Analysis of a fuzzy economic order quant (3)
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Mathematical and Computer Modelling 53 (2011) 1621–1636
Contents lists available at ScienceDirect
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,1
a
Department of Mathematics, Sitananda College, P. O. + P. S. - Nandigram, Dist. - Purba Medinipur, PIN - 721631, West Bengal, India
b
Department of Commerce, Srikrishna College, P.O. + P.S. - Bagula, Dist. - Nadia, PIN - 741502, West Bengal, India
article
info
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
abstract
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.
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mcm.2010.12.028
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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 twolevel 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
in Fig. 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 about h’’, or ‘‘unit
purchase cost is approximately c 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 this view, several researchers developed fuzzy inventory models in situations where these parameters are described
imprecisely. Several authors namely Chang et al. [18], Lee and Yao [19], Lin and Yao [20], Yao et al. [21], Mahata et al. [22],
Mahata and Goswami [11] developed inventory models in the fuzzy sense by considering different parameters as fuzzy
parameters.
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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
R is a fuzzy subset of R that 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
A is a generalized fuzzy number as shown in Fig. 2 and is described as any fuzzy subset of the real line R, whose
membership function µ
A satisfies the following conditions:
1.
2.
3.
4.
5.
6.
µA (x) is continuous mapping from R to the closed interval [0, 1],
µA (x) = 0, − ∞ < x ≤ a1 ,
µA (x) = L(x) is strictly increasing on [a1 , a2 ],
µA (x) = wA , a2 ≤ x ≤ a3 ,
µA (x) = R(x) is strictly decreasing on [a3 , a4 ],
µA (x) = 0, a4 ≤ x < ∞,
where 0 < wA ≤ 1, and a1 , a2 , a3 and a4 are real numbers. Also this type of generalized fuzzy number may be denoted as
A = (a1 , a2 , a3 , a4 ; wA )LR . When wA = 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 mean h-level of 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 mean h-level value of the generalized fuzzy number
A = (a1 , a2 , a3 , a4 ; wA )LR is h(L−1 (h) +
−1
R (h))/2 (see Fig. 2). Then the Graded Mean Integration Representation of A is P (A) with grade wA , where
P (
A) =
∫
wA
h
0
(L−1 (h) + R−1 (h))
2
with 0 < h ≤ wA and 0 < wA ≤ 1.
∫
dh
0
wA
hdh,
(1)
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Fig. 2. The graded mean h-level value of generalized fuzzy number
A = (a1 , a2 , a3 , a4 ; wA )LR .
The generalized triangular fuzzy number (GTFN)
B is a special case of the generalized fuzzy number and be denoted as
B = (a, b, c ; w). Its corresponding graded mean integration representation is
w
h{a + (b − a)h/w + c − (c − b)h/w}/2dh
w
P (B) = 0
0
=
hdh
a + 4b + c
,
(2)
6
where h lies 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 mean h-level value of the fuzzy number A˜ = (a, b, c )TFN is assumed to be of the form h[β L−1 (h) + (1 − β)R−1 (h)],
where β 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 of β closer to 0 implies that the decision maker is more pessimistic while a value of β closer to 1 means that the
decision maker is more optimistic.
Therefore, the formula (1) is modified as below:
Now,
P (
A) =
w
0
L(x) = w
(h[β L−1 (h) + (1 − β)R−1 ])dh
w
.
0
x−a
b−a
and R(x) = w
Thus,
,
(3)
hdh
a≤x≤b
c−x
c−b
b ≤ x ≤ c.
,
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 of A˜ is given by
P (A˜ ) =
=
w
0
h[β a + (1 − β)c + {b − β a − (1 − β)c }h/w]
w
0
β a + 2b + (1 − β)c
hdh
.
(4)
3
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 GTFN B˜ is called a triangular fuzzy number (TFN) denoted by
B = (a, b, c )TFN .
The defuzzification of
B = (a, b, c )TFN can be found by the centroid or the graded mean integration method. The centroid
of the TFN
B = (a, b, c )TFN is C (
B) = 13 (a + b + c ) and the graded mean integration representation of
B = (a, b, c )TFN is
P (B˜ ) =
1
6
(a + 4b + c ). The mid-point of the interval [a, c ] is M =
C (B˜ ) − P (B˜ ) =
1
3
(M − b),
P (B˜ ) − b =
1
3
a+c
.
2
Thus
(M − b) and M − C (B˜ ) =
1
3
(M − b).
(5)
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Fig. 3. Case M > b.
Fig. 4. Case M < b.
From Eq. (5), we notice that
(i) If M > b, then a < b < P (B˜ ) < C (B˜ ) < M < c.
(ii) If M < b, then a < M < C (B˜ ) < P (B˜ ) < b < c.
(iii) If M = b, then a < M = C (B˜ ) = P (B˜ ) = b < c.
From (i) and (ii), it is clear that P (B˜ ) is near b, and C (B˜ ) is near M. From Figs. 3 and 4, we observe that the membership grade
of B˜ at b is 1 and the membership grade of B˜ at P (B˜ ) is greater than that at C (B˜ ). Then, we have µB˜ (P (B˜ )) > µB˜ (C (B˜ )).
Property 1. From the membership grade viewpoint, it will be efficient to defuzzify the fuzzy number B˜ = (a, b, c )TFN by 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
A and
B is
A
B = (a1 + b1 , a2 + b2 , a3 + b3 ), where a1 , a2 , a3 , b1 , b2 and b3 are any real numbers.
2. −B = (−b3 , −b2 , −b1 ), then the substraction of
A and
B is
A ⊖
B = (a1 − b3 , a2 − b2 , a3 − b1 ), where a1 , a2 , a3 , b1 , b2
and b3 are any real numbers.
3. The multiplication of
A and
B is
A
B = (c1 , c2 , c3 ), where T = {a1 b1 , a1 b4 , a4 b1 , a4 b4 }, T1 = {a2 b2 , a2 b3 , a3 b2 , a3 b3 },
c1 = min T , c2 = min T1 , c3 = max T1 , c4 = max T .
A
B = (a1 b1 , a2 b2 , a3 b3 ), where
A
B
Also, if a1 , a2 , a3 , b1 , b2 and b3 are all nonzero positive real numbers, then
is a triangular fuzzy number.
1
4.
=
B−1 = (1/b3 , 1/b2 , 1/b1 ), where b1 , b2 and b3 are all nonzero positive real numbers.
B
If a1 , a2 , a3 , b1 , b2 and b3 are all nonzero positive real numbers, then the division of
A and
B is
A ⊘
B =
(a1 /b3 , a2 /b2 , a3 /b1 ), where
A ⊘
B is a triangular fuzzy number.
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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 as Fig. 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 as Fig. 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
D is 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 as h˜ = (h − δh1 , h, h + δh2 ), A˜ = (A − δA1 , A, A + δA2 ), c˜ = (c − δc1 , c , c + δc2 ), and s˜ = (s − δs1 , s, s + δs2 ).
7. Ic : the interest charged per $ in stocks per year by the supplier; Ie : the interest earned per $ per year where Ic ≥ Ie .
8. M: the retailer’s trade credit period offered by the supplier in years and N: the customer’s trade credit period offered by
the retailer in years. It is assumed that M ≥ 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.
10. The supplier offers the full trade credit to the retailer. When T ≥ M, the account is settled at T = M, the retailer pays
off all units sold and keeps his/her profits and the retailer starts paying for the interest charges on the items in stock
with rate Ic . When T ≤ M, the account is settled at time T = M and the retailer does not pay any interest charge.
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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 rate Ie .
3.2. The mathematical model formulation and its analysis
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 both until it becomes zero at time t = T . The differential equation governing the system in the
interval (0, T ) is
dq(t )
+ θ q(t ) = −D, 0 ≤ t ≤ T ,
(6)
dt
with the boundary conditions q(0) = Q and q(T ) = 0. The solution of the differential equation (6) with the boundary
condition q(T ) = 0 is
D
q(t ) =
θ
(eθ (T −t ) − 1),
0 ≤ t ≤ T.
(7)
Using the boundary condition q(0) = Q , the order quantity can be obtained as
Q =
D
θ
(eθ T − 1).
(8)
Total demand during one cycle is DT .
The number of units deteriorated during one cycle is
Q − DT =
D
θ
(eθ T − 1 − θ T ).
The total annual cost due to deterioration of items during the cycle, denoted by DC , is
DC =
cD
θT
(eθ T − 1 − θ T ).
The total annual inventory holding cost (excluding interest charges) per cycle, denoted by HC , is given by
HC =
h
T
∫
T
q(t )dt =
0
hD
θ 2T
(eθ T − 1 − θ T ).
According to assumption (10), three cases may occur in the calculation of interest charges for the items kept in stock per
year.
Case 1. M ≤ T .
Annual interest payable =
cIc
∫
T
q(t )dt
T M
cIc D
= 2 (eθ (T −M ) − θ (T − M ) − 1).
θ T
Case 2. N ≤ T ≤ M.
In this case, annual interest payable = 0.
Case 3. T ≤ N.
Similar as case 2, annual interest payable = 0.
According to assumption (11), three cases will occur in interest earned per year.
Case 1. M ≤ T , (shown in Fig. 7)
Annual interest earned =
=
sIe
T
[∫
2T
Case 2. N ≤ T ≤ M, (shown in Fig. 8)
Annual interest earned =
=
T
∫
M
Dtdt
N
]
2
[M − (1 − α)N 2 ].
[∫
sIe D
2T
α Dtdt +
0
sIe D
sIe
N
0
N
α Dtdt +
∫
N
T
]
Dtdt + DT (M − T )
[2MT − (1 − α)N 2 − T 2 ].
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sDT
α sDN
Time
0
N
M
T
Fig. 7. Total amount of interest earned when M ≤ T .
sDT
α sDN
Time
0
N
T
M
Fig. 8. Total amount of interest earned when N ≤ T ≤ M.
