# Decision-Making Techniques of the Consumer Behaviour Optimisation of the Product Own Price
## 3 Methodology
As such, in a certain observed period, the stock level \(Z\) in the warehouse is as follows:  
\[Z=z_{0}+\Delta Z\]  
\[Z=z_{0}+Y-X\]  
\[\Delta Z=Y-X \tag{1}\]0  
\[\Delta Z=Y-X \tag{1}\]1  
The random variables \(Z\) and \(X\) are interconnected, and if the probability density function of the consumer demand is known, the probability density function of the stock fluctuation in the warehouse can be determined. \[\Delta Z=Y-X \tag{1}\]2  
\[\Delta Z=Y-X \tag{1}\]3  
\[\Delta Z=Y-X \tag{1}\]4  
\[\Delta Z=Y-X \tag{1}\]5  
\[\Delta Z=Y-X \tag{1}\]6  
\[\Delta Z=Y-X \tag{1}\]7  
Let us determine the upper limit of the function \(f_{Z}(z)\):  
\[\Delta Z=Y-X \tag{1}\]8  
\[\Delta Z=Y-X \tag{1}\]9  
Figure 4 shows the probability density function of the product storage. The mathematical expectations \(E[X]\) and \(E[Z]\) are defined by integrating the functions \(f_{X}(x)\) and \(f_{Z}(z)\):  
\[E[\Delta Z]=E[Y]-E[X] \tag{2}\]0  
\[E[\Delta Z]=E[Y]-E[X] \tag{2}\]1  
\[E[\Delta Z]=E[Y]-E[X] \tag{2}\]2 Probability density function of the product storage. For a constant production flow \(y_{0}\) that will equal the theoretical mean value (the mathematical expectation) \(E[X]\),  
\[E[\Delta Z]=E[Y]-E[X] \tag{2}\]3  
The parameter \(a\) is determined by equating the theoretical mean value \(E[X]\) with the mean value \(\overline{x}\) obtained from real consumer demand data with descriptive statistics. \[E[\Delta Z]=E[Y]-E[X] \tag{2}\]4  
The determination of the storage costs depending on the stock level is as follows:  
\[E[\Delta Z]=E[Y]-E[X] \tag{2}\]5  
\(g(z)\)--the specific cost of the product [EURO/PC], \(z\)--the number of product pieces in the warehouse [pc], \(p\)--the storage cost of a product when it is not in stock [EURO/PC], \(h\)--the storage cost of a product when it is in stock [EURO/PC], \(C(z)\)--the storage cost [EURO], \(z_{0}\)--the optimum initial stock level [pc], \(C[O_{ij}]\)--the cost of operation \(i\) for product \(j\) [EURO/PC]. \[ \begin{array}{l}g(z)=p\to z<0\\ g(z)=h\to z>0\end{array}  \tag{15}\]  
Figure 5 shows the storage cost per unit of product \(p\) and \(h\) depending on the stock level in the warehouse. The cost \(p>h\) because the product is not in stock, and additional effort is needed to meet the demand; this costs money. The mathematical expectation or mean value for the storage cost of the product \(E[C]\) as a function of \(z_{0}\) is  
\[E[C]=\int_{-\infty}^{y_{0}+z_{0}}g(z)\cdot z\cdot f_{Z}(z)\;dz \tag{16}\]  
\[E[C]=\int_{-\infty}^{0}p\cdot z\cdot f_{Z}(z)\cdot dz+\int_{0}^{y_{0}+z_{0}}h \cdot z\cdot f_{Z}(z)\cdot dz \tag{17}\]  
\[E[C]=h\cdot z_{0}+\frac{p+h}{a\cdot e^{2}}\;(a\cdot z_{0}+4)\cdot e^{-a\cdot z _{0}} \tag{18}\]  
The optimum level \(z_{0}\) of products or semi-products equals  
\[\frac{dE[C]}{dz_{0}}=h-\frac{p+h}{e^{2}}\;\big{[}(a\cdot z_{0}+3)\cdot e^{-a \cdot z_{0}}\big{]} \tag{19}\]  
\[z_{0}=\frac{1}{a}\;ln\frac{p+h}{h\cdot e^{2}}\;(a\cdot z_{0}+3) \tag{20}\]  
Figure 5: Storage cost per unit of product \(p\) and \(h\) depending on the stock level in the warehouse. By inserting the values of \(p\), \(h\) and \(a\) into transcendent Equation (21), the optimal \(z_{0}\) can be calculated using the numerical method. If the obtained \(z_{0}\) is then inserted into Equation (19), we obtain the optimal cost of the stock (Figure 6). The movement of level \(z_{0}\) can be controlled using a control chart. An example of a control chart can be found in Figure 7.  
Figure 6 shows a control chart \(z_{0}\)(t), which allows us to monitor the trend of the stock level movement \(z_{0}\) in time intervals (weeks, months or years). Finally, according to the previous equations, the optimal average cost of product _ACP_ equals  
\[ACP=\frac{Q_{IN}\cdot C[O]+E[C]}{Q_{OUT}} \tag{21}\]  
\(Q_{IN}\)--the number of pieces that enter into the operation \([O]\), \(C[O]\)--the cost of the operation \([O]\), \(E[C]\)--the storage cost of the semi-finished product before entering into the operation, \(Q_{OUT}\)--the number of pieces that leave the operation \([O]\).