**INTRODUCTION**

A vendor-buyer channel coordinates to achieve better joint profit by optimizing
the integrated inventory policy. This vendor-buyer coordination policy has received
significant attention among researchers over the past decades. Goyal
(1977) first developed an integrated inventory model for a single supplier
- single customer problem. Later, Banerjee (1986) developed
a joint economic-lot-size model for a single-buyer single-vendor system, where
the vendor has a finite production rate. Goyal (1988)
extended Banerjee’s model by relaxing the lot-for-lot policy and suggested
that the vendor’s economic production quantity should be an integer multiple
of the buyer’s purchase quantity, Goyal and Gupta (1989)
reviewed related literature, afterward. Since then, Goyal
and Srinivasan (1992), Lu (1995), Ha
and Kim (1997), Hill (1997, 1999),
Goyal and Nebebe (2000), Hoque and
Goyal (2000), Wu and Ouyang (2003), Wu
* et al*. (2007), Hsiao (2008) and Ben-Daya
*et al*. (2008) have been devoted to developing integrated vendor-buyer
inventory models under a variety of circumstances. Recently, Yang
* et al*. (2007) proposed a global optimal policy for vendor-buyer
integrated inventory system within just in time environment.

In a earlier study,

Grubbstrom and Erdem (1999) showed
that the formula for the Economic Order Quantity (EOQ) with backlogging could
be derived algebraically without reference to derivatives.

Cardenas-Barron (2001) extended the mentioned algebraic approach to the Economic
Production Quantity (EPQ) model with shortage.

Grubbstrom and
Erdem (1999) stated that the algebraic approach had to be considered as a
pedagogical advantage for explaining the EOQ concepts to the students who lacked
the knowledge of derivatives, simultaneous equations and the procedure to construct
and examine the Hessian matrix. Following, there is a vast inventory literature
on algebraically method, the outline which can be found in review study by

Chou
*et al*. (2006),

Lai *et al*. (2006),

Shyu
*et al*. (2006),

Leung (2008) and others.

Recently, Yang *et al*. (2007) investigated a global
optimal policy for vendor-buyer integrated inventory system within just in time
environment using differential calculus to find the optimal solution. In this
note, we refer to the algebraic approach method by Grubbstrom
and Erdem’s (1999) and solve the Yang *et al*.
(2007) inventory problem without using derivatives (neither applying the
first-order nor the second-order differentiations). Based on this approach,
the exact expression of the number of deliveries and integrated total cost in
optimum are obtained directly. As a result, students who are unfamiliar with
differential calculus may be able to understand the solution procedure with
ease.

**ASSUMPTIONS AND NOTATIONS**

The assumptions made in the research are defined as follows:

• |
Both the production and demand rate are constant |

• |
The integrated system of single-vendor single-buyer is considered |

• |
The vendor and the buyer have complete knowledge of each other’s
information |

• |
Shortage is not allowed |

The following notations are used throughout the study:

Q |
: |
Buyer’s lot size per delivery |

n |
: |
Number of deliveries from the vendor to the buyer per vendor’s
replenishment interval |

S |
: |
Vendor’s setup cost per setup |

A |
: |
Buyer’s ordering cost per order |

C_{v} |
: |
Vendor’s unit production cost |

C_{b} |
: |
Unit purchase cost paid by the buyer |

r |
: |
Annual inventory carrying cost per dollar invested in stocks |

P |
: |
Annual production rate |

D |
: |
Annual demand rate |

TC |
: |
Integrated total cost of the vendor and the buyer when both the
vendor and the buyer collaborate instead of being independent |

**ALGEBRAIC DERIVATION OF THE INTEGRATED INVENTORY MODEL**

From Eq. 1 of Yang *et al*. (2007),
the vendor’s average inventory level I_{v} is defined:

and the integrated total cost of the vendor and the buyer per year is
denoted as:

Referring to Grubbström and Erdem’s (1999) algebraic method,
Eq. 2 can be rewritten as:

Where:

and

For fixed integer n, Eq. 3 has a minimum value when
the quadratic non-negative term is made equal to zero. Therefore, the
optimal solution of Q (denoted by Q*) is:

Substituting Eq. 6 into Eq. 3, we have
the optimal integrated total cost for fixed n is:

The results of Eq. 6 and Eq. 7 are
the same as Yang *et al*.’s (2007) model.

Next, for convenience, we let:

Furthermore, in order to find the optimal value of n, the Eq.
7 can be rearranged as:

Where:

Then, there are two cases to occur.

