Papers by Matthias Ehrgott
arXiv (Cornell University), Oct 23, 2016
We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization... more We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions.
This chapter addresses two complementary themes in relation to problem structuring and MCDA. The ... more This chapter addresses two complementary themes in relation to problem structuring and MCDA. The first and primary theme highlights the nature and importance of problem structuring for MCDA and then reviews suggested ways for how this process may be approached. The integrated use of Problem Structuring Methods (PSMs) and MCDA is one such approach; this potential is explored in greater depth and illustrated by four short case studies. In reflecting on these and other experiences we conclude with a brief discussion of the complementary theme, that MCDA can also be viewed as creating a problem structure within which many other standard tools of OR may be applied, and could therefore also be viewed as a PSM. Identification of the problem/issue Problem structuring Model building Using the model to inform and challenge thinking Developing an action plan Values Goals Constraints
International series in management science/operations research, 2005
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.
HAL (Le Centre pour la Communication Scientifique Directe), Apr 20, 2006
In this paper we present a synthesis of the two phase method for the biobjective assignment probl... more In this paper we present a synthesis of the two phase method for the biobjective assignment problem. The method, which is a general technique to solve multiobjective combinatorial optimization (MOCO) problems, has been introduced by Ulungu in 1993. However, no complete description of the method to find all efficient solutions of the biobjective assignment problem (that allows an independent implementation) has been published. First, we present a complete description of a two phase method for the biobjective assignment problem, with an improved upper bound. Second, a combination of this method with a population based heuristic using path relinking is proposed to improve computational performance. Third, we propose a new technique for the second phase with a ranking approach. All of the methods have been tested on instances of varying size and range of objective function coefficients. We discuss the obtained results and explain our observations based on the distribution of objective function values.
arXiv (Cornell University), Jul 14, 2020
Data Envelopment Analysis (DEA) is a nonparametric, data driven technique used to perform relativ... more Data Envelopment Analysis (DEA) is a nonparametric, data driven technique used to perform relative performance analysis among a group of comparable decision making units (DMUs). Efficiency is assessed by comparing input and output data for each DMU via linear programming. Traditionally in DEA, the data are considered to be exact. However, in many real-world applications, it is likely that the values for the input and output data used in the analysis are imprecise. To account for this, we develop the uncertain DEA problem for the case of box uncertainty. We introduce the notion of DEA distance to determine the minimum amount of uncertainty required for a DMU to be deemed efficient. For small problems, the minimum amount of uncertainty can be found exactly, for larger problems this becomes computationally intensive. Therefore, we propose an iterative method, where the amount of uncertainty is gradually increased. This results in a robust DEA problem that can be solved efficiently. This study of uncertainty is motivated by the inherently uncertain nature of the radiotherapy treatment planning process in oncology. We apply the method to evaluate the quality of a set of prostate cancer radiotherapy treatment plans relative to each other.
Journal of Multi-criteria Decision Analysis, Nov 23, 2016
Building on previous work of the authors, this paper formally defines and reviews the first appro... more Building on previous work of the authors, this paper formally defines and reviews the first approach, referred to as navigation, towards a common understanding of search and decision making strategies to identify the mostpreferred solution among the Pareto set for a multiobjective optimization problem. In navigation methods, the decision maker interactively learns about the problem, while the decision support system learns about the preferences of the decision maker. This work introduces a detailed view on navigation leading to the identification of integral components and features. A number of different existing navigation methods are reviewed and characterized. Finally, an overview of applications involving navigation is given, and promising future research direction are discussed.
