The objective of this thesis is to demonstrate the use of the Activity-Based Costing (ABC) approach together with the Theory of Constraints (TOC) philosophy in determining the optimal product-mix and restrictive bottlenecks of a company. The contribution of this thesis is a new product-mix decision model that uses activity-based cost information. This new model is proposed to be used with the TOC philosophy in order to improve the financial performance of a company.</p> Four case studies, all of which are based on hypothetical data, are prepared in this research to show the applicability of the proposed model in different manufacturing environments. Specifically, the first case study shows that the conventional product-mix decision model and the model developed in this thesis can give significantly different results regarding the best product-mix and associated bottlenecks of a company. The second case study demonstrates the use of the proposed product-mix decision model in a complex manufacturing environment. Specifically, this case study shows how companies should consider alternatives such as activity flexibility and outsourcing to improve their profitability figures. The third case study is an extension of the second case study, and it is prepared to illustrate that the proposed model can be extended to include more than one time period. The final case study demonstrates the applicability of the proposed model in a lean manufacturing environment.</p> Using the proposed model developed in this research will give managers more accurate information regarding the optimum product-mix and critical bottlenecks of their companies. By applying the TOC philosophy based on this information, managers will be able to take the right actions that will improve the profitability of their companies. Specifically, they will be able to observe the effects of several alternatives, such as activity flexibility and outsourcing, on the throughput of the whole system. In addition, the proposed model should help managers to prevent making decisions that sub-optimize the system. This may occur, for example, when using only the most efficient methods to produce each product even though the capacities of these methods are limited and some other less efficient methods are currently available in the company. By extending the model to include more than one time period, managers will be able to estimate the potential bottlenecks and the amount of idle capacities of each non-bottleneck activity performed in the company ahead of time. This information is powerful and can give companies a substantial advantage over their competitors because the users of the new model will have enough time to improve the performance of their potential bottlenecks and to search for more profitable usage of excess capacities before the actual production takes place. / Master of Science

Workplaces operating around the clock often use rotating shift schedules that repeat throughout the year, except for the summer schedule, which is non-cyclic due to vacations resulting in reduced staff availability. This thesis presents the development of an integer programming model designed to combine the formulation of a year-long rotating workforce schedule with special adjustments for the summer months. Both hard and soft constraints, the latter represented as objective functions, are integrated into the model to ensure schedule feasibility and quality. A case study tests various objective functions with different prioritising orders and analyses their impacts on each other. This reveals that the optimisation order, whether prioritising the summer or rotating schedule first, does not significantly affect the outcome. Comparing the generated schedule to a manually created one demonstrates that the model produces a high-quality schedule, successfully meeting the annual rotating requirements and the special adjustments needed for the summer period. Additionally, the model offers significant time savings and increased efficiency in schedule creation.

Approved for public release, distribution is unlimited / We optimize long-term project schedules subject to annual budget constraints, where the duration and cost of each task may increase as the project progresses. Initially, tasks are scheduled without regard to budgets and the project completion time is minimized. Treating the task durations as random variables, we then use simulation to describe the distribution of the project completion time. Next, we minimize the completion time under budget constraints with fixed task durations, where budget violations are tolerated albeit with penalties. Annual reviews are then introduced, which allow underway tasks to be delayed or monthly budgets to be increased. We obtain estimates of the completion time of the project and its final cost under each of these scenarios. The U.S. Army Future Combat Systems (FCS) is used for illustration. FCS is a suite of information technologies, sensors, and command systems with an estimated acquisition cost of over $90 billion. The U.S. General Accounting Office found that FCS is at risk of substantial cost overrun and delay. We analyze three schedule plans for FCS to identify which can be expected to deliver the earliest completion time and the least cost. / Major, Australian Army

