DWPVDWPV > , and
)()/( *
22 TPVDWPV >
So we conclude that
)(TPV∞ has the minimum value at
*
2TT = on ),0( ∞ . Consequently,
*
T is
*
2T .
(5) If
0,0 21 <∆≤∆ and
03 ≤∆
, which gives that
0)/(/
1 ≤DWPV ,
0)/(/
2 <DWPV ,
0)(/
2 ≤MPV and
0)(/
3 ≤MPV and which imply that DWT /*
1 ≥ , DWT /*
2 > , MT ≥*
2 and
MT ≥*
3 respectively. Furthermore, from the lemma it implies that
(i)
)(3 TPV is decreasing on
),( *
3TM and increasing on
),[ *
3 ∞T
(ii)
)(2 TPV is decreasing on ],/[ MDW](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-9-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 36
(iii)
)(1 TPV is decreasing on )/,0( DW .
Since
)/()/( 21 DWPVDWPV > , combining the above we conclude that
)(TPV∞ has the
minimum value at
*
3TT = on ),0( ∞ . Consequently,
*
T is
*
3T . This completes the proof.
(B) Suppose DWM /≤ .
Here
)(TPV∞ can be expressed as follows:
≤
<<
=∞
TDWifTPV
DWTifTPV
TPV
/)(
/0)(
)(
2
1
and let
4
/
/
3 )(
∆=
∂
∂
= DWT
T
TP
(14)
By using proposition 1, we have
041 >∆−∆ , which leads to 41 ∆>∆ .
From (14), we also find that
04 <∆ iff
0)/(/
3 <DWPV iff
DWT /*
3 > (15)
Lemma-4
(a) If 04 ≤∆ , then the present value of
)(3 TPV possesses the unique minimum value at the
point 3TT = , where
),/[3 ∞∈ DWT and satisfies
0
)(3
=
∂
∂
T
TP
.
(b) If 04 >∆ , then the present value of
)(3 TPV possesses a minimum value at the boundary
point DWT /= .
Proof: The proof is similar to that of Lemma-2.
Theorem-2
(1) If 01 >∆ and 04 ≥∆ , then { })/(),(min)( *
1
*
DWPVTPVTPV ∞∞∞ = . Hence
*
T is
*
1T or
W/D is associated with the least cost.
(2) If 01 >∆ and 04 <∆ , then
{ })(),(min)( *
3
*
1
*
TPVTPVTPV ∞∞∞ = . Hence
*
T is
*
1T or
*
3T
is associated with the least cost.
(3) If 01 ≤∆ and 04 <∆ , then
)()( *
3
*
TPVTPV ∞∞ = . Hence
*
T is
*
3T .
Proof:
(1) If 01 >∆ and 04 >∆ , which imply that 0)/(/
1 >DWPV and
0)/(/
3 ≥DWPV , and also
DWT /*
1 < and
DWT /*
3 ≤ . Furthermore, we have
(i)
)(3 TPV is increasing on ),/[ ∞DW
(ii)
)(1 TPV is decreasing on ],0( *
1T and increasing on )/[ *
1 DWT < . Combining the
above we conclude that
)(TPV∞ has the minimum value at
*
1TT = on )/,0( DW and
)(TPV∞
has the minimum value at DWT /= on ),/[ ∞DW . Hence,
{ })/(),(min)( *
1
*
DWPVTPVTPV ∞∞∞ = . Consequently,
*
T is
*
1T or W/D associated with the least
cost.](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-10-320.jpg)
![P.K. Tripathy & S. Pradhan
International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 37
(2) If 01 >∆ and 04 <∆ , which imply that 0)/('
1 >DWPV and
0)(/
3 <MPV which implies
that DWT /*
1 < and
DWT /*
3 > and also
(i)
)(3 TPV is decreasing on
],/[ *
3TDW and increasing on
),[ *
3 ∞T
(ii)
)(1 TPV is decreasing on ],0( *
1T and increasing on ]/,( *
1 DWT . Combining (i) and
(ii) we conclude that
)(TPV∞ has the minimum value at
*
1TT = on )/,0( DW and
)(TPV∞ has the
minimum value at
*
3TT = on ),/[ ∞DW . Hence,
{ })(),(min)( *
3
*
1
*
TPVTPVTPV ∞∞∞ = .
Consequently,
*
T is
*
1T or
*
3T associated with the least cost.
(3) If
01 ≤∆ and
04 ≤∆ , which implies that
0)/(/
1 ≤DWPV and
0)(/
3 <MPV and
DWT /*
1 ≥ and
DWT /*
3 > and also
(i)
)(3 TPV is decreasing on.
],/[ *
3TDW and increasing on
),[ *
3 ∞T .
