## 4 Methodology
### Discrete-Time System Control--System Dynamics Modelling  
System dynamics (SD) is a modelling approach widely used in logistics and supply chain management [45]. System dynamics (SD) relates the modelling of processes over time. SD is often considered a modelling technique based on continuous time [46]. In this paper, we consider SD modelling also for the discrete concept of time. Finite changes over time mode means dealing with changes over time in a manner that resembles our everyday experience. It is based on a simple principle: \"In order to notify some change, some time has to pass by\" [46]. Changes are related to time intervals, whereas the state of a variable is specified for specific points of time. The distinction between time-points and time-intervals yields two distinct types of data: data related to time-points and data related to time-intervals. The data relating to time-points will be named stocks and the data related to time-intervals flows. If the stocks are considered as state variables for time-points and the flows as changes of the stocks for certain time-intervals, the relation between the stocks and the flows is trivial arithmetic. For a given time interval (\(t_{0}\), \(t_{1}\)) and given flows for that time interval we can calculate the \"new\" value of the stock at the end of the time interval according to the following equation [46]:  
\[stock(t_{1})=stock(t_{0})+\mathit{inflows}(t_{0},t_{1})-\mathit{outflows}(t_{0},t_{1})\]  
A stock accumulates its flows. According to the stock-flow principle [46], the new \(stock(t)\) is defined through the initial \(stock(t_{0})\) plus all inflow(s) subtracted by all the outflow(s) between the time \(t_{0}\) and time \(t\). The core idea of SD modelling is the accumulation of flows over finite time intervals of duration discrete time periods, \(t\). On the stock-flow diagram, there are two main types of elements: flow regulators (actions) and stock as state variable (accumulations) [47]. The control model of the problem defined in this paper implies two flows (Figure 3), the information flow of the planned inventory state and the material flow of the actual state of inventory. Each flow has three phases, one phase of accumulation and two phases of action. The action phases represent input and output flow regulators, respectively. The accumulation, representing inventory level, for each flow, is calculated with the quantity of the inventory from the end of the previous time period and is increased by the planned or realized quantity for delivery over the observed period of time, and it is reduced by the planned or realized sales over the period of time. For example, Figure 3 shows the stock-flow diagram for one item (one stock keeping unit-SKU, \(t=1\dots 24\) months). In order to define a discrete-time mathematical model of the problem, a notation will be defined per type of element of the system and from the position of the element on stock-flow diagram  
Figure 2: Elements of the feedback loop for the inventory control model. (from top to bottom, and from left to right). For the stock-flow diagram presented by Figure 3, the following notations will be used:  
* \(X^{1}_{t}\)--State variable. Planned stock level of a product on hand at the end of period \(t\). * \(X^{2}_{t}\)--State variable. Realized stock level of a product on hand at the end of period \(t\). * \(Y^{1}_{t}\)--Inflow regulator variable (left upper). Planned stock input of a product at the beginning of period \(t\). * \(Y^{2}_{t}\)--Inflow regulator variable (left lower). Realized stock input of a product at the beginning of period \(t\). * \(Y^{3}_{t}\)--Outflow regulator variable (right upper). Forecasted sales plan for a product for period \(t\). * \(Y^{4}_{t}\)--Outflow regulator variable (right lower). Realized sales of a product at period \(t\). The regulator variables have the same notation \(Y\), because of the same nature of the flow, but have different superscripts suitable for the writing of equations describing the system.