Gesellschaft für Operations Research e.V.
Operations Research Proceedings
Marco LübbeckeArie KosterPeter LetmatheReinhard MadlenerBritta PeisGrit Walther Editors
Operations Research Proceedings 2014Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), RWTH Aachen University, Germany, September 2–5, 2014
Operations Research Proceedings
GOR (Gesellschaft für Operations Research e.V.)
More information about this series at http://www.springer.com/series/722
Marco Lübbecke • Arie KosterPeter Letmathe • Reinhard MadlenerBritta Peis • Grit WaltherEditors
Operations ResearchProceedings 2014Selected Papers of the Annual InternationalConference of the German OperationsResearch Society (GOR), RWTH AachenUniversity, Germany, September 2–5, 2014
123
EditorsMarco LübbeckeOperations ResearchRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
Arie KosterLehrstuhl II für MathematikRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
Peter LetmatheManagement AccountingRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
Reinhard MadlenerE.ON Energy Research CenterRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
Britta PeisManagement ScienceRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
Grit WaltherOperations ManagementRWTH Aachen UniversityAachen, Nordrhein-WestfalenGermany
ISSN 0721-5924 ISSN 2197-9294 (electronic)Operations Research ProceedingsISBN 978-3-319-28695-2 ISBN 978-3-319-28697-6 (eBook)DOI 10.1007/978-3-319-28697-6
Library of Congress Control Number: 2015960228
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Preface
OR2014, the German Operations Research Society’s (GOR) annual internationalscientific conference was held at RWTH Aachen University on September 2–5,2014. The conference’s general theme was “business analytics and optimization”that reflected the growing importance of data-driven applications and the underlyingdecision support by operations research (OR) models and methods. This volumecontains a selection of extended abstracts of papers presented at OR2014. Eachof the 124 submissions was reviewed by two or three editors and we decided toaccept 90 of them, solely on the basis of scientific merit. In particular, the GORMaster’s thesis and dissertation price winners summarize their work in this volume.The convenient EasyChair system was used for submission and paper handling.
OR2014 was a truly interdisciplinary conference, as is OR itself, and it reflectedalso the great interest from the natural sciences. Researchers and practitioners frommathematics, computer science, business and economics, and engineering (indecreasing order of their share) attended. The conference Web site www.or2014.decontains additional materials such as slides and videos of several of the presenta-tions, in particular the talks given by practitioners at the business day.
We would like to thank the many people who made the conference a tremendoussuccess, in particular the program committee, the 35 stream chairs, our 12 invitedplenary and semi-plenary speakers, our exhibitors and sponsors, the 60 personsorganizing behind the scenes, and, last but not least, the 881 participants from47 countries. We hope that you enjoyed the conference as much as we did.
Aachen Marco LübbeckeNovember 2015 Arie Koster
Peter LetmatheReinhard Madlener
Britta PeisGrit Walther
v
Contents
Experimental Validation of an Enhanced SystemSynthesis Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Lena C. Altherr, Thorsten Ederer, Ulf Lorenz, Peter F. Pelzand Philipp Pöttgen
Stochastic Dynamic Programming Solution of a Risk-AdjustedDisaster Preparedness and Relief Distribution Problem. . . . . . . . . . . . . 9Ebru Angün
Solution Approaches for the Double-Row EquidistantFacility Layout Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Miguel F. Anjos, Anja Fischer and Philipp Hungerländer
Simulation of the System-Wide Impact of Power-to-Gas EnergyStorages by Multi-stage Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . 25Christoph Baumann, Julia Schleibach and Albert Moser
An Approximation Result for Matchings in PartitionedHypergraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Isabel Beckenbach and Ralf Borndörfer
On Class Imbalance Correction for Classification Algorithmsin Credit Scoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Bernd Bischl, Tobias Kühn and Gero Szepannek
The Exact Solution of Multi-period Portfolio Choice Problemwith Exponential Utility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Taras Bodnar, Nestor Parolya and Wolfgang Schmid
On the Discriminative Power of Tournament Solutions . . . . . . . . . . . . . 53Felix Brandt and Hans Georg Seedig
An Actor-Oriented Approach to Evaluate Climate Policieswith Regard to Resource Intensive Industries . . . . . . . . . . . . . . . . . . . . 59Patrick Breun, Magnus Fröhling and Frank Schultmann
vii
Upper Bounds for Heuristic Approaches to the StripPacking Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Torsten Buchwald and Guntram Scheithauer
An Approximative Lexicographic Min-Max Approachto the Discrete Facility Location Problem . . . . . . . . . . . . . . . . . . . . . . . 71Ľuboš Buzna, Michal Koháni and Jaroslav Janáček
Effects of Profit Taxes in Matrix Games. . . . . . . . . . . . . . . . . . . . . . . . 77Marlis Bärthel
Extension and Application of a General Pickup and DeliveryModel for Evaluating the Impact of Bundling Goods in an UrbanEnvironment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Stephan Bütikofer and Albert Steiner
Optimizing Time Slot Allocation in Single Operator HomeDelivery Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Marco Casazza, Alberto Ceselli and Lucas Létocart
Robust Scheduling with Logic-Based Benders Decomposition . . . . . . . . 99Elvin Coban, Aliza Heching, J.N. Hooker and Alan Scheller-Wolf
Optimal Adaptation Process of Emergency Medical Services Systemsin a Changing Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Dirk Degel
Strategic Network Planning of Recycling of Photovoltaic Modules . . . . . 115Eva Johanna Degel and Grit Walther
Designing a Feedback Control System via Mixed-IntegerProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Lena C. Altherr, Thorsten Ederer, Ulf Lorenz, Peter F. Pelzand Philipp Pöttgen
Fair Cyclic Roster Planning—A Case Study for a Large EuropeanAirport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Torsten Fahle and Wolfgang Vermöhlen
Why Does a Railway Infrastructure Company Need an OptimizedTrain Path Assignment for Industrialized Timetabling? . . . . . . . . . . . . 137Matthias Feil and Daniel Pöhle
A Polyhedral Study of the Quadratic Traveling Salesman Problem . . . . 143Anja Fischer
New Inequalities for 1D Relaxations of the 2D Rectangular StripPacking Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Isabel Friedow and Guntram Scheithauer
