An EOQ model for deteriorating items und
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An EOQ model for deteriorating items under trade credit
financing in the fuzzy sense
G. C. Mahata a; A. Goswami a
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
a
Online Publication Date: 01 December 2007
To cite this Article: Mahata, G. C. and Goswami, A. (2007) 'An EOQ model for
deteriorating items under trade credit financing in the fuzzy sense', Production
Planning & Control, 18:8, 681 - 692
To link to this article: DOI: 10.1080/09537280701619117
URL: http://dx.doi.org/10.1080/09537280701619117
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Production Planning & Control,
Vol. 18, No. 8, December 2007, 681–692
An EOQ model for deteriorating items under
trade credit financing in the fuzzy sense
G. C. MAHATA and A. GOSWAMI*
Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, India
This paper deals with the problem of determining the economic order quantity (EOQ) for
deteriorating items in the fuzzy sense where delay in payments for retailer and customer are
permissible and generalizes the earlier published results in this direction. The demand rate,
holding cost, ordering cost and purchasing cost are taken as fuzzy numbers. We also
assume that the supplier would offer the retailer a delay period for payment and the retailer
would also offer the trade credit period to the customer. The total variable cost in fuzzy sense
is defuzzified using the graded mean integration representation method. Then we have
shown that the defuzzified total variable cost is convex, that is, a unique solution exists.
For determination of optimal ordering policies, with the help of theorems we have
developed the neccessary algorithms. Finally, the theorems and the algorithms are
illustrated with the help of numerical examples.
Keywords: EOQ; Fuzzy annual total cost; Trade credit; Graded mean integration
representation
1. Introduction
The basic economic order quantity (EOQ) model is
based on the implicit assumption that the retailer must
pay for the items as soon as he receives them from a
supplier. However, in practice, the supplier will allow a
certain fixed period (credit period) for settling the
amount that the supplier owes to retailer for the items
supplied. Before the end of the trade credit period, the
retailer can sell the goods and accumulate revenue and
earn interest. A higher interest is charged if the
payment is not settled by the end of the trade credit
period. In the real world, the supplier often makes use of
this policy to promote his commodities. In this regard, a
number of research papers appeared which deal with the
EOQ problem under fixed credit period. Goyal (1985)
first studied an EOQ model under the conditions
*Corresponding
ernet.in
author.
Email:
[email protected].
of permissible delay in payments. Chand and Ward
(1987) analysed Goyal’s (1985) problem under assumptions of the classical EOQ model, obtaining different
results. Chung (1998) presented the DCF (discounted
cash flow) approach for the analysis of the optimal
inventory policy in the presence of trade credit. Later,
Shinn et al. (1996) extended Goyal’s (1985) model and
considered quantity discount for freight cost. Recently,
to accommodate more practical features of the real
inventory systems, Aggarwal and Jaggi (1995), Shah
(1993), Hwang and Shinn (1997) extended Goyal’s
(1985) model to consider the deterministic inventory
model with a constant deterioration rate. Shah and
Shah (1998) developed a probabilistic inventory model
when delay in payment is permissible. They developed
an EOQ model for deteriorating items in which time
and deterioration of units are treated as continuous
variables and demand is a random variable. Later on,
Jamal et al. (1997) extended Aggarwal and Jaggi’s
(1995) model to allow for shortages and make it more
Production Planning and Control
ISSN 0953–7287 print/ISSN 1366–5871 online ß 2007 Taylor & Francis
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DOI: 10.1080/09537280701619117
Downloaded By: [The Indian Institute of Technology] At: 11:53 27 December 2007
682
G. C. Mahata and A. Goswami
applicable in the real world. Shawky and Abou-El-Ata
(2001) investigated the production lot-size model
with both restrictions on the average inventory level
and trade credit policy using geometric programming
and Lagrange approaches. All the above models
assumed that the supplier would offer the retailer a
delay period, but the retailer would not offer the trade
credit period to his/her customer. Huang (2003)
assumed that the retailer should also adopt the
trade credit policy to stimulate his/her customer
demand to develop the retailer’s replenishment model.
On the other hand, most inventory systems are
developed without considering the effects of deterioration. However, in real life situations there are products
such as volatile liquids, medicines, materials, etc., in
which the rate of deterioration is very large. Therefore,
the loss of items due to deterioration should not be
neglected. Inventory models for different deteriorating
items have been developed by several researchers in past
and recent years. Ghare and Schrader (1963) developed
an EOQ model for items with an exponentially decaying
inventory. An EOQ model for items with variable rate
of deterioration has been developed by Covert and
Philip (1973) by introducing a two-parameter Weibull
distribution for the time to deterioration. Philip (1974)
developed a three parameter Weibull distribution for the
deterioration time. Many more papers have been
published in this direction.
Usually researchers consider different parameters of
an inventory model either as constant or dependent on
time or probabilistic in nature for the development of
the economic order quantity model. But, in the real life
situations, these parameters may have little deviations
from the exact value which may not follow any
probability distribution. In these situations, if these
parameters are treated as fuzzy parameters, then it
will be more realistic. These types of problems are
defuzzified first using a suitable fuzzy technique and
then the solution procedure can be obtained in the usual
manner. Several authors, namely Chang et al. (1998),
Lee and Yao (1998), Lin and Yao (2000), Yao et al.
(2000), developed inventory models in fuzzy sense by
considering different parameters as fuzzy parameters.
