Mathematics
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Anokić, Ana (Beograd , 2017)[more][less]
Abstract: Optimization problems arise from many real-life situations. The development of adequate mathematical models of optimization problems and appropriate solution methods are of great importance for performance of real systems. The subject of this doctoral dissertation is a novel vehicle scheduling problem (VSP) that arises from optimizing the transport of agricultural raw materials. The organization of transport of raw materials is of great importance in the initial phase of production. This is particularly evident in the case of agricultural raw materials, because their price in the market is very low, and therefore, the costs of their transport represent the largest part of the total production cost. For this reason, any reduction of time and money spent in this early production stage directly increases the company’s profitability. The considered variant of VSP arises from optimizing the transport of sugar beet in a factory for sugar production in Serbia, but it can also be applied in a wider context, i.e., to optimize the transport of raw materials or goods in large companies under the same or similar conditions. The considered problem involves a number of specific constraints that distinguish it from existing variants of the vehicle scheduling problem. Therefore, mathematical models proposed in the literature for other variants of VSP do not describe adequately the considered problem. The complexity of the newly introduced VSP is analyzed. It is proven that the introduced VSP belongs to the class of NP-hard problems by comparing its relaxation with the Parallel Machine Scheduling Problem (PMSP). PMSP is known to be NP-hard, as it is equivalent to the Partitioning problem. From the established analogy between the relaxation of the considered VSP and PMSP, it is concluded that the VSP introduced in this dissertation is NP-hard. New mathematical models of the considered problem that involve all problem specific properties, are developed. The proposed mathematical models are compared in sense of efficiency by using Lingo 17 and CPLEX MIP 12.6.2 solvers. Experimental results showed that both exact solvers provided optimal or feasible solutions only for small-size real-life problem instances. However, this was expectable, having in mind the NP-hardness of the considered problem. Therefore, heuristic and metaheuristic method seem to be appropriate approaches for solving problem instances of larger dimension. Due to specific properties of the considered problem, the existing implementations of heuristic and metaheuristic methods for vehicle routing and scheduling problems can not be directly applied. For this reason, different variants of well-known Variable Neighborhood Search (VNS) metaheuristic, as well as Greedy Randomized Adaptive Search Procedure (GRASP), are designed. The constructive elements of the proposed VNS and GRASP implementations are adapted to the characteristics of the considered vehicle scheduling problem. A subproblem of the proposed variant of vehicle scheduling problem, denoted as VSP-P is considered first. VSP-P is obtained from the initial VSP by excluding problem specific constraints regarding vehicle arriving times to each location and to the factory area. Two metaheuristic algorithms are designed as solution methods for this subproblem: Basic Variable Neighborhood Search - BVNS, and Greedy Randomized Adaptive Search Procedure - GRASP. Both proposed approaches were tested on instances based on real-life data and on the set of generated instances of lager dimensions. Experimental results show that BVNS and GRASP reached all optimal solutions obtained by exact solvers on small-size real-life problem instances. On medium-size real-life instances, BVNS reached or improved upper bounds obtained by CPLEX solver under time limit of 5 hours. BVNS showed to be superior compared to GRASP in the sense of solution quality on medium size real-life instances, as well as on generated large-size problem instances. However, general conclusion is that both proposed methods represent adequate solution approaches for the subproblem VSP-P. BVNS provides solutions of better quality compared to GRASP, while GRASP outperforms BVNS regarding the average CPU time required to produce its best solutions. For the initial vehicle scheduling problem (VSP) that includes all problem specific constraints, three VNS-based metaheuristic methods are designed and implemented: Basic Variable Neighborhood Search - BVNS, Skewed Variable Neighborhood Search - SVNS, and Improved Basic Variable Neighborhood Search - BVNSi. BVNS and SVNS use the same neighborhood structures, but different search strategies in local