Abstract
Generalized algorithms for solving problems of discrete, integer, and Boolean programming are discussed. These algorithms are associated with the method of normalized functions and are based on a combination of formal and heuristic procedures. This allows one to obtain quasi-optimal solutions after a small number of steps, overcoming the NP-completeness of discrete optimization problems. Questions of constructing so-called “duplicate” algorithms are considered to improve the quality of discrete problem solutions. An approach to solving discrete problems with fuzzy coefficients in objective functions and constraints on the basis of modifying the generalized algorithms is considered. Questions of applying the generalized algorithms to solve multicriteria discrete problems are also discussed. The results of the paper are of a universal character and can be applied to the design, planning, operation, and control of systems and processes of different purposes. The results of the paper are already being used to solve power engineering problems.
Highlights
Discrete, integer, and Boolean optimization problems have important applications in many fields [1] [2]
Before starting to discuss questions of multiattribute decision making in a fuzzy environment, it is necessary to note that considerable contraction of the decision uncertainty regions may be obtained by formulating and solving one and the same problem within the framework of mutually interrelated models: 1) model of maximization (29) with the constraints (30) approximated by (3); 2) model of minimization (31) with the constraints (30) approximated by (5)
The generalized algorithms for solving problems of discrete, integer, and Boolean programming are discussed. These algorithms are based on a combination of formal procedures and informal procedures
Summary
Integer, and Boolean (in the general case, discrete) optimization problems have important applications in many fields [1] [2] Taking this into account, it should be stressed that direct determination of discrete solutions to problems of discrete character is necessary. The algorithms permit one to obtain quasioptimal solutions after a small number of steps, overcoming the problem NP-completeness They do not require an analytical specification of objective functions and constraints, ensuring the flexibility and the possibility to solve problems, for which adequate analytical descriptions are difficult or impossible. It should be stressed that taking into account the uncertainty factor in shaping mathematical models is to be inherent to the practice of systems analysis This serves as a means for increasing the adequacy of the built models and, as a consequence, the credibility and factual effectiveness of solutions based on their analysis. The present paper reflects results related to modifying the generalized algorithms to solve discrete problems with fuzzy coefficients in objective functions and constraints as well as multiobjective discrete problems
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