In this paper, we propose a deteriorating inventory
model under the condition of permissible delay in
payments in the fuzzy sense. Most of the inventory
models on this topic assumed that the supplier would
offer the retailer a delay period and the retailer could sell
the goods and accumulate revenue and earn interest
within the trade credit period. They implicitly assumed
that the customer would pay for the items as soon as the
items are received from the retailer. That is, the
retailer would not offer the trade credit period to
his/her customer. In most of the business transactions,
this assumption is debatable. In this paper, we assume
that the supplier would offer the retailer a delay period
and the retailer would also offer the trade credit period
to his/her customer. Furthermore, the demand rate and
the inventory costs (namely holding cost, purchase cost
and ordering cost) may be flexible with some vagueness
for their values. In real life situations, all these
parameters in an inventory model are uncertain,
imprecise and the determination of optimum cycle
time becomes a non-stochastic vague decision-making
process. Again, for this type of model, statistical
estimations proved to be inefficient because of the lack
of statistical observations. In this situation, a suitable
way to model these imprecise data is to use fuzzy sets.
The ill-formed vagueness in the above parameters are
introduced making them fuzzy in nature and then
the model is formulated in a fuzzy environment.
We use the graded mean integration representation
method for defuzzifying fuzzy total average cost. In this
paper, it is shown that the total variable cost per
unit time after defuzzification is convex. Then, with
convexity, a simple optimisation procedure is developed.
Numerical examples are used to illustrate the results
given in this paper. Finally, the results in this
paper generalise some already published results in the
crisp sense.
2. Assumptions and notations
The mathematical model is developed on the basis
of the following assumptions and notations:
(1) The demand rate D is assumed to be constant
e is the fuzzy demand
for the crisp model whereas D
rate for the fuzzy model.
(2) Replenishment rate is infinite and lead time
is zero.
(3) Shortage is not allowed.
(4) A constant fraction , assumed to be small,
of the on-hand inventory gets deteriorated per
unit time.
(5) h, the holding cost per unit; C, the purchasing
cost per unit and A, ordering cost per order, are
e C,
e and
known and constant in the crisp model. h,
e are the fuzzy holding cost, fuzzy purchasing
A
cost and fuzzy ordering cost respectively in fuzzy
model.
(6) Ic, the interest charged per $ in stocks per year by
the supplier; Ie, the interest earned per $ per year
where Ic Ie.
(7) M, the retailer’s trade credit period offered by
supplier in years and N, the customer’s trade
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683
An EOQ model for deteriorating items
credit period offered by retailer in years. It is
assumed that M N.
(8) When T M, the account is settled at time T ¼ M
and retailer starts paying for the interest charges
on the items in stock with rate Ic. When T M,
the account is settled at T ¼ M and the retailer
does not need to pay interest charge.
(9) The retailer can accumulate revenue and earn
interest after his/her customer pays for the
amount of purchasing cost to the retailer until
the end of the trade credit period offered by the
supplier. That is, the retailer can accumulate
revenue and earn interest during the period N to
M with rate Ie under the condition of trade credit.
Case 1.
MT
Total interest payable
ZT
qðtÞdt
¼ CIc
M
CIc D
¼ 2 eðTMÞ ðT MÞ 1 :
ð8Þ
Case 2. N T M
In this case, total interest payable ¼ 0.
Case 3. T N
Similar as case 2, total interest payable ¼ 0.
According to assumption (9), three cases will occur
in interest earned per year.
3. Crisp mathematical model
Let q(t) be the inventory level at any time t(0 t T ).
Initially, the stock level is Q. The inventory level
decreases due to demand and deterioration until it
becomes zero at time t ¼ T. The differential equation
governing the system in the interval (0, T ) is
dqðtÞ
þ qðtÞ ¼ D,
dt
According to assumption (8), three cases may occur in
calculation of interest charges for the items kept in stock
per year.
0 t T,
M T, (shown in figure 1)
Case 1.
Total interest earned ¼ CIe
¼
M
Dtdt
N
CIe DðM2 N2 Þ
:
2
ð9Þ
ð1Þ
N T M, (shown in figure 2)
Case 2.
with the initial condition
qðT Þ ¼ 0:
Z
ð2Þ
Total interest earned ¼ CIe
Z
T
Dtdt þ DTðM T Þ
N
The solution of (1) is
D ðTtÞ
qðtÞ ¼
e
1 ,
¼
0 t T:
ð3Þ
Q ¼ qð0Þ ¼
D T
e 1 :
ð4Þ
Total demand during one cycle is DT.
The number of units deteriorated during one cycle is:
Q DT ¼
D T
ðe 1 T Þ:
ð10Þ
T N, (shown in figure 3)
Case 3.
Using the condition (2), the order quantity can be
obtained as
CIe D
2MT N2 T2 :
2
Total interest earned ¼ CIe DTðM NÞ:
ð11Þ
From the above arguments, the annual total relevant
cost for the retailer can be expressed as, TVC(T ) ¼
ordering cost þ holding cost þ deterioration cost þ
Inventory level
ð5Þ
Q
DT
The total cost due to deterioration of items during the
cycle, denoted by DC, is
DC ¼
CD T
e 1 T :
ð6Þ
The total inventory holding cost per cycle, denoted by
HC, is given by
ZT
hD
qðtÞdt ¼ 2 eT 1 T :
ð7Þ
HC ¼ h
0
0
N
M
T
t
Figure 1. The total accumulation of interest earned when
M T.
