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Mathematical and Computer Modelling
journal homepage:www.elsevier.com/locate/mcm
Analysis of a fuzzy economic order quantity model for deteriorating
items under retailer partial trade credit financing in a supply chain
Gour Chandra Mahata
a,∗, Puspita Mahata
b,1aDepartment of Mathematics, Sitananda College, P. O. + P. S. - Nandigram, Dist. - Purba Medinipur, PIN - 721631, West Bengal, India
bDepartment of Commerce, Srikrishna College, P.O. + P.S. - Bagula, Dist. - Nadia, PIN - 741502, West Bengal, India
a r t i c l e
i n f o
Article history:
Received 1 January 2010
Received in revised form 11 December 2010 Accepted 15 December 2010
Keywords: Inventory EOQ model Partial trade credit Supply chain Deteriorating items
a b s t r a c t
This paper investigates the economic order quantity (EOQ) — based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier yet the retailer just offers a partial trade credit to customers. The demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers. Under these conditions, the retailer can obtain the most benefits. Study also investigates the retailer’s inventory policy for deteriorating items in a supply chain management situation as a cost minimization problem in the fuzzy sense. The annual total variable cost for the retailer in fuzzy sense is defuzzified using Graded Mean Integration Representation method. Then the present study shows that the defuzzified annual total variable cost for the retailer is convex, that is, a unique solution exists. Mathematical theorems and algorithms are developed to efficiently determine the optimal inventory policy for the retailer. Numerical examples are given to illustrate the theorems and the algorithms. Finally, the results in this paper generalize some already published results in the crisp sense.
©2010 Elsevier Ltd. All rights reserved.
1. Introduction
The basic 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 a 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 [1] first studied an EOQ model under the conditions of permissible delay in payments. Chand and Ward [2] analyzed Goyal’s [1] problem under assumptions of the classical EOQ model, obtaining different results. Chung [3] 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. [4] extended Goyal’s [1] model and considered quantity discount for freight cost. Recently, to accommodate more practical features of the real inventory systems, Aggarwal and Jaggi [5], Shah [6], Hwang and Shinn [7] extended Gayal’s [1] model to consider the deterministic inventory model with a constant deterioration rate. Shah and Shah [8] developed a probabilistic inventory model when delay in payment is permissible. They developed an EOQ model for deteriorating items in which
∗ Corresponding author. Tel.: +91 9474190816; fax: +91 3224232295.
E-mail addresses:[email protected](G.C. Mahata),[email protected](P. Mahata). 1 Tel.: +91 9433557570.
Fig. 1. Two-level trade credit policy with partial trade credit financing to common customers.
time and deterioration of units are treated as continuous variables and demand is a random variable. Later on, Jamal et al. [9] extended Aggarwal and Jaggi’s [5] model to allow for shortages and make it more applicable in the real world. Shawky and Abou-El-Ata [10] investigated the production lot-size model with both restrictions on the average inventory level and trade credit policy using geometric programming and Lagrange approaches. Mahata and Goswami [11] presented a fuzzy EPQ model for deteriorating items when delay in payment is permissible. Huang [12] assumed that retailer would adopt a similar trade credit policy to stimulate demand from customer to develop the retailer’s replenishment method. There are several interesting and relevant papers related to trade credit such as Chung et al. [13], Chung and Liao [14], Mahata and Mahata [15] and Huang [16] and their references.
All the above articles 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, they assumed that the supplier would offer the retailer a delay period but the retailer would not offer any delay period to his/her customer. That is one level of trade credit. In most business transactions, this assumption is unrealistic. Usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on this credit period to his/her customers. Recently, Huang [12] modified this assumption to assume that the retailer will adopt the trade credit policy to stimulate his/her customers’ demand to develop the retailer’s replenishment model. That is two levels of trade credit. Haung [17] incorporated Haung’s [12] model to investigate the two-level trade credit policy in the EPQ framework. This new viewpoint is more matched to real-life situations in the supply chain model. Therefore, we want to extend Huang’s model [12] to investigate the situation under which the retailer has the powerful decision-making right. That is, we want to assume that the retailer can obtain the full trade credit offered by the supplier and the retailer just offers a partial trade credit to his/her customer. The path of the trade credit policy is illustrated inFig. 1. In practice, this circumstance is very realistic.
For example, in India, the TATA Company can require his supplier to offer the full trade credit period to him and just offer a partial trade credit to his dealership. That is, the TATA Company can delay the full amount of the purchase cost until the end of the delay period offered by his supplier. But the TATA Company only offers a partial delay payment to his dealership on the permissible credit period and the rest of the total amount is payable at the time the dealership places a replenishment order.