Case 3. T ≤ N, (shown in Fig. 9)
]
α Dtdt + α DT (N − T ) + DT (M − N )
T
0
]
[
αT
.
= sIe D M − (1 − α)N −
Annual interest earned =
sIe
[∫
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 );
TVC2 (T );
TVC3 (T );
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N ,
(9)
where
(h + θ c )D θ T
cIc D
sIe D
(e − 1 − θ T ) + 2 [eθ (T −M ) − θ (T − M ) − 1] −
[M 2 − (1 − α)N 2 ],
θ 2T
θ T
2T
(h + θ c )D θ T
sIe D
A
+
(e − 1 − θ T ) −
[2MT − (1 − α)N 2 − T 2 ]
TVC2 (T ) =
2
T
θ T
2T
]
[
A
(h + θ c )D θ T
αT
and TVC3 (T ) =
.
+
(e − 1 − θ T ) − sIe D M − (1 − α)N −
T
θ 2T
2
TVC1 (T ) =
A
T
+
(10)
(11)
(12)
Now, we develop the above retailer’s inventory costs in a fuzzy environment. In the present day scenario, it is very difficult
or simply it is not possible to determine a precise value of D. In this case, the retailer needs to collect the demand information
from customers. When the customers’ opinions are vague or linguistic like ‘‘demand is about D’’, the average annual demand
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G.C. Mahata, P. Mahata / Mathematical and Computer Modelling 53 (2011) 1621–1636
1629
sDT
α sDT
Time
0
T
N
M
Fig. 9. Total amount of interest earned when T ≤ N.
can be vaguely expressed. So, it is more suitable to describe it as fuzzy in nature. Therefore, the total annual demand is treated
˜ = (D −δD1 , D, D +δD2 ). Also,
as a fuzzy variable. Here we consider the fuzzy annual demand D as a triangular fuzzy number D
inventory costs are normally assumed to be constant but this is not always true. In a perfect competitive market, ordering
cost A, unit holding cost h, unit purchasing cost c, unit selling price s etc. per day in a plan period T may fluctuate a little. For
example, ‘‘unit holding cost is around h’’, ‘‘unit selling price is about s’’, etc. Suppose these cost parameters ordering cost,
holding cost, purchase cost, selling price lie in the intervals [A −δA1 , A +δA2 ], [h −δh1 , h +δh2 ], [c −δc1 , c +δc2 ], [s −δs1 , s +δs2 ].
Similarly, corresponding to these intervals, we set the following triangular fuzzy numbers: A˜ = (A − δA1 , A, A + δA2 ),
h˜ = (h − δh1 , h, h + δh2 ), c˜ = (c − δc1 , c , c + δc2 ) and s˜ = (s − δs1 , s, s + δs2 ). 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 any T > 0, we get fuzzy annual total variable costs
1 (T ) = X11 ⊗
TVC
A ⊕ X12 ⊗
h ⊗
D ⊕ X13 ⊗
c ⊗
D ⊖ X14 ⊗
s ⊗
D
2 (T ) = X21 ⊗
TVC
A ⊕ X22 ⊗
h ⊗
D ⊕ X23 ⊗
c ⊗
D ⊖ X24 ⊗
s ⊗
D
3 (T ) = X31 ⊗
and TVC
A ⊕ X32 ⊗
h ⊗
D ⊕ X33 ⊗
c ⊗
D ⊖ X34 ⊗
s ⊗
D,
(13)
(14)
(15)
where ⊗, ⊕ and ⊖ are the fuzzy arithmetical operations under the Function Principle and
X13 =
eθ T
1
1
(eθ T − 1 − θ T ),
θ 2T
Ic (eθ (T −M ) − θ (T − M ) − 1)
eθ T − 1 − θ T
− 1 − θT
+
,
X
=
X
=
,
23
33
θT
θ 2T
θT
X11 = X21 = X31 =
T
,
X12 = X22 = X32 =
Ie
Ie
[M 2 − (1 − α)N 2 ],
X24 =
[2MT − (1 − α)N 2 − T 2 ]
2T
2T
]
[
αT
.
and X34 = Ie M − (1 − α)N −
X14 = −
2
˜ = (D1 , D, D2 ), A˜ = (A1 , A, A2 ), h˜ = (h1 , h, h2 ), c˜ = (c1 , c , c2 ), and s˜ = (s1 , s, s2 ) are nonnegative
Here we assume that D
triangular fuzzy numbers, where D1 = D − δD1 , D2 = D + δD2 , A1 = A − δA1 , A2 = A + δA2 , h1 = h − δh1 , h2 = h + δh2 ,
1 (T ) by Eq. (13) as
c1 = c − δc1 , c2 = c + δc2 , s1 = s − δs1 , s2 = s + δs2 . Then we get the fuzzy annual total variable cost TVC
1 (T ) = [X11 A1 + X12 h1 D1 + X13 c1 D1 − X14 s2 D2 , X11 A + X12 hD + X13 cD − X14 sD,
TVC
X11 A2 + X12 h2 D2 + X13 c2 D2 − X14 s1 D1 ].
1 (T ) by formula (2) and obtain the graded mean integration
We defuzzify the fuzzy annual total variable cost TVC
representation of TVC 1 (T ) as
1 (T )) =
P (TVC
1
6
[(X11 A1 + X12 h1 D1 + X13 c1 D1 − X14 s2 D2 ) + 4(X11 A + X12 hD + X13 cD − X14 sD)
+ (X11 A2 + X12 h2 D2 + X13 c2 D2 − X14 s1 D1 )]
= X11 F1 + X12 F2 + X13 F3 − X14 F4
F1
F2 + θ F3 θ T
F3 Ic
F4 Ie
=
+
(e − 1 − θ T ) + 2 (eθ (T −M ) − θ (T − M ) − 1) −
[M 2 − (1 − α)N 2 ],
2
T
θ T
θ T
2T
where F1 =
A1 +4A+A2
,
6
F2 =
h1 D1 +4hD+h2 D2
,
6
F3 =
c1 D1 +4cD+c2 D2
6
and F4 =
s1 D1 +4sD+s2 D2
.
6
(16)
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2 (T ) and TVC
3 (T ) are respectively,
Similarly, the graded mean integration representation of TVC
2 (T )) =
P (TVC
F1
F2 + θ F3
F4 Ie
[2MT − (1 − α)N 2 − T 2 ],
θ
2T
]
[
3 (T )) = F1 + F2 + θ F3 (eθ T − 1 − θ T ) − F4 Ie M − (1 − α)N − α T .
P (TVC
T
θ 2T
2
T
+
2T
(eθ T − 1 − θ T ) −
(17)
(18)
Thus the annual total cost in the fuzzy sense based on the graded mean integration representation is
1 (T ));
P (TVC
(T )) = P (TVC
2 (T ));
P (TVC
P (TVC 3 (T ));
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N ,
(19)
1 (M )) = P (TVC
2 (M )) and P (TVC
2 (N )) =
1 (T )), P (TVC
2 (T )) and P (TVC
3 (T )) are given by (16)–(18). Since P (TVC
where P (TVC
3 (N )), P (TVC
(T )) is continuous and well-defined on T > 0.
P (TVC
Then, we have
1 (T ))′ = −
P (TVC
F1
F2 + θ F3
F3 Ic
(θ T eθ (T −M ) − eθ (T −M ) + 1 − θ M )
θ
θ 2T 2
F4 Ie
+ 2 [M 2 − (1 − α)N 2 ],
2T
2F1
1 2 2 θT
2(F2 + θ F3 )
′′
θT
θT
P (TVC 1 (T )) =
e − θTe − 1 + θ T e
+
2 3
T3
2
θ T
F4 Ie
1
2F3 Ic
θ ( T −M )
θ (T −M )
− θTe
− 1 + θ M + θ 2 T 2 eθ (T −M ) − 3 [M 2 − (1 − α)N 2 ]
+ 2 3 e
θ T
2
T
2 (T ))′ = −
P (TVC
T2
F1
+
2T 2
(θ T eθ T − eθ T + 1) +
F 2 + θ F3
F4 Ie
(θ T eθ T − eθ T + 1) + 2 [T 2 − (1 − α)N 2 ]
θ
2T
2F1
F4 Ie (1 − α)N 2
1 2 2 θT
2(F2 + θ F3 )
′′
θT
θT
P (TVC 2 (T )) = 3 +
+
e
−
θ
T
e
−
1
+
θ
T
e
T
θ 2T 3
2
T3
T2
+
F1
T2
+
(21)
(22)
2T 2
(23)
F 2 + θ F3
F4 Ie α
(θ T eθ T − eθ T + 1) +
θ 2T 2
2
1 2 2 θT
2(F2 + θ F3 )
2F1
θT
θT
′′
e − θTe − 1 + θ T e
P (TVC 3 (T )) = 3 +
T
θ 2T 3
2
3 (T ))′ = −
P (TVC
(20)
(24)
(25)
where (′ ) represents differentiation with respect to T .
3.2.1. The convexity
1 (T )), P (TVC
2 (T )) and P (TVC
3 (T )) are convex on their appropriate domains.
In this section, we shall show that P (TVC
1 (T )) is convex on [M , ∞).
Theorem 1. (1) P (TVC
(2) P (TVC 2 (T )) is convex on [0, ∞).
3 (T )) is convex on [0, ∞).
(3) P (TVC
Before proving Theorem 1, we need the following lemmas.
Lemma 1. eθ T − θ T eθ T − 1 + 12 θ 2 T 2 eθ T > 0 for all T > 0.
Proof. Let f (x) = ex − 1 − xex + 21 x2 ex if x > 0, then f ′ (x) =
x
x
f (x) > f (0) = 0 if x > 0. We have f (x) = e − 1 − xe +
if T > 0. This completes the proof.
