## 6 Computational Results
### TSP Implementation in GAMS
Similarly, when solving differential equations, e.g., to calculate the differential equation \(y^{\prime}(x)=4 xy+x^{3}\) with initial condition \(y(4)=2\), it is enough to enter dsolve('Dy=4*x*y+x^3','y(4)=2','x'). In MATHEMATICA, too, to obtain the impulse function of the system described by a differential equation, it is enough to rewrite it in the form of a Laplace transfer and use a single command InverseLaplaceTransform. In contrast, the code notation in GAMS is similar to code in programming languages with the definition of constants, the declaration of variables, and the body of the program. Again, there are assignment statements, conditional statements, and loop statements. For example, the binary values of the reachability matrix \(\mathbf{A}\) from the distance matrix \(\mathbf{D}\) and the defined reachable distance threshold \(D_{max}\) are determined in GAMS as follows:

LOOP(I,  LOOP(J,  IF (D(I,J) <= Dmax,  A(I,J)=1;  ELSE  A(I,J)=0;  );  );

The only disadvantage of GAMS is that it has no graphical tools, and the results of the calculations are only in text form. This requires exporting them to a suitable program and postprocessing. In [13] we used MATLAB, here Figures 4 and 5 are generated in the MATHEMATICA environment. Only where for extremely large instances of problems of exponential complexity we cannot obtain an exact solution using GAMS in a reasonable amount of time (e.g., no more than in tens of minutes), do we use one of the many heuristic methods. Given the No Free Lunch Theorem [74; 75], none of them can be recommended as the best in the general case, since finding the optimal solution is not guaranteed and the result is always an approximation of the optimum, so our modification of the genetic algorithm, implemented in Java and described in more detail in [79], can be used without loss of generality. One-point heuristics (hill climbing, tabu search, simulated annealing) in solving problems where in each iteration the neighborhood operation often generates tens of infeasible solutions and it is necessary to use a repair operator for them (here it concerns the coverage problem), and slow down the computation considerably, so in these cases we prefer, e.g., a genetic algorithm that generates only two new solutions in each iteration. The model in Section 4.4 is original with another possible modification proposed at the end. Its model has already been built and verified on smaller-scale instances so far and will be investigated in more complex cases. In future research, we expect to focus on the Quadratic Assignment Problem, the Vehicle Routing Problem and its solvability using GAMS, and applications in agriculture with consideration of data uncertainty using probabilistic models or fuzzy modeling, since yields can only be estimated. Although the Quadratic Assignment Problem has a non-linear objective function with quadratic terms, it can be converted to a mixed integer programming problem using [PERSON]'s linearization [7] and the MIP solver of GAMS can be used again. ## Funding: This research received no external funding. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} AP & Assignment Problem \\ TSP & Travelling Salesman Problem \\ VRP & Vehicle Routing Problem \\ PFSSP & Permutation Flow Shop Scheduling Problem \\ GAMS & General Algebraic Modelling System \\ GA & Genetic Algorithm \\ ANN & Artificial Neural Network \\ \end{tabular} 

## References

* (1) [PERSON] _Linear Programming. Methods and Applications_; Dover Books on Computer Science; Courier Corporation: North Chelmsford, MA, USA, 2010. * (2) [PERSON]; [PERSON] _Handbook of Combinatorial Optimization. Volume A_; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999. * (3) [PERSON]; [PERSON] _Handbook of Combinatorial Optimization. Volume B_; Springer: Berlin/Heidelberg, Germany, 2005. * (4) [PERSON] The Hungarian Method for the Assignment Problem. _Nav. Res.