It is a formidable task for the retailers to estimate different inventory parameters as crisp or stochastic as due to rapid changes of product specification and the introduction of new products; in the market sufficient past data is not available for such an estimation. In the above inventory models, it was assumed that the demand rate and the inventory costs are constant in nature. Due to various uncertainties, the annual demand rate may have a little fluctuation, especially in a perfect competitive market. For developing inventory models, a major difficulty faced by a decision maker (retailer) is to forecast the demand. In the present day scenario, it is tough to decide the exact annual demand rate, namely, how many items customers will purchase during the whole year. Also the cost parameters such as the purchase cost, the holding cost and the ordering cost are constants. These kinds of assumptions are not always true. It may not be possible to specify the values of these cost parameters precisely but they may contain some uncertain values such as ‘‘unit holding cost is abouth’’, or ‘‘unit purchase cost is approximatelyc or more’’, etc. In another sense, these parameters may contain some uncertain values. In these circumstances it is better to model these parameters as fuzzy because the estimation (fuzzy) is done by experts’ opinions and salesmens’/representatives’ experience. Again in the present competitive market along with the profit/cost function, customer service also becomes a crucial factor. Due to high bank interest, and limitation of resources, profit with respect to investment is also important. So the goal of present day inventory problems are multiple rather than single. As a result, retailers of all corners in the World very frequently face non-linear optimization problems whose objective involves fuzzy parameters. The significance of this study is to develop an inventory model incorporating the above-mentioned real life situations that will help the retailers to survive in the market.
On the other hand, most inventory systems are developed without considering the effects of deterioration. However, in real life situations there are perishable products commonly found in commerce and industry. Sometimes the rate of deterioration is too low, for items such as steel, hardware, glassware and toys, to cause a consideration of deterioration in the determination of economic lot sizes. However, some items have a significant rate of deterioration, such as fruits, volatiles, liquids, medicines, materials, fresh fish, perfumes, alcohol, etc., in which the rate of deterioration is very large. Therefore, the loss of items due to deterioration should not be neglected in the decision making process of ordering the lot size. The traditional EOQ models for perishable items can be found in the literature. Ghare and Schrader [23] developed an EOQ model for items with an exponentially decaying inventory. An EOQ model for items with a variable rate of deterioration has been developed by Covert and Philip [24] by introducing two parameter Weibull distribution for the time to deterioration. Philip [25] developed a three parameter Weibull distribution for the deterioration time. Many more papers have been published in this direction.
The present study investigates the economic order quantity (EOQ)-based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by supplier and the retailer just offers partial trade credit to customers. Furthermore, the demand rate and the inventory costs namely holding cost, ordering cost, purchase cost and selling price may be flexible with some vagueness as to their values. In real life situations, all these parameters in an inventory model are uncertain, imprecise and the determination of the optimum cycle time becomes a non-stochastic vague decision making process. Again, for this type of models, 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 the fuzzy annual total variable cost. In this paper, it is shown that the annual total variable cost per unit time after defuzzification is convex. Then, with convexity, a simple optimization procedure is developed. Numerical examples are used to illustrate the results given in this paper. Finally, the results in this paper generalize some already published results in the crisp sense.
2. Methodology
2.1. Development of a modified graded mean integration representation of a generalized fuzzy number
More recently, additional important works on the concept of fuzzy numbers have been written by Dubois and Prade [26,27], and by several other authors. Kaufmann and Gupta [28] give a definition for a fuzzy number that a fuzzy number in Ris a fuzzy subset ofRthat is convex and normal. Thus a fuzzy number can be considered a generalization of the interval of confidence. However, it is not a random variable. A random variable is defined in terms of the theory of probability, which has evolved from theory of measurement. A random variable is an objective datum, whereas a fuzzy number is a subjective datum. It is a valuation, not a measure.
For achieving computational efficiency, we use the method of defuzzification of a generalized fuzzy number by its graded mean integration representation. Throughout this paper, we only use the popular triangular fuzzy number (TFN) as the type of all fuzzy parameters in our proposed fuzzy inventory model.
Here, we first describe the generalized fuzzy number as follows:
Suppose
Ais a generalized fuzzy number as shown inFig. 2and is described as any fuzzy subset of the real lineR, whose membership functionµ
Asatisfies the following conditions:1.
µ
A(
x)
is continuous mapping fromRto the closed interval [0, 1], 2.µ
A(
x)
=
0,
− ∞
<
x≤
a1,3.
µ
A(
x)
=
L(
x)
is strictly increasing on[
a1,
a2]
, 4.µ
A(
x)
=
w
A,
a2≤
x≤
a3,5.
