Matthias Ehrgott

An MCDM approach to portfolio optimization

Abstract We propose a model for portfolio optimization extending the Markowitz mean–variance model. Based on cooperation with Standard and Poor’s we use five specific objectives related to risk and return and allow consideration of individual preferences through the construction of decision-maker specific utility functions and an additive global utility function. Numerical results using customized local search, simulated annealing, tabu search and genetic algorithm heuristics show that problems of practically relevant size can be solved quickly.

Railway track allocation: models and methods

Efficiently coordinating the movement of trains on a railway network is a central part of the planning process for a railway company. This paper reviews models and methods that have been proposed in the literature to assist planners in finding train routes. Since the problem of routing trains on a railway network entails allocating the track capacity of the network (or part thereof) over time in a conflict-free manner, all studies that model railway track allocation in some capacity are considered relevant. We hence survey work on the train timetabling, train dispatching, train platforming, and train routing problems, group them by railway network type, and discuss track allocation from a strategic, tactical, and operational level.

Approximative solution methods for multiobjective combinatorial optimization

In this paper we present a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO). First, we discuss questions related to approximation in this context, such as performance ratios, bounds, and quality measures. We give some examples of heuristics proposed for the solution of MOCO problems. The main part of the paper covers metaheuristics and more precisely non-evolutionary methods. The pioneering methods and their derivatives are described in a unified way. We provide an algorithmic presentation of each of the methods together with examples of applications, extensions, and a bibliographic note. Finally, we outline trends in this area.

Trends in multiple criteria decision analysis

Dynamic MCDM, Habitual Domains and Competence Set Analysis for Effective Decision Making in Changeable Spaces.- The Need for and Possible Methods of Objective Ranking.- Preference Function Modelling: The Mathematical Foundations of Decision Theory.- Robustness in Multi-criteria Decision Aiding.- Preference Modelling, a Matter of Degree.- Fuzzy Sets and Fuzzy Logic-Based Methods in Multicriteria Decision Analysis.- Argumentation Theory and Decision Aiding.- Problem Structuring and Multiple Criteria Decision Analysis.- Robust Ordinal Regression.- Stochastic Multicriteria Acceptability Analysis (SMAA).- Multiple Criteria Approaches to Group Decision and Negotiation.- Recent Developments in Evolutionary Multi-Objective Optimization.- Multiple Criteria Decision Analysis and Geographic Information Systems.

A comparison of solution strategies for biobjective shortest path problems

We consider the biobjective shortest path (BSP) problem as the natural extension of the single-objective shortest path problem. BSP problems arise in various applications where networks usually consist of large numbers of nodes and arcs. Since obtaining the set of efficient solutions to a BSP problem is more difficult (i.e. NP-hard and intractable) than solving the corresponding single-objective problem there is a need for fast solution techniques. Our aim is to compare different strategies for solving the BSP problem. We consider a standard label correcting and label setting method, a purely enumerative near shortest path approach, and the two phase method, investigating different approaches to solving problems arising in phases 1 and 2. In particular, we investigate the two phase method with ranking in phase 2. In order to compare the different approaches, we investigate their performance on three different types of networks. We employ grid networks and random networks, as is generally done in the literature. Furthermore, road networks are utilized to compare performance on networks with a structure that is more likely to actually arise in applications.

Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.

Approximation algorithms for combinatorial multicriteria optimization problems

Abstract The computational complexity of combinatorial multiple objective programming problems is investigated. NP -completeness and #P-completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well-known tree and Christofides’ heuristics for the traveling salesman problem is investigated in the multicriteria case with respect to the two definitions of approximability.

A discussion of scalarization techniques for multiple objective integer programming

In this paper we consider solution methods for multiobjective integer programming (MOIP) problems based on scalarization. We define the MOIP, discuss some common scalarizations, and provide a general formulation that encompasses most scalarizations that have been applied in the MOIP context as special cases. We show that these methods suffer some drawbacks by either only being able to find supported efficient solutions or introducing constraints that can make the computational effort to solve the scalarization prohibitive. We show that Lagrangian duality applied to the general scalarization does not remedy the situation. We also introduce a new scalarization technique, the method of elastic constraints, which is shown to be able to find all efficient solutions and overcome the computational burden of the scalarizations that use constraints on objective values. Finally, we present some results from an application in airline crew scheduling as evidence.

Computation of ideal and Nadir values and implications for their use in MCDM methods

Abstract In this paper we investigate the problem of finding the Nadir point for multicriteria optimization problems (MOP). The Nadir point is characterized by the componentwise maximal values of efficient points for MOP. It can be easily computed in the bicriteria case. However, in general this problem is very difficult. We review some existing methods and heuristics and also discuss some new ones. We propose a general method to compute Nadir values based on theoretical results on Pareto optimal solutions of subproblems with fewer criteria. The general scheme is valid for any number of criteria and practical for the case of three objectives. We also investigate the use of the Nadir point for compromise programming, when the goal is to be as far away as possible from the worst outcomes. We prove some results about (weak) Pareto optimality of the resulting solutions and present examples, which show the limitations of this idea, and therefore limitations of multicriteria methods using this type of problems.

Mathematical optimization in intensity modulated radiation therapy

The design of an intensity modulated radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of an intensity map for each selected beam angle (intensity problem), and finding a sequence of configurations of a multileaf collimator to deliver the treatment (realization problem). Until the end of the last century research on radiotherapy treatment design has been published almost exclusively in the medical physics literature. However, since then, the attention of researchers in mathematical optimization has been drawn to the area and important progress has been made. In this paper we survey the use of optimization models, methods, and theories in intensity modulated radiotherapy treatment design.