A Fuzzy Arithmetic Approach for Perishable Items in Discounted Entropic Order Quantity Model

P.K. Tripathy & M. Pattnaik
International Journal of Scientific and Statistical Computing (IJSSC), Volume (1): Issue (2) 7
A Fuzzy Arithmetic Approach for Perishable Items in Discounted
Entropic Order Quantity Model
P.K. Tripathy msccompsc@gmail.com
P.G. Dept. of Statistics, Utkal University,
Bhubaneswar-751004, India.
M. Pattnaik monalisha_1977@yahoo.com
Dept. of Business Administration , Utkal University,
Bhubaneswar-751004, India.
Abstract
This paper uses fuzzy arithmetic approach to the system cost for perishable items
with instant deterioration for the discounted entropic order quantity model. Traditional
crisp system cost observes that some costs may belong to the uncertain factors. It is
necessary to extend the system cost to treat also the vague costs. We introduce a
new concept which we call entropy and show that the total payoff satisfies the
optimization property. We show how special case of this problem reduce to perfect
results, and how post deteriorated discounted entropic order quantity model is a
generalization of optimization. It has been imperative to demonstrate this model by
analysis, which reveals important characteristics of discounted structure. Further
numerical experiments are conducted to evaluate the relative performance between
the fuzzy and crisp cases in EnOQ and EOQ separately.
Key Words: Discounted Selling Price, Fuzzy, Instant Deterioration, Inventory.
1. INTRODUCTION
In this paper we consider a continuous review, using fuzzy arithmetic approach to the system cost for
perishable items. In traditional inventory models it has been common to apply fuzzy on demand rate,
production rate and deterioration rate, whereas applying fuzzy arithmetic in system cost usually
ignored in [5] and [13]. From practical experience, it has been found that uncertainty occurs not only
due to lack of information but also as a result of ambiguity concerning the description of the semantic
meaning of declaration of statements relating to an economic world. The fuzzy set theory was
developed on the basis of non-random uncertainties. For this reason, we consider the same since no
researcher have discussed EnOQ model by introducing the holding cost and disposal cost as the
fuzzy number. The model provides an approach for quantifying these benefits which can be
substantial, and should be reflected in fuzzy arithmetic system cost. Our objective is to find optimal
values of the policy variables when the criterion is to optimize the expected total payoff over a finite
horizon.
In addition, the product perishability is an important aspect of inventory control. Deterioration in
general, may be considered as the result of various effects on stock, some of which are damage,
decreasing usefulness and many more. While kept in store fruits, vegetables, food stuffs, bakery
items etc. suffer from depletion by decent spoilage. Lot of articles are available in inventory literature
considering deterioration. Interested readers may consult the survey paper of [10], [18], [16], [15], [4]
and [9] classified perishability and deteriorating inventory models into two major categories, namely
decay models and finite lifetime models. Products which deteriorate from the very beginning and the
products which start to deteriorate after a certain time. Lot of articles are available in inventory
literature considering deterioration. If this product starts to deteriorate as soon as it is received in the
stock, then there is no option to provide pre-deterioration discount. Only we may give post
deterioration discount on selling price.
Every organisation dealing with inventory faces a numbers of fundamental problems. Pricing decision
is one of them. In the development of an EOQ system, we usually omit the case of discounting on