1 2 x
x e
2
1 2 x
x e
2
> 0. Hence f (x) is increasing for all x > 0. Consequently,
> 0 if x > 0. Let x = θ T . Then eθ T −θ T eθ T − 1 + 21 θ 2 T 2 eθ T > 0
2 2
Lemma 2. If T ≥ M, then eθ (T −M ) − 1 − θ T eθ (T −M ) + θ M + 12 θ 2 T 2 eθ (T −M ) − θ 2M ≥ 0.
2
2
Proof. Let g (T ) = eθ (T −M ) − 1 − θ T eθ (T −M ) + θ M + 12 θ 2 T 2 eθ (T −M ) − θ 2M . Then we have g ′ (T ) =
g (T ) is increasing on [M , ∞). So g (T ) > g (M ) = 0 if T > M. This completes the proof.
1
2
θ 3 T 2 eθ (T −M ) > 0. Hence
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The proof of Theorem 1. (1) From Eq. (21), we have
2F1
1 (T )) =
P (TVC
′′
T3
2F3 Ic
+
θ 2T 3
2F1
>
+
T3
+
θ 2T 3
− θTe
θT
1
2
2
e
θ (T −M )
1
eθ T − θ T eθ T − 1 +
2T 3
θ ( T −M )
− θTe
2 θT
−1+ θ T e
eθ (T −M ) − θ T eθ (T −M ) − 1 + θ M +
θ
θT
e
θ 2T 3
2(F2 + θ F3 )
2F3 Ic
+
2(F2 + θ F3 )
2
F4 Ie
θ 2 T 2 eθ (T −M ) − 3 [M 2 − (1 − α)N 2 ]
2
T
1
θ 2 T 2 eθ T
1
2
2 θ ( T −M )
− 1 + θM + θ T e
2
−
θ 2M 2
2
+
F4 Ie (1 − α)N 2
T3
.
(26)
1 (T ))′′ > 0 if T ≥ M. Hence P (TVC
1 (T )) is convex on [M , ∞).
Lemmas 1 and 2 imply that P (TVC
(2) From Eq. (23), we have
F4 Ie (1 − α)N 2
1 2 2 θT
2(F2 + θ F3 )
2F1
θT
θT
′′
+
e
−
θ
T
e
−
1
+
θ
T
e
.
P (TVC 2 (T )) = 3 +
T
θ 2T 3
2
T3
2 (T ))′′ > 0 if T > 0. Consequently, P (TVC
2 (T )) is convex on [0, ∞).
Lemma 1 implies that P (TVC
(3) From Eq. (25), we have
2F1
3 (T )) =
P (TVC
′′
+
T3
2(F2 + θ F3 )
θ 2T 3
e
θT
θT
− θTe
1
2
2 θT
−1+ θ T e
2
.
3 (T ))′′ > 0 if T > 0. Consequently, P (TVC
3 (T )) is convex on [0, ∞).
From Lemma 1, P (TVC
This complete the proof of Theorem 1.
From Eqs. (20), (22) and (24), we have
1 (M ))′ = P (TVC
2 (M ))′
P (TVC
=−
F1
F2 + θ F3
+
M2
θ
2M 2
(θ Meθ M − eθ M + 1) +
F4 Ie
2M 2
[M 2 − (1 − α)N 2 ]
(27)
and
2 (N ))′ = P (TVC
3 (N ))′
P (TVC
=−
F1
N2
+
F2 + θ F3
θ
2N 2
(θ Neθ N − eθ N + 1) +
F4 Ie α
.
(28)
[M 2 − (1 − α)N 2 ]
(29)
.
(30)
2
Furthermore, we let
△1 = −
F1
M2
+
F2 + θ F3
θ
2M 2
(θ Meθ M − eθ M + 1) +
F4 Ie
2M 2
and
△2 = −
F1
N2
+
F2 + θ F3
θ
2N 2
(θ Neθ N − eθ N + 1) +
F 4 Ie α
2
Therefore,
△1 − △2 = F1
Let, K (x) =
xex −ex +1
x2
1
N2
−
1
M2
+
F4 Ie (1 − α)
2M 2
, so that by Lemma 1, k′ (x) =
θ Meθ M − eθ M + 1 θ Neθ N − eθ N + 1
.
(M − N ) + (F2 + θ F3 )
−
θ 2M 2
θ 2N 2
2
2
x3
2
(31)
(ex − 1 − xex + 12 x2 ex ) > 0 if x > 0. Hence k(x) is increasing if x > 0.
θM
θM
θN
θN
Consequently, k(x) ≥ k(y) if x > y. So that, k(θ M ) > k(θ N ). This gives θ Me θ 2−Me2 +1 ≥ θ Ne θ 2−Ne2 +1 . From (31), we have
∆1 ≥ ∆2 .
Consider the following equations:
1 (T ))′ = 0,
P (TVC
2 (T )) = 0,
P (TVC
′
3 (T ))′ = 0.
P (TVC
(32)
(33)
(34)
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If the root of each of Eqs. (32), (33) or (34) exists, then it is unique. Let T1∗ denote the root of Eq. (32), T2∗ denote the root of
i (T )) (i = 1, 2, 3), we see that
Eq. (33) and T3∗ denote the root of Eq. (34). By the convexity of P (TVC
< 0;
i (T )) = 0;
P (TVC
> 0;
′
if T < Ti∗ ,
if T = Ti∗ ,
if T > Ti∗ .
(35)
i (T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞) for all i = 1, 2, 3.
Eq. (35) implies that P (TVC
i
i
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.
(T ∗ )) = P (TVC
3 (T ∗ )) and T ∗ = T ∗ .
Theorem 2. (A) If ∆1 > 0 and ∆2 ≥ 0, then P (TVC
3
3
∗
∗
(B) If ∆1 > 0 and ∆2 < 0, then P (TVC (T )) = P (TVC 2 (T2 )) and T ∗ = T2∗ .
(T ∗ )) = P (TVC
1 (T ∗ )) and T ∗ = T ∗ .
(C) If ∆1 ≤ 0 and ∆2 < 0, then P (TVC
1
1
1 (M ))′ = P (TVC
2 (M ))′ > 0
Proof. (A) If ∆1 > 0 and ∆2 ≥ 0, then T1∗ < M, T2∗ < M, T3∗ ≤ N and T2∗ ≤ N. We have P (TVC
′
′
2 (N )) = P (TVC
3 (N )) ≥ 0. Eq. (35) imply that
and P (TVC
1 (T )) is increasing on [M , ∞),
(i) P (TVC
2 (T )) is increasing on [N , M ] and
(ii) P (TVC
(iii) P (TVC 3 (T )) is decreasing on (0, T3∗ ] and is increasing [T3∗ , N ].
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
3
3
∗
∗
∗
∗
(T )) = P (TVC
3 (T )).
Consequently, T = T3 and P (TVC
3
1 (M ))′ = P (TVC
2 (M ))′ > 0 and
(B) If ∆1 > 0 and ∆2 < 0, then T3∗ > N, T2∗ > N, T1∗ < M and T2∗ < M. We have P (TVC
′
′
P (TVC 2 (N )) = P (TVC 3 (N )) < 0. Eq. (35) imply that
1 (T )) is increasing on [M , ∞),
(i) P (TVC
2 (T )) is decreasing on [N , T ∗ ] and increasing on [T ∗ , M ],
(ii) P (TVC
2
2
3 (T )) is decreasing on (0, N ].
(iii) P (TVC
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
2
2
∗
∗
∗
∗
(T )) = P (TVC
2 (T )).
Consequently, T = T2 and P (TVC
2
1 (M ))′ = P (TVC
2 (M ))′ ≤ 0 and
(C) If ∆1 ≤ 0 and ∆2 < 0, then T3∗ > N, T2∗ > N, T1∗ ≥ M and T2∗ ≥ M. We have P (TVC
′
′
P (TVC 2 (N )) = P (TVC 3 (N )) < 0. Eq. (35) imply that
1 (T )) is decreasing on [M , T ∗ ] and increasing on [T ∗ , ∞),
(i) P (TVC
1
1
(ii) P (TVC 2 (T )) is decreasing on [N , M ] and
3 (T )) is decreasing on (0, N ].
(iii) P (TVC
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
1
1
(T ∗ )) = P (TVC
1 (T ∗ )).
Consequently, T ∗ = T1∗ and P (TVC
1
3.2.3. The algorithms for the determination of T ∗ , T1∗ , T2∗ and T3∗
We first describe the following theorem:
Intermediate value theorem: Let f (x) be a continuous function on [a, b] and f (a).f (b) < 0, then there exists a number
c ∈ (a, b) such that f (c ) = 0.
(i) Suppose that ∆1 > 0 and ∆2 ≥ 0, then T3∗ exists, T ∗ = T3∗ and 0 < T3∗ < N. Recall T3∗ to denote the unique root of the
F +θ F
F I α
Eq. (34). Let f3 (T ) = −F1 + 2 θ 2 3 (θ T eθ T − eθ T + 1) + 42e T 2 . Then f3′ (T ) = (F2 + θ F3 )T eθ T + F4 Ie α T ≥ 0. Hence f3 (T ) is
increasing on T ≥ 0. We see that f3 (0) = −F1 < 0 = f3 (T3∗ ) < f3 (N ). So f3 (0)f3 (N ) < 0. Consequently, we are in a position
to outline the algorithm to find T3∗ .
Algorithm 1. Step 1. Let ϵ > 0.
Step 2. Let TL = 0 and TU = N.
Step 3. Let Topt =
TL +TU
2
.
Step 4. If |f3 (Topt )| < ϵ , go to step 6. Otherwise, go to step 5.
Step 5. If f3 (Topt ) > 0, Set TU = Topt . If f3 (Topt ) < 0, Set TL = Topt . Then go to step 3.
