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Scope of Various Random Number Generators in Ant System Approach for Tsp

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Scope of Various Random Number Generators in Ant System Approach for Tsp AC 2007-458: SCOPE OF VARIOUS RANDOM NUMBER GENERATORS IN ANT SYSTEM APPROACH FOR TSP S.K. Sen, Florida Institute of Technology Syamal K Sen ([email protected]) is currently a professor in the Dept. of Mathematical Sciences, Florida Institute of Technology (FIT), Melbourne, Florida. He did his Ph.D. (Engg.) in Computational Science from the prestigious Indian Institute of Science (IISc), Bangalore, India in 1973 and then continued as a faculty of this institute for 33 years. He was a professor of Supercomputer Computer Education and Research Centre of IISc during 1996-2004 before joining FIT in January 2004. He held a Fulbright Fellowship for senior teachers in 1991 and worked in FIT. He also held faculty positions, on leave from IISc, in several universities around the globe including University of Mauritius (Professor, Maths., 1997-98), Mauritius, Florida Institute of Technology (Visiting Professor, Math. Sciences, 1995-96), Al-Fateh University (Associate Professor, Computer Engg, 1981-83.), Tripoli, Libya, University of the West Indies (Lecturer, Maths., 1975-76), Barbados.. He has published over 130 research articles in refereed international journals such as Nonlinear World, Appl. Maths. and Computation, J. of Math. Analysis and Application, Simulation, Int. J. of Computer Maths., Int. J Systems Sci., IEEE Trans. Computers, Internl. J. Control, Internat. J. Math. & Math. Sci., Matrix & Tensor Qrtly, Acta Applicande Mathematicae, J. Computational and Applied Mathematics, Advances in Modelling and Simulation, Int. J. Engineering Simulation, Neural, Parallel and Scientific Computations, Nonlinear Analysis, Computers and Mathematics with Applications, Mathematical and Computer Modelling, Int. J. Innovative Computing, Information and Control, J. Computational Methods in Sciences and Engineering, and Computers & Mathematics with Applications. Besides, he has coauthored seven books including the most recent one entitled “Computational Error and Complexity in Science and Engineering (with V. Lakshmikantham), Elsevier, Amsterdam, 2005. He had also authored several book chapters. All his research and book publications are in several areas mainly in computational science. He has been teaching several courses in areas such as stochastic and deterministic operations research, applied statistical analysis, and computational mathematics since late sixties. Further, he has been a member of the editorial board of international journals such as Computer Science and Informatics (India), and Neural, Parallel and Scientific Computations (USA). He has also been cited in Marquis Whos Who (Sep 2005). Gholam Ali Shaykhian, NASA Gholam “Ali” Shaykhian Gholam Ali Shaykhian ([email protected]) is a software engineer with the National Aeronautics and Space Administration (NASA), Kennedy Space Center (KSC), Engineering Directorate. He is a National Administrator Fellowship Program (NAFP) fellow and served his fellowships at Bethune Cookman College in Daytona Beach, Florida. Ali is currently pursing a Ph.D. in Operations Research at Florida Institute of Technology. He has received a Master of Science (M.S.) degree in Computer Systems from University of Central Florida in 1985 and a second M.S. degree in Operations Research from the same university in 1997. His research interests include object-oriented methodologies, design patterns, software safety, and genetic and optimization algorithms. He teaches graduate courses in Computer Information Systems at Florida Institute of Technology’s University College. Mr. Shaykhian is a senior member of the Institute of Electrical and Electronics Engineering (IEEE) and is the Vice-Chair (2005-2007), Education Chair (2003-2007) and Awards Chair of the IEEE Canaveral section. He is a professional member of the American Society for Engineering Education (ASEE), serving as the Program Chair and Web Master for the Minorities in Page 12.1256.1 Page Engineering Division of ASEE (2006-2008). He was an assistant professor and coordinator of the Information Systems program at the University of Central Florida prior to his full time appointment at NASA KSC. © American Society for Engineering Education, 2007 Scope of various random number generators in ant system approach for TSP Abstract Random numbers are essential ingredients to all stochastic methods including probabilistic heuristic ones. There are several random number generators. Given a method to solve a traveling salesman problem (TSP), which generator should one use to obtain the best result in terms of quality/accuracy and cost/computational complexity? The article attempts to seek an answer to this question when the specified method is the ant algorithm. Nature and characteristics of random numbers, their generators, and TSPs are specially stressed for a better appreciation. 1. Introduction A given number cannot be just termed random unless we check/test the sequence which it belongs to. This is unlike the transcendental number r … .3 14159265358 or the algebraic number l ? 