Dynamic Programming is an optimization technique used in computer science and mathematics. Introduced in the 1950s, it has been applied to many classic combinatorial optimization problems, such as the Shortest Path Problem, the Knapsack Problem, and the Traveling Salesman Problem, with varying degrees of practical success. In this thesis, we present two applications of dynamic programming to optimization problems. The first application is as a method to compute the Branch-Cut-and-Price (BCP) family of lower bounds for the Traveling Salesman Problem (TSP), and several vehicle routing problems that generalize it. We then prove that the BCP family provides a set of lower bounds that is at least as strong as the Approximate Linear Program (ALP) family of lower bounds for the TSP. The second application is a novel dynamic programming model used to determine the placement of cuts for a particular form of skull surgery called Cranial Vault Remodeling.

Thesis (M.S.)--Cleveland State University, 2009. / Abstract. Title from PDF t.p. (viewed on Sept. 8, 2009). Includes bibliographical references (p. 70-73). Available online via the OhioLINK ETD Center and also available in print.

Indiana University-Purdue University Indianapolis (IUPUI) / Brain connectome network visualization could help the neurologists inspect the brain structure easily and quickly. In the thesis, the model of the brain connectome network is visualized in both three dimensions (3D) environment and two dimensions (2D) environment. One is named “Brain Explorer for Connectomic Analysis” (BECA) developed by the previous research already. It could present the 3D model of brain structure with region of interests (ROIs) in different colors [5]. The other is mainly for the information visualization of brain connectome in 2D. It adopts the force-directed layout to visualize the network. However, the brain network visualization could not bring the user intuitively ideas about brain structure. Sometimes, with the increasing scales of ROIs (nodes), the visualization would bring more visual clutter for readers [3]. So, brain connectome network properties visualization becomes a useful complement to brain network visualization. For a better understanding of the effect of Alzheimer’s disease on the brain nerves, the thesis introduces several methods about the brain graph properties visualization. There are the five selected graph properties discussed in the thesis. The degree and closeness are node properties. The shortest path, maximum flow, and clique are edge properties. Except for clique, the other properties are visualized in both 3D and 2D. The clique is visualized only in 2D. For the clique, a new hypergraph visualization method is proposed with three different algorithms. Instead of using an extra node to present a clique, the thesis uses a “belt” to connect all nodes within the same clique. The methods of node connections are based on the traveling salesman problem (TSP) and Law of cosines. In addition, the thesis also applies the result of the clique to adjust the force-directed layout of brain graph in 2D to dramatically eliminate the visual clutter. Therefore, with the support of the graph properties visualization, the brain connectome network visualization tools become more flexible.

We study the problem of viewpoint planning in static and dynamic scenes using multi-robot teams. This work is motivated by two applications: bridge inspection and environmental monitoring using Unmanned Aerial Vehicles. For static scenes, we are given a set of target points in a polygonal environment that must be monitored using robots with cameras. The goal is to compute a tour for all the robots such that every target is visible from at least one tour. We solve this problem optimally by reducing it to Generalized Travelling Salesman Problem. For dynamic scenes, we study the multi-robot assignment problem for multi-target tracking. The problem can be viewed as the mixed packing and covering problem. We optimally solve the problem using Mixed Quadratic Integer Linear Program to maximize the total number of targets covered. In addition to theoretical contribution, we also present our hardware system design and findings from field experiments. / Master of Science / We study the problem of viewpoint planning in static and dynamic scenes using multi-robot teams. This work is motivated by two applications: bridge inspection and environmental monitoring using Unmanned Aerial Vehicles. For static scenes, we are given a set of target points in a static 2D or 3D environment such as a bridge. Target points are key locations that we are interested to monitor using cameras on the robots. The goal is to compute a tour for all the robots such that every target location is visible from at least one robot’s tour. We want to minimize the sum of lengths of all the robot’s tours combined. We find the best possible solution for this problem. For dynamic scenes, we study the multi-robot trajectory assignment problem for multi-target tracking. Here, the target points may be moving, e.g., expanding plumes in an oil spill. The goal in this is to maximize the total number of targets covered at each time step. We provide the best possible solution in this case. In addition to theoretical contribution, we also present our hardware system design and findings from field experiments.

