Abstract -Lateral transshipment within an inventory system reallocate the inventory of locations to match the demand and supply again when demand is completely observed. This paper addresses issues about the coordination mechanism for the lateral transshipment problem in two-location system with identical product. Firstly, we introduce the conditions for coordination mechanism proposed by Rudi et al. Then, we propose the Mutual Compensation contract considering arbitrage behavior at shipping stage and ordering stage. We find that there exist Nash equilibrium and the Mutual Compensation contract can coordinate the two-location system.

Keywords– Emergency lateral transshipment, two-location system, coordination mechanism, inventory management

I. INTRODUCTION

With uncertain demand in many manufacturing, distribution, and retailing businesses, stock-out are endemic like the retail industry which lead to lost sales of $7-12 billion annually[1]and overstocks are common like the products sold with a markdown price at outlet store which lead to new business model like overstock.com. In order to reducing the under and over stock cost, Bosch, one of the world’s largest producers of power tools and electric parts for automobiles make the production decision considering the distributor who has surplus stock transfer their product to the distributor who is stock out when demand is known[2]. Balancing the supply and demand after ordering period, such lateral transshipment behavior is desirable to improve the service level and make more profit and reduce the system cost. Other examples in the published literature are: flu shots at U.S.A.[3]; formal dress ware at Giorgiou in New York City[4] and various other industries[5],[6].

The lateral transshipment after demand is fully realized is called emergency lateral transshipment (ELT)in [7].Focusing on the research on coordination mechanism about ELT problem in decentralized system, reference [2] firstly considers decentralized decision makers and prove that ex ante transfer price mechanism

can coordinate the single-period two-location ELT problem. Thereafter, reference [8]presents a counterexample of [2] and show that the predetermined transshipment price may easily fail to exist in a range of cases. They derive the sufficient and necessary condition for the existence of coordinating price. Their counterexample is used by [9] to test their pricing mechanism based on General Nash Bargaining Solution and also used in this paper to test the Mutual Compensation contract proposed by us. Reference [10] compares ex ante and ex post transfer price policy and get the ex ante and ex post transfer price policy cannot coordinate the two-location system. For multi-location system, ex post transfer price policy can coordinate the system well (Please see [10], [11]). Reference [12] proposes a transshipment fund mechanism to coordinate multiple locations.

The review paper about the lateral transshipment research done by [13] says[8]“highlights an area of future research which could consider more complex pricing structures.” ([13], P131).Actually, the papers proposed coordination mechanism for the two-location system we are aware of till now are only[9], [12]. Our paper is most closely related to [2], and we are interesting to find a coordination mechanism. We analysis the condition of ex ante transfer price mechanism and then propose the Mutual Compensation contract which is sharing the risk within the two locations for the unsold products. We prove that our policy can coordinate the ELT problem in two-location system.

The rest of this paper is organized as follows. In section 2, we present the model, notation and analysis the condition of ex ante transfer price mechanism. Section 3 is devoted to analyze the Mutual Compensation contract we proposed and get the Nash equilibrium existence of ordering and prove that the policy can coordinate the ELT problem in two-location system. In section 4, we use the counterexample proposed by [8] to test our contract and find the coordination solution. Section 5 contains concluding remarks.

Inventory Coordinating Contract for the Two-location Supply Chain with Lateral Transshipment

Guangtao Guo, Xiaohong Li Da Yang Jiaying Yuan School of Management

Xi’an Jiaotong University Xi’an, 710049, China

E-mail:guoguangtao@gmail.com

School of Economics and Finance Xi’an Jiaotong University

Xi’an, 710049, China E-mail:xuanxuan.07.22@stu.xjtu.edu.cn

China Mobile Group Design Institute Co.,Ltd Shaanxi Branch

Xi’an, 710077, China E-mail:jiayingyuan@gmail.com

2012 Fifth International Joint Conference on Computational Sciences and Optimization

978-0-7695-4690-2/12 $26.00 © 2012 IEEE

DOI 10.1109/CSO.2012.158

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II. MODEL

In this paper, we focus on finding new coordination mechanism on the model in [2]. We share the same model as it and do not present it again here briefly.