A Planning Tool for IMRT
Australasian Physical & Engineering Sciences in Medicine, Dec 1, 2002
Modelling stochastic route choice with bi-objective traffic assignment
In this paper, we propose a novel approach to model stochastic route choice in a tolled road netw... more In this paper, we propose a novel approach to model stochastic route choice in a tolled road network. First of all, we assume that all users have two objectives: (1) minimise travel time; and (2) minimise toll cost. Users are all rational in the sense that given a choice set, they will only choose one of the efficient paths. This will result in a bi-objective user equilibrium (BUE) condition such that traffic arranges itself in such a way that no individual trip maker can improve either his/her toll or travel time or both without worsening the other objective by unilaterally switching routes. We assume further that users have different preferences in the sense that for any given path with a specific toll, there is a limit on the time that an individual would be willing to spend. Each individual can have his/her own preference represented by this indifference function between toll and time. Time surplus is defined as the maximum time minus the actual time. Given a set of paths, the one with the highest (or least negative) time surplus will be the preferred path for the individual. As a result, for a specific origin-destination (O-D) pair, each individual can have a different preferred path, even though all individuals are considering the same choice set. In this way, based on the distribution of individual indifference curves, we can deduce the bi-objective equilibrium solution satisfying the time surplus maximisation bi-objective user equilibrium (TSMaxBUE) condition. That is, for each O-D pair, all individuals are travelling on the path with the highest time surplus value among all the efficient paths between this O-D pair. The philosophy behind the TSMaxBUE model is to overcome the restrictions that came with the two most commonly applied methods in tolling analysis, namely, user equilibrium (UE) and stochastic user equilibrium (SUE). UE assumes that all individuals behave the same way, i.e. to minimise generalised cost. The difference in individual preferences is modelled by creating user classes with different values of time (VOT). As a result, UE is restricted by the assumption that all users with the same VOT behave in exactly the same way, which can be dictated by a generalised cost function. SUE assumes that drivers choose their preferred routes based on perceived costs on different routes. Some formulations of this model come with the well known independence of irrelevant alternatives (IIA) property and assumptions on the probability density function of the error term of the utility function. Both the IIA property and assumptions on the error term have imposed limitations on the capability to replicate travel behaviour in reality. The model proposed here can overcome all these difficulties by introducing bi-objective traffic assignment and an indifference function between toll and time which can vary between individuals with no restrictions.
In this paper we will introduce the concept of lexicographic max-ordering solutions for multicrit... more In this paper we will introduce the concept of lexicographic max-ordering solutions for multicriteria combinatorial optimization problems. Section 1 provides the basic notions of multicriteria combinatorial optimization and the definition of lexicographic max-ordering solutions. In Section 2 we will show that lexicographic max-ordering solutions are pareto optimal as weil as max-ordering optimal solutions. Furthermore lexicographic max-ordering solutions can be used to characterize the set of pareto solutions. Further properties of lexicographic max-ordering solutions are given. Section 3 will be devoted to algorithms. We give a polynomial time algorithm for the two criteria case where one criterion is a sum and one is a bottleneck objective function, provided that the one criterion sum problem is solvable in polynomial time. For bottleneck functions an algorithm for the general case of Q criteria is presented.
Inverse planning has the ability to dramatically improve treatment outcomes for cancer patients u... more Inverse planning has the ability to dramatically improve treatment outcomes for cancer patients undergoing radiation treatment. One of the major challenges in inverse planning is the development of effective tools to support the planning process. Not only must a large amount of patient data be collected and analysed, but a significant number of output variables can be changed and has to be determined to optimise treatment. These variables include irradiation directions, intensity profiles and beam parameters. We present a prototype decision tool for planners which optimises irradiation direction. This decision tool also incorporates intensity profile optimisation, to create an integrated approach to planning. Optimisation of irradiation directions is a computationally difficult problem. A number of strategies are explored which will produce good treatment plans within a realistic time frame. These strategies incorporate heuristic optimisation techniques, aiming to minimise both tumour underdosing and overdosing of healthy organs simultaneously. Initial results indicate that significant improvements can be made in primary tumour treatment over current approaches.
