## 5 Mathematical Formulation, Assumptions and Notation
### Mathematical Formulation of the Inventory Control Problem  
The problem described in the above sections is considered as a time-continuous multiproduct inventory control problem. For each product \(i\) the total deterministic demand should be satisfied within a finite time horizon \(T=2\) years (24 months) in the following manner:  
* The same or variable amount \(Q_{t}^{i}\) of product \(i\), depending on the ordering policy, is ordered with no constant time \(t_{i}\) between the two orders.
* The ordered amount \(Q_{t}^{i}\) arrives on the stock simultaneously and immediately after \(LT\), while products are withdrawn from the stock continuously by the rate of the sales plan.  
The problem is modelled as the corresponding discrete-time system control process. Rather than continuous time periods for products ordering, the entire time horizon \([0,T]\) is divided into \(n\) periods \(t\) with the same length \(T/n\), where \(t=1,\ldots,n\). For example, if \(T\) is a year then \(t\) is a month. Ordering of any product can be realized only at the beginning of a period \(t\). During this period of length \(T/n\) the products are withdrawn from the stock continuously in accordance with the sales plan \(Y_{t}^{i3}\).  
The decision variable \(Q_{t}^{i}\) is generated in the model for all time periods. The inventory state of the product \(i\) changes during the entire period of time. The inventory system can be formally represented as a discrete-time system control process with the following elements: \(X_{t}^{i1},\ X_{t}^{i2},\ Y_{t}^{i1},\ Y_{t}^{i2},\ Y_{t}^{i3}\) and \(Y_{t}^{i4}\). If we consider \(X_{t}^{i1}\) and \(X_{t}^{i2}\) as the state of a process at the end of each period \(t\), then Equations (1)-(3) describe the low of behaviour of the discrete-time process, defined as follows. At the beginning of the time horizon, the initial inventory level is known and has to be a non-negative value (\(\geq\)0).  
\[X_{0}^{i1}=known,\ X_{0}^{i2}=known,\ i=1,\ 2,...,\ \text{m} \tag{1}\]  
Inventory dynamics of accumulation \(X_{t}^{i1}\) are described by Equation (2) and appoint that the planned stock level of product \(i\) on hand at the end of period \(t\) depends on the stock level of product \(i\) at the end of the previous time period \(t-1\), increased by all inputs and decreased by all outputs during the time period \(t\).  
\[X_{t}^{i1}=\left\{\begin{array}{cc}0,&X_{t-1}^{i1}+Y_{t}^{i1}-Y_{t}^{i3}<0 \\ X_{t-1}^{i1}+Y_{t}^{i1}-Y_{t}^{i3},&otherwise\end{array}\right\}\,\ S_{t}^{i}=0\right\},\ t= \overline{1,\ 24},\ \ i=\overline{1,m} \tag{2}\]  
(LT/2). The amount of average stock corresponds to the average sale for at least half of a lead time period. It is expressed as:  
\[AS_{t}^{i}=SS_{t}^{i}*\frac{LT}{2},\ t=\overline{1,\ 24},\ i=\overline{1,m} \tag{11}\]  
The planned re-order quantity \(Q_{t}^{i}\) realizes at the beginning of a period \(t\) only in the case when the stored quantity of product \(i\) remaining at the end of the previous time period \(t-1\) is not greater than the average stock on hand in time horizon \(T\). This decision variable is generated by relation (12) and depends on the ordering policy. For FOQ policy, quantity of product \(i\) is defined by  
to obtain the demand (e.g., demand for three months at the end of the time horizon) in the case of LFL policy. The discrete controlled object model, with the previously described law of dynamics, control domain, performance criterion, and all discrete equations and inequalities, remains the same as the model of the first instance, but the \(minQ^{i}\) for FOQ policy will be changed with the LFL order quantity \(LQ^{i}_{t}\). The new decision variable \(LQ^{i}_{t}\) for period \(t\) can be defined as:  
\[\begin{split}\Delta t^{i}=known,& i=\overline{1,m}\\ LQ^{i}_{t}=\left\{\begin{array}{l}\Delta t^{i}\cdot SS^{i}_{t},Y^{ 3}_{t+LT+\Delta t^{i}}=0\\ \sum_{t+LT+\Delta t^{i}}^{t+LT+\Delta t^{i}}Y^{i3}_{t},\text{ otherwise}\end{array}\right\},t=\overline{1,\ 24},\ \ i=\overline{1,m}\end{split} \tag{17}\]