# Generalized Reachable Sets Method in Multiple Criteria Problems

Conference paper

## Abstract

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.

### Keywords

Income Hull Convolution## Preview

Unable to display preview. Download preview PDF.

### References

- Bushenkov, V.A. (1985). Iteration method for construction of orthogonal projections of convex polyhedral sets.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 25 (9).Google Scholar - Bushenkov, V.A., Ereshko, F., Kindler, J., Lotov, A. and de Mare, L. (1982). Application of the GRS Method to Water Resources Problems in Southwestern Skane, Sweden. WP-82–120. International Institute for Applied Systems Analysis, Laxenburg, Austria.Google Scholar
- Bushenkov, V.A., Kamenev, G.K., Lotov, A.V., and Chernykh, O.L. (1986). Applications of the geometric method for the investigation of economic-ecological systems. In A.A. Samarsky, N.N. Moiseev and A.A. Petrov, eds., Mathematical Modelling. Processes in Complicated Economic and Ecological Systems. Nauka Publishing House, Moscow (in Russian).Google Scholar
- Bushenkov, V.A. and Lotov, A.V. (1980a). An algorithm for analysis of independents of inequalities in linear system.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 20 (3).Google Scholar - Bushenkov, V.A. and Lotov, A.V. (1980b). Methods and algorithms for analysis of linear systems based on constructing of GRS.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 20 (5).Google Scholar - Bushenkov, V.A. and Lotov, A.V. (1982). Methods of Construction and Application of GRS. Reports on Applied Mathematics. Computing Center of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Bushenkov, V.A. and Lotov, A.V. (1983). Analysis of potential possibilities of a region in multiregional multiindustrial model of world economy. In A.G. Granberg and S.M. Menshikov, eds., Multiregional Multiindustrial Models of World Economy. Nauka Publishing House, Novosibirsk (in Russian).Google Scholar
- Bushenkov, V.A. and Lotov, A.V. (1984). POTENTIAL applied programs system. In Applied Programs Systems. Algorithms and Algorithmic Languages. Nauka Publishing House, Moscow (in Russian).Google Scholar
- Chernikov, S.N. (1965). Convolution of finite systems of linear inequalities.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 5 (1).Google Scholar - Chernykh, O.L. (1984). Analysis of Potential Possibilities of the Development of Economic Systems Taking Pollution Into Account. Reports on Applied Mathematics. Computing Center of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Cohon, J. (1978). Multiobjective Programming and Planning. Academic Press, New York.Google Scholar
- Egorova, N.E., Kamenev, G.K. and Lotov, A.V. (1985). Application of the GRS method in simulation system for industrial planning. In K.A. Bagrinovsky and T.I. Konnik, eds., Methods of Simulation of Economic Systems. Central Economic—Mathematical Institute of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Fourier, J.B. (1826). Solution d’une question particulière du calcul des inégalités. In Nouveau Bulletin des Sciences par la Societé philomatique de Paris.Google Scholar
- Kamenev, G.K. (1986). Investigation of Iteration Methods for Approximation of Convex Sets by Polyhedral Sets. Reports on Applied Mathematics. Computing Center of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Kamenev, G.K. and Lotov, A.V. (1985). Interactive structured MCDM procedure based on GRS method. In Multicriteria Mathematical Programming Problems. Proc. of Seminar. Institute for Systems Studies, Moscow.Google Scholar
- Kamenev, G.K., Lotov, A.V. and van Walsum, P. (1986). Application of the GRS Method to Water Resources Problems in the Southern Peel Region of the Netherlands. CP-86–19. International Institute for Applied Systems Analysis, Laxen-burg, Austria.Google Scholar
- Lotov, A.V. (1972). A numerical method of constructing reachable sets for linear systems.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 12 (3).Google Scholar - Lotov, A.V. (1973a). An approach to perspective planning in the case of absence of unique objective. Proc. of Conf. on Systems Approach and Perspective Planning (Moscow, May 1972 ). Computing Center of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Lotov, A.V. (1973b). A numerical method of studying the optimal-time continuity in linear systems and the solution of the Cauchy problem for Bellmans equation.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 13 (5).Google Scholar - Lotov, A.V. (1975a). Numerical method for reachable sets construction in the case of linear controlled systems with phase constraints.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 15 (1).Google Scholar - Lotov, A.V. (1975b). Economic Systems Analysis by means of Reachable sets. Proc. Conf. on Economic Processes Modelling. (Erevan, 1974 ). Computing Center of the USSR Academy of Sciences, Moscow (in Russian).Google Scholar
- Lotov, A.V. (1979). On convergence of methods for numerical approximation of reachable sets for linear differential systems with convex phase constraints.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 19 (1).Google Scholar - Lotov, A.V. (1980). On the concept of GRS and its construction for linear controlled systems.
*Soviet Physics Doklady*, 25 (2).Google Scholar - Lotov, A.V. (1981a). Reachable Sets Approach to Multiobjective Problems and its Possible Application to Water Resources Management in the Skane Region. WP-81–145. International Institute for Applied Systems Analysis, Laxenburg, Austria.Google Scholar
- Lotov, A.V. (1981b). On the concept and construction of GRS for linear controlled systems described by partial differential equations.
*Soviet Physics Doklady*, 26 (11).Google Scholar - Lotov, A.V. (1982). Aggregation as approximation of the GRS.
*Soviet Physics Doklady*, 27 (8).Google Scholar - Lotov, A.V. (1983). Coordination of economic models with the help of GRS. In E.L. Berlyand and S.B. Barabash, eds., Mathematical Methods of Analysis of Interaction between Industrial and Regional Systems. Nauka Publishing House, Novosibirsk (in Russian).Google Scholar
- Lotov, A.V. (1984a). Introduction into Mathematical Modelling of Economic Systems. Nauka Publishing House, Moscow (in Russian).Google Scholar
- Lotov, A.V. (1984b). On evaluation of stability and on the condition number for the solutions set of the linear inequalities systems.
*USSR Computational Mathematics and Mathematical Physics*, Pergamon Press, 24 (12).Google Scholar - Lotov, A.V. and Ognivtsev, S.B. (1984). Application of coordination methods based on GRS in goal-program approach to economic planning.
*Technical Cybernetics*, 2 (in Russian).Google Scholar - Moiseev, N.N. (1981). Mathematical Problems of Systems Analysis. Nauka Publishing House, Moscow (in Russian).Google Scholar
- Moiseev, N.N. et al. (1983). Global Models. The Biospheric Approach. CP-83–33. International Institute for Applied Systems Analysis, Laxenburg, Austria.Google Scholar
- Motzkin, T.S. et al. (1953). The double description method. In Contributions to the Theory of Games, vol. 2. Prinston University Press, Prinston.Google Scholar
- Steuer, R. (1986). Multiple Criteria Optimization. Wiley, New York.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1989