Step 6. Set T3∗ = Topt .
(ii) Suppose that ∆1 > 0 and ∆2 < 0, then T2∗ exists, T ∗ = T2∗ and N < T2∗ < M. T2∗ denotes the root of the Eq. (33). We have
2 (N ))′ < 0 = P (TVC
2 (T ∗ ))′ < P (TVC
2 (T ))′ is increasing on [N , M ].
2 (M ))′ . By (23) and Lemma 1, we see that P (TVC
P (TVC
2
′
′
Hence P (TVC 2 (N )) P (TVC 2 (M )) < 0. Consequently, we are in a position to outline the algorithm to find T2∗ .
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Algorithm 2. Step 1. Let ϵ > 0.
Step 2. Let TL = N and TU = M.
Step 3. Let Topt =
TL +TU
2
.
2 (Topt ))′ | < ϵ , go to step 6. Otherwise, go to step 5.
Step 4. If |P (TVC
2 (Topt ))′ > 0, Set TU = Topt . If P (TVC
2 (Topt ))′ < 0, Set TL = Topt . Then go to step 3.
Step 5. If P (TVC
Step 6. Set T2∗ = Topt .
(iii) Suppose that ∆1 ≤ 0 and ∆2 < 0, then T1∗ exists, T ∗ = T1∗ . T1∗ denotes the root of the Eq. (32). Let
T1U =
2(F2 + θ F3 ) + 2F3 Ic (e−θ M + θ M ) + θ 2 F4 Ie (1 − α)N 2 + 2F1 θ 2
2θ{(F1 + θ F3 ) + F3 Ic e−θ M }
(36)
.
Then we have the following lemma:
Lemma 3. Suppose that ∆1 ≤ 0 and ∆2 < 0, then T1U > T1∗ > M, where T1U is given by (36).
Proof. Since 0 < e−θ T < 1,
1 (T ))′ =
P (TVC
F2 + θ F3
θ
+
2T 2
F4 Ie
2T 2
(θ T eθ T − eθ T + 1) +
[M 2 − (1 − α)N 2 ] −
F3 Ic
θ 2T 2
(θ T eθ (T −M ) − eθ (T −M ) + 1 − θ M )
F1
T2
F3 Ic
F2 + θ F3 θ T
e (θ T − 1 + e−θ T ) + 2 2 eθ T (θ T e−θ M − e−θ M + e−θ T − θ Me−θ T )
=
2
2
θ T
θ T
[
]
2
F
I
M
F1
F
I
(
1
−
α)N 2
4 e
4 e
−θ T
−θ T
+ eθ T
e
e
−
+
2T 2
2T 2
T2
eθ T
[
F2 + θ F3
F3 Ic
−θ M
F4 Ie (1 − α)N 2
]
(θ T − 1) + 2 {(θ T − 1)e
− θ M} −
+ F1
θ2
θ
2
[
]
eθ T
F2 + θ F3
F2 + θ F3
F3 Ic −θ M
F3 Ic −θ M
F4 Ie (1 − α)N 2
= 2
θ
T
−
+
e
+
(
e
+
θ
M
)
+
+
F
.
1
T
θ2
θ2
θ2
θ2
2
>
T2
1 (T U ))′ > 0 = P (TVC
1 (T ∗ ))′ . Since P (TVC
1 (T ))′ is increasing on [M , ∞) and P (TVC
1 (T U ))′ > P (TVC
1 (T ∗ ))′ >
Hence P (TVC
1
1
U
1 (M ))′ , we obtain T > T ∗ > M. Consequently, we have completed the proof. By Lemma 3, we are in a position to
P (TVC
1
1
outline the algorithm to find T1∗ .
Algorithm 3. Step 1. Let ϵ > 0.
Step 2. Let TL = M and TU = T1U .
Step 3. Let Topt =
TL +TU
2
.
1 (Topt ))′ | < ϵ , go to step 6. Otherwise, go to step 5.
Step 4. If |P (TVC
1 (Topt ))′ > 0, Set TU = Topt . If P (TVC
1 (Topt ))′ < 0, Set TL = Topt . Then go to step 3.
Step 5. If P (TVC
Step 6. Set T1∗ = Topt .
4. Numerical examples
To illustrate the results, let us apply the proposed method to solve the following numerical examples. The following
parameters
D = (2500, 3000, 3500) units/year,
A = $(120, 130, 140)/order,
h = $(5, 7, 9)/unit/year,
c =
$(10, 15, 20)/unit, Ic = $0.15/$/year, Ie = $0.12/$/year, m = 0.1 year are used from Examples 1, 2, 3.
Example 1. If N = 0.08 year,
s = $(15, 20, 25)/unit, α = 0.1, θ = 0.02, then ∆1 = −312.5555 < 0, ∆2 = −8810.626 <
(T ∗ )) = 2258.316.
0. Using Algorithm 3, we get T ∗ = T1∗ = 0.1010656, Q ∗ = 303.5034 and P (TRC
Example 2. If N = 0.05 year,
s = $(25, 30, 35)/unit, α = 0.5, θ = 0.05, then ∆1 = 2045.699 > 0, ∆2 = 38342.79 < 0.
(T ∗ )) = 2156.621.
Using Algorithm 2, we get T ∗ = T2∗ = 0.09317759, Q ∗ = 280.1849 and P (TRC
Example 3. If N = 0.08 year,
s = $(45, 50, 55)/unit, α = 0.9, θ = 0.08, then ∆1 = 8037.667 > 0, ∆2 = 385.9616 > 0.
(T ∗ )) = 1567.746.
Using Algorithm 1, we get T ∗ = T3∗ = 0.07058884, Q ∗ = 212.3656 and P (TRC
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5. Special cases
In this section, we obtain some previously published results of other authors as special cases.
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, when s = c, θ → 0
(it means that the deterioration rate is ignored) and α → 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,
K (T ) =
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N .
K1 (T );
K2 (T );
K3 (T );
(37)
where,
K1 (T ) =
K2 (T ) =
K3 (T ) =
A
T
A
T
A
T
K1′ (T ) = −
′
K2 (T ) = −
K3′ (T ) = −
+
+
+
DTh
2
DTh
2
DTh
2
+
−
cIc D(T − M )2
2T
cIe D(M 2 − N 2 )
−
cIe D(2MT − N 2 − T 2 )
− cIe D(M − N )
(40)
2T 2
2A + cDN 2 Ie
2T 2
T2
+
(39)
2T
2A + cDM 2 (Ic − Ie ) + cDN 2 Ie
A
(38)
2T
+D
h + cIe
2
+D
h + cIc
2
(41)
(42)
Dh
(43)
2
¯ 1 = −2A + DM 2 (h + cIe ) − cDN 2 Ie
∆
(44)
2
¯ 2 = −2A + DN h.
∆
∗
∗
∗
(45)
′
′
′
Let, T4 , T5 and T6 be the roots of equations K1 (T ) = 0, K2 (T ) = 0 and K3 (T ) = 0 respectively, where
T4∗ =
T5∗ =
∗
T6 =
2A + cD[M 2 (Ic − Ie ) + N 2 Ie ]
D(h + cIc )
2A + cDN 2 Ie
D(h + cIe )
2A
.
Dh
Theorem 2 can be modified as follows:
(46)
(47)
(48)
¯ 1 > 0 and ∆
¯ 2 ≥ 0, then K (T ∗ ) = K3 (T6∗ ) and T ∗ = T6∗ .
Theorem 3. (A) If ∆
¯
¯
(B) If ∆1 > 0 and ∆2 < 0, then K (T ∗ ) = K2 (T5∗ ) and T ∗ = T5∗ .
¯ 1 ≤ 0 and ∆
¯ 2 < 0, then K (T ∗ ) = K1 (T4∗ ) and T ∗ = T4∗ .
(C) If ∆
Theorem 3 has 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, when N = 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 and s = c, then the cost function (19) reduces to,
H (T ) =
H1 (T );
H2 (T );
if T ≥ M ,
if 0 ≤ T ≤ M ,
(49)
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G.C. Mahata, P. Mahata / Mathematical and Computer Modelling 53 (2011) 1621–1636
1635
where,
(h + c θ )D θ T
cIc D θ (T −M )
cIe DM 2
(
e
−
1
−
θ
T
)
+
(
e
−
θ
(
T
−
M
)
−
1
)
−
,
T
θ 2T
θ 2T
2T
(h + c θ )D θ T
A
+
(e − 1 − θ T ) − cIe D(M − T /2).
H2 (T ) =
T
θ 2T
H1 (T ) =
A
+
(50)
(51)
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, when N = 0, s = c,
α → 0, θ → 0 (it means that the deterioration rate is ignored), let
G1 (T ) =
G2 (T ) =
∗
T7 =
A
T
A
T
+
+
DTh
2T
DTh
2T
+
cIc D(T − M )2
2T
−
cIe DM 2
2T
,
− cIe D(M − T /2),
2A + DM 2 c (Ic − Ie )
D(h + cIc )
(52)
(53)
(54)
and
∗
T8 =
2A
D(h + cIe )
.
(55)
Then G′i (Ti∗ ) = 0 for i = 7, 8. Eq. (19) will be reduced as follows:
G(T ) =
G1 (T );
G2 (T );
if T ≥ M , (a)
if 0 ≤ T ≤ M . (b)
(56)
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 + DM 2 (h + cIe). If we let ∆ = −2A + DM 2 (h + cIe), Theorem 2 can be modified as follows:
Theorem 4. (A) If ∆ > 0, then T ∗ = T8∗ .
(B) If ∆ < 0, then T ∗ = T7∗ .
(C) If ∆ = 0, then T ∗ = T7∗ = T8∗ = M.
