Online Assignment- SIMULATION
- 1. 1 ON-LINE ASSIGNMENT SIMULATION Submitted by, Supriya.M Reg No: 13971020 Mathematics optional K U C T E, Kumarapuram
- 2. 2 INTRODUCTION A broad collection of methods used to study and analyze the behavior and performance of actual or theoretical systems. Simulation studies are performed, not on the real-world system, but on a (usually computer-based) model of the system created for the purpose of studying certain system dynamics and characteristics. The purpose of any model is to enable its users to draw conclusions about the real system by studying and analyzing the model. The major reasons for developing a model, as opposed to analyzing the real system, include economics, unavailability of a “real” system, and the goal of achieving a deeper understanding of the relationships between the elements of the system. Many topics in mathematics that have immediate utility value can be best introduced using the technique of simulation that is enacting a real situation in the class. Topics that have commercial concern is an example. Simulation can be used in task or situational training areas in order to allow humans to anticipate certain situations and be able to react properly; decision-making environments to test and select alternatives based on some criteria; scientific research contexts to analyze and interpret data; and understanding and behavior prediction of natural systems, such as in studies of stellar evolution or atmospheric conditions. The word “system” refers to a set of elements (objects) interconnected so as to aid in driving toward a desired goal. This definition has two connotations: First, a system is made of parts (elements) that have relationships between them (or processes that link them together). These relationships or processes can range from relatively simple to extremely complex. One of the necessary requirements for creating a “valid” model of a system is to capture, in as much detail as possible, the nature of these interrelationships. Second, a system constantly seeks to be improved. Feedback (output) from the system must be used to measure the performance of the system against its desired goal. Both of these elements are important in simulation. With simulation a decision maker can try out new designs, layouts, software programs, and systems before committing resources to their acquisition or implementation; test why certain
- 3. 3 phenomena occur in the operations of the system under consideration; compress and expand time; gain insight about which variables are most important to performance and how these variables interact; identify bottlenecks in material, information, and product flow; better understand how the system really operates (as opposed to how everyone thinks it operates); and compare alternatives and reduce the risks of decisions. Systems can be classified in three major ways. They may be deterministic or stochastic (depending on the types of elements that exist in the system), discrete-event or continuous (depending on the nature of time and how the system state changes in relation to time), and static or dynamic (depending on whether or not the system changes over time at all). This categorization affects the type of modeling that is done and the types of simulation tools that are used. Models, like the systems they represent, can be static or dynamic, discrete or continuous, and deterministic or stochastic. Simulation models are composed of mathematical and logical relations that are analyzed by numerical methods rather than analytical methods. Numerical methods employ computational procedures to run the model and generate an artificial history of the system. Observations from the model runs are collected, analyzed, and used to estimate the true system performance measures. See Model theory. There is no single prescribed methodology in which simulation studies are conducted. Most simulation studies proceed around four major areas: formulating the problem, developing the model, running the model, and analyzing the output. Statistical inference methods allow the comparison of various competing system designs or alternatives. For example, estimation and hypothesis testing make it possible to discuss the outputs of the simulation and compare the system metrics. Many of the applications of simulation are in the area of manufacturing and material handling systems. Simulation is taught in many engineering and business curricula with the focus of the applications also being on manufacturing systems. The characteristics of these systems, such as physical layout, labor and resource utilization, equipment usage, products, and supplies, are extremely amenable to simulation modeling methods. See Computer-integrated manufacturing, Flexible manufacturing system.
