This paper provides a survey of the research in and an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. The main parts of the paper are a section on the review of the available solution methodology, both exact and heuristic, and a section on the annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research.

A survey and annotated bibliography of multiobjective combinatorial optimization

The properties of efficient (admissible) points of subsets of $R^n $ are discussed in the case when the space is ordered by a convex cone. It is demonstrated that the notion of cone-compactness (a generalization of compactness) is sufficient to guarantee the existence of an efficient point. The relationship between the set of efficient points and the optimal sets of certain linear functions is elucidated in a series of theorems extending previous results, when the underlying set satisfies a weakened convexity assumption (cone-convexity). Geoffrion’s concept of proper efficiency is generalized and after showing that our definition is a valid generalization, the appropriate characterization theorem is established.

On Cone-Efficiency, Cone-Convexity and Cone-Compactness

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, 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. Examples are provided to illustrate these results.

Existence of efficient solutions for vector maximization problems

This chapter provides an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe their main characteristics, and review the main properties and theoretical results. One section is devoted to a brief description of the available solution methodology, both exact and heuristic. The main part of the chapter consists of an annotation of the existing literature in the field organized problem by problem. We conclude the chapter by stating open questions and areas of future research. The list of references comprises more than 400 entries.

Multiobjective Combinatorial Optimization — Theory, Methodology, and Applications

This paper provides an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. One section is devoted to a brief description of the available solution methodology, both exact and heuristic. The main part of the paper is devoted to an annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research. The list of references comprises more than 350 entries.

https://kluedo.ub.uni-kl.de/files/1130/gelb_62.pdf

An Annotated Bibliography of Multiobjective Combinatorial Optimization

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.

Proper Efficiency and the Linear Vector Maximum Problem

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Proper Efficiency and the Linear Vector Maximum Problem