selling price. But in real world, it exists and is quite flexible in nature. On the other hand, in order to
motivate customers to order more quantities for instant deterioration model usually supplier offers
discount on selling prices. [2] developed an inventory model under continuous discount pricing. [11]
studied an inventory problem under the condition that multiple discounts can be used to sell excess
inventory. [14] mentioned that discount is considered temporarily for exponentially decaying inventory
model. However, most of the studies except few, do not attempt to unify the two research streams:
temporary price reductions and instant deterioration. This paper outlines the issue in details.
The awareness of the importance of including entropy cost is increasing everyday. Indeed, entropy is
frequently defined as the amount of disorder in a system. The above consideration leads us to some
important points. First, the use of entropy must be carefully planned, taking into account the
multiplicity of objectives inherent in this kind of decision problem. Second, these are several economic
strategies with conflicting objectives in this kind of decision making process. [12] proposed an analogy
between the behaviour of production system and the behaviour of physical system. The main purpose
of this research is to introduce the concept of entropy cost to account for hidden cost such as the
additional managerial cost that is needed to control the improvement process.
In last two decades the variability of inventory level dependent demand rate on the analysis of
inventory system was described by researchers like [17], [1] and [3]. They described the demand rate
as the power function of on hand inventory. There is a vast literature on stock development inventory
and its outline can be found in the review article by [19] where he unified two types of inventory level
dependent demand by considering a periodic review model. Researchers such as [1], [16], [17], [4],
[6] and [8] discussed the EOQ model assuming time value of money, demand rate, deterioration rate,
shortages and so on a constant or probabilistic number or an exponential function. In this paper we
consider demand as a constant function for instant deterioration model.
The paper tackles to investigate the effect of the approximation made by using the average payoff
when determining the optimal values of the policy variables. The problem consists of the simultaneous
optimization of fuzzy entropic EOQ and crisp entropic EOQ model, taking into account the conflicting
payoffs of the different decision makers involved in the process. A policy iteration algorithm is
designed with the help of [7] and optimum solution is obtained through LINGO software. In order to
make the comparisons equitable a particular evaluation function based on discount is suggested.
Numerical experiments are carried out to analyse the magnitude of the approximation error. However,
a discount during post deterioration time, fuzzy system cost which might lead to a non-negligible
approximation errors. The remainder of this paper is organised as follows. In section 2 assumptions
and notations are provided for the development of the model. Section 3 describes the model
formulation. Section 4 develops the fuzzy model. Section 5 provides mathematical analysis. In section
6, an illustrative numerical experiment is given to illustrate the procedure of solving the model. Finally
section 7 concludes this article with a brief summary and provides some suggestions for future
research.
TABLE-1: Major Characteristics of Inventory Models on Selected Researches.
Author(s) and
published Year
Structure
of the
Model
Deterioration Inventory
Model Based
on
Discoun
t
allowed
Demand Back-
logging
allowed
Mahata et al. (2006) Fuzzy Yes
(constant)
EOQ No Constant No
Panda et al. (2009) Crisp Yes
(constant)
EOQ Yes Stock
dependent
Yes
(partial)
Jaber et al. (2008) Crisp Yes (on hand
inventory)
EnOQ No Unit selling
price
No
Vujosevic et al.
(1996)
Fuzzy No EOQ No Constant No
Skouri et al. (2007) Crisp Yes (Weibull) EOQ No Ramp Yes
(partial)
Present paper
(2010)
Fuzzy Yes
(constant)
EnOQ Yes Constant No

2. NOTATIONS AND ASSUMPTIONS
Notations
C0 : set up cost
c : per unit purchase cost of the product
s : constant selling price of the product per unit (s>c)
h : holding cost per unit per unit time
d : disposal cost per unit.
r : discount offer per unit after deterioration.
Q1 : order level for post deterioration discount on selling price with instant deterioration.
Q2 : order level for no discount on selling price with instant deterioration.
T1, T2 : cycle lengths for the above two respective cases.
Assumptions
Replenishment rate is infinite.
The deterioration rate θ is constant and (0 <θ < 1)
3. Demand is constant and defined as follows.
R (I (t)) = a
Where a>0 is the demand rate independent of stock level.
r,
( )10 ≤≤ r is the percentage discount offer on unit selling price during instant deterioration.
( ) n
r
−
−= 1α ( Rn∈ the set of real numbers) is the effect of discounting selling price on demand
during deterioration. α is determined from priori knowledge of the seller with constant demand.
5. The entropy generation must satisfy dt
td
S
)(σ
=
where,
)(tσ is the
total entropy generated by time t and S is the rate at which entropy is generated. The entropy cost is
computed by dividing the total commodity flow in a cycle of duration Ti. The total entropy generated
over time Ti as
( )
s
a
s
tIR
SSdtT
iT
O
i === ∫
))((
,σ
Entropy cost per cycle is
EC (Ti) = (EC) With deterioration
( )i
i
T
Q
σ
=
(i=1,2)
3. MATHEMATICAL MODEL
At the beginning of the replenishment cycle the inventory level raises to Q1. As the time progresses it
is decreased due to instantaneous stock with constant demand. Ultimately inventory reaches zero
level at T1. As instant deterioration starts from origin, r% discount on selling price is provided to
enhance the demand of decreased quality items. This discount is continued for the rest of the
replenishment cycle. Then the behaviour of inventory level is governed by the following system of linear
differential equation.
( )[ ]tIa
dt
tdI
θα +−=
)(
10 Tt ≤≤ (1)
with the initial boundary condition
1
1
1
0
0)(and
)0(
Tt
TI
QI
≤≤



