# Modeling and Optimization

A model is, generally defined on the basis of, “a simplified mapping, created for a certain objective, of a detail of reality perceived as a system”. It is necessary as the original requires a reduction or expansion to be descriptive, the original is not accessible, or the original is too complex in its variety of characteristics to explain its inner interrelations or to predict its behavior.

This definition contains the three following fundamental properties that are part of every model –

• Mapping: Models are representations of originals (which may be models themselves). Model and original are in an analogy relation, i.e. their attributes are assigned to one another, allowing a purposeful and conclusive system correlation.
• Reduction: Models do not capture all the attributes of the original, only those attributes that are of interest to the user. The relevance of characteristics is determined by the model’s objective.
• Pragmatism: The model-original relation is determined by an objective, i.e. it is not a natural or general assignment. Models serve as substitutes for the original for a limited time, serve certain persons for certain purposes and are restricted to certain operations.

The modeling or mapping is therefore a first opportunity for problem simplification since it includes selection and transfer – diminishing complexity. It is nonetheless due to complexity that in many cases a model’s optimal solution is unreachable, as the knowledge of the (global) optimum depends on the knowledge of the entire solution space, and can therefore frequently not be determined efficiently; e.g., the combinatorial problem of the traveling salesman with cities yields solutions.

Hence, optimization must be based on the premise of maintainable effort – the objective must be a solution that suffices. Essentially, problems may have to be reduced furthermore. The consideration of logistics problems undertaken in this work is based on mathematical modeling, a mapping of reality (object) to mathematics (model). The derivation of an objective insight relies on the conversion of input to output. Besides the model itself, algorithms, as the method of examination, play an essential role in the context of the optimization.

Optimization deals with the determination of the admissible course of action which is best, according to an objective. Its execution does not necessarily yield a (globally) optimal solution. One can differentiate between four solution methods

• Complete enumeration means the consideration and evaluation of all alternatives, among them the optimal choice. This is only advisable in the case of a small input size.
• Exact algorithms do not completely enumerate yet still compute the optimal solution of an optimization problem. Due to the polynomial time restriction, they are not always applicable.
• Approximation algorithms are efficient procedures that a) always give a feasible solution, b) in polynomial time and c) of a certain assured quality level. Often they are characterized by an , meaning the algorithm’s solution is “at most times the optimum” (
• Metaheuristics / heuristics offer no guarantee that an optimal solution will be found or identified as such. They embrace “certain courses of action for solution finding or improvement” that are “reasonable, appropriate, and promising” in regard to the objective and the problem structure. While metaheuristics are based on a general approach which is easily adaptable to a specific problem, heuristics are usually formulated problem-specifically