## 6 Computational Results
### TSP Implementation in GAMS
Among these circles, we determine the one with the largest diameter, and its center is then the optimal position for the location of the new center, see Figure 7.

In fact, the calculated position may not be available, the cost of building here may be too high, thus a suitable nearby location must be found, or the center of one of the other empty circles must be chosen in descending order of diameters. Figure 6: Lin-2-Opt change neighbourhood operation,

Another issue is the amount of inventory in logistics operations. The stock changes over time according to demand and needs to be replenished accordingly. We speak about _inventory management_ and the resulting sustainability [88; 89; 90; 91]. However, demands are stochastic in nature and, in addition, inventory management must take into account the cost of maintaining inventory and losses from premature depletion of inventory for undleivered goods. In [92], a new mathematical model is derived, the properties of the profit function are proved, and the profitability in a two-channel production system considering carbon emissions and green technology is numerically verified on specific data. While artificial neural networks (ANN) have very little application in combinatorial optimization, their main use is in cluster analysis, pattern recognition, image processing and prediction, in [93] the authors present an efficient implementation of ANNs in an inventory management model under uncertainty and inflation. In [94] the unreliability of the supply chain and methods to eliminate this unreliability are explored, and the required mathematical equations are derived and verified by numerical experiments, including sensitivity analysis. However, all these aspects are beyond the scope of this paper and can be the subject of separate texts as also evidenced by the papers mentioned. ## 7 Conclusions

This paper studies the assignment problem and its modifications with logistics applications, in routing, distribution, and scheduling tasks. Its first contribution is the correlation of the problem models, which are often distant in nature and time complexity. It has also shown how the described models can be directly transferred to the GAMS environment. NP-hard Permutation Flow Shop Scheduling Problem (PFSSP) and the Travelling Salesman Problem are used to show that the optimal solution can be determined in the available time of a few minutes for instances with 20 jobs on 10 machines in the case of PFSSP, and for 100 cities in the case of TSP. Previously, these boundaries were inaccessible with mixed integer programming solvers, but with the new version of GAMS, they have been significantly extended. This of course means first to build the appropriate model (and this is not always a simple matter, as the informal derivation of the PFSSP model in Section 5 showed) and then, for instance, for benchmark libraries (e.g., OR-Library or TSPLIB), to search individually for the appropriate bounds. The findings from PFSSP and TSP are not isolated examples of the successful application of GAMS in solving large instances of optimization versions of NP-complete/NP-hard problems. We have already validated it in [13] in solving the covering problem with matrices of hundreds of thousands of elements, and more recently in solving the Steiner problem in graphs in [95], where first using the terminology of network flows

Figure 7: Finding a new location using the largest empty circles. a mixed integer programming model was derived, then modified for GAMS, and finally exact results for a representative class of benchmarks from OR-Library were obtained. Another goal of this paper was to introduce code generation in GAMS on non-trivial tasks because in the manuals [76; 77] we can find only a description of individual elements of this tool, but not the codes of complete task models. In MATLAB, running the computation of an optimization program means writing just a single command (intlinprog or linprog with the appropriate parameters).