A. Notation

We also share the same definition of all the parameters if there is no special explanation. We simply introduce all the parameters for location I as fixed unit cost like production cost themselves or wholesales price setting by supplier , unit revenue , unit penalized cost , unit salvage value , and the marginal value of additional sales . The transshipment price paid by location i to location j is and the transshipment cost per unit transshipped from location i to location j is which is assumed it is incurred by location i. and is the demand and inventory decision of location i respectively. We assume the demand is continuous and twice differentiable density function with positive values. At the shipping stage, the transshipment quantities from location i to location j is min , , retail sales at location i are min , , the unsold stock is

, and the unmet demand is

Different from definition of , , in the table 1([2], P1672), we set α Pr , Pr , Pr

For the situation that the demand follows continuous distribution, there is no significance difference between the definitions of , , of [2]and our definition, but it work not very well for the situation that the demand follows discrete distribution. Taking the counterexample of [8], the location 2 meets the deterministic demand as 1 unit. We have 1,3Pr 1 2, 3 3 0.7 with one or two transshipment unit, but 1,3 Pr 1 2 3 0.32only with one unit. Clearly, our definition is more accurate for the demand with discrete distribution. Same as [2], we assume that c c τ , and .

B. The ex ante transfer price mechanism

We show the reason why ex ante transfer price mechanism[2]does not work for some situation. According to[2] presenting in Proposition 5, the coordinating transshipment prices expression (17) at page 1675 is

Reference [2] doesn’t pay attention to is that0 at some cases. Clearly, we cannot get as the above expression when 0 like special case in [8]. Reference [2] extend the condition and find that the existence of linear coordinating transshipment price. Be back to special case in[8] (we show the case in section 4 in this paper). Intuitively, as the inventory decision {1, 3} in centralized system, retailer 2 takes more risk than retailer 1. The Theorem 1 of [8] tells us that there is no linear transshipment price when one retailer will take more risk and the other will take less. It arise us that the coordination mechanism should balance the risk.

III. MUTUAL COMPENSATIONCONTRACT A. The Mutual Compensation contract

In this section, we propose the Mutual Compensation contract to share the risk for the two locations which is the transshipment price per unit pays by location j to i, the bonus price per unit is paid by retail j to i when location i has unsold stock. Then we have the expected profit for location i is, , ,

(1)

Avoiding the arbitrary behavior, the contract must satisfy that the location i cannot benefit from ordering more at the ordering stage firstly which is,

for i=1,2 (2)

At the shipping stage, the contract must incentive the location willingness to transship. Then benefit from transshipment must more than no transshipment if one location has excess inventory and another location are stock-out so that the contract coordinate the shipping stage which is,

for i=1,2 (3)

for i=1,2 (4)

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for i=1,2 (5)

Simplify the above expression (2-5), we have

for i=1,2 (6)

for i=1,2 (7)

for i=1,2 (8)

Theorem 1, There exists Nash equilibrium of ordering decision , .

PROOF. According to the expected profit of location i(expression (1)), we have the expected marginal profit of an additional unit is,

, 1 α ,, ,α , α, . (9)

Define the following marginal probabilities: Pr | , Pr | , Pr | , Pr | ,

and .

Take expression (9) as the implicit function of , , we can get

. Because of n, and , we then

know 1 0 . Then there exists the Nash

equilibrium of ordering decision , . □

B. The coordination solution

As a benchmark, we study the conditions characterizing the optimal inventory order in the centralized system

firstly. We have the expected marginal profit of an additional unit is,

, , 1 α ,, ,α , . (10).

We know the optimal ordering , satisfy , , 0 for location i, then we have,

α ,, (11)

Theorem 2. There exists the Mutual Compensation contract { , , } that coordinate the two-location

system where , and

PROOF. Combine expression (9) and (11), then we have the solution for the { , , , } are,

We set . According to the definition of , , , we have 0 if there is transshipment with probability, and 0(if not, then all the time. It is not the optimal ordering decision.)Then, we get

□

Corollary 1: For the case the two locations are identical, we have 0 , and the transshipment price itsself can coordinate the system.

IV. NUMERICAL TEST

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Different from the assumptions of the demand in this paper, we use the counterexample proposed in [8] which shows the contract in [2] does not work. It is worth us to use as the test of our contract. The same as [8], let unit revenues r={11, 11}, transshipment cost ={4, 1}, salvage value per unit s={1, 4}, cost like production or raw material whole sell price c={5, 5},and unit penalty costs can be negligible or are including in unit revenues r. The bold symbol here without any subscript represents a vector, e.g. r={r1, r2}. For the demand information, Location 1’s demand D1 is {1, 2, 3} with probabilities {0.3, 0.32, 0.38} respectively, and location 2 has deterministic demand D2 is 1.