This thesis focuses on the challenges that arise from the scheduling of heat exchanger network maintenance problems which undergo fouling and run continuously over time. The original contributions of the current research consist of the development of novel optimisation methodologies for the scheduling of cleaning actions in heat exchanger network problems, the application of the novel solution methodology developed to other general maintenance scheduling problems, the development of a stochastic programming formulation using this optimisation technique and its application to these scheduling problems with parametric uncertainty. The work presented in this thesis can be divided into three areas. To efficiently solve this non-convex heat exchanger network maintenance scheduling problem, new optimisation strategies are developed. The resulting contributions are outlined below. In the first area, a novel methodology is developed for the solution of the heat exchanger network maintenance scheduling problems, which is attributed towards a key discovery in which it is observed that these problems exhibit bang-bang behaviour. This indicates that when integrality on the binary decision variables is relaxed, the solution will tend to either the lower or the upper bound specified, obviating the need for integer programming solution techniques. Therefore, these problems are in ac- tuality optimal control problems. To suitably solve these problems, a feasible path sequential mixed integer optimal control approach is proposed. This methodology is coupled with a simple heuristic approach and applied to a range of heat exchanger network case studies from crude oil refinery preheat trains. The demonstrated meth- odology is shown to be robust, reliable and efficient. In the second area of this thesis, the aforementioned novel technique is applied to the scheduling of the regeneration of membranes in reverse osmosis networks which undergo fouling and are located in desalination plants. The results show that the developed solution methodology can be generalised to other maintenance scheduling problems with decaying performance characteristics. In the third and final area of this thesis, a stochastic programming version of the feasible path mixed integer optimal control problem technique is established. This is based upon a multiple scenario approach and is applied to two heat exchanger network case studies of varying size and complexity. Results show that this methodology runs automatically with ease without any failures in convergence. More importantly due to the significant impact on economics, it is vital that uncertainty in data is taken into account in the heat exchanger network maintenance scheduling problem, as well as other general maintenance scheduling problems when there is a level of uncertainty in parameter values.

The main subject of this thesis is the development of new geometric tools and techniques for solving classic problems within the geometry of numbers and convex geometry. At a high level, the problems considered in this thesis concern the varied interplay between the continuous and the discrete, an important theme within computer science and operations research. The first subject we consider is the study of cutting planes for non-linear integer programs. Cutting planes have been implemented to great effect for linear integer programs, and so understanding their properties in more general settings is an important subject of study. As our contribution to this area, we show that Chvatal-Gomory closure of any compact convex set is a rational polytope. As a consequence, we resolve an open problem of Schrijver (Ann. Disc. Math. `80) regarding the same question for irrational polytopes. The second subject of study is that of ellipsoidal approximation of convex bodies. Different such notions have been important to the development of fundamental geometric algorithms: e.g. the ellipsoid method for convex optimization (enclosing ellipsoids), or random walk methods for volume estimation (inertial ellipsoids). Here we consider the construction of an ellipsoid with good "covering" properties with respect to a convex body, known in convex geometry as the M-ellipsoid. As our contribution, we give two algorithms for constructing M-ellipsoids, and provide an application to near-optimal deterministic volume estimation in the oracle model. Equipped with this new geometric tool, we move to the study of classic lattice problems in the geometry of numbers, namely the Shortest (SVP) and Closest Vector Problems (CVP). Here we use M-ellipsoid coverings, combined with an algorithm of Micciancio and Voulgaris for CVP in the ℓ₂ norm (STOC `10), to obtain the first deterministic 2^O(ⁿ) time algorithm for the SVP in general norms. Combining this algorithm with a novel lattice sparsification technique, we derive the first deterministic 2^O(ⁿ)(1+1/ϵ)ⁿ time algorithm for (1+ϵ)-approximate CVP in general norms. For the next subject of study, we analyze the geometry of general integer programs. A central structural result in this area is Kinchine's flatness theorem, which states that every lattice free convex body has integer width bounded by a function of dimension. As our contribution, we build on the work Banaszczyk, using tools from lattice based cryptography, to give a new and tighter proof of the flatness theorem. Lastly, combining all the above techniques, we consider the study of algorithms for the Integer Programming Problem (IP). As our main contribution, we give a new 2^O(ⁿ)nⁿ time algorithm for IP, which yields the fastest currently known algorithm for IP and improves on the classic works of Lenstra (MOR `83) and Kannan (MOR `87).