The Tree and Christofides heuristic are weil known 1-and ta.pproximate a.lgorithms for the 6.-TSP... more The Tree and Christofides heuristic are weil known 1-and ta.pproximate a.lgorithms for the 6.-TSP. In thi.s note their performance for the multicriteria. ca.se ia described, depending on the norm in JRQ in case of Q criteria. Let G be a complete graph on n nodes and w : E( G) -R~ be a Q-criteria weight function. We assume that the triangle inequality is fulfilled, i.e. w(ik) $ w(ij) + w(jk) for all nodes i,j,k of G. Furthermore we assume that 11•11: IRQ -IR is ' a monotonous norm on IRQ. Hence !lall$ llbll whenever a $ b for a, b E IRQ, where the order on IRQ is the commonly used componentwise order. We first state the extensions of the two algorithms for the case of Q criteria and will then investigate their peformance. In the following text we will always abbreviate feasible Travelling Salesman tours by TS-tours. The weight of a TS-tour T is w(T) = (w1(T), ... wQ(T)) where w;(T) = LeeE(T) w;(e).
Cologne Twente Workshop on Graphs and Combinatorial Optimization, 2013
It is well known that the greedy algorithm solves matroid base problems for all linear cost funct... more It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or k-sum objective functions. In this paper, we address matroid base problems with a more general -"universal" -objective function which contains the previous ones as special cases. This universal objective function is of the sum type and associates multiplicative weights with the ordered cost coefficients of the elements of matroid bases such that, by choosing appropriate weights, many different -classical and new -objectives can be modeled. We show that the greedy algorithm is applicable to a larger class of objective functions than commonly known and, as such, it solves universal matroid base problems with non-negative or non-positive weight coefficients. Based on problems with mixed weights and a single (-, +)-sign change in the universal weight vector, we give a characterization of uniform matroids. In case of multiple sign changes, we use partition matroids. For non-uniform matroids, single sign change problems can be reduced to problems in minors obtained by deletion and contraction. Finally, we discuss how special instances of universal bipartite matching and shortest
Lecture Notes in Computer Science, 2009
On finding representative non-dominated points for bi-objective integer network flow problems
Computers & Operations Research, Aug 1, 2014
ABSTRACT This paper proposes a new algorithm to find a representation of the set of all non-domin... more ABSTRACT This paper proposes a new algorithm to find a representation of the set of all non-dominated points of the bi-objective integer network flow problem. The algorithm solves a sequence of ε-constraint problems with a branch-and-bound algorithm to find a subset of non-dominated points that represents the set of all non-dominated points well in the sense of coverage or uniformity. At each iteration of the algorithm, one non-dominated point, determined by solving one ε-constraint problem, is added to the representation until it is guaranteed that the representation has the desired quality. Computational experiments on different problem types show the efficacy of the algorithm.
Journal of Rail Transport Planning & Management
Using the output of optimisation models to make real-time changes to railway timetables can be an... more Using the output of optimisation models to make real-time changes to railway timetables can be an effective way to reduce the propagation of delay. In this study, we develop a methodology for evaluating the fairness of such optimisation models with respect to competing train operators. Whilst both fairness and optimisation-based railway timetable rescheduling have both been widely studied, they have not previously been studied together. We propose definitions of fairness and efficiency for timetable rescheduling, and analyse the fairness of efficiency-maximising solutions for a case study with seven train operators. We also investigate the pairwise trade-offs between operators and show that the priority given to different train classes has an important impact on fairness.
An infeasible interior-point technique to generate the nondominated set for multiobjective optimization problems
Computers & Operations Research
In this paper, an infeasible interior-point technique is proposed to generate the nondominated se... more In this paper, an infeasible interior-point technique is proposed to generate the nondominated set of nonlinear multi-objective optimization problems with the help of the direction-based cone method. We derive the proposed method for both convex and nonconvex problems. In order to solve the parametric optimization problems of the cone method, the infeasible interior-point method starts with an initial iterate outside the feasible region, and then gradually reduces the primal and dual infeasibility measures and the objective function value across the iterations with the help of a merit function. Estimates of the reduction of primal and dual infeasibility parameters per iteration are given. The convergence analysis of the method and an estimate of the number of iterations to reach an ϵ-precise solution are also provided. We provide the performance of the proposed methods on a variety of convex and nonconvex multi-objective test problems. Performance comparison between the proposed method and popular existing solvers is provided with respect to two performance measures and the corresponding relative efficiency measures. The reduction of a combined infeasibility measure, as the iterations progress, on the test problems is also shown graphically.
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Papers by Matthias Ehrgott