This dissertation is devoted to the development of new flow-based formulations for the asymmetric traveling salesman problem (ATSP) and to the demonstration of their applicability in effectively solving some scheduling problems. The ATSP is commonly encountered in the areas of manufacturing planning and scheduling, and transportation logistics. The integration of decisions pertaining to production and shipping, in the supply chain context, has given rise to an additional and practical relevance to this problem especially in situations involving sequence-dependent setups and routing of vehicles. Our objective is to develop new ATSP formulations so that algorithms can be built by taking advantage of their relaxations (of integer variables, thereby, resulting in linear programs) to effectively solve large-size problems. In view of our objective, it is essential to have a formulation that is amenable to the development of an effective solution procedure for the underlying problem. One characteristic of a formulation that is helpful in this regard is its tightness. The tightness of a formulation usually refers to the quality of its approximation to the convex hull of integer feasible solutions. Another characteristic is its compactness. The compactness of a formulation is measured by the number of variables and constraints that are used to formulate a given problem. Our formulations for the ATSP and the scheduling problems that we address are both tight and compact. We present a new class of polynomial length formulations for the asymmetric traveling salesman problem (ATSP) by lifting an ordered path-based model using logical restrictions in concert with the Reformulation-Linearization Technique (RLT). We show that a relaxed version of this formulation is equivalent to a flow-based ATSP model, which, in turn, is tighter than the formulation based on the exponential number of Dantzig-Fulkerson-Johnson (DFJ) subtour elimination constraints. The proposed lifting idea is applied to derive a variety of new formulations for the ATSP, and a detailed analysis of these formulations is carried out to show that some of these formulations are the tightest among those presented in the literature. Computational results are presented to exhibit the relative tightness of our formulations and the efficacy of the proposed lifting process.> While the computational results demonstrate the efficacy of employing the proposed theoretical RLT and logical lifting ideas, yet it remains of practical interest to take due advantage of the tightest formulations. The key requirement to accomplish this is the ability to solve the underlying LP relaxations more effectively. One approach, to that end, is to solve these LP relaxations to (near-) optimality by using deflected subgradient methods on Lagrangian dual formulations. We solve the LP relaxation of our tightest formulation, ATSP6, to (near-) optimality by using a deflected subgradient algorithm with average direction strategy (SA_ADS) (see Sherali and Ulular [69]). We also use two nondifferentiable optimization (NDO) methods, namely, the variable target value method (VTVM) presented by Sherali et al. [66] and the trust region target value method (TRTV) presented by Lim and Sherali [46], on the Lagrangian dual formulation of ATSP6. The preliminary results show that the near-optimal values obtained by the VTVM on solving the problem in the canonical format are the closest to the target optimal values. Another approach that we use is to derive a set of strong valid inequalities based on our tighter formulations through a suitable surrogation process for inclusion within the more compact manageable formulations. Our computational results show that, when the dual optimal solution is available, the associated strong valid inequalities generated from our procedure can successfully lift the LP relaxation of a less tight formulation, such as ATSP2R¯, to be as tight as the tightest formulation, such as ATSP6. We extend our new formulations to include precedence constraints in order to enforce a partial order on the sequence of cities to be visited in a tour. The presence of precedence constraints within the ATSP framework is encountered quite often in practice. Examples include: disassembly optimization (see Sarin et al. [62]), and scheduling of wafers/ ICs on automated testing equipments in a semiconductor manufacturing facility (see Chen and Hsia [17]); among others. Our flow-based ATSP formulation can very conveniently capture these precedence constraints. We also present computational results to depict the tightness of our precedence-constrained asymmetric traveling salesman problem (PCATSP) formulations. We, then, apply our formulations to the hot strip rolling scheduling problem, which involves the processing of hot steel slabs, in a pre-specified precedence order, on one or more rollers. The single-roller hot strip rolling scheduling problem can be directly formulated as a PCATSP. We also consider the multiple-roller hot strip rolling scheduling problem. This gives rise to the multiple-asymmetric traveling salesman problem (mATSP). Not many formulations have been presented in the literature for the mATSP, and there are none for the mATSP formulations involving a precedence order among the cities to be visited by the salesmen, which is the case for the multiple-roller hot strip rolling scheduling problem. To begin with, we develop new formulations for the mATSP and show the validity of our formulations, and present computational results to depict their tightness. Then, we extend these mATSP formulations to include a pre-specified, special type of precedence order in which to process the slabs, and designate the resulting formulations as the restricted precedence-constrained multiple-asymmetric traveling salesman problem (rPCmATSP) formulations. We directly formulate the multiple-roller hot strip rolling scheduling problem as a rPCmATSP. Furthermore, we consider the hot strip rolling scheduling problem with slab selection in which not all slabs need to be processed. We model the single-roller hot strip rolling scheduling problem with slab selection as a multiple-asymmetric traveling salesman problem with exactly two traveling salesmen. Similarly, the multiple-roller hot strip rolling scheduling problem with slab selection is modeled as a multiple-asymmetric traveling salesman problem with (m+1) traveling salesmen. A series of computational experiments are conducted to exhibit the effectiveness of our formulations for the solution of hot strip rolling scheduling problems. Furthermore, we develop two mixed-integer programming algorithms to solve our formulations. These are based on Benders΄ decomposition [13] and are designated Benders΄ decomposition and Modified Benders΄ methods. In concert with a special type of precedence order presented in the hot strip rolling scheduling problems, we further introduce an adjustable density ratio of the associated precedence network and we use randomly generated test problems to study the effect of various density ratios in solving these scheduling problems. Our experimentation shows the efficacy of our methods over CPLEX. Finally, we present a compact formulation for the job shop scheduling problem, designated as JSCD (job shop conjunctive-disjunctive) formulation, which is an extension of our ATSP formulations. We use two test problems given in Muth and Thompson [53] to demonstrate the optimal schedule and the lower bound values obtained by solving the LP relaxations of our formulations. However, we observe that the lower bound values obtained by solving the LP relaxations of all variations of our JSCD formulation equal to the maximum total processing time among the jobs in the problem. / Ph. D.

