# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 3 Solution Algorithms
### Hmpga-Vns Algorithm
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Range** & **Parameter** & **Range** & **Parameter** & **Range** \\ \hline \(f_{i}^{1}\) & \(10^{5}\) U (3.0, 4.0) & \(f_{j}^{2}\) & \(10^{5}\) U (1.5, 2.0) & \(rc_{p}\) & U (30, 50) \\ \(pc_{ip}\) & U (80, 100) & \(tc_{Invp}^{2}\) & U (0.4, 0.5) & \(d_{kpt}\) & U (30, 60) \\ \(tc_{ip}^{1}\) & U (0.2, 0.3) & \(hc_{jp}\) & U (30, 40) & \(v_{vp}^{cap}\) & U (200, 300) \\ \(m_{jpt}^{cap}\) & U (1000, 1500) & A & \% U (10, 30) & \(\tau_{p}\) & U (3, 5) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uniform distributions of random parameters. measure of variation. The term \"signal\" represents the desired value, the mean response variable, while \"noise\" refers to the undesired value associated with the standard deviation. The S/N ratio quantifies the variation present in the response variable. Undoubtedly, a larger S/N ratio indicates a better outcome. This approach includes three types of response: \"the smaller the better\", \"the larger the better\", and \"the closer to the objective value the better\". In our study, the criterion \"the smaller the better\" is consistent with our objective value. The S/N value is defined by Equation (32):

\[S/N=-10\times log\left(\sum_{i=1}^{n}Y_{i}^{2}/n\right), \tag{32}\]

where \(Y_{i}\) represents the response value for the \(i^{th}\) orthogonal array and \(n\) denotes the number of the orthogonal arrays. The Taguchi experimental design method effectively reduces the number of experimental tests within a reasonable time by utilizing orthogonal arrays. These orthogonal arrays can be generated using Minitab software. The factors (i.e., parameters of the algorithms) and their corresponding levels are described in Table 4. The details of the orthogonal arrays L9 for GA are provided in Table 5. It should be noted that the orthogonal arrays for HGA-VNS, MPGA, and HMPGA-VNS are L27. To ensure more reliable results, five replications for each trial are implemented considering the inherent randomness of the metaheuristic algorithm. \begin{table}
\begin{tabular}{c c c c} \hline
**GA Factors** & **GA Levels** & **HGA-VNS Factors** & **HGA-VNS Levels** \\ \hline \(P\) & A1: 120 A2: 240 A3: 360 & \(P\) & A1: 120 A2: 240 A3: 360 \\ \(Max\_iter\) & B1: 400 B2: 600 B3: 800 & \(Max\_iter\) & B1: 400 B2: 600 B3: 800 \\ \(P_{c}\) & C1: 0.70 C2: 0.75 C3: 0.80 & \(P_{c}\) & C1: 0.70 C2: 0.75 C3: 0.80 \\ \(P_{m}\) & D1: 0.10 D2: 0.15 D3: 0.20 & \(P_{m}\) & D1: 0.10 D2: 0.15 D3: 0.20 \\ \(SP\) & E1: 20 E2: 30 E3: 40 & \(SP\) & E1: 20 E2: 30 E3: 40 \\ \(M_{p}\) & F1: 10 F2: 20 F3: 40 & \(M_{p}\) & F1: 10 F2: 20 F3: 40 \\ \(M_{s}\) & G1: 4 G2: 6 G3: 8 & \(M_{s}\) & G1: 4 G2: 6 G3: 8 \\ \multicolumn{3}{c}{_Maxiter_} & H1: 100 H2: 200 H: 300 \\ \(N_{l}\) & \multicolumn{1}{c}{I1: 3 I2: 4 I3: 5} \\ \hline \end{tabular}
\end{table} 
Table 4: Factors and their levels. \begin{table}
\begin{tabular}{c c c c} \hline
**Instance No. ** & \(1\mathbf{I}1-\mathbf{J}1-\mathbf{I}\mathbf{K}1-\) & **Instance No. ** & \(1\mathbf{I}1-\mathbf{J}1-\mathbf{K}1-\) \\  & \(1\mathbf{P}1-\mathbf{V}1-\mathbf{I}\mathbf{T}1\) & \(1\mathbf{P}1-\mathbf{V}1-\mathbf{I}\mathbf{T}1\) & \(1\mathbf{P}1-\mathbf{V}1-\mathbf{I}\mathbf{T}1\) \\ \hline P1 & \(2-2-5-2-2-2\) & P6 & \(3-4-18-4-4-4\) \\ P2 & \(2-2-8-2-2-2\) & P7 &Due to the differing scales of objective functions across experiments, the Relative Percent Deviation (RPD) is used to normalize the data. The RPD value for each experiment is calculated using Equation (33). \[RPD=\frac{Alg_{sol}-Min_{sol}}{Min_{sol}}\times 100\%, \tag{33}\]

where the objective function values obtained for each iteration of the algorithm and the best solution found by the algorithm are denoted as \(Alg_{sol}\) and \(Min_{sol}\), respectively. The calculation of RPDs for each experiment involves the conversion of objective function values, followed by the computation of the mean RPD.