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But these methods usually face difficulties to succeed when noise appears in the input data or during the search, specially in large instances.
A genetic algorithm, along with tuned parameters and a simulated annealing algorithm, are proposed to solve the model in large instances.
Both two approaches are applied to several groups of instances with real-world data, and results show that the GASA approach is efficient even in large instances.
Unfortunately, it has been demonstrated that classical algorithms for generating Erdős Rényi based random graphs do not scale well in large instances and, in addition to this, fail to make use of the parallel processing capabilities of modern hardware.
Reported results in Table 3 confirm that the proposed algorithm is efficient to solve the main problem in large instances while it contains acceptable gap from the lower bound.
To enhance run time for in large instances, we applied PSO algorithm to generate some initial solutions and found the best time to combine these two algorithms, so the run time was reduced by more than 40%.
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Finally, we want to show to what extent the set-covering model can be optimally solved in larger instances taken from VRP benchmarks, in order to investigate a potential increase in the carrier's container volumes.
In larger instances, a solution may call for filling in a cavity, getting braces or dentures, or require more in-depth surgery depending on the situation.
It means that the computed lower bound can be considered in the large instances.
This problem is NP-hard, although in practice fairly large instances can be solved to optimality within reasonable running time.
Although it is an NP-hard problem, exact solutions can be found in reasonable time even for large instances (see in Section 6 the computing times for instances up to 5000 items).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com