INTRODUCTION
In the current supply chain management environment, Stefan et al.
(2004) have demonstrated that buyers and vendors can both obtain greater
benefit through strategic collaboration with each other.
Ha and Kim (1997)^{ }proposed a singlebuyer and a singlevendor
deterministic model with a single product integrated strategy that sends
the first shipment as the product arrives at the transported quantity
in a simple JIT environment. Pan and Yang (2004) extended Goyal’s
model^{ } (1988)^{ } by relaxing the production assumption
and presented an integrated inventory model with controllable lead time.
Huang (2002) developed an integrated vendorbuyer cooperative inventory
model for items with imperfect quality and assumed that the number of
defective items followed a given probability density function. Yang and
Pan (2004)^{ }developed an integrated inventory model involving
deterministic variable lead time and quality improvement investment.
Many researchers have being applied fuzzy theory and techniques to develop
and solve production/inventory problems. For example, Park^{ }(1987)
considered fuzzy inventory costs by using arithmetic operations of the Extension
Principle. Chen and Wang (1996)^{ }fuzzified the demand, ordering cost,
inventory cost and backorder cost into trapezoidal fuzzy numbers in an EOQ model
with backorder consideration. Roy and Maiti (1997) presented a fuzzy EOQ model
with demanddependent unit cost under limited storage capacity. Lee and Yao
(1998) fuzzified the demand quantity and production quantity per day with EPQ
model. Yao et al. (2000) proposed an EOQ model where both order quantity
and total demand were fuzzified as triangular fuzzy numbers. Chang (2004) applied
fuzzy method for both imperfect quality items and annual demand to the EOQ model.
Building upon the work of Yang (2006), this paper proposes the model
incorporates the fuzziness of annual demand and adjusted production rate.
For the model, Signed distance’s ranking method (Yao and Wu, 2000)^{
}for fuzzy number is employed to find the estimation of the joint
total expected annual cost in the fuzzy sense and the corresponding order
quantity of the buyer is derived accordingly.
A FUZZY INTEGRATED INVENTORY MODEL
Consider the model fuzzy D and θ c to triangular fuzzy number and
,
where =
(D  Δ_{1}, D, D+Δ_{2}), 0<Δ_{1}<D,
0<Δ_{2} =
(θ c Δ_{3}, θ c, θ c+Δ_{4}),
0 < Δ_{3}<θ c, 0<Δ_{4}, >1
and Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4
}are both determined by decisionmakers. Modify Yang’s model
(2006), the joint total expected annual cost is a fuzzy function and can
be expressed as
The objective of this problem is to determine the optimal order quantity
of the purchaser Q^{*} and the optimal integer number of lots
in which the items are delivered from the vendor to the purchaser such
that (Q, m) achieves its minimum value. Utilizing classical optimization,
we take the first and second derivatives of (Q, m) with respect to Q and
obtain
and
Since ∂^{2} (Q,
m)/∂Q^{2} > 0, i.e., (Q,
m) is convex in Q and hence the minimum value of (Q,
m) will occur at the point that satisfies ∂
(Q, m)/∂Q = 0. Setting (2) equal to zero and solving for Q, we obtain
the optimal order quantity of the purchaser as:
Definition 1: From Kaufmann and Gupta (1991), Zimmermann (1996),
Yao and Wu (2000), for any a and 0εR, define the signed distance
from a to 0 as d_{0 }(a, 0) = a. If a > 0, a is on the right hand side of origin 0; and the distance
from a to 0 is d_{0 }(a, 0) = a. If a<0, a is on the left hand
side of origin 0; and the distance from a to 0 is  d_{0 }(a,
0) =  a. This is the reason why d_{0 }(a, 0) = a is called the
signed distance from a to 0.
Let Ω be the family of all fuzzy sets defined
on R, the αcut of is
C(α) = [C_{L}(α), C_{U} (α)], 0≤α≤1,
and both C_{L}(α) and C_{U}(α) are continuous
functions on α ε [0,1]. Then, for any εΩ,
we have;
Besides, for every α ε[0,1], the αlevel fuzzy inter val
[C_{L}(α)_{α}, C_{U}(α)_{α}]
has a onetoone correspondence with the crisp interval [A_{L}(α),
A_{U}(α)], that is, [C_{L} (α)_{α},
C_{U} (α)_{α}] ↔ [C_{L}(α),C_{U}(α)]
is onetoone mapping. From definition 1, the signed distance of two end
points, C_{L}(α) and C_{U}(α) to 0 are d_{0}(C_{L}(α),0)
= C_{L}(α) and d_{0}(C_{U}(α),0) = C_{U}(α),
respectively.
Hence, the signed distance of interval [C_{L}(α),C_{U}(α)]
to 0 can be represented by their average, [C_{L}(α)+C_{U}(α)/2.
Therefore, the signed distance of interval [C_{L}(α),C_{U}(α)]
to 0 can be represented as;
Further, because of the 1level fuzzy point is mapping to the real number
,
the signed distance of [C_{L}(α),C_{U}(α)] to
can
be defined as
Thus, from (5) and (6), since the above function is continuous on 0≤α≤1
for εΩ, we can use the following equation to define the signed
distance of to
as follows.