µ
A(
x)
=
R(
x)
is strictly decreasing on[
a3,
a4]
, 6.µ
A(
x)
=
0,
a4≤
x<
∞
,where 0
< w
A≤
1, anda1,a2,a3anda4are real numbers. Also this type of generalized fuzzy number may be denoted as
A=
(
a1,
a2,
a3,
a4;
w
A)
LR. Whenw
A=
1, it can be simplified as
A=
(
a1,
a2,
a3,
a4)
LR.In addition, Chen and Hsieh [29] introduced the Graded Mean Integration Representation method based on the integral value of the graded meanh-levelof the generalized fuzzy number for defuzzifying a generalized fuzzy number. This method is reasonable in that it adopts a grade as the important degree of each point of the support set of a fuzzy number for representing the fuzzy number.
Second, by the Graded Mean Integration Representation method L−1 and R−1 are the inverse functions of L and R respectively, and the graded meanh-level value of the generalized fuzzy number
A=
(
a1,
a2,
a3,
a4;
w
A)
LRish(
L−1(
h)
+
R−1
(
h))/
2 (seeFig. 2). Then the Graded Mean Integration Representation of
AisP(
A)
with gradew
A, where
P
(
A)
=
∫
wA0 h
(
L−1
(
h)
+
R−1(
h))
2 dh
∫
wA0
hdh
,
(1)Fig. 2. The graded meanh-level value of generalized fuzzy numberA=(a1,a2,a3,a4;wA)LR.
The generalized triangular fuzzy number (GTFN)
Bis a special case of the generalized fuzzy number and be denoted as
B=
(
a,
b,
c;
w)
. Its corresponding graded mean integration representation isP
(
B)
=
w0 h
{
a+
(
b−
a)
h/w
+
c−
(
c−
b)
h/w
}
/
2dhw
0 hdh
=
a+
4b+
c6
,
(2)wherehlies between 0 and
w
, 0< w
≤
1.It is to be noted here that in(1), equal weight has been given to the left and right parts of the membership function. But the weight actually depends on the attitude or optimism of the decision maker. So, the formula used, in this paper, as the graded meanh-level value of the fuzzy numberA
˜
=
(
a,
b,
c)
TFNis assumed to be of the formh[
β
L−1(
h)
+
(
1−
β)
R−1(
h)
]
, whereβ
is called the decision maker’s attitude or optimism parameter.β
can take values between 0 and 1 i.e., 0≤
β
≤
1. A value ofβ
closer to 0 implies that the decision maker is more pessimistic while a value ofβ
closer to 1 means that the decision maker is more optimistic.Therefore, the formula(1)is modified as below:
P
(
A)
=
w0
(
h[
β
L−1
(
h)
+
(
1−
β)
R−1]
)
dh
w0 hdh
.
(3)Now,
L
(
x)
=
w
x
−
a b−
a
,
a≤
x≤
band R
(
x)
=
w
c
−
x c−
b
,
b≤
x≤
c.
Thus,
L−1
(
h)
=
a+
(
b−
a)
h/w
and R−1(
h)
=
c−
(
c−
b)
h/w.
Now, using the formula(3), the graded mean integration representation ofA
˜
is given byP
(
A˜
)
=
w0 h
[
β
a+
(
1−
β)
c+ {
b−
β
a−
(
1−
β)
c}
h/w
]
w
0 hdh
=
β
a+
2b+
(
1−
β)
c3
.
(4)It is to be noted that when
β
=
0.
5 i.e., when equal weight is given to the left and right parts of the membership function, then, the formula(4)reduces to the formula(2).Remark 1. By formula(4), it is easy to observe that the graded mean integration representation of the GTFN
B=
(
a,
b,
c;
w)
is independent of
w
.If
w
=
1, the GTFNB˜
is called a triangular fuzzy number (TFN) denoted by
B=
(
a,
b,
c)
TFN.The defuzzification of
B=
(
a,
b,
c)
TFNcan be found by the centroid or the graded mean integration method. The centroid of the TFN
B=
(
a,
b,
c)
TFNisC(
B)
=
31(
a+
b+
c)
and the graded mean integration representation of
B=
(
a,
b,
c)
TFNis P(
B˜
)
=
16(
a+
4b+
c)
. The mid-point of the interval[
a,
c]
isM=
a+2c. ThusC
(
B˜
)
−
P(
B˜
)
=
13
(
M−
b),
P(
˜
B
)
−
b=
13
(
M−
b)
and M−
C(
˜
B)
=
1Fig. 3. CaseM>b.
Fig. 4. CaseM<b.