Downloaded By: [The Indian Institute of Technology] At: 11:53 27 December 2007
684
G. C. Mahata and A. Goswami
Inventory level
Q
~
A
1
wA
DT
h
0
t
0
N
T
a1 L−1(h) a2
h(L−1(h)+R−1(h))
2
R−1(h) a4
a3
Figure 4. The graded mean h-level value of generalized fuzzy
number A~ ¼ ða1 , a2 , a3 , a4 ; wA Þ.
M
Figure 2. The total accumulation of interest earned when
N T M.
4. Fuzzy methodology
Inventory level
Q
4.1 Graded mean integration representation method
DT
t
0
T
N
M
Figure 3. The total accumulation of interest earned when
T N.
interest payable interest earned.
8
TVC1 ðT Þ; if T M,
>
>
<
TVCðT Þ ¼ TVC2 ðT Þ; if N T M,
>
>
:
TVC3 ðT Þ; if 0 5 T N,
ð12Þ
A ðh þ CÞD T
þ
e 1 T
2
T
T
CIc D ðTMÞ
þ 2
ðT MÞ 1
e
T
CIe D 2
M N2
2T
TVC2 ðT Þ ¼
and
TVC3 ðT Þ ¼
A ðh þ CÞD T
þ
e 1 T
T
2 T
CIe D
2MT N2 T2
2T
A ðh þ CÞD T
þ
e 1 T
2
T
T
CIe DðM NÞ:
(1) A~ ðxÞ is continuous mapping from R to the
closed interval [0, 1]
(2) eðxÞ ¼ 0, 1
< PðTVC1 ðT ÞÞ;
g ÞÞ ¼ PðTVC
g 2 ðT ÞÞ;
PðTVCðT
>
:
g 3 ðT ÞÞ;
PðTVC
if T M,
if N T M,
if 0 5 T N,
which is the defuzzified equation corresponding to the
equation (19). Then, we have
F1 F2 þ F3
ðTeT eT þ 1Þ
þ
T2
2 T2
F3 Ic
þ 2 2 ðTeðTMÞ eðTMÞ þ 1 MÞ
T
F3 Ie ðM2 N2 Þ
þ
ð34Þ
2T2
1 2 2 T
00 2F1 2ðF2 þF3 Þ
T
T
g
e Te 1þ T e
PðTVC1 ðTÞÞ ¼ 3 þ
T
2 T3
2
2F3 Ie ðTMÞ
þ 2 3 e
TeðTMÞ
T
F I ðM2 N2 Þ
1
3 e
1þMþ 2 T2 eðTMÞ
2
T3
ð35Þ
F1 F2 þ F3
0
T
T
g
ðTe e þ 1Þ
PðTVC2 ðT ÞÞ ¼ 2 þ
T
2 T2
F3 Ie ðT2 N2 Þ
ð36Þ
þ
2T2
g 2 ðT ÞÞ00 ¼ 2F1 þ 2ðF2 þ F3 Þ eT TeT
PðTVC
2 T3
T3
1
F 3 Ie N 2
1 þ 2 T2 eT þ
ð37Þ
2
T3
g 1 ðT ÞÞ0 ¼
PðTVC
g 3 ðT ÞÞ0 ¼ F1 þ F2 þ F3 ðTeT eT þ 1Þ ð38Þ
PðTVC
T2
2 T2
g 3 ðT ÞÞ00 ¼ 2F1 þ 2ðF2 þ F3 Þ
PðTVC
2 T3
T3
1
ð39Þ
eT TeT 1 þ 2 T2 eT
2
where (0 ) represents differentiation with respect to T.
6. The convexity
g 1 ðT ÞÞ,
In this section, we shall show that PðTVC
g
g
PðTVC2 ðT ÞÞ and PðTVC3 ðT ÞÞ are convex on their
appropriate domains.
Theorem 1
ð31Þ
ð33Þ
g 1 ðT ÞÞ is convex on ½M, 1Þ.
(1) PðTVC
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687
An EOQ model for deteriorating items
g 2 ðT ÞÞ is convex on ½0, 1Þ.
(2) PðTVC
g 3 ðT ÞÞ is convex on ½0, 1Þ.
(3) PðTVC
on ½0, 1Þ. From equations (34), (36) and (38),
we have
Before proving Theorem 1, we need the following lemmas.
Lemma 1: eT TeT 1 þ 122 T2 eT 4 0 for all T>0.
Proof:
0
f ðxÞ ¼
x
x
Let fðxÞ ¼ e 1 xe þ
1 2 x
2x e
1 2 x
2x e
if x>0, then
g 2 ðMÞÞ0
g 1 ðMÞÞ0 ¼ PðTVC
PðTVC
¼
4 0. Hence f(x) is increasing for all x>0.
þ
Consequently, f(x)>f(0) ¼ 0 if x>0. We have fðxÞ ¼
ex 1 xex þ 12x2 ex 4 0 if x>0. Let x ¼ T. Then eT
TeT 1 þ 122 T2 eT 4 0 if T>0. This completes the
proof.
œ
Lemma 2: If T M, then eðTMÞ 1 TeðTMÞ þ
M þ 122 T2 eðTMÞ ð2 M2 Þ=2 0.
Proof: Let
gðT Þ ¼ eðTMÞ 1 TeðTMÞ þ Mþ
1 2 2 ðTMÞ
ð2 M2 Þ=2.