Theorem 4 has 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 off
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Mathematical and Computer Modelling 53 (2011) 1621–1636
Contents lists available at ScienceDirect
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,1
a
Department of Mathematics, Sitananda College, P. O. + P. S. - Nandigram, Dist. - Purba Medinipur, PIN - 721631, West Bengal, India
b
Department of Commerce, Srikrishna College, P.O. + P.S. - Bagula, Dist. - Nadia, PIN - 741502, West Bengal, India
article
info
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
abstract
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.
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mcm.2010.12.028
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G.C. Mahata, P. Mahata / Mathematical and Computer Modelling 53 (2011) 1621–1636
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 twolevel 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
in Fig. 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 about h’’, or ‘‘unit
purchase cost is approximately c 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 this view, several researchers developed fuzzy inventory models in situations where these parameters are described
imprecisely. Several authors namely Chang et al. [18], Lee and Yao [19], Lin and Yao [20], Yao et al. [21], Mahata et al. [22],
Mahata and Goswami [11] developed inventory models in the fuzzy sense by considering different parameters as fuzzy
parameters.
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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
R is a fuzzy subset of R that 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
A is a generalized fuzzy number as shown in Fig. 2 and is described as any fuzzy subset of the real line R, whose
membership function µ
A satisfies the following conditions:
1.
2.
3.
4.
5.
6.
µA (x) is continuous mapping from R to the closed interval [0, 1],
µA (x) = 0, − ∞ < x ≤ a1 ,
µA (x) = L(x) is strictly increasing on [a1 , a2 ],
µA (x) = wA , a2 ≤ x ≤ a3 ,
µA (x) = R(x) is strictly decreasing on [a3 , a4 ],
µA (x) = 0, a4 ≤ x < ∞,
where 0 < wA ≤ 1, and a1 , a2 , a3 and a4 are real numbers. Also this type of generalized fuzzy number may be denoted as
A = (a1 , a2 , a3 , a4 ; wA )LR . When wA = 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 mean h-level of 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 mean h-level value of the generalized fuzzy number
A = (a1 , a2 , a3 , a4 ; wA )LR is h(L−1 (h) +
−1
R (h))/2 (see Fig. 2). Then the Graded Mean Integration Representation of A is P (A) with grade wA , where
P (
A) =
∫
wA
h
0
(L−1 (h) + R−1 (h))
2
with 0 < h ≤ wA and 0 < wA ≤ 1.
∫
dh
0
wA
hdh,
(1)
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Fig. 2. The graded mean h-level value of generalized fuzzy number
A = (a1 , a2 , a3 , a4 ; wA )LR .
The generalized triangular fuzzy number (GTFN)
B is a special case of the generalized fuzzy number and be denoted as
B = (a, b, c ; w). Its corresponding graded mean integration representation is
w
h{a + (b − a)h/w + c − (c − b)h/w}/2dh
w
P (B) = 0
0
=
hdh
a + 4b + c
,
(2)
6
where h lies 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 mean h-level value of the fuzzy number A˜ = (a, b, c )TFN is assumed to be of the form h[β L−1 (h) + (1 − β)R−1 (h)],
where β 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 of β closer to 0 implies that the decision maker is more pessimistic while a value of β closer to 1 means that the
decision maker is more optimistic.
Therefore, the formula (1) is modified as below:
Now,
P (
A) =
w
0
L(x) = w
(h[β L−1 (h) + (1 − β)R−1 ])dh
w
.
0
x−a
b−a
and R(x) = w
Thus,
,
(3)
hdh
a≤x≤b
c−x
c−b
b ≤ x ≤ c.
,
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 of A˜ is given by
P (A˜ ) =
=
w
0
h[β a + (1 − β)c + {b − β a − (1 − β)c }h/w]
w
0
β a + 2b + (1 − β)c
hdh
.
(4)
3
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 GTFN B˜ is called a triangular fuzzy number (TFN) denoted by
B = (a, b, c )TFN .
The defuzzification of
B = (a, b, c )TFN can be found by the centroid or the graded mean integration method. The centroid
of the TFN
B = (a, b, c )TFN is C (
B) = 13 (a + b + c ) and the graded mean integration representation of
B = (a, b, c )TFN is
P (B˜ ) =
1
6
(a + 4b + c ). The mid-point of the interval [a, c ] is M =
C (B˜ ) − P (B˜ ) =
1
3
(M − b),
P (B˜ ) − b =
1
3
a+c
.
2
Thus
(M − b) and M − C (B˜ ) =
1
3
(M − b).
(5)
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1625
Fig. 3. Case M > b.
Fig. 4. Case M < b.
From Eq. (5), we notice that
(i) If M > b, then a < b < P (B˜ ) < C (B˜ ) < M < c.
(ii) If M < b, then a < M < C (B˜ ) < P (B˜ ) < b < c.
(iii) If M = b, then a < M = C (B˜ ) = P (B˜ ) = b < c.
From (i) and (ii), it is clear that P (B˜ ) is near b, and C (B˜ ) is near M. From Figs. 3 and 4, we observe that the membership grade
of B˜ at b is 1 and the membership grade of B˜ at P (B˜ ) is greater than that at C (B˜ ). Then, we have µB˜ (P (B˜ )) > µB˜ (C (B˜ )).
Property 1. From the membership grade viewpoint, it will be efficient to defuzzify the fuzzy number B˜ = (a, b, c )TFN by 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
A and
B is
A
B = (a1 + b1 , a2 + b2 , a3 + b3 ), where a1 , a2 , a3 , b1 , b2 and b3 are any real numbers.
2. −B = (−b3 , −b2 , −b1 ), then the substraction of
A and
B is
A ⊖
B = (a1 − b3 , a2 − b2 , a3 − b1 ), where a1 , a2 , a3 , b1 , b2
and b3 are any real numbers.
3. The multiplication of
A and
B is
A
B = (c1 , c2 , c3 ), where T = {a1 b1 , a1 b4 , a4 b1 , a4 b4 }, T1 = {a2 b2 , a2 b3 , a3 b2 , a3 b3 },
c1 = min T , c2 = min T1 , c3 = max T1 , c4 = max T .
A
B = (a1 b1 , a2 b2 , a3 b3 ), where
A
B
Also, if a1 , a2 , a3 , b1 , b2 and b3 are all nonzero positive real numbers, then
is a triangular fuzzy number.
1
4.
=
B−1 = (1/b3 , 1/b2 , 1/b1 ), where b1 , b2 and b3 are all nonzero positive real numbers.
B
If a1 , a2 , a3 , b1 , b2 and b3 are all nonzero positive real numbers, then the division of
A and
B is
A ⊘
B =
(a1 /b3 , a2 /b2 , a3 /b1 ), where
A ⊘
B is a triangular fuzzy number.
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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 as Fig. 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 as Fig. 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
D is 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 as h˜ = (h − δh1 , h, h + δh2 ), A˜ = (A − δA1 , A, A + δA2 ), c˜ = (c − δc1 , c , c + δc2 ), and s˜ = (s − δs1 , s, s + δs2 ).
7. Ic : the interest charged per $ in stocks per year by the supplier; Ie : the interest earned per $ per year where Ic ≥ Ie .
8. M: the retailer’s trade credit period offered by the supplier in years and N: the customer’s trade credit period offered by
the retailer in years. It is assumed that M ≥ 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.
10. The supplier offers the full trade credit to the retailer. When T ≥ M, the account is settled at T = M, the retailer pays
off all units sold and keeps his/her profits and the retailer starts paying for the interest charges on the items in stock
with rate Ic . When T ≤ M, the account is settled at time T = M and the retailer does not pay any interest charge.
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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 rate Ie .
3.2. The mathematical model formulation and its analysis
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 both until it becomes zero at time t = T . The differential equation governing the system in the
interval (0, T ) is
dq(t )
+ θ q(t ) = −D, 0 ≤ t ≤ T ,
(6)
dt
with the boundary conditions q(0) = Q and q(T ) = 0. The solution of the differential equation (6) with the boundary
condition q(T ) = 0 is
D
q(t ) =
θ
(eθ (T −t ) − 1),
0 ≤ t ≤ T.
(7)
Using the boundary condition q(0) = Q , the order quantity can be obtained as
Q =
D
θ
(eθ T − 1).
(8)
Total demand during one cycle is DT .
The number of units deteriorated during one cycle is
Q − DT =
D
θ
(eθ T − 1 − θ T ).
The total annual cost due to deterioration of items during the cycle, denoted by DC , is
DC =
cD
θT
(eθ T − 1 − θ T ).
The total annual inventory holding cost (excluding interest charges) per cycle, denoted by HC , is given by
HC =
h
T
∫
T
q(t )dt =
0
hD
θ 2T
(eθ T − 1 − θ T ).
According to assumption (10), three cases may occur in the calculation of interest charges for the items kept in stock per
year.
Case 1. M ≤ T .
Annual interest payable =
cIc
∫
T
q(t )dt
T M
cIc D
= 2 (eθ (T −M ) − θ (T − M ) − 1).
θ T
Case 2. N ≤ T ≤ M.
In this case, annual interest payable = 0.
Case 3. T ≤ N.
Similar as case 2, annual interest payable = 0.
According to assumption (11), three cases will occur in interest earned per year.
Case 1. M ≤ T , (shown in Fig. 7)
Annual interest earned =
=
sIe
T
[∫
2T
Case 2. N ≤ T ≤ M, (shown in Fig. 8)
Annual interest earned =
=
T
∫
M
Dtdt
N
]
2
[M − (1 − α)N 2 ].
[∫
sIe D
2T
α Dtdt +
0
sIe D
sIe
N
0
N
α Dtdt +
∫
N
T
]
Dtdt + DT (M − T )
[2MT − (1 − α)N 2 − T 2 ].
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sDT
α sDN
Time
0
N
M
T
Fig. 7. Total amount of interest earned when M ≤ T .
sDT
α sDN
Time
0
N
T
M
Fig. 8. Total amount of interest earned when N ≤ T ≤ M.