TABLE I THE EXPECTED PROFITS WITH MUTUAL COMPENSATION CONTRACT

Q1 Q2

1 2 3

Expected profit of location 1 with different ordering combination

1 6 8.57 9.74

2 8.21 9.38 8.88

3 6.39 5.89 5.39

Expected profit of location 2 with different ordering combination

1 6 6.63 6.74

2 6.79 6.90 6.40

3 8.41 7.91 7.41

According to the value of , , , we have the coordination solution 7.11 , and 0.30.62 0.7 2.11. Set , , as {7.11, 0.5, -2.624} respectively, test the value and we get there is no arbitrage behavior. We have the expected profit of each location for the different ordering decision combination (see Table 1).

In table 1, the ordering decision (1, 3) is the Nash equilibrium solution and is the centralized optimal solution with total expected profit of 16.48.

IV. CONCLUSION

We consider a single-period two-location ELT problem with uncertain demand based on model of [2]. We focus on the problem why ex ante transfer price mechanism proposed by [2] cannot coordinate the ELT problem in two-location system. We then proposed the Mutual

Compensation contract. Within the contract, we get the Nash equilibrium of ordering and find it can coordinate the ELT problem in two-location system. We also get the coordination solution for counterexample in [8]. It is worth mentioning that the positive or negative value of bonus { , } for the unsold unit at the end of sales. For the negative value just like the solution in the numerical test, it is practical especially when all the locations use electronic data interchange to share data or they employ the third supply chain auditor.

ACKNOWLEDGMENT

The authors thank Prof. Schrage’s valuable comments and discussions which greatly improve this paper.

REFERENCES

[1] Z.Ton, The Role of Store Execution in Managing Product Availability. Harvard University Press, Cambridge, MA. 2002.

[2] N. Rudi, S. Kapur, D. F. Pyke, “A two-location inventory model with transshipment and local decision making”,Management Science, vol. 47 no. 12, pp. 1668-1680, 2001.

[3] R. Rubin, “Imports may boost USA’s flu shot doses”, USA Today. Oct. 2004.

[4] L.Dong, N. Rudi, “Who benefits from transshipment? Exogenous vs. endogenous wholesale prices”,Management Science, vol.210, no. 5, pp. 645–657, 2004.

[5] H. Zhao, V. Deshpande, J. K. Ryan, “Inventory sharing and rationing in decentralized dealer networks”, Management Science, vol. 51, no. 4, pp. 531-547, 2005.

[6] H. Zhao, J. K. Ryan, V. Deshpande, “Optimal dynamic production and inventory transshipment policies for a two-location make-to-stock system”, Operations Research, vol. 56, no. 2, pp. 400-410, 2008.

[7] G. Tagaras, “Effects of pooling on the optimization and service levels of two-location inventory systems”,IIE Trans, vol. 21, pp. 250-257, 1989.

[8] X. Hu,I. Duenyas, R. Kapuscinski, “Existence of coordinating transshipment prices in a two-location inventory model”, Management Science, vol. 53, no. 8, pp. 1289-1302, 2007.

[9] B. Hezarkhani, W. Kubiak, “A coordinating contract for transshipment in a two-company supply chain”,European Journal of Operational Research,vol. 207, pp. 232-237, 2010.

[10] X. Huang, G. Sošic, “Transshipment of inventories: Dual allocations vs. transshipment prices”, Manufacturing & Service Operations Management, vol. 12 no. 2, pp. 299-318, 2010.

[11] R. Anupindi, Y. Bassok, E. Zemel, “A general framework for the study of decentralized distribution Systems”,Manufacturing & Service Operations Management, 3(4), 349-368, 2001.

[12] E. Hanany, M. Tzur, A. Levran, “The transshipment fund mechanism: Coordinating the decentralized multi-location Transshipment Problem”, Naval Research Logistics, vol. 57, no. 4, pp. 342-353, 2010.

[13] C. Paterson, G. Kiesmuller, R. Teunter, K.Glazebrook,“Inventory models with lateral transshipments: A review”,European Journal of Operational Research,vol. 210, no. 2, pp.125-136, 2010.

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