(ii) )(1 TPV is decreasing on )/,0( DW
From which we conclude that )(TPV∞ is decreasing on )/,0( DW and )(TPV∞ hs the
minimum value at
*
3TT = on ),/[ ∞DW . Since,
)/()/( 31 DWPVDWPV > , and conclude that
)(TPV∞ has minimum value at
*
3TT = on ),0( ∞ . Consequently
*
T is
*
3T .
This completes the proof.
NUMERICAL EXAMPLES
The followings are considered to be its base parameters A=$5/order, r=0.3/$, C=$1. a=1, k=2.4,
P0(Market Price)=$3, Price at the beginning of a cycle P(0)=$1, D=-k(P(0)-P0)=-2.4(1-3)=4.8
Example-1
If M=2, W=2, W/D<M
1∆ =31.47>0, 2∆ =-2.5020<0, 3∆ =-1.520559<0
*
T =T3=2.55,
)(3 TPV =36.910259
Example-2
If M=2, W=3, W/D<M
1∆ =48.416874>0, 2∆ =-0.112<0, 3∆ =-1.52<0
*
T =T3=2.55,
)(3 TPV =36.910259
Example-3
If M=5, W=3, W/D<M
1∆ =48.51>0, 2∆ =-0.112<0, 3∆ =2.1715>0
*
T =T2=3.1, )(2 TPV =36.302795
Example-4
If M=20, W=6, W/D<M
1∆ =83.5186>0, 2∆ =1.47>0, 3∆
=4.001>0
*
T =
*
1T =0.7442, )(1 TPV =48.134529
Example-5
If M=5, W=2, W/D<M
1∆ =36.5465>0, 2∆ =-2.5<0, 3∆
=2.738>0](https://image.slidesharecdn.com/ijssc-11-160310183547/85/A-Retail-Category-Inventory-Management-Model-Integrating-Entropic-Order-Quantity-and-Trade-Credit-Financing-11-320.jpg)
A Retail Category Inventory Management Model Integrating Entropic Order Quantity and Trade Credit Financing
- 1. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 27 A Retail Category Inventory Management Model Integrating Entropic Order Quantity and Trade Credit Financing P.K. Tripathy [email protected] P.G. Dept. of Statistics, Utkal University, Bhubaneswar-751004, India. S. Pradhan [email protected] Dept. of Mathematics, Orissa Engineering College, Bhubaneswar-751007, India. Abstract A retail category inventory management model that considers the interplay of entropic product assortment and trade credit financing is presented. The proposed model takes into consideration of key factors like discounted-cash-flow. We establish a stylized model to determine the optimal strategy for an integrated supplier-retailer inventory system under the condition of trade credit financing and system entropy. This paper applies the concept of entropy cost estimated using the principles of thermodynamics. The classical thermodynamics reasoning is applied to modelling such systems. The present paper postulates that the behaviour of market systems very much resembles those of physical systems. Such an analogy suggests that improvements to market systems might be achievable by applying the first and second laws of thermodynamics to reduce system entropy(disorder). This paper synergises the above process of entropic order quantity and trade credit financing in an increasing competitive market where disorder and trade credit have become the prevailing characteristics of modern market system. Mathematical models are developed and numerical examples illustrating the solution procedure are provided. Key words: Discounted Cash-Flow, Trade Credit, Entropy Cost. 1. INTRODUCTION In the classical inventory economic order quantity (EOQ) model, it was tacitly assumed that the customer must pay for the items as soon as the items are received. However, in practice or when the economy turns sour, the supplier allows credit for some fixed time period in settling the payment for the product and does not charge any interest from the customer on the amount owned during this period.Goyal(1985)developed an EOQ model under the conditions of permissible delay in payments.Aggarwal and Jaggi(1995) extended Goyal’s(1985) model to consider the deteriorating items .Chung(1999) presented an EOQ model by considering trade credit with DCF approach.Chang(2004) considered the inventory model having deterioration under inflation when supplier credits linked to order quantity. Jaber et al.(2008) established an entropic order quantity (EnOQ) model for deteriorating items by applying the laws of thermodynamics.Chung and Liao(2009) investigated an EOQ model by using a discounted-cash-flows(DCF) approach and trade credit depending on the quantity ordered. The specific purpose of this paper is to trace the development of entropy related thought from its thermodynamic origins through its organizational and economic application to its relationship to discounted-cash-flow approach.
