New PDF release: Periodic Optimization: Volume I: Course Held at the

By Angelo Marzollo

ISBN-10: 3211811354

ISBN-13: 9783211811351

ISBN-10: 3709126525

ISBN-13: 9783709126523

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

Example text

Moreover, allowing the t~~·s , not to be equal to. one, the multigraph case can be reduced to the simpler linear case. In fact, if r arcs link node i. to node ~ to substitute every arc, but one, from i. to , then it suffices ~ with a couple of arcs passing through an artificial node, with costs, times and constraint values which have half the value of the original ones. ~'s propo~ to be integers. This is not a se- vere restriction, since the problem with rational lastings can be easily reformulated in terms of integer lastings.

2 •... L. 1) imply implicit fun£ tiona! relationships between the variables Ford 11 [8] c ~ . 2) { (m+i) c. ~ Because c~~ - • { m~n •0 , (m) ck + cki. (m+i). ~ L ~ ~ ~ We can understand the equations (2. 2), and demonstrate convergence to c~. by applying the following interpretation: c {m) ~ = the cost of a least-costly path from node 1 to node ~ , subject to the constraint that the path contains no more than m arcs. In the absence of negative cycles, there exists an optimal path to any node i.

If the lasting of a cycle is defined as 't (r) ~ ~ t.. r' ~( ' then an 66 A. Locatelli - P. Migliarese - S. Rinaldi optimal cycle (if any) is still constitued by an elementary cycle. Moreover, allowing the t~~·s , not to be equal to. one, the multigraph case can be reduced to the simpler linear case. In fact, if r arcs link node i. to node ~ to substitute every arc, but one, from i. to , then it suffices ~ with a couple of arcs passing through an artificial node, with costs, times and constraint values which have half the value of the original ones.

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Periodic Optimization: Volume I: Course Held at the Department of Automation and Information, June 1972 by Angelo Marzollo


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