## 5 Experimental Study
### Methodology Applied to Analyze the GGA Performance  
Three indicators are evaluated in the experimental study: Latency, speedup, and solution quality. Latency measures the time delay experienced in one distributed environment, and it is analyzed to determine the effect of using the two-stage scheme to migrate information between the nodes of the MiniGrid, and its value is based on the transfer rates between the two clusters in the MiniGrid.  
Two processes are used in these experiments: First, 120 processes were used in the MiniGrid (one process for each core), and then an overload of up to 500 uniformly distributed processes was used to evaluate the latency generated by GGA in the MiniGrid. On the other hand, speedup measures the gain achieved by using the parallel program over the sequential version, and its value is computed in Equation (14).  
\[\textit{speedup}=\frac{T_{1}}{T_{n}} \tag{14}\]  
where \(T_{1}\) is the total running time of the sequential version of the algorithm and \(T_{n}\) is the running time of the parallel version using \(n\) processes. The speedup was analyzed to determine the effect of increasing the number of nodes for the distributed application of mutation operator.  
Furthermore, to obtain reliable estimates of the results quality, the relative error (RE) is used to compare the results of this experimental study with those produced by other approaches to solve the VRPTW. RE is a factor measuring the difference between the result generated by one experiment and the best-known result value in the existing literature. RE is defined in Equation (15).  
\[RE=100\frac{f(x)-f\Big{(}x^{opt}\Big{)}}{f(x^{opt})} \tag{15}\]where \(f(x)\) is the fitness value result of the current experiment and \(f(x^{opt})\) is the best-known fitness value reported in the existent literature.  
GGA results are compared with those achieved by the following approaches described in the existing literature:  
* GA-VRPTW [1]: Sequential GA-based algorithm with specialized operators to solve VRPTW instances. These operators are used in the GGA. Its experimental study is conducted using a computer with one 1.6 GHz Pentium processor, and the algorithm is implemented in Microsoft Visual C++ 6.0.
* KDMSS [11]: This method is identified in the existing literature using the initials of its authors ([PERSON], [PERSON], [PERSON], [PERSON], and [PERSON]). Its experimental study is carried out in one HP9000/829 computer with a 100 MHz PA7200 processor. The KDMSS algorithm is implemented in ANSI C.
* CPLA (cooperative population learning algorithm) [22]: Author uses a cluster HOLK of 256 computers with one Intel Itanium 2 Dual Core. The proposed algorithm has been implemented in Java and uses a software framework to peer-to-peer applications.
* BCO-SIH (a bee colony optimization algorithm with a sequential insertion heuristic) [23]: This algorithm is implemented using Java and performed on one computer with an Intel Core I3 processor.
* MOGA (multi-objective genetic algorithm) [24]: The algorithm is implemented in C/C++ and run on a Silicon Graphics machine with 128 CPUs.
* ACO-TS (ant colony optimization and Tabu search) [25]: The algorithm has been programmed in C++, and the experimental study is carried out in an Intel Core I3 processor machine.
* HM4 (Gehring and Homberger 4) [15]: Experiments are conducted in one computer with a Pentium processor.
* LC03 (Le Bouthillier and Crainic version 03) [17]: The algorithm run in one cluster of five Pentium III computers.
* RP (Ropke and Pisinger) [18]: Experiments are performed on a Pentium IV processor machine, and the heuristic is implemented in C++.  
Furthermore, the GGA results with the Gehring and Homberger benchmark problems are compared with the best-known values obtained by other implementations such as Q (Quintiq's optimization technology) [62], CAINIAO (Cainiao network technology) [63], and SCR (Emapa Inc.) [64].