- 2. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 28 2. Model Development 2.1 Basis of the Model: The classical economic order quantity (EOQ) or lot sizing model chooses a batch size that minimises the total cost calculated as the sum of two conflicting cost functions, the order/setup cost and the inventory holding costs. The entropic order quantity (EnOQ) is derived by determining a batch size that minimizes the sum of the above two cost and entropic cost. Notations T : the inventory cycle time, which is a decision variable; C,A : the purchase cost and ordering cost respectively h : the unit holding cost per year excluding interest change; D : the demand rate per unit time. D(T) : the demand rate per unit time where cycle length is T. r : Discount rate (opportunity cost) per time unit. Q : Procurement quantity; M : the credit period; W : quantity at which the delay payment is permitted; σ(t) : total entropy generated by time t. S : rate of change of entropy generated at time t. E(t) : Entropy cost per cycle; PV1(T) : Present value of cash-out-flows for the basic EnOQ model; PV2(T) : present value of cash-out-flows for credit only on units in stock when MT ≤ . PV3(T) : Present value of cash-out-flows for credit only on units in stock when MT ≥ . ))),( 0 tPtP : unit price and market equilibrium price at time t respectively. (T)PV∞ : the present value of all future cash-flows. * T : the optimal cycle time of (T)PV∞ when T>0. ASSUMPTIONS (1) The demand is constant. (2) The ordering lead time is zero. (3) Shortages are not allowed. (4) Time period is infinite. (5) If Q<W, the delay in payment is not permitted, otherwise, certain fixed trade credit period M is permitted. That is, Q<W holds if and only if T<W/D. (6) During the credit period, the firm makes payment to the supplier immediately after use of the materials. On the last day of the credit period, the firms pays the remaining balance. 2.2 Commodity flow and the entropy cost: The commodity flow or demand/unit time is of the form D= -k(P(t)-P0(t))(1) The concept represented by equation (1) is analogous to energy flow (heat or work) between a thermodynamics system and its environment where k (analogous to a thermal capacity) represents the change in the flow for the change in the price of a commodity and is measured in additional units sold per year per change in unit price e.g. units/year/$. Let P(t) be the unit price at time t and P0(t) the market equilibrium price at time t, where P(t)<P0(t) for every ],0[ Tt ∈ . At constant demand rate P(t)=P and P0(t)=P0 noting that when P<P0 , the direction of the commodity flow is from the system to the surroundings. The entropy generation rate must satisfy −+== 2 )( 0 0 P P P P k dt td S σ To illustrate, assume that the price of a commodity decreases according to the following relationship