Many companies have traveling salesmen that market and sell their products.This results in much traveling by car due to the daily customer visits. Thiscauses costs for the company, in form of travel expenses compensation, and environmentaleffects, in form of carbon dioxide pollution. As many companies arecertified according to environmental management systems, such as ISO 14001,the environmental work becomes more and more important as the environmentalconsciousness increases every day for companies, authorities and public.The main task of this thesis is to compute reasonable limits on the mileage ofthe salesmen; these limits are based on specific conditions for each salesman’sdistrict. The objective is to implement a heuristic algorithm that optimizes thecustomer tours for an arbitrary chosen month, which will represent a “standard”month. The output of the algorithm, the computed distances, will constitute amileage limit for the salesman.The algorithm consists of a constructive heuristic that builds an initial solution,which is modified if infeasible. This solution is then improved by a local searchalgorithm preceding a genetic algorithm, which task is to improve the toursseparately.This method for computing mileage limits for traveling salesmen generates goodsolutions in form of realistic tours. The mileage limits could be improved if theinput data were more accurate and adjusted to each district, but the suggestedmethod does what it is supposed to do.

<p>Many companies have traveling salesmen that market and sell their products.This results in much traveling by car due to the daily customer visits. Thiscauses costs for the company, in form of travel expenses compensation, and environmentaleffects, in form of carbon dioxide pollution. As many companies arecertified according to environmental management systems, such as ISO 14001,the environmental work becomes more and more important as the environmentalconsciousness increases every day for companies, authorities and public.The main task of this thesis is to compute reasonable limits on the mileage ofthe salesmen; these limits are based on specific conditions for each salesman’sdistrict. The objective is to implement a heuristic algorithm that optimizes thecustomer tours for an arbitrary chosen month, which will represent a “standard”month. The output of the algorithm, the computed distances, will constitute amileage limit for the salesman.The algorithm consists of a constructive heuristic that builds an initial solution,which is modified if infeasible. This solution is then improved by a local searchalgorithm preceding a genetic algorithm, which task is to improve the toursseparately.This method for computing mileage limits for traveling salesmen generates goodsolutions in form of realistic tours. The mileage limits could be improved if theinput data were more accurate and adjusted to each district, but the suggestedmethod does what it is supposed to do.</p>