viii Contents
Representing Production Scheduling with Constraint Answer SetProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Gerhard Friedrich, Melanie Frühstück, Vera Mersheeva, Anna Ryabokon,Maria Sander, Andreas Starzacher and Erich Teppan
Day-Ahead Versus Intraday Valuation of Flexibility for Photovoltaicand Wind Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Ernesto Garnier and Reinhard Madlener
A Real Options Model for the Disinvestment in Conventional PowerPlants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Barbara Glensk, Christiane Rosen and Reinhard Madlener
Political Districting for Elections to the German Bundestag:An Optimization-Based Multi-stage Heuristic RespectingAdministrative Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Sebastian Goderbauer
Duality for Multiobjective Semidefinite Optimization Problems . . . . . . . 189Sorin-Mihai Grad
Algorithms for Controlling Palletizers . . . . . . . . . . . . . . . . . . . . . . . . . 197Frank Gurski, Jochen Rethmann and Egon Wanke
Capital Budgeting Problems: A Parameterized Point of View . . . . . . . . 205Frank Gurski, Jochen Rethmann and Eda Yilmaz
How to Increase Robustness of Capable-to-Promise . . . . . . . . . . . . . . . 213Ralf Gössinger and Sonja Kalkowski
An Iterated Local Search for a Re-entrant Flow Shop SchedulingProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Richard Hinze and Dirk Sackmann
Selecting Delivery Patterns for Grocery Chains . . . . . . . . . . . . . . . . . . 227Andreas Holzapfel, Michael Sternbeck and Alexander Hübner
Towards a Customer-Oriented Queuing in Service IncidentManagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Peter Hottum and Melanie Reuter-Oppermann
A Lagrangian Relaxation Algorithm for Modularity MaximizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Kotohumi Inaba, Yoichi Izunaga and Yoshitsugu Yamamoto
Row and Column Generation Algorithm for Maximizationof Minimum Margin for Ranking Problems . . . . . . . . . . . . . . . . . . . . . 249Yoichi Izunaga, Keisuke Sato, Keiji Tatsumi and Yoshitsugu Yamamoto
Contents ix
Dual Sourcing Inventory Model with Uncertain Supplyand Advance Capacity Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Marko Jakšič
Integrated Line Planning and Passenger Routing: Connectivityand Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Marika Karbstein
Robust Discrete Optimization Problems with the WOWA Criterion . . . 271Adam Kasperski and Paweł Zieliński
Robust Single Machine Scheduling Problem with Weighted Numberof Late Jobs Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Adam Kasperski and Paweł Zieliński
Price of Anarchy in the Link Destruction (Adversary) Model . . . . . . . . 285Lasse Kliemann
Decision Support System for Week Program Planning . . . . . . . . . . . . . 293Benjamin Korth and Christian Schwede
On Multi-product Lot-Sizing and Scheduling with Multi-machineTechnologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Anton V. Eremeev and Julia V. Kovalenko
Condition-Based Maintenance Policies for Modular MonoticMulti-state Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Michael Krause
An Optimization Model and a Decision Support System to OptimizeCar Sharing Stations with Electric Vehicles . . . . . . . . . . . . . . . . . . . . . 313Kathrin S. Kühne, Tim A. Rickenberg and Michael H. Breitner
Optimising Energy Procurement for Small and Medium-SizedEnterprises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Nadine Kumbartzky and Brigitte Werners
Approximation Schemes for Robust MakespanScheduling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Adam Kurpisz
The Price of Fairness for a Small Number of Indivisible Items . . . . . . . 335Sascha Kurz
Robustness Concepts for Knapsack and Network Design ProblemsUnder Data Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Manuel Kutschka
An Insight to Aviation: Rostering Ground Personnel in Practice . . . . . . 349Manuel Kutschka and Jörg Herbers
x Contents
Consensus Information and Consensus Rating . . . . . . . . . . . . . . . . . . . 357Christoph Lehmann and Daniel Tillich
Integration of Prospect Theory into the Outranking ApproachPROMETHEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Nils Lerche and Jutta Geldermann
Multistage Optimization with the Help of Quantified LinearProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369T. Ederer, U. Lorenz, T. Opfer and J. Wolf
Analysis of Ambulance Location Models Using Discrete EventSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Pascal Lutter, Dirk Degel, Lara Wiesche and Brigitte Werners
Tight Upper Bounds on the Cardinality Constrained Mean-VariancePortfolio Optimization Problem Using TruncatedEigendecomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Fred Mayambala, Elina Rönnberg and Torbjörn Larsson
Scheduling Identical Parallel Machines with a Fixed Numberof Delivery Dates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Arne Mensendiek and Jatinder N.D. Gupta
Kinetic Models for Assembly Lines in Automotive Industries . . . . . . . . 399Lena Michailidis, Michael Herty and Marcus Ziegler
Optimization of Vehicle Routes with Delivery and Pickupfor a Rental Business: A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . 407Susumu Morito, Tatsuki Inoue, Ryo Nakahara and Takuya Hirota
An Ant Colony System Adaptation to Deal with Accessibility IssuesAfter a Disaster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415Héctor Muñoz, Antonio Jiménez-Martín and Alfonso Mateos
Modelling and Solving a Train Path Assignment Model . . . . . . . . . . . . 423Karl Nachtigall and Jens Opitz
A New Approach to Freight Consolidation for a Real-WorldPickup-and-Delivery Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429Curt Nowak, Felix Hahne and Klaus Ambrosi
High Multiplicity Scheduling with Switching Costsfor Few Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437Michaël Gabay, Alexander Grigoriev, Vincent J.C. Kreuzenand Tim Oosterwijk
Contents xi
Sampling-Based Objective Function Evaluation Techniquesfor the Orienteering Problem with Stochastic Travel and ServiceTimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445Vassilis Papapanagiotou, Roberto Montemanniand Luca Maria Gambardella
Optimized Pattern Design for Photovoltaic Power Stations . . . . . . . . . . 451Martin Bischoff, Alena Klug, Karl-Heinz Küfer, Kai Plociennikand Ingmar Schüle
What Are New Insights Using Optimized Train Path Assignmentfor the Development of Railway Infrastructure? . . . . . . . . . . . . . . . . . . 457Daniel Pöhle and Matthias Feil
The Cycle Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Ralf Borndörfer, Marika Karbstein, Julika Mehrgardt, Markus Reutherand Thomas Schlechte
Dispatch of a Wind Farm with a Battery Storage . . . . . . . . . . . . . . . . . 473Sabrina Ried, Melanie Reuter-Oppermann, Patrick Jochemand Wolf Fichtner