From Eq.(5), we notice that
(i) IfM
>
b, thena<
b<
P(
B˜
) <
C(
B˜
) <
M<
c. (ii) IfM<
b, thena<
M<
C(
B˜
) <
P(
B˜
) <
b<
c. (iii) IfM=
b, thena<
M=
C(
B˜
)
=
P(
B˜
)
=
b<
c.From (i) and (ii), it is clear thatP
(
B˜
)
is nearb, andC(
B˜
)
is nearM. FromFigs. 3and4, we observe that the membership grade ofB˜
atbis 1 and the membership grade ofB˜
atP(
B˜
)
is greater than that atC(
B˜
)
. Then, we haveµ
B˜(
P(
B˜
)) > µ
˜B(
C(
B˜
))
.Property 1. From the membership grade viewpoint, it will be efficient to defuzzify the fuzzy numberB
˜
=
(
a,
b,
c)
TFNby P(
B˜
)
instead of C(
B˜
)
.2.2. The fuzzy arithmetical operations under function principle
In this paper, we use the Function Principle to simplify the calculation. Function Principle [30] in fuzzy theory is used as the computational model avoiding the computations which can be caused by the operations using the Extension Principle. We describe some fuzzy arithmetical operations under the Function Principle as follows:
Suppose
A=
(
a1,
a2,
a3)
and
B=
(
b1,
b2,
b3)
are two triangular fuzzy numbers. Then,1. The addition of
Aand
Bis
A
B=
(
a1+
b1,
a2+
b2,
a3+
b3)
, wherea1,
a2,
a3,
b1,
b2andb3are any real numbers. 2.−
B=
(
−
b3,
−
b2,
−
b1)
, then the substraction of
Aand
Bis
A⊖
B=
(
a1−
b3,
a2−
b2,
a3−
b1)
, wherea1,
a2,
a3,
b1,
b2andb3are any real numbers.
3. The multiplication of
Aand
Bis
A
B=
(
c1,
c2,
c3)
, whereT= {
a1b1,
a1b4,
a4b1,
a4b4}
,T1= {
a2b2,
a2b3,
a3b2,
a3b3}
, c1=
minT,c2=
minT1,c3=
maxT1,c4=
maxT.Also, ifa1
,
a2,
a3,
b1,
b2andb3are all nonzero positive real numbers, then
A
B=
(
a1b1,
a2b2,
a3b3)
, where
A
B is a triangular fuzzy number.4. 1 B
=
B−1
=
(
1/
b3
,
1/
b2,
1/
b1)
, whereb1,
b2andb3are all nonzero positive real numbers.If a1
,
a2,
a3,
b1,
b2 and b3 are all nonzero positive real numbers, then the division of
A and
B is
A⊘
B=
Fig. 5. The fuzzy addition operation of Function Principle and Extension Principle.
Fig. 6. The comparing of fuzzy multiplication operation under the Function Principle and Extension Principle.
5. Let
α
∈
R. Then(
i) α
⊗
A=
(α
a1, α
a2, α
a3),
α
≥
0(
ii) α
⊗
A=
(α
a3, α
a2, α
a1),
α <
0.
Note. We do not introduce a new addition symbol, as the sum under the Extension Principle is the same asFig. 5. For a mathematically minded reader, we observe that the Extension Principle is a form of convolution [31] while the Function Principle is akin to a pointwise multiplication asFig. 6.
3. Modelling of EOQ-based inventory problem with fuzzy variables
This study develops a retailer’s EOQ-based inventory model under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain full trade credit offered by the supplier and the retailer just offers partial trade credit to customers. Here, we also consider that the demand rate and the inventory costs namely the selling price, purchasing cost, holding cost, ordering cost are all fuzzy numbers.
3.1. Assumptions and notation
The mathematical model is developed on the basis of the following assumptions and notation:
1. The demand rate,D, is assumed to be known and constant for the crisp model whereas
Dis the fuzzy demand rate for the fuzzy model.2. Replenishments are instantaneous, the rate is infinite and the lead time is zero. 3. Shortage is not allowed.
4. Time horizon is infinite.
5. A constant fraction
θ
, assumed to be small, of the on-hand inventory gets deteriorated per unit time.6. h: inventory holding cost per item per unit time;A: the replenishment (ordering) cost per order;c: the unit purchase cost; and s: the unit selling price of items of good quality, where s
≥
c. In fuzzy sense, these quantities may be represented ash˜
=
(
h−
δ
h1,
h,
h+
δ
h2)
,A˜
=
(
A−
δ
A1,
A,
A+
δ
A2)
,c˜
=
(
c−
δ
c1,
c,
c+
δ
c2)
, and˜
s=
(
s−
δ
s1,
s,
s+
δ
s2)
. 7. Ic: the interest charged per $ in stocks per year by the supplier;Ie: the interest earned per $ per year whereIc≥
Ie.8. M: the retailer’s trade credit period offered by the supplier in years andN: the customer’s trade credit period offered by the retailer in years. It is assumed thatM
≥
N.9.
