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Extra info for Periodic Optimization: Volume I: Course Held at the Department of Automation and Information, June 1972

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1) call for minimization over two alternatives. It is not clear that the first of these alternatives sired path does not pass through node native ( c~<;:> + c~)) (c<~:>) is that the da~ m and the second altar- is that it does. i- • We estimate the complexity of the Floyd-Warshall algorithm as follows. It is necessary to compute ~ • 1,2 ••.. 2, ... ,n and for c (m) .. ~. for m • 2,3, ... ,n+1. 1), requires one addition and one comparison. Thus, exactly n 3 additions and n 3 comparisons are required overall.

Edmonds, ''Path, Trees, and Flowers tt, Can. ]. , 17 (1965) 449-467. [6] M. Florian and P. Robert, "A Direct Search Method to Locate Negative Cycles in a Graph", Management Science, 17 (1971) 307-310. W. Floyd, nAlgorithm 97, Shortest Path", Comm. of the ACM; 5 (1962) 345. [s] L. R. , P-923 (Aug. 1965). L. Fox, 1 ~inding Minimal Cost-Time Ratio Circuits", Operations Research, 17 (1969) 546-551. L. Lawler, "Optimal Cycles in Doubly Weighted Directed Linear Graphs r;, Theory of Graphs, International Symposium, Dunod, Paris, and Gordon and Breach, New York (1966) 209-213.

5) As it is well known, such a problem can be described by means of a finite and oriented graph: every node represents a state and every arc ( ~· . ~4 i! 0 and q (k) constraint values \)' .. are associated to every arc ( ~ , 1 ) ~t from state ~ and the problem consists ;n the determination of the cycle with minimum cost per unit time. r) and '(:; S be the (r) = ~ ci.. "' , the loss rate (cost to time ratio) c (r) = is associated to every element ~(r) 't(l"") 1"" of S . 5). 5) are those in which only some of the constraints are p1·esent.

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