Multiobjectieve

Multiobjectieve, in Dutch often used to refer to multiobjective optimization, concerns problems that involve optimizing more than one objective function simultaneously. The term is the Dutch equivalent of "multiobjective" or "multi-criteria" optimization.

Formal framework involves a vector of objective functions f(x) = (f1(x), ..., fm(x)) that are to be optimized over a feasible set X. In practice, the objectives are typically conflicting, so improving one objective may worsen another. The aim is not a single best solution but a set of preferred trade-offs, captured by Pareto optimality.

A solution x* is Pareto optimal if there is no x in X such that fi(x) ≤ fi(x*) for all i and fi(x) < fi(x*) for at least one i. The collection of all Pareto optimal solutions forms the Pareto frontier, which represents the trade-offs among objectives. Decision-makers often select a specific point on this frontier based on preferences.

Solution methods include scalarization techniques (for example, weighted sums or epsilon-constraint methods), goal programming, and deviation measures. Heuristic and exact algorithms exist, with evolutionary multiobjective optimization (EMO) methods, such as NSGA-II and SPEA2, widely used for complex, nonlinear problems. The choice of method depends on the problem structure, including differentiability, constraints, and computational budget.

Applications span engineering design, logistics and supply chain, energy systems, environmental planning, finance, and healthcare. Key challenges include incorporating subjective preferences, handling high-dimensional objective spaces, managing uncertainty, and ensuring convergence to a well-distributed Pareto frontier.