1( - 2/)5 … 1.61803398874989 (golden ratio) or the Hilbert number 2 2 … 2.66514414269023. The word random implies that the predictability (probability of correct prediction) is low and never 100%. As long as there is a finite number of outcomes, the predictability is never zero. In the case of tossing a fair coin, the predictability is 50% while that 2 of rolling a six-faced fair die, it is 16 %. However, an approximate global prediction with high 3 predictability is possible so far as the character of the sequence is concerned. After a large number, say 500, of tosses of a coin, we may say that approximately 250 heads will be obtained. This number could be 245 or 256. By a statistical test, say e 2 -test, we may conclude that the coin is fair (unbiased) statistically/probabilistically. These heads and tails or, equivalently, 1s and 0s constitute a uniformly distributed random sequence. If we consider a class of 60 students and note their heights or weights one after the other, it is not possible to predict what the height/weight of the next student would be. These 60 heights or 60 weights are statistically random. But these random heights/weights are not uniformly distributed. We have invariably discovered that such heights/weights are normally distributed. Thus we have random sequences which are exponentially distributed or log normally distributed or distributed in any of the infinite possible ways. Thus the uniform distribution of random numbers is one of infinite possible distribution. Once we have a procedure to produce a uniformly distributed random sequence, we can easily produce a random sequence having any other distribution from this sequence. One may distinguish between global randomness and local randomness although most randomness concepts are global. In a near-infinite random sequence, there could be long sequences (stretches) of the same digit(s), say, 1, the overall sequence might be random though. 12.1256.2 Page Long sequences of the same digits, even though generated by a random process would reduce the local randomness of a sample. That is, a sample could only be globally random for sequences of, say, 100,000 digits while it might not appear at all random when a sequence of less than 500 digits is considered. Usually in a statistical environment, the numeric sequence need to be a large one (30 or more entries) before we could talk about whether the sequence is random or not. For example, in a tossing of a coin denoting a head by 1 and a tail by 0, if we get 15 0’s successively, can we say that the coin is biased statistically? The answer is no. However, if we get say 25 0’s out of 30 first throws then we have a statistical reason by, say, e 2 -test, to believe that the coin is biased assuming that the tossing is done totally in an unbiased way. When we talk about the human population of a country or the red blood cell count in a human body, we do not normally use the unit as one human being or one red blood cell. If we do so, then too large numbers need to be written or need to be used in a computation. These may not be readily comprehensible by a human. Also, these large numbers occupy more space. So, a more convenient unit, say, million is usually employed. Thus, by an appropriate choice of the unit in a given environment, a sample size of 30 or larger but comparable could be considered, as a rule of thumb, large in statistics. Moreover, these sizes are easily comprehensible. Statistical randomness does not necessarily mean true randomness [1, 2]. Strictly speaking, any number which is the outcome of a process / artificial or natural / can be at best called a pseudo-random number (pseudo implying false) and can never be a true random number. In fact, it is strictly impossible to generate a true random number since if we have to generate by any physical means or any artificial means such as by a computer, we must pass through a natural process or an artificial process. Thus the output of such a process is related to the input through the process and cannot be random. Even if we produce a number through a combination of a natural process and an artificial process, the number cannot again be a true random number. Strictly speaking, if any natural process, which we may term as chaotic or possibly random, is exactly modeled, then the predictability is and has to be always 100%. Such an exact modeling, however, is impossible by human beings/computers. Nature only does exact modeling which may be very much beyond human sphere of capturing the exactness of all parameters known as well as unknown to us. The pseudo-random numbers are good enough to be used in ant system approach, genetic algorithms, Monte Carlo methods, any evolutionary process or any other randomized algorithm to solve many vitally important computational problems, specifically NP-hard [1, 3] problems, such as the TSP, in polynomial-time. It is, however, possible to compare random number generators statistically based on the large sample theory with respect to a given type of problems and rank them.
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