- 3. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 29 t T a PtP −= )0()( where, )()0( TPPa −= as linear form of price being time dependent. =− − −== TP TPTP aT a TP T D tE 0 0 2 0 )0( 1ln )( )( σ Entropy cost per cycle. Case-1:Instantaneous cash-flows (the case of the basic EnOQ model) The components of total inventory cost of the system per cycle time are as follows: (a) Ordering cost = rT rTrT e A AeAeA − −− − =+++ 1 ....2 (b) Present value of the purchase cost can be shown as −+ − =+++ − −− ∫∫∫ )0( 21 ........ 0 2 2 pP a e kCT dtDeCTdtDeCTDdtCT rT rtrt (c) The present value of the out-of-pocket inventory carrying cost can be shown as +−+−+− ∫ ∫∫ +−+−− .........)()()( 0 0 )2()( 0 T T rTrtTr T rt dtetTDdtetTDdtetTDhc = ( ) ( ) ( ) − + + + + + −− −− − )0( 11 12 )0( 0202 Pp r kT e hc er ehcka rT a PP r hck rTrT rT So, the present value of all future cash-flow in this case is −−+ −+ − + − = −− rT a PP r hck Pp a e kcT e A TPV rTrT 2 )0()0( 211 )( 020 2 1 ( ) ( ) ( ) TP Pp a a TP Pp r kT e hc er ehcka rTrT rT 0 0 2 0 02 )0( 1ln)0( 11 1 − − −+ − − + − + + −− − (1) Case -2:Credit only on units in stock when MT ≤ . During the credit period M, the firm makes payment to the supplier immediately after the use of the stock. On the last day of the credit period, the firm pays the remaining balance. Furthermore, the credit period is greater than the inventory cycle length. The present value of the purchase cost can be shown as +++ ∫∫∫ +−+−− ........ 0 )2( 0 )( 0 T tTr T tTr T rT dtDedtDedtDeC ( ) ( ) ( ) ( )rT rT rT rTrT rT ert kaCTe Rr cka r PPck r Te e rT Ka e r PpK e C − − − −− − − −+ −− = −−+− − − = 1 )0( 1 1 1 ))0(( 1 2 0 2 0 The present value of all future cash flows in this case is ( ) ( ) −−+ − −+ − − − = − − − rT a PP r hck er ckae Tr cka r PPck e A TPV rT rT rT 2 )0( 1 )0( 1 )( 022 0 2 ( ) ( ) ( ) − −+ − − + − + + −− − TPPT aT a TP PP r kT e hc er ehcka rTrT rT 0 2 0 02 )0(( 1ln)0( 11 1 (2) Case-3: Credit only on units in stock when MT ≥ The present value only on units in stock can be shown as
- 4. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 30 +−++−+ +−+−−− ∫∫ ...)()( )( 0 )( 0 TMr M tTrMr M rt eMTDdtDeeMTDdtDeC ( ) ( ) ( )[ ]kPkPrTc eTr e rT rM 02 )0( 1 1 − − − = − − ( ) ( )[ ]TPatTPeMTrkccaMe erT rMrM rT 0)0()( 1 1 −−−+ − − −− − . Therefore the present value of all future cash-flows in this case is ( ) ( ) ( ) − − + − + + −−+ − = −− − − )0( 11 12 )0( 1 )( 02023 PP r kT e hc er ehcka rT a PP r hck e A TPV rTrT rT rT ( ) ( ) ( )[ ]cakkPkPrTc eTr e rT rM −− − − + − − 02 )0( 1 1 ( ) ( )[ ]TPatTPeMTrkccaMe erT rMrM rT 0)0()( 1 1 −−−+ − − −− − TP Pp aT a TP 0 0 2 0 )0( 1ln − − −+ (3) Now our main aim is to minimize the present value of all future cash-flow cost )(TPV∞ . That is Minimize )(TPV∞ subject to T>0. We will discuss the situations of the two cases, (A) Suppose M>W/D In this case we have ≤ <≤ << =∞ .)( /)( /0)( )( 3 2 1 TMifTPV MTDWifTPV DWTifTPV TPV It was found that 0)()( 21 >− TPVTPV and 0)()( 31 >− TPVTPV for T>0 and MT ≥ respectively. which implies )()( 21 TPVTPV > and )()( 31 TPVTPV > for T>0 and MT ≥ respectively. Now we shall determine the optimal replenishment cycle time that minimizes present value of cash- out-flows. The first order necessary condition for )(1 TPV in (1) to be minimized is expressed as 0 )(1 = ∂ ∂ T TPV which implies ( ) ( ) ( ) −+ − −− + − − − −− − − )0( 21 12 1 02 2 2 kPkP ak e CTreeCT e Are rT rTrT rT rT ( ) ( ) ( ) − +−−− + − −−−− 22 1 11 rT rTrTrTrT e reeere r hcka
