## 3 Presentation of the Proposed Model
### Sub-Modules of the Presented Model
#### 3.3.1 Warehouse Structure Generation Module  
The proposed model can be implemented in most general warehouses, i.e., those with a storage rack system for storing products on pallets. The parameters for the storage compartments have to have the same dimension due to a matrix-parametric description of the warehouse structure. Multiplying the storage compartments both in the horizontal and the vertical direction gives a storage rack with columns and tiers. In this particular model, it is necessary to define additional parameters for the dimensions of the storage racks and picking aisles, as well as locations where the picking aisles between the storage racks are situated. The warehouse structure is described by a transition matrix, the so-called incident matrix. Therefore, the structure of the warehouse storage area can be written in the form of a \(\mathbf{W}\) transition matrix. \[W\ =\ \left[ \begin{array}{ccccc}r_{11}&r_{12}&\ldots&r_{1y}\\ r_{21}&r_{22}&\ldots&r_{2y}\\ \vdots&\vdots&\ddots&\vdots\\ r_{x1}&r_{x2}&\ldots&r_{xy}\end{array} \right] \tag{1}\]  
For example, for a warehouse with eight (8) storage racks, four (4) picking aisles and one (1) cross-warehouse aisle in the middle, the warehouse structure in the form of a matrix may be presented as follows:  
\[W=\left[ \begin{array}{ccccccccccccc}1&1&1&1&1&1&0&1&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&1&0&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ \end{array} \right] \tag{2}\]  
The \(W\) sample matrix includes four picking aisles along the storage rack and one cross-warehouse aisle. The packing area always has the location coordinates \(r\) (0,1). This location lies outside the \(W\) matrix so that it cannot be switched with the location in the storage area. This solution is quite common in small and medium-sized warehouses. In addition to the warehouse layout reflecting the product storage area, it is also important to consider the number of storage tiers. They can be represented with a vector showing the access (travel) time to each tier in the storage rack. This (travel) time depends on the transport means (forklift truck) used to store/retrieve the products to/from the warehouse. The \(W\) matrix, together with the parameters describing the warehouse structure, fully reflects its actual parameters. Based on the matrix \(W\), a matrix \(D\) is created representing the access (travel) times from the packing location \(r\) (0,1) to each single (selected) tier (storage compartment) of the storage rack. The matrix \(D\) contains values with access times for pallets' places in racks, and a \"0\" value for the locations of aisles. For example, for 12 matrix rows (8 of them are representing racks rows), 12 racks in each, and one cross-warehouse aisle, the \(D\) matrix for the first level of storage will be as follows:  
\[D=\left[ \begin{array}{cccccccccccc}6,6&8,4&10,2&12,0&13,8&15,6&0&19,2&21,0&22,8 &24,6&26,4&28,2\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 6,6&8,4&10,2&12,0&13,8&15,6&0&19,2&21,0&22,8&24,6&26,4&28,2\\ 17,8&19,6&21,4&23,2&25,0&26,8&0&30,4&32,2&34,0&35,8&37,6&39,4\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ 17,8&19,6&21,4&23,2&25,0&26,8&0&30,4&32,2&34,0&35,8&37,6&39,4\\ 29,0&30,8&32,6&34,4&36,2&38,0&0&41,6&43,4&45,2&47,0&48,8&50,6\\ 0&0&0&0&0&0&0&0&0&0&0&0&0\\ 29,0&30,8&32,6&34,4&36,2&38,0&0&41,6&43,4&45,2&47,0&48,8&50,6\\ 40,2&42,0&43,8&45,6&47,4&49,2&0&52,8&54,6&56,4&58,2&60,0&61,8\\ 0&0&0&0&0&0&0&0&0&0&0&0&0\\ 40,2&42,0&43,8&45,6&47,4&49,2&0&52,8&54,6&56,4&58,2&60,0&61,8\\ \end{array} \right] \tag{3}\]  
Relying on the determined matrix \(D\), it is possible to order individual pallets according to their importance by sorting them according to access times. The racks with the shortest access time will be able to store major goods. The lifting time is defined by the type of forklift.