Then we have g0 ðT Þ ¼
2 T e
1 3 2 ðTMÞ
4 0. Hence g(T ) is increasing on ½M, 1Þ.
2 T e
So g(T )>g(M) ¼ 0 if T>M. This completes the
proof.
œ
The proof of Theorem 1
F1 F2 þ F3
þ 2 2 ðMeM eM þ 1Þ
M2
M
F3 Ie ðM2 N2 Þ
2M2
ð41Þ
and
g 2 ðNÞÞ0 ¼ PðTVC
g 3 ðNÞÞ0
PðTVC
¼
F1 F2 þ F3
þ
ðNeN eN þ 1Þ ð42Þ
N2
2 N2
Furthermore, we let
1 ¼
F1 F2 þ F3
F3 Ie ðM2 N2 Þ
þ 2 2 ðMeM eM þ 1Þ þ
,
2
2M2
M
M
ð43Þ
and
(1) From equation (35), we have
2 ¼
g 1 ðT ÞÞ00
PðTVC
2F1 2ðF2 þ F3 Þ T
1 2 2 T
T
¼ 3 þ
e Te 1 þ T e
2 T3
2
T
2F3 Ic ðTMÞ
þ 2 3 e
TeðTMÞ 1 þ M
T
1
F3 Ie ðM2 N2 Þ
þ 2 T2 eðTMÞ
2
T3
2F1 2ðF2 þ F3 Þ T
1 2 2 T
T
4 3 þ
e
Te
1
þ
T
e
2 T3
2
T
2F3 Ic ðTMÞ
þ 2 3 e
TeðTMÞ 1 þ M
T
1
2 M2
F3 Ie N2
þ 2 T2 eðTMÞ
:
ð40Þ
þ
2
2
T3
g 1 ðT ÞÞ00 4 0 if
Lemmas 1 and 2 imply that PðTVC
g 1 ðT ÞÞ is convex on ½M, 1Þ.
T M. Hence PðTVC
g 2 ðT ÞÞ00 ¼
(2) From equation (37), we have PðTVC
ð2F1 =T3 Þ þ ð2ðF2 þ F3 Þ=2 T3 Þ ðeT TeT 1þ
1 2 2 T
þ ðF3 Ie N2 =T3 Þ. Lemma 1 implies that
2 T Þe
g 2 ðT ÞÞ00 4 0 if T>0. Consequently,
PðTVC
g 2 ðT ÞÞ is convex on ½0, 1Þ.
PðTVC
g 3 ðT ÞÞ00 ¼
(3) From equation (39), we have PðTVC
ð2F1 =T3 Þ þ ð2ðF2 þ F3 Þ=2 T3 ÞðeT TeT 1 þ
00
1 2
2 T
g
2 T e Þ. From lemma 1 PðTVC3 ðT ÞÞ 4 0 if
g 3 ðT ÞÞ is convex
T>0. Consequently, PðTVC
F1 F2 þ F3
þ
ðNeN eN þ 1Þ:
N2
2 N2
ð44Þ
Therefore,
1
1
F3 Ie
1 2 ¼ F1 2 2 þ
ðM2 N2 Þ þ ðF2 þ F3 Þ
N
M
2M2
MeM eM þ 1 NeN eN þ 1
:
2 M2
2 N2
ð45Þ
Let, KðxÞ ¼ ððxex ex þ 1Þ=x2 Þ, so that by lemma 1,
Hence
k ðxÞ ¼ x23 ðex 1 xex þ 12x2 ex Þ 4 0 if x 4 0.
k(x) is increasing if x>0. Consequently, k(x) k(y) if
x>y. So that, k(M)>k(N).
This gives ððMeM eM þ 1Þ=2 M2 Þ ððNeN
N
e þ 1Þ=2 N2 Þ. From (45), we have 1 2.
Consider the following equations:
0
g 1 ðT ÞÞ0 ¼ 0
PðTVC
g 2 ðT ÞÞ0 ¼ 0
PðTVC
g 3 ðT ÞÞ0 ¼ 0
PðTVC
ð46Þ
ð47Þ
ð48Þ
If the root of each of equations (46), (47) or (48)
exists, then it is unique. Let T1 denote the root of
equation (46), T2 denote the root of equation (47) and
Downloaded By: [The Indian Institute of Technology] At: 11:53 27 December 2007
688
G. C. Mahata and A. Goswami
T3 denote the root of equation (48). By the convexity of
g i ðT ÞÞ ði ¼ 1, 2, 3Þ, we see that
PðTVC
8
5 0; if T 5 Ti ,
>
>
<
g i ðT ÞÞ0 ¼ 0; if T ¼ T ,
PðTVC
ð49Þ
i
>
>
:
4 0; if T 4 Ti ,
g
Equation
(49) implies that PðTVCi ðT ÞÞ is decreasing
on 0, Ti and increasing on Ti , 1 for all i ¼ 1, 2, 3.
Decision rule of the optimal cycle time T*:
Theorem 2:
g ÞÞ ¼
A. If 1>0 and 2 0, then PðTVCðT
g 3 ðT ÞÞ and T ¼ T .
