Elektronika 2009-11.pdf - Instytut SystemÃ³w Elektronicznych

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In the average individual classification probability function the following relation are found: • average individual classification probability function presents classification probability to linguistic quantifiers one of the average person of the group (a person typical of the group), • in many cases quantifier supports overlap, • the beginning and the end of the average quantifier supports is a mean value of the beginnings and ends values the appropriate individual quantifiers, • the height of the average linguistic quantifiers usually increase from medium quantifiers to border quantifiers, • because values of quantifiers need to add up to one, they are strongly dependent on the quantifier supports, • surface area of the average individual quantifiers equals the average surface area of individual quantifiers, a) Fig. 2. The quantifiers of classification function F1 and F2 in the form of step function and calculated on the basis of F1 and F2 average individual quantifier F_aver of classification function Rys. 2. Kwantyfikatory funkcji klasyfikacji F1 i F2 w formie funkcji schodkowej oraz obliczony na podstawie F1 i F2 przeciętny indywidualny kwantyfikator F_aver funkcji klasyfikacji b) a) Fig. 4. The examples of the individual classification probability function (dotted line) and the average individual classification probability function (continuous line) received from the individual models, the functions of 5-element dictionaries Rys. 4. Przykłady indywidualnych funkcji prawdopodobieństwa klasyfikacji (linia przerywana) i przeciętna indywidualna funkcja prawdopodobieństwa klasyfikacji (linia ciągła) otrzymana z modeli indywidualnych, funkcje o słowniku 5-elementowym b) Fig. 3. The examples of the individual classification probability function (dotted line) and the average individual classification probability function (continuous line) received from the individual models, the functions of 3-element dictionaries Rys. 3. Przykłady indywidualnych funkcji prawdopodobieństwa klasyfikacji (linia przerywana) i przeciętna indywidualna funkcja prawdopodobieństwa klasyfikacji (linia ciągła) otrzymana z modeli indywidualnych, funkcje o słowniku 3-elementowym Fig. 5. The examples of the 40 individual classification probability function, (dotted line) and the average individual classification probability function (thick line) received on the basis of the experiment Rys. 5. Przykłady 40 indywidualnych funkcji prawdopodobieństwa klasyfikacji (linia przerywana) i przeciętna indywidualna funkcja prawdopodobieństwa klasyfikacji (linia ciągła) otrzymana na podstawie eksperymentu 14 ELEKTRONIKA 11/<strong>2009</strong>
• in some cases in the methods of determining the average individual classification function it is necessary to normalize the average individual quantifiers, so that they ad up to one, • it is crucial that the identification of the probability identification function is made with the whole dictionaries because the shapes and supports of the received models of linguistics concept depend on quantity of elements in dictionaries. The question arises, when do we apply the group definition of linguistic quantifiers and when average individual definitions? It depend on the examined problem. If the problem has to be based on the understanding of linguistic concept by the whole group we apply the group model of the concept. If the solution is to be based on the average individual understanding of the concept we need to apply the average individual model. The person solving the problem decides on the model to be used. Solving the problem using the group model we receive the safer but more uncertain result. When we solve the problem using the average individual model the obtained result is less uncertain. The result will be correct in most cases however there is a limited risk that the result has a large error. The problem is typical of the probability theory and is similar to the problem of calculating the results of final values y on the bases of whole distributions of probable input data of the variable x i or only on the basis of the center of gravity of these distributions. Conclusion In the article the method identifying the average individual classification probability function is presented as well as the relations that occur in the average individual function. The received on the basis of the presented method functions are vital in probabilistic reasoning and in the decision making theory because the received probabilistic model of linguistic concept describes perception of an average person in a group (population). We need to state that it is not the average model of the group of people. According to the author’s knowledge the concept of the individual classification probability function is new in the world scientific literature. References [1] Zadeh L. A.: Fuzzy Sets, Information and Control 8 (1965) 338- 353. [2] Hisdal E.: Are Grades of Membership Probabilities. Fuzzy Sets and Systems 25 (1988), 325-348, North-Holland. [3] Cheeseman P.: An inquiry into computer understanding. Comput. Intell. 1988, vol. 4, 58-66. [4] Kandel A., Martins A., Pacheco R.: On the Very Real Distinction Between Fuzzy and Statistical Methods. Technometrics, August 1995, vol. 37, no 3, 276-281. [5] Piegat A., A New Definition of the Fuzzy Set, Int. J. Appl. Math. Comput. Sci. 2005, vol. 15. no 1, 125-140. Evolutionary sets of cooperating ship trajectories: open waters and restricted waters (Ewolucyjne zbiory bezpiecznych trajektorii statków dla wód otwartych i obszarów ograniczonych) dr inż. RAFAŁ SZŁAPCZYŃSKI Gdańsk University of Technology, Faculty of Ocean Engineering and Ship Technology The paper contains a description of an already developed, revised version of the method, which has been first presented by the author in its early version in [1]. The method combines some of the assumptions of game theory approach to this problem [2] with evolutionary programming approach [3] and aims to find an optimal set of cooperating trajectories of all ships involved in an encounter situation, by means of evolutionary algorithms. Depending, whether we are focusing on restricted water regions or open waters, specific methods of generating the initial population, various ship domains and different sets of specialised operators have been used throughout the research presented here. Optimisation problem It is assumed that we are given the following data: stationary constraints (obstacles and other constraints modelled as polygons), positions, courses and speeds of all ships involved, ship domains and times necessary for accepting and executing the proposed manoeuvres. A ship domain can be computed, based on the ship’s length, its motion parameters and the type of a water region. Ship domain has been first defined as a two-dimensional area surrounding a ship, which other ships must avoid. Since a domain shape depends on the type of a water region, the author has decided to use a domain model by Davis [4] for open waters and a domain model by Coldwell [5] for restricted waters. Knowing all the parameters values, the goal is to find a set of trajectories, which minimizes the average way loss spent on manoeuvring, while fulfilling the following conditions: • none of the stationary constraints are violated (restricted waters only), • none of the ship domains are violated, • the minimal acceptable course alteration is not lesser than 15 degrees, • the maximal acceptable course alteration is not larger than 60 degrees, • speed alterations are not to be applied unless necessary, • ship manoeuvres only when she is obliged to, • manoeuvres to starboard are favoured over manoeuvres to port board. As for the first two conditions, all obstacles have to be avoided and the domains should not be violated by definition. All other conditions are imposed by COLREGS [6] or by the economics. ELEKTRONIKA 11/<strong>2009</strong> 15
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Elektronika 2009-11.pdf - Instytut SystemÃ³w Elektronicznych

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