The knapsack problem manifests itself in many domains like cryptography, financial domain and bio-informatics. Knapsack problems are often inside optimization loops in system-level design and analysis of embedded systems as well. Given a set of items, each associated with a profit and a weight, the knapsack problem deals with how to choose a subset of items such that the profit is maximized and the total weight of the chosen items is less than the capacity of the knapsack. There exists several variants and extensions of this knapsack problem. In this thesis, we focus on the multiple-choice knapsack problem, where the items are grouped into disjoint classes. However, the multiple-choice knapsack problem is known to be NP-hard. While many different heuristics and approximation schemes have been proposed to solve the problem in polynomial-time, such techniques do not return the optimal solution. A dynamic programming algorithm to solve the problem optimally is known, but has a pseudo-polynomial running time. This leads to high running times of tools in various application domains where knapsack problems must be solved. Many system-level design tools in the embedded systems domain, in particular, would suffer from high running when such a knapsack problem must be solved inside a larger optimization loop. To mitigate the high running times of such algorithms, in this thesis, we propose a GPU-based technique to solve the multiple-choice knapsack problem. We study different approaches to map the dynamic programming algorithm on the GPU and compare their performance in terms of the running times. We employ GPU specific methods to further improve the running times like exploiting the GPU on-chip shared memory. Apart from results on synthetic test-cases, we also demonstrate the applicability of our technique in practice by considering a case-study from system-level design. Towards this, we consider the problem of instruction-set selection for customizable processors.

The thesis deals with knapsack problems variants and possibility of their solving, furthermore with the impact of particular task (instance) special structure on the effciency of tested approach. The thesis also proposes conversion possibility between described tasks and their continuous extension (continuous relaxation). It describes L3 algorithm and superdecreasing knapsack problem solving from the common sort of algorithms and Monte Carlo Method, simulated annealing and genetic algorithms from the sort of probability ones. Other possibilities are also discussed. Integral part of this thesis is the accompanying application, which was used to create groundwork used in the text and which can be also used to solve other instances.

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

Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)

Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)

The rapid developments in the field of DNA computing reflects two substantial questions: 1. Which models for DNA based computation are really universal? 2. Which model fulfills the requirements to a universal lab-practicable programmable DNA computer that is based on one of these models? This paper introduces the functional model DNA-HASKELL focussing its lab-practicability. This aim could be reached by specifying the DNA based operations in accordiance to an analysis of molecular biological processes. The specification is determined by an abstraction level that includes nucleotides and strand end labels like 5'-phosphate. Our model is able to describe DNA algorithms for any NP-complete problem - here exemplified by the knapsacik problem - as well as it is able to simulate some established mathematical models for computation. We point out the splicing operation as an example. The computational completeness of DNA-HASKELL can be supposed. This paper is based on discussions about the potenzial and limits of DNA computing, in particular the practicability of a universal DNA computer.

The rapid developments in the field of DNA computing reflects two substantial questions: 1. Which models for DNA based computation are really universal? 2. Which model fulfills the requirements to a universal lab-practicable programmable DNA computer that is based on one of these models? This paper introduces the functional model DNA-HASKELL focussing its lab-practicability. This aim could be reached by specifying the DNA based operations in accordiance to an analysis of molecular biological processes. The specification is determined by an abstraction level that includes nucleotides and strand end labels like 5'-phosphate. Our model is able to describe DNA algorithms for any NP-complete problem - here exemplified by the knapsacik problem - as well as it is able to simulate some established mathematical models for computation. We point out the splicing operation as an example. The computational completeness of DNA-HASKELL can be supposed. This paper is based on discussions about the potenzial and limits of DNA computing, in particular the practicability of a universal DNA computer.

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / Recently, interdisciplinary (management, engineering, science, and economics) collaboration research has been growing to achieve the synergy and to reinforce the weakness of each discipline. Along this trend, this research combines three topics: mathematical programming, data mining, and supply chain management. A new pegging algorithm is developed for solving the continuous nonlinear knapsack problem. An efficient solving approach is proposed for solving the ν-support vector machine for classification problem in the field of data mining. The new pegging algorithm is used to solve the subproblem of the support vector machine problem. For the supply chain management, this research proposes an efficient integrated solving approach for the supplier selection problem. The support vector machine is applied to solve the problem of selecting potential supplies in the procedure of the integrated solving approach. In the first part of this research, a new pegging algorithm solves the continuous nonlinear knapsack problem with box constraints. The problem is to minimize a convex and differentiable nonlinear function with one equality constraint and box constraints. Pegging algorithm needs to calculate primal variables to check bounds on variables at each iteration, which frequently is a time-consuming task. The newly proposed dual bound algorithm checks the bounds of Lagrange multipliers without calculating primal variables explicitly at each iteration. In addition, the calculation of the dual solution at each iteration can be reduced by a proposed new method for updating the solution. In the second part, this research proposes several streamlined solution procedures of ν-support vector machine for the classification. The main solving procedure is the matrix splitting method. The proposed method in this research is a specified matrix splitting method combined with the gradient projection method, line search technique, and the incomplete Cholesky decomposition method. The method proposed can use a variety of methods for line search and parameter updating. Moreover, large scale problems are solved with the incomplete Cholesky decomposition and some efficient implementation techniques. To apply the research findings in real-world problems, this research developed an efficient integrated approach for supplier selection problems using the support vector machine and the mixed integer programming. Supplier selection is an essential step in the procurement processes. For companies considering maximizing their profits and reducing costs, supplier selection requires seeking satisfactory suppliers and allocating proper orders to the selected suppliers. In the early stage of supplier selection, a company can use the support vector machine classification to choose potential qualified suppliers using specific criteria. However, the company may not need to purchase from all qualified suppliers. Once the company determines the amount of raw materials and components to purchase, the company then selects final suppliers from which to order optimal order quantities at the final stage of the process. Mixed integer programming model is then used to determine final suppliers and allocates optimal orders at this stage.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IPs) are mathematical models that can provide organizations with the ability to optimally obtain their goals through appropriate utilization and allocation of available resources. Unfortunately, IPs are NP-complete in the strong sense, and many integer programs cannot be solved. Introduced by Gomory, lifting is a technique that takes a valid inequality and strengthens it. Lifting can result in facet defining inequalities, which are the theoretically strongest inequalities; because of this, lifting techniques are commonly used in commercial IP software to reduce the time required to solve an IP. This thesis introduces two new algorithms for exact simultaneous up lifting multiple sets into binary knapsack problems and introduces sequential simultaneous lifting. The Dynamic Programming Multiple Lifting Set Algorithm (DPMLSA) is a pseudopolynomial time algorithm bounded by O(nb) effort that can exactly uplift an arbitrary number of sets. The Three Set Simultaneous Lifting Algorithm (TSSLA) is a polynomial time algorithm bounded by O(n2) and can exact simultaneously up lift three sets. The simultaneously lifted inequalities generated by the DPMLSA and the TSSLA can be facet defining, and neither algorithm requires starting with a minimal cover. A brief computational study shows that the DPMLSA is fast and required an average of only 0.070 seconds. The computational study also shows these sequential simultaneously lifted inequalities are useful, as the solution time decreased by an overall average of 18.4%. Therefore, implementing the DPMLSA should be beneficial for large IPs.