Designing Satellite Communication Networks by Zero-One Quadratic Programming

In satellite communications networks, distinctive facilities called homing stations perform special transmission functions. Local demand nodes clustered around each homing station communicate with each other via a local switch at the homing station; demand nodes in different clusters communicate with each other via satellite earth stations at the homing stations. Designing such a communication network requires choices on the locations of the earth stations and on the assignments of demand nodes to the local clusters at the earth stations. We formulate this problem as a zero-one quadratic facility location problem and transform it into an equivalent zero-one integer linear program. Computational experience on real data shows that a branch and bound procedure is effective in solving problems with up to forty demand nodes (major cities) and that the solutions that this algorithm finds improve considerably upon management generated solutions. We also show that a greedy add heuristic, as implemented on an IBM PC, consistently generates optimal or near-optimal solutions

A Strong Cutting Plane Algorithm for Production Scheduling with Changeover Costs

Changeover costs (and times) are central to numerous manufacturing operations. These costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another. Although many researchers have contributed to the solution of scheduling problems that include changeover costs, due to the problem's combinatorial explosiveness, optimization-based methods have met with limited success. In this paper, we develop and apply polyhedral methods from integer programming for a dynamic version of the problem. Computational tests with problems containing one to five products (and up to 225 integer variables) show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer program formulation of the problem and its linear programming relaxation (by a factor of 94 to 100 per cent). These results suggest the use of a combined cutting plane/branch and bound procedure as a solution approach. In a test with a five product problem, this procedure, when compared with a standard linear programming-based branch and bound approach, reduced computation time by a factor of seven

Facets and Algorithms for Capacitated Lot Sizing

The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use these results to develop solution methods for multi-item applications. In particular, we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron. Computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane/branch and bound procedure. Moreover, our results show that in many instances adding certain of these inequalities a priori to the problem formulation, and avoiding the generation of cutting planes, can be equally effective

Adaptive Branch and Bound for Efficient Solution of Mixed-Integer Programs Formulated with Big-M

This thesis describes three specialized branch-and-bound (B and B) algorithms for solving a mixed-integer program (MIP) that incorporates standard big-M constructs. The goal is to identify valid values for M that also lead to short solution times. One algorithm initializes large instances of M (giving a weak relaxation of the MIP), and decreases these as required to increase efficiency of the standard B and B. Two algorithms initialize small and possibly invalid instances of M, and subsequently increase those values in an attempt to ensure solution validity. Each algorithm requires a model-specific test condition to detect weak or invalid Ms. We test all algorithms on an uncapacitated k-median problem (a variant of the uncapacitated facility location problem), and one algorithm on a shortest-path interdiction problem (SPIP). We observe substantial reduction in run times in almost all cases tested. When solving for exact solutions, computational results show that the proposed algorithms may reduce solution times by up to 75 per cent for the uncapacitated k-median problem and 99 per cent for the SPIP. When the algorithms yield marginally suboptimal solutions, substantial solution-time improvements are also recorded. While testing is limited, this thesis serves as a proof-of-concept that the proposed adaptive algorithms can be effective in reducing solution times and producing optimal or nearly optimal solutions.http://archive.org/details/adaptivebranchnd1094517370Major, Singapore ArmyApproved for public release; distribution is unlimited

Developing New Multidimensional Knapsack Heuristics Based on Empirical Analysis of Legacy Heuristics

The multidimensional knapsack problem (MKP) has been used to model a variety of practical optimization and decision-making applications. Due to its combinatorial nature, heuristics are often employed to quickly find good solutions to MKPs. While there have been a variety of heuristics proposed for the MKP, and a plethora of empirical studies comparing the performance of these heuristics, little has been done to garner a deeper understanding of heuristic performance as a function of problem structure. This dissertation presents a research methodology, empirical and theoretical results explicitly aimed at gaining a deeper understanding of heuristic procedural performance as a function of test problem characteristics. This work first employs an available, robust set of two-dimensional knapsack problems in an empirical study to garner performance insights. These performance insights are tested against a larger set of problems, five-dimensional knapsack problems specifically generated for empirical testing purposes. The performance insights are found to hold in the higher dimensions. These insights are used to formulate and test a suite of three new greedy heuristics for the MKP, each improving upon its successor. These heuristics are found to outperform available legacy heuristics across a complete spectrum of test problems. Problem reduction heuristics are examined and the subsequent performance insights garnered are used to derive a new problem reduction heuristic, which is then further extended to employ a local improvement phase. These problem reduction heuristics are also found to outperform currently available approaches. Available problem test sets are shown lacking along multiple dimensions of importance for viable empirical testing. A new problem generation methodology is developed and shown to overcome the current limitations in available problem test sets. This problem generation methodology is used to generate a new set of empirical test problems specifically designed for competitive computational tests. This new test set is shown to stress existing heuristics; not only does the computational time required by these legacy heuristics increase with problem size, but solution quality is found to decrease with problem size. However, the solution quality obtained by the suite of heuristics developed in this dissertation are shown to be unaffected by problem size thereby providing a level of robust solution quality not previously seen in heuristic development for the MKP. This research demonstrates that the test problems can have a profound, and sometimes misleading, impact on the general insights gained via empirical testing, provides six new quality heuristics, and two new robust sets of test problems, one focused on empirical testing, the other focused on competitive testing

Strategic Surveillance System Design for Ports and Waterways

The purpose of this dissertation is to synthesize a methodology to prescribe a strategic design of a surveillance system to provide the required level of surveillance for ports and waterways. The method of approach to this problem is to formulate a linear integer programming model to prescribe a strategic surveillance system design (SSD) for ports or waterways, to devise branch-and-price decomposition (