- 5. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 31 ( ) ( ) ( ) 0 )0( 1ln 2 1 1 )0( 0 0 2 0 2320 =− − −++ − −− −+ − −− p PP a a P Tr hcka e Tree PP r hck rT rTrT (4) Similarly, )(2 TPV in equation(2) to be minimized is 0 )(2 = ∂ ∂ T TPV which implies ( ) ( ) ( ) ( ) ( ) ( ) − +−−− + − −−− − − − − −−−− − −−− − − 222 2 2 1 11 1 1 1 rT rTrTrTrT rT rTrTrT rT rT e reeere r hcka e reere r cka e Are ( ) ( ) ( ) ( ) 0 )0( 1ln 2 1 1)0( 0 0 2 0 2322 0 =− − −+++ − −−− + − −− p PP a a P Tr hcka Tr cka e rerTe r PPhck rT rTrT (5) Likewise, the first order necessary condition for )(3 TPV in equation(3) to be minimized is 0 )(3 = ∂ ∂ T TPV which leads ( ) ( ) ( )( ) ( )222 1 11 1 rT rTrTrTrT rT rT er reeerehcka e Are − −−−− − − − +−−− − − − ( )( ) ( ) ( )( )cakPKPerc er rTeePPhcrk rM rT rTrT −−−+ − −−− + − − −− 02 0 )0(1 1 1)0( 0 2 )0( 1ln 230 0 2 0 =+− − − Tr hcka p PP a a P (6) Furthermore, we let DWTT TPV / 1 1 )( =∂ ∂ =∆ (7) DWTT TPV / 2 2 )( =∂ ∂ =∆ (8) MTT TPV =∂ ∂ =∆ )(3 3 (9) Lemma-1 (a) If 01 ≥∆ , then the total present value of PV1(T) has the unique minimum value at the point T=T1 where )/,0(1 DWT ∈ and satisfies 01 = ∂ ∂ T PV . (b) If 01 <∆ , then the value of )/,0(1 DWT ∈ which minimizes PV1(T) does not exist. Proof: Now taking the second derivative of PV1(T) with respect to )/,0(1 DWT ∈ , we have ( ) ( ) ( )4 222 1 121 rT rTrTrTrT e eereer − −−−− − −+−
- 6. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 32 ( ) ( ) ( ) ( ) ( ) − + −+ − + − − −− + − − − −− − − 2203 222 2 1 2 )0( 21 1 1 412 rT rT rT rTrT rT rT e e r hcka kPkP a e eeTcr e rcTec ( ) ( ) [ ] ( )3 0 2 2 0 332 2 1 .))0((2 2)1( 1 ))0((4 1 2 rT rT rT rTrT rT e eTPPhckr rrTe e PPhck Tr hcka er hckae − − − −− − − − +−+ − − ++ − + We obtain from the above expression 0 )( 2 1 2 > ∂ ∂ T TPV , which implies 0 )(1 > ∂ ∂ T TPV is strictly increasing function of T in the interval (0,W/D). Also we know that, 0 )( 0 lim 1 <−= ∂ ∂ → rA T TPV T and 1 1 )( / lim ∆= ∂ ∂ → T TPV DWT Therefore, if 0 )( / lim 1 1 ≥∆= ∂ ∂ → T TPV DWT , then by applying the intermediate value theorem, there exists a unique value )/,0(1 DWT ∈ such that 0 )(1 = ∂ ∂ T TPV and 2 1 2 )( T TPV ∂ ∂ at the point T1 is greater than zero. Thus )/,0(1 DWT ∈ is the unique minimum solution to )(1 TPV . However, if 0 )( / lim 1 1 <∆= ∂ ∂ → T TPV DWT for all ( )DWT /,0∈ and also we find 0 )( 2 1 2 < ∂ ∂ T TPV for all ( )DWT /,0∈ . Thus )(1 TPV is a strictly decreasing function of T in the interval ( )DW /,0 . Therefore, we can not find a value of T in the open interval ( )DW /,0 that minimizes )(1 TPV . This completes the proof. Lemma-2 (a) If 12 0 ∆≤≤∆ , then the total present value of PV2(T) has the unique minimum value at the point T=T2 where ),/(2 MDWT ∈ and satisfies 02 = ∂ ∂ T PV . (b) If 02 >∆ , then the present value PV2(T) has a minimum value at the lower boundary point DWT /= . (c) If 02 <∆ , then the present value PV2(T) has a minimum value at the upper boundary point T=M. Proof: ( ) ( ) ( )4 222 2 2 2 1 121)( rT rTrTrTrT e eereer T TPV − −−−− − −+− = ∂ ∂
- 7. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 33 ( ) ( ) ( ) ( ) ( ){ }rrTe e PPhck er hckae e eckare rT rTrT rT rT rTrT 21 1 ))0(( 11 1 2 0 4 2 3 −+ + − + + + − + − − −− − − −− ( ) 33323 0 2 42 1 ))0(((2 Tr hcka Tr cka e TePPckhr rT rT −+ − − + − − which is >0 where [ ]MDWT ,/∈ . Which implies T PV ∂ ∂ 2 is strictly increasing function of T in the interval ),/( MDW . Also we have 2 / 2 )( ∆= ∂ ∂ = DWTT TPV . If 02 ≤∆ , then by applying the intermediate value theorem there exists a unique value [ ]MDWT ,/2 ∈ so that 0 )( 2 2 = ∂ ∂ =TTT TPV . Moreover, by taking the second derivative of PV2(T) with respect