- 4. 4 Simulation is the imitation of the operation of a real-world process or system over time. The act of simulating something first requires that a model be developed; this model represents the key characteristics or behaviors/functions of the selected physical or abstract system or process. The model represents the system itself, whereas the simulation represents the operation of the system over time. Simulation is used in many contexts, such as simulation of technology for performance optimization, safety engineering, testing, training, education, and video games. Often, computer experiments are used to study simulation models. Simulation is also used with scientific modelling of natural systems or human systems to gain insight into their functioning. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it may simply not exist. Key issues in simulation include acquisition of valid source information about the relevant selection of key characteristics and behaviours, the use of simplifying approximations and assumptions within the simulation, and fidelity and validity of the simulation outcomes. The process of imitating a real phenomenon with a set of mathematical formulas. Advanced computer programs can simulate weather conditions, chemical reactions, atomic reactions, even biological processes. In theory, any phenomena that can be reduced to mathematical data and equations can be simulated on a computer. In practice, however, simulation is extremely difficult because most natural phenomena are subject to an almost infinite number of influences. One of the tricks to developing useful simulations, therefore, is to determine which the most important factors. The functioning of a co-operative be society or bank cited as examples. First the students may be taken to such institutions to observe the nature and techniques of the various activities going on there. Notes may be taken. In order to reinforce and to make the activity more familiar the working of such institutions may be enacted in the class. The simulation should be carefully arranged so as to make the insight as
- 5. 5 meaningful as possible. For example, there is a school co-operative society. The working of the society may be observes and the salient features of how it was organized and what the activities taken up are noted. Then imagine that the learners are planning to start a class co-operative society. The steps such as selling of shares to pool the capital required, election of various office bearers, nature of transaction involved the style of keeping records concerning the various aspects including the Account book, the technique of preparing a balance sheet, calculation and dispersal of dividends to the share holders, etc. may be simulated. This will not only help in realizing the utility value of mathematics, but also will give realistic insights into the related commercial mathematics. Further the roles played in simulation will create interest among the learners.
- 6. 6 APPLICATION Natural numbers The natural numbers are 1, 2, 3, 4, 5,……… in the set of natural numbers, which in math- Numbers ematics is referred to as N. Additions can be executed without limit as well as multiplications, which are to be understood as multiple additions: 3. 4 = 4 + 4 + 4. In using number notation, one differentiates between ordinal numbers (the third – in an imagined sequence) and cardinal numbers (three pieces). Toddlers of 3–4 years often know the ordinal numbers up to 10 and they can also execute simple additions via counting. The more abstract notion of the cardinal number children mostly understand only when they start school; in addition, even for the adult, the number of units that can Simulation. Spontaneous grasping of the number of elements in a set (cardinal numbers) A random number generator produces red points, whose number lies between 1 and the maximum number in the number field (in the figure the maximum number is 5, 5 are shown). The sets change with a frequency that can be adjusted with the slider from 1 to 10 per second be grasped at a glance is quite limited (to around 5–7, which is also what intelligent animals are capable of); for fast calculations with cardinal numbers, the relationship is memorized or simplified in our thoughts (5+7 = 5+5+2 = 10+2 = 12). If one realizes this fact, one gains a deeper understanding of the difficulty that children have with learning the elementary rules of arithmetic. Simply assuming the memorized routines, which are present in an educated adult, leads to severely underestimating
- 7. 7 the natural hurdles of understanding that the children have to overcome when they learn arithmetic. The simulation in Figure visualizes the sharp threshold that nature imposes for spontaneously grasping the number of elements of a set. In this simulation, points are shown in a random arrangement that can be spontaneously grasped as a group. The number changes with a frequency that can be specified between 1 and a maximum number. Even numbers are a multiple of the number 2; a prime number cannot be decomposed into a product of natural numbers, excluding 1. The lower limit of the natural numbers is the unity 1. This number had a close to mystical meaning for number theoreticians of antiquity, as the symbol for the unity of the computable and the cosmos. It also has a special meaning in modern arithmetic as that number which, when multiplied with another number, produces the same number again. There is, however, no upper limit of the natural numbers: for each number there exists an even larger number. As a token for this boundlessness, the notion of infinity developed, with the symbol∞, which does not represent a number in the usual sense. Already, the preplatonic natural philosophers (Plato himself lived from 427–347 BC) worked on the question of the infinite divisibility of matter (If one divides a sand grain infinitely often, is it then still sand?) and time (if one adds to a given time interval infinitely often half of itself, will that take infinitely long?) Zenon of Elea (490–430 BC) showed in his astute paradoxes, Achilles and the tortoise and the arrows,11 that the ideas of movement and number theory at the time were in contradiction to each other. Subtraction is the logical inversion of addition: for natural numbers it is only permissible if the number to subtract is smaller than the original number by at least 1. Division is the natural inversion of multiplication. For natural numbers it is permissible if the dividend is an integer multiple of the divisor – 6 : 2=3.
- 8. 8 CONCLUSION Simulation - Attempting to predict aspects of the behaviour of some system by creating an approximate (mathematical) model of it. This can be done by physical modelling, by writing a special-purpose computer program or using a more general simulation package, probably still aimed at a particular kind of simulation (e.g. structural engineering, fluid flow). *************************************** REFERENCES Mathematics in Education- Dr .K Sivarajan Net Reference-Wikipedia Teaching of Mathematics –Anice James
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