Influence of Lateral Transshipment Policy on Supply Chain Performance : A Stochastic Demand Case

Considering the supply chain consists of one supplier and two retailers, we construct the system’s dynamic models which face stochastic demand in the case of non-lateral transshipment (NLT), unidirectional lateral transshipment (ULT) and bidirectional lateral transshipment (BLT). Numerical example simulation experiments of these models were run on Venple. We adopt customer demand satisfaction rate and total inventory as performance indicators of supply chain. Through the comparative of the simulation results with the NLT policy, we analyze the influence of ULT policy and BLT policy on system performance. It shows that, if retailers face the same random distribution demand, lateral transshipment policy can effectively improve the performance of supply chain system; if the retailers face different random distribution demand, lateral transshipment policy cannot effectively improve the performance of supply chain systems, even reduce system’s customer demand satisfaction rate, and increase system inventory variation.


Introduction
Lateral transshipment, an important inventory replenishment policy, has gained a common concern of the academics and business managers in recent years.There are numerous researches about this issue.Lateral transshipment is defined as the redistribution of stock from retailers with stock on hand to retailers that cannot meet customer demands or to retailers that expect significant losses due to high risk [1].The early pioneering work of Krishnan and Rao [2] examine a periodic review policy in a single-echelon, single-periodic setting [2].Robinson [3] extends the research to a multi-period case, and establishes the system's lateral transshipment model [3].The emergency lateral transshipment model of repairable product was analyzed by Lee [4] for the two-echelon inventory system case [4].Axsäter [5] studies the emergency lateral transshipment problem of the multi-level repairable product inventory system, and gets some interesting conclusions different from Lee's [5].Archibald et al. [6] develop a lateral transshipment model among the multi-retailer based on Markov decision-making methods [6].Grahovac and Chakravarty [7] limit the research object to low demanded expensive product and analyze the lateral transshipments model in a multiechelon supply chain system [7].Kukreja et al. [8] consider a single-echelon continuous review inventory system which contains n depots, and takes the expensive consumable product as object, study the one-to-one lateral transshipment model [8].Rudi et al. [9] work on optimal order policy of the vendors in the existence of lateral transshipment circumstances [9].Minner and Silver [10] provide a new decision rule of system's lateral transshipment, and prove it can figure out the size of transshipment as well as some important problems [10].Xu et al. [11] analyze emergency lateral transshipment policy between the two-echelon continuous review inventory system that use   , Q R policy [11].Banerjee et al. [12] study lateral transshipment of two-echelon supply chain systems which include multiple retailers and single supplier based on DOE [12].Xu and Luo [13] use Expect Cost Method to analyze lateral transshipment policy in cross-docking system [13].Xu and Xiong [14] analyze the best time for one-off transshipments in a cross-docking system with stochastic demand [14].Wang et al. [15] conduct a quantitative analysis to the value of lateral transshipment policy of regional inventory distribution systems, which consist of a distribution center and multiple retail points [15].Huo and Li [16] develop batch ordering policy in a single-echelon, multi-location transshipment inventory system [16].Li et al. [17] study inventory management model of the cluster supply chain system with the existence of emergency lateral transshipment [17].
Most of the papers above dealing with transshipment assume that lateral transshipment already exists in system.However, lateral transshipment will make the problem complicate and tend to be very difficult to analyze analytically, especially BLT [18].Hence, will lateral transshipment really need?This paper handles this problem from the performance measurement point.We consider one supply chain system consists of one supplier and two retailers, allowing a retailer transship from the other one for inventory replenishment besides order from supplier.System's models were developed by system dynamics assume that all the members use the order-up-to policy, and numerical experiment was run on Venple platform.
The paper is organized as follows: in Section 2, the models with lateral transshipment were developed, as well as without lateral transshipment.The accuracy of the model is tested against simulation in Section 3; Section 4 deals with the influence of ULT and BLT.The conclusion of this paper was presented in Section 5.

Model Description
We consider two retailers facing independent stochastic customer demand and one supplier (Figure 1).Without lateral transshipment, retailers order from suppliers to replenish inventory in case of stock out.In order to respond to customer demand quickly, they can use lateral transshipment policy besides order from the supplier, which means they replenish inventory from the other one if there exist surplus stock on hand.

Model Assumption
Development of the model needs the following assumptions.
 Customer1 and Customer2 face independent stochastic demand;  Both retailers adopt order-up-to policy, the ordering period is constant;  Lateral transshipments take no time;  Transshipments take place when there are surplus stocks.That is, if retailer 1 needs transshipment from retailer 2, retailer 2 only transships the redundant stock.

System Model
As a modeling and simulation technology, system dynamics has a wide range of applications since its birth, especially in dealing with long-term, chronic, dynamic management problems [19].Forrester [20] applies system dynamics in industrial business management, addressing issues such as fluctuations in production and employees, instability of market shares and market growth [20].Logistics and Supply Chain Management is an important area of System Dynamics.Sterman [21]designs the well-known beer game by System Dynamics, and carries out detailed analysis on feedback loops, nonlinear, time-delay and management behavior in the system [21].Diseny et al. [22] analyze VMI in transport operation by system dynamics [22].Marquez [23] establishes a model for measuring financial and operational performance in the supply chain based on System Dynamics [23], and so on.
Generally speaking, a complete system dynamics model usually consists of three parts: model variables, causal loop diagrams and mathematical description.We analyze the three part of model in turn as follows.