PðTVC
3
3
g ÞÞ ¼
B. If 1>0 and 20 and 20, Set TU ¼ Topt. If f(Topt)
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Production Planning & Control
The Management of Operations
Publication details, including instructions for authors and subscription information:
http://www.informaworld.com/smpp/title~content=t713737146
An EOQ model for deteriorating items under trade credit
financing in the fuzzy sense
G. C. Mahata a; A. Goswami a
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
a
Online Publication Date: 01 December 2007
To cite this Article: Mahata, G. C. and Goswami, A. (2007) 'An EOQ model for
deteriorating items under trade credit financing in the fuzzy sense', Production
Planning & Control, 18:8, 681 - 692
To link to this article: DOI: 10.1080/09537280701619117
URL: http://dx.doi.org/10.1080/09537280701619117
PLEASE SCROLL DOWN FOR ARTICLE
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This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction,
re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly
forbidden.
The publisher does not give any warranty express or implied or make any representation that the contents will be
complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be
independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings,
demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or
arising out of the use of this material.
Downloaded By: [The Indian Institute of Technology] At: 11:53 27 December 2007
Production Planning & Control,
Vol. 18, No. 8, December 2007, 681–692
An EOQ model for deteriorating items under
trade credit financing in the fuzzy sense
G. C. MAHATA and A. GOSWAMI*
Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, India
This paper deals with the problem of determining the economic order quantity (EOQ) for
deteriorating items in the fuzzy sense where delay in payments for retailer and customer are
permissible and generalizes the earlier published results in this direction. The demand rate,
holding cost, ordering cost and purchasing cost are taken as fuzzy numbers. We also
assume that the supplier would offer the retailer a delay period for payment and the retailer
would also offer the trade credit period to the customer. The total variable cost in fuzzy sense
is defuzzified using the graded mean integration representation method. Then we have
shown that the defuzzified total variable cost is convex, that is, a unique solution exists.
For determination of optimal ordering policies, with the help of theorems we have
developed the neccessary algorithms. Finally, the theorems and the algorithms are
illustrated with the help of numerical examples.
Keywords: EOQ; Fuzzy annual total cost; Trade credit; Graded mean integration
representation
1. Introduction
The basic economic order quantity (EOQ) model is
based on the implicit assumption that the retailer must
pay for the items as soon as he receives them from a
supplier. However, in practice, the supplier will allow a
certain fixed period (credit period) for settling the
amount that the supplier owes to retailer for the items
supplied. Before the end of the trade credit period, the
retailer can sell the goods and accumulate revenue and
earn interest. A higher interest is charged if the
payment is not settled by the end of the trade credit
period. In the real world, the supplier often makes use of
this policy to promote his commodities. In this regard, a
number of research papers appeared which deal with the
EOQ problem under fixed credit period. Goyal (1985)
first studied an EOQ model under the conditions
*Corresponding
ernet.in
author.
Email:
[email protected].
of permissible delay in payments. Chand and Ward
(1987) analysed Goyal’s (1985) problem under assumptions of the classical EOQ model, obtaining different
results. Chung (1998) presented the DCF (discounted
cash flow) approach for the analysis of the optimal
inventory policy in the presence of trade credit. Later,
Shinn et al. (1996) extended Goyal’s (1985) model and
considered quantity discount for freight cost. Recently,
to accommodate more practical features of the real
inventory systems, Aggarwal and Jaggi (1995), Shah
(1993), Hwang and Shinn (1997) extended Goyal’s
(1985) model to consider the deterministic inventory
model with a constant deterioration rate. Shah and
Shah (1998) developed a probabilistic inventory model
when delay in payment is permissible. They developed
an EOQ model for deteriorating items in which time
and deterioration of units are treated as continuous
variables and demand is a random variable. Later on,
Jamal et al. (1997) extended Aggarwal and Jaggi’s
(1995) model to allow for shortages and make it more
Production Planning and Control
ISSN 0953–7287 print/ISSN 1366–5871 online ß 2007 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/09537280701619117
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682
G. C. Mahata and A. Goswami
applicable in the real world. Shawky and Abou-El-Ata
(2001) investigated the production lot-size model
with both restrictions on the average inventory level
and trade credit policy using geometric programming
and Lagrange approaches. All the above models
assumed that the supplier would offer the retailer a
delay period, but the retailer would not offer the trade
credit period to his/her customer. Huang (2003)
assumed that the retailer should also adopt the
trade credit policy to stimulate his/her customer
demand to develop the retailer’s replenishment model.
On the other hand, most inventory systems are
developed without considering the effects of deterioration. However, in real life situations there are products
such as volatile liquids, medicines, materials, etc., in
which the rate of deterioration is very large. Therefore,
the loss of items due to deterioration should not be
neglected. Inventory models for different deteriorating
items have been developed by several researchers in past
and recent years. Ghare and Schrader (1963) developed
an EOQ model for items with an exponentially decaying
inventory. An EOQ model for items with variable rate
of deterioration has been developed by Covert and
Philip (1973) by introducing a two-parameter Weibull
distribution for the time to deterioration. Philip (1974)
developed a three parameter Weibull distribution for the
deterioration time. Many more papers have been
published in this direction.
Usually researchers consider different parameters of
an inventory model either as constant or dependent on
time or probabilistic in nature for the development of
the economic order quantity model. But, in the real life
situations, these parameters may have little deviations
from the exact value which may not follow any
probability distribution. In these situations, if these
parameters are treated as fuzzy parameters, then it
will be more realistic. These types of problems are
defuzzified first using a suitable fuzzy technique and
then the solution procedure can be obtained in the usual
manner. Several authors, namely Chang et al. (1998),
Lee and Yao (1998), Lin and Yao (2000), Yao et al.
