One of the most frequently occurring substructures in integer linear programs (ILPs) is the knapsack constraint. In this paper, we study ways to deal with uncertainty in the coefficients of such constraints. We combine the budget uncertainty set of Bertsimas and Sim (Math Program Ser B 98:49–71, 2003; Oper Res 52(1):35–53, 2004) with a recovery action, i.e., in order to restore feasibility up to k items may be removed when the actual coefficients are known. We present three different approaches to formulate this recoverable robust knapsack (rrKP) as ILP, including a novel compact reformulation of quadratic size. The other two formulations have exponentially many variables and/or constraints. To keep the ILPs small in practice, we develop separation algorithms, not only for the exponential formulations, but also for the compact reformulation. An experimental comparison of six different approaches to solve the rrKP on a carefully designed set of benchmark instances reveals that a lazy constraint-and-variables approach for the compact reformulation outperforms other alternatives.

We analyze the computational complexity of the synthesis problem of decentralized energy systems. This synthesis problem consists of combining various types of energy conversion units and determining their sizing as well as operations in order to meet time-varying energy demands while maximizing an objective function, e.g., the net present value. In this paper, we prove that the synthesis problem of decentralized energy systems is strongly NP-hard. Furthermore, we prove a strong inapproximability result. This paper provides the first complexity findings in the long scientific history of the synthesis problem of decentralized energy systems. (C) 2019 Elsevier Ltd. All rights reserved.

Electoral districts have great significance for many democratic parliamentary elections. Voters of each district elect a number of representatives into parliament. The districts form a partition of the electoral territory, meaning each part of the territory and population is represented. The problem of partitioning a territory into a given number of electoral districts, meeting various criteria specified by laws, is known as the Political Districting Problem. In this paper, we review solution approaches proposed in the literature and survey districting software, which provides assistance with interactive districting by hand or even decision support in the form of optimization-based automated districting. As a specific application, we consider the Political Districting Problem for the federal elections in Germany. Regarding the present requirements and objectives, we discuss and examine the applicability of the approaches mentioned in the literature to this specific German Political Districting Problem.

We generalize extensively studied apportionment methods to apply them to the political districting problem. For the generalization, we prove NP-hardness and develop a mixed-integer linear program.

About half of the seats in German Parliament (Bundestag) are assigned through relative majority vote in each of the 299 constituencies in German Federal Elections. Legal requirements and jurisprudence of courts regulate the characteristics and principles that have to or rather should be satisfied by constituencies in Germany. We investigate how well these requirements are met and whether some legal guidelines are given preferential treatment. We further analyze if, and to what extent, the decision-maker of the constituencies, i.e., the legislator, adopts proposals made by an independent Constituency Commission. No systematic and numerical study of constituency delimitation laws and practices in Germany has been conducted to date. This paper rectifies that shortcoming and provides the basis to prepare substantive arguments for upcoming delimitation debates in Germany. Our work is based on an extensive set of geographical and population data of the last five German Federal Elections, including the last one in September 2017.

DESSLib provides benchmark instances obtained by real world data for synthesis problems of decentralized energy supply systems (DESS). In this paper, the considered optimization problem is described in detail. Decentralized Energy Supply Systems An energy system consists of a subsystem of energy consumers and a subsystem of energy suppliers. In this case the different forms of final energy consumed, are satisfied by different supplier technologies. Since the supplier subsystem is represented by multiple decentralized, on-site components, we speak of a decentralized energy supply system (DESS). The application of DESS encompasses, e.g., chemical parks, urban districts, hospitals or research complexes. Besides climatic goals, energy costs usually match the companies’ profits in magnitude and energy efficient DESS can reduce energy cost significantly [2]. Thus, optimally designed decentralized energy supply systems can lead to a considerable increase of profits. DESS can consist of several energy conversion components (e.g., boilers and chillers) providing different utilities (e.g., heating, cooling, electricity, steam). DESS are highly integrated and complex systems due to the integration between different energy forms and connection to the gas and electricity market as well as the energy consumers. An example for a DESS is shown in Figure 1. The target of optimal synthesis of DESS is the identification of an (economically) optimal structure and optimal component dimensions, while simultaneously considering the optimal operation of the selected components [4]: 1. Structure: Which energy conversion components and how many of each type? 2. Dimension: How big should these components be? 3. Operation: Which components are operated at which level at what time? These three decision-levels could be considered sequentially. However, the levels influence each other, thus only a simultaneous optimization will find a global optimal solution. For some literature about synthesis of DESS see [6], [7], [5].

Sebastian Goderbauer zeigt, dass das Einteilen von Wahlkreisen für die Deutsche Bundestagswahl aufgrund der gesetzlichen Vorgaben als ein mathematisches Optimierungsproblem angesehen werden kann. Er g

DESSLib provides benchmark instances obtained by real world data for synthesis problems of decentralized energy supply systems (DESS). This paper contains information about name convention, structure and generation of the DESSLib benchmark instances. Moreover, all parameters and functions needed to describe the component models and the overall system parameters are defined. See [2] for the description of the considered optimization problem. Benchmark Instances for Synthesis of DESS DESSLib contains categorized problem instances for synthesis problems of decentralized energy supply systems based on the original real world example stated by Voll et al. (2013) [6]. The raw data is given in hourly power demand levels. In order to reduce the computational complexity, the historical data was divided into representative load cases. Those cases represent the most common loads occuring per year. Moreover the peak load demands of the thermal energy demand are added as seperate loads. Note that peak load cases have a duration of zero. Thereby, the energy system will be designed to meet all occuring energy demands. The instances are characterized by two dimensions: (i) the number of considered components in the superstructure S and (ii) the number of considered load cases L. A file representing an DESSLib instance is named, e.g., as follows:

According to the legal requirements for Elections to the German Bundestag the problem of partitioning Germany into electoral districts can be formulated as a multi-criteria graph partition problem. To solve this regularly current problem, an optimization-based heuristic is introduced and successfully applied to German population data.

Decentralized energy supply systems (DESS) are highly integrated and complex systems designed to meet time-varying energy demands, e.g., heating, cooling, and electricity. The synthesis problem of DESS addresses combining various types of energy conversion units, choosing their sizing and operations to maximize an objective function, e.g., the net present value. In practice, investment costs and part-load performances are nonlinear. Thus, this optimization problem can be modeled as a nonconvex mixed-integer nonlinear programming (MINLP) problem. We present an adaptive discretization algorithm to solve such synthesis problems containing an iterative interaction between mixed-integer linear programs (MIPs) and nonlinear programs (NLPs). The proposed algorithm outperforms state-of-the-art MINLP solvers as well as linearization approaches with regard to solution quality and computation times on a test set obtained from real industrial data, which we made available online.