The thesis describes the principles of dynamic programming and it's application to concrete problems. (The travelling salesman problem, the knapsack problem, the shortest path priblem,the set covering problem.)

The diploma thesis focuses on tracing in layout by handlers between assembly lines in new plant for corporation Continental Automotive Czech Republic ltd, where boosters are produced. The theoretical part involves definitions of logistics, supply chain, material flow and handling equipment. Furthermore, methods of mathematic programming and software equipment are described, such as quadratic assignment problem, knapsack problem, travelling salesman problem from graph theory. In the practical part the situation in corporation has been analyzed and the data prepared for further examination. Then layout of plant and internal processes are evaluated and an appropriate model or concept of solution is selected. Subsequently, application in MS Excel is created with support of VBA scripts (3 kinds of layouts). The user manipulates with application followed by Solver for implementation of a new solution into practice. Finally, the models are interpreted and verified by Lingo. The focus of the thesis is the design of a layout change of a new plant including the description of tracing.

In which ways can the process of trimming dispersion coated board products be optimized so as to reduce material waste and increase production efficiency? This is the question that this master thesis report seeks to answer. In paper production, alot of waste is generated when cutting production reels into customer reels. Some material waste are necessary in order to ensure good quality, however a large amount of the wastecould be reduced if the cutting process was to be optimized. During this project, carried out at a forest company, a mathematical optimization model was developed in order to reduce waste and save costs. This model is based on a cutting stock problem using column generation approach. It provides as its output cutting patterns and an optimal allocation of rolls for production purposes, which implies minimizing the number production rolls needed.The visualization of the results could also be used to achieve optimal stock levels, and easier keep track on how to use the material available in stock. Findings show that there are potential savings to be done. Simulations suggest an implementation of this model could result in material savings of around 7 %. This could also translateto environmental savings in CO2, where every decrease of one tonne material equals to adecrease in CO2emissions of 500 kg

No description available.

Engine-triggered tasks are real-time tasks that are released when the crankshaft arrives at certain positions in its path of rotation. This makes the rate of release of these jobs a function of the crankshaft's angular speed and acceleration. In addition, several properties of the engine triggered tasks like the execution time and deadlines are dependent on the speed profile of the crankshaft. Such tasks are referred to as adaptive-variable rate (AVR) tasks. Existing methods to calculate the worst-case demand of AVR tasks are either inaccurate or computationally intractable. We propose a method to efficiently calculate the worst-case demand of AVR tasks by transforming the problem into a variant of the knapsack problem. We then propose a framework to systematically narrow down the search space associated with finding the worst-case demand of AVR tasks. Experimental results show that our approach is at least 10 times faster, with an average runtime improvement of 146 times for randomly generated task sets when compared to the state-of-the-art technique. / Master of Science / Real-time systems require temporal correctness along with accuracy. This notion of temporal correctness is achieved by specifying deadlines to each of the tasks. In order to ensure that all the deadlines are met, it is important to know the processor requirement, also known as demand, of a task over a given interval. For some tasks, the demand is not constant, instead it depends on several external factors. For such tasks, it becomes necessary to calculate the worst-case demand. Engine-triggered tasks are activated when the crankshaft in an engine is at certain points in its path of rotation. This makes their activation rate dependent on the angular speed and acceleration of the crankshaft. In addition, several properties of the engine triggered tasks like the execution time and deadlines are dependent on the speed profile of the crankshaft. Such tasks are referred to as adaptive-variable rate (AVR) tasks. Existing methods to calculate the worst-case demand of AVR tasks are either inaccurate or computationally intractable. We propose a method to efficiently calculate the worst-case demand of AVR tasks by transforming the problem into a variant of the knapsack problem. We then propose a framework to systematically narrow down the search space associated with finding the worst-case demand of AVR tasks. Experimental results show that our approach is at least 10 times faster, with an average runtime improvement of 146 times for randomly generated task sets when compared to the state-of-the-art technique.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IPs) are mathematical models that can provide an optimal solution to a variety of different problems. They have been used to reduce costs and optimize organizations. Additionally, IPs are NP-complete resulting in many IPs that cannot be solved. Cutting planes or valid inequalities have been used to decrease the time required to solve IPs. Lifting is a technique that strengthens existing valid inequalities. Lifting inequalities can result in facet defining inequalities, which are the theoretically strongest valid inequalities. Because of these properties, lifting procedures are used in software to reduce the time required to solve an IP. The thesis introduces a new algorithm for exact synchronized simultaneous uplifting over an arbitrary initial inequality for knapsack problems. Synchronized Simultaneous Lifting (SSL) is a pseudopolynomial time algorithm requiring O(nb+n[superscript]3) effort to solve. It exactly uplifts two sets simultaneously into an initial arbitrary valid inequality and creates multiple inequalities of a particular form. This previously undiscovered class of inequalities generated by SSL can be facet defining. A small computational study shows that SSL is quick to execute, requiring on average less than a quarter of a second. Additionally, applying SSL inequalities to a knapsack problem enabled commercial software to solve problems that it could not solve without them.

Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick's Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick's Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick's Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research.

In this study cost-benefits problems concerning the knapsack problem of limited resources is studied and how this relates to an attacker perspective when choosing defense strategies. This is accomplished by adopting a cost-benefit method and merging it with game theory. The cost-benefit method chosen for this study is the Analytical Hierarchy Process and from the field of game theory the Bayesian Nash Equilibrium is used. The Analytical Hierarchy Process allows the user to determine internally comparable weights between elements, and to bring in a security dimension to the Analytical Hierarchy Process a sub category consisting of confidentiality, integrity and availability is used. To determine the attacker strategy and, in effect, determine the best defense strategy the Bayesian Nash Equilibrium is used.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.