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Tue, 29 Aug 2000 00:00:00 +0200Tue, 29 Aug 2000 00:00:00 +0200A Fuzzy Programming Approach to Multicriteria Facility Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1124
Facility Location Problems are concerned with the optimal location of one or several new facilities, with respect to a set of existing ones. The objectives involve the distance between new and existing facilities, usually a weighted sum or weighted maximum. Since the various stakeholders (decision makers) will have different opinions of the importance of the existing facilities, a multicriteria problem with several sets of weights, and thus several objectives, arises. In our approach, we assume the decision makers to make only fuzzy comparisons of the different existing facilities. A geometric mean method is used to obtain the fuzzy weights for each facility and each decision maker. The resulting multicriteria facility location problem is solved using fuzzy techniques again. We prove that the final compromise solution is weakly Pareto optimal and Pareto optimal, if it is unique, or under certain assumptions on the estimates of the Nadir point. A numerical example is considered to illustrate the methodology.Matthias Ehrgott; Rakesh Vermapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1124Tue, 29 Aug 2000 00:00:00 +0200A Characterization of Lexicographic Max-Ordering Solutions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/484
In this paper we give the definition of a solution concept in multicriteria combinatorial optimization. We show how Pareto, max-ordering and lexicographically optimal solutions can be incorporated in this framework. Furthermore we state some properties of lexicographic max-ordering solutions, which combine features of these three kinds of optimal solutions. Two of these properties, which are desirable from a decision maker" s point of view, are satisfied if and only of the solution concept is that of lexicographic max-ordering.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/484Mon, 03 Apr 2000 00:00:00 +0200On the number of Criteria Needed to Decide Pareto Optimality
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/492
In this paper we prove a reduction result for the number of criteria in convex multiobjective optimization. This result states that to decide wheter a point x in the decision space is pareto optimal it suffices to consider at most n? criteria at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of pareto, strict pareto and weak pareto solutionsMatthias Ehrgott; Stefan Nickelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/492Mon, 03 Apr 2000 00:00:00 +0200Geometric Methods to Solve Max-Ordering Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502
Location problems with Q (in general conflicting) criteria are considered. After reviewing previous results of the authors dealing with lexicographic and Pareto location the main focus of the paper is on max-ordering locations. In these location problems the worst of the single objectives is minimized. After discussing some general results (including reductions to single criterion problems and the relation to lexicographic and Pareto locations) three solution techniques are introduced and exemplified using one location problem class, each: The direct approach, the decision space approach and the objective space approach. In the resulting solution algorithms emphasis is on the representation of the underlying geometric idea without fully exploring the computational complexity issue. A further specialization of max-ordering locations is obtained by introducing lexicographic max-ordering locations, which can be found efficiently. The paper is concluded by some ideas about future research topics related to max-ordering location problems.Matthias Ehrgott; Horst W. Hamacher; Stefan Nickelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502Mon, 03 Apr 2000 00:00:00 +0200Saddle Points and Pareto Points in Multiple Objective Programming
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/504
In this paper relationships between Pareto points and saddle points in multiple objective programming are investigated. Convex and nonconvex problems are considered and the equivalence between Pareto points and saddle points is proved in both cases. The results are based on scalarizations of multiple objective programs and related linear and augmented Lagrangian functions. Partitions of the index sets of objectives and constranints are introduced to reduce the size of the problems. The relevance of the results in the context of decision making is also discussed.Matthias Ehrgott; Margaret M. Wiecekpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/504Mon, 03 Apr 2000 00:00:00 +0200Discrete Decision Problems, Multiple Criteria Optimization Classes and Lexicographic Max-Ordering
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/505
The topic of this paper are discrete decision problems with multiple criteria. We first define discrete multiple criteria decision problems and introduce a classification scheme for multiple criteria optimization problems. To do so we use multiople criteria optimization classes. The main result is a characterization of the class of lexicographic max-ordering problems by two very useful properties, reduction and regularity. Subsequently we discuss the assumptions under which the application of this specific MCO class is justified. Finally we provide (simple) solution methods to find optimal decisions in the case of discrete multiple criteria optimization problems.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/505Mon, 03 Apr 2000 00:00:00 +0200Approximation Algorithms for Combinatorial Multicriteria Optimization Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/512
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 TSP is investigated in the multicriteria case with respect to the two definitions of approximability.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/512Mon, 03 Apr 2000 00:00:00 +0200Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/514
The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.Alexander L. Topchishvili; Vilhelm G. Maisuradze; Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/514Mon, 03 Apr 2000 00:00:00 +0200Min-Max Formulation of the Balance Number in Multiobjetive Global Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/515
The notion of the balance number introduced in [3,page 139] through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria.Matthias Ehrgott; Efim A. Galperinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/515Mon, 03 Apr 2000 00:00:00 +0200Fixed Cardinality Combinatorial Optimization Problems - A Survey
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/971
Matthias Ehrgott; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/971Fri, 18 Feb 2000 00:00:00 +0100Multicriteria Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/838
Life is about decisions. Decisions, no matter if taken by a group or an individual, involve several conflicting objectives. The observation that real world problems have to be solved optimally according to criteria, which prohibit an "ideal" solution - optimal for each decisionmaker under each of the criteria considered - , has led to the development of multicriteria optimization. From its first roots, which where laid by Pareto at the end of the 19th century the discilpine has prospered and grown, especially during the last three decades. Today, many decision support systems incorporate methods to deal with conflicting objectives. The foundation for such systems is a mathematical theory of optimaztion under multiple objectives. With this manuscript, which is based on lectures I taught in the winter semester 1998/99 at the University of Kaiserslautern, I intend to give an introduction to and overview of this fascinating field of mathematics. I tried to present theoretical questions such as existence of solutions as well as methodological issues and hope the reader finds the balance not too heavily on one side. The interested reader should be able to find classical results as well as up to date research. The text is accompanied by exercises, which hopefully help to deepen students' understanding of the topic.Matthias Ehrgottlecturehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/838Thu, 29 Apr 1999 00:00:00 +0200