Exact Algorithms for the Vehicle Routing Problem with Soft TimeWindows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Matteo Salani, Maria Battarra and Luca Maria Gambardella
Impact of Heat Storage Capacity on CHP Unit CommitmentUnder Power Price Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487Matthias Schacht and Brigitte Werners
Optimizing Information Security Investments with Limited Budget . . . . 493Andreas Schilling and Brigitte Werners
Cut-First Branch-and-Price Second for the Capacitated Arc-RoutingProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Claudia Schlebusch and Stefan Irnich
Solving a Rich Position-Based Model for Dairy Products . . . . . . . . . . . 507Karl Schneeberger, Michael Schilde and Karl F. Doerner
A European Investment and Dispatch Model for Determining CostMinimal Power Systems with High Shares of Renewable Energy. . . . . . 515Angela Scholz, Fabian Sandau and Carsten Pape
Pre-selection Strategies for Dynamic Collaborative TransportationPlanning Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523Kristian Schopka and Herbert Kopfer
xii Contents
Optimal Operation of a CHP Plant for the EnergyBalancing Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531Katrin Schulz, Bastian Hechenrieder and Brigitte Werners
Gas Network Extension Planning for Multiple Demand Scenarios . . . . . 539Jonas Schweiger
Impacts of Electricity Consumers' Unit Commitmenton Low Voltage Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545Johannes Schäuble, Patrick Jochem and Wolf Fichtner
Congestion Games with Multi-Dimensional Demands . . . . . . . . . . . . . . 553Andreas Schütz
Unit Commitment by Column Generation . . . . . . . . . . . . . . . . . . . . . . 559Takayuki Shiina, Takahiro Yurugi, Susumu Morito and Jun Imaizumi
Parallel Algorithm Portfolio with Market Trading-Based TimeAllocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567Dimitris Souravlias, Konstantinos E. Parsopoulos and Enrique Alba
Global Solution of Bilevel Programming Problems . . . . . . . . . . . . . . . . 575Sonja Steffensen
Robustness to Time Discretization Errors in Water NetworkOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581Nicole Taheri, Fabian R. Wirth, Bradley J. Eck, Martin Mevissenand Robert N. Shorten
Forecasting Intermittent Demand with GeneralizedState-Space Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589Kei Takahashi, Marina Fujita, Kishiko Maruyama, Toshiko Aizonoand Koji Ara
Optimal Renewal and Electrification Strategy for Commercial CarFleets in Germany. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597Ricardo Tejada and Reinhard Madlener
A Time-Indexed Generalized Vehicle Routing Model for MilitaryAircraft Mission Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605Jorne Van den Bergh, Nils-Hassan Quttineh, Torbjörn Larssonand Jeroen Beliën
Adapting Exact and Heuristic Procedures in Solving an NP-HardSequencing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613Andreas Wiehl
Strategic Deterrence of Terrorist Attacks . . . . . . . . . . . . . . . . . . . . . . . 621Marcus Wiens, Sascha Meng and Frank Schultmann
Contents xiii
Optimal Airline Networks, Flight Volumes, and the Numberof Crafts for New Low-Cost Carrier in Japan . . . . . . . . . . . . . . . . . . . 627Ryosuke Yabe and Yudai Honma
Variable Speed in Vertical Flight Planning . . . . . . . . . . . . . . . . . . . . . . 635Zhi Yuan, Armin Fügenschuh, Anton Kaier and Swen Schlobach
A Fast Greedy Algorithm for the Relocation Problem. . . . . . . . . . . . . . 643Rabih Zakaria, Laurent Moalic, Mohammad Diband Alexandre Caminada
Multicriteria Group Choice via Majority Preference RelationBased on Cone Individual Preference Relations . . . . . . . . . . . . . . . . . . 649Alexey Zakharov
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
xiv Contents
Experimental Validation of an EnhancedSystem Synthesis Approach
Lena C. Altherr, Thorsten Ederer, Ulf Lorenz, Peter F. Pelzand Philipp Pöttgen
Abstract Planning the layout and operation of a technical system is a common taskfor an engineer. Typically, the workflow is divided into consecutive stages: First,the engineer designs the layout of the system, with the help of his experience or ofheuristic methods. Secondly, he finds a control strategy which is often optimizedby simulation. This usually results in a good operating of an unquestioned systemtopology. In contrast, we apply Operations Research (OR) methods to find a cost-optimal solution for both stages simultaneously via mixed integer programming(MILP). Technical Operations Research (TOR) allows one to find a provable globaloptimal solution within the model formulation. However, the modeling error dueto the abstraction of physical reality remains unknown. We address this ubiquitousproblem of ORmethods by comparing our computational results with measurementsin a test rig. For a practical test case we compute a topology and control strategy viaMILP and verify that the objectives are met up to a deviation of 8.7%.
1 Introduction
Mixed-integer linear programming (MILP) [4] is the outstandingmodeling techniquefor computer-aided optimization of real-world problems, e.g. logistics, flight or pro-duction planning. Regarding the successful application in other fields, it is desirable
L.C. Altherr (B) · U. Lorenz (B) · P.F. Pelz (B) · P. Pöttgen (B)Chair of Fluid Systems, TU Darmstadt, Darmstadt, Germanye-mail: [email protected]
U. Lorenze-mail: [email protected]
P.F. Pelze-mail: [email protected]
P. Pöttgene-mail: [email protected]
T. Ederer (B)Discrete Optimization, TU Darmstadt, Darmstadt, Germanye-mail: [email protected]
© Springer International Publishing Switzerland 2016M. Lübbecke et al. (eds.), Operations Research Proceedings 2014,Operations Research Proceedings, DOI 10.1007/978-3-319-28697-6_1
1
2 L.C. Altherr et al.
to transfer Operations Research (OR) methods to the optimization of technical sys-tems.
The design process of a technical system is typically divided into two consecutivestages: First, the engineer designs the layout of the system, with the help of hisexperienceor of heuristicmethods. Secondly, hefinds a control strategywhich is oftenoptimized by simulation. This usually results in a good operating of an unquestionedsystem topology.
In order to provide engineers with a methodical procedure for the design of newtechnical systems, we strive to establish Technical Operations Research (TOR) inengineering sciences. The TOR approach allows one to find an optimal solution forboth the topologydecision and the usage strategy simultaneously viaMILP [3].Whilethis formulation enables us to prove global optimality and to assess feasible solutionsusing the global optimality gap, the modeling error often cannot be quantified.