Theorem 1:
Proof:For a fuzzy set εΩ and αε[0,1], the
αcut of the fuzzy set is C (α) = {x ε Ω ׀μ_{C}
(x) > α} = [C_{L} (α), C_{U} (α)],
where C_{L} (α) = a + (ba) α and C_{U} (α)
= c  (cb) α. From definition 1, we can obtain the following equation.
The signed distance of to is defined as
Substituting the result of (7) into (1) and (4), we have
and the optimal order quantity of the purchaser as:
For a particular value of m, the joint total expected annual cost
in fuzzy sense is described by:
We can ignore the terms that are independent of m and take the square
of (m). Then, minimizing (m) is equivalent to minimizing
Once again, while ignoring the terms that are independent of t,
the minimization of the problem can be reduced to the minimization of
The optimal value of m = m^{*} is
obtained when (m^{*})
≤ (m^{*}1)
and (m^{*})
≤(m^{*}+1).
On substituting relevant values inequality, we get:
and
Then,
and
Finally, it is concluded that
Thus, we can use the following procedure to find the optimal values of
Q and m.
Step 1: 
Obtain Δ_{1}, Δ_{2, }Δ_{3
}and Δ_{4} from the decisionmakers. 
Step 2: 
Compute the optimal integer number of lots in which the items are
delivered from the vendor to the purchaser by equation
(10). 
Step 3: 
Compute the optimal order quantity of the purchaser by equation
(9). 
Step 4: 
The (Q^{*}, m^{*}) is the optimal joint total expected
annual cost. 
Remark 1. If Δ_{1} = Δ_{2} = Δ_{3}
= Δ_{4} = Δ, then and reduces to D and θ c; thus
the estimate of the joint total expected annual cost in fuzzy sense (1)
is identical to the crisp case . Hence, the crisp average demand per year
model is a special case of the fuzzy model presented here. Besides, for
the optimal order quantity of the purchaser (9), when Δ_{1}
= Δ_{2} = Δ_{3} = Δ_{4} = Δ,
it reduces to
and the derivation of equation (10) reduces to
NUMERICAL EXAMPLES
To illustrate the results of the proposed models, consider an inventory
system with data: annual demand D = 1,500 unit/year, production rate θ
c = 2, purchaser’s ordering cost per order A = $25/order, vendor’s
setup cost S = $400/setup, lead time L = 8weeks, purchase cost C_{P}
= $25/unit, production cost C_{V} = $20/unit, annual inventory
holding cost per dollar invested in stock r = 0.2, safety stock factor
k = 2.33, standard deviation σ = 7 unit/week.
To solve for the optimal order quantity of purchaser and find the optimal
joint total expected annual cost J(Q^{*}, m^{*}) in the
fuzzy sense for various given sets of (Δ_{1}, Δ_{2})
and (Δ_{3}, Δ_{4}) . Note that in practical
situations, Δ_{1}, Δ_{2,}, Δ_{3}
and Δ_{4 }are determined by the decisionmakers due to the
uncertainty of the problem. The results are summarized in Table
1.
Table 1: 
Optimal solutions for the fuzzy integrated vendorbuyer
inventory model 

CONCLUSIONS
Uncertainties of annual demand and adjustable production rate are inherented
in real supply chain inventory systems. However, in practice, there may
be a lack of historical data to estimate the annual demand and adjustable
production rate. In this situation, using a crisp value is not appropriate.
This paper proposes a fuzzy model for the integrated vendorbuyer inventory
problem. For the fuzzy model, a method of defuzzification, namely the
signed distance, is employed to find the estimation of total profit per
unit time in the fuzzy sense and then the corresponding optimal m and
Q are derived to minimize the total cost. In addition, it is shown that
in some cases, the proposed fuzzy model can be reduced to a crisp problem
and the optimal order quantity of purchaser in the fuzzy sense can be
reduced to that of the classical integrated vendorbuyer inventory model.
Although we are not sure the solution obtained from a fuzzy model is better
than that of the crisp one, the advantage of the fuzzy approach is that
it keeps the uncertainties which always fits real situations better than
the crisp approach does.
NOTATION AND ASSUMPTIONS
To develop the proposed model, the following notation is used:
D 
: 
average demand per year; 
θ c 
: 
adjustable production rate; 
Q 
: 
order quantity of the buyer; 
A 
: 
purchaser’s ordering cost per order; 
S 
: 
vendor’s setup cost per setup; 
L 
: 
length of lead time; 
C_{V} 
: 
unit production cost paid by the vendor; 
C_{p} 
: 
unit purchase cost paid by the purchaser; 
m 
: 
an integer representing the number of lots in which the items are
delivered from the vendor to the buyer; 
r 
: 
annual inventory holding cost per dollar invested in stocks; 
K 
: 
safety stock factor. 
The assumptions made in this paper are as follows.
• 
The adjustable production rate, θ c, is given by
production rate = θ cD, where θ is greater than 1 and fixed. 
• 
The demand X during lead time L follows a normal distribution with
mean μL and standard deviation . 
• 
The reorder point (ROP) equals the sum of the expected demand during
lead time and safety stock. 
• 
Inventory is continuously reviewed. 