to T at the point T2 we have 02 2 2 > ∂ ∂ T PV . Thus [ ]MDWT ,/2 ∈ is the unique solution to )(2 TPV . Now if 0 )( 2 / 2 >∆= ∂ ∂ = DWTT TPV which implies 02 > ∂ ∂ T PV for all ( )MDWT ,/2 ∈ . Therefore, )(2 TPV is strictly increasing function of T in the interval ( )MDW ,/ . Therefore )(2 TPV has a minimum value at the lower boundary point DWT /= and similarly we can prove )(2 TPV has a minimum value at upper boundary point T=M. Lemma-3 (a) If 03 ≤∆ , then the total present value of )(3 TPV has the unique minimum value at the point T=T3 on ),(31 ∞∈ MT and satisfies 03 = ∂ ∂ T PV . (b) If 03 >∆ , then the present value of )(3 TPV has a minimum value at the boundary T=M. Proof: The proof is same to the lemma-2. Proposition 1 (i) 022 >−− rTert (ii) 0312 >−+ rTrT ee Proof: (i) ( ) rTerTe rTrT −−=−−− 1222 ( ) rT rT rt − ++++= 1..... !2 12 2 ( ) ( ) +++= ..... !3!2 2 32 rTrT rt which is always +ve as value of r and T are always positive. (ii) rTrT ee 312 −+
- 8. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 34 ( ) ( ) ( ) ( ) ++++−+++++= ... !3!2 131..... !3 3 !2 2 21 3232 rTrT rT rTrT rT ( ) ( ) .... 6 6 2 1 32 +++−−= rTrT rT ( ) ( ) ( )43 2 5.10 2 1 rTrT rT rT +++−−= which is also give a positive value a r and T are always positive. By this two position it is easy to say that 21 ∆>∆ . Then the equations (7) – (9) yield 01 <∆ iff 0)/(/ 1 <DWPV iff DWT /* 1 > (10) 02 <∆ iff 0)/(/ 2 <DWPV iff DWT /* 2 > (11) 03 <∆ iff 0)(/ 2 <MPV iff MT >* 2 (12) 03 <∆ iff 0)(/ 3 <MPV iff MT >* 3 (13) From the above equations we have the following results. Theorem-1 (1) If 0,0 21 ≥∆>∆ and 03 >∆ , then { })/(),(min)( * 1 * DWPVTPVTPV ∞∞∞ = . Hence * T is * 1T or W/D associated with the least cost. (2) If 0,0 21 <∆>∆ and 03 >∆ , then )()( * 2 * TPVTPV ∞∞ = . Hence * T is * 2T . (3) If 0,0 21 <∆>∆ and 03 ≤∆ , then )()( * 3 * TPVTPV ∞∞ = . Hence * T is * 3T . (4) If 0,0 21 <∆≤∆ and 03 >∆ , then )()( * 2 * TPVTPV ∞∞ = . Hence * T is * 2T . (5) If 0,0 21 <∆≤∆ and 03 ≤∆ , then )()( * 3 * TPVTPV ∞∞ = . Hence * T is * 3T . Proof: (1) If 0,0 21 ≥∆>∆ and 03 >∆ , which imply that 0)/(/ 1 >DWPV , 0)/(/ 2 ≥DWPV , 0)(/ 2 >MPV and 0)(/ 3 >MPV . From the above lemma we implies that (i) )(3 TPV is increasing on ),[ ∞M (ii) )(2 TPV is increasing on ),/[ MDW (iii) )(1 TPV is increasing on )/,[ * 1 DWT and decreasing on ],0( * 1T Combining above three, we conclude that )(TPV∞ has the minimum value at * 1TT = on )/,0( DW and )(TPV∞ has the minimum value at DWT /= . Hence, { })/(),(min)( * 1 * DWPVTPVTPV ∞∞∞ = . Consequently, * T is * 1T or W/D associated with the least cost. (2) If 0,0 21 <∆>∆ and 03 >∆ , which imply that 0)/(/ 1 >DWPV , 0)/(/ 2 <DWPV , 0)(/ 2 >MPV and 0)(/ 3 >MPV which implies that DWT /* 1 ≤ , DWT /* 2 > , MT <* 2 and MT <* 3 respectively. Furthermore from the lemma (i) )(3 TPV is increasing on ),[ ∞M