**Case I:** X_{1} ≤ 0 (1.e., Cb+C_{v}(2D/p-1) ≤
0).

The right hand side of Eq. 10 is a function of n; we
denoted it by f(n), that is:

For any two positive integers n_{1} and n_{2} (where,
n_{1} < n_{2}), we have:

Since V > 0 and W ≤ 0 (from Eq. 12 and 13).
It implies that f(n), or equivalently, (TC*)^{2} and thereby TC*
is a strictly increasing function of n. Therefore, for optimal solution
of n such that TC* has a minimum value, is n* = 1 and thus, from Eq.
6 and 7, we obtain the following results:

and

The results of Eq. 14 and 15 are the same
as the models proposed by Banerjee (1986) and Yang
*et al*. (2007).

**Case II:** X_{1} > 0 (1.e., C_{b}+C_{v}(2D/p-1)
> 0).

From Eq. (10), it can be rearranged as:

where, V > 0 and W > 0 (from Eq. 12 and (13)).

As a result, the value of n that minimizes the Eq. 16
is:

**Property 1:** If, X_{1} = C_{b}+C_{v} (2D/p-1)
> 0, TC* is decreasing on (0, n] and increasing on [0, ∞) where
.

**Proof:** From Eq. 16, for any two positive integers
n_{1} and n_{2} (where, n_{1} < n_{2}),
we have:

When, ,
we have ,
i.e.,
and .
Thus, we obtain .
Hence, from Eq. 18, we have f(n_{2})-f(n_{1})
< 0, i.e., f(n_{2}) < f(n_{1}). Therefore, f(n),
or equivalently, (TC*)^{2} and thereby TC* is decreasing on (0,
n).

Furthermore, for ,
we have ,
i.e.,
and ,
which implies .
Therefore, from Eq. 18, we have ,f(n_{2})-f(n_{1})
> 0, i.e., f(n_{2}) > f(n_{1}). Therefore, f(n),
or equivalently, (TC*)^{2} and thereby TC* is increasing on [n,
∞). The proof is completed.

Since the value of n is a positive integer, from Property 1, we can obtain
the optimal value of n (denote by n*) as:

Now, we can establish the following algorithm to obtain solution of (Q,
n).

**Algorithm**

**Step 1:** |
Calculate X_{1} = C_{b}+C_{v}(2D/p-1).
If X_{1} > 0, go to Step 2; otherwise, go to Step 3 |

**Step 2:** |
Determine Y_{1} = C_{v} (1-D/P) and then n from
Eq. 17 and compute the corresponding
from Eq. 7. If ,
then ;
otherwise .
Go to Step 4 |

**Step 3:** |
The optimal solution n is obtained; i.e., n* = 1 |

**Step 4:** |
Substituting n* into Eq. 6 to evaluate Q* |

Once n* is obtained the optimal integrated total cost TC*(n*) can be
found from 7.

**Example 1:** In order to show the above solution procedure, let us consider
an inventory system with the data as in Goyal (1988)
and Yang * et al*. (2007). D = 1,000 units year^{-1},
P = 3,200 units year^{-1}, C_{b} = $25 per unit, C_{v}
= $20 per setup, A = $25 per order, S = $400 per set-up, r = 0.2.

Applying the proposed algorithm, we check the condition:

Then, determine Y_{1} = C_{v} (1-D/P) = 13.75 and finding the
value of n from Eq. 17, we get n = 4.5126. Since Tc (n = 5)-TC
(n = 4) = 1,903.287-1,903.943 < 0, the optimal No. of deliveries is n* =
5. Substituting this value into Eq. 6, we obtain the optimal
order quantity Q = 110 units. These results are the same as Yang
*et al*. (2007).

**Example 2:** The data are the same as in Example 1, expect C_{v}
= $70. Under the parameter values given above, we check the condition:

Therefore, by using the proposed algorithm, the optimal No. of delivery
is n* = 1. Next, utilizing Eq. 14 and 15,
we have the optimal order quantity Q* = 301 units and the integrated total
cost is $2,822.9.

**CONCLUSION**

In this study, without reference to the use of derivatives, we algebraically
derive the optimal order quantity and optimal number of delivery for the
integrated single-buyer single-vendor inventory. Using this approach,
the optimal solution of proposed model can be asserted without the need
to apply the first-order and the second-order differentiations. Also,
the exact expression of the number of deliveries and integrated total
cost in optimum are obtained directly. As a result, students who are unfamiliar
with differential calculus may be able to understand the solution procedure
with ease.