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 (

Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

AbstractExtending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope

Polyhedral techniques in combinatorial optimization II: applications and computations

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems

Strong Bounds for Resource Constrained Project Scheduling: Preprocessing and Cutting Planes

Resource Constrained Project Scheduling Problems (RCPSPs) without preemption are well-known NP-hard combinatorial optimization problems. A feasible RCPSP solution consists of a time-ordered schedule of jobs with corresponding execution modes, respecting precedence and resources constraints. In this paper, we propose a cutting plane algorithm to separate five different cut families, as well as a new preprocessing routine to strengthen resource-related constraints. New lifted versions of the well-known precedence and cover inequalities are employed. At each iteration, a dense conflict graph is built considering feasibility and optimality conditions to separate cliques, odd-holes and strengthened Chv\'atal-Gomory cuts. The proposed strategies considerably improve the linear relaxation bounds, allowing a state-of-the-art mixed-integer linear programming solver to find provably optimal solutions for 754 previously open instances of different variants of the RCPSPs, which was not possible using the original linear programming formulations.Comment:

Resource constrained shortest paths and extensions

In this thesis, we use integer programming techniques to solve the resource constrained shortest path problem (RCSPP) which seeks a minimum cost path between two nodes in a directed graph subject to a finite set of resource constraints. Although NP-hard, the RCSPP is extremely useful in practice and often appears as a subproblem in many decomposition schemes for difficult optimization problems. We begin with a study of the RCSPP polytope for the single resource case and obtain several new valid inequality classes. Separation routines are provided, along with a polynomial time algorithm for constructing an auxiliary conflict graph which can be used to separate well known valid inequalities for the node packing polytope. We establish some facet defining conditions when the underlying graph is acyclic and develop a polynomial time sequential lifting algorithm which can be used to strengthen one of the inequality classes. Next, we outline a branch-and-cut algorithm for the RCSPP. We present preprocessing techniques and branching schemes which lead to strengthened linear programming relaxations and balanced search trees, and the majority of the new inequality classes are generalized to consider multiple resources. We describe a primal heuristic scheme that uses fractional solutions, along with the current incumbent, to search for new feasible solutions throughout the branch-and-bound tree. A computational study is conducted to evaluate several implementation choices, and the results demonstrate that our algorithm outperforms the default branch-and-cut algorithm of a leading integer programming software package. Finally, we consider the dial-a-flight problem (DAFP), a new vehicle routing problem that arises in the context of on-demand air transportation and is concerned with the scheduling of a set of travel requests for a single day of operations. The DAFP can be formulated as an integer multicommodity network flow model consisting of several RCSPPs linked together by set partitioning constraints which guarantee that all travel requests are satisfied. Therefore, we extend our branch-and-cut algorithm for the RCSPP to solve the DAFP. Computational experiments with practical instances provided by the DayJet Corporation verify that the extended algorithm also outperforms the default branch-and-cut algorithm of a leading integer programming software package.Ph.D.Committee Co-Chair: George L. Nemhauser; Committee Co-Chair: Shabbir Ahmed; Committee Member: Martin W. P. Savelsbergh; Committee Member: R. Gary Parker; Committee Member: Zonghao G