## 3 Presentation of the Proposed Model
### Sub-Modules of the Presented Model
#### 3.3.7 Efficiency Analysis Module
**Step 1**--In the first step of calculating picking time, an absolute value is calculated, from a difference between the initial starting point and the ending one (19) for racks and rows. \[ \begin{array}{rcl}x_{temp}&=&\left|x_{pocz}-x_{konc}\right|\\ y_{temp}&=&\left|y_{pocz}-y_{konc}\right|\end{array}  \tag{19}\]  
**Step 2**--In the second step, it is verified whether the racks are in the same row, and whether or not it is the row coming out directly from the packaging area--the starting point. \[y_{temp}\ \
eq\ 0\ or\ y_{temp}\ \
eq\ 2 \tag{20}\]  
**Step 3**--If the condition of step 2 was fulfilled, then in the calculation of the time it is important to take two curves into account, \(l_{luk}\) = 2. Besides, as in the contemplated warehouse, where two-row racks with constant access provided are used (every third row is a corridor between the racks), it is necessary to contemplate several longitudinal corridors (21)  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]  
In this case, the length of a road across is described by formula (22). \[dr_{poprz}\ \ =\ \ D_{wr}(y)\left(y_{temp}-l_{koruz}\right)+\ d_{cr}l_{koruz} \tag{22}\]  
**Step 4**--The analyzed warehouse has one transversal corridor; therefore, before calculating the road along the warehouse, it is important to verify whether the two points are in the same or different sections of the warehouse, meaning they are separated by a corridor (23). \[\left(x_{pocz}\ \leq\ \frac{l_{reg}(y)}{2}orazx_{konc}\ >\ \frac{l_{reg}(y)}{2} \right)or\left(x_{konc}\ \leq\ \frac{l_{reg}(y)}{2}orazx_{pocz}\ >\ \frac{l_{ reg}(y)}{2}\right) \tag{23}\]  
**Step 5**--If the condition in step 4 is satisfied, then the road along the warehouse (\(d_{ruzd}\)) should be calculated from the formula (24). \[dr_{wzd}\ =\ D_{lr}(y)\cdot d_{cr}\Big{(}[x_{pocz}-x_{konc}]-1\Big{)} \tag{24}\]  
**Step 6--**If the condition in step 4 is not satisfied, then it is important to verify in which part of the warehouse (before or after the transversal corridor) both considered points are located (25). \[x_{pccz}\ <\ \frac{l_{reg}(y)}{2} \tag{25}\]  
**Step 7--**If the condition in step 6 is satisfied, then both points are located in the first area of the warehouse, so if  
\[D_{lr}d_{cr}\Big{(}x_{pccz}+x_{konc}\Big{)}<D_{lr}d_{cr}\Big{(}l_{reg}-x_{pocz} -x_{konc}\Big{)} \tag{26}\]  
then  
\[dr_{vzd}\ =\ D_{lr}\cdot d_{cr}\Big{(}x_{pocz}+x_{konc}\Big{)} \tag{27}\]  
otherwise  
\[dr_{vzd}\ =\ D_{lr}\cdot d_{cr}\Big{(}l_{reg}-x_{pocz}-x_{konc}\Big{)} \tag{28}\]  
**Step 8--**If the condition in step 6 is not fulfilled then both points are located in a second zone of the warehouse, so if  
\[D_{lr}d_{cr}\Big{(}x_{pocz}+x_{konc}-l_{reg}\Big{)}\ <D_{lr}\cdot d_{cr}\Big{(}2l_{ reg}-x_{pocz}-x_{konc}\Big{)} \tag{29}\]  
then  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]0  
otherwise  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]1  
**Step 9--**If the condition in step 2 was not satisfied then the calculation of time should not take curves into account, \(l_{lnk}=0\), nor the length of the road in the transverse corridor \(dr_{poprz}=0\). Therefore, the length of the road in the longitudinal corridor depends on whether both points are located in this part of the warehouse, or if they are separated by a corridor (32). So, if  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]2  
then the points are located in different parts of the warehouse, so  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]3  
otherwise  
\[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]4  
**Step 10--**Based on the calculated picking route, it is important to calculate its duration based on formula (35). \[l_{koruz}\ \ =\ \ \frac{y_{temp}}{3} \tag{21}\]5