Mathematics
Recent Submissions

Stančić, Olivera (Beograd , 2018)[more][less]
Abstract: Hub Location Problems (HLP) represent an important class of optimiza tion problems due to their numerous applications in many areas of real life. They often arise from practical situations that require routing of the flow from origin node (supplier) to the destination node (customer) under given conditions, such that the value of considered objective function is optimal. Hubs are special objects (nodes in the network) that represent centres for consolidation and flow collection between two selected locations  suppliers and customers. As transportation costs (per unit of flow) along the links that connect hub nodes are lower compared to other links in the network, directing the flow to hubs may lead to significant reductions of transportation cost in the network. The subject of this doctoral dissertation is one class of hub location problems, denoted as Hub Maximal Covering Problems (HMCPs) in the literature. The goal of HMCPs is to determine optimal locations for establishing certain number of hubs in order to maximize the total flow between all the covered origindestination pairs, under the assumption of binary or partial covering. Three variants of the hub maximal covering problem are considered: uncapacitated single allocation p hub maximal covering problem (USApHMCP), uncapacitated multiple allocation p hub maximal covering problem (UMApHMCP) and uncapacitated r allocation p hub maximal covering problem (UrApHMCP). Note that the UrApHMCP has not been studied in the literature so far. All three considered problems are proven to be NP hard. In case of USApHMCP, for the given set of hubs, the obtained subproblem of optimal allocation of nonhub nodes by established hubs is also NPhard. In this dissertation, new mathematical models for USApHMCP with binary and partial covering are proposed. The main advantage of the newly proposed models, in respect to existing ones from the literature, is the fact that small modifications of the new models enable their transformation to new models for p hub maximal covering problems with different allocation schemes. More precisely, new models for UMApHMCP and UrApHMCP can be obtained from the newly proposed mod els for USApHMCP in both coverage cases. All proposed models for USApHMCP and UMApHMCP are compared with the existing ones from the literature in the terms of efficiency within the framework of exact CPLEX 12.6 solver. Several hub data sets from the literature are used in numerical experiments when comparing the formulations. The obtained experimental results indicate that new models for UMApHMCP with both binary and partial coverage show the best performance in terms of solutions’ quality and execution times. For UrApHMCP and both coverage criteria, three mathematical models are proposed, and compared in terms of effi ciency using the exact CPLEX 12.6 solver. It turns out that the exact solver finds optimal or feasible solutions only for smallsize problem instances. Having in mind the complexity of all three problems under consideration and the results obtained by CPLEX 12.6 solver, the conclusion is that, in practice, exact methods can not provide solutions for large problem dimensions. For this reason, it was necessary to implement adequate heuristic or metaheuristic methods, in order to obtain highquality solutions in short execution times, even in the case of large problem dimensions. Up to now, only simple but insufficiently effective heuris tic methods for solving USApHMCP and UMApHMCP with binary coverage have been proposed in the literature, while the HMCP variants with partial coverage have not been previosly solved by using metaheuristic methods. As UrApHMCP with binary and partial coverage has not been previously considered in the litera ture, no solution methods suggested for this problem existed up to now. Inspired by previous successful applications of variable neighborhood search method (VNS) to other hub location problems from the literature, this metaheuristic approach is applied to the considered HMCP problems. In this dissertation, several variants of VNS metaheuristic are designed and implemented: General Variable Neighborhood Search (GVNS) for USApHMCP, Basic Variable Neighborhood Search (BVNS) for UMApHMCP and a variant of General Variable Neighborhood Search (GVNSR) for UrApHMCP. In the case of UrApHMCP, two additional metaheuristic meth ods are proposed: Greedy Randomized Adaptive Search Procedure with Variable Neighborhood Descent (GRASPVND) and Genetic Algorithm (GA). Constructive components of all proposed metaheuristics are