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Medium distances of probability-fuzzy points and an application to linear programming. (English) Zbl 0715.90074
According to the author’s concept a probability-fuzzy point a in X is characterized by a probability measure $$\mu_ a$$ on (X,A), where A is a $$\sigma$$-algebra of subsets in the space X. In the space X, provided with a metric $$\rho$$ (x,y), the medium distance between two probability-fuzzy points a and b is defined as the mean value of $$\rho$$ with respect to the measure being the product of the probability measures $$\mu_ a$$ and $$\mu_ b$$. Similarly, the medium distance from a probability-fuzzy point a to a crisp set $$B\subset X$$ is defined as the expected valued of $$\rho$$ (x,B) with respect to $$\mu_ a$$, $$\rho (x,B)=\inf \{\rho (x,b)|$$ $$b\in B\}$$. Next, the author investigates a specific case of probability- fuzzy point $$p_ a$$ defined by the uniform probability distribution on the set $S_ a=\{x=(x_ 1,...,x_ n)\in R^ n| \quad x_ i\leq c\quad \forall i\text{ and } \sum^{n}_{i=1}x_ i=a\},$ where c is a fixed positive constant and a is a parameter satisfying $$| a| \leq cn$$. Then, the properties of medium distance of $$p_ a$$ from the set $$R^ n_+$$ are analyzed. Using this distance the author derives a new rule, called the Maximum-Sum Rule, of choosing the basic variable removed in the simplex procedure iteration, being of importance in the case of a degenerate linear programming problem. The author presents two small numerical examples of linear programs in which the application of the maximum-sum rule decreases the number of simplex iterations if compared to the smallest index and lexicographical rules. In this moment one must notice that these examples cannot be treated as a proof of the assumption that the maximum-sum rule is generally better than the latter two rules.
And one more remark. It is not clear why the term “probability-fuzzy point” is used in the paper. The notion “probability point” is more natural. The introduced definition of probability-fuzzy point has nothing in common with fuzzy set theory!
Reviewer: S.Chanas

##### MSC:
 90C05 Linear programming
##### Keywords:
probability-fuzzy point; Maximum-Sum Rule
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##### References:
 [1] A. N. Kolmogorov, S. V. Fomin: Elements of Function Theory and Functional Analysis. Nauka, Moscow 1972. In Russian. · Zbl 0235.46001 [2] M. Mareš: How to handle fuzzy-quantities. Kybernetika 13 (1977), 1, 23-40. [3] M. Mareš: Fuzzy-quantities with real and integer values. Kybernetika 13 (1977), 1, 41-56. [4] S. I. Gass: Linear Programming – Methods and Applications. McGraw-Hill, New York 1969.
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