(2000), developed inventory models in fuzzy sense by
considering different parameters as fuzzy parameters.
In this paper, we propose a deteriorating inventory
model under the condition of permissible delay in
payments in the fuzzy sense. Most of the inventory
models on this topic assumed that the supplier would
offer the retailer a delay period and the retailer could sell
the goods and accumulate revenue and earn interest
within the trade credit period. They implicitly assumed
that the customer would pay for the items as soon as the
items are received from the retailer. That is, the
retailer would not offer the trade credit period to
his/her customer. In most of the business transactions,
this assumption is debatable. In this paper, we assume
that the supplier would offer the retailer a delay period
and the retailer would also offer the trade credit period
to his/her customer. Furthermore, the demand rate and
the inventory costs (namely holding cost, purchase cost
and ordering cost) may be flexible with some vagueness
for their values. In real life situations, all these
parameters in an inventory model are uncertain,
imprecise and the determination of optimum cycle
time becomes a non-stochastic vague decision-making
process. Again, for this type of model, statistical
estimations proved to be inefficient because of the lack
of statistical observations. In this situation, a suitable
way to model these imprecise data is to use fuzzy sets.
The ill-formed vagueness in the above parameters are
introduced making them fuzzy in nature and then
the model is formulated in a fuzzy environment.
We use the graded mean integration representation
method for defuzzifying fuzzy total average cost. In this
paper, it is shown that the total variable cost per
unit time after defuzzification is convex. Then, with
convexity, a simple optimisation procedure is developed.
Numerical examples are used to illustrate the results
given in this paper. Finally, the results in this
paper generalise some already published results in the
crisp sense.
2. Assumptions and notations
The mathematical model is developed on the basis
of the following assumptions and notations:
(1) The demand rate D is assumed to be constant
e is the fuzzy demand
for the crisp model whereas D
rate for the fuzzy model.
(2) Replenishment rate is infinite and lead time
is zero.
(3) Shortage is not allowed.
(4) A constant fraction , assumed to be small,
of the on-hand inventory gets deteriorated per
unit time.
(5) h, the holding cost per unit; C, the purchasing
cost per unit and A, ordering cost per order, are
e C,
e and
known and constant in the crisp model. h,
e are the fuzzy holding cost, fuzzy purchasing
A
cost and fuzzy ordering cost respectively in fuzzy
model.
(6) Ic, the interest charged per $ in stocks per year by
the supplier; Ie, the interest earned per $ per year
where Ic Ie.
(7) M, the retailer’s trade credit period offered by
supplier in years and N, the customer’s trade
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683
An EOQ model for deteriorating items
credit period offered by retailer in years. It is
assumed that M N.
(8) When T M, the account is settled at time T ¼ M
and retailer starts paying for the interest charges
on the items in stock with rate Ic. When T M,
the account is settled at T ¼ M and the retailer
does not need to pay interest charge.
(9) The retailer can accumulate revenue and earn
interest after his/her customer pays for the
amount of purchasing cost to the retailer until
the end of the trade credit period offered by the
supplier. That is, the retailer can accumulate
revenue and earn interest during the period N to
M with rate Ie under the condition of trade credit.
Case 1.
MT
Total interest payable
ZT
qðtÞdt
¼ CIc
M
CIc D
¼ 2 eðTMÞ ðT MÞ 1 :
ð8Þ
Case 2. N T M
In this case, total interest payable ¼ 0.
Case 3. T N
Similar as case 2, total interest payable ¼ 0.
According to assumption (9), three cases will occur
in interest earned per year.
3. Crisp mathematical model
Let q(t) be the inventory level at any time t(0 t T ).
Initially, the stock level is Q. The inventory level
decreases due to demand and deterioration until it
becomes zero at time t ¼ T. The differential equation
governing the system in the interval (0, T ) is
dqðtÞ
þ qðtÞ ¼ D,
dt
According to assumption (8), three cases may occur in
calculation of interest charges for the items kept in stock
per year.
0 t T,
M T, (shown in figure 1)
Case 1.
Total interest earned ¼ CIe
¼
M
Dtdt
N
CIe DðM2 N2 Þ
:
2
ð9Þ
ð1Þ
N T M, (shown in figure 2)
Case 2.
with the initial condition
qðT Þ ¼ 0:
Z
ð2Þ
Total interest earned ¼ CIe
Z
T
Dtdt þ DTðM T Þ
N
The solution of (1) is
D ðTtÞ
qðtÞ ¼
e
1 ,
¼
0 t T:
ð3Þ
Q ¼ qð0Þ ¼
D T
e 1 :
ð4Þ
Total demand during one cycle is DT.
The number of units deteriorated during one cycle is:
Q DT ¼
D T
ðe 1 T Þ:
ð10Þ
T N, (shown in figure 3)
Case 3.
Using the condition (2), the order quantity can be
obtained as
CIe D
2MT N2 T2 :
2
Total interest earned ¼ CIe DTðM NÞ:
ð11Þ
From the above arguments, the annual total relevant
cost for the retailer can be expressed as, TVC(T ) ¼
ordering cost þ holding cost þ deterioration cost þ
Inventory level
ð5Þ
Q
DT
The total cost due to deterioration of items during the
cycle, denoted by DC, is
DC ¼
CD T
e 1 T :
ð6Þ
The total inventory holding cost per cycle, denoted by
HC, is given by
ZT
hD
qðtÞdt ¼ 2 eT 1 T :
ð7Þ
HC ¼ h
0
0
N
M
T
t
Figure 1. The total accumulation of interest earned when
M T.