adapted to the characteristics of the considered problems. Experimental study was conducted on the existing hub data sets from the lit erature, which include instances with up to 1000 nodes in the network. The ob tained results show that the proposed metaheuristics for the considered problems reach all known optimal solutions previously obtained by CPLEX 12.6 solver or establish new bestknown solutions in significantly shorter CPU time compared to CPLEX 12.6. The proposed GVNS and BVNS metaheuristics quickly reach all known optimal solutions on smallsize problem instances when solving USApHMCP and UMApHMCP, respectively. In the case of largesize problem instances, which have not been previously used for testing purposes for these problems, the proposed GVNS and BVNS return their best solutions in short execution times. The results obtained by the proposed GVNSR and GRASPVND for UrApHMCP on largesize problem instances indicate their effectiveness in both coverage cases. The proposed GA method showed to be successful only for UrApHMCP in binary covering, on instances up to 200 nodes. The variants of hub maximal covering problems considered in this dissertation are important from both theoretical and practical points of view. The new mathe matical models proposed in this dissertation for the considered variants of HMCP, represent a scientific contribution to the theory of hub location problems, mathemat ical modeling and optimization. Designed and implemented metaheuristic methods for solving the studied variants of HMCP are the scientific contribution to the field of optimization methods for solving location problems, as well as the development of software. The considered variants of HMCP have numerous applications in the optimization of telecommunication and transport systems, air passenger and goods transport, emergency services, postal and other delivery systems, so that the results obtained in this doctoral dissertation can be applied in practice, partially or com pletely. URI: http://hdl.handle.net/123456789/4750 Files in this item: 1
StancicOliveradisertacija.pdf ( 1.688Mb ) 
Melentijević, Petar (Beograd , 2018)[more][less]
Abstract: In this thesis we study sharp estimates of gradients and operator norm estimates in harmonic function theory. First, we obtain Schwarztype inequalities for holomorphic mappings from the unit ball B n to the unit ball B m , and then analoguous inequalities for holomorphic functions on the disk D without zeros and pluriharmonic functions from the unit ball B n to ( − 1 , 1) . These extend results from [ 32 ] and [ 18 ]. Also, we give a new proof of the fact that positive harmonic function in the upperhalf plane is a contraction with resprect to hyperbolic metrics on both H and R + ([ 47 ]). Besides that, in the second chapter, we construct the examples to show that the analoguous does not hold for the higherdimensional upperhalf spaces. All mentioned results are from the authors’ paper [55]. In the third chapter we intend to calculate the exact seminorm of the weighted Berezin transform considered as an operator from L ∞ ( B n ) to the ”smooth” Bloch space ([57]). The fourth chapter contains results concerning Bergman projection. We solve the problem posed by Kalaj and Marković in [ 28 ] on determining the exact seminorm of the Bergman projections from L ∞ ( B n ) to the B ( B n ) . The crucial obstacle is the fact that B ( B n ) is equipped with M− invariant gradient seminorm. Also, we provide the sharp gradient estimates of the Bergman projection of an L p function in the unit ball B n , as well as its consequences on Cauchy projection and certain gradient estimates for the functions from the Hardy and Bergman spaces.We obtain the exact values of the Bloch’s seminorms and norms for the Cauchy projection on L ∞ ( S n ) . These results are based on the papers [56] and [58]. The last chapter contains the proof of the one part of HollenbeckVerbitsky conjecture from [ 26 ]. Exactly, we find the exact norms of (  P +  s +  P −  s ) 1 s for 0 < s ≤ 2 on L p ( T ) , where P + is the Riesz projection and P − = I − P + . Also we give the appropriate dual estimates and prove that they are sharp. The paper [ 45 ] is motivated by the results from [25] and [33]. URI: http://hdl.handle.net/123456789/4749 Files in this item: 1
doktorat_Petar_merged.pdf ( 1.507Mb ) 
Lazović, Bojana (Beograd , 2018)[more][less]
URI: http://hdl.handle.net/123456789/4748 Files in this item: 1
B_Lazovic_Doktorska_disertacija.pdf ( 2.269Mb ) 
Kovač, Nataša (Beograd , 2018)[more][less]
