For more than 35 years now, George B. Dantzig's Simplex-Method has
been the most efficient mathematical tool for solving linear
programming problems. It is proba bly that mathematical algorithm
for which the most computation time on computers is spent. This
fact explains the great interest of experts and of the public to
understand the method and its efficiency. But there are linear
programming problems which will not be solved by a given variant of
the Simplex-Method in an acceptable time. The discrepancy between
this (negative) theoretical result and the good practical behaviour
of the method has caused a great fascination for many years. While
the "worst-case analysis" of some variants of the method shows that
this is not a "good" algorithm in the usual sense of complexity
theory, it seems to be useful to apply other criteria for a
judgement concerning the quality of the algorithm. One of these
criteria is the average computation time, which amounts to an anal
ysis of the average number of elementary arithmetic computations
and of the number of pivot steps. A rigid analysis of the average
behaviour may be very helpful for the decision which algorithm and
which variant shall be used in practical applications. The subject
and purpose of this book is to explain the great efficiency in prac
tice by assuming certain distributions on the "real-world"
-problems. Other stochastic models are realistic as well and so
this analysis should be considered as one of many possibilities."
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