Case 3. T ≤ N, (shown in Fig. 9)
]
α Dtdt + α DT (N − T ) + DT (M − N )
T
0
]
[
αT
.
= sIe D M − (1 − α)N −
Annual interest earned =
sIe
[∫
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 );
TVC2 (T );
TVC3 (T );
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N ,
(9)
where
(h + θ c )D θ T
cIc D
sIe D
(e − 1 − θ T ) + 2 [eθ (T −M ) − θ (T − M ) − 1] −
[M 2 − (1 − α)N 2 ],
θ 2T
θ T
2T
(h + θ c )D θ T
sIe D
A
+
(e − 1 − θ T ) −
[2MT − (1 − α)N 2 − T 2 ]
TVC2 (T ) =
2
T
θ T
2T
]
[
A
(h + θ c )D θ T
αT
and TVC3 (T ) =
.
+
(e − 1 − θ T ) − sIe D M − (1 − α)N −
T
θ 2T
2
TVC1 (T ) =
A
T
+
(10)
(11)
(12)
Now, we develop the above retailer’s inventory costs in a fuzzy environment. In the present day scenario, it is very difficult
or simply it is not possible to determine a precise value of D. In this case, the retailer needs to collect the demand information
from customers. When the customers’ opinions are vague or linguistic like ‘‘demand is about D’’, the average annual demand
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sDT
α sDT
Time
0
T
N
M
Fig. 9. Total amount of interest earned when T ≤ N.
can be vaguely expressed. So, it is more suitable to describe it as fuzzy in nature. Therefore, the total annual demand is treated
˜ = (D −δD1 , D, D +δD2 ). Also,
as a fuzzy variable. Here we consider the fuzzy annual demand D as a triangular fuzzy number D
inventory costs are normally assumed to be constant but this is not always true. In a perfect competitive market, ordering
cost A, unit holding cost h, unit purchasing cost c, unit selling price s etc. per day in a plan period T may fluctuate a little. For
example, ‘‘unit holding cost is around h’’, ‘‘unit selling price is about s’’, etc. Suppose these cost parameters ordering cost,
holding cost, purchase cost, selling price lie in the intervals [A −δA1 , A +δA2 ], [h −δh1 , h +δh2 ], [c −δc1 , c +δc2 ], [s −δs1 , s +δs2 ].
Similarly, corresponding to these intervals, we set the following triangular fuzzy numbers: A˜ = (A − δA1 , A, A + δA2 ),
h˜ = (h − δh1 , h, h + δh2 ), c˜ = (c − δc1 , c , c + δc2 ) and s˜ = (s − δs1 , s, s + δs2 ). 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 any T > 0, we get fuzzy annual total variable costs
1 (T ) = X11 ⊗
TVC
A ⊕ X12 ⊗
h ⊗
D ⊕ X13 ⊗
c ⊗
D ⊖ X14 ⊗
s ⊗
D
2 (T ) = X21 ⊗
TVC
A ⊕ X22 ⊗
h ⊗
D ⊕ X23 ⊗
c ⊗
D ⊖ X24 ⊗
s ⊗
D
3 (T ) = X31 ⊗
and TVC
A ⊕ X32 ⊗
h ⊗
D ⊕ X33 ⊗
c ⊗
D ⊖ X34 ⊗
s ⊗
D,
(13)
(14)
(15)
where ⊗, ⊕ and ⊖ are the fuzzy arithmetical operations under the Function Principle and
X13 =
eθ T
1
1
(eθ T − 1 − θ T ),
θ 2T
Ic (eθ (T −M ) − θ (T − M ) − 1)
eθ T − 1 − θ T
− 1 − θT
+
,
X
=
X
=
,
23
33
θT
θ 2T
θT
X11 = X21 = X31 =
T
,
X12 = X22 = X32 =
Ie
Ie
[M 2 − (1 − α)N 2 ],
X24 =
[2MT − (1 − α)N 2 − T 2 ]
2T
2T
]
[
αT
.
and X34 = Ie M − (1 − α)N −
X14 = −
2
˜ = (D1 , D, D2 ), A˜ = (A1 , A, A2 ), h˜ = (h1 , h, h2 ), c˜ = (c1 , c , c2 ), and s˜ = (s1 , s, s2 ) are nonnegative
Here we assume that D
triangular fuzzy numbers, where D1 = D − δD1 , D2 = D + δD2 , A1 = A − δA1 , A2 = A + δA2 , h1 = h − δh1 , h2 = h + δh2 ,
1 (T ) by Eq. (13) as
c1 = c − δc1 , c2 = c + δc2 , s1 = s − δs1 , s2 = s + δs2 . Then we get the fuzzy annual total variable cost TVC
1 (T ) = [X11 A1 + X12 h1 D1 + X13 c1 D1 − X14 s2 D2 , X11 A + X12 hD + X13 cD − X14 sD,
TVC
X11 A2 + X12 h2 D2 + X13 c2 D2 − X14 s1 D1 ].
1 (T ) by formula (2) and obtain the graded mean integration
We defuzzify the fuzzy annual total variable cost TVC
representation of TVC 1 (T ) as
1 (T )) =
P (TVC
1
6
[(X11 A1 + X12 h1 D1 + X13 c1 D1 − X14 s2 D2 ) + 4(X11 A + X12 hD + X13 cD − X14 sD)
+ (X11 A2 + X12 h2 D2 + X13 c2 D2 − X14 s1 D1 )]
= X11 F1 + X12 F2 + X13 F3 − X14 F4
F1
F2 + θ F3 θ T
F3 Ic
F4 Ie
=
+
(e − 1 − θ T ) + 2 (eθ (T −M ) − θ (T − M ) − 1) −
[M 2 − (1 − α)N 2 ],
2
T
θ T
θ T
2T
where F1 =
A1 +4A+A2
,
6
F2 =
h1 D1 +4hD+h2 D2
,
6
F3 =
c1 D1 +4cD+c2 D2
6
and F4 =
s1 D1 +4sD+s2 D2
.
6
(16)
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2 (T ) and TVC
3 (T ) are respectively,
Similarly, the graded mean integration representation of TVC
2 (T )) =
P (TVC
F1
F2 + θ F3
F4 Ie
[2MT − (1 − α)N 2 − T 2 ],
θ
2T
]
[
3 (T )) = F1 + F2 + θ F3 (eθ T − 1 − θ T ) − F4 Ie M − (1 − α)N − α T .
P (TVC
T
θ 2T
2
T
+
2T
(eθ T − 1 − θ T ) −
(17)
(18)
Thus the annual total cost in the fuzzy sense based on the graded mean integration representation is
1 (T ));
P (TVC
(T )) = P (TVC
2 (T ));
P (TVC
P (TVC 3 (T ));
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N ,
(19)
1 (M )) = P (TVC
2 (M )) and P (TVC
2 (N )) =
1 (T )), P (TVC
2 (T )) and P (TVC
3 (T )) are given by (16)–(18). Since P (TVC
where P (TVC
3 (N )), P (TVC
(T )) is continuous and well-defined on T > 0.
P (TVC
Then, we have
1 (T ))′ = −
P (TVC
F1
F2 + θ F3
F3 Ic
(θ T eθ (T −M ) − eθ (T −M ) + 1 − θ M )
θ
θ 2T 2
F4 Ie
+ 2 [M 2 − (1 − α)N 2 ],
2T
2F1
1 2 2 θT
2(F2 + θ F3 )
′′
θT
θT
P (TVC 1 (T )) =
e − θTe − 1 + θ T e
+
2 3
T3
2
θ T
F4 Ie
1
2F3 Ic
θ ( T −M )
θ (T −M )
− θTe
− 1 + θ M + θ 2 T 2 eθ (T −M ) − 3 [M 2 − (1 − α)N 2 ]
+ 2 3 e
θ T
2
T
2 (T ))′ = −
P (TVC
T2
F1
+
2T 2
(θ T eθ T − eθ T + 1) +
F 2 + θ F3
F4 Ie
(θ T eθ T − eθ T + 1) + 2 [T 2 − (1 − α)N 2 ]
θ
2T
2F1
F4 Ie (1 − α)N 2
1 2 2 θT
2(F2 + θ F3 )
′′
θT
θT
P (TVC 2 (T )) = 3 +
+
e
−
θ
T
e
−
1
+
θ
T
e
T
θ 2T 3
2
T3
T2
+
F1
T2
+
(21)
(22)
2T 2
(23)
F 2 + θ F3
F4 Ie α
(θ T eθ T − eθ T + 1) +
θ 2T 2
2
1 2 2 θT
2(F2 + θ F3 )
2F1
θT
θT
′′
e − θTe − 1 + θ T e
P (TVC 3 (T )) = 3 +
T
θ 2T 3
2
3 (T ))′ = −
P (TVC
(20)
(24)
(25)
where (′ ) represents differentiation with respect to T .
3.2.1. The convexity
1 (T )), P (TVC
2 (T )) and P (TVC
3 (T )) are convex on their appropriate domains.
In this section, we shall show that P (TVC
1 (T )) is convex on [M , ∞).
Theorem 1. (1) P (TVC
(2) P (TVC 2 (T )) is convex on [0, ∞).
3 (T )) is convex on [0, ∞).
(3) P (TVC
Before proving Theorem 1, we need the following lemmas.
Lemma 1. eθ T − θ T eθ T − 1 + 12 θ 2 T 2 eθ T > 0 for all T > 0.
Proof. Let f (x) = ex − 1 − xex + 21 x2 ex if x > 0, then f ′ (x) =
x
x
f (x) > f (0) = 0 if x > 0. We have f (x) = e − 1 − xe +
if T > 0. This completes the proof.