=
=
Solving the equations,
( ) [ ] 1
)(
011
Tte
a
tI tT
≤≤−= −θ
θ
α
(2)
[ ]11
1 −= T
e
a
Q θ
θ
α
(3)
Holding cost and disposal cost of inventories in the cycle is,

( ) ( )∫+=+
1
0
T
dttIdhDCHC θ
Purchase cost in the cycle is given by PC = cQ1.
Entropy cost in the cycle is
EC=(EC)With deterioration=
( )1
1
1
det
)( T
Q
T
Q eriorationwithi
σσ
=
1
11
00
1 ;)(
11
aT
sQ
EC
s
aT
dt
s
a
SdtT
TT
==== ∫∫σ
Total sales revenue in the order cycle can be found as
( )








−= ∫
1
0
1
T
dtarsSR α
Thus total profit per unit time of the system is
( )
1
1
11 ,
T
TP
Tr =π
[ ]OCECDCHCPCSR
T
−−−−−=
1
1
On integration and simplification of the relevant costs, the total profit per unit time becomes
( )








−−−





−
−
−





−
−
−−= 01
1
1
111
1
1
11
1
1 11
CcQ
aT
sQ
T
edaa
T
e
haTrs
T
TT
θθ
αθ
θ
α
θ
απ
θθ
(4)
If the product starts to deteriorate as soon as it is received in the stock, then there is only one option
we may give post deterioration discount. The post deterioration discount on selling price is to be given
in such a way that the discounted selling price is not less that the unit cost of the product, i.e. s(1-r)-
c>0. Applying this constraint on unit total profit function we have the following maximization problem.
Maximize ),( 11 Trπ
Subject to
s
c
r −<1
0, 1 ≥∀ Tr (5)
dFhFF 3211 ++=π
where ( ) 





−−−−= 01
1
1
1
1
1 1
1
CcQ
aT
sQ
aTrs
T
F α (6)
θ
α
θ
θ
a
T
e
T
F
T






−
−
−= 1
1
2
11 1
(7)












−
−
−= 1
1
3
11 1
T
e
a
T
F
T
θ
α
θ
(8)
4. FUZZY MODEL
We replace the holding cost and disposal cost by fuzzy numbers h
~
and d
~
respectively. By
expressing h
~
and d
~
as the normal triangular fuzzy numbers (h1, h0, h2) and (d1, do, d2) ,
where, h1=h- 1∆ , ho = h, h2= h+ 420312 ,,, ∆+==∆−=∆ dddddd such that

,0,0,0 321 dh <∆<∆<<∆< 43214 and,,,0 ∆∆∆∆∆< are determined by the decision maker
based on the uncertainty of the problem.
The membership function of fuzzy holding cost and fuzzy disposal cost are considered as:









≤≤
−
−
≤≤
−
−
=
otherwise
hhh
hh
hh
hhh
hh
hh
hh
,0
,
,
)( 20
02
2
01
10
1
~µ
(9)









≤≤
−
−
≤≤
−
−
=
otherwise
ddd
dd
dd
ddd
dd
dd
dd
,0
,
,
)( 20
02
2
01
10
1
~µ
(10)
Then the centroid for h
~
and d
~
are given by
33
1221
~
∆−∆
+=
++
= h
hhh
M o
h
and 33
3421
~
∆−∆
+=
++
= d
ddd
M o
d
respectively.
For fixed values of r and 1T , let
ydTrFhTrFTrFdhZ =++= ),(),(),(),( 131211
Let 2
31
F
dFFy
h
−−
=
,
1
12
3
ψ=
∆−∆
and
2
34
3
ψ=
∆−∆
By extension principle the membership function of the fuzzy profit function is given by
{ }