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684
G. C. Mahata and A. Goswami
Inventory level
Q
~
A
1
wA
DT
h
0
t
0
N
T
a1 L−1(h) a2
h(L−1(h)+R−1(h))
2
R−1(h) a4
a3
Figure 4. The graded mean h-level value of generalized fuzzy
number A~ ¼ ða1 , a2 , a3 , a4 ; wA Þ.
M
Figure 2. The total accumulation of interest earned when
N T M.
4. Fuzzy methodology
Inventory level
Q
4.1 Graded mean integration representation method
DT
t
0
T
N
M
Figure 3. The total accumulation of interest earned when
T N.
interest payable interest earned.
8
TVC1 ðT Þ; if T M,
>
>
<
TVCðT Þ ¼ TVC2 ðT Þ; if N T M,
>
>
:
TVC3 ðT Þ; if 0 5 T N,
ð12Þ
A ðh þ CÞD T
þ
e 1 T
2
T
T
CIc D ðTMÞ
þ 2
ðT MÞ 1
e
T
CIe D 2
M N2
2T
TVC2 ðT Þ ¼
and
TVC3 ðT Þ ¼
A ðh þ CÞD T
þ
e 1 T
T
2 T
CIe D
2MT N2 T2
2T
A ðh þ CÞD T
þ
e 1 T
2
T
T
CIe DðM NÞ:
(1) A~ ðxÞ is continuous mapping from R to the
closed interval [0, 1]
(2) eðxÞ ¼ 0, 1
< PðTVC1 ðT ÞÞ;
g ÞÞ ¼ PðTVC
g 2 ðT ÞÞ;
PðTVCðT
>
:
g 3 ðT ÞÞ;
PðTVC
if T M,
if N T M,
if 0 5 T N,
which is the defuzzified equation corresponding to the
equation (19). Then, we have
F1 F2 þ F3
ðTeT eT þ 1Þ
þ
T2
2 T2
F3 Ic
þ 2 2 ðTeðTMÞ eðTMÞ þ 1 MÞ
T
F3 Ie ðM2 N2 Þ
þ
ð34Þ
2T2
1 2 2 T
00 2F1 2ðF2 þF3 Þ
T
T
g
e Te 1þ T e
PðTVC1 ðTÞÞ ¼ 3 þ
T
2 T3
2
2F3 Ie ðTMÞ
þ 2 3 e
TeðTMÞ
T
F I ðM2 N2 Þ
1
3 e
1þMþ 2 T2 eðTMÞ
2
T3
ð35Þ
F1 F2 þ F3
0
T
T
g
ðTe e þ 1Þ
PðTVC2 ðT ÞÞ ¼ 2 þ
T
2 T2
F3 Ie ðT2 N2 Þ
ð36Þ
þ
2T2
g 2 ðT ÞÞ00 ¼ 2F1 þ 2ðF2 þ F3 Þ eT TeT
PðTVC
2 T3
T3
1
F 3 Ie N 2
1 þ 2 T2 eT þ
ð37Þ
2
T3
g 1 ðT ÞÞ0 ¼
PðTVC
g 3 ðT ÞÞ0 ¼ F1 þ F2 þ F3 ðTeT eT þ 1Þ ð38Þ
PðTVC
T2
2 T2
g 3 ðT ÞÞ00 ¼ 2F1 þ 2ðF2 þ F3 Þ
PðTVC
2 T3
T3
1
ð39Þ
eT TeT 1 þ 2 T2 eT
2
where (0 ) represents differentiation with respect to T.
6. The convexity
g 1 ðT ÞÞ,
In this section, we shall show that PðTVC
g
g
PðTVC2 ðT ÞÞ and PðTVC3 ðT ÞÞ are convex on their
appropriate domains.
Theorem 1
ð31Þ
ð33Þ
g 1 ðT ÞÞ is convex on ½M, 1Þ.
(1) PðTVC
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687
An EOQ model for deteriorating items
g 2 ðT ÞÞ is convex on ½0, 1Þ.
(2) PðTVC
g 3 ðT ÞÞ is convex on ½0, 1Þ.
(3) PðTVC
on ½0, 1Þ. From equations (34), (36) and (38),
we have
Before proving Theorem 1, we need the following lemmas.
Lemma 1: eT TeT 1 þ 122 T2 eT 4 0 for all T>0.
Proof:
0
f ðxÞ ¼
x
x
Let fðxÞ ¼ e 1 xe þ
1 2 x
2x e
1 2 x
2x e
if x>0, then
g 2 ðMÞÞ0
g 1 ðMÞÞ0 ¼ PðTVC
PðTVC
¼
4 0. Hence f(x) is increasing for all x>0.
þ
Consequently, f(x)>f(0) ¼ 0 if x>0. We have fðxÞ ¼
ex 1 xex þ 12x2 ex 4 0 if x>0. Let x ¼ T. Then eT
TeT 1 þ 122 T2 eT 4 0 if T>0. This completes the
proof.
œ
Lemma 2: If T M, then eðTMÞ 1 TeðTMÞ þ
M þ 122 T2 eðTMÞ ð2 M2 Þ=2 0.
