# Hybrid Algorithms for Inbound and Outbound Truck Scheduling in Cross-Docking Systems
## 5 Computational Experiment  
In this study, we would like to use three metaheuristics, i.e., PSO, VNS, and SA, to construct three kinds of hybrid metaheuristic approaches for solving the truck scheduling problem. Eventually, we are trying to find out the best approach for the problem. Before comparing these three approaches, we attempt to conduct some experiments for adjusting the parameters of PSO, VNS, and SA to make sure these metaheuristics all have the best functionalities. First, we design a scenario with 14 trucks for both inbound and outbound areas and 13 kinds of product types, and the total number of products is equal to 5040. Next, we randomly generate the number of products for each product type to construct several experiments. Finally, we use these experiments to test the three metaheuristics in 20 s of computation time. The final parameters of the three metaheuristics are illustrated as follows. * Number of particles, \(N_{k}=20\); Initial iteration, \(t=0\); Inertia weight, \(w(t)=(w(0)-w(n_{t}))^{\star}[(n_{t}-t)/(n_{t})]+w(n_{t})\); (\(n_{t}\) is the max number of iterations, \(w(0)=0.9\), \(w(n_{t})=0.4\)) Acceleration constant of the local particle, \(c_{1}(t)=(c_{1,min}-c_{1,max})^{\star}(t/n_{t})+c_{1,max}\) Acceleration constant of the global particle, \(c_{2}(t)=(c_{2,max}-c_{2,min})^{\star}(t/n_{t})+c_{2,min}\) (\(c_{1,max}=c_{2,max}=2.5\), \(c_{1,min}=c_{2,min}=0.5\))
* The classification method can be seen in Engelbrecht [23]. * K\({}_{2}\))*N, where K is an experimental variable, N is the number of truck pairs obtained by the six algorithms by [PERSON] (2002), and K\({}_{1}\)-K\({}_{2}\) are the upper-bound and lower-bound, respectively, to generate a random integer. The experimental result is shown in Table 1, and it reveals that the K*N2 design with K = 20 can obtain the minimal objective value, so we use the result as the final parameter setting in the VNS. * **SA** In the proposed SA, there are two parameters that need to be determined--length and stop criteria. Length is used to decide on the maximum accepted number of solutions in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Counter Size** & **K** & **Objective Value** \\ \hline \multirow{5}{*}{K*N} & 10 & 2132.59 \\  & 20 & 2033.01 \\  & 30 & 2021.59 \\  & 40 & 2003.85 \\  & 50 & 2001.61 \\ \hline \multirow{5}{*}{K*N2} & 10 & 1996.11 \\  & **20** & **1933.63** \\ \cline{1-1}  & 30 & 1972.59 \\ \cline{1-1}  & 40 & 1965.52 \\ \cline{1-1}  & 50 & 1959.59 \\ \hline \multirow{5}{*}{(K\({}_{1}\) - K\({}_{2}\))*N} & 1-10 & 2283.13 \\  & 1-20 & 2132.74 \\ \cline{1-1}  & 1-30 & 2072.20 \\ \cline{1-1}  & 1-40 & 2038.38 \\ \cline{1-1}  & 1-50 & 2002.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Result of the VNS parameter experiment. each temperature, and the stop criteria in this study are designed as the iteration times after which the solution makes no improvement. We design two methods to generate the length--by a constant and by a random number. Three values of no improvement are also adopted to test in Table 2. The result shows that when \(L_{max}\) is equal to 10 and the stop criteria use 500 times of no improvement made, a better objective value can be obtained. After the parameters of each of the three metaheuristics are determined, three hybrid metaheuristics, i.e., PSO + VNS, PSO + SA, and PSO + SA + SA, will be constructed. Next, we attempt to design some experiments to test the effectiveness of the proposed approaches, so we adopt a small-sized experiment with 20 small examples--the same as those proposed by [PERSON] [4]. In addition, we would like to test if our approaches can solve problems with more complexity. We design a large-sized experiment with 25 large-scale examples which