Model Variables
The structure of a system dynamics model contains stock, smoothed stock, flow rate, auxiliary variables and constants.Stock variables are used to describe the cumulative effect of the system.Smoothed stock variables are the expected values of specific variables obtained by exponential smoothing techniques.Flow rate describes the rate of the cumulative effect of the system.Auxiliary variables are the middle variables which express the decision-making process.Constants change little or relatively do not change during the study period.The fundamental notations of the model are following:

Causal Loop Diagrams
Causal loop diagram is a tool that expresses the structure of the system, playing an extremely important role in system dynamics.There are two reasons for that.First, during model development, they serve as preliminary sketches of causal hypotheses and secondly, they can simplify the representation of a model.
The first step of our analysis is to capture the relationship among the system operations in a system dynamics manner and to construct the appropriate causal loop diagram.Figure 2 describes the causal loop of the supply chain without lateral transshipment.
The system structure in Figure 2

Mathematical Description
The next step of system dynamics the development of the mathem presented as a stock-flow diagram that captures the model structure and the interrelationships among the variables.Combining mathematical description and causal loop diagram of variables, we use visual simulation software (Venple) to reflect the behavior of the system and provide a basis for decision-making.We analyze the mathematical description of the main variables in the system under the three cases of NLT, ULT and BLT so as to lay a good foundation for model simulation.
NLT case (1) is the inventory dynamics equation o is the inventory dynamics equation of retailer (3) is the average order quantity equation o (4) is t f retailer i , he average sales rate equation of retailer i .

 
(5) and ( 6) are the order rate equations of suppliers and retailers, in which are the delay function of the correspondg order quantities in a given period of time.(7) . ( 14)～( 16) express the auxiliary variables equations of retailers.( 14) and ( 15) express the desired inventory equation and the inventory gap equation of retailer .( 16) is the order quantity equation of retailer .When is the total inventory equation of the supply chain system.( 18) is the customer demand satisfaction rate equation of the supply chain system.When is a random variable, and it obeys a certain random distribution.9) is red inventory eq 0) is the order rate equation of supplier.(11) is the inventory gap equation of supplier.( 12) is the inventory adjustment rate equation of supplier.( 13) is the ordquan ULT case.
0)～(23 and their e nations are similar with NLT.BLT case. (2 ) express the variables changed in ULT, xpla  29) are the equations of variables in the BLT situation, and the other variables which were not included in the above expression are the same as the varibles in the NLT.The explanations of above equations r ich is the validity of the set of relampared with the real process.aws in system dynamics mod-and behavior sensitivity tests [24].Generally, the goal of extreme test lies in whether the equations a ingful or not and whether the model condi reasonable or not under the situation of using extreme in-( 27) a a e the same as the ULT case, and the concrete explanations will be omitted.

Models Validation
The main criterion for system dynamics models validation is structure validity, wh tion used in the model, as co For detection of structural fl els, certain procedures and tests are used.These structure validity tests are grouped as direct structure tests and indirect structure tests.Direct structure tests involve comparative evaluation of each model equation against its counterpart in the real system (or in the relevant literature).Direct structure testing is important, yet it evolves a very qualitative, subjective process that needs comparing the forms of equations against real relationship.It is therefore, very hard to communicate to others in a quantitative and structured way.Indirect structure testing, on the other hand, is a more quantitative and structured method of testing the validity of the model structure.The two most significant and practical indirect structure tests are extreme-condition put.Behavior sensitivity test consists of determining those parameters to which the model is highly sensitive and re still meantions are still asking if these sensitivities would make sense in the real system.If we discover certain parameters to which the model behavior is surprisingly sensitive, it may indicate a flaw in the model equations.
In view of the deficient of real data and the difficulty of gaining mass data, we only use the extreme condition test and the sensitive test in the indirect structure tests to carry on the validation for the models of this paper.We use the total inventory which is a main output of the system to confirm the performance in the extreme test and the sensitive test.Suppose that the system is facing an extreme demand, we use Venple to test the models which are NLT, ULT and BLT, and compare the simulation results with the normal demand case.The results are shown in Figure 5.Although the total inventory becomes very big with the sharply increasing demand, the tendency is similar with the normal demand case.This indicates that the variables and the equations in the system still have effects, and the model displays a good robustness under the extreme input.Through changing the demand parameters, we use Poisson distribution to test the models' sensitivity, and the simulation experiment is also carried on the Venple.Fig- ure 6 presents the result that compare with the normal parameters case.We can see although fluctuation range of the total inventory is inconsistent under the two kinds of demand with different parameters, the tendency displays as basically consistent.This explains that the models don't sensitively response to the parameters, and this is conducive for the model's practical application.