1 2 x
x e
2
1 2 x
x e
2
> 0. Hence f (x) is increasing for all x > 0. Consequently,
> 0 if x > 0. Let x = θ T . Then eθ T −θ T eθ T − 1 + 21 θ 2 T 2 eθ T > 0
2 2
Lemma 2. If T ≥ M, then eθ (T −M ) − 1 − θ T eθ (T −M ) + θ M + 12 θ 2 T 2 eθ (T −M ) − θ 2M ≥ 0.
2
2
Proof. Let g (T ) = eθ (T −M ) − 1 − θ T eθ (T −M ) + θ M + 12 θ 2 T 2 eθ (T −M ) − θ 2M . Then we have g ′ (T ) =
g (T ) is increasing on [M , ∞). So g (T ) > g (M ) = 0 if T > M. This completes the proof.
1
2
θ 3 T 2 eθ (T −M ) > 0. Hence
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The proof of Theorem 1. (1) From Eq. (21), we have
2F1
1 (T )) =
P (TVC
′′
T3
2F3 Ic
+
θ 2T 3
2F1
>
+
T3
+
θ 2T 3
− θTe
θT
1
2
2
e
θ (T −M )
1
eθ T − θ T eθ T − 1 +
2T 3
θ ( T −M )
− θTe
2 θT
−1+ θ T e
eθ (T −M ) − θ T eθ (T −M ) − 1 + θ M +
θ
θT
e
θ 2T 3
2(F2 + θ F3 )
2F3 Ic
+
2(F2 + θ F3 )
2
F4 Ie
θ 2 T 2 eθ (T −M ) − 3 [M 2 − (1 − α)N 2 ]
2
T
1
θ 2 T 2 eθ T
1
2
2 θ ( T −M )
− 1 + θM + θ T e
2
−
θ 2M 2
2
+
F4 Ie (1 − α)N 2
T3
.
(26)
1 (T ))′′ > 0 if T ≥ M. Hence P (TVC
1 (T )) is convex on [M , ∞).
Lemmas 1 and 2 imply that P (TVC
(2) From Eq. (23), we have
F4 Ie (1 − α)N 2
1 2 2 θT
2(F2 + θ F3 )
2F1
θT
θT
′′
+
e
−
θ
T
e
−
1
+
θ
T
e
.
P (TVC 2 (T )) = 3 +
T
θ 2T 3
2
T3
2 (T ))′′ > 0 if T > 0. Consequently, P (TVC
2 (T )) is convex on [0, ∞).
Lemma 1 implies that P (TVC
(3) From Eq. (25), we have
2F1
3 (T )) =
P (TVC
′′
+
T3
2(F2 + θ F3 )
θ 2T 3
e
θT
θT
− θTe
1
2
2 θT
−1+ θ T e
2
.
3 (T ))′′ > 0 if T > 0. Consequently, P (TVC
3 (T )) is convex on [0, ∞).
From Lemma 1, P (TVC
This complete the proof of Theorem 1.
From Eqs. (20), (22) and (24), we have
1 (M ))′ = P (TVC
2 (M ))′
P (TVC
=−
F1
F2 + θ F3
+
M2
θ
2M 2
(θ Meθ M − eθ M + 1) +
F4 Ie
2M 2
[M 2 − (1 − α)N 2 ]
(27)
and
2 (N ))′ = P (TVC
3 (N ))′
P (TVC
=−
F1
N2
+
F2 + θ F3
θ
2N 2
(θ Neθ N − eθ N + 1) +
F4 Ie α
.
(28)
[M 2 − (1 − α)N 2 ]
(29)
.
(30)
2
Furthermore, we let
△1 = −
F1
M2
+
F2 + θ F3
θ
2M 2
(θ Meθ M − eθ M + 1) +
F4 Ie
2M 2
and
△2 = −
F1
N2
+
F2 + θ F3
θ
2N 2
(θ Neθ N − eθ N + 1) +
F 4 Ie α
2
Therefore,
△1 − △2 = F1
Let, K (x) =
xex −ex +1
x2
1
N2
−
1
M2
+
F4 Ie (1 − α)
2M 2
, so that by Lemma 1, k′ (x) =
θ Meθ M − eθ M + 1 θ Neθ N − eθ N + 1
.
(M − N ) + (F2 + θ F3 )
−
θ 2M 2
θ 2N 2
2
2
x3
2
(31)
(ex − 1 − xex + 12 x2 ex ) > 0 if x > 0. Hence k(x) is increasing if x > 0.
θM
θM
θN
θN
Consequently, k(x) ≥ k(y) if x > y. So that, k(θ M ) > k(θ N ). This gives θ Me θ 2−Me2 +1 ≥ θ Ne θ 2−Ne2 +1 . From (31), we have
∆1 ≥ ∆2 .
Consider the following equations:
1 (T ))′ = 0,
P (TVC
2 (T )) = 0,
P (TVC
′
3 (T ))′ = 0.
P (TVC
(32)
(33)
(34)
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If the root of each of Eqs. (32), (33) or (34) exists, then it is unique. Let T1∗ denote the root of Eq. (32), T2∗ denote the root of
i (T )) (i = 1, 2, 3), we see that
Eq. (33) and T3∗ denote the root of Eq. (34). By the convexity of P (TVC
< 0;
i (T )) = 0;
P (TVC
> 0;
′
if T < Ti∗ ,
if T = Ti∗ ,
if T > Ti∗ .
(35)
i (T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞) for all i = 1, 2, 3.
Eq. (35) implies that P (TVC
i
i
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.
(T ∗ )) = P (TVC
3 (T ∗ )) and T ∗ = T ∗ .
Theorem 2. (A) If ∆1 > 0 and ∆2 ≥ 0, then P (TVC
3
3
∗
∗
(B) If ∆1 > 0 and ∆2 < 0, then P (TVC (T )) = P (TVC 2 (T2 )) and T ∗ = T2∗ .
(T ∗ )) = P (TVC
1 (T ∗ )) and T ∗ = T ∗ .
(C) If ∆1 ≤ 0 and ∆2 < 0, then P (TVC
1
1
1 (M ))′ = P (TVC
2 (M ))′ > 0
Proof. (A) If ∆1 > 0 and ∆2 ≥ 0, then T1∗ < M, T2∗ < M, T3∗ ≤ N and T2∗ ≤ N. We have P (TVC
′
′
2 (N )) = P (TVC
3 (N )) ≥ 0. Eq. (35) imply that
and P (TVC
1 (T )) is increasing on [M , ∞),
(i) P (TVC
2 (T )) is increasing on [N , M ] and
(ii) P (TVC
(iii) P (TVC 3 (T )) is decreasing on (0, T3∗ ] and is increasing [T3∗ , N ].
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
3
3
∗
∗
∗
∗
(T )) = P (TVC
3 (T )).
Consequently, T = T3 and P (TVC
3
1 (M ))′ = P (TVC
2 (M ))′ > 0 and
(B) If ∆1 > 0 and ∆2 < 0, then T3∗ > N, T2∗ > N, T1∗ < M and T2∗ < M. We have P (TVC
′
′
P (TVC 2 (N )) = P (TVC 3 (N )) < 0. Eq. (35) imply that
1 (T )) is increasing on [M , ∞),
(i) P (TVC
2 (T )) is decreasing on [N , T ∗ ] and increasing on [T ∗ , M ],
(ii) P (TVC
2
2
3 (T )) is decreasing on (0, N ].
(iii) P (TVC
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
2
2
∗
∗
∗
∗
(T )) = P (TVC
2 (T )).
Consequently, T = T2 and P (TVC
2
1 (M ))′ = P (TVC
2 (M ))′ ≤ 0 and
(C) If ∆1 ≤ 0 and ∆2 < 0, then T3∗ > N, T2∗ > N, T1∗ ≥ M and T2∗ ≥ M. We have P (TVC
′
′
P (TVC 2 (N )) = P (TVC 3 (N )) < 0. Eq. (35) imply that
1 (T )) is decreasing on [M , T ∗ ] and increasing on [T ∗ , ∞),
(i) P (TVC
1
1
(ii) P (TVC 2 (T )) is decreasing on [N , M ] and
3 (T )) is decreasing on (0, N ].
(iii) P (TVC
(T )) is decreasing on (0, T ∗ ] and increasing on [T ∗ , ∞).
Combining (i)–(iii) and Eq. (19), we have that P (TVC
1
1
(T ∗ )) = P (TVC
1 (T ∗ )).
Consequently, T ∗ = T1∗ and P (TVC
1
3.2.3. The algorithms for the determination of T ∗ , T1∗ , T2∗ and T3∗
We first describe the following theorem:
Intermediate value theorem: Let f (x) be a continuous function on [a, b] and f (a).f (b) < 0, then there exists a number
c ∈ (a, b) such that f (c ) = 0.
(i) Suppose that ∆1 > 0 and ∆2 ≥ 0, then T3∗ exists, T ∗ = T3∗ and 0 < T3∗ < N. Recall T3∗ to denote the unique root of the
F +θ F
F I α
Eq. (34). Let f3 (T ) = −F1 + 2 θ 2 3 (θ T eθ T − eθ T + 1) + 42e T 2 . Then f3′ (T ) = (F2 + θ F3 )T eθ T + F4 Ie α T ≥ 0. Hence f3 (T ) is
increasing on T ≥ 0. We see that f3 (0) = −F1 < 0 = f3 (T3∗ ) < f3 (N ). So f3 (0)f3 (N ) < 0. Consequently, we are in a position
to outline the algorithm to find T3∗ .
Algorithm 1. Step 1. Let ϵ > 0.
Step 2. Let TL = 0 and TU = N.
Step 3. Let Topt =
TL +TU
2
.
Step 4. If |f3 (Topt )| < ϵ , go to step 6. Otherwise, go to step 5.
Step 5. If f3 (Topt ) > 0, Set TU = Topt . If f3 (Topt ) < 0, Set TL = Topt . Then go to step 3.
Step 6. Set T3∗ = Topt .
