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Research Article
 

Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment



Fang-Kuo Wang and Kun-Shan Wu
 
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ABSTRACT

The present study was carried out to investigate Yang’s model by using an algebraic method (neither applying the first-order nor the second-order differentiations) to determine the optimal replenishment policy. The number of delivery and the integrated total cost is immediately provided by the proposed algebraic derivation as well. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure with ease.

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  How to cite this article:

Fang-Kuo Wang and Kun-Shan Wu, 2009. Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment. Journal of Applied Sciences, 9: 1194-1197.

DOI: 10.3923/jas.2009.1194.1197

URL: https://scialert.net/abstract/?doi=jas.2009.1194.1197
 

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
Cv : Vendor’s unit production cost
Cb : 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 Iv is defined:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(1)

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(2)

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(3)

Where:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(4)

and

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(5)

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:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(6)

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(7)

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

Next, for convenience, we let:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(8)

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(9)

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(10)

Where:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(11)

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(12)

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(13)

Then, there are two cases to occur.

Case I: X1 ≤ 0 (1.e., Cb+Cv(2D/p-1) ≤ 0).

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment

For any two positive integers n1 and n2 (where, n1 < n2), we have:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment

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:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(14)

and

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(15)

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

Case II: X1 > 0 (1.e., Cb+Cv(2D/p-1) > 0).

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(16)

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

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

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(17)

Property 1: If, X1 = Cb+Cv (2D/p-1) > 0, TC* is decreasing on (0, n] and increasing on [0, ∞) where Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment.

Proof: From Eq. 16, for any two positive integers n1 and n2 (where, n1 < n2), we have:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment
(18)

When, Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, we have Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, i.e., Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment and Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment. Thus, we obtain Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment. Hence, from Eq. 18, we have f(n2)-f(n1) < 0, i.e., f(n2) < f(n1). Therefore, f(n), or equivalently, (TC*)2 and thereby TC* is decreasing on (0, n).

Furthermore, for Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, we have Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, i.e., Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment and Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, which implies Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment. Therefore, from Eq. 18, we have ,f(n2)-f(n1) > 0, i.e., f(n2) > f(n1). 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:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment

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

Algorithm

Step 1: Calculate X1 = Cb+Cv(2D/p-1). If X1 > 0, go to Step 2; otherwise, go to Step 3
Step 2: Determine Y1 = Cv (1-D/P) and then n from Eq. 17 and compute the corresponding Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment from Eq. 7. If Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment, then Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment; otherwise Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment. 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, Cb = $25 per unit, Cv = $20 per setup, A = $25 per order, S = $400 per set-up, r = 0.2.

Applying the proposed algorithm, we check the condition:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment

Then, determine Y1 = Cv (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 Cv = $70. Under the parameter values given above, we check the condition:

Image for - Using the Complete Squares Method to Analysis the Global Optimal Policy for Vendor-Buyer Integrated Inventory System Within Just in Time Environment

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.

REFERENCES
1:  Banerjee, A., 1986. A joint economic lot size model for purchaser and vendor. Dec. Sci., 17: 292-311.
CrossRef  |  

2:  Ben-Daya, M., M. Darwish and K. Ertogral, 2008. The joint economic lot sizing problem: Review and extensions. Eur. J. Operat. Res., 185: 726-742.
CrossRef  |  

3:  Cardenas-Barron, L.E., 2001. The economic production quantity (EPQ) with shortage derived algebraically. Int. J. Prod. Econ., 70: 289-292.
CrossRef  |  

4:  Chou, C.L., Y.F. Huang and H.F. Huang, 2006. A note on the deterministic inventory models with shortage and defective items derived without derivatives. J. Applied Sci., 6: 325-327.
CrossRef  |  Direct Link  |  

5:  Goyal, S.K., 1977. An integrated inventory model for a single supplier-single customer problem. Int. J. Prod. Res., 15: 107-111.
CrossRef  |  Direct Link  |  

6:  Goyal, S.K., 1988. Joint economic lot size model for purchaser and vendor: A comment. Dec. Sci., 19: 236-241.
CrossRef  |  

7:  Goyal, S.K. and Y.P. Gupta, 1989. Integrated inventory models: The buyer-vendor coordination. Eur. J. Operat. Res., 41: 261-269.
CrossRef  |  

8:  Goyal, S.K. and F. Nebebe, 2000. Determination of economic production-shipment policy for a single-vendor-single-buyer system. Eur. J. Operat. Res., 121: 175-178.
CrossRef  |  

9:  Goyal, S.K. and G. Srinivasan, 1992. The individually responsible and rational decision approach to economic lot size for one vendor and many purchasers: A comment. Dec. Sci., 23: 777-784.
CrossRef  |  

10:  Grubbstrom, R.W. and A. Erdem, 1999. The EOQ with backlogging derived without derivatives. Int. J. Prod. Econ., 59: 529-530.
CrossRef  |  

11:  Ha, D. and S.L. Kim, 1997. Implementation of JIT purchasing: An integrated approach. Prod. Plan. Control, 8: 152-157.
CrossRef  |  

12:  Hill, R.M., 1997. The single-vendor single-buyer integrated production-inventor model with a generalized policy. Eur. J. Operat. Res., 97: 493-499.
CrossRef  |  

13:  Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory model. Int. J. Prod. Res., 37: 2463-2475.
CrossRef  |  

14:  Hoque, M.A. and S.K. Goyal, 2000. An optimal policy for a single-vendor single-buyer integrated production-inventory system with capacity constraint of the transport equipment. Int. J. Prod. Econ., 65: 305-315.
CrossRef  |  

15:  Hsiao, Y.C., 2008. A note on integrated single vendor single buyer model with stochastic demand and variable lead time. Int. J. Prod. Econ., 114: 294-297.
CrossRef  |  

16:  Lai, C.S., Y.F. Huang and H.F. Huang, 2006. A deterministic inventory model under quantity-depended payments delay policy using algebraic method. J. Applied Sci., 6: 406-410.
CrossRef  |  Direct Link  |  

17:  Leung, K.N., 2008. Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain. Eur. J. Operat. Res., 187: 19-30.
CrossRef  |  

18:  Lu, L., 1995. A one-vendor multi-buyer integrated inventory model. Eur. J. Operat. Res., 81: 312-323.
CrossRef  |  

19:  Shyu, M.L., Y.F. Huang and H.F. Huang, 2006. Technical note-an integrated single-buyer inventory system with shortage derived algebraically. J. Applied Sci., 6: 1628-1630.
CrossRef  |  Direct Link  |  

20:  Wu, K.S. and L.Y. Ouyang, 2003. An integrated single-vendor single-buyer inventory system with shortage derived algebraically. Prod. Plan. Control, 14: 555-561.
CrossRef  |  

21:  Wu. K.S., L.Y. Ouyang and C.H. Ho, 2007. Integrated vendorâ€`buyer inventory Int. J. Syst. Sci., 38: 339-350.
CrossRef  |  

22:  Yang, P.C., H.M. Wee and H.J. Yang, 2007. Global optimal policy for vendor-buyer integrated inventory system within just in time environment. J. Glob. Optim., 37: 505-511.
CrossRef  |  

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