No description available.

Dissertation (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (p.94-95).

Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2010. / Committee Chair: Ellis L. Johnson; Committee Member: Brady Hunsaker; Committee Member: George Nemhauser; Committee Member: Jozef Siran; Committee Member: Shabbir Ahmed; Committee Member: William Cook. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (Ph. D.)--Lehigh University, 2005. / Includes vita. Includes bibliographical references (leaves 164-166).

Thesis (Ph.D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2009. / Committee Chair: William J. Cook; Committee Member: Ellis Johnson; Committee Member: George Nemhauser; Committee Member: Robin Thomas; Committee Member: Zonghao Gu

Thesis (M. Sc.)--University of Canterbury, 2006. / Typescript (photocopy). Includes bibliographical references (p. [93]-98). Also available via the World Wide Web.

Thesis (PhD)--University of Stellenbosch, 2007. / Bibliography. Also availabe via the Internet.

In this thesis, we propose and study discrete, multi-period extensions of classical packing problems, a fundamental class of models in combinatorial optimization. Those extensions fall under the general name of incremental packing problems. In such models, we are given an added time component and different capacity constraints for each time. Over time, capacities are weakly increasing as resources increase, allowing more items to be selected. Once an item is selected, it cannot be removed in future times. The goal is to maximize some (possibly also time-dependent) objective function under such packing constraints. In Chapter 2, we study the generalized incremental knapsack problem, a multi-period extension to the classical knapsack problem. We present a policy that reduces the generalized incremental knapsack problem to sequentially solving multiple classical knapsack problems, for which many efficient algorithms are known. We call such an algorithm a single-time algorithm. We prove that this algorithm gives a (0.17 - ⋲)-approximation for the generalized incremental knapsack problem. Moreover, we show that the algorithm is very efficient in practice. On randomly generated instances of the generalized incremental knapsack problem, it returns near optimal solutions and runs much faster compared to Gurobi solving the problem using the standard integer programming formulation. In Chapter 3, we present additional approximation algorithms for the generalized incremental knapsack problem. We first give a polynomial-time (½-⋲)-approximation, improving upon the approximation ratio given in Chapter 2. This result is based on a new reformulation of the generalized incremental knapsack problem as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Using the same sequencing reformulation, combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we give a quasi-polynomial time approximation scheme for the problem, thus ruling out the possibility that the generalized incremental knapsack problem is APX-hard under widely-believed complexity assumptions. In Chapter 4, we first turn our attention to the submodular monotone all-or-nothing incremental knapsack problem (IK-AoN), a special case of the submodular monotone function subject to a knapsack constraint extended to a multi-period setting. We show that each instance of IK-AoN can be reduced to a linear version of the problem. In particular, using a known PTAS for the linear version from literature as a subroutine, this implies that IK-AoN admits a PTAS. Next, we study special cases of the generalized incremental knapsack problem and provide improved approximation schemes for these special cases. In Chapter 5, we give a polynomial-time (¼-⋲)-approximation in expectation for the incremental generalized assignment problem, a multi-period extension of the generalized assignment problem. To develop this result, similar to the reformulation from Chapter 3, we reformulate the incremental generalized assignment problem as a multi-machine sequencing problem. Following the reformulation, we show that the (½-⋲)-approximation for the generalized incremental knapsack problem, combined with further randomized rounding techniques, can be leveraged to give a constant factor approximation in expectation for the incremental generalized assignment problem. In Chapter 6, we turn our attention to the incremental knapsack polytope. First, we extend one direction of Balas's characterization of 0/1-facets of the knapsack polytope to the incremental knapsack polytope. Starting from extended cover inequalities valid for the knapsack polytope, we show how to strengthen them to define facets for the incremental knapsack polytope. In particular, we prove that under the same conditions for which these inequalities define facets for the knapsack polytope, following our strengthening procedure, the resulting inequalities define facets for the incremental knapsack polytope. Then, as there are up to exponentially many such inequalities, we give separation algorithms for this class of inequalities.