Proof: Let
gðT Þ ¼ eðTMÞ 1 TeðTMÞ þ Mþ
1 2 2 ðTMÞ
ð2 M2 Þ=2.
Then we have g0 ðT Þ ¼
2 T e
1 3 2 ðTMÞ
4 0. Hence g(T ) is increasing on ½M, 1Þ.
2 T e
So g(T )>g(M) ¼ 0 if T>M. This completes the
proof.
œ
The proof of Theorem 1
F1 F2 þ F3
þ 2 2 ðMeM eM þ 1Þ
M2
M
F3 Ie ðM2 N2 Þ
2M2
ð41Þ
and
g 2 ðNÞÞ0 ¼ PðTVC
g 3 ðNÞÞ0
PðTVC
¼
F1 F2 þ F3
þ
ðNeN eN þ 1Þ ð42Þ
N2
2 N2
Furthermore, we let
1 ¼
F1 F2 þ F3
F3 Ie ðM2 N2 Þ
þ 2 2 ðMeM eM þ 1Þ þ
,
2
2M2
M
M
ð43Þ
and
(1) From equation (35), we have
2 ¼
g 1 ðT ÞÞ00
PðTVC
2F1 2ðF2 þ F3 Þ T
1 2 2 T
T
¼ 3 þ
e Te 1 þ T e
2 T3
2
T
2F3 Ic ðTMÞ
þ 2 3 e
TeðTMÞ 1 þ M
T
1
F3 Ie ðM2 N2 Þ
þ 2 T2 eðTMÞ
2
T3
2F1 2ðF2 þ F3 Þ T
1 2 2 T
T
4 3 þ
e
Te
1
þ
T
e
2 T3
2
T
2F3 Ic ðTMÞ
þ 2 3 e
TeðTMÞ 1 þ M
T
1
2 M2
F3 Ie N2
þ 2 T2 eðTMÞ
:
ð40Þ
þ
2
2
T3
g 1 ðT ÞÞ00 4 0 if
Lemmas 1 and 2 imply that PðTVC
g 1 ðT ÞÞ is convex on ½M, 1Þ.
T M. Hence PðTVC
g 2 ðT ÞÞ00 ¼
(2) From equation (37), we have PðTVC
ð2F1 =T3 Þ þ ð2ðF2 þ F3 Þ=2 T3 Þ ðeT TeT 1þ
1 2 2 T
þ ðF3 Ie N2 =T3 Þ. Lemma 1 implies that
2 T Þe
g 2 ðT ÞÞ00 4 0 if T>0. Consequently,
PðTVC
g 2 ðT ÞÞ is convex on ½0, 1Þ.
PðTVC
g 3 ðT ÞÞ00 ¼
(3) From equation (39), we have PðTVC
ð2F1 =T3 Þ þ ð2ðF2 þ F3 Þ=2 T3 ÞðeT TeT 1 þ
00
1 2
2 T
g
2 T e Þ. From lemma 1 PðTVC3 ðT ÞÞ 4 0 if
g 3 ðT ÞÞ is convex
T>0. Consequently, PðTVC
F1 F2 þ F3
þ
ðNeN eN þ 1Þ:
N2
2 N2
ð44Þ
Therefore,
1
1
F3 Ie
1 2 ¼ F1 2 2 þ
ðM2 N2 Þ þ ðF2 þ F3 Þ
N
M
2M2
MeM eM þ 1 NeN eN þ 1
:
2 M2
2 N2
ð45Þ
Let, KðxÞ ¼ ððxex ex þ 1Þ=x2 Þ, so that by lemma 1,
Hence
k ðxÞ ¼ x23 ðex 1 xex þ 12x2 ex Þ 4 0 if x 4 0.
k(x) is increasing if x>0. Consequently, k(x) k(y) if
x>y. So that, k(M)>k(N).
This gives ððMeM eM þ 1Þ=2 M2 Þ ððNeN
N
e þ 1Þ=2 N2 Þ. From (45), we have 1 2.
Consider the following equations:
0
g 1 ðT ÞÞ0 ¼ 0
PðTVC
g 2 ðT ÞÞ0 ¼ 0
PðTVC
g 3 ðT ÞÞ0 ¼ 0
PðTVC
ð46Þ
ð47Þ
ð48Þ
If the root of each of equations (46), (47) or (48)
exists, then it is unique. Let T1 denote the root of
equation (46), T2 denote the root of equation (47) and
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688
G. C. Mahata and A. Goswami
T3 denote the root of equation (48). By the convexity of
g i ðT ÞÞ ði ¼ 1, 2, 3Þ, we see that
PðTVC
8
5 0; if T 5 Ti ,
>
>
<
g i ðT ÞÞ0 ¼ 0; if T ¼ T ,
PðTVC
ð49Þ
i
>
>
:
4 0; if T 4 Ti ,
g
Equation
(49) implies that PðTVCi ðT ÞÞ is decreasing
on 0, Ti and increasing on Ti , 1 for all i ¼ 1, 2, 3.
Decision rule of the optimal cycle time T*:
Theorem 2:
g ÞÞ ¼
A. If 1>0 and 2 0, then PðTVCðT
g 3 ðT ÞÞ and T ¼ T .
PðTVC
3
3
g ÞÞ ¼
B. If 1>0 and 20 and 20, Set TU ¼ Topt. If f(Topt)