# B1 | Optimisation of Process Chains under Uncertainty

The aim of subproject B1 was to control uncertainty in process chains using mathematical models and optimization methods. Uncertainty was taken into account in the production phase, as well as in the using phase of the product. On the one had, components underly random influences in the raw material, on the other hand, uncertainty results from unpredictable process behavior or due to the fact that the behavior of the end customers is difficult to predict. The optimization methods used in the project are based on quantified (mixed-integer) linear programs. Together with engineering projects, the following applications were examined within the framework of the project:

### Control of a process chain under material uncertainty on the example of tumbling

Together with project B2, it was investigated how a forward-looking information survey on the uncertain material properties of semi-finished products can be combined with a robust production planning. For the model it was assumed that the height of the semi-finished products is uncertain and that the height distribution can be anticipated in advance by the use of sensors in each case for a semi-finished batch. The adaptive forming process can be adjusted to different heights, but not every height can be produced. A multi-stage optimization model was developed to calculate an optimal production strategy. Calculation with realistic data showed that a robust solution has similar average costs as the stochastic. However, it shows significantly better worst-case behavior and is thus to be preferred.

### Production planning under uncertainty in the virtual SFB demonstrator

In the first funding period, a production and lot size planning model was developed, which was taken up in the second funding period and modified for the production planning of the virtual demonstrator. Quantified integer optimization models were developed to calculate robust batch sizes under uncertainty. Sources of uncertainty include time-dependent demand, energy costs, and machine failures. The production process chain of the SFB demonstrator was represented by a Gozinto graph. The model can be used to evaluate the impact of design changes at the SFB demonstrator at an early stage on production costs and to assess the uncertainty in production. In particular, the calculations show that energy costs are less critical for planning compared to the influence of changing demand or machine failures.

### Optimal energy application planning of an actively damped electric vehicle

Active systems such as the active air spring damper (aLFD), which is investigated within the framework of project C4, are able to regulate the build-up acceleration and wheel load fluctuations of a vehicle by using energy in such a way that the driving safety compared to a vehicle with a passive damping is increased. In collaboration with transfer project T2, a dynamic program for the calculation of an optimal energy application strategy was developed.

### Mathematical results

In the first funding period it was shown that quantified programs are well suited to model multi-stage decision-making under uncertainty. In the second funding period the definition of the quantified linear program (QLP) was extended by a target function according to the minimax principle. This allowed us to treat problems with a quality measure. In particular, multi-stage robust optimization problems resulting from CRC 805 applications and having target functions such as cost or efficiency can be modeled.

The conceptually simplest way to solve quantified programs is the formulation of a deterministic equivalent (DEP) and the use of standard solvers for linear (mixed-integer) optimization problems. In the second funding period, a QLP solver established during the first funding period was further developed. On the one hand, the core became faster and more numerically stable, in particular on multi-stage problems. A technique called Quantifier Shifting has been integrated which generates both feasible solutions and bounds with a small computational effort. NETLIB generated test instances with up to 25 universally quantified variables (over 30 million scenarios) can be solved in less than 24 hours.

## Subproject Managers

Name |
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Prof. Dr. rer. nat. Ulf Lorenz |

Prof. Dr. rer. nat. Alexander Martin |

Prof. Dr. Marc Pfetsch |