search phase: BVNS uses Best improvement strategy, while SVNS uses First improvement strategy. All three VNS-based methods are tested on real-life and generated problem instances. As it was expected, experimental results showed that BVNS outperformed SVNS regarding solution quality, while SVNS running time was significantly shorter compared to BVNS. The third designed algorithm BVNSi represents a variant of BVNS that uses more general neighborhood structures compared to the ones used in BVNS and SVNS. The use of such neighborhood structures lead to the simplicity of BVNSi and shorter running times compared to BVNS. Two variants of BVNSi method that exploit different strategies in Local search phase are designed: BVNSiB with best improvement strategy and BVNSiF with First improvement strategy. The results of computational experiments for all proposed VNS-based methods for VSP are analyzed and compared. Regarding the quality of the obtained solutions, BVNS method shows the best performance, while SVNS needed the shortest average running times to produce its best solutions. Two variants of BVNSi method succeeded to find new best solutions on two medium size real life instances and to solve large size instances in shorter running time compared to BVNS and SVNS, respectively. However, both BVNSiB and BVNSiF turn out to be less stabile than BVNS and SVNS on real-life and generated inatances. In the case of one large-size generated instance, both BVNSi variants had significantly worse performance compared to BVNS and SVNS, which had negative impact on their average objective values and average running times. The proposed vehicle scheduling problem is of great practical importance for optimizing the transport of agricultural raw materials. It is planned to use the obtained results in practice (partially or completely), as a support to decision makers who organize transportation of in the early production phase. From the theoretical point of view, the developed mathematical models represent a scientific contribution to the fields of optimization and mathematical modeling. The variants of VNS methods that are developed and adapted to the problem, as well as comparison of their performances, represent a scientific contribution to the field of metaheuristic methods for solving NP-hard optimization problems. URI: http://hdl.handle.net/123456789/4664 Files in this item: 1
Anokic_Ana_disertacija.pdf ( 2.688Mb ) -
Milošević, Stefan (Beograd , 2017)[more][less]
Abstract: In this paper we present some norm inequalities for certain elementary operators and inner product type transformers, specially for Schatten norms, if the families of operators generating those transforms consists of arbitrary operators, and Q norms if at least one of those families consists of mutually commuting normal operators. Among others, we present inequalities that are generalizing the inequality p IA AX p IB B 6 X AXB ; from [11, Th. 2.3], for normal contractions and arbitrary unitarily invariant norm, to the case of Schatten norms and arbitrary contractions, as well as Q norms if at least one of the contractions A or B is normal. Also, by applying norm inequalities for operator monotone and operator convex functions, some refined Cauchy - Schwarz operator inequalities, as well as Minkowski and Landau - Gruss norm inequalities for operators are obtained as well. URI: http://hdl.handle.net/123456789/4656 Files in this item: 1
Stefan_Milosevic_Disertacija.pdf ( 2.544Mb ) -
Mikić, Marija (Beograd , 2017)[more][less]
Abstract: The subject of this dissertation is the investigation of asymptotic properties of solutions for di erential equations of Emden-Fowler type and their generalizations. The eld to which this dissertation belongs is a Qualitative theory of ordinary di e- rential equations. Emden-Fowler di erential equation has the form (t u0(t))0 t u (t) = 0 ; where ; ; 2 R. With some changes of the variables, this di erential equation can be reduced to the equations y00 xay = 0 and y00 + xay = 0. Firstly, in this dissertation, the di erential equation y00 = xay ; where a; 2 R was observed. The conditions, which provide that this equation has in nitely many solutions de ned in some neighborhood of zero, were described here, both with the conditions, which guarantee the existence of in nitely many solutions with certain asymptotic behavior. Also, a complete picture of asymptotic behavior of solutions of equation along the positive parts of both axes is given. The conditions, which assure existence and unique solvability of solution of the Cauchy problem for this equation, were shown in the cases when the familiar theory can't be applied. In some cases, asymptotic formulas for solutions were obtained. The di erential equation y00 = xay ; where a 2 R i < 0 ; has also been taken into consideration. The conditions, which assure the existence of in nitely many solutions of observed equation tending to zero as x ! 