- 9. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 35 (ii) )(2 TPV is decreasing on ],/[ * 2TDW and increasing on ],( * 2 MT (iii) )(1 TPV is decreasing on ],0( * 1T and increasing on ]/,( * 1 DWT . From the above we conclude that )(TPV∞ has the minimum value at * 1TT = on )/,0( DW and )(TPV∞ has the minimum value at * 2TT = on ),/[ ∞DW . Since )(1 TPV > )(2 TPV and T>0. Then )()( * 2 * TPVTPV ∞∞ = and * T is * 2T (3) If 0,0 21 <∆>∆ and 03 ≤∆ , which implies that 0)/(/ 1 >DWPV , 0)/(/ 2 <DWPV , 0)(/ 2 ≤MPV and 0)(/ 3 ≤MPV which imply that DWT /* 1 < , DWT /* 2 > , MT ≥* 2 and MT >* 3 respectively. Furthermore, from the lemma it implies that (i) )(3 TPV is decreasing on ),( * 3TM and increasing on ),( * 3 ∞T (ii) )(2 TPV is decreasing on ),/( MDW (iii) )(1 TPV is decreasing on ),0( * 1T and increasing on ]/,( * 1 DWT . Combining all above, we conclude that )(TPV∞ has the minimum value at * 1TT = on )/,0( DW and )(TPV∞ has the minimum value at * 3TT = on ),/[ ∞DW . Since, )(2 TPV is decreasing on ),0( * 2T , DWT /* 1 < and DWMT /* 2 >≥ we have )()( * 22 * 11 TPVTPV > , )()( 2 * 12 MPVTPV > and )()( * 333 TPVMPV > . Hence we conclude that )(TPV∞ has the minimum value at * 3TT = on ),0( ∞ . Consequently, * T is * 3T . (4) If 0,0 21 <∆≤∆ and 03 >∆ , which implies that 0)/(/ 1 ≤DWPV , 0)/(/ 1 ≤DWPV , 0)(/ 2 >MPV and 0)(/ 3 >MPV which imply that DWT /* 1 ≥ , DWT /* 2 > , MT <* 2 and MT <* 3 . Furthermore, we have (i) )(3 TPV is increasing on ),[ ∞M (ii) )(2 TPV is decreasing on ],/[ * 2TDW and increasing on ],( * 2 MT (iii) )(1 TPV is decreasing on )/,0( DW . Since )/()/( 21 DWPVDWPV > , and )()/( * 22 TPVDWPV > So we conclude that )(TPV∞ has the minimum value at * 2TT = on ),0( ∞ . Consequently, * T is * 2T . (5) If 0,0 21 <∆≤∆ and 03 ≤∆ , which gives that 0)/(/ 1 ≤DWPV , 0)/(/ 2 <DWPV , 0)(/ 2 ≤MPV and 0)(/ 3 ≤MPV and which imply that DWT /* 1 ≥ , DWT /* 2 > , MT ≥* 2 and MT ≥* 3 respectively. Furthermore, from the lemma it implies that (i) )(3 TPV is decreasing on ),( * 3TM and increasing on ),[ * 3 ∞T (ii) )(2 TPV is decreasing on ],/[ MDW
- 10. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 36 (iii) )(1 TPV is decreasing on )/,0( DW . Since )/()/( 21 DWPVDWPV > , combining the above we conclude that )(TPV∞ has the minimum value at * 3TT = on ),0( ∞ . Consequently, * T is * 3T . This completes the proof. (B) Suppose DWM /≤ . Here )(TPV∞ can be expressed as follows: ≤ << =∞ TDWifTPV DWTifTPV TPV /)( /0)( )( 2 1 and let 4 / / 3 )( ∆= ∂ ∂ = DWT T TP (14) By using proposition 1, we have 041 >∆−∆ , which leads to 41 ∆>∆ . From (14), we also find that 04 <∆ iff 0)/(/ 3 <DWPV iff DWT /* 3 > (15) Lemma-4 (a) If 04 ≤∆ , then the present value of )(3 TPV possesses the unique minimum value at the point 3TT = , where ),/[3 ∞∈ DWT and satisfies 0 )(3 = ∂ ∂ T TP . (b) If 04 >∆ , then the present value of )(3 TPV possesses a minimum value at the boundary point DWT /= . Proof: The proof is similar to that of Lemma-2. Theorem-2 (1) If 01 >∆ and 04 ≥∆ , then { })/(),(min)( * 1 * DWPVTPVTPV ∞∞∞ = . Hence * T is * 1T or W/D is associated with the least cost. (2) If 01 >∆ and 04 <∆ , then { })(),(min)( * 3 * 1 * TPVTPVTPV ∞∞∞ = . Hence * T is * 1T or * 3T is associated with the least cost. (3) If 01 ≤∆ and 04 <∆ , then )()( * 3 * TPVTPV ∞∞ = . Hence * T is * 3T . Proof: (1) If 01 >∆ and 04 >∆ , which imply that 0)/(/ 1 >DWPV and 0)/(/ 3 ≥DWPV , and also DWT /* 1 < and DWT /* 3 ≤ . Furthermore, we have (i) )(3 TPV is increasing on ),/[ ∞DW (ii) )(1 TPV is decreasing on ],0( * 1T and increasing on )/[ * 1 DWT < . Combining the above we conclude that )(TPV∞ has the minimum value at * 1TT = on )/,0( DW and )(TPV∞ has the minimum value at DWT /= on ),/[ ∞DW . Hence, { })/(),(min)( * 1 * DWPVTPVTPV ∞∞∞ = . Consequently, * T is * 1T or W/D associated with the least cost.
