Robust Optimization

Robust Optimization Models Robust Optimization is an approach to ad- dress optimization problems with uncertain input data. The goal of Robust Optimization is to find a solution to the optimization problem, which can cope best with all possible realizations of the uncertain data. In general, this solution is not optimal for every realization of the uncertain input data, but performs well even in the worst case. In many cases, Robust Optimization deals with uncertain input parameters which are known only within certain bounds. The first robust approach for linear optimization problems was presented by A.L. Soyster in 1973 (see [90]). Soyster proposes a linear optimization model which leads to a solution that is feasible for all input data. However, this approach generates rather over-conservative solutions.

In [72], J.M. Mulvey et al. present a scenario based robust approach. The model combines solution robustness and model robustness. A main feature of

CHAPTER 3. PROBLEM FORMULATION AND OBJECTIVES

the approach of Mulvey et al. is the fact that the robust solutions contain common components for all scenarios as well as components which are scenario specific. However, the computational efforts for large problem instances can be very high.

Less conservative robust models were proposed by A. Ben-Tal and A. Nemirovski starting from 1998 (see [11], [12] and [13], for example) as well as L. El-Ghaoui et al. (see [39] and [40]). In these papers, uncertain linear problems with ellipsoidal uncertainties are considered, which allow the approximation of more complex uncertainty sets. The resulting robust optimization problems, called robust counterparts, involve conic quadratic problems.

A robust approach which allows to control the level of conservatism in the robust solution was proposed by D. Bertsimas and M. Sim (see [14] and [15]). A robustness parameter specifies the number of coefficients that are allowed to change such that the optimal solution can be guaranteed to be feasible. Moreover, the approach of Bertsimas and Sim is directly applicable to discrete optimization problems. In [16], an application on optimal control of supply chains subject to stochastic demand is presented.

P. Kouvelis and G. Yu present a scenario-based robust framework specially designed for discrete optimization problems (see [62]). A drawback of their approach is the fact, that the robust counterpart of many polynomially solvable discrete optimization problems is NP-hard.

In [64] and [91], the new approach recoverable robustness of S. Stiller et al. is presented. In recoverable robustness, the aim is to find solutions to an opti- mization problem with uncertain input data, which can be made feasible, or recovered, within a given budget for all possible situations. The optimization process consists of two phases, a planning phase and a recovery phase.

M. Fischetti and M. Monaci propose a robustness concept called Light Robust- ness in [42]. Light Robustness is based on the approach given by Bertsimas and Sim ([14]). The aim is to balance a possible violation of the constraints and the quality of the solution concerning the objective function value.

In [85], A. Sch¨obel and A. Kratz analyze the so-called price of robustness, i.e. the trade-off between the best possible solution and a robust solution. A bicriteria problem is formulated by adding the trade-off as an additional objective function to the original problem. The bicriteria approach is developed in the context of aperiodic timetabling problems.

For a detailed survey on the major existing robust approaches in the context of Network Flow Problems, see [46].

3.3. LITERATURE REVIEW

Supply Chain Problems In the context of supply chain problems, Robust Optimization has been successfully applied. We give some example references here.

In [98], C.-S. Yu and H.-L. Li propose a Robust Optimization model for stochas- tic logistic problems, inspired by the approach of Mulvey et al. ([72]). Yu and Li focus on linear logistic models with a scenario-based data set. In comparison to the original approach of Mulvey et al., the number of variables and constraints of the robust problem is decreased, resulting in a lower running time of the method of Yu and Li. The computational performance of the model is demon- strated by means of two logistic management examples. However, even though the resulting robust counterpart of a linear problem remains linear when apply- ing the method of Yu and Li, the robust formulation of a network flow problem can no longer be represented as a network flow problem.

In [16], D. Bertsimas and A. Thiele present a Robust Optimization approach for the problem of optimally controlling a supply chain in discrete time. Uncertain demand values in the supply chain without specific distribution are considered. The optimization method is based on the robust optimization framework of Bertsimas and Sim ([14], [15]). The resulting optimization model is a linear programming problem in case that no fixed costs occur along the supply chain and a mixed integer programming problem otherwise. In contrast to the ap- proach presented in this thesis, the supply chain problem is not modeled as a network flow problem.

In [1], E. Adida and G. Perakis examine demand uncertainty in dynamic pricing and inventory control problems. A robust approach based on a demand-based fluid model is presented. The optimization model includes linear and convex functions. As a dynamic problem setting is considered, the coefficients of the input data are time-dependent, such that the optimal pricing and production policy in the fluid model are determined for a time horizon. In contrast to the time-discrete network flow problem representation in this thesis, the model of Adida and Perakis is a time-continuous one.