α
: the customer’s fraction of the total amount owed payable at the time of placing an order within the delay period to the retailer, where 0≤
α
≤
1.11. The retailer just offers a partial trade credit to his/her customer. Hence, his/her customer must make a partial payment to the retailer when the item is sold. Then his/her customer must pay off the remaining balance at the end of the trade credit period offered by the retailer. That is, the retailer can accumulate interest from his/her customer partial payment on
(
0,
N]
and from the total amount of payment on[
N,
M]
with rateIe.3.2. The mathematical model formulation and its analysis
Letq
(
t)
be the inventory level at any timet(
0≤
t≤
T)
. Initially, the stock level isQ. The inventory level decreases due to demand and deterioration both until it becomes zero at timet=
T. The differential equation governing the system in the interval(
0,
T)
isdq
(
t)
dt
+
θ
q(
t)
= −
D,
0≤
t≤
T,
(6)with the boundary conditionsq
(
0)
=
Q andq(
T)
=
0. The solution of the differential equation(6)with the boundary conditionq(
T)
=
0 isq
(
t)
=
Dθ
(
eθ (T−t)
−
1),
0≤
t≤
T.
(7)Using the boundary conditionq
(
0)
=
Q, the order quantity can be obtained asQ
=
Dθ
(
eθT
−
1).
(8)Total demand during one cycle isDT.
The number of units deteriorated during one cycle is
Q
−
DT=
Dθ
(
eθT
−
1−
θ
T).
The total annual cost due to deterioration of items during the cycle, denoted byDC, is
DC
=
cDθ
T(
eθT
−
1−
θ
T).
The total annual inventory holding cost (excluding interest charges) per cycle, denoted byHC, is given by
HC
=
hAccording to assumption(10), three cases may occur in the calculation of interest charges for the items kept in stock per year.
In this case, annual interest payable
=
0. Case3.T≤
N.Similar as case 2, annual interest payable
=
0.sDT
0 sDN
T
N M
α
Time
Fig. 7. Total amount of interest earned whenM≤T.
sDT
0 sDN
T
N M
α
Time
Fig. 8. Total amount of interest earned whenN≤T≤M.
Case3.T
≤
N, (shown inFig. 9)Annual interest earned
=
sIe T[∫
T0
α
Dtdt+
α
DT(
N−
T)
+
DT(
M−
N)
]
=
sIeD[
M
−
(
1−
α)
N−
α
T 2]
.
From the above arguments, the annual total relevant cost for the retailer can be expressed as,TVC
(
T)
=
ordering cost+
holding cost+
deterioration cost+
interest payable−
interest earned.TVC
(
T)
=
TVC1
(
T)
;
ifT≥
M,
TVC2(
T)
;
ifN≤
T≤
M,
TVC3(
T)
;
if 0<
T≤
N,
(9)
where
TVC1
(
T)
=
A T+
(
h+
θ
c)
Dθ
2T(
eθT
−
1−
θ
T)
+
cIcDθ
2T[
eθ (T−M)
−
θ (
T−
M)
−
1] −
sIeD 2T[
M2
−
(
1−
α)
N2]
,
(10)TVC2
(
T)
=
A T+
(
h+
θ
c)
Dθ
2T(
eθT
−
1−
θ
T)
−
sIeD2T
[
2MT−
(
1−
α)
N 2−
T2
]
(11)and TVC3
(
T)
=
A T+
(
h+
θ
c)
Dθ
2T(
eθT
−
1−
θ
T)
−
sIeD[
M
−
(
1−
α)
N−
α
T 2]
.
(12)sDT
0 sDT
T N M
α
Time
Fig. 9. Total amount of interest earned whenT≤N.
