A Two-Echelon Inventory Model for Single-Vender and Multi-Buyer System Through Common Replenishment

Epochs

Wen-Jen Chang

Instructor

and Chih-Hung Tsai

Associate Professor

Abstract

To cope with the competitiveness of global market, the enterprise’s decision has now been based on seeking a global optimum of supply chain. The inventory policy of enterprise can be made by the all members of supply chain in order to reach the win-win for both venders and buyers. This study considers the problem of one-vender supplying a product to multi-buyer (customers). The objective is to minimize the vender’s total costs of order processing and transportation subject to the maximum costs which buyers are prepared to incur. In the proposed model, the vender offers a price discount to entice the buyers to accept the policy of common replenishment epochs (CRE). Finally, this study evaluates the benefit of the proposed coordinated strategy with a numerical study.

Keywords: Supply Chain, Common Replenishment Epochs, Replenishment Policy

1. Introduction

Loosely speaking, a supply chain

management is a management tool that

involves three main factors such as

procurement, production, and distribution.

In the past, the enterprise experienced a

stage marked by the producer-oriented

manufacturing environment. In that stage,

a single individual generally limits the

enterprise decisions. Since the

responsibilities of the three factors are all

independent, the enterprise must adopt an

appropriate inventory control to pursue its

normal operations. According to a

statistical data from U.S. manufacturers,

the average inventory cost shares about

thirty to thirty-five percent of their annual

incomes. The inventory is regarded as a

very important asset at that time.

However, the inventory is no longer to be

a good management strategy due to the

International Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp. 48- 61

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Department of Industrial Engineering and ManagementTa-Hwa Institute of Technology

Hsin-Chu, Taiwan, ROCE-mail:ietch@thit.edu.tw

rapid product development. Since the new

innovative technology has made the product

life cycle become shorter and shorter, the

excess inventory will block the cash flow

and indeed gives an adversely effect on the

enterprise. It is a common belief that an

optimum inventory policy is the only way to

lower the impact to a lowest level.

Therefore, the supply chain management

has become a focus issue in the enterprise.

There are many scholars and owners of

enterprises have devoted their time and

money to improve the supply chain

management techniques.

As the popularity of the Internet system, the distance between the supplier and the customer is greatly shortened. As a result, the globalization has changed the trend of the commercial behavior between two organizations (Thomas and Griffin, 1996). To cope with the competitiveness of global market, the enterprise’s decisions have now been based on seeking a global optimum of supply chain instead of the traditional local optimum. The enterprise owner also recognizes the fact that the integrating and coordinating with all members of upstream and downstream of supply chain will result in more effective management, and reach the win-win for both venders and buyers. Therefore, this research will study the problem of one-vender supplying a product to multi-buyer (customers). Through the mutual agreement between buyers and vender, the vender offers a price discount to entice the buyers to accept the policy of common replenishment epochs (CRE). The objective is to determine the most optimum ordering and replenishment epochs by minimizing the vender’s total costs of order processing

and transportation subjected to the maximum costs which buyers are prepared to incur.

2. Literature Review

Supply chain management arises in many physical situations. It has become a hot issue because of the rapid development of the Internet. The so-called supply chain provides a vocabulary to describe the activities that may be encountered in the enterprise during the period between order receiving and product delivery. These activities include obtaining materials, product design, manufacturing and distribution. If different units perform these activities simultaneously, the system formed by these units is called supply chain. The main factors that supply chain planning must consider is to conduct integration. Banerjee (1986) first proposed a two-echelon inventory model for vender and buyer. These after, there are several researchers who have explored many problems on the integration of vender and buyer. Hill (1997), Viswanathan (1999), and Hill (1999) investigated the inventory policy for single-vender and single-buyer respectively. Lu (1995), Viswanathan and Piplani (2001) evolved a model, which considered single-vender and multi-buyers.

In Lu’s study (1995), previous purchasing information and highest expected acceptable cost were utilized to determine the minimum value of the sum of the inventory cost and setup cost. This paper mainly proposed a solution method to seek the best solution of inventory for the single-vender and single-buyer. In addition, Lu (1995) also developed a heuristic model to discuss the coordinating mechanism

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among single-vender and multi-buyer. Hill [4] constructed a production and inventory model with integration as a whole. In dealing with the model, he assumed that the venders produced the product at their greatest rate. The objective was to minimize the total inventory costs per unit time. This model was expected to provide the venders with the optimum production and delivery scheduling. Viswanathan and Piplani (2001) further proposed a model, which allowed the venders to offer a price discount so as to entice the buyers to accept the common replenishment epochs. Furthermore, the problem considered that there was only one delivery when buyers placed an order each time. The most optimum common replenishment epochs were then obtained by minimizing the price discounts and the order processing cost.

