# A Survey on Variable Neighborhood Search for Sustainable Logistics
## 4 Results and Findings
### Multiobjective Analysis  
Sustainable logistics inherently involves addressing multiple, often conflicting objectives, such as minimizing costs, reducing environmental impact, and ensuring social equity. These objectives reflect the three pillars of sustainability, which must be balanced to achieve effective and responsible logistics solutions. Multiobjective optimization methodologies are particularly relevant in this context, as they enable decision-makers to systematically explore and navigate trade-offs between these competing goals. Figure 5: Methods combined with VNS to solve sustainable logistic problems. Of the 173 studies collected, 37 address multiobjective problems. In particular, studies focused on scheduling and network design are more likely to adopt multiobjective versions. These kinds of problems can be approached using various methods, each with its advantages and limitations. One of the simplest and most common techniques is to transform a multiple-objective problem into a single-objective one. This is typically accomplished through scalarization, where a weighted sum of the objectives is created, allowing for the adjustment of importance between the objectives by assigning different weights. By fine-tuning these weights, a decision-maker can explore different trade-offs between objectives, ultimately guiding the solution towards an optimal balance. Due to its simplicity, the number of studies using this kind of approach is among the highest in the context of this review [16; 29; 61; 62; 63; 64; 65; 66; 67]. Another approach is the \(\epsilon\)-constraint method, which involves selecting one of the objectives as the primary goal to be optimized while treating the other objectives as constraints with varying upper bounds (right-hand side values). By systematically adjusting these constraints, multiple Pareto-optimal points can be generated, each representing a different compromise between competing objectives. This method provides a structured way to explore the Pareto front, offering a range of solutions that reflect different priorities. Mathematically, the \(\epsilon\)-constraint method can be expressed as:  
\[\min f_{1}(x),\quad\text{subject to: }f_{i}(x)\leq\epsilon_{i},\,\forall i\in \{2,\dots,k\},\,x\in X \tag{1}\]  
where \(f_{1}(x)\) is the primary objective and \(\epsilon_{i}\) represents upper bounds for the secondary objectives \(f_{i}(x)\). In the context of VNS for sustainable logistics, it is less frequently used [68; 69]. This may be due to the high number of executions necessary to obtain the Pareto front changing the corresponding constraint, which makes it a laborious task. A powerful technique for handling multiobjective optimization is the use of achievement scalarizing functions (ASFs). This method allows decision-makers to specify a target or aspiration level for each objective, and then the optimization process seeks to minimize the distance between the current solution and this desired achievement. Mathematically, the ASF can be represented as:  
\[\min S(x)=\max_{i\in\{1,\dots,k\}}\left\{w_{i}\cdot\frac{f_{i}(x)-z_{i}}{f_{i} ^{\max}-f_{i}^{\min}}\right\},\,x\in X \tag{2}\]  
where \(z_{i}\) is the aspiration level for the i-th objective and \(w_{i}\) represents weights reflecting the decision-maker's preferences. The ASF generates a scalar value for each solution, representing how well it aligns with the defined aspiration levels, and the solution that minimizes this value is selected as optimal. The flexibility of this method makes it useful for problems where decision-makers have specific performance targets for each objective. However, it is rarely applied in VNS-related studies for sustainable logistics [70]. In addition to approaches that transform the problem into a single-objective one, there is the option of addressing the problem as truly multiobjective from the outset. This direct approach allows the simultaneous optimization of all objectives without reducing them to a single aggregated function.