(ii) Suppose that ∆1 > 0 and ∆2 < 0, then T2∗ exists, T ∗ = T2∗ and N < T2∗ < M. T2∗ denotes the root of the Eq. (33). We have
2 (N ))′ < 0 = P (TVC
2 (T ∗ ))′ < P (TVC
2 (T ))′ is increasing on [N , M ].
2 (M ))′ . By (23) and Lemma 1, we see that P (TVC
P (TVC
2
′
′
Hence P (TVC 2 (N )) P (TVC 2 (M )) < 0. Consequently, we are in a position to outline the algorithm to find T2∗ .
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Algorithm 2. Step 1. Let ϵ > 0.
Step 2. Let TL = N and TU = M.
Step 3. Let Topt =
TL +TU
2
.
2 (Topt ))′ | < ϵ , go to step 6. Otherwise, go to step 5.
Step 4. If |P (TVC
2 (Topt ))′ > 0, Set TU = Topt . If P (TVC
2 (Topt ))′ < 0, Set TL = Topt . Then go to step 3.
Step 5. If P (TVC
Step 6. Set T2∗ = Topt .
(iii) Suppose that ∆1 ≤ 0 and ∆2 < 0, then T1∗ exists, T ∗ = T1∗ . T1∗ denotes the root of the Eq. (32). Let
T1U =
2(F2 + θ F3 ) + 2F3 Ic (e−θ M + θ M ) + θ 2 F4 Ie (1 − α)N 2 + 2F1 θ 2
2θ{(F1 + θ F3 ) + F3 Ic e−θ M }
(36)
.
Then we have the following lemma:
Lemma 3. Suppose that ∆1 ≤ 0 and ∆2 < 0, then T1U > T1∗ > M, where T1U is given by (36).
Proof. Since 0 < e−θ T < 1,
1 (T ))′ =
P (TVC
F2 + θ F3
θ
+
2T 2
F4 Ie
2T 2
(θ T eθ T − eθ T + 1) +
[M 2 − (1 − α)N 2 ] −
F3 Ic
θ 2T 2
(θ T eθ (T −M ) − eθ (T −M ) + 1 − θ M )
F1
T2
F3 Ic
F2 + θ F3 θ T
e (θ T − 1 + e−θ T ) + 2 2 eθ T (θ T e−θ M − e−θ M + e−θ T − θ Me−θ T )
=
2
2
θ T
θ T
[
]
2
F
I
M
F1
F
I
(
1
−
α)N 2
4 e
4 e
−θ T
−θ T
+ eθ T
e
e
−
+
2T 2
2T 2
T2
eθ T
[
F2 + θ F3
F3 Ic
−θ M
F4 Ie (1 − α)N 2
]
(θ T − 1) + 2 {(θ T − 1)e
− θ M} −
+ F1
θ2
θ
2
[
]
eθ T
F2 + θ F3
F2 + θ F3
F3 Ic −θ M
F3 Ic −θ M
F4 Ie (1 − α)N 2
= 2
θ
T
−
+
e
+
(
e
+
θ
M
)
+
+
F
.
1
T
θ2
θ2
θ2
θ2
2
>
T2
1 (T U ))′ > 0 = P (TVC
1 (T ∗ ))′ . Since P (TVC
1 (T ))′ is increasing on [M , ∞) and P (TVC
1 (T U ))′ > P (TVC
1 (T ∗ ))′ >
Hence P (TVC
1
1
U
1 (M ))′ , we obtain T > T ∗ > M. Consequently, we have completed the proof. By Lemma 3, we are in a position to
P (TVC
1
1
outline the algorithm to find T1∗ .
Algorithm 3. Step 1. Let ϵ > 0.
Step 2. Let TL = M and TU = T1U .
Step 3. Let Topt =
TL +TU
2
.
1 (Topt ))′ | < ϵ , go to step 6. Otherwise, go to step 5.
Step 4. If |P (TVC
1 (Topt ))′ > 0, Set TU = Topt . If P (TVC
1 (Topt ))′ < 0, Set TL = Topt . Then go to step 3.
Step 5. If P (TVC
Step 6. Set T1∗ = Topt .
4. Numerical examples
To illustrate the results, let us apply the proposed method to solve the following numerical examples. The following
parameters
D = (2500, 3000, 3500) units/year,
A = $(120, 130, 140)/order,
h = $(5, 7, 9)/unit/year,
c =
$(10, 15, 20)/unit, Ic = $0.15/$/year, Ie = $0.12/$/year, m = 0.1 year are used from Examples 1, 2, 3.
Example 1. If N = 0.08 year,
s = $(15, 20, 25)/unit, α = 0.1, θ = 0.02, then ∆1 = −312.5555 < 0, ∆2 = −8810.626 <
(T ∗ )) = 2258.316.
0. Using Algorithm 3, we get T ∗ = T1∗ = 0.1010656, Q ∗ = 303.5034 and P (TRC
Example 2. If N = 0.05 year,
s = $(25, 30, 35)/unit, α = 0.5, θ = 0.05, then ∆1 = 2045.699 > 0, ∆2 = 38342.79 < 0.
(T ∗ )) = 2156.621.
Using Algorithm 2, we get T ∗ = T2∗ = 0.09317759, Q ∗ = 280.1849 and P (TRC
Example 3. If N = 0.08 year,
s = $(45, 50, 55)/unit, α = 0.9, θ = 0.08, then ∆1 = 8037.667 > 0, ∆2 = 385.9616 > 0.
(T ∗ )) = 1567.746.
Using Algorithm 1, we get T ∗ = T3∗ = 0.07058884, Q ∗ = 212.3656 and P (TRC
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G.C. Mahata, P. Mahata / Mathematical and Computer Modelling 53 (2011) 1621–1636
5. Special cases
In this section, we obtain some previously published results of other authors as special cases.
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, when s = c, θ → 0
(it means that the deterioration rate is ignored) and α → 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,
K (T ) =
if T ≥ M ,
if N ≤ T ≤ M ,
if 0 < T ≤ N .
K1 (T );
K2 (T );
K3 (T );
(37)
where,
K1 (T ) =
K2 (T ) =
K3 (T ) =
A
T
A
T
A
T
K1′ (T ) = −
′
K2 (T ) = −
K3′ (T ) = −
+
+
+
DTh
2
DTh
2
DTh
2
+
−
cIc D(T − M )2
2T
cIe D(M 2 − N 2 )
−
cIe D(2MT − N 2 − T 2 )
− cIe D(M − N )
(40)
2T 2
2A + cDN 2 Ie
2T 2
T2
+
(39)
2T
2A + cDM 2 (Ic − Ie ) + cDN 2 Ie
A
(38)
2T
+D
h + cIe
2
+D
h + cIc
2
(41)
(42)
Dh
(43)
2
¯ 1 = −2A + DM 2 (h + cIe ) − cDN 2 Ie
∆
(44)
2
¯ 2 = −2A + DN h.
∆
∗
∗
∗
(45)
′
′
′
Let, T4 , T5 and T6 be the roots of equations K1 (T ) = 0, K2 (T ) = 0 and K3 (T ) = 0 respectively, where
T4∗ =
T5∗ =
∗
T6 =
2A + cD[M 2 (Ic − Ie ) + N 2 Ie ]
D(h + cIc )
2A + cDN 2 Ie
D(h + cIe )
2A
.
Dh
Theorem 2 can be modified as follows:
(46)
(47)
(48)
¯ 1 > 0 and ∆
¯ 2 ≥ 0, then K (T ∗ ) = K3 (T6∗ ) and T ∗ = T6∗ .
Theorem 3. (A) If ∆
¯
¯
(B) If ∆1 > 0 and ∆2 < 0, then K (T ∗ ) = K2 (T5∗ ) and T ∗ = T5∗ .
¯ 1 ≤ 0 and ∆
¯ 2 < 0, then K (T ∗ ) = K1 (T4∗ ) and T ∗ = T4∗ .
(C) If ∆
Theorem 3 has 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, when N = 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 and s = c, then the cost function (19) reduces to,
H (T ) =
H1 (T );
H2 (T );
if T ≥ M ,
if 0 ≤ T ≤ M ,
(49)
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G.C. Mahata, P. Mahata / Mathematical and Computer Modelling 53 (2011) 1621–1636
1635
where,
(h + c θ )D θ T
cIc D θ (T −M )
cIe DM 2
(
e
−
1
−
θ
T
)
+
(
e
−
θ
(
T
−
M
)
−
1
)
−
,
T
θ 2T
θ 2T
2T
(h + c θ )D θ T
A
+
(e − 1 − θ T ) − cIe D(M − T /2).
H2 (T ) =
T
θ 2T
H1 (T ) =
A
+
(50)
(51)
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, when N = 0, s = c,
α → 0, θ → 0 (it means that the deterioration rate is ignored), let
G1 (T ) =
G2 (T ) =
∗
T7 =
A
T
A
T
+
+
DTh
2T
DTh
2T
+
cIc D(T − M )2
2T
−
cIe DM 2
2T
,
− cIe D(M − T /2),
2A + DM 2 c (Ic − Ie )
D(h + cIc )
(52)
(53)
(54)
and
∗
T8 =
2A
D(h + cIe )
.
(55)
Then G′i (Ti∗ ) = 0 for i = 7, 8. Eq. (19) will be reduced as follows:
G(T ) =
G1 (T );
G2 (T );
if T ≥ M , (a)
if 0 ≤ T ≤ M . (b)
(56)
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 + DM 2 (h + cIe). If we let ∆ = −2A + DM 2 (h + cIe), Theorem 2 can be modified as follows:
Theorem 4. (A) If ∆ > 0, then T ∗ = T8∗ .
(B) If ∆ < 0, then T ∗ = T7∗ .
(C) If ∆ = 0, then T ∗ = T7∗ = T8∗ = M.
Theorem 4 has 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 off