## 4 A Mathematical Model
### Remarks for Restricting a Possible Application of the Mathematical Model
In the considered case, when the function \(g_{j}\) is determined by Formula (4), one can assume that  
\[f_{j}(X)=\sum_{i=1}^{I}\sum_{k=1}^{K}\sum_{t=1}^{T}y_{ijkt}\cdot x_{ijkt}\cdot D _{t}-f_{j}\cdot l_{j},j=1,\stackrel{{-}}{{J}}, \tag{7}\]  
where the parameters \(a\), \(c\) and \(d\) are equal to \(0\), \(1\) and \(g_{0}\), respectively:  
\[a=0,\ c=1,\ d=g_{0} \tag{8}\]  
In this case, array \(X\) consists of variables \(x_{ijkt}\) and \(y_{ijkt}\). According to Equations (7) and (8), the Formula (4) can be written as follows:  
\[g_{j}=h\big{[}f_{j}(X)\big{]} \tag{9}\]  
In the next step, the task is, in some way, to approximate the discontinuous function \(h(y)\) with the continuous function \(\stackrel{{\sim}}{{h}}(y)\) in order to be able to approximate the discontinuous Function (4) by the following smooth function:  
\[\stackrel{{\sim}}{{g}}_{j}=\stackrel{{\sim}}{{h}} \big{[}f_{j}(X)\big{]} \tag{10}\]  
As a basis for the approximation of the function \(h(y)\), one can take the logistic Function (5). Note that \(\varphi(-\infty)=0\) and \(\varphi(\infty)=1\). Moreover, with a sufficiently large value of \(s\), it can be assumed that \(\varphi(-s)\approx 0\) and \(\varphi(s)\approx 1\). For example, for \(s=6\), the values of \(\varphi(-s)\) and \(\varphi(s)\) are already close enough to \(0\) and \(1\), respectively, as can be seen from Figure 1, namely, \(\varphi(-6)=0.0025\) and \(\varphi(6)=0.9975\). We take the value \(b>a\). Then, based on the Function (5), we define the function \(\psi(y)\) such that \(\psi(a)=\varphi(-s)\) and \(\psi(b)=\varphi(s)\). Therefore, \(\psi(a)\approx 0\) and \(\psi(b)\approx 1\). As a result, we build a linear function that corresponds to the values \(a\) and \(b\) of the values of \(-s\) and \(s\), respectively. Such a linear function is given by the following formula:  
\[z=\frac{2y-a-b}{b-a}s \tag{11}\]  
In Function (11), the variable \(y\) acts as an argument. Substituting the Formula (11) into the Equality (5), we obtain the desired function \(\psi(y)\) as follows:  
\[\psi(y)=\frac{1}{1+\exp\Bigl{(}-\frac{2y-a-b}{b-a}s\Bigr{)}} \tag{12}\]  
Figure 1: The graph of the logistic function. Based on Function (8) (such that \(\psi(a)\approx 0\) and \(\psi(b)\approx 1\)), we construct the function \(h(y)\) such that \(\widetilde{h}(a)\approx c\) and \(\widetilde{h}(b)\approx d\). To do this, one can build a linear function that corresponds to the values 0 and 1 of the variables \(c\) and \(d\), respectively. Such a function is determined by the following equality:  
\[\chi=(d-c)\psi+c \tag{13}\]  
In Function (13), the variable \(\psi\) acts as an argument. Substituting the Formula (12) into the Equality (13), we obtain the desired function \(\widetilde{h}(y)\) as follows:  
\[\widetilde{h}(y)=(d-c)\psi(y)+c=\frac{d-c}{1+\exp\Bigl{(}-\frac{2y-a-b}{b-a}s \Bigr{)}}+c=\frac{d+c\cdot\exp\Bigl{(}-\frac{2y-a-b}{b-a}s\Bigr{)}}{1+\exp \Bigl{(}-\frac{2y-a-b}{b-a}s\Bigr{)}}\]  
Thus, we obtain the following equality:  
\[\widetilde{h}(y)=\frac{d+c\cdot\exp\Bigl{(}-\frac{2y-a-b}{b-a}s\Bigr{)}}{1+ \exp\Bigl{(}-\frac{2y-a-b}{b-a}s\Bigr{)}} \tag{14}\]  
Note that in accordance with the above smoothing procedure used, Function (14) is increasing, and the following conditions hold:  
\[a=0,\ c=1,\ d=g_{0} \tag{8}\]0  
Therefore, Function (14) approximates Function (6). In such a case, the accuracy of the approximation increases when the value of the parameter \(b\) decreases (recall that a value of the parameter \(b\) must be greater than a value of the parameter \(a\)). In the case of interest (i.e., when performing Equalities (7) and (8)), by virtue of the Formula (14), Equality (10) takes the following form:  
\[a=0,\ c=1,\ d=g_{0} \tag{8}\]1  
In Equality (16), \(b\) denotes the step of the smoothing procedure and \(s\) is a scale factor. It should be noted that in accordance with the above arguments, the accuracy of the approximation increases with an increase in the value of the parameter \(s\) and a decrease in the value of the parameter \(b\), where \(s>0\) and \(b>1\).