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Methods for semi-infinite programs
Peinlich, Jiří ; Grygarová, Libuše (advisor) ; Zimmermann, Karel (referee)
The aim of this work is to give an overview of methods for solving linear semi-infinite programming problems. The work also discusses various types of cutting plane method for sdemi-infinite programming problems. The work involves implementation of two types of this method in programming language Octave and the behavior of these methods is shown on examples.
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Duality in multiple criteria optimization problems
Kůrka, Michal ; Grygarová, Libuše (advisor) ; Zimmermann, Karel (referee)
This diploma thesis comprises of theoretical and practical part. In the theoretical part, we present comparison of di erent approaches to duality in multiobjective programming. We focus on dual problem formulated by Bitran and generalize the assumptions of strong duality theorem for this type of problem. We show that this dual problem is a special case of concept of exact duality developed by Dolecki and that it can be viewed as a generalization of Wolfe type dual problem presented by Nehse. In the practical part, we present algorithm for generating set of weak efficient solutions of concave multiobjective maximization problem with compact convex set of feasible solutions. This algorithm is based on construction of scalarized problem to the original multiobjective problem and makes use of the properties of its dual problem. We describe implementation of this algorithm and illustrate its usage on an example.
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Methods for solving selected vehicle routing problems and their implementation.
Drobný, Michal ; Grygarová, Libuše (advisor) ; Zimmermann, Karel (referee)
Various types of transportation issues are a common practice. The issue may be approached mainly as the distribution of products from suppliers to consumers while minimising distribution costs. The difference of real transportation issues predominantly relates to the considered restrictions, such as capacities of vehicles and orders, time windows and other special distribution restrictions. Transportation issues were already defined by F.L. Hitchcock in 1941 and since then, a wide range of stochastic and non- determinist methods providing solutions to transportation issues have been developed. Nevertheless, introducing distribution restrictions in resolving real-life problems makes it difficult for such methods to be applied. This thesis provides a compilation of the well-known determinist methods that may be used to resolve transportation issues. The methods that are appropriate for resolving real issues are discussed in more detail. The solution procedure of the selected method is demonstrated using simple examples and the results are compared with the results of other methods. An analysis of the above methods is used to design and implement new methods to resolve real transportation issues, their results being compared with the methods provided by the commercial software product.
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Optimization Problems under (max; min) - Linear Constraint and Some Related Topics
Gad, Mahmoud Attya Mohamed ; Zimmermann, Karel (advisor) ; Gavalec, Martin (referee) ; Grygarová, Libuše (referee)
Title: Optimization Problems under (max, min)-Linear Constraints and Some Related Topics. Author: Mahmoud Gad Department/Institue: Department of Probability and Mathematical Statis- tics Supervisor of the doctoral thesis: 1. Prof. RNDr. Karel Zimmermann,DrSc 2. Prof. Dr. Assem Tharwat, Cairo University, Egypt Abstract: Problems on algebraic structures, in which pairs of operations such as (max, +) or (max, min) replace addition and multiplication of the classical linear algebra have appeared in the literature approximately since the sixties of the last century. The first publications on these algebraic structures ap- peared by Shimbel [37] who applied these ideas to communication networks, Cunninghame-Green [12, 13], Vorobjov [40] and Gidffer [18] applied these alge- braic structures to problems of machine-time scheduling. A systematic theory of such algebraic structures was published probable for the first time in [14]. In recently appeared book [4] the readers can find latest results concerning theory and algorithms for (max, +)-linear systems of equations and inequalities. Since operation max replacing addition in no more a group, but a semigroup oppera- tion, it is a substantial difference between solving systems with variables on one side and systems with variables occuring on both sides of the equations....
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Methods for solving selected vehicle routing problems and their implementation.
Drobný, Michal ; Grygarová, Libuše (advisor) ; Zimmermann, Karel (referee)
Various types of transportation issues are a common practice. The issue may be approached mainly as the distribution of products from suppliers to consumers while minimising distribution costs. The difference of real transportation issues predominantly relates to the considered restrictions, such as capacities of vehicles and orders, time windows and other special distribution restrictions. Transportation issues were already defined by F.L. Hitchcock in 1941 and since then, a wide range of stochastic and non- determinist methods providing solutions to transportation issues have been developed. Nevertheless, introducing distribution restrictions in resolving real-life problems makes it difficult for such methods to be applied. This thesis provides a compilation of the well-known determinist methods that may be used to resolve transportation issues. The methods that are appropriate for resolving real issues are discussed in more detail. The solution procedure of the selected method is demonstrated using simple examples and the results are compared with the results of other methods. An analysis of the above methods is used to design and implement new methods to resolve real transportation issues, their results being compared with the methods provided by the commercial software product.
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Duality in multiple criteria optimization problems
Kůrka, Michal ; Zimmermann, Karel (referee) ; Grygarová, Libuše (advisor)
This diploma thesis comprises of theoretical and practical part. In the theoretical part, we present comparison of di erent approaches to duality in multiobjective programming. We focus on dual problem formulated by Bitran and generalize the assumptions of strong duality theorem for this type of problem. We show that this dual problem is a special case of concept of exact duality developed by Dolecki and that it can be viewed as a generalization of Wolfe type dual problem presented by Nehse. In the practical part, we present algorithm for generating set of weak efficient solutions of concave multiobjective maximization problem with compact convex set of feasible solutions. This algorithm is based on construction of scalarized problem to the original multiobjective problem and makes use of the properties of its dual problem. We describe implementation of this algorithm and illustrate its usage on an example.
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