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.
2.1 Model AssumptionDevelopment 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.
2.2 System ModelAs 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.
2.2.1 Model VariablesThe 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:
2.2.2 Causal Loop DiagramsCausal 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 contains supplier and retailers. For the supplier, 
is decided by 
and
. 
is determined commonly by 
and
. Delivery rate 
is determined by 
and
. Supplier adjust inventory level by setting
, together with 
determine
. 
and 
determine
, in turn, 
has a direct impact on
and an indirect impact on
. For retailers, 
is determined by 
and
. 
is the delay of
, delay time is
. 
is decided by 
and
. 
is obtained from 
after the time
. Retailers also adjust inventory level by setting
. 
is decided by 
and
. 
and 
jointly determine
. 
and 
commonly determine
. If 
is greater than 0, Retailer sent orders to suppliers, order quantity 
is decided by
.
, 
, and 
determine
. There are two performance variables, customer demand satisfaction rate 
and total inventory
. 
is decided by inventory 
and customer demands
, 
is a accumulation sum of supplier inventory 
and
.
Figure 3 describes the causal loop diagram of supply chain with ULT. 
means the transshipment from retailer 2 to retailer 1. 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. 
is decided by 
and
, and influence
.
policy [