Our aim in this paper is to quantify the modelling error for a MILP of a boosterstation with accumulators based on [1, 2]. We examine a practical test case andcompare the computed results with measurements in a test rig.
2 Problem Description
We replicate MILP predictions for the topology and operating of a technical systemin a test rig and compare the computed optimal solution to experimental results.A manageable test case is a water-conveying system, in which a certain amount ofwater per time has to be pumped from the source to the sink. Such a time-dependentvolume flow demand can for example be observed when people shower in a multi-story building. To fulfill this time-varying load, a system designer may choose onesingle speed-controlled pump dimensioned to meet the peak demand.
Another option is a booster station. It consists of an optional accumulator and a setof pumps which are able to satisfy the peak load in combined operation. Comparedto the single pump, this set-up allows for a more flexible operating that may lead tolower energy consumption. The speed of each active pump can be adjusted accordingto the demand, so that they may operate near their optimal working point and thuswith higher efficiency. The designer’s challenging task is to trade off investment costsand energy efficiency while considering all possible topology and operating options.
3 Mixed Integer Linear Program
Our model consists of two stages: First, find a low-priced investment decision inan adequate set of pumps, pipes, accumulators and valves. Secondly, find energy-optimal operating settings for the selected components. The goal is to compare allpossible systems that fulfill the load and to minimize the sum of investment andenergy costs over a given depreciation period.
Experimental Validation of an Enhanced System Synthesis Approach 3
All possible systems can be modelled by a graph G = (V, E) with edges Ecorresponding to possible components, and verticesV representing connection pointsbetween these components. A binary variable pi, j for each optional component(i, j) ∈ V indicates the purchase decision. Since accumulators can store volume,we generate a time-expansion G = (V,E) of the system graph G by copying it oncefor every time step [1]. Each edge (i, ti , j, t j ) ∈ E connects vertices (i, ti ) ∈ V attime ti and ( j, t j ) ∈ V at time t j . An accumulator is represented by edges in time,connecting one point in time with the next, while the other components are edgesin space, representing quasi-static behavior. Binary variables ai, ti , j, t j for each edgeof the expanded graph allow to deactivate purchased components during operation.The conservation of the volume flow Qi, ti , v, tv in space and time is given by
∀ v ∈ V :∑
(i, ti , v, tv) ∈ E
Qi, ti , v, tv · �t =∑
(v, tv, j, t j ) ∈ E
Qv, tv, j, t j · �t (1)
with time step�t . An additional condition with an adequate upper limit Qmax makessure that only active components contribute to the volume flow conservation:
∀ e ∈ E : Qi, ti , j, t j ≤ Qmax · ai, ti , j, t j (2)
Another physical constraint is the pressure propagation
∀ (i, ti , j, ti ) ∈ E : p j,ti ≤ pi, ti + �p + M · ai, ti , j, ti (3)
p j,ti ≥ pi, ti + �p − M · ai, ti , j, ti (4)
which has to be fulfilled along each edge in each time step, if the component is active.Regarding pumps, the resulting increase of pressure depends on the rotational speedof the pump and on the volume flow that is conveyed, cf. Fig. 1b. For pipes and valves,pressure loss increasing with the volume flow is observed, cf. Fig. 1a, c and d. All ofthe measured characteristic curves were linearly approximated and included in themodel by a convex combination formulation [5].
4 Experimental Validation
To validate our mathematical model, we consider three test cases with different time-dependent demand profiles. To assess the modeling error, the computed optimalcombination of the available components is replicated in an experimental setup,and the settings of the system (e.g. the speed of the used pumps or the valve lift)are adjusted according to the computed optimal variable assignment. Subsequently,we verify if the demand profiles are met in each time step. Moreover, the energyconsumption of the setup is measured and the resulting energy costs are calculatedand compared to the objective value of the mathematical model.
4 L.C. Altherr et al.
0 20 40 60 80 1000
10
20
30
40
VALVE LIFT in %
FLOW
FACTORKvin
l/min
discharge
charge
0 0.4 0.8 1.20
2
4
6
8
10
1400 rpm
2162 rpm
2925 rpm
3687 rpm
4450 rpm
VOLUME FLOW in m3/h
PRESS
UREHEADin
mH2O
0 0.2 0.4 0.6 0.8 1 1.2
0
2
4
6
8
10
VOLUME FLOW in m3/h
HEADin
mH2O
0 0.2 0.4 0.6 0.8 1 1.2
0
2
4
6
8
10
VOLUME FLOW in m3/h
HEADin
mH2O
(a) (b)
(c) (d)
Fig. 1 Input data for themodel are themeasured characteristic curves of the components of the fluidsystem. Each data point is themean value of 10,000 samples. The error bars depict the correspondingstandard deviation. a Characteristic curves of the valve. Discharging the accumulator causes morepressure loss than charging it. b Characteristic curve of the most powerful of the available threepumps with a speed range of 1400–4500 rpm. c Characteristic curve of the system for the sectionfrom the source to the junction, cf. Fig. 2. A geodetic offset of around 0.5m has to be overcomefrom the source to the junction. d Characteristic curve from junction to sink. Since the geodeticheight of the sink is around 0.5m lower than that of the junction, this curve starts with negativepressure values
4.1 The Test Rig
Figure2 shows the modular test rig used for validation measurements. It consistsof a combination of up to three speed-controlled centrifugal pumps in a row andan optional acrylic barrel which serves as volume and pressure accumulator. Thethree pumps differ in their maximum rotating speed (S: 2800 rpm, M: 3400 rpm,L: 4450 rpm) and power consumption. Figure1b depicts the characteristic curves ofpump L. The accumulator has a maximum volume of 50 l and a maximum storablepressure of ≈0.2 bar. The barrel can be charged and discharged via a controllablevalve, cf. Fig. 1a. Closing the ball valve allows to charge the accumulator withoutconveying water to the sink. The volume flow is measured by a magnetic flow meterwith a tolerance of ± 0.1 l/min = ± 0.006m3/h. Pressure measurements are per-formed by manometers with a tolerance range of ± 0.01 bar ≈ ± 0.1mH2O. Alldata points represent the mean value of 10,000 samples, collected within 10s.
