Generalized Reachable Sets Method in Multiple Criteria Problems
The Generalized Reachable Sets (GRS) method was developed as a method for investigation of open models, i.e. models with exogenous variables. Development of the GRS method started in late 60’s and the first results have been published in early 70’s (Lotov
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Introduction
The Generalized Reachable Sets (GRS) method was developed as a method for investigation of open models, i.e. models with exogenous variables. Development of the GRS method started in late 60's and the first results have been published in early 70's (Lotov, 1972). The basic idea of the method can be formulated as follows. The properties of the open model under study are investigated by means of aggregated variables. The set of all combinations of values of aggregated variables which are accessible (or reachable) using feasible combinations of values of original variables is constructed. This set should be described in explicit form. The GRS method was applied for investigation of dynamic models (Lotov, 1973b), for aggregation of economic models (Lotov, 1982), for coordination of economic decision (Lotov, 1983) and so on. Application of the GRS method to multiple criteria problems started in early 70's (Lotov, 1973a). This publication is devoted to application of this method to the multiple criteria decision making (MCDM) problems. The exogenous variables hereafter are treated as decisions and the aggregated variables are treated as decision criteria. The GRS method in MCDM problems employs an explicit representation of the accessible set, i.e. the set of all accessible values of criteria. Construction of the accessible set in explicit form provides the possibility to study the MCDM problems in terms of a small number of criteria instead of hundreds and thousands of decision variables. IT the number of criteria is greater than two it is hardly possible to imagine a practical form of explicit representation of nonconvex sets in criteria space. Therefore we study the MCDM problems in the case when accessible set is convex and can be approximated by polyhedral set. Note that in the multiple attribute decision making problems dealing with finite and relatively small number of alternatives usually apply decision matrix. The decision matrix describes the alternatives in terms of criteria values. The accessible set plays the same role for the DM problems with infinite number of alternatives. Construction of the accessible set permits to reformulate various existing MCDM dialogue procedures in simplified form and to suggest some new ones based mostly on the visualization of the accessible set. A. Lewandowski et al. (eds.), Methodology and Software for Interactive Decision Support © Springer-Verlag Berlin Heidelberg 1989
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Construction of accessible sets GRS method for linear models
We start with the mathematical formulation of the problem for the most simple case, i.e. for linear problems. Let the mathematical model of the system under study be x EG
= {x
(1)
E R" : Ax ~ b },
where R" is n-dimensional linear space of variables x (controls), G is the set of all feasible values of variables x, while A and b are the given matrix and vector respectively. Let y E Rm. be the vector of criteria. The criteria yare connected with variables x by mapping 1 : R" => Rm.. If the mapping is linear it can be
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