- 11. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 37 (2) If 01 >∆ and 04 <∆ , which imply that 0)/(' 1 >DWPV and 0)(/ 3 <MPV which implies that DWT /* 1 < and DWT /* 3 > and also (i) )(3 TPV is decreasing on ],/[ * 3TDW and increasing on ),[ * 3 ∞T (ii) )(1 TPV is decreasing on ],0( * 1T and increasing on ]/,( * 1 DWT . Combining (i) and (ii) we conclude that )(TPV∞ has the minimum value at * 1TT = on )/,0( DW and )(TPV∞ has the minimum value at * 3TT = on ),/[ ∞DW . Hence, { })(),(min)( * 3 * 1 * TPVTPVTPV ∞∞∞ = . Consequently, * T is * 1T or * 3T associated with the least cost. (3) If 01 ≤∆ and 04 ≤∆ , which implies that 0)/(/ 1 ≤DWPV and 0)(/ 3 <MPV and DWT /* 1 ≥ and DWT /* 3 > and also (i) )(3 TPV is decreasing on. ],/[ * 3TDW and increasing on ),[ * 3 ∞T . (ii) )(1 TPV is decreasing on )/,0( DW From which we conclude that )(TPV∞ is decreasing on )/,0( DW and )(TPV∞ hs the minimum value at * 3TT = on ),/[ ∞DW . Since, )/()/( 31 DWPVDWPV > , and conclude that )(TPV∞ has minimum value at * 3TT = on ),0( ∞ . Consequently * T is * 3T . This completes the proof. NUMERICAL EXAMPLES The followings are considered to be its base parameters A=$5/order, r=0.3/$, C=$1. a=1, k=2.4, P0(Market Price)=$3, Price at the beginning of a cycle P(0)=$1, D=-k(P(0)-P0)=-2.4(1-3)=4.8 Example-1 If M=2, W=2, W/D<M 1∆ =31.47>0, 2∆ =-2.5020<0, 3∆ =-1.520559<0 * T =T3=2.55, )(3 TPV =36.910259 Example-2 If M=2, W=3, W/D<M 1∆ =48.416874>0, 2∆ =-0.112<0, 3∆ =-1.52<0 * T =T3=2.55, )(3 TPV =36.910259 Example-3 If M=5, W=3, W/D<M 1∆ =48.51>0, 2∆ =-0.112<0, 3∆ =2.1715>0 * T =T2=3.1, )(2 TPV =36.302795 Example-4 If M=20, W=6, W/D<M 1∆ =83.5186>0, 2∆ =1.47>0, 3∆ =4.001>0 * T = * 1T =0.7442, )(1 TPV =48.134529 Example-5 If M=5, W=2, W/D<M 1∆ =36.5465>0, 2∆ =-2.5<0, 3∆ =2.738>0
- 12. P.K. Tripathy & S. Pradhan International Journal of Computer Science and Security, (IJCSS), Volume (1): Issue (2) 38 * T = * 2T =3.1, )(2 TPV =36.302795 Example-6 If M=2, W=1, W/D<M 1∆ =-1.546<0, 2∆ =-15.079<0, 3∆ =-1.52<0 * T = * 3T =2.55, )(3 TPV =36.910259 Example-7 If M=10, W=1, W/D<M 1∆ =-1.546<0, 4∆ =-15.079<0, 3∆ =3.728554>0 * T = * 2T =3.1, )(2 TPV =36.302795 Example-8 If M=1, W=5, W/D<M 1∆ =164..83>0, 4∆ =2.946>0 * T = 1T =0.7442, )(1 TPV =48.134529 Example-9 If M=10, W=5, W/D<M 1∆ =83.518661>0, 4∆ =-1.837413<0 * T = * 3T =2.66, )(3 TPV =34.98682 Based on the above computational result of the numerical examples, the following managerial insights are obtained and following comparative evaluation are observed. If the supplier does not allow the delay payment, cash-out-flow is more but practically taking in view of real-world market, to attract the retailer(customer)credit period should be given and it observed that it should be less than equal to the inventory cycle to achieve the better goal. Furthermore, it is preferable for the supplier to opt a credit period which is marginally small. CONCLUSION AND FUTURE RESEARCH This paper suggested that it might be possible to improve the performance of a market system by applying the laws of thermodynamics to reduce system entropy (or disorder). It postulates that the behaviours of market systems very much resembles that of physical system operating within surroundings, which include the market and supply system. In this paper, the suggested demand is price dependent. Many researchers advocated that the proper estimation of input parameters in EOQ models which is essential to produce reliable results. However, some of those costs may be difficult to quantify. To address such a problem, we propose in this paper accounting for an additional cost (entropy cost) when analysing EOQ systems which allow a permissible delay payments if the retail orders more than or equal to a predetermined quantity. The results from this paper suggest that the optimal cycle time is more sensitive to the change in the quantity at which the fully delay payment is permitted. An immediate extension is to investigate the proposed model to determine a retailer optimal cycle time and the optimal payment policy when the supplier offers partially or fully permissible delay in payment linking to payment time instead of order quantity. ACKNOWLEDGEMENTS The authors sincerely acknowledge the anonymous referees for their constructive suggestion. REFERENCES (1) Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 335-338. (2) Khouja, M., Mehrez, A., 1996. Optimal Inventory policy under different supplier credits. Journal of manufacturing Systems, 15, 334-339.
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