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

This research addresses sourcing decisions and how those decisions can affect the management of a company's assets. The study begins with a single-plant problem, in which one facility chooses, from a list of parts, which parts to bring in-house. The selection is based on maximizing the value of the selected parts, while remaining within the plant's capacity. This problem is defined as the insourcing problem and modeled as a multidimensional knapsack problem (MKP). The insourcing model is extended to address outsourcing and multiple plants. This multi-plant model, also modeled as an MKP, enables the movement of parts from one plant to another and consideration of a company-wide objective function (as opposed to a single-plant objective function as in the insourcing model). The sourcing problem possesses characteristics that distinguish it from the standard MKP. One such characteristic is what we define as multiple attributes. To understand the multiple attribute characteristic, we compare the various dimensions in the multidimensional knapsack problem. A classification is given for an MKP as either having a single attribute (SA) or multiple attributes (MA). Mathematically, the problems of each attribute classification can be modeled in the same way with simply a different interpretation of the knapsack constraints. However, experimentation indicates that the MA-MKP is more difficult to solve than the SA-MKP. For small problems, with 100 variables and 5 constraints, the CPU time required to find the optimal solution for MA-MKP to SA-MKP problems has a ratio of 32:1. To determine effective methods for addressing the MA-MKP, standard mixed integer programming techniques are tested. The results of this testing are that the exact approaches are not successful in dramatically reducing the solution time to the level of the SA problems. However, a simple heuristic that performs very well on the MA-MKP is presented. The heuristic utilizes variations on the benefit-to-cost ratio and strongest surrogate constraints. The results from experimentation for MA-MKP problem sets, generated using the methods for standard MKP test data sets in the literature, are presented and indicate that the heuristic performs well and improves with larger problems. The average gap between the heuristic solution and the optimal solution is 1.39% for 200-part problems and is reduced to 0.69% when the size of the problem is increased to 298 parts. Although the MA characteristic reflects the sourcing problem, the actual data used in the eperimentation is generated with techniques presented in the literature for standard MKP test problems. Therefore, to more accurately represent the sourcing problem, industry data from a manufacturing facility is studied to identify further sourcing problem characteristics. As a result, industry-motivated data sets are generated that reflect the characteristics of industry data, yet maintain the structure of literature data sets to allow for easy comparison. It is found that both industry and industry-motivated data sets, although possessing the MA characteristic, are much easier to solve than SA problems. Indicators of difficulty appear to be the constraint tightness and a measure of the matrix sparsity. The sparsity is a significant factor because industry data tends to be very sparse, while data sets generated in the literature are completely dense. Another interesting result from the industry-motivated data sets with the single-plant problem is the tendency for a facility to prefer currently produced parts over insourcing new parts from outside the facility. It is not uncommon for a company to have more than one facility with a particular capability. Therefore, the sourcing model is extended to include multiple facilities. With multiple-facilities, effectively all the parts are removed to form one list, and then each part is assigned to one of the facilities or outsourced externally. The multi-facility model is similar to the single-facility model with the addition of assignment constraints enforcing that each part can be assigned to only one facility. Experimentation is performed for the two-, three-, and four-facility models. The problem gets easier to solve as the number of facilities increases. With a greater number of facilities, it is likely that for each part one of facilities will dominate as the best option. Therefore, other solutions can quickly be eliminated and the problem solved more quickly. The two-facility problem is the most difficult; however, the heuristic performs well with an average gap of 0.06% between the heuristic and optimal solutions. We conclude with a summary on experiences with modeling and solving the sourcing problem for a sheet metal fabrication facility. The model solved for this problem had over 1857 parts with 19 machines, which translates to over 70,000 variables and 38 constraints. Although extremely large compared to problems solved in the literature, this problem was solvable because of the unique structure of industry data. Our work with the facility saved the parent organization up to $4.16M per year and provided a tool that encourages a systematic and quantitative process for evaluating decisions related to sheet metal fabrication capacity. / Ph. D.

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

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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.

Nowadays all existing public key cryptosystems are classified into three categories relied on different mathematical foundations. The first one is based on the difficulty of factoring the product of two big prime numbers. The representatives are the RSA and the Rabin cryptosystems. The second one such as the ElGamal cryptosystem is based on the discrete logarithm problem. The last one is based on the NP-completeness of the knapsack problem. The first two categories survived crypto attacks, whereas the last one was broken and there has been no attempt to use such a cryptosystem. In order to save the last category, Kiriyama proposed a new public key cryptosystem based on the non-linear knapsack problem, which is an NP-complete problem. Due to the non-linear property of the non-linear knapsack problem, this system resists all known attacks to the linear knapsack problem. Based on his work, we extend our research in several ways. Firstly, we propose an encrypted secret sharing scheme. We improve the security of shares by our method over other existing secret sharing schemes. Simply speaking, in our scheme, it would be hard for outsiders to recover a secret even if somehow they could collect all shares, because each share is already encrypted when it is generated. Moreover, our scheme is efficient. Then we propose a multiple identities authentication scheme, developed on the basis of the non-linear knapsack scheme. It verifies the ownership of an entity's several identities in only one execution of our scheme. More importantly, it protects the privacy of the entities from outsiders. Furthermore, it can be used in resource-constrained devices due to low computational complexity. We implement the above schemes in the C language under the Linux system. The experimental results show the high efficiency of our schemes, due to low computational complexity of the non-linear knapsack problem, which works as the mathematical foundation of our research.

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.

Designers, operators, and users of multiple-device, access control security systems are challenged by the false alarm, false clear tradeoff. Given a particular access control security system, and a prespecified false clear standard, there is an optimal (minimal) false alarm rate that can be achieved. The objective of this research is to develop methods that can be used to determine this false alarm rate. Meeting this objective requires knowledge of the joint conditional probability density functions for the security device responses. Two sampling procedures, the static grid estimation procedure and the dynamic grid estimation procedure, are proposed to estimate these functions. The concept of a system response function is introduced and the problem of determining the optimal system response function that minimizes the false alarm rate, while meeting the false clear standard, is formulated as a decision problem and proven to be NP-complete. Two heuristic procedures, the Greedy algorithm and the Dynamic Programming algorithm, are formulated to address this problem. Computational results using simulated security data are reported. These results are compared to analytical results, obtained for a prespecified system response function form. Suggestions for future research are also included. / Master of Science