∨




 −−
=
∨=
≤≤
∈ −
)(
)()(
~
2
31
~
~~
)(),(
)(
)
~
,
~
(~
21
1
d
F
dFFy
Sup
dhSup
dh
ddd
dh
yZdh
y
dhz
µµ
µµµ
(11)
Now,
( )
( )









≤≤
−
−++
≤≤
−
−−−
=




 −−
otherwise
udu
hhF
ydFhFF
udu
hhF
dFhFFy
F
dFFy
h
,0
,
,
23
022
3221
12
102
3121
2
31
~µ
(12)
where,
3
121
1
F
hFFy
u
−−
=
, 3
021
2
F
hFFy
u
−−
=
and 3
221
3
F
hFFy
u
−−
=
when du ≤2 and 1ud ≤ then 03021 dFhFFy ++≤ and 13121 dFhFFy ++≥ . It is clear that for
every
[ ] ')(,, 0302113121 PPydFhFFdFhFFy y =++++∈ µ
. From the equations (9) and (12) the
value of 'PP may be found by solving the following equation:

( )102
3121
10
`1
hhF
dFhFFy
dd
dd
−
−−−
=
−
−
or
( )( ) ( )
( ) ( )103102
101210121
ddFhhF
hhdFddhFFy
d
−+−
−+−−−
=
Therefore,
( ) ( )
)(' 1
103102
3121
10
`1
y
ddFhhF
dFhFFy
dd
dd
PP µ=
−+−
−−−
=
−
−
=
, (say). (13)
When
du ≤3 and 2ud ≤ then 23221 dFhFFy ++≤ and 03021 dFhFFy ++≥ . It is evident that for
every
[ ] ")(,, ~2322103021 PPydFhFFdFhFFy y =++++∈ µ
. From the equations (9) and (12), the
value of "PP may be found by solving the following equation:
( )022
3221
02
2
hhF
ydFhFF
dd
dd
−
−++
=
−
−
or,
( ) ( )( )
( ) ( )023022
022210222
ddFhhF
ddyhFFhhdF
d
−+−
−−+−−
=
Therefore,
( ) ( )
)(" 2
023022
23221
02
`2
y
ddFhhF
ydFhFF
dd
dd
PP µ=
−+−
−++
=
−
−
=
, (say). (14)
Thus the membership function for fuzzy total profit is given by
otherwise;
;
;
0
)(
)(
)( 2322103021
0302113121
2
1
)
~
,
~
(~ dFhFFydFhFF
dFhFFydFhFF
y
y
ydhz
++≤≤++
++≤≤++





= µ
µ
µ
(15)
Now, let
∫
∞
∞−
= )()
~
,
~
(~1 yP dhz
µ
dy and
∫
∞
∞−
= )()
~
,
~
(~1 yyR dhz
µ
dy
Hence, the centroid for fuzzy total profit is given by
1
1
11 ),(~
~
P
R
TrM
TP
==π
),(),(
),(),(),(
122121
131211
TrFTrF
dTrFhTrFTrF
ψψ ++
++=
(16)
322111 )()(),(~ FdFhFTrM
TP
ψψ ++++=
(17)
where, ),( 11 TrF , ),( 12 TrF and
),( 13 TrF are given by equations (6), (7) and (8).
The post-deterioration discount on selling price is to be given in such a way that the
discounted selling price is not less that the unit cost of the product, i.e. s(1-r)-c>0.
Applying this constraint on the unit total profit function in equation (17) we have the following
maximization problem.
Maximize
),( 1~
1
TrM
TP
Subject to, s
c
r −<1
(18)
0, 1 ≥∀ Tr
Our objective here is to determine the optimal values of r and 1T to maximize the unit profit function. It
is very difficult to derive the results analytically. Thus some numerical methods must be applied to
derive the optimal values of r and 1T , hence the unit profit function. There are several methods to

cope with constraint optimization problem numerically. But here we use penalty function method [7]
and LINGO software to derive the optimal values of the decision variables.
a. Special Case
b. I Model for instant deterioration with no discount
In this case order level and unit profit function for model with constant deterioration and
constant demand with no discount are obtained from (3) and (4) by substituting r=0 as
( )12
2 −= T
e
a
Q θ
θ (19)
From equation (4) total profit per unit time becomes
( ) ( ) 