La 1ere partie de ce travail traite l'optimisation des tournées sous forme d'un problème d'optimisation nommé Le problème de Voyageur de Commerce. Dans cette partie nous nous intéressons à faire une riche présentation du problème de Voyageur de Commerce, ses variantes, puis nous proposons une stratégie de génération de contrainte pour la résolution du TSP. Ensuite on traite sa version stochastique : le problème de Voyageur de commerce Probabiliste. Nous proposons une formulation mathématique du PTSP et nous présentons des résultats numériques obtenus par résolution exacte pour une série d'instances de petite taille. Dans la seconde partie, nous proposons une méthode d'approximation générale permettant d'approcher différents type de données, d'abord nous traitons l'approximation d'un signal de vent (cas simple, ID), ensuite l'approximation d'un champ de vecteurs avec prise en compte de la topographie qui constitue la principale contribution de cette partie. / The first part of this work deals with route optimization in the form of an optimization problem named The Traveler's Business Problem. In this part we are interested to make a rich presentation of the problem of Traveler Commerce, its variants, then we propose a strategy of constraint generation for the resolution of the TSP. Then we treat its stochastic version : the probabilistic business traveler problem. We propose a mathematical formulation of the PTSP and we present numerical results obtained by exact resolution for a series of small instances. In the second part, we propose a method of general approximation to approximate different type of data, first we treat the approximation of a wind signal (simple case, 1D), then the approximation of a vector field taking into account the topography which is the main contribution of this part.

Ant Colony Optimization (ACO) algorithms have been successful in solving a wide variety of NPhard optimization problems. The Traveling Salesman Problem (TSP) has served as a benchmarking problem for many novel ACO algorithms. The slightly harder Dynamic Traveling Salesman Problem (DTSP) is more realistic in the sense that real-time changes happen in the graph belonging to a TSP instance. This thesis studied the original ACO algorithm: the Ant System, and how the amount of pheromone deposited by the ants within the algorithm affected the performance when solving both TSP and DTSP problems. Additionally, 2-opt local search was added to the algorithm, to see how it impacted the performance. We found that when the ants deposited a greater amount of pheromone, the performance for TSP increased, while the performance for DTSP decreased. We concluded that the Ant System in its original form is unsuitable for solving the DTSP. 2-opt local search improved the performance in all instances. / Ant Colony Optimization-algoritmer (ACO) har visat sig vara bra på att lösa många olika NP-svåra optimeringsproblem. För att mäta prestandan för nya ACO-algoritmer har i många fall Handelsresandeproblemet (eng. TSP) använts. Den dynamiska varianten av TSP (eng. DTSP), är ett något svårare problem då förändringar i grafen kan ske i realtid. Denna uppsats utredde hur olika mängder feromon som avges av myrorna inuti algoritmen Ant System, påverkade prestandan för både TSPoch DTSP-instanser. Utöver detta studerades hur den lokala sökningsheuristiken 2-opt påverkade prestandan. Resultaten visade att om myrorna tilläts släppa mer feromoner, ökade prestantan för TSP, men minskade för DTSP. Därav drog vi slutsatsen att algoritmen Ant System i sin ursprungliga form ej är lämplig för att lösa DTSP. Den lokala söknigsheuristiken 2-opt förbättrade prestandan i alla tester.

This thesis addresses the problem of setting up a surface mount placement machine for production. The objective is to minimize the number of machine changeovers made during a production run consisting of a number of circuit cards. The solution to the problem involves two separate decisions. The first decision considers determining how to combine feeders together in "bays" or groups of feeders, and how to assign the bays to the circuit cards. The second decision considers the circuit card production sequence. A mathematical programming formulation is given, however, its solution is very difficult for problems of a realistic size. Several heuristic approaches are suggested and used to solve actual and test problems. The heuristic for bay design uses clustering techniques used in Group Technology while the sequencing problem is solved using heuristics based on solution techniques for the Traveling Salesman problem.