Experimental Validation of an Enhanced System Synthesis Approach 5
TANK (SOURCE)PUMPS PROPORTIONAL VALVE
TANK (SINK)
ACCUMULATOR
BALL VALVE
JUNCTION
Fig. 2 The test rig consists of a combination of up to three out of three different speed-controlledcentrifugal pumps. An optional accumulator can be used to fulfill the demand at the sink. It can becharged and discharged via a controllable valve
Fig. 3 Test case 1. Onepump fulfills the load
0 50 100 150 200 250 300
0.2
0.4
0.6
TIME in s
VOLUMEFL
OW
inm
3/h
measured predicted
4.2 Comparison of Optimization Results and Measurements
Three different load profiles are given as an input to the optimization program. Webuilt every calculated first-stage solution on our test rig, set up the control strategyand measured the volume flows at the sink and the power consumption of the pumps.The measurement results are given in Figs. 3, 4 and 5.
The time-varying flow demand of the first test case is between 0.25m3/h and0.6m3/h. It can be fulfilled by pumpM and pump L, but not by pump S. As pumpMis at a lower price than pump L, the optimal result via MILP is to buy pump M. Themeasured flow is in good agreement with the demand profile, cf. Fig. 3, if the pump isdriven with the predicted control settings. The computed total energy consumptionfor a recovery period of 10 years is 1.9126 × 103 kWh, corresponding to energycosts of e478.14, and total costs of e923.14. The measured energy consumptionfor one repetition of the load cycle is (663 ± 11.2) × 103 kWh, which sums up to(1.9369 ± 0.1117) × 103 kWh and e(484.23 ± 9.57) within 10 years.
The second test case contains higher flow demands than the first one: 0.4m3/hto 0.9m3/h. The optimization result is to use pumps S and M to cover the load. Thedemanded andmeasured volumeflow ratesmatch, cf. Fig. 4.During a recovery period
6 L.C. Altherr et al.
Fig. 4 Test case 2. Twopumps fulfill the load
0 50 100 150 200 250 300
0.4
0.6
0.8
TIME in s
VOLUMEFL
OW
inm
3/h
measured predicted
Fig. 5 Test case 3. Onepump in combination with anaccumulator fulfills the load
0 50 100 150 200 250 3000
0.2
0.4
0.6
0.8
1
TIME in s
VOLUMEFL
OW
inm
3/h
measured predicted
of 20 years the pumps consume 1.0565 × 104 kWh according to the optimizationresult, compared to (1.0765 ± 0.0239) × 104 kWh derived from the measurements.This leads to total optimal costs of e3436.27, compared to e(3486.32 ± 59.85).Pump L could have also been used, but its energy consumption is higher for flowdemands around 0.7m3/h.
The flow demands in the third test case range from 0.1m3/h to 0.8m3/h. Theoptimal topology consists of pump L, the accumulator and the valve. Pump L cannotconvey volume flows as low as 0.05m3/h in the test rig configuration. The optimiza-tion model correctly predicts the usage of the accumulator during time steps withthese small demands. The accumulator starts with a positive water level. To satisfythe conservation of energy, the water level at the last time step has to be equal to thisstarting value.
In Fig. 5 the measured data is in satisfying agreement with the time-varyingdemand. The optimal energy costs are e537.57. Compared to e(584.27 ± 27.91)derived from the measurements this corresponds to the highest observed deviationof 8.7%. For all test cases a delayed step response of around 5–10s to the changedrotational speed settings is observed.
Experimental Validation of an Enhanced System Synthesis Approach 7
5 Conclusion
In this paper, we presented a MILP model for a system synthesis problem. We areable to find the best combination out of a set of pumps, valves and accumulatorsto satisfy a given time-dependent flowrate demand with minimal weighted purchaseand energy costs. The predicted topology and operating decisions were validated inan experimental setup for three different load demands. The measured volume flowsand the power consumption of the pumps match the predicted values with satisfyingaccuracy. The observed deviations could be caused by the delayed response of thepumps when changing their speed settings.We plan to investigate the influence of thetime step size on the modeling error in a future research project. This will allow usto determine to which degree the components’ start-up characteristics and deferredadaptation should be included into our model formulation.
Acknowledgments This work is partially supported by the German Federal Ministry for Eco-nomic Affairs and Energy funded project “Entwicklung hocheffizienter Pumpensysteme” and bythe German Research Foundation (DFG) funded SFB 805.
References
1. Dörig,B., Ederer, T.,Hedrich, P., Lorenz,U., Pelz, P. F., Pöttgen, P.: Technical operations research(TOR) exemplified by a hydrostatic power transmission system. In: 9. IFK-Proceedings, vol. 1(2014)
2. Pelz, P. F., Lorenz, U., Ederer, T., Lang, S., Ludwig, G.: Designing pump systems by discretemathematical topology optimization: the artificial fluid systems designer (AFSD). In: IREC,Düsseldorf, Germany (2012)
3. Pelz, P. F., Lorenz, U., Ederer, T., Metzler, M., Pöttgen, P.: Global system optimization andscaling for turbo systems and machines. In: ISROMAC-15, Honolulu, USA (2014)
4. Schrijver, A.: Theory of Linear and Integer Programming. Wiley (1998)5. Vielma, J. P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-
linear optimization: unifying framework and extensions. Oper. Res. 58(2), 303–315 (2010)
Stochastic Dynamic Programming Solutionof a Risk-Adjusted Disaster Preparednessand Relief Distribution Problem
Ebru Angün
Abstract This chapter proposes a multistage stochastic optimization frameworkthat dynamically updates the purchasing and distribution decisions of emergencycommodities in the aftermath of an earthquake. Furthermore, the models considerthe risk of exceeding the budget levels at any stage through chance constraints,which are then converted to Conditional Value-at-Risk constraints. Compared tothe previous papers, our framework provides the flexibility of adjusting the level ofconservativeness to the users by changing risk related parameters. Under some con-ditions, the resulting linear programming problems are solved through the Stochas-tic Dual Dynamic Programming algorithm. The preliminary numerical results areencouraging.
