# A Survey on Variable Neighborhood Search for Sustainable Logistics
## 4 Results and Findings
### VNS Version Analysis  
A search on Scopus for the term \"Variable Neighborhood Search\" yields over 3000 works referencing this metaheuristic. However, many of these works do not explicitly mention the specific version of VNS used, making it necessary to perform a detailed content analysis to determine the version applied. Among the explicitly mentioned versions, BVNS is identified in approximately 80 articles. However, as observed in this study on sustainable logistics, it is likely that many of the works that do not specify a version implicitly use the basic version. The most frequently cited variant is VND, appearing in 540 articles. This popularity is due to its natural derivation from local search methods when different types of movements are considered. GVNS is explicitly mentioned in 213 works, while RVNS appears in 88. Notably, adaptive versions of VNS, such as AVNS or AGVNS, are referenced in 40 studies, with a significant concentration in recent years. Other notable variants include VNDS (37 articles) and SVNS (26 articles). Similar trends are reported in other surveys on VNS applications [49; 50], further confirming these observations. As illustrated in Figure 4, the most widely used VNS variants in sustainable logistics are BVNS and GVNS. VND is also frequently applied, as it naturally extends local search methods by incorporating changes in the movement applied. Several studies have used  
Figure 3: Number of publications per general decision problem. Figure 2: A cloud of concepts with the journals according to the papers. RVNS, which simplifies the search process by omitting the local search component. In recent years, the adaptability of metaheuristic designs has led to the emergence of various AVNS proposals. Other VNS versions have been applied in a limited number of studies, reflecting their more specialized or niche use cases. The most challenging approach is the AVNS. Table 2 presents a chronological overview of the various studies on sustainable logistics that incorporate adaptability. The table categorizes the adaptability features based on their implementation in different phases of the VNS process or inside other optimization methods: in the shaking phase of the VNS, in the improvement phase of the VNS, in the selection of the next neighborhood, in the order in which neighborhoods are applied, in hybrid approaches deciding when VNS is applied, and in the search strategy (best improvement or first improvement). This table demonstrates that adaptability is most commonly utilized (marked with X) in the shaking and improvement phases of the VNS, with a particular emphasis on selecting the order in which neighborhoods are applied. Specifically, in [19], a Memetic Algorithm (MA) incorporated AVNS as its local search heuristic to design a robust closed-loop global supply chain network for the medical device \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Paper** & **Shake** & **Improve** & **Select** & **Order** & **Hybrid** & **Best/First** \\ \hline
[27] & - & - & - & - & - & X & - \\
[28] & - & - & - & - & - & X & - \\
[19] & X & X & X & - & - & - \\
[14] & - & X & X & X & - & - \\
[20] & - & X & X & X & - & - \\
[29] & - & - & - & - & X & - \\
[31] & - & X & X & X & - & X \\
[23] & X & X & - & - & - & - \\
[24] & X & X & - & - & - & - \\
[15] & X & - & - & X & - & - \\
[21] & X & - & - & X & - & - \\
[25] & X & - & - & X & - & - \\
[22] & X & X & - & X & - & - \\
[16] & X & - & X & - & - & - \\
[30] & - & X & X & - & X & - \\
[17] & X & - & - & X & - & - \\
[18] & X & - & X & X & - & - \\
[26] & - & - & - & - & X & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ways to include adaptability in AVNS. Figure 4: Distribution of publications per most common VNS versions. industry under uncertainty. Another hybrid heuristic combining AVNS with Tabu Search (TS) was developed in [14] to solve the problem of routing and locating multiple charging stations for electric vehicles within time windows.