can be vaguely expressed. So, it is more suitable to describe it as fuzzy in nature. Therefore, the total annual demand is treated as a fuzzy variable. Here we consider the fuzzy annual demandDas a triangular fuzzy numberD
˜
=
(
D−
δ
D1,
D,
D+
δ
D2)
. Also, inventory costs are normally assumed to be constant but this is not always true. In a perfect competitive market, ordering costA, unit holding costh, unit purchasing costc, unit selling pricesetc. per day in a plan periodTmay fluctuate a little. For example, ‘‘unit holding cost is aroundh’’, ‘‘unit selling price is abouts’’, etc. Suppose these cost parameters ordering cost, holding cost, purchase cost, selling price lie in the intervals[
A−
δ
A1,
A+
δ
A2]
,[
h−
δ
h1,
h+
δ
h2]
,[
c−
δ
c1,
c+
δ
c2]
,[
s−
δ
s1,
s+
δ
s2]
. Similarly, corresponding to these intervals, we set the following triangular fuzzy numbers:A˜
=
(
A−
δ
A1,
A,
A+
δ
A2)
,˜
h
=
(
h−
δ
h1,
h,
h+
δ
h2)
,c˜
=
(
c−
δ
c1,
c,
c+
δ
c2)
and˜
s=
(
s−
δ
s1,
s,
s+
δ
s2)
. Here we consider that all the (fuzzy) observations of a fuzzy variable as triangular fuzzy numbers. This consideration does not restrict the solution procedures for other fuzzy numbers. Through(10)–(12), for anyT>
0, we get fuzzy annual total variable costs
TVC1
(
T)
=
X11⊗
A⊕
X12⊗
h⊗
D⊕
X13⊗
c⊗
D⊖
X14⊗
s⊗
D (13)
TVC2
(
T)
=
X21⊗
A⊕
X22⊗
h⊗
D⊕
X23⊗
c⊗
D⊖
X24⊗
s⊗
D (14)and TVC
3(
T)
=
X31⊗
A⊕
X32⊗
h⊗
D⊕
X33⊗
c⊗
D⊖
X34⊗
s⊗
D,
(15)where
⊗
,⊕
and⊖
are the fuzzy arithmetical operations under the Function Principle andX11
=
X21=
X31=
1T
,
X12=
X22=
X32=
1θ
2T(
eθT
−
1−
θ
T),
X13
=
eθT
−
1−
θ
Tθ
T+
Ic
(
eθ (T−M)−
θ (
T−
M)
−
1)
θ
2T,
X23=
X33=
eθT
−
1−
θ
Tθ
T,
X14
= −
Ie2T
[
M2
−
(
1−
α)
N2]
,
X 24=
Ie
2T
[
2MT−
(
1−
α)
N 2−
T2]
and X34
=
Ie[
M
−
(
1−
α)
N−
α
T 2]
.
Here we assume thatD
˜
=
(
D1,
D,
D2)
,A˜
=
(
A1,
A,
A2)
,h˜
=
(
h1,
h,
h2)
,c˜
=
(
c1,
c,
c2)
, and˜
s=
(
s1,
s,
s2)
are nonnegative triangular fuzzy numbers, whereD1=
D−
δ
D1,D2=
D+
δ
D2,A1=
A−
δ
A1,A2=
A+
δ
A2,h1=
h−
δ
h1,h2=
h+
δ
h2, c1=
c−
δ
c1,c2=
c+
δ
c2,s1=
s−
δ
s1,s2=
s+
δ
s2. Then we get the fuzzy annual total variable costTVC
1(
T)
by Eq.(13)as
TVC1
(
T)
= [
X11A1+
X12h1D1+
X13c1D1−
X14s2D2,
X11A+
X12hD+
X13cD−
X14sD,
X11A2+
X12h2D2+
X13c2D2−
X14s1D1]
.
We defuzzify the fuzzy annual total variable cost TVC
1(
T)
by formula (2) and obtain the graded mean integration representation ofTVC
1(
T)
asP
(
TVC
1(
T))
=
16
[
(
X11A1+
X12h1D1+
X13c1D1−
X14s2D2)
+
4(
X11A+
X12hD+
X13cD−
X14sD)
+
(
X11A2+
X12h2D2+
X13c2D2−
X14s1D1)
]
=
X11F1+
X12F2+
X13F3−
X14F4=
F1T
+
F2
+
θ
F3θ
2T(
eθT
−
1−
θ
T)
+
F3Icθ
2T(
eθ (T−M)
−
θ (
T−
M)
−
1)
−
F4Ie 2T[
M2
−
(
1−
α)
N2]
,
(16)P
(
TVC
3(
T))
=
Thus the annual total cost in the fuzzy sense based on the graded mean integration representation is
P
(
TVC
(
T))
=
where
(
′)
represents differentiation with respect toT.3.2.1. The convexity
In this section, we shall show thatP
(
TVC
1(
T))
,P(
TVC
2(
T))
andP(
TVC
3(
T))
are convex on their appropriate domains.Theorem 1. (1) P
(
TVC
1(
T))
is convex on[
M,
∞
)
. (2) P(
TVC
2(
T))
is convex on[
0,
∞
)
.(3) P
(
TVC
3(
T))
is convex on[
0,
∞
)
.Before provingTheorem 1, we need the following lemmas.
Lemma 1. eθT
−
θ
TeθT−
1+
1ifT
>
0. This completes the proof.The proof of Theorem 1. (1) From Eq.(21), we have
This complete the proof ofTheorem 1.
P
(
TVC
i(
T))
=
0;
ifT=
Ti,
>
0;
ifT>
Ti∗.
