In this chapter, we gave an analytical characterization of the tradeoff curve between the competitive ratio and the additive term for the linear search problem with turn costs (note that the result matches the experimental characterization given in (Demaine et al., 2006)). Furthermore, the optimal search strategy corresponding to a competitive ratio
˜
c≥9 is given. Therefore, we now have a complete characterization of the linear search problem with turn costs.
5
Optimization in the Wood Cutting Industry
5.1 Introduction
The problem considered in this chapter is a real-world application from the veneer cutting industry, which we were introduced to by Fritz Becker KG, a manufacturer of shaped wood components from Northern Germany, who also provided us with real-world data.
In the application problem, tree trunks are peeled into thin veneer strips which are cut, glued together and pressed into bentwood pieces for seats, backrests, armrests, chair legs, etc. The production process of these veneers is to be optimized with respect to a minimal wood offcut.
Currently, the production process is planned manually. On the one hand, this en-ables the planner to utilize his experience and certain rules of thumb, especially with respect to the wood quality, which is an important aspect of the problem. On the other hand, with an increasing number of orders, the problem becomes hardly comprehensible and understandable and, consequently, optimization tools have the potential to increase the quality of the production process significantly, especially with respect to long term planning periods. In the first part of this chapter, we develop a model for the problem at hand that both computes an optimal solution in reasonable time and incorporates all restrictions and features of the production process.
In the second part of this chapter, we have a closer look at the inherent uncertainties of the problem attributable to the quality of the wood. Before production, the quality of the wood is not known with certainty, but can only be estimated. In general, dealing with uncertainties in optimization problems is an important issue as disturbances or fluctuations in the problem formulation might significantly change the value of a formerly optimal solution. In fact, formerly feasible solutions may even become infeasible. With respect to the considered problem, the quality of the used wood is subject to fluctuations and it is only possible to determine the quality during the production process itself, which makes it necessary to take these uncertainties into account already during the planning of the production process. Due to the complex structure of the problem and for the sake of practicability, we depart from the concept of online optimization in this chapter.
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Different ways of dealing with uncertain input data are commonly known throughout the literature such as stochastic optimization (for an overview, we refer to (Birge and Louveaux, 2011)). While stochastic optimization assumes some kind of probability dis-tribution for the realizations of the uncertain parameters, the problem at hand calls for a different approach since the manufacturer could not provide any probabilistic informa-tion about the quality distribuinforma-tion of the wood. Therefore, we decided to approach the problem via robust optimization where we do not need such probabilistic information.
The aim of robust optimization is to find solutions which remain feasible and of good quality in all scenarios, whereby a scenario is a realization of the uncertain input data.
For single-objective optimization problems several definitions of robustness, i.e., when a solution is seen as robust against uncertainties, have been analyzed thoroughly. One of these concepts is the concept of minmax robustness, introduced by Soyster (1973) and extensively researched by Ben-Tal and Nemirovski (1998, 1999); Ben-Tal et al. (2009).
Here, a solution is called robust if it is feasible for every scenario and minimizes the objective function in the worst case. Very close to this concept is the concept of regret robustness, suggested, for example, by Kouvelis and Yu (1997), where the worst case regret is to be minimized and the solution has to be feasible in every scenario. Both of these concepts are quite strict with respect to the requirement that a solution has to be feasible in every scenario. To loosen this strict requirement, several other concepts have been proposed, such as the concept of light robustness (see, for example, (Fischetti and Monaci, 2009; Schöbel, 2014)) or the concept of recovery robustness (see, for ex-ample, (Liebchen et al., 2009; Erera et al., 2009; Goerigk and Schöbel, 2011)). Since the manufacturer’s goal is to hedge against the worst case, we will follow the concept of minmax robustness throughout this chapter.
In applications of mathematical optimization and especially in the application pre-sented in this chapter, there is often more than just one objective to consider. Therefore, we have to deal with uncertain multi-objective optimization for which several definitions of robustness have been presented in the literature, see for instance (Branke, 1998; Deb and Gupta, 2006).
Since we would like to hedge against theworst case, we follow the concept of minmax robust efficiency for multi-objective optimization problems. This concept is an extension of the concept of minmax robustness for single-objective optimization problems and has been presented by Ehrgott et al. (2014). Since in multi-objective optimization the term worst case is not that clear, as there is no total order on Rk, the authors replace the worst case with a multi-objective maximization problem and define a dominance relation between the resulting sets, namely a set dominates another if it is completely contained in the other set minus the positive orthant ofRk.
The rest of this chapter is organized as follows: In Section 5.2, the problem is pre-sented explicitly and classified with respect to cutting stock problems. Then, in Sec-tion 5.3, the real-world cutting problem is modeled as a deterministic single-objective optimization problem and results for instances with practical relevance are presented.
Then, we apply the concept of minmax robust efficiency to the cutting problem in Sec-tion 5.4. After clarifying the notaSec-tion for uncertain multi-objective optimizaSec-tion and
recalling the concept of minmax robust efficiency in Section 5.4.1, we simplify the ap-plication problem in Section 5.4.2 in order to be able to apply the concept of minmax robust efficiency properly in Section 5.4.3. We discuss the value of the minmax robust efficient solutions in practice and, finally, give concluding remarks and an outlook to future research in Section 5.5.