System Analysis and Design
Discreet Math In Analysis Work
Topic
Uses
Set theory
Combinatorics
Graph theory
Probability
Algebra
Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.
Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for countingpermutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties.Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.
Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[17] Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g. knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs, however for the most part research in graph theory falls within the domain of discrete mathematics.
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.
Topic
Uses
Set theory
Combinatorics
Graph theory
Probability
Algebra
For organizations operating with number of products/services and number of suppliers, to select the right supplier meeting all their requirements will be a challenging job. Such organizations need a good decision support system to evaluate the suppliers effectively. Several decision support systems have been reported to deal with complex selection process to decide the right supplier. Many mathematical models have also been developed. Help of Boolean logic and Boolean algebra is taken to assign binary digit values to the selection criteria and generate mathematical equations that correlate the inputs to the output at each stage of decision making.
Most areas of engineering, science, and management use important tools based on probabilistic methods. The common thread of the entire spectrum of these tools is aiding in decision making under uncertainty: the choice of an interpretation of reality or the choice of a course of action. Although the importance of dealing with uncertainty in decision making is widely acknowledged, dissemination of probabilistic and decision-theoretic methods in Artificial Intelligence has been surprisingly slow. Opponents of probability theory have pointed out three major obstacles to applying it in computerized decision aids: (1) the counterintuitiveness of probabilistic inference, which makes it hard for system builders, experts, and users to translate knowledge into probabilistic form, create knowledge bases, and to interpret results; (2) the quantitative character of probability theory, which implies collection or assessment of vast quantities of numbers and, since these are not always readily available, raises questions about their quality; and (3) closely related to its quantitative character, the computational complexity of probabilistic inference.
The operational monitoring and control involves detection of pipe leakages. The training data for the GFMMNN is obtained through simulation of leakages in a water network for a given operational period. The training data generation scheme includes a simulator algorithm based on loop corrective flows equations, a Least Squares (LS) loop flows state estimator and a Confidence Limit Analysis (CLA) algorithm for uncertainty quantification entitled Error Maximization (EM) algorithm. These three numerical algorithms for modeling and simulation of water networks are based on loop corrective flows equations and graph theory. It is shown that the detection of leakages based on the training and testing of the GFMMNN with patterns of variation of nodal consumptions with or without confidence limits is computational superior to the training based on patterns of nodal heads and pipe flows state estimates with or without confidence limits and to the original recognition system trained with patterns of data obtained with the LS nodal heads state estimator
A container terminal is a facility where cargo containers are transshipped between different transport vehicles. We focus our attention on the transshipment between vessels and land vehicles, in which case the terminal is described as a maritime container terminal. In these container terminals, many combinatorial related problems appear and the solution of one of the problems may affect to the solution of other related problems. For instance, the berth allocation problem can affect to the crane assignment problem and both could also affect to the Container Stacking Problem. Thus, terminal operators normally demand all containers to be loaded into an incoming vessel should be ready and easily accessible in the yard before vessel’s arrival. Similarly, customers (i.e., vessel owners) expect prompt berthing of their vessels upon arrival. However the efficiency of the loading/unloading tasks of containers in a vessel depends on the number of assigned cranes and the efficiency of the container yard logistic. In this paper, we present a decision support system to guide the operators in the development of these typical tasks. Due to some of these problems are combinatorial, some analytical formulas are presented to estimate the behavior of the container terminal.
Rough set theory as a new mathematical tool is used to deal with the imprecise, incomplete and uncertain knowledge for pattern recognition. In current information explosion era, the powerful functions of agricultural decision support system(ADSS) on assistant decision-making and rough set on information processing is becoming increasingly obviously, and have called people's attention. According to the characteristics of the area of agriculture data, this thesis proposed the model of DSS for agriculture application based on rough set theory.