(35)
Eq.(35)implies thatP
(
TVC
i(
T))
is decreasing on(
0,
Ti∗]
and increasing on[
T∗
i
,
∞
)
for alli=
1,
2,
3.
3.2.2. Decision rule of the optimal cycle time T∗
In this section, we develop efficient decision rules to find the optimal cycle time for the retailer.
Theorem 2. (A) If
∆
1>
0and∆
2≥
0, then P(
TVC
(
T∗))
=
P(
TVC
3(
T3∗))
and T∗=
T3∗. (B) If∆
1>
0and∆
2<
0, then P(
TVC
(
T∗))
=
P(
TVC
2(
T2∗))
and T∗=
T2∗.(C) If
∆
1≤
0and∆
2<
0, then P(
TVC
(
T∗))
=
P(
TVC
1(
T1∗))
and T∗=
T1∗.Proof. (A) If
∆
1>
0 and∆
2≥
0, thenT1∗<
M,T2∗<
M,T3∗≤
NandT2∗≤
N. We haveP(
TVC
1(
M))
′=
P(
TVC
2(
M))
′>
0 andP(
TVC
2(
N))
′=
P(
TVC
3(
N))
′≥
0. Eq.(35)imply that(i) P
(
TVC
1(
T))
is increasing on[
M,
∞
)
, (ii) P(
TVC
2(
T))
is increasing on[
N,
M]
and(iii) P
(
TVC
3(
T))
is decreasing on(
0,
T3∗]
and is increasing[
T3∗,
N]
.Combining (i)–(iii) and Eq. (19), we have that P
(
TVC
(
T))
is decreasing on(
0,
T3∗]
and increasing on[
T3∗,
∞
)
. Consequently,T∗=
T3∗andP(
TVC
(
T∗))
=
P(
TVC
3(
T3∗))
.(B) If
∆
1>
0 and∆
2<
0, thenT3∗>
N,T2∗>
N,T1∗<
MandT2∗<
M. We haveP(
TVC
1(
M))
′=
P(
TVC
2(
M))
′>
0 and P(
TVC
2(
N))
′=
P(
TVC
3(
N))
′<
0. Eq.(35)imply that(i) P
(
TVC
1(
T))
is increasing on[
M,
∞
)
,(ii) P
(
TVC
2(
T))
is decreasing on[
N,
T2∗]
and increasing on[
T2∗,
M]
, (iii) P(
TVC
3(
T))
is decreasing on(
0,
N]
.Combining (i)–(iii) and Eq. (19), we have that P
(
TVC
(
T))
is decreasing on(
0,
T2∗]
and increasing on[
T2∗,
∞
)
. Consequently,T∗=
T2∗andP(
TVC
(
T∗))
=
P(
TVC
2(
T2∗))
.(C) If
∆
1≤
0 and∆
2<
0, thenT3∗>
N,T2∗>
N,T1∗≥
MandT2∗≥
M. We haveP(
TVC
1(
M))
′=
P(
TVC
2(
M))
′≤
0 and P(
TVC
2(
N))
′=
P(
TVC
3(
N))
′<
0. Eq.(35)imply that(i) P
(
TVC
1(
T))
is decreasing on[
M,
T1∗]
and increasing on[
T1∗,
∞
)
, (ii) P(
TVC
2(
T))
is decreasing on[
N,
M]
and(iii) P
(
TVC
3(
T))
is decreasing on(
0,
N]
.Combining (i)–(iii) and Eq. (19), we have that P
(
TVC
(
T))
is decreasing on(
0,
T∗1
]
and increasing on[
T1∗,
∞
)
. Consequently,T∗=
T∗1 andP
(
TVC
(
T∗))
=
P(
TVC
1(
T1∗))
.3.2.3. The algorithms for the determination of T∗, T1∗, T2∗and T3∗ We first describe the following theorem:
Intermediate value theorem:Letf
(
x)
be a continuous function on[
a,
b]
andf(
a).