Abstract: Dissertation title : Metaheuristic approach for solving one class of optimization problems in transp ort Abstract : Berth Allo cation Problem incorp orates some of the most imp ortant de cisions that have to b e made in order to achieve maximum e ciency in a p ort. Terminal manager of a p ort has to assign incoming vessels to the available b erths, where they will b e loaded/unloaded in such a way that some ob jective function is optimized. It is well known that even the simpler variants of Berth Allo cation Problem are NPhard, and thus, metaheuristic approaches are more convenient than exact metho ds, b ecause they provide high quality solutions in reasonable compu tational time. This study considers two variants of the Berth Allo cation Problem: Minimum Cost Hybrid Berth Allo cationProblem (MCHBAP) and Dynamic Mini mum Cost Hybrid Berth Allo cationProblem (DMCHBAP), b oth with xed handling times of vessels. Ob jective function to b e minimized consists of the following com p onents: costs of p ositioning, sp eeding up or waiting of vessels, and tardiness of completion for all vessels. Having in mind that the sp eed of nding highquality solutions is of crucial imp ortance for designing an e cient and reliable decision supp ort system in container terminal, metaheuristic metho ds represent the natural choice when dealing with MCHBAP and DMCHBAP. This study examines the fol lowing metaheuristic approaches for b oth typ es of a given problem: two variants of the Bee Colony Optimization (BCO), two variants of the Evolutionary Algorithm (EA), and four variants of Variable Neighb orho o d Search (VNS). All metaheuristics are evaluated and compared against each other and against exact metho ds inte grated in commercial CPLEX solver on reallife instances from the literature and randomly generated instances of higher dimensions. The analysis of the obtained results shows that on reallife instances all metaheuristics were able to nd optimal solutions in short execution times. Randomly generated instances were out of reach for exact solver due to time or memory limits, while metaheuristics easily provided highquality solutions in short CPU time in each run. The conducted computational analysis indicates that metaheuristics represent a promising approach for MCHBAP and similar problems in maritime transp ortation. The results presented in this pap er represent a contribution to the elds of combinatorial optimization, op erational research, metaheuristic metho ds, and b erth allo cation problem in the container terminals. URI: http://hdl.handle.net/123456789/4747 Files in this item: 1
N_Kovacdoktorska_disertacija.pdf ( 3.540Mb ) 
Makragić, Milica (Beograd , 2018)[more][less]
Abstract: This doctoral dissertation comprises two parts. Trigonometric polynomial rings are the central topic of the first part of the dissertation. It is presented that the ring of complex trigonometric polynomials, C [cos x, sin x ], is a unique factorization domain, and that the ring of real trigonometric polynomials, R [cos x, sin x ], is not a unique factorization domain. Necessary and sufficient conditions for the case when in the ring C [cos x, sin x ], unlike the ring R [cos x, sin x ], the degree of the product of two trigonometric polynomials is not equal to the sum of degrees of its factors, are given. The theory of trigonometric polynomials is extended to hyperbolictrigonometric polynomials, or HTpolynomials for short, which are defined similarly to trigonome tric polynomials. Real or complex HTpolynomials form a ring and even an integral domain R [cosh x, sinh x ], or C [cosh x, sinh x ]. Factorization in these domains is con sidered, and it is shown that these are unique factorization domains. The irreducible elements, as well as the form of the maximal ideals of both these domains are deter mined. The algorithms for dividing, factoring, computing greatest common divisors, as well as the algorithms for simplifying ratios of two HTpolynomials are considered over the field of rational numbers. In the second part of the dissertation, related to applications, two methods of proving inequalities of the form f ( x ) > 0 are described over the given finite in terval ( a,b ) ⊂ R , a ≤ 0 ≤ b , which by using the finite Maclaurin series expan sion generate polynomial approximations, when the function f ( x ) is element of the ring extension of R [cos x, sin x ], or R [cosh x, sinh x ], denoted by R [ x, cos x, sin x ], or R [ x, cosh x, sinh x ]. The completeness of the presented methods is proved and the concrete results of these methods are illustrated through examples of proving actual inequalities. URI: http://hdl.handle.net/123456789/4745 Files in this item: 1
Disertacija_Milica_Makragic.pdf ( 2.169Mb )