## 5 Mathematical Formulation, Assumptions and Notation
### Notation  
In order to explain the model, the following notations are used:  
* \(m\)-Total number of products (\(i=1,\,2,\,\ldots,\,m\)).
* \(T\)-Finite time horizon \(T=24\) months (\(t=1,\,2,\,\ldots,\,T\)).
* LT-Delivery lead time.
* \(X_{t}^{i1}\)-Planned stock level of product \(i\) on hand at the end of period \(t\). This phase of accumulation represents the total amount of product \(i\) remaining on the stock at the end of period \(t\) on the flow of the planned state of inventory \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(X_{t}^{i2}\)-Realized stock level of product \(i\) on hand at the end of period \(t\). This phase of accumulation represents the total amount of product \(i\) remaining in the stock at the end of period \(t\), on the flow of the realized state of inventory \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(Y_{t}^{i1}\)-Planned stock input of product \(i\) at the beginning of period \(t\). This inflow regulator represents the amount of product \(i\) expected to be delivered after the lead time at the beginning of period \(t\). It is the regulator of the flow of the planned inventory state \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(Y_{t}^{i2}\)-Realized stock input of product \(i\) at the beginning of period \(t\). The inflow regulator represents the input variable that relates to the inventory fulfilment at the beginning of the month. This flow regulator represents the amount of product \(i\) delivered to the warehouse after the lead time expiry. It pertains to the flow of the realized state of inventory. The variable value is confirmed in the company's software when order quantity arrives in the stock \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(Y_{t}^{i3}\)-Forecasted sales plan for product \(i\) for period \(t\). This outflow regulator represents the amount of product \(i\) planned for withdrawing from the accumulation of the planned inventory state continuously per month. The sales manager forecasts the sales plan for \(24\) months \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(Y_{t}^{i4}\)-Realized sales of product \(i\) at period \(t\). This outflow regulator represents the amount of product \(i\) actually withdrawn from the accumulation of the realized state of inventory continuously per month \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).
* \(\min\)\(Q_{t}\)-FQQ for the product \(i\) in period \(t\), defined by manufacturer \(\big{(}t=\overline{1,\,24},\,i=\overline{1,m}\big{)}\).  
* \(LQ_{t}^{i}\)-LFL order quantity for each product \(i\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\).
* \(S_{t}^{i}\)-The auxiliary variable representing value from a stock level report for product \(i\) at the beginning of period \(t\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\), and representing value obtained from stock level report, which is provided each month from a warehouse of a distributor. This variable presents an exact level of inventory for product \(i\).
* \(A_{t}^{i}\)-The auxiliary variable indicates if the planned stock level on hand \(X_{t}^{i1}\) is greater than security stock \(SS_{t}^{i}\) for product \(i\) in period \(t\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\).
* \(Q_{t}^{i}\)-The planned re-order quantity for product \(i\) in period \(t\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\).
* \(SS_{t}^{i}\)-The security stock for product \(i\) in period \(t\) of time horizon \(T\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\).
* \(AS_{t}^{i}\)-The planned average level of stock for product \(i\) in period \(t\) of time horizon \(T\)\(\left(t=\overline{1,\ 24},\ i=\overline{1,m}\right)\).
* \(\Delta t^{i}\)-The auxiliary variable for calculation of the \(LQ_{t}^{i}\) in the case of a lot-for-lot ordering policy (\(i=\overline{1,m}\)).