In real situations, the vender pursue batch delivery with one ordering by buyers is very common. This paper will take account this situation, and will utilize the Viswanathan and Piplani’s model (2001) to investigate the most optimum common replenishment epochs for single-vender and multi-buyer under the condition of batch delivery. In this study, the objective is to minimize the total costs of order processing, price discount, and transportation so as to determine the most optimum replenishment epochs, price discount, and replenishment.

3. Mathematical Model

3.1 Notations

As: vender’s order setup cost when vender receiving an order at each time.

Ai: vender’s order processing cost for processing a specific order from buyer i.

Ci: vender’s delivery cost to buyer i ($/each time).

Di: demand quantity for buyer i.Gi0

b: total inventory costs for buyer i before reaching agreement with the members of supply chain (including the cost of order processing, inventory holding, and replenishment).

G0v: vender’s total costs for order processing and delivery before reaching agreement with members of supply chain.

Gicb: total inventory costs for buyer i after reaching agreement with members of supply chain (including the cost of order processing, inventory holding, and replenishment).

Gcv: vender’s total costs for order

processing and delivery after reaching agreement with members of supply chain.

hi: unit inventory holding cost for buyer i.Ki: ordering cost for buyer i. ni: a positive integer, where the

replenishment period for buyer i is niTo.

Ni: a positive integer, where the order period for buyer i is NiTo.

Qi: order quantity for buyer i before reaching agreement with members of supply chain.

Ri: replenishment processing cost for buyer i.

S: vender’s compensation rate for buyers.T0: vender’s replenishment period before

reaching agreement with members of supply chain.

Ti: replenishment period for buyer i before reaching agreement with members of supply chain.

Ti’: ordering period for buyer i after

reaching agreement with members of supply chain.

Ti”: replenishment period for buyer i after

reaching agreement with members of supply chain.

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TCi: total inventory costs for buyer i.xi: an positive integer, where xi = ( Ni / ni).Zi: lowest price discount accepted by

buyer i.Z: vender’s final proposed price discount,

where Z = max{Zi}.

3.2 Assumptions

This study will develop a two-echelon inventory model for single product, single-vender and multi-buyer for supply chain management problems. The assumptions are listed as following:

1. The market demand is known and fixed for buyers.

2. Buyer doesn’t allow any material shortage.

3. Vender’s replenishment epochs must satisfy the condition: ,

, where a is a positive

integer.4. The vender’s replenishment period (T0)

remains fixed. Buyers’ ordering must be at the period of vender’s replenishment.

5. There must be batch delivery for each buyer ordering, and the condition should be met: niT0 = (Ni / xi)T0.

3.3 Mathematical Modeling

The total inventory costs for buyer to be considered in this study include the cost of ordering, inventory holding, and the processing of replenishment. Hence, the total inventory costs (TCi) for buyer i are given by:

(1)

Minimizing the total costs and letting

, the most optimum economical

ordering quantity (Qi*) and replenishment

epoch (Ti) for buyer i can be expressed by Eq. (2) and (3).

(2)

(3)

Substituting Eq. (2) and (3) into (1) and noting that each party can make their own decisions; the total inventory costs (Gi0

b) for buyer i can be simplified as:

(4)

When vender accepts buyers’ ordering, the costs of the order processing and transportation for delivery would be incurred. These costs for buyer i are (Ai + As

+ Ci). In order to satisfy the demand for buyers, the total costs (G0

v) for vender including the cost of order processing and delivery are represented by Eq. (5):

(5)

However, from previous ordering information the vender could know the buyer’s acceptable least price discount (Zi) under both agreement condition. Based on that, vender then proposes a Common Replenishment Epochs or Periods (CRE) or (T0), and determines the final price discount (Z) and cost compensation rate (S) that could offer to buyers. This would entice the buyer to accept the strategy of the fixed

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replenishment epochs. At this study, vender will adopt batch delivery in order to reduce the inventory holding and increase the flexibility for changing the specification. Under the strategy of CRE, the buyer can pursue replenishment at the time of supplier’s delivery. The ordering (Ti

’) and replenishment period (Ti”) for buyer i, and the total inventory costs (Gic

b) before discount can be expressed by Eq. (6) and (7), respectively.

