In this thesis we address three related topic in the field of Operations Research. Firstly we discuss the problems and limitation of most common solvers for linear programming, precision. We then present a solver that generate rational optimal solutions to linear programming problems by solving a succession of (increasingly more precise) floating point approximations of the original rational problem until the rational optimality conditions are achieved. This method is shown to be (on average) only 20% slower than the common pure floating point approach, while returning true optimal solutions to the problems. Secondly we present an extension of the Local Cut procedure introduced by Applegate et al, 2001, for the Symmetric Traveling Salesman Problem (STSP), to the general setting of MIP problems. This extension also proves finiteness of the separation, facet and tilting procedures in the general MIP setting, and also provides conditions under which the separation procedure is guaranteed to generate cuts that separate the current fractional solution from the convex hull of the mixed-integer polyhedron. We then move on to explore some configurations for local cuts, realizing extensive testing on the instances from MIPLIB. Those results show that this technique may be useful in general MIP problems, while the experience of Applegate et al, shows that the ideas can be successfully applied to structures problems as well. Thirdly we present an extensive computational experiment on the TSP and Domino Parity inequalities as introduced by Letchford, 2000. This work also include a safe-shrinking theorem for domino parity inequalities, heuristics to apply the planar separation algorithm introduced by Letchford to instances where the planarity requirement does not hold, and several practical speed-ups. Our computational experience showed that this class of inequalities effectively improve the lower bounds from the best relaxations obtained with Concorde, which is one of the state of the art solvers for the STSP. As part of these experience, we solved to optimality the (up to now) largest two STSP instances, both of them belong to the TSPLIB set of instances and they have 18,520 and 33,810 cities respectively.

In this thesis I focus on cutting planes for large Mixed Integer Programming (MIP) problems. More specifically, I focus on two independent cutting planes studies. The first of these deals with cutting planes for the Traveling Salesman Problem (TSP), and the second with cutting planes for general MIPs. In the first study I introduce a new class of cutting planes which I call the Generalized Domino Parity (GDP) inequalities. My main achievements with regard to these are: (1) I show that these are valid for the TSP and for the graphical TSP. (2) I show that they generalize most well-known TSP inequalities (including combs, domino-parity constraints, clique-trees, bipartitions, paths and stars). (3) I show that a sub-class of these (which contains all clique-tree inequalities w/ a fixed number of handles) can be separated in polynomial time, on planar graphs. My second study can be subdivided in two parts. In the first of these I study the Mixed Integer Knapsack Problem (MIKP) and develop a branch-and-bound based algorithm for solving it. The novelty of the approach is that it exploits the notion of "dominance" in order to effectively prune solutions in the branch-and-bound tree. In the second part, I develop a Mixed Integer Rounding (MIR) cut separation heuristic, and embed the MIKP solver in a column generation algorithm in order to assess the performance of said heuristic. The goal of this study is to understand why no other class of inequalities derived from single-row systems has been able to outperform the MIR. Computational results are presented.

This study deals with chocolate production line scheduling. The particular production line allows producing multiple items at the same time. Another distinguishing property affecting the planning methodology is that an item can have different production capacities when produced in different product combinations which are called production patterns in this study. Planning is done on a 12 weeks rolling horizon. There are 21 products and 103 production patterns covering all the production possibilities. The subject of the study is to construct an algorithm that gives 12 weeks&rsquo / production values of each product and to construct the shift based scheduling of the first week of the planning horizon. The first part is Master Production Scheduling (MPS) and the objective is minimizing the shortage and overage costs. A mathematical modeling approach is used to solve the MPS problem. The second part is the scheduling part which aims to arrange the production patterns obtained from the MPS module within the shifts for the first week of the planning horizon considering the setup times. The MPS module is a large integer programming model. The challenge is finding a reasonable lower bound whenever possible. If it is not possible, finding a reasonable upper bound and seeking solutions better than that is the main approach. The scheduling part, after solving MPS, becomes a TSP and the setup times are sequence independent. In this part, the challenge is solving TSP with an appropriate objective function.

In this study, a new selective and time-window routing problem is defined for the first time in the literature, which is called the campaign routing problem (CRP). The two special cases of the CRP correspond to the two real-life problems, namely political campaign routing problem (PCRP) and the experiments on wheels routing problem (EWRP). The PCRP is based on two main decision levels. In the first level, a set of campaign regions is selected according to a given criteria subject to the special time-window constraints. In the second level, a pair of selected regions or a single region is assigned to a campaign day. In the EWRP, a single selected region (school) is assigned to a campaign day. These two problems are modeled using classical mathematical programming and bi-level programming methods, and a two-step heuristic approach is developed for the solution of the problems. Implementation of the solution methods is done using the test instances that are compiled from the real-life data. Computational results show that the solution methods developed generate good solutions in reasonable time.