−−−





−
−
+−== 02
2
2
22
22
2
22
11 2
CcQ
aT
sQ
T
ea
dhsaT
TT
TP
T
T
θθ
θπ
θ
(20)
dFhFF 654 ++=
where,






−−−= 02
2
2
2
2
4
1
CcQ
aT
sQ
saT
T
F (21)






−
−−
= 2
2
5
12
T
e
T
a
F
T
θθ
θ
(22)






−
−
−= 2
2
6
12
T
e
T
a
F
T
θθ
θ
(23)
Thus we have to determine T2 from the fuzzy maximization problem
maximize )( 2~
2
TM
TC
(24)
02 ≥∀ T
where, ( ) ( ) 2625142
~)(~
2
πψψ =++++= FdFhFTM
TC
. (25)
5. MODEL ANALYSIS THEOREM
For 21
~~,1 ππ >≠n if














−






+
−−<
)1(
1,1min 2
ns
aT
s
cn
s
c
r .
Proof:
The values of 1
~π for fixed r are always less than optimal value of r. Thus it is sufficient to show that
21
~~ ππ > for fixed r. Here, 1T is the cycle length when post deterioration discount is applied on unit
selling price to enhance the demand of decreased quality items. For the enhancement of demand the
inventory depletion rate will be higher and consequently the cycle time will reduce. T2 is the cycle
length when no discount is applied on selling price. Obviously T2 is greater than T1. Without loss of
generality let both the profit function 21
~and~ ππ are positive.
2
~
2
~
1
2
~
2
1
~
1
21
~~
T
TPTP
T
TP
T
TP −
≥−=−ππ

It is sufficient to show that
0
2
~
2
~
1
>
−
T
TPTP
. If it can be shown that
0
2
~
2
~
1
>
−
T
TPTP
is an increasing
function of r then our purpose will be served. Now differentiating it with respect to r we have,
( )
( ) 







−
−






−
+
−−+
−





 +
+−−=
∂
−∂
++ 1
2
1
2
2
21
1
1
)1(
)1)(1(
1)~~( 2
n
T
n
r
ean
aT
sdh
c
r
aTdh
nrns
Tr θθ
θ
θ
θππ θ
Therefore, 0
2
~
2
~
1
>
−
T
TPTP
0
)~~( 21
>
∂
−∂
⇒
r
ππ
,
i.e. if
( ) 0
1
)1)(1(
22
2
>
−
×





−
+
−−+


 +
+−−
T
en
aT
sdh
c
dh
nrns
T
θθ
θ
θ
θ θ
(26)
Now,
( ) 1
1
2
2
>
−
T
e T
θ
θ
.
we have,
0)1)(1(
2
>





−−+−−
aT
s
cnrns
i.e.
)1(
)1(
2
−






−−+−
<
ns
aT
s
cnns
r
We have the restriction
s
c
r −<1 .
Therefore, 21
~~ ππ > if














−






+
−−<
)1(
1,1min 2
ns
aT
s
cn
s
c
r (27)
Theorem indicates that for n ≠ 1 post instant deterioration discount on unit selling price produces
higher profit than that of instant deterioration with no discount on unit selling price in fuzzy
environment, if the percentage of post deterioration discount on unit selling price is less than min














−






+
−−
)1(
1,1 2
ns
aT
s
cn
s
c
.
A simple managerial indication is that in pure inventory scenario if the product deteriorates after a
certain time then it is always more profitable to apply only post deterioration discount on unit selling
price and the amount of percentage discount must be less than the limit provided in equation (27) for
the post deterioration discount.
6. NUMERICAL EXAMPLE
LINGO software is used to solve the aforesaid numerical example.

