## 2 Problem Description and Related Works  
As mentioned before, in the case of transport companies, the issue of WFM is related to the proper assignment of resource space elements (vehicles, drivers) to the request space elements (orders, requests) in such a way as to minimize the defined cost function. Figure 1 shows the general concept of the optimization module. It consists of the generation of sequences service records (denoted as SRs), representing the assignment of each of the cumulated requests (orders) to one of the resources (vehicles). Each set of SRs represents one of the possible constraints solutions. Every set can be assigned to a specific cost, typically represented by total mileage (distance). The aim is to find the solution closest to the unknown optimum (lowest cost) in the shortest possible time. The proposed model was developed in consultation with transport companies. As a result, it is adapted to real-life forwarding and logistics problems and is primarily based on real data. The collective list of orders (indicated as Requests) is provided by the customer (i.e., a shipping company). The list of potential vehicles (indicated as TrucksList) is also known from the same data source. In turn, information on recipient locations (indicated as DestPos) comes from GIS (GPS Tracking) systems. An acceptable, quasi-optimized solution should be found within several minutes. It must also be remembered that after planning, the goods are loaded in a given order, consequently, subsequent corrections may be limited. On the other hand, one vehicle can do two to three routes per day in real life. Therefore, frequent updating of data may affect the proposed planning of subsequent routes. The challenge encountered belongs to the capacitated, split delivery, and heterogeneous fleet variants of the VRP problem, discussed in [31; 32; 33; 34]. Generally, the set of destinations is represented by a directed graph, G = (V, A), where V = {1... D, D + 1... N + D} is a set of D + N nodes, where D is the number of the so-called depots, and N--the number of destinations, the A denotes the set of weighted edges of the graph. In the case of Single Depot VRP, the graph should be connected, and in this case, it is always possible to create a complete graph, in which the missing branch values can be calculated by finding the shortest path, using the well-known Dijkstra algorithm [35]. Therefore, the completeness of the graph is assumed, with the edge weights d (\(i,j\)) usually representing the conventional distances between particular pairs of destinations, known further as DistMx. Detailed descriptions of the model under consideration can be found in [36; 37; 38]. However, this article is not only focused on the mentioned above VRP problem. Although the considered scenario belongs to the rarely discussed variant of the heterogeneous  
Figure 1: Optimization system including upgrading of geopositioning data. (mixed) fleet, its approximate solution in a reasonable time is possible, for example, with the use of heuristic algorithms (greedy, clustering, etc.) [39; 40; 41]. A separate issue is the influence of outdated or approximated edge weights \(\text{d}\left(i,j\right)\) on the correctness of the solution returned by this type of algorithms. This remains an open problem. The significance of the distance matrix in transport issues have been discussed (e.g., in [42] where the authors emphasized the advantages of a distance matrix for a country to facilitate the survey for Austrian road freight transport statistics). The distance matrix turned out to be a helpful instrument to decrease the burden of the respondents. In [43], several different models that allow you to estimate the value of the distance based on the knowledge of the location of the destinations were presented. The authors considered the use of basic Euclidean and Manhattan metrics in the process of actual distance estimation.