Open Access
Proper Efficiency and the Linear Vector Maximum Problem
W. E. Walker,H. Isermann +1 more
- 01 Jan 2016
TL;DR: In this article, it was shown that each efficient solution of a linear vector maximum problem is also properly efficient, and the notion of proper efficiency has been proposed to exclude efficient solutions of a certain anomalous type.
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Abstract: V ECTOR MAXIMUM problems arise when K>1 scalar-valued objective functions are to be maximized simultaneously over a given range of definition. Though the concept of efficiency has proved to be of great significance in the discussion of decision problems with multiple criteria, a restricted definition of efficiency, the notion of proper efficiency, has been proposed (see references 3, 4, and 5) in order to exclude efficient solutions of a certain anomalous type. This note shows that each efficient solution of a linear vector maximum problem is also properly efficient.
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Citations
A survey and annotated bibliography of multiobjective combinatorial optimization
TL;DR: The main parts of the paper are a section on the review of the available solution methodology, both exact and heuristic, and a sections on the annotation of the existing literature in the field organized problem by problem.
784
On Cone-Efficiency, Cone-Convexity and Cone-Compactness
TL;DR: In this article, it is shown that the notion of cone-compactness (a generalization of compactness) is sufficient to guarantee the existence of an efficient point and the relationship between the set of efficient points and the optimal sets of certain linear functions is elucidated.
203
Existence of efficient solutions for vector maximization problems
TL;DR: In this article, the existence of efficient and properly efficient solutions for the vector maximization problem is examined, and conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points.
159
Multiobjective Combinatorial Optimization — Theory, Methodology, and Applications
Matthias Ehrgott,Xavier Gandibleux +1 more
- 01 Jan 2003
TL;DR: This chapter provides an annotated bibliography of multiple objective combinatorial optimization, M OCO, and presents a general formulation of MOCO problems, describe their main characteristics, and review the main properties and theoretical results.
146
An Annotated Bibliography of Multiobjective Combinatorial Optimization
Matthias Ehrgott,Xavier Gandibleux +1 more
- 01 Jan 2000
TL;DR: This paper provides an annotated bibliography of multiple objective combinatorial optimization, MOCO, and an annotation of the existing literature in the field organized problem by problem.
137
References
•Book
Management Models and Industrial Applications of Linear Programming
Abraham Charnes,William W. Cooper +1 more
- 01 Jan 1961
TL;DR: In place of a survey or evaluation of industrial studies, two broad issues which are relevant to all such applications will be discussed, including the use of linear programming models as guides to data collection and analysis and prognosis of fruitful areas of additional research, especially those which appear to have been opened by industrial applications.
1.9K
Letter to the Editor—Improper Solutions of the Vector Maximum Problem
TL;DR: The purpose of this note is to show that all ‘improper’ solutions have this undesirable property, thus justifying the calculation of only the ‘proper” solutions.
31
Proper efficiency and the theory of vector maximization
TL;DR: In this paper, the concept of proper efficiency was introduced to eliminate efficient points of a certain anomalous nature in the problem of vector maximization, which is related in spirit to the notion of "proper" efficiency introduced by Kuhn and Tucker in their celebrated paper of 1950.