We redo the same example of [18] to see the optimal replenishment policy while considering the fuzzy
holding cost, fuzzy disposal cost and entropy cost. The parameter values are a=80, b=0.3, h=0.6,
d=2.0, s=10.0, C0=100.0, c=4.0, θ=0.03, n =2.0, ∆1=0.1, ∆2=0.2, ∆3=0.5, ∆4=0.8.
After 185 and 50 iterations in Table 2 we obtain the optimal replenishment policy for instant
deterioration fuzzy entropic order quantity models with post deterioration discount and no discount
respectively. The total profits for both the cases obtained here is at least 4.12% and 3.76%
respectively less than that in [18] i.e. our CEOQ models. This is because we modified the model by
introducing the hidden cost that is entropy cost where the optimal values for both the cases are
21.03623 and 20.28649 respectively. In Tables 3 and 4 we obtain the numerical results of different
models like FEnOQ, FEOQ, CEnOQ and CEOQ for above two cases separately. The behaviour of the
total profit to the lot size and the cycle length of post deterioration discounted model is shown in
Figure 1.
TABLE-2: The Numerical Results of the Instant Deterioration Fuzzy Entropic Order Quantity (FEnOQ) Models
(i=1,2)
Model Local optimal
solution found at
iteration
r Ti Qi EC
iπ
FEnOQ (Only post
deterioration
discount)
185 0.0350 1.8221 160.8798 21.0362 354.1393
FEnOQ (No
discount)
50 - 1.8814 154.8204 20.28649 353.6979
% change - - -3.1457 3.9136 3.6958 0.1248
TABLE-3: Comparison of Results for the different Post Deterioration Discount Models
Model Local optimal
solution found
at iteration
r Ti Qi EC
1π
FEnOQ 185 0.0350 1.8221 160.8798 21.0362 354.1393
FEOQ 193 0.0673 1.6063 151.3270 - 366.6226
CEnOQ 105 0.0392 1.8561 165.4009 21.1392 357.0641
CEOQ 196 0.0708 1.6367 155.4112 - 369.3739

FIGURE1: The behaviour of the total profit to the lot size and the cycle length of post deterioration discounted
model.
7. COMPARATIVE EVALUATION
Table 2 shows that 3.4% discount on post deterioration model is provided on unit selling price to earn
0.12% more profit than that with no discounted instant deterioration model. From Table 5 it indicates
that the uncertainty and entropy cost are provided on the post deterioration discount model to lose
3.4%, 0.81% and 4.12% less profits for FEOQ, CEnOQ and CEOQ models respectively than that with
FEnOQ model. Similarly it shows that the no discounted deterioration model to lose 3.08%, 0.78%
and 3.76% less profits for FEOQ, CEnOQ and CEOQ models respectively than that with FEnOQ
model. This paper investigates a computing schema for the EOQ in fuzzy sense. From Tables 3 and 4
it shows that the fuzzy and crisp results are very approximate, i.e. it permits better use of EOQ as
compared to crisp space arising with the little change in holding cost and in disposal cost respectively.
It indicates the consistency of the crisp case from the fuzzy sense.
TABLE-4: Comparison of Results for the different No Discounted Instant Deterioration Models
Model Local optimal solution found
at iteration
T2 Q2 EC
2π
FEnOQ 50 1.8814 154.8207 20.28649 353.6979
FEOQ 32 1.7203 141.2193 - 364.9558
CEnOQ 48 1.9239 158.4196 20.2931 356.5054
CEOQ 34 1.7592 144.4996 - 367.5178
TABLE-5: Relative Error (RE) of Post Deterioration Discount and No Discounted Deterioration FEnOQ Models
with the different Models

