Застосування прямокутних стохастичних матриць до задачі оцінювання та ранжування альтернатив
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Date
2021-12-10
Authors
Олецький, Олексій
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Abstract
Досліджено можливість застосування моделі "стан–імовірність вибору" на основі прямокутних стохастичних матриць до задачі ранжування альтернатив агентом або колективом агентів. Показано, що отримані раніше результати для задачі вибору можуть бути з деякими змінами перенесені на задачу ранжування. Висвітлено також, як на основі розподілів мір важливості для задачі оцінювання та ранжування альтернатив можна отримати аналогічну модель для задачі вибору.
Для формування матриці станів, рядки якої відповідають розподілам мір важливості альтернатив, використано метод попарних порівнянь у рамках методу аналізу ієрархій Сааті. Запропоновано розглядати транзитивні шкали з параметром, який впливає на розкид мір важливості. Введено також параметр, який задає міру впевненості агента.
Наведено результати численних експериментів.
The paper investigates the issue related to a possible generalization of the "state-probability of choice" model so that the generalized model could be applied to the problem of ranking alternatives, either individual or by a group of agents. It is shown that the results obtained before for the problem of multi-agent choice and decision making by majority of votes can be easily transferred to the problem of multi-agent alternatives ranking. On the basis of distributions of importance values for the problem of ranking alternatives, we can move on to similar models for the choice and voting with the help of well-known exponential normalization of rows. So we regard two types of matrices, both of which belonging to the sort of matrices named balanced rectangular stochastic matrices. For such matrices, sums of elements in each row equal 1, and all columns have equal sums of elements. Both types are involved in a two-level procedure regarded in this paper. Firstly a matrix representing all possible distributions of importance among alternatives should be formed, and secondly a "state-probability of choice" matrix should be obtained on its base. For forming a matrix of states, which belongs and the rows of which correspond to possible distributions of importance, applying pairwise comparisons and the Analytic Hierarchy Method is suggested. Parameterized transitive scales with the parameter affecting the spread of importance between the best and the worst alternatives are regarded. For further getting the matrices of choice probabilities, another parameter which reflects the degree of the agent’s decisiveness is also introduced. The role of both parameters is discussed and illustrated with examples in the paper. The results are reported regarding some numerical experiments which illustrate getting distributions of importance on the basis of the Analytic Hierarchy Process and which are connected to gaining the situation of dynamic equilibrium of alternatives, i.e. the situation when alternatives are considered as those of equal value.
The paper investigates the issue related to a possible generalization of the "state-probability of choice" model so that the generalized model could be applied to the problem of ranking alternatives, either individual or by a group of agents. It is shown that the results obtained before for the problem of multi-agent choice and decision making by majority of votes can be easily transferred to the problem of multi-agent alternatives ranking. On the basis of distributions of importance values for the problem of ranking alternatives, we can move on to similar models for the choice and voting with the help of well-known exponential normalization of rows. So we regard two types of matrices, both of which belonging to the sort of matrices named balanced rectangular stochastic matrices. For such matrices, sums of elements in each row equal 1, and all columns have equal sums of elements. Both types are involved in a two-level procedure regarded in this paper. Firstly a matrix representing all possible distributions of importance among alternatives should be formed, and secondly a "state-probability of choice" matrix should be obtained on its base. For forming a matrix of states, which belongs and the rows of which correspond to possible distributions of importance, applying pairwise comparisons and the Analytic Hierarchy Method is suggested. Parameterized transitive scales with the parameter affecting the spread of importance between the best and the worst alternatives are regarded. For further getting the matrices of choice probabilities, another parameter which reflects the degree of the agent’s decisiveness is also introduced. The role of both parameters is discussed and illustrated with examples in the paper. The results are reported regarding some numerical experiments which illustrate getting distributions of importance on the basis of the Analytic Hierarchy Process and which are connected to gaining the situation of dynamic equilibrium of alternatives, i.e. the situation when alternatives are considered as those of equal value.
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Keywords
ранжування альтернатив, стохастичні прямокутні матриці, попарні порівняння, метод аналізу ієрархій, агенти, стаття, ranking of alternatives, rectangular stochastic matrices, pairwise comparisons, Analytic Hierarchy Process, agents, article
Citation
Олецький О. В. Застосування прямокутних стохастичних матриць до задачі оцінювання та ранжування альтернатив / Олецький О. В. // Наукові записки НаУКМА. Комп'ютерні науки. - 2021. - Т. 4. - С. 4-9. - https://doi.org/10.18523/2617-3808.2021.4.4-9