f(
b) <
0, then there exists a number c∈
(
a,
b)
such thatf(
c)
=
0.(i) Suppose that
∆
1>
0 and∆
2≥
0, thenT3∗exists,T∗=
T3∗and 0<
T3∗<
N. RecallT3∗to denote the unique root of the Eq.(34). Letf3(
T)
= −
F1+
F2+θ2θF3(θ
TeθT−
eθT+
1)
+
F4Ieα
2 T
2. Thenf′
3
(
T)
=
(
F2+
θ
F3)
TeθT+
F4Ieα
T≥
0. Hencef3(
T)
is increasing onT≥
0. We see thatf3(
0)
= −
F1<
0=
f3(
T3∗) <
f3(
N)
. Sof3(
0)
f3(
N) <
0. Consequently, we are in a position to outline the algorithm to findT3∗.Algorithm 1. Step1. Let
ϵ >
0. Step2. LetTL=
0 andTU=
N.Step3. LetTopt
=
TL+2TU.Step4. If
|
f3(
Topt)
|
< ϵ
, go to step 6. Otherwise, go to step 5.Step5. Iff3
(
Topt) >
0, SetTU=
Topt. Iff3(
Topt) <
0, SetTL=
Topt. Then go to step 3.Step6. SetT∗ 3
=
Topt.Algorithm 2. Step1. Let
ϵ >
0.5.1. Huang’s model
If
D=
(
D,
D,
D)
,
A=
(
A,
A,
A)
,
h=
(
h,
h,
h)
,
c=
(
c,
c,
c)
,
s=
(
s,
s,
s)
, that is, in the crisp sense, whens=
c,θ
→
0 (it means that the deterioration rate is ignored) andα
→
0 (it means that the retailer also offers the full trade credit to his/her customer), the inventory model is identical to that of Huang [12] model. We notice that the Eqs.(19),(16),(17),(18), (20),(22),(24),(29),(30)become Eqs.(37)–(45)respectively. That is,Theorem 2can be modified as follows:
Theorem 3. (A) If
∆
¯
1>
0and∆
¯
2≥
0, then K(
T∗)
=
K3(
T6∗)
and T∗=
T ∗ 6. (B) If∆
¯
1>
0and∆
¯
2<
0, then K(
T∗)
=
K2(
T5∗)
and T∗=
T5∗.(C) If
∆
¯
1≤
0and∆
¯
2<
0, then K(
T∗)
=
K1(
T4∗)
and T∗=
T4∗.Theorem 3has been discussed in Theorem 1 of Huang [12] model. Hence Huang [12] model will be a special case of this paper.
5.2. Shah’s model
When
D=
(
D,
D,
D)
,
A=
(
A,
A,
A)
,
h=
(
h,
h,
h)
,
c=
(
c,
c,
c)
,
s=
(
s,
s,
s)
, that is, in the crisp sense, whenN=
0 (it means that the supplier would offer the retailer a delay period but the retailer would not offer the delay period to his/her customer) that is one level trade credit,α
→
0 ands=
c, then the cost function(19)reduces to,H
(
T)
=
H1
(
T)
;
ifT≥
M,
where,
Eqs.(50)and(51)are consistent with the Shah [6] model. Hence, the Shah [6] model will be a special case of this paper.
5.3. Goyal’s model
If
D=
(
D,
D,
D)
,
A=
(
A,
A,
A)
,
h=
(
h,
h,
h)
,
c=
(
c,
c,
c)
,
s=
(
s,
s,
s)
, that is, in the crisp sense, whenN=
0,s=
c,α
→
0,θ
→
0 (it means that the deterioration rate is ignored), letG1
(
T)
=
Eqs. (56)(a, b) will be consistent with Eqs. (1) and (4) in Goyal [1] model, respectively. Eq. (29) can be modified as
∆
1= −
2A+
DM2(
h+
cIe)
. If we let∆
= −
2A+
DM2(
h+
cIe)
,Theorem 2can be modified as follows:Theorem 4. (A) If
∆
>
0, then T∗=
T∗ 8. (B) If∆
<
0, then T∗=
T7∗.(C) If
∆
=
0, then T∗=
T7∗=
T8∗=
M.Theorem 4has been discussed in Theorem 1 of Chung [3] model. Hence, the Goyal [1] model will be a special case of this paper.
6. Summary
In this paper, we have developed an EOQ-based inventory model for deteriorating items to determine the optimal ordering policies of a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier and the retailer just offers partial trade credit to customers. Furthermore, the demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers.
Based on above situations, we investigate the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory policy under the supply chain management in the fuzzy sense. The annual total variable cost for the retailer in the fuzzy sense is defuzzified using the Graded Mean Integration Representation method. From the viewpoint of the costs, decision rules to find the optimal cycle timeT∗contains three cases: (i)T
≤
N(ii)N≤
T≤
Mand (iii)T≥
M. In order to obtain the optimal ordering policy, we propose two theorems and three algorithms. UsingTheorem 1, it is proved that the defuzzified annual total variable cost for the retailer is convex. Numerical examples are given to illustrate Theorem 2. Furthermore, three algorithms to find the optimal replenishment cycle time are presented. Finally, there are three special cases which are discussed in this paper.supported by the University Grants Commission (UGC), INDIA, for providing a Minor Research Project (MRP, UGC) under the research grant no. PSW — 150/09-10, SNO: 95958.
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