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# Optimization

These methods formulate the practical problem as a mathematical model, which maximizes or minimizes an objective function. Furthermore, constrains are added to these problems in oder to define the feasible region. Efficient algorithms evaluate all possible solutions to return the best solution in the end.

Mixed integer linear programming:
Mixed integer linear programming (MILP) is  a mathematical modeling approach to find a best outcome of a system which has certain constraints. It is widely used in many optimization areas like production planning, transportation, network desgin, etc. MILP consists of a linear objective function and a certain number of linear constraints built by continuous and integer variables. The aim is to find the optimal value of the objective function (either the maximum or minimum) without violating any of the imposed constraints. MILP is a special case of linear programming with binary variables. MILP models are considerably harder to solve than normal linear programming models. MILP models are usually solved by commercial and noncommercial solvers like Fico Xpress or SCIP.

Stochastic modeling:
A certain degree of randomness and unpredictability is part of most real-world situations. Stochastic modeling is a mathematical approach of presenting data or predicting outcomes in these situations. For example, in a manufacturing process, there are usually unknown parameters like quality of the input materials, reliability of the machines, and competence among employees which affect the outcome of the manufacturing process but cannot be measured with exact values. Stochastic modeling enables predicting the outcome of this process with a certain error rate by taking into account the unpredictability of these factors.

Uncertainty modeling:
A realistic modeling of a system requires taking the inherent uncertainties into account. The uncertainty is quantified to the extent that the uncertain features of the system are modeled with probabilistic nature. Characterizing the uncertain parameters with probability distributions, modeling as a Markov Chain to take dependencies easily into account or using queuing theory for modeling the systems where ‘waiting’ plays an important role are common ways of modeling uncertainty.

Bilevel optimization:
Bilevel problems can come up in practice whenever decentralized or hierarchical decisions have to be made. In these problems several parties take decisions after each other, which influence their respective profit. So far Bilevel problems are only heuristically solvable for realistic sizes. We are trying to improve the optimal methods to get optimal solutions for real problems, as well.

### Exemplary publications

• Fontaine, P., Minner, S. (2014), Benders Decomposition for Discrete-Continuous Linear Bilevel Problems with Application to Traffic Network Design, Transportation Research Part B: Methodological 70: 163-172.