Solving real world problems with mixed integer programming (MIP) involves efforts in modeling and efficient algorithms. To solve a minimization MIP problem, a lower bound is needed in a branch-and-bound algorithm to evaluate the quality of a feasible solution and to improve the efficiency of the algorithm. This thesis develops a new MIP model and studies algorithms for obtaining lower bounds for MIP. The first part of the thesis is dedicated to a new production planning model with pricing decisions. To increase profit, a company can use pricing to influence its demand to increase revenue, decrease cost, or both. We present a model that uses pricing discounts to increase production and delivery flexibility, which helps to decrease costs. Although the revenue can be hurt by introducing pricing discounts, the total profit can be increased by properly choosing the discounts and production and delivery decisions. We further explore the idea with variations of the model and present the advantages of using flexibility to increase profit. The second part of the thesis focuses on solving integer programming(IP) problems by improving lower bounds. Specifically, we consider obtaining lower bounds for the multi- dimensional knapsack problem (MKP). Because MKP lacks special structures, it allows us to consider general methods for obtaining lower bounds for IP, which includes various relaxation algorithms. A problem relaxation is achieved by either enlarging the feasible region, or decreasing the value of the objective function on the feasible region. In addition, dual algorithms can also be used to obtain lower bounds, which work directly on solving the dual problems. We first present some characteristics of the value function of MKP and extend some properties from the knapsack problem to MKP. The properties of MKP allow some large scale problems to be reduced to smaller ones. In addition, the quality of corner relaxation bounds of MKP is considered. We explore conditions under which the corner relaxation is tight for MKP, such that relaxing some of the constraints does not affect the quality of the lower bounds. To evaluate the overall tightness of the corner relaxation, we also show the worst-case gap of the corner relaxation for MKP. To identify parameters that contribute the most to the hardness of MKP and further evaluate the quality of lower bounds obtained from various algorithms, we analyze the characteristics that impact the hardness of MKP with a series of computational tests and establish a testbed of instances for computational experiments in the thesis. Next, we examine the lower bounds obtained from various relaxation algorithms com- putationally. We study methods of choosing constraints for relaxations that produce high- quality lower bounds. We use information obtained from linear relaxations to choose con- straints to relax. However, for many hard instances, choosing the right constraints can be challenging, due to the inaccuracy of the LP information. We thus develop a dual heuristic algorithm that explores various constraints to be used in relaxations in the Branch-and- Bound algorithm. The algorithm uses lower bounds obtained from surrogate relaxations to improve the LP bounds, where the relaxed constraints may vary for different nodes. We also examine adaptively controlling the parameters of the algorithm to improve the performance. Finally, the thesis presents two problem-specific algorithms to obtain lower bounds for MKP: A subadditive lifting method is developed to construct subadditive dual solutions, which always provide valid lower bounds. In addition, since MKP can be reformulated as a shortest path problem, we present a shortest path algorithm that uses estimated distances by solving relaxations problems. The recursive structure of the graph is used to accelerate the algorithm. Computational results of the shortest path algorithm are given on the testbed instances.

Thesis (PhD (Logistics))--University of Stellenbosch, 2007. / In this dissertation original algorithms are introduced to solve separable resource allocation problems (RAPs) with increasing nonlinear functions in the objective function, and lower and upper bounds on each variable. Algorithms are introduced in three special cases. The first case arises when the objective function of the RAP consists of the sum of convex functions and all the variables for these functions range over the same interval. In the second case RAPs with the sum of convex functions in the objective function are considered, but the variables of these functions can range over different intervals. In the last special case RAPs with an objective function comprising the sum of convex and concave functions are considered. In this case the intervals of the variables can range over different values. In the first case two new algorithms, namely the fraction and the slope algorithm are presented to solve the RAPs adhering to the conditions of the case. Both these algorithms yield far better solution times than the existing branch and bound algorithm. A new heuristic and three new algorithms are presented to solve RAPs falling into the second case. The iso-bound heuristic yields, on average, good solutions relative to the optimal objective function value in faster times than exact algorithms. The three algorithms, namely the iso-bound algorithm, the branch and cut algorithm and the iso-bound branch and cut algorithm also yield considerably beter solution times than the existing branch and bound algorithm. It is shown that, on average, the iso-bound branch and cut algorithm yields the fastest solution times, followed by the iso-bound algorithm and then by die branch and cut algorithm. In the third case the necessary and sufficient conditions for optimality are considered. From this, the conclusion is drawn that search techniques for points complying with the necessary conditions will take too long relative to branch and bound techniques. Thus three new algorithms, namely the KL, SKL and IKL algorithms are introduced to solve RAPs falling into this case. These algorithms are generalisations of the branch and bound, branch and cut, and iso-bound algorithms respectively. The KL algorithm was then used as a benchmark. Only the IKL algorithm yields a considerable improvement on the KL algorithm.