Ti’= NiT0

Ti”= niT0, where Ni, ni 1 (6)

(7)

Vender proposes a fixed replenishment period based on previous ordering information before buyers and vender reaching an agreement. The lowest acceptable price discount (Zi) can be estimated by Eq. (8):

(8)

Let the final price discount . If the total price discount is

higher than the increase amount of total inventory costs, buyers can then accept the CRE strategy proposed by the vender. Therefore, the total price discount ( ) gained by buyer i must satisfy Eq. (9):

(9)

Moreover, the ordering and replenishment period based on assumption 4

and 5 must be a multiple of a positive integer. Since the total price of the buyer possesses the characteristic of convex function, a variation method can be used to solve Eq. (7). Hence the corresponding Ni, ni

to have a lowest total price must satisfy Eq. (10):

(10)

Therefore, the total costs of order processing and delivery (Gc

v) and the total inventory costs (Gic

b) after price discount for buyer i can be expressed by Eq. (11) and (12), respectively:

(11)

(12) This research intends to pursue

minimization of vender’s total cost of ordering processing and delivery so as to obtain the most optimum replenishment period ( ), price discount (Z), and the buyer’s ordering and replenishment period. All equations involved in the above derivation are summarized by Eq. (13):

Min.

(13)

s.t.

, where both are positive integer.

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, , where a is a

positive integer.

4. Model Verification and Comparison

This section will justify the applicability of proposed method and make a comparison for each result. In this model, a two-echelon inventory model for a single-vender and five buyers will be used. The annual demand quantity (Di), cost of order processing (Ki), and replenishment processing (Ri)

are shown in the Table 1. The delivery cost (Ci) for vender is assumed to be 1,000 dollars and the unit inventory holding cost (hi) for buyers is fixed. When a cost compensation rate (S =10%) offered by vender, the vender must absorb the amount of the increased inventory cost. In other word, vender offers an appropriate price discounts (Z) to buyers.

Table 1: Numerical data for Relevant Cost

Buyer No.

i

Annual Demand

Di

Order Processing

CostKi

Replenish-ment

Processing Cost

Ri

1 200,000 100 2002 400,000 200 3003 600,000 500 1,0004 800,000 200 5005 100,000 100 200

Since the vender’s order processing cost is different from the cost of buyers’ unit inventory holding cost. The vender processing cost for As are 0, 100, 500 and 1000, Ai are 100, 500 and 1000, and buyers’ unit inventory holding cost for hi are 1, 2

and 3, will be utilized in this study. A comparison with Viswanathan and Piplani’s model (2001) is shown in Table 2. A numerical example for As = 100, Ai = 500 and hi = 1, the resulting total costs for vender and system with making decision independently are $247,716 and $362,309 respectively. If the buyers and vender reach an agreement, and buyers are willing to accept vender’s proposed replenishment period, the resulting optimum replenishment period will be five weeks. The total cost savings for vender and system are $151,303 and $175,132 respectively. The buyers will have 14.1% cost savings.

A numerical data is shown in Table 2. When vender takes buyers’ ordering, and decide to batch delivery upon both side agreements, the favorable price discounts offered by vender will entice the buyers to accept fixed replenishment period. The vender’s replenishment period (T0) will be decreased with the increasing of unit inventory holding cost. Also from Table 2, making decision based on this model can have a tremendous savings for both members. But in Viswanathan and Piplani’s model (2001), only the buyers can have benefits. In other conditions, the vender should take more costs. In the case of As = 0, Ai = 100, and hi = 1, the buyer would have 14.1% cost savings, but the cost of vender would increase to 55.7%.

Figure 1 and 2 show the correlation among vender order processing cost, vender savings, system savings and buyers’ unit inventory holding cost (hi = 1) with the condition of vender delivery cost, buyers’ unit inventory holding cost, and replenishment processing cost remained constant. It is shown by the figure when As is kept constant and Ai is increased, this will result in more cost savings for vender and system. The cost

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savings for vender and system will gradually be increased if Ai is kept at a specific value and As is increased. Figure 3 and 4 show the correlation among vender order processing cost, vender savings percentage, system savings percentage and buyers’ unit inventory holding cost (hi = 1) with the condition of vender

delivery cost, buyer unit inventory holding cost, and replenishment processing cost remained constant. From the figure, it is obvious that this model will increase the cost savings percentage for vender and system when As is fixed and Ai is increased.

