# Multi-Objective Decision-Making Meets Dynamic Shortest Path: Challenges and Prospects +
## 3 Multi-Objective Decision-Making  
Multi-objective decision-making resembles many real-world problems. This is due to the fact that decisions are often weighted against more than one objective. When the objectives involved in decision-making are conflicting, one objective may be optimized at the expense of another one. In other words, there is a trade-off between objectives. In this scenario, all trade-offs can be optimal solutions, and this condition turns the problem into a Multi-Objective Optimization Problem (MOOP). An MOOP is composed of \(d\) objectives \(\mathbf{c}=[c_{1},c_{2},\ldots,c_{d}]\) and \(q\) decision variables \(\mathbf{w}=[w_{1},w_{2},\ldots w_{q}]\), with \(\chi=\{\mathbf{w}|\,\mathbf{w}\in\mathbb{R}^{q}\}\) denoting the decision space [25]. Equation (1) shows the vector of objective functions to be optimized for the MOOP resolution. The decision space \(\chi\) regulates the objective vector \(\mathbf{f}(\mathbf{w})\) with \(\mathbf{f}:\chi\rightarrow\mathbb{R}^{d}\), where \(\mathbb{R}^{d}\) represents the objective space [26].  
\[\mathbf{c}(\mathbf{w})=[c_{1}(\mathbf{w}),c_{2}(\mathbf{w}),\ldots,c_{d}( \mathbf{w})] \tag{1}\]  
In multi-objective optimization, there are several possible solutions considering the trade-off between objectives. Comparing possible solutions is a way to indicate which ones stand out. Using the technique of comparing solutions, it is possible to look for a solution that is as satisfactory as any other considering most of the objectives, but strictly better in at least one of the objectives of the problem [27]. This solution is also called a non-dominated solution.  
A set of non-dominated solutions in the objective space produces a boundary called the Pareto front. Figure 2 presents a Pareto front example with non-dominated solutions (also known as optimal solutions) and dominated solutions (also known as sub-optimal solutions). Analyzing Figure 2 from the objective \(c_{2}\) perspective, solutions \(c(\mathbf{w})\) and \(c(\mathbf{w}^{\prime})\) have the same value, i.e., \(c_{2}(\mathbf{w})=c_{2}(\mathbf{w}^{\prime})\). Now, observing from the objective \(c_{1}\) perspective, solution \(c(\mathbf{w})\) is better than solution \(c(\mathbf{w}^{\prime})\), i.e., \(c_{1}(\mathbf{w})\prec c_{1}(\mathbf{w}^{\prime})\). With both objectives \(c_{1}\) and \(c_{2}\) perspectives, the solution \(c(\mathbf{w})\) weakly dominates the solution \(c(\mathbf{w}^{\prime})\). However, the \(c(\mathbf{w})\) and \(c(\mathbf{w}^{\prime})\) solutions are not present in the Pareto front, making both sub-optimal solutions. Looking at the Pareto front, the \(c(\mathbf{w}^{\prime\prime})\) is an optimal solution since it dominates the solutions \(c(\mathbf{w})\) and \(c(\mathbf{w}^{\prime})\).  
An MOOP non-dominated solution enables an optimal choice for the decision-making process regarding the trade-offs between the objectives. Therefore, finding the Pareto front representing the best trade-off for the objectives solves an MOOP. However, computing an exact Pareto front can be unfeasible due to the huge number or the infinite number of non-dominated solutions [28]. Then, a Pareto front approximation is a good approach that requires less effort and can provide a set of non-dominated solutions accurately enough to represent the exact Pareto front [29]. With the Pareto front of the MOOP, it is possible to proceed with the decision-making process, selecting the optimal solution regarding the objectives' importance [24]. Then, an MODM problem is the composition of an MOOP and the decision-making process.  
Figure 2: Bi-dimensional objective space with a Pareto front using standard nomenclature on the solutions.