## 5 Discussion  
The correct form of the distance matrix is a crucial parameter influencing the effectiveness of the optimization algorithm used in the optimizer. Many previous papers discussed the possibility of obtaining distance matrices from the corrected Euclidean distance. To approximate the actual distance, [PERSON] [56] proposed using a factor of the curvature of the road, which can be calculated as the ratio between the actual distance and the Euclidean distance [57]. For UK roads, a value of 1.2 was proposed by [PERSON] [58] for the correction factor, which has subsequently been widely accepted and used in the scientific community. According to [59], correction factors of 1.25 for the long haul and 1.30 for an urban drayage area are used. The listed values are slightly lower than those proposed in this article. Based on the approximations made, the correction coefficient can be assumed at the level of 1.287 for distances above 10 km, which is a value comparable to the proposals appearing in the cited papers. Note that in the case of the space represented by the Manhattan metric, the value of the correction factor should not exceed \(\sqrt{2}\approx 1.414\). In reality, however, a more significant correction should be made, especially for very short distances below 3 km. As shown in Figure 3 the correction factor can be as high as 1.7 for ultra-short distances. Figure 5: Evolution of cost reduction in case dynamic matrix upload (starting from highest and lowest matrix values, respectively). The discrepancies between values proposed in this work and values proposed in mentioned above papers suggest that receiving more real data obtained from Mapping API services is highly recommended. Moreover, the presented results indicate that the evaluation of the algorithm's effectiveness may be determined by the smallest distance values (the lowest weights of the graph representing the transport problem). As shown in Figure 5 (trend \"from highest\"), the last 10% of the lowest coefficients significantly affect the calculated _Gain_ value. It makes sense as a more significant relative error characterizes these values. On the other hand, the absolute error may be more significant for longer distances. It shows the advantage of the reverse approach, where the update should start with the highest values and was confirmed by analyzing the correlations between mixed-matrix and _DistOSRM_. The observed trend of the _Gain_ coefficient should be linked by a change in the rank of individual elements of the distance matrix. The behavior of two correlation coefficients ([PERSON]'s linear and [PERSON]'s rank) was analyzed. The appropriate values were calculated between the final _DistOSRM_ matrix and successively updated mix-matrices for both the \"from lowest\" and \"from highest\" variants. Figure 6 shows all four corresponding graphs. In the case of [PERSON]'s correlation, significant differences in the formation of the correlation are observed during the update of the distance matrix between the \"from lowest\" and \"from highest\" approach. Moreover, the chart suggests that during the update of the values from the highest, the correlation sooner tends to 1, which negates the earlier thesis. However, a more reliable parameter is the rank correlation coefficient. In this case, the changes in the trend are similar. In fact, the correlation analysis does not provide backgrounds for selecting the \"from lowest\" variant. However, the choice of the \"from lowest\" approach can be defended by looking at the histogram representing the distribution of the cost values of single jumps proposed by the optimization algorithm. A histogram represents a distribution of single-hop OSRM distances reported in optimizer solution was calculated (Figure 7A). As shown in the proposed solutions, shorter distances, below 30 km, are reused more often. It indicates that it is the shorter distances that should be updated first. On the other hand, the share of short distances in the total cost is lower.