Academic literature on the topic 'LINEAR PROGRAMMING PROBLEM (LPP)'

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.

The aim of this thesis is to deepen the Containership Stowage Planning Problem (CSPP). In general terms, this problem consists of finding optimal plans for stowing containers into a containership, satisfying several restrictions. This topic has a lot of variations regarding the objective functions and the constraints required, depending on the situation taken into account. This dissertation is developed by referring to a real case, with its specific objective function and restrictions. In the first part of this thesis, an overview on the different approaches given by the literature is provided. After this outline, a Mixed Integer Linear Programming Model is proposed with the goal of finding feasible solutions for the CSPP. A consistent number of instances is generated to analyze how the model performs depending on the input parameters. The model is then tested by using CPLEX Solver in the mathematical programming and optimization modelling language AMPL. Finally, the importance of the stability of the vessel is underlined. Constraints concerning the ship stability are added to the model and, throughout other tests in AMPL, the computational results and the comparison to the results previously obtained are evaluated.

The travelling salesman problem is a well known optimization problem. The goal is to nd theshortest tour that visits each city in a given list exactly once. Despite the simple problem statementit belongs to the class of NP-complete problems. Its importance arises from a plethora of applicationsas well as a theoretical appeal. The asymmetric TSP is not as well researched as the symmetric TSP,in this paper we focus on a solution approach suitable for the asymmetric case. We demonstrate howa linear programming formulation can be used to solve the problem. We also show the limitationsof this solution approach and provide suggestions for improving it.i

Thesis (M.S.)--Georgia Southern University, 2009.<br>"A thesis submitted to the Graduate Faculty of Georgia Southern University in partial fulfillment of the requirements for the degree Master of Science." Directed by Goran Lesaja. ETD. Includes bibliographical references (p. 76) and appendices.

Knapsack problem has been widely studied in computer science for years. There exist several variants of the problem, with zero-one maximum knapsack in one dimension being the simplest one. In this thesis we study several existing approximation algorithms for the minimization version of the problem and propose a scaling based fully polynomial time approximation scheme for the minimum knapsack problem. We compare the performance of this algorithm with existing algorithms. Our experiments show that, the proposed algorithm runs fast and has a good performance ratio in practice. We also conduct extensive experiments on the data provided by Canadian Pacific Logistics Solutions during the MITACS internship program. We propose a scaling based e-approximation scheme for the multidimensional (d-dimensional) minimum knapsack problem and compare its performance with a generalization of a greedy algorithm for minimum knapsack in d dimensions. Our experiments show that the e- approximation scheme exhibits good performance ratio in practice.<br>x, 85 leaves ; 29 cm

This research effort is concerned with a class of mathematical programming problems referred to as Mixed-Integer Bilinear Programming Problems. This class of problems, which arises in production, location-allocation, and distribution-application contexts, may be considered as a discrete version of the well-known Bilinear Programming Problem in that one set of decision variables is restricted to be binary valued. The structure of this problem is studied, and special cases wherein it is readily solvable are identified. For the more general case, a new linearization technique is introduced and demonstrated to lead to a tighter linear programming relaxation than obtained through available linearization methods. Based on this linearization, a composite Lagrangian relaxation-implicit enumeration-cutting plane algorithm is developed. Extensive computational experience is provided to test the efficiency of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm. The solution strategy developed for the Mixed-Integer Bilinear Programming Problem may be applied, with suitable modifications,. to other classes of mathematical programming problems: in particular, to the Zero-One Quadratic Programming Problem. In what may be considered as an extension to the work performed on the Mixed-Integer Bilinear Programming Problem, a solution strategy based on an equivalent linear reformulation is developed for the Zero-One Quadratic Programming Problem. The strategy is essentially an implicit enumeration algorithm which employs Lagrangian relaxation, Benders' cutting planes, and local explorations. Computational experience for this problem class is provided to justify the worth of the proposed linear reformulation and algorithm.<br>Ph. D.