Numerical Simulation
We employ a numerical example to examine the influence of three different policies on system performance, and further analyze the simulation results and provide reasonable suggestions.In order to facilitate the numerical simulation, we set the initial value of the constants, . Where T is the simulation time.

Same Distribution Demand Case
We assume retailer i face same distribution demand that is Poisson distribution with range from 50 units to 150 units.Simulation results are shown in Figure 7, Figure 8 and Table 1.
From the simulation results, the system has the lowest total inventory in BLT case, the average size is 540.2057 units and the standard deviation is 192.2115units; The following is that in ULT case with its mean is 546.1087 units and the standard deviation is 199.2071units; Under the situation of NLT, the system has the largest total inventory, its average size is 671.9624 units and the standard deviation is 215.6611 units.It can be seen, lateral transshipments policy reduces the system's total inventory.However, by comparing with BLT, we can find that ULT policy reduce more total inventory.In addition, from the standard deviation of the total inventory we can see that lateral transshipment policies make the system's total inventory stabilized.Comparing with ULT, BLT cannot make significantly advantages.Thus it can be seen, in terms of the system's total inventory, lateral transshipment policy can effectively reduce the size and fluctuation of total inventory, but if we view from effect, little difference can be seen between BLT and ULT. From

Different Distribution Demand Case
We assume retailer face different distribution demand.Retailer 1 is still subject to the Poisson distribution as described above.Retailer 2 is subject to the Normal distribution with the range from 50 units to 150 units.Simulation results are shown in Figure 9, Figure 10 and Table 2.
Comparing with that in NLT situation, the system total inventory decreases in BLT and ULT cases, but slightly.In addition, viewing the standard deviation of total inventory in three varieties of policies, we can see that it decreases slightly in BLT case.It shows th with the different distribution demand case.usion of this paper, we believe that ts make the system handle inventory ses the total inventory, and improves demand.The main reason may be that different distribution demand will make ordering and replenishment become extremely complex.Moreover, if the retailers still use a separate order-up-to policy, lateral transshipments may becomes impossible and difficult to improve system performance.Hence, considering lateral transshipment, how to find the optimal inventory control policy rather than simply use order-up-to policy in different distribution case will be our further research problems.

Conclusions
Taking the supply chain system that includes a supplier and two r the influence of lateral transshipments policy on supply chain performance based on system dynamics.We established NLT model, ULT model and BLT model.Through the simulation analysis of these three different models of the supply chain system by Venple, we found that: first, if the two retailers are facing the same distribution demand, lateral transshipments not only reduce total invenry but also increase the customer demand satisfaction rate.Moreover, the effect is more obvious in ULT case; secondly, if the two retailers are facing with the different distribution nd, lateral transshipments reduce total inventory of the s but the extent is not obvious.However, it decreases the customer demand satisfaction rate of the supply chain system.
As to the first concl lateral transshipmen rationally.It decrea customer demand satisfaction rate.It is an inventory control policy that is worth popularizing.For the second conclusion, we question the suitability of the lateral transshipments policy under the different distribution 1992.
[2] K. S. Krishnan and V. R. K. Rao, "Inventory control in n warehouses," Journal of Industrial Engineering, Vol.

Figure 1
Figure 1.2-echelon supply chain.Filled arrows represent the flow of regular replenishments while dashed arrows represent the lateral transshipment flow

Figure 3 .
Figure 3. Causal loop diagram of the supply chain with unidirectional lateral transshipment

Figure 5 .
Figure 5.Total inventory in normal condition and extreme condition

Figure 7 .Figure 8 .
Figure 7.Total inventory in the same distribution demand case

igure 9 .Figure 10 .
Figure 10.Customer satisfaction rate in the different distribution demand case

Figure lateral t 2. Causal loop pment diagram of the supply chain without ransshi
IGRi , IARRi , and AOQSTRi determine AOQRi There are tw .rforman s, customer satisfaction rate CSRRi and total inventory T CSRRi is decided by inventory IRi and customer demands CDRi , TI is a accumulation sum of supplier inventory SI and IRi .It is a flow rate variable and means that when retailer 1 out of stock, retailer 2 will replenish retailer 1 by transshipment on condition that it has surplus stock.CDRi and IRi , and influence OQRi .
cided by and an indirect impact on ORS .For retailers, is determined by ORRi and . is the delay of , delay time is .is de-SRRi after e time etailers also adjust .DIRi is decided by ASRRi and DICTRi .DIR jointly deter-IGRi .Ri com nly determine IARRi G greater 0, Retailer sent orders der quantity OQRi is decided by CDRi .ASRRi ASRSTRi .Copyright © 2009 SciRes iB ASRRi .T is decided by 12 L T is a flow rate variable and means that when retailer 2 is out of stock, retailer 1 will replenish retailer 2 ansshipment on condition that it has surplus stock.by tr 12 LT is decided by CDRi and IRi, and influence OQRi .
is the equation of supplier shipment rate to retailers.
CDRi , meaning that customers' demand will be m ly,

Table 2
CSRR are 0.618, 0.594 an respectively.From this point, we deduce th transshipment may be not compatible with the different distribution demand.