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Table 2: Detailed results of numerical study

As Ai Unit

holding

cost,

hi

Discou

nt Z

(%)

CRE

T0

(weeks

)

System without

coordination

　

Savings with CRE strategy 　

Viswanathan and

Piplani’s model

Vender

cost

System

cost 　

Buyers’

savings

$

Vender

savings

$

System

savings

$

Buyer

s’

savin

gs %

Vend

er

savin

gs %

Syste

m

savin

gs %

　

Buyers’

savings

%

Vende

r

savin

gs %

Syste

m

savin

gs %

0 100 1 0.92 4 170,305 284,898 　 16,695 83,676 100,372 14.6 49.1 35.2 14.1 -55.7 0.8

0 500 1 1.72 5 232,234 346,827 　 25,700 141,541 167,241 22.4 60.9 48.2 14.1 -5.5 3.5

0 1000 1 1.72 5 309,645 424,238 　 25,700 216,352 242,053 22.4 69.9 57.1 14.1 0.8 4.8

100 100 1 1.95 5 185,787 300,380 　 28,742 90,284 119,027 25.1 48.6 39.6 14.1 -4.4 8.2

100 500 1 1.72 5 247,716 362,309 　 23,828 151,303 175,132 20.8 61.1 48.3 14.1 3.2 7.8

100 1000 1 1.72 5 325,127 439,720 　 25,700 230,795 256,495 22.4 71.0 58.3 14.1 5.0 7.5

500 100 1 0.92 4 247,716 362,309 　 21,200 155,540 176,740 18.5 62.8 48.8 14.1 29.8 23.3

500 500 1 1.72 5 309,645 424,238 　 25,700 213,752 239,453 22.4 69.0 56.4 14.1 20.7 18.7

500 1000 1 1.72 5 387,056 501,650 　 25,700 288,564 314,264 22.4 74.6 62.6 14.1 16.2 15.7

1000 100 1 0.92 4 325,127 439,720 　 21,200 226,452 247,651 18.5 69.7 56.3 14.1 37.6 31.0

1000 500 1 1.72 5 387,056 501,650 　 25,700 285,964 311,664 22.4 73.9 62.1 14.1 29.4 26.0

1000 1000 1 1.72 5 464,468 579,061 　 25,700 360,775 386,475 22.4 77.7 66.7 14.1 23.8 22.1

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0 100 2 1.55 3 240,847 402,907 　 31,827 119,936 151,763 19.6 49.8 37.7 13.9 -53.8 1.1

0 500 2 1.55 3 328,428 490,487 　 31,827 201,039 232,866 19.6 61.2 47.5 13.9 -4.9 3.8

0 1000 2 1.55 3 437,904 599,963 　 31,854 302,553 334,408 19.7 69.1 55.7 13.9 1.3 5.0

100 100 2 1.55 3 262,743 424,802 　 31,854 140,111 171,966 19.7 53.3 40.5 13.9 -14.4 4.9

100 500 2 1.55 3 350,323 512,383 　 31,854 221,269 253,123 19.7 63.2 49.4 13.9 0.1 5.8

100 1000 2 1.55 3 459,799 621,859 　 31,854 322,715 354,569 19.7 70.2 57.0 13.9 3.4 6.3

500 100 2 1.55 3 350,323 512,383 　 31,854 220,759 252,613 19.7 63.0 49.3 13.9 11.8 12.7

500 500 2 1.55 3 437,904 599,963 　 31,854 301,916 333,770 19.7 68.9 55.6 13.9 10.0 11.2

500 1000 2 3.21 4 547,380 709,440 　 38,133 403,568 441,701 23.5 73.7 62.3 13.9 9.1 10.2

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Table 2: Detailed results of numerical study (continued)

As Ai Unit

holdin

g cost,

hi

Discou

nt Z

(%)

CRE

T0

(weeks

)

System without

coordination

　Savings with CRE strategy 　

Viswanathan and

Piplani’s model

Vender

cost

System

cost 　

Buyers’

savings

$

Vender

savings

$

System

savings

$

Buyers’

savings

%

Vender

savings

%

Syste

m

savin

gs %

　

Buyers’

savings

%

Vender

savings

%

Syste

m

savin

gs %

1000 100 2 1.55 3 459,799 621,859 　 31,799 321,513 353,311 19.6 69.9 56.8 13.9 17.7 16.7