FEnOQ Q1 160.8798 Q2 154.8207
1π 354.1393
2π 353.6979
FEOQ Q11 151.3270 Q21 141.2193
11π 366.6226
21π 364.9558
RE % change 6.3127 % change 9.6314
% change -3.4050 % change -3.0847
CEnOQ Q12 165.4009 Q22 158.4196
12π 357.0641
22π 356.5054
RE % change -2.7334 % change -2.2718
% change -0.8191 % change -0.7878
CEOQ Q13 155.4112 Q23 144.4996
13π 369.3739
23π 367.5178
RE % change 3.5188 % change -6.6665
% change -4.1244 % change -3.7603
8. CRITICAL DISCUSSION
When human originated data like holding cost and disposal cost which are not precisely known but
subjectively estimated or linguistically expressed is examined in this paper. The mathematical model
is developed allowing post deterioration discount on unit selling price in fuzzy environment. It is found
that, if the amount of discount is restricted below the limit provided in the model analysis, then the unit
profit is higher. It is derived analytically that the post deterioration discount on unit selling price is to
earn more revenue than the revenue earned for no discount model. The numerical example is
presented to justify the claim of model analysis. Temporary price discount for perishable products to
enhance inventory depletion rate for profit maximization is an area of interesting research. This paper
introduces the concept of entropy cost to account for hidden cost such as the additional managerial
cost that is needed to control the improvement of the process. This paper examines the idea by
extending the analysis of [18] by introducing fuzzy approach and entropy cost to provide a firm its
optimum discount rate, replenishment schedule, replenishment order quantity simultaneously in order
to achieve its maximum profit.
Though lower amount of percentage discount on unit selling price in the form of post deterioration
discount for larger time results in lower per unit sales revenue, still it is more profitable. Because the
inventory depletion rate is much higher than for discount with enhanced demand resulting in lower
amount inventory holding cost and deteriorated items. Thus it can be conjectured that it is always
profitable to apply post deterioration discount on unit selling price to earn more profit. Thus the firm in
this case can order more to get earn more profit.
These models can be considered in a situation in which the discount can be adjusted and number of
price changes can be controlled. Extension of the proposed model to unequal time price changes and
other applications will be a focus of our future work.
9. CONCLUSION
This paper provides an approach to extend the conventional system cost including fuzzy arithmetic
approach for perishable items with instant deterioration for the discounted entropic order quantity
model in the adequacy domain. To compute the optimal values of the policy parameters a simple and
quite efficient policy model was designed. Theorem determines effectively the optimal discount rate r
for post deterioration discount. Finally, in numerical experiments the solution from the instant
deteriorated model evaluated and compared to the solutions of other different EnOQ and traditional
EOQ policies.
However, we saw few performance differences among a set of different inventory policies in the
existing literature. Although there are minor variations that do not appear significant in practical terms,
at least when solving the single level, incapacitated version of the lot sizing problem. From our
analysis it is demonstrated that the retailer’s profit is highly influenced by offering post discount on
selling price. The results of this study give managerial insights to decision maker developing an
optimal replenishment decision for instant deteriorating product. Compensation mechanism should

also be included to induce collaboration between retailer and dealer in a meaningful supply chain. We
conclude this paper by summarizing some of the managerial insights resulting from our work.
In general, for normal parameter values the relative payoff differences seem to be fairly small. The
optimal solution of the suggested post deterioration discounted model has a higher total payoff as
compared with no discounted model. Conventional wisdom suggests that workflow collaboration in a
fuzzy entropic model in a varying deteriorating product in market place are promising mechanism and
achieving a cost effective replenishment policy. Theoretically such extensions would require analytical
paradigms that are considerably different from the one discussed in this paper, as well as additional
assumptions to maintain tractability.
The approach proposed in the paper based on EnOQ model seems to be a pragmatic way to
approximate the optimum payoff of the unknown group of parameters in inventory management
problems. The assumptions underlying the approach are not strong and the information obtained
seems worthwhile. Investigating optimal policies when demand are generated by other process and
designing models that allow for several orders outstanding at a time, would also be challenging tasks
for further developments. Its use may restrict the model’s applicability in the real world. Future
direction may be aimed at considering more general deterioration rate or demand rate. Uses of other
demand side revenue boosting variables such as promotional efforts are potential areas of future
research. There are numerous ways in which one could consider extending our model to encompass
a wider variety of operating environments. The proposed paper reveals itself as a pragmatic
alternative to other approaches based on constant demand function with very sound theoretical
underpinnings but with few possibilities of actually being put into practice. The results indicate that this
can become a good model and can be replicated by researchers in neighbourhood of its possible
extensions. As regards future research, one other line of development would be to allow shortage and
partial backlogging in the discounted model.
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