1 Introduction
This chapter proposes a dynamic and stochastic methodology to generate a risk-averse disaster preparedness and logistics plan that can mitigate demand and roadcapacity uncertainties. More specifically, we apply multistage stochastic optimiza-tion for dynamically purchasing and distributing emergency commodities with timedependent demands and road capacities. Several authors have dealt with problemssimilar to ours, but [2, 4] are the most related papers. In many cases, our approachcan give less conservative solutions than [2], which considers a robust dynamic opti-mization framework. Furthermore, our approach gives more conservative solutionsthan [4], which considers a risk-neutral dynamic stochastic optimization frameworkwith a finite number of scenarios.
This researchwith the project number 13.402.005 has been financially supported byGalatasarayUniversity Research Fund.
E. Angün (B)Department of Industrial Engineering, Galatasaray University,Ciragan Cad. Ortaköy, 34349 Istanbul, Turkeye-mail: [email protected]; [email protected]
© Springer International Publishing Switzerland 2016M. Lübbecke et al. (eds.), Operations Research Proceedings 2014,Operations Research Proceedings, DOI 10.1007/978-3-319-28697-6_2
9
10 E. Angün
The structure of the chapter is as follows. In Sect. 2, we introduce multistagestochastic programming models that take risk into account. Section3 presents thenovelty in our application of the risk-averse Stochastic Dual Dynamic Program-ming (SDDP) algorithm, and Sect. 3.1 presents some preliminary numerical results.Finally, Sect. 4 summarizes the chapter and presents a few future research directions.
2 Risk-Adjusted Multistage Stochastic ProgrammingModel
We formulate the problem through a risk-adjusted, T -stage stochastic programmingmodel, where the decisions at the first-stage belong to the preparedness phase, andthe decisions at later stages belong to the response phase of a disaster. The risk adjust-ments are achieved by adding probabilistic constraints to the risk-neutral formulationat stages t = 1, . . . , T − 1. A risk-neutral formulation and solution of this problemis given in [1].
We make the following two assumptions for the random vector ξ t whose com-ponents are the demands and the road capacities: i—The distribution Pt of ξ t is
known, and this Pt is supported on a set �t ⊂ Rdt ; ii—The random process
{ξ t
}T
t=2is stage-wise independent.
We formulate the T -stage problem through the following dynamic programmingequations. At stage t = 1, the problem is
Min∑i∈I
[∑l∈L
fil yil + ∑k∈K
qk1rk
1i
]+ E
[Q2
(x1, ξ 2
)]
s.t∑
k∈Kbkrk
1i ≤ ∑l∈L
Ml yil ∀i ∈ I∑l∈L
yil ≤ 1 ∀i ∈ I
Prob{
Q2(x1, ξ 2
) ≤ η2} ≥ 1 − α2
yil ∈ {0, 1} , rk1i ≥ 0,∀i ∈ I, l ∈ L , k ∈ K
(1)
where I , L , and K are the set of potential nodes to open storage facilities, the set ofsize categories of the facilities, and the set of commodity types, respectively, fil isthe fixed cost of opening a facility of size l in location i , qk
t is the unit acquisitioncost of commodity k at stage t , bk is the unit space requirement for commodity k,Ml is the overall capacity of a facility of size l, rk
ti is the amount of commodity kpurchased at stage t in location i , yil is the location i and the size l of a facility, ηt
and αt are the known budget limit and the significance level at stage t , respectively,and x1 is the vector with the components yil ’s and rk
1i ’s. Furthermore, in (1), thefirst set of constraints limits the capacity of a facility, the second set of constraintsrestricts the number of facilities per node, and the chance constraint ensures thatthe second-stage cost-to-go function Q2
(x1, ξ 2
)does not exceed the budget limit η2
with high probability.
Stochastic Dynamic Programming Solution … 11
For later stages t = 2, . . . , T − 1 and for a realization ξ st of ξ t , the cost-to-go
functions Qt(xt−1, ξ
st
)are given by
Min∑
k∈K
[∑i∈I
qkt r k
ti + ∑(i ′, j ′)∈A
ckti ′ j ′mk
ti ′ j ′ + ∑j∈J
pkt w
kt j
]+ E
[Qt+1
(xt , ξ t+1
)]
s.t zkti + ∑
(i, j ′)∈Amk
ti j ′ − ∑( j ′,i)∈A
mkt j ′i = rk
t−1,i + zkt−1,i ∀i ∈ I, k ∈ K
∑(i ′, j)∈A
mkti ′ j − ∑
( j,i ′)∈Amk
t ji ′ + wkt j = νks
t j ∀ j ∈ J, k ∈ K
∑k∈K
bk(
mkti ′ j ′ + mk
t j ′i ′
)≤ κs
ti ′ j ′ ∀ (i ′, j ′) ∈ A
∑k∈K
bk(zk
ti + rkti
) ≤ ∑l∈L
Ml yil ∀i ∈ I
Prob{
Qt+1(xt , ξ t+1
) ≤ ηt+1} ≥ 1 − αt+1
rkti , mk
ti ′ j ′ , wkt j , zk
ti ≥ 0∀i ∈ I, j ∈ J, k ∈ K , (i ′, j ′) ∈ A
(2)
where J and A are the set of nodes that represent shelters and the set of arcs thatrepresent roads in the network, respectively, ck
ti ′ j ′ is the unit transportation cost ofcommodity k through arc
(i ′, j ′), pk
t is the unit shortage cost of commodity k,mk
ti ′ j ′ is the amount of commodity k transported through arc(i ′, j ′), wk
t j and zkti are
the shortage amount of commodity k in shelter j and the amount of commodity kstored in location i , respectively, νks
t j and κsti ′ j ′ are the demand for the commodity
k in shelter j and the road capacity of arc(i ′, j ′) for a realization s, respectively,
and xt is the vector with components rkti ’s and zk
ti ’s; all values depend on stage t .Moreover, in (2), the first set of constraints represents the flow conservation withzk1,i = 0∀i ∈ I, k ∈ K , the second set of constraints is for the demand satisfaction,and the third set of constraints is for the road capacity. The stage T problem has thesame three sets of constraints as in (2), but there are no more acquisition decisionsand the remaining inventories are penalized through a unit holding cost hk
T . Hence,the objective function at t = T becomes
Min∑
k∈K
⎡
⎣∑
i∈I
hkT zk
T i +∑
(i ′, j ′)∈A
ckT i ′ j ′mk
T i ′ j ′ +∑
j∈J
pkT wk
T j
⎤
⎦ .