Ammunition Transfer System (ATS) is the electro-mechanical system of the Ammunition Resupply Vehicle (ARV) which will be used to meet T-155 mm Firtina howitzers&rsquo / ammunition demand for tactical requirements of higher firing rate by off-road mobility and survivability. The transfer of ammunitions from ARV to Firtina is to be optimized for an effective improvement of firing rate. In this thesis the transferring order of carried ammunitions is being optimized to minimize the total ammunition transferring time. This transfer problem is modeled as a modification of Travelling Salesman Problem (TSP). The given locations of the ammunitions are treated as cities to be visited and the gripper of ATS is treated as the traveling salesman. By GAMS / the small-size problems are solved optimally but large-size ones get only local optimum. A heuristic algorithm that contains nearest neighbor heuristics as construction method and 2-opt exchange heuristic as improvement method is developed to obtain same or better solutions obtained by GAMS with less computational time.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed January 5, 2010) Includes bibliographical references (p. 152-159).

The traveling salesperson or salesman problem (TSP) is a well known and important combinatorial optimization problem. The goal is to find the shortest tour that visits each city in a given list exactly once and then returns to the starting city. Despite this simple problem statement, solving the TSP is difficult since it belongs to the class of NP-complete problems. The importance of the TSP arises besides from its theoretical appeal from the variety of its applications. In addition to vehicle routing, many other applications, e.g., computer wiring, cutting wallpaper, job sequencing or several data visualization techniques, require the solution of a TSP. In this paper we introduce the R package TSP which provides a basic infrastructure for handling and solving the traveling salesperson problem. The package features S3 classes for specifying a TSP and its (possibly optimal) solution as well as several heuristics to find good solutions. In addition, it provides an interface to Concorde, one of the best exact TSP solvers currently available. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

In this thesis we focus on two NP-hard and intensively studied problems: The travelling salesman problem (TSP), which aims to find a minimum cost tour that visits every node exactly once in a complete weighted graph, and the 2-edge-connected spanning subgraph problem (2ECSS), which aims to find a minimum size 2-edge-connected spanning subgraph in a given graph. TSP and 2ECSS have many real world applications. However, both problems are NP-hard which means it is unlikely that polynomial time algorithms exist to solve them, so methods that return close to optimal solutions are sought. In this thesis we mainly focus on k-approximation algorithms for the two problems, which efficiently return a solution within k times of the optimal solution. For a special case of TSP called graph TSP, using ideas from Momke and Svensson, we present a 25/18-approximation algorithm for a special class of graphs using circulations and T-joins, which improves the previous known best bound of 7/5 for such graphs. Moreover, if the graph does not contain special nodes, our algorithm ensures the ratio of 4/3. For 2ECSS, given any k-edge-connected graph G=(V,E), |V|=n, |E|=m, we present an approximation algorithm that gives a 2-edge-connected spanning subgraph with the number of edges at most n+(m-n)/(k-1)-(k-2)/(k-1) with a novel use of circulations, which improves both the approximation ratio and the simplicity of the proof compared to a result by Huh in 2004.

Constructing the genomic median of several given genomes is crucial in developing evolutionary trees, since the genomic median provides an estimate for the ordering of the genes in a common ancestor of the given genomes. This is due to the fact that the content of DNA molecules is often similar, but the difference is mainly in the order in which the genes appear in various genomes. The mutations that affect this ordering are called genome rearrangements, and many structural differences between genomes can be studied using genome rearrangements. In this thesis our main focus is on applying combinatorial optimization techniques to genomic median problems, with particular emphasis on the breakpoint distance as a measure of the difference between two genomes. We will study different variations of the breakpoint median problem from signed to unsigned, unichromosomal to multichromosomal, and linear to circular to mixed. We show how these median problems can be formulated in terms of problems in combinatorial optimization, and take advantage of well-known combinatorial optimization techniques and apply these powerful methods to study various median problems. Some of these median problems are polynomial and many are NP-hard. We find efficient algorithms and approximation methods for median problems based on well-known combinatorial optimization structures. The focus is on algorithmic and combinatorial aspects of genomic medians, and how they can be utilized to obtain optimal median solutions.

Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Thomas, Robin; Committee Co-Chair: Cook, William J.; Committee Member: Dvorak, Zdenek; Committee Member: Parker, Robert G.; Committee Member: Yu, Xingxing.