# Recent Application of Dijkstra's Algorithm in the Process of Production Planning
## 3 Results and Discussion  
Based on the rules for creating a matrix of unit prices and Relations (1) and (2), (Table 2), we created matrices of JC-type 8 \(\times\) 8 unit prices for road transport [19]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(v_{1}\) & \(v_{2}\) & \(v_{3}\) & \(v_{4}\) & \(v_{5}\) & \(v_{6}\) & \(v_{7}\) & \(v_{8}\) \\ \hline \(v_{1}\) & 0 & 5 & 9 & - & 22 & - & - & - \\ \(v_{2}\) & 5 & 0 & 5 & 8 & - & - & - & - \\ \(v_{3}\) & 9 & 5 & 0 & - & 13 & - & - & - \\ \(v_{4}\) & - & 8 & - & 0 & - & 8 & - & - \\ \(v_{5}\) & 22 & - & 13 & - & 0 & 9 & - & 25 \\ \(v_{6}\) & - & - & - & 8 & 9 & 0 & 25 & - \\ \(v_{7}\) & - & - & - & - & - & 25 & 0 & 3 \\ \(v_{8}\) & - & - & - & - & 25 & - & 3 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Distances of neighbouring places along the road. In our case, we propose an optimal solution of a model in which the network of mining and processing sites and customer sites, including their transport interconnection, represents the graph G = {V, H} and has the following properties [20,21]:  
* G is a continuous undirected graph without multiple edges and loops;
* V = {v1, v2, \(\cdots\), vm} is a set of graph vertices that represent mining and processing and consumption points;
* H = {h1, h2, \(\cdots\), hn} is a set of graph edges that represent the transport network (road or rail) between individual places; the vertices of the graph, which correspond to the mining processing, are evaluated, and the evaluation of such a peak vi is defined by [22]: \[\text{O}(\text{v}_{\text{i}})=\text{PTN}_{\text{i}}+\text{NTN}_{\text{i}}+ \text{PSN}_{\text{i}}+\text{NSN}_{\text{i}}\] (1) where \(\text{PTN}_{\text{i}}\) is the direct extraction costs per tonne of raw material, \(\text{NTN}_{\text{i}}\) is the indirect extraction costs per tonne of raw material, \(\text{PSN}_{\text{i}}\) is the direct processing costs per tonne of material, and \(\text{NSN}_{\text{i}}\) is the indirect processing costs per 1 tonne ton of material; the vertices of the graph, which correspond to those customer points, are not simultaneously mining-processing points and are not evaluated, and all of the edges of the graph are evaluated. The evaluation of the edge hi connecting the vertices vk and vl is defined by equation [21]: \[\text{O}(\text{h}_{\text{i}})=\text{PN}\times\text{VZD}_{\text{kl}}\] (2)  
The parameters of the distance of peaks (cities) and parameters of the financial costs of transportation are presented in Table 2.  
It should be noted that for the value of \(\infty\), the unit price matrices mean that the respective vertices of the graph are not adjacent. In computer processing, the value \(\infty\) is replaced by a sufficiently large number: infinity [22, 23, 24]. We will optimize the path in the graph for road traffic (Figure 2) for the peaks corresponding to the individual mining and processing sites, i.e., for the peaks v1 and v2, using the described algorithm [25,26]. The search procedure and its results are clear in Table 3, in which the states of the vertices of the graph are indicated as follows: the permanent top state is in blue, permanent peaks that are also the working peaks in a given step are in green, and temporary peaks with minimum values are in red. In step 0 (Figure 3), we set the evaluation of the source, the peak v1, to the value of the price of the raw material after production behind the plant gates, i.e., the work area. We set the predecessor as ourselves, and marked it as permanent (green). The other vertices on the map only form customer sites (white). We marked the transport costs between the individual peaks in blue. A sufficiently significant number of reserves in the deposit is required for a long-term project [6].