It was suggested in [5] to replace the chance constraint by the CV@Rα-typeconstraint, where CV@Rα is given by
V@Rα
[Qt
(xt−1, ξ t
)] + α−1E
[Qt
(xt−1, ξ t
) − V@Rα
[Qt
(xt−1, ξ t
)]]+ (3)
where the Value-at-Risk (V@Rα) in (3) is, by definition, the left-side (1 − α)-quantile of the distribution of Qt
(xt−1, ξ t
), and
[Qt − V@Rα (Qt )
]+ = max {Qt − V@Rα (Qt ) , 0} .
12 E. Angün
A problem with a CV@R-type constraint is that it can make the problem infea-sible. Consequently, it could be convenient to move the CV@R-type constraint intothe objective function; that is, we redefine the cost-to-go function as
Vλt
[Qt
(xt−1, ξ t
)] := (1 − λt )E[Qt
(xt−1, ξ t
)] + λtCV@Rαt
[Qt
(xt−1, ξ t
)]
(4)
where λt ∈ [0, 1] is a parameter that can be tuned for a tradeoff between minimizingon average and risk control.
The expectation and the CV@R in (4) usually make the problem analyticallyuntractable. A possible way to deal with this problem is to use Sample Aver-age Approximation (SAA). That is, sample ξ t from its distribution Pt to obtainSt := {
ξ 1t , . . . , ξ
Ntt
}, where Nt is the sample size at stage t . Then, setting λt = 0
in (4) and for a fixed xt−1, solve the stage t problem to obtain the Nt opti-mal values Qt
(xt−1, ξ
1t
), . . . , Qt
(xt−1, ξ
Ntt
). Let Qt,(1) < Qt,(2) < · · · < Qt,(ι) <
· · · < Qt,(Nt ) be the order statistics obtained from these optimal values, and ι bethe smallest integer that satisfies ι ≥ Nt (1 − αt ). This Qt,(ι) is an estimate ofV@R
[Qt
(xt−1, ξ t
)]so that (4) is estimated through
(1 − λt )
Nt
Nt∑
s=1
Qt(xt−1, ξ
st
) + λt Qt,(ι) + λt
Ntαt
Nt∑
s=1
[Qt
(xt−1, ξ
st
) − Qt,(ι)]+ .
3 Stochastic Dual Dynamic Programming Applications
The Stochastic Dual Dynamic Programming (SDDP) algorithm was introduced in[3], and the risk-averse SDDP algorithm was applied to an SAA problem in [6].Furthermore, a detailed description of the risk-neutral SDDP algorithm applied to anSAA problemwas given in [1]. We do not give further detail on the SDDP algorithm,but refer to the papers above.
The novelty in our application of the risk-averse SDDP follows from the followingproposition.
Proposition 1 For a realization ξ st of ξ t and at a given xt−1, a subgradient gs
t ofVλt
[Qt
(xt−1, ξ t
)]is computed through
gst =
{−
(1 − λt + λtα
−1t
)BsT
t π st −
(λt − λtα
−1t
)B(ι)T
t π(ι)t if Qt
(xt−1, ξ
st)
> Qt,(ι)
−(1 − λt )BsTt π s
t − λt B(ι)Tt π
(ι)t if Qt
(xt−1, ξ
st) ≤ Qt,(ι)
where π st is the vector of dual variables corresponding to the first set of constraints
for t = 3, . . . , T , and to the first and the second set of constraints for t = 2, Bst is the
Stochastic Dynamic Programming Solution … 13
matrix whose entries are given by the coefficients of rkt−1,i and zk
t−1,i for t = 3, . . . , T ,
and by the coefficients of rkt−1,i and yil for t = 2, and B(ι)
t and π(ι)t correspond
to Qt,(ι).
Then, a subgradient gt of (4) is estimated through gt = 1Nt
Nt∑s−1
gst .
3.1 Numerical Results
We consider three consumable emergency commodity types, 10 potential locationsfor facilities, and 30 shelters in the two boroughs of Istanbul. The data for costsand volumes of commodities, the data for costs and capacities of facilities, and thepopulation data are the same as in [1]. Furthermore, [7] estimated the total numbersof buildings that are prone to be damaged at various levels for an earthquake ofmagnitude 7.3 on the Richter scale; these data are also summarized in [1].
We model the random demand νkt j for commodity k at shelter j and random
capacity κt i ′ j ′ for any arc (i ′, j ′) at stage t (t = 2, . . . , T ) as follows:
νkt j = δk
t
(ςt−1, j + ςt, j
) ∀ j ∈ J and κti ′ j ′ = η ∗ τ(t)ω(i ′, j ′)/γti ′ j ′
∀(i ′, j ′) ∈ A
where δkt is the amount of commodity k needed by a single individual during stage t ,
ςt−1, j is the number of evacuees who were expected to arrive at shelter j by the endof stage (t − 1), and ςt, j is the random additional number of evacuees who arrive atshelter j at stage t . Moreover, η is the capacity of a single vehicle, τ(t) is the lengthof stage t , ω(i ′, j ′) is the actual distance between nodes i ′ and j ′, and γti ′ j ′ is therandom speed of the vehicle. Both ςt, j and γti ′ j ′ are assumed to be normal; see [1].
We consider T = 6 stages, and concentrate on the first 72h in the aftermathof an earthquake. The stopping criterion of the SDDP algorithm is the maximumnumber of iterations, which is 100. All computational experiments are conductedon a workstation with Windows 2008 Server, three Intel(R) Xeon(R) CPU E5-2670CPUs of 2.60GHz, and 4GB RAM. The linear programming problems are solvedby ILOG CPLEX Callable Library 12.2.
So far we have only experimented with risk-related parameters, namely λ andα. Values of λ closer to 1 and values of α closer to 0 make the 6-stage problemsmore risk-averse. In Fig. 1, for α = 1% (on the left) the lower bounds on the 6-stage costs for λ = 0.4 and λ = 0.5 stabilize at almost the same value. For α = 5%(on the right), however, the lower bound for the more risk-averse case (λ = 0.5)stabilizes at a value which is much lower than the lower bound of the less risk-aversecase (λ = 0.4); this is due to the fact that for the λ = 0.5 case, facilities store moreemergency commodities, and hence the shortage amounts and the penalty costs aremuch lower compared to the λ = 0.4 case.