Recent developments in mixed integer programming have highlighted the need for multi-row cuts. To this day, the performance of such cuts has typically fallen short of the single-row Gomory mixed integer cut. This disparity between the theoretical need and the practical shortcomings of multi-row cuts motivates the study of both the mixed integer cut and multi-row cuts. In this thesis, we build on the theoretical foundations of the mixed integer cut and develop techniques to derive multi-row cuts. The first chapter introduces the mixed integer programming problem. In this chapter, we review the terminology and cover some basic results that find application throughout this thesis. Furthermore, we describe the practical solution of mixed integer programs, and in particular, we discuss the role of cutting planes and our contributions to this theory. In Chapter 2, we investigate the Gomory mixed integer cut from the perspective of group polyhedra. In this setting, the mixed integer cut appears as a facet of the master cyclic group polyhedron. Our chief contribution is a characterization of the adjacent facets and the extreme points of the mixed integer cut. This provides insight into the families of cuts that may work well in conjunction with the mixed integer cut. We further provide extensions of these results under mappings between group polyhedra. For the remainder of this thesis we explore a framework for deriving multi-row cuts. For this purpose, we favor the method of superadditive lifting. This technique is largely driven by our ability to construct superadditive under-approximations of a special value function known as the lifting function. We devote our effort to precisely this task. Chapter 3 reviews the theory behind superadditive lifting and returns to the classical problem of lifted flow cover inequalities. For this specific example, the lifting function we wish to approximate is quite complicated. We overcome this difficulty by adopting an indirect method for proving the validity of a superadditive approximation. Finally, we adapt the idea to high-dimensional lifting problems, where evaluating the exact lifting function often poses an immense challenge. Thus we open entirely unexplored problems to the powerful technique of lifting. Next, in Chapter 4, we consider the computational aspects of constructing strong superadditive approximations. Our primary contribution is a finite algorithm that constructs non-dominated superadditive approximations. This can be used to build superadditive approximations on-the-fly to strengthen cuts derived during computation. Alternately, it can be used offline to guide the search for strong superadditive approximations through numerical examples. We follow up in Chapter 5 by applying the ideas of Chapters 3 and 4 to high-dimensional lifting problems. By working out explicit examples, we are able to identify non-dominated superadditive approximations for high-dimensional lifting functions. These approximations strengthen existing families of cuts obtained from single-row relaxations. Lastly, we show via the stable set problem how the derivation of the lifting function and its superadditive approximation can be entirely embedded in the computation of cuts. Finally, we conclude by identifying future avenues of research that arise as natural extensions of the work in this thesis.

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

This research shows optimization approaches to protein folding. The protein folding problem is to predict the compact three dimensional structure of a protein based on its amino acid sequence. This research focuses on ab-initio mathematical models to find provably optimal solutions to the 2D HP-lattice protein folding model. We built two integer programming (IP) models and five constraint programming (CP) models. All the models give provably optimal solutions. We also developed some CP techniques to solve the problem faster and then compared their computational times. We tested the models with several protein instances. My models, while they are probably too slow to use in practice, are significantly faster than the alternatives, and thus are mathematically relevant. We also provided reasons why protein folding is hard using complexity analysis. This research will contribute to showing whether CP can be an alternative to or a complement of IP in the future. Moreover, figuring out techniques combining CP and IP is a prominent research issue and our work will contribute to that literature. It also shows which IP/CP strategies can speed up the running time for this type of problem. Finally, it shows why a mathematical approach to protein folding is especially hard not only mathematically, i.e. NP-hard, but also practically.

Many supply chain problems involve discrete decisions in a dynamic environment. The inventory routing problem is an example that combines the dynamic control of inventory at various facilities in a supply chain with the discrete routing decisions of a fleet of vehicles that moves product between the facilities. We study these problems modeled as mixed-integer programs and propose a time decomposition based on approximate inventory valuation. We generate the approximate value function with an algorithm that combines data fitting, discrete optimization and dynamic programming methodology. Our framework allows the user to specify a class of piecewise linear, concave functions from which the algorithm chooses the value function. The use of piecewise linear concave functions is motivated by intuition, theory and practice. Intuitively, concavity reflects the notion that inventory is marginally more valuable the closer one is to a stock-out. Theoretically, piecewise linear concave functions have certain structural properties that also hold for finite mixed-integer program value functions. (Whether the same properties hold in the infinite case is an open question, to our knowledge.) Practically, piecewise linear concave functions are easily embedded in the objective function of a maximization mixed-integer or linear program, with only a few additional auxiliary continuous variables. We evaluate the solutions generated by our value functions in a case study using maritime inventory routing instances inspired by the petrochemical industry. The thesis also includes two other contributions. First, we review various data fitting optimization models related to piecewise linear concave functions, and introduce new mixed-integer programming formulations for some cases. The formulations may be of independent interest, with applications in engineering, mixed-integer non-linear programming, and other areas. Second, we study a discounted, infinite-horizon version of the canonical single-item lot-sizing problem and characterize its value function, proving that it inherits all properties of interest from its finite counterpart. We then compare its optimal policies to our algorithm's solutions as a proof of concept.