Medium distances of probability-fuzzy points and an application to linear programming.

*(English)*Zbl 0715.90074According 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!

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 |

##### 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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