# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 3 Solution Algorithms
### Hmpga-Vns Algorithm  
The MPGA is an extended version of GA that enhances its capabilities. MPGA incorporates multiple independent populations that operate in parallel to perform search and optimization. Each population comprises multiple individuals that perform genetic operations, including selection, crossover, mutation, and fitness evaluation. Specific communication mechanisms, such as information sharing and individual migration, are implemented to facilitate information exchange and maintain diversity among the populations. This enables MPGA to conduct a more comprehensive and efficient search. MPGA effectively tackles challenges encountered in single-population GA, such as convergence and being trapped in local optima. The computational aspects of MPGA are summarized in the following section. ``` Step 1: Initialize a population and divide it into \(n\) subpopulations. Step 2: Independently perform selection, crossover, mutation, and evaluation operations on each subpopulation. Step 3: Conduct migration operations among the subpopulations. Step 4: Repeat steps 2 to 3 until the termination criteria are satisfied. Step 5: Merge each subpopulation to form a population. Step 6: Select the optimal individual from the population. The MPGA has been successfully applied to solve NP-hard global optimization problems such as transportation scheduling, parallel machine scheduling, dynamic facility layout, and multi-energy complementary system optimization [36; 37; 38; 39]. The HMPGA-VNS employs a distinct strategy where the population is divided into \(P/SP\) subpopulations that perform independent genetic operations. After generating offspring subpopulations, a subset of the top-performing individuals is randomly migrated to other subpopulations to promote diversity. Subsequently, the subpopulations are merged. The integration of VNS as a local search technique facilitates the improvement of individuals within the population and the achievement of optimal solutions at each iteration. Algorithm 3 presents the pseudocode for the HMPGA-VNS algorithm. ## 4 Computation Results

To validate the reliability of our model and evaluate the effectiveness of our proposed approach, we conducted a series of experiments. These experiments involve generating experimental data and utilizing the Taguchi experimental design method. Subsequently, a comprehensive comparison among HGA-VNS, HMP-GA-VNS, GA, and MPGA were performed across all experiments. The evaluation of these algorithms was based on the statistical analysis of experimental outcomes, specifically focusing on the means and variances. The proposed algorithms were implemented by MATLAB(r) 2020a software on a PC with an Intel(R) Core(TM) i5-8265U CPU @1.60 GHz. Finally, a sensitivity analysis of the model's key parameters was conducted and valuable insights for management from the obtained results were derived. ### Instances

Due to the lack of existing datasets on the PPSCM, 10 randomly generated instances with different sizes are utilized. The details regarding the random parameters can be found in Table 2. For the numerical test instances, the information on manufacturers (I), DCs (J), retailers (K), products (P), vehicles (V), and periods (T) is presented in Table 3.

### Parameters Setting

Proper parameters significantly impact the output of a metaheuristic algorithm [40; 41]. Configuring algorithm parameters effectively during the design phase can greatly enhance performance and problem-solving capabilities. In this study, the Taguchi experimental design method is employed to train the parameters of the metaheuristic algorithms. [PERSON] introduced a method to convert repetitive data into the signal-to-noise (S/N) ratio, a robust