0+, were obtained. The conditions, which assure the unique solvability of the Cauchy problem for generalized Emden-Fowler equation y00 = q(x)f(y(x)); lim x!0+ y(x) = 0; lim x!0+ y0(x) = ; were described, for any > 0 and functions f and q which satisfy certain conditions. The given results generalize the results both for sublinear Emden-Fowler di erential equation (i.e. case when 0 < < 1) and the case when < 0. In literature, it is very rare to nd the conditions for di erent values of the para- metar which appears in the equations of Emden-Fowler type. In this dissertation, the results for sublinear and superlinear di erential equation Emden-Fowler, as well as the case when < 0, are presented. Therefore, the story of the asymptotic be- havior of solutions of the observed equation is "almost complited". URI: http://hdl.handle.net/123456789/4655 Files in this item: 1
Marija_Mikic_disertacija_MTF.pdf ( 1.461Mb ) -
Nikolić, Jovana (Beograd , 2017)[more][less]
Abstract: In this doctoral dissertation we de ne and investigate spectral invariants in Floer homology for conormal bundle and Floer homology of an open sub- set. As a key step to well de ned spectral invariants we give a construction of Piunikhin-Salamon-Schwarz isomorphism in both of these homologies. Ad- ditional algebraic structures, such as a product on Floer homology, give us various inequalities between spectral invariants. We can compare spectral in- variants from di erent Floer homologies by observing appropriate perturbed holomorphic Riemmanian surfaces with boundary. URI: http://hdl.handle.net/123456789/4506 Files in this item: 1
J_Nikolic_doktteza.pdf ( 2.076Mb ) -
Muzika Dizdarević, Manuela (Beograd , 2017)[more][less]
Abstract: Subject of this doctoral thesis is the application of algebraic techniques on one of the central topics of combinatorics and discrete geometry - polyomino tiling. Polyomino tilings are interesting not only to mathematicians, but also to physicists and biologists, and they can also be applied in computer science. In this thesis we put some emphasis on possibility to solve special class of tiling problems, that are invariant under the action of nite group, by using theory of Gr obner basis for polynomial rings with integer coe cients. Method used here is re ecting deep connection between algebra, geometry and combinatorics. Original scienti c contribution of this doctoral thesis is, at the rst place, in developing a techniques which enable us to consider not only ordinary Z?tiling problems in a lattice but the problems of tilings which are invariant under some subgroups of the symmetry group of the given lattice. Besides, it provides additional generalizations, originally provided by famous mathematicians J. Conway and J. Lagarias, about tiling of the triangular region in hexagonal lattice. Here is a summary of the content of the theses. In the rst chapter we give an exposition of the Gr obner basis theory. Especially, we emphasize Gr obner basis for polynomial rings with integer coe cients. This is because, in this thesis, we use algorithms for determining Gr obner basis for polynomials with integer coe cients. Second chapter provides basic facts about regular lattices in the plane. Also, this chapter provides some fundamental terms of polyomino tiling in the square and hexagonal lattice. Third chapter of this thesis is about studying Ztilings in the square lattice, which are invariant under the subgroup G of the group of all isometric transformations of the lattice which is generated by the central symmetry. One of the steps to resolve this problem was to determine a ring of invariants PG and its generators and relations among them. We use Gr obner basis theory to achieve this. Forth chapter covers the analysis of Ztilings in the hexagonal lattice which are symmetric with respect to the rotation of the plane for the angle of 120 . Main result of the fourth chapter is the theorem which gives conditions for symmetric tiling of the triangular region in plane TN, where N is the number of hexagons on each side of triangle. This theorem is one of the possible generalizations of the well known result, provided by Conway and Langarias. Fifth chapter provides another generalization of Conway and Lagarias result, but this time it is about determining conditions of tiling of triangular region TN in the hexagonal lattice not only with tribones, but with nbones. nbone is basic shape of of n connected cells in the hexagonal lattice, where n is arbitrary integer. URI: http://hdl.handle.net/123456789/4503 Files in this item: 1
muzikadizarevic.manuela.pdf ( 33.23Mb )