## 4 Results  
In the ABC algorithm, the following parameters can be specified:  
* Size of the workers population--30;
* Number of iterations--17000;
* Maximum number of iterations without improvement for a single bee--40;
* Weight of penalty factor that is penalizing not completely full cars--16.6%. We have used various population sizes from \(n=15\) to 100, and found that in most cases, it is sufficient to use \(n=15\) to 50. Therefore, we have used population a size of \(n=30\) in all our simulations. We tested the number of iterations in the range (500-2500), and the value of 1700 turned out to be the best compromise between the computation time and the quality of the obtained solution. Other parameters were also selected through empirical research. The Evolutionary Algorithm worked for 50,000 iterations. The total number of objective function calculations was then almost the same for both tested algorithms. After a series of experiments, ReplaceCarsOperator and ExchangeRealisationOperator parameters values were determined as:  
* Minimal part of cars/streams modified during a single call--6.25%;
* Maximal part of cars/streams modified during a single call--12.5%. The calculations were performed using a linear solver engine of CPLEX 12.8.0.0 on a 2.1 GHz Xeon E7-4830 v.3 processor with 256 GB RAM, running under the Linux Debian operating system. Figures 1 and 2 show the convergence curves of the considered algorithms for BLP and ELP models, respectively. Each curve represents an average of 20 independent runs. Inboth cases, logistic networks with NetworkSize = 5, 10, 15, 20, 25 and 30 were considered. The following observations can be made:  
* For the BLP model (Figure 1), the MIP method found a solution in every case, with only NetworkSize = 5 being the optimal solution;
* In the case of the ELP model (Figure 2), the MIP method only found an optimum for the five-node network. However, for NetworkSize = 20, 25 and 30, it did not find an integer solution at all;
* Both heuristic methods found a sub-optimal solution in every case considered;
* In most cases, EA gives better results. However, for smaller network instances, the ABC method is comparable or slightly better. Such observation is strengthened by the results of the Mann-Whitney test (column T in Tables 1 and 3). In most cases, for both problem variants, there is a significant difference in a location shift in favour of the EA algorithm. Figure 1: Convergence curves plotted for the algorithms considered, for a logistic network of different sizes, for the BLP model. Each curve is an average of 20 independent runs. (**a**) 5-node. (**b**) 10-node. (**c**) 15-node. (**d**) 20-node. (**e**) 25-node. (**f**) 30-node. Figure 2: _Cont._  
Tables 1 and 3 show the mean and best values of the objective function in each test case. Whereas, Tables 2 and 4 show the mean and standard deviation values of the computational time in each test case. The last column in each table contains the result of the Wilcoxon-Mann-Whitney test [17] with a confidence level of 0.95 applied to the outcomes of the two metaheuristics (EA and ABC). We used symbols \(+\), \(-\) and \(=\) to indicate whether the EA algorithm was--respectively--better, worse or equal to the ABC algorithm. Figure 2(a),b depict the calculation time achieved by heuristics and the MIP solvers before termination for networks with 5, 10, 20 and 30 nodes for BLP and ELP variants. One may notice that for the BLP variant, the MIP solver on small networks (NetworkSize = 5 and 15) outperformed the ABC algorithm and was slightly worse than the EA. However, for the ELP variant, the MIP achieved a comparable calculation time to the heuristics on the network only with five nodes. For each bigger network, i.e., with 20 or 30 nodes, the MIP solver reached the total calculation time stop condition without a feasible solution, while the heuristics returned a suboptimal albeit feasible solution in total calculation time smaller by a factor of 1 or 2 than the MIP runtime.