14 E. Angün
0 10 20 30 40 50 60 701.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8x 107
CPU time in seconds
Low
er b
ound
on
the
6−stag
e co
sts
λ = 0.4λ = 0.5
0 10 20 30 40 50 60 70 800.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8x 107
CPU time in seconds
Low
er b
ound
on
the
6−stag
e co
sts
λ = 0.4λ = 0.5
Fig. 1 Changes in the lower bound on the 6-stage costs for α = 1% on the left and α = 5% onthe right
4 Conclusions
In this chapter, we formulate a short-term disaster management problem through amultistage stochastic programming model. The model takes the risk of exceedingthe budget level at that stage into account through a chance constraint, which is thenconverted into a CV@R-type constraint. Because the CV@R-type constraint canmake the problem infeasible, that constraint is further added to the objective function.Under some assumptions, the resulting problem is solved through the StochasticDual Dynamic Programming (SDDP) algorithm. The numerical results are verypreliminary, but nevertheless encouraging; the model responds to the risk factors,namely λ and α, as it should. Furthermore, the solution time of a risk-adjustedproblem is not worse than a risk-neutral one.
Future research should include the derivation of a stopping rule for the risk-adjusted SDDP. Moreover, more numerical experiments should be done concerningthe risk factors and testing the model sensitivities to various cost parameters.
References
1. Angun, E.: Stochastic dual dynamic programming solution of a short-term disaster managementproblem. In: Dellino, G.,Meloni, C. (eds.) UncertaintyManagement in Simulation-Optimizationof Complex Systems: Algorithms and Applications. Operations Research/Computer ScienceSeries. Springer, New York, 2015, pp. 225–250
2. Ben-Tal, A., Chung, B.D., Mandala, S.R., Yao, T.: Robust optimization for emergency logisticsplanning: risk mitigation in humanitarian relief supply chains. Transp. Res. Part B: Methodol.45, 1177–1189 (2011)
3. Pereira,M.V.F., Pinto, L.M.V.G.:Multi-stage stochastic optimization applied to energy planning.Math. Program. 52, 359–375 (1991)
4. Rawls, C.G., Turnquist, M.A.: Pre-positioning and dynamic delivery planning for short-termresponse following a natural disaster. Socio. Econ. Plan. Sci. 46, 46–54 (2012)
Stochastic Dynamic Programming Solution … 15
5. Rockafellar, R.T., Uryasev, S.P.: Optimization of conditional value-at-risk. J. Risk 2, 21–41(2000)
6. Shapiro, A., Tekaya, W., da Costa, J.P., Soares, M.P.: Risk neutral and risk averse stochastic dualdynamic programming method. Eur. J. Oper. Res. 224, 375–391 (2013)
7. Unpublished data obtained through private communication from Ansal, A., Bogaziçi UniversityKandilli Observatory and Earthquake Research Center, Istanbul, Turkey
Solution Approaches for the Double-RowEquidistant Facility Layout Problem
Miguel F. Anjos, Anja Fischer and Philipp Hungerländer
Abstract We consider the Double-Row Equidistant Facility Layout Problem andshow that the number of spaces needed to preserve at least one optimal solution ismuch smaller compared to the general double-row layout problem. We exploit thisfact to tailor exact integer linear programming (ILP) and semidefinite programming(SDP) approaches that outperform other recent methods for this problem. We reportcomputational results on a variety of benchmark instances showing that the ILP ispreferable for small and medium instances whereas the SDP yields better results onlarge instances with up to 60 departments.
1 Introduction
An instance of the Double-Row Equidistant Facility Layout Problem (DREFLP)consists of d one-dimensional departments with equal lengths and pairwise non-negative weights wi j . The objective is to find an assignment r : [d] → {1, 2} of thedepartments to the rows and feasible horizontal positions p for the centers of thedepartments minimizing the total weighted sum of the center-to-center distancesbetween all pairs of departments:
M.F. AnjosCanada Research Chair in Discrete Nonlinear Optimization in Engineering,GERAD & École Polytechnique de Montréal, Montreal, QC H3C 3A7, Canadae-mail: [email protected]
A. Fischer (B)Department of Mathematics, TU Dortmund, Dortmund, Germanye-mail: [email protected]
P. HungerländerDepartment of Mathematics, Alpen-Adria Universität Klagenfurt,Klagenfurt am Wörthersee, Austriae-mail: [email protected]
© Springer International Publishing Switzerland 2016M. Lübbecke et al. (eds.), Operations Research Proceedings 2014,Operations Research Proceedings, DOI 10.1007/978-3-319-28697-6_3
17
18 M.F. Anjos et al.
minr,p
∑
i, j∈[d]i< j
wi j |p(i) − p( j)|, s.t. |p(i) − p( j)| ≥ 1 if i �= j and r(i) = r( j).
In this short paper we summarize our new results on the structure of the optimallayouts, present the ILP and SDP models that follow from those results and pro-vide representative computational results. Formal proofs and detailed computationalresults are omitted due to space limitations and are provided in the full paper [3].
2 The Structure of Optimal Layouts
The definition of the (DREFLP) suggests that the spaces between the departmentscan be of arbitrary lengths in a feasible layout. Hence it is intuitive to model theproblem using continuous variables. But in fact the (DREFLP) has some hiddenunderlying combinatorial structure that makes it possible to model it using onlybinary variables. The following theorem is a special case of Theorem 2 from [8]:
Theorem 1 There is always an optimal solution to the (DREFLP) on the grid.
Note that for such layouts all departments and spaces have equal size. An illustrationof this result is provided in Fig. 1:
Note that the grid property is implicitly fulfilled for layouts corresponding to thegraph version of the(DREFLP), i.e. the extension of the linear arrangement problemwhere two or more nodes can be assigned to the same position. Hence in particularthe Minimum Duplex Arrangement Problem considered by Amaral [2] is a specialcase of the (DREFLP) by Theorem1.
For layouts fulfilling the grid property we say that department i lies in column jif the center of department i is located at the j th grid point. For example, department5 lies in column 4 in Fig. 1.
In the next theorem we make three assumptions.
Assumption 1: Columns that contain only spaces can be deleted.
d1
d2
d3 d4
d5
d1
d2
d3 d4
d5
Assumption 2: If two non-empty adjacent columns both contain only one depart-ment, then the two departments can be assigned to the left column and the rightcolumn can be deleted.
Fig. 1 Illustration of thegrid property of the(DREFLP) s
s
d3
sd1
d2
d4
d5
d6
d7
d8
s
s
s
Row 1
Row 2
