Study of hybrid methods for approximating the Edgeworth-Pareto hull in nonlinear multicriteria optimization problems

Methods for approximating the Edgeworth-Pareto hull (EPH) of the set of feasible criteria vectors in nonlinear multicriteria optimization problems are examined. The relative efficiency of two EPH approximation methods based on classical methods of searching for local extrema of convolutions of criteria is experimentally studied for a large-scale applied problem (with several hundred variables). A hybrid EPH approximation method combining classical and genetic approximation methods is considered.     

Convergence analysis of two-phase methods for approximating the Edgeworth-Pareto hull in nonlinear multicriteria optimization problems

The convergence of two-phase methods for approximating the Edgeworth-Pareto hull (EPH) in nonlinear multicriteria optimization problems is analyzed. The methods are based on the iterative supplement of the finite set of feasible criteria vectors (approximation basis) whose EPH approximates the desired set. A feature of two-phase methods is that the criteria images of randomly generated points of the decision space approach the Pareto frontier via local optimization of adaptively chosen convolutions of criteria. The convergence of two-phase methods is proved for both an abstract form of the algorithm and for a two-phase method based on the Germeier convolution.     

Insoluble multicriteria linear programming problems

This note concerns a method for analyzing insoluble multicriteria linear programming problems.     

An approximate solution of linear multicriteria control problems through the multicriteria simplex method

This paper deals with multicriteria, linear, continuous optimal control problems through the application of the multicriteria simplex method. Since the system is linear, the state variables at a given time can be expressed in terms of definite integrals. Upon using standard numerical integration formulas, the definite integrals are approximately expressed by means of weighted sums of the integrands. Then, after introducing some suitable auxiliary variables, approximate linear multicriteria programming problems are formulated. Illustrative numerical examples are provided to indicate the efficiency of the proposed method.The author is indebted to Professor Y. Sawaragi of Kyoto University for his constant encouragement. The author also wishes to thank T. Suwa for his cooperation in preparing the programming for the digital computer.     

Stability of compromise solution in multicriteria decision-making problems

Stability is a desirable attribute of the nondominated solution in the decision-making problem with multiple noncommensurable objectives. It is assumed that the decision space is given by a system of inequalitiesG(x)≤b, b ∈ R ^N . In this paper, sufficient conditions are determined for a compromise solution to be stable with respect to the changes ofb.     

Solving large-scale multicriteria problems by the decomposition method

The problem of constructing a set of effective vector estimates under a large number of partial performance criteria is examined.     

Comparative analysis of UTA multicriteria methods

This paper reviews the main variants of the utility additive (UTA) multicriteria method, and systematically compares their predictive performance on two sets of data. It analyses both those cases where the model provides a ranking with errors and those without errors. First, it shows that the reference projects should be chosen carefully in order to elicit as much information as possible from the decision maker: a set of projects satisfying a fractional factorial plan is recommended. Second, it discusses the use of alternative post-optimality methods for solving the problem of multiple solutions and their different predictive performances. Third, it presents the results of simulations based on utility functions involving interdependence between criteria, and shows that UTA handles this problem effectively by an adjustment of its coefficients. Finally, the influence of the model's parameters on the predictive performance is also investigated.     

Using mathematical programming heuristics in a multicriteria network flow context

In this paper, we propose a local search procedure to test the robustness of a specific ‘satisfying point’ neighbourhood. It consists of the following steps: (1) build an indifference area around the satisfying point in the criteria space by using thresholds (this takes into account the possible uncertainty, vagueness and/or inaccuracy of data); (2) find some points in the satisfying point neighbourhood and the corresponding solutions in the decision variables space; (3) test the quality of these solutions from the point of view of user preference. The indifference area is defined by adding constraints to the network model. This approach, which allows us to verify the adequacy of the model, has been applied to a set of multicriteria network flow problems. A heuristic method, based on Lagrangian duality and subgradient techniques, exploits the combinatorial structure of network flow problems in order to find certain feasible points.     

Weakly-efficient solutions of limiting multicriteria optimization problems

This paper concerns the connection among different sets of multicriteria optimization problem solutions. For the family of bicriteria optimization problems, the limiting properties of the sets of weakly-efficient solutions are determined.     

A special class of extended multicriteria location problems

This paper is devoted to the study of extended multicriteria location problems, which are obtained from a given planar single-facility multicriteria location problem with respect to the maximum norm by adding new cost functions. By means of an appropriate decomposition approach, we develop an implementable algorithm for generating an efficient solution of such extended problems.     

A note on the use of PROMETHEE multicriteria methods

The PROMETHEE methods are rather popular in the world of outranking methods. One of the reasons for this popularity is the existence of the very userfriendly software, called PROMCALC-PROMethee CALCulation. More and more practitioners are using PROMCALC to handle their multiple criteria problems. However, not all users are aware of the consequences of the model assumptions made in PROMETHEE methods. A short overview of some drawbacks of PROMETHEE methods that users must know and avoid are described in this paper.     

On finding compromise solutions in multicriteria problems using the fuzzy min-operator

This paper presents an application of fuzzy approaches to the linear vectormaximum problem. It shows that using the fuzzy min-operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem.     

Regularized extragradient method in multicriteria control problems with inaccurate data

The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.     

A note on global Pareto optimality in multicriteria optimization problems

We establish a range of sufficient conditions for (proper) Pareto optimality of all points in natural domains of multicriteria optimization problems.     

Stability and accuracy functions in multicriteria linear combinatorial optimization problems

We consider a vector linear combinatorial optimization problem in which initial coefficients of objective functions are subject to perturbations. For Pareto and lexicographic principles of efficiency we introduce appropriate measures of the quality of a given feasible solution. These measures correspond to so-called stability and accuracy functions defined earlier for scalar optimization problems. Then we study properties of such functions and calculate the maximum norms of perturbations for which an efficient solution preserves the efficiency. This work was partially supported through NATO Science Fellowship grant.