1000 500 2 1.55 3 547,380 709,440 　 31,854 402,726 434,580 19.7 73.6 61.3 13.9 15.0 14.8

1000 1000 2 3.21 4 656,856 818,916 　 38,078 506,180 544,259 23.5 77.1 66.5 13.9 13.1 13.2

0 100 3 0.90 2 294,977 493,458 　 30,310 142,892 173,202 15.3 48.4 35.1 13.2 -53.7 0.5

0 500 3 3.24 3 402,241 600,722 　 45,568 244,952 290,521 23.0 60.9 48.4 14.4 -14.5 -1.2

0 1000 3 3.24 3 536,321 734,802 　 45,568 376,484 422,052 23.0 70.2 57.4 14.4 -8.4 -1.6

100 100 3 0.90 2 321,792 520,274 　 30,310 167,108 197,418 15.3 51.9 37.9 13.2 7.8 11.5

100 500 3 3.24 3 429,057 627,538 　 45,568 270,035 315,603 23.0 62.9 50.3 14.4 -5.7 2.7

100 1000 3 3.24 3 563,137 761,618 　 45,568 401,566 447,134 23.0 71.3 58.7 14.4 -4.1 1.1

500 100 3 3.24 3 429,057 627,538 　 45,356 265,098 310,454 22.9 61.8 49.5 14.4 21.6 18.6

500 500 3 3.24 3 536,321 734,802 　 45,568 370,366 415,934 23.0 69.1 56.6 14.4 12.1 12.8

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500 1000 3 3.24 3 670,401 868,882 　 45,568 501,897 547,465 23.0 74.9 63.0 14.4 7.3 8.9

1000 100 3 3.24 3 563,137 761,618 　 45,568 390,555 436,123 23.0 69.4 57.3 14.4 29.4 25.2

1000 500 3 3.24 3 670,401 868,882 　 45,568 495,779 541,348 23.0 74.0 62.3 14.4 20.9 19.5

1000 1000 3 3.24 3 804,481 1,002,963 　 45,568 627,311 672,879 23.0 78.0 67.1 14.4 15.2 15.0

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Figure 1: Vender’s cost savings

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Figure 2: System Cost Savings

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Figure 3: The percentage of cost savings for vender

Figure 4: The percentage of cost savings for System

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5. Conclusions

This research explores a two-echelon inventory model for a single-vender and multi-buyer. By way of mutual agreement, vender can propose the most optimum fixed replenishment period, a reasonable price discount, and cost compensation rate in order to entice the buyers to accept the proposed strategy. From the model and associated data, we obtain the following conclusions:

1. Under the CRE strategy, the vender’s replenishment period (T0) will decrease with increasing buyers’ unit inventory holding cost (hi) when both sides reach agreement.

2. There would be a significant cost saving for each member if the upstream and downstream members of supply chain can communicate and integrate well.

3. If the vender order processing cost increases, the cost savings for both sides will also increase.

4. The main advantage of batch delivery to the buyers is cost savings. In addition, it is possible to have a specification modification in order to respond customer requirement in a short time.

5. This model can save a significant cost for vender and buyers at any time, but Viswanathan and Piplani’s model (2001) is only beneficial to buyers. There are only a few specific cases that are beneficial to both buyers and vender.

References

1. Banerjee, A., 1986, “A joint economic lot size model for purchaser and vender,” Decision Sciences 17, 292-311.

2. Hill, R.M., 1997, “The single-vender single-buyer integrated production-inventory model with a generalized policy,” European Journal of Operational Research 97, 493-499.

3. Hill, R.M., 1999, “The optimal production and shipment policy for the single-vender single-buyer integrated production-inventory problem,” International Journal of Production Research 37, 2463-2475.

4. Lu, L., 1995, “A one-vender multi-buyer integrated inventory model,” European Journal of Operational Research 81, 312-323.

5. Thomas, D.J. and Griffin, P.M., 1996, “Coordinated supply chain management,” European Journal of Operational Research 94, 1-15.

6. Viswanathan, S., 1998, “Optimal strategy for the integrated vender-buyer inventory model,” European Journal of Operational Research 105, 38-42.

7. Viswanathan, S., and Piplani, R., 2001, “Coordinating supply chain inventories through common replenishment epochs,” European Journal of Operational Research 129, 277-286.

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