Telecommunicatie en Informatieverwerking UNIVERSITEIT GENT Didactisch materiaal bij de cursus Academiejaar Tel: 09/ Fax: 09/ Prof. dr. ir. W. Philips Optimalisatietechnieken

© W. Philips, Universiteit Gent, versie: 14/3/ c. 2 Copyright notice This powerpoint presentation was developed as an educational aid to the renewed course “Optimisation Techniques” (Optimalisatietechnieken), taught at the University of Gent, Belgium as of This presentation may be used, modified and copied free of charge for non-commercial purposes by individuals and non-for-profit organisations and distributed free of charge by individuals and non-for-profit organisations to individuals and non-for-profit organisations, either in electronic form on a physical storage medium such as a CD-rom, provided that the following conditions are observed: 1.If you use this presentation as a whole or in part either in original or modified form, you should include the copyright notice “© W. Philips, Universiteit Gent, 1998” in a font size of at least 10 point on each slide; 2.You should include this slide (with the copyright conditions) once in each document (by which is meant either a computer file or a reproduction derived from such a file); 3. If you modify the presentation, you should clearly state so in the presentation; 4.You may not charge a fee for presenting or distributing the presentation, except to cover your costs pertaining to distribution. In other words, you or your organisation should not intend to make or make a profit from the activity for which you use or distribute the presentation; 5. You may not distribute the presentations electronically through a network (e.g., an HTTP or FTP server) without express permission by the author. In case the presentation is modified these requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the presentation, and can be reasonably considered independent and separate works in themselves, then these requirements do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the presentation, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. In particular note that condition 4 also applies to the modified work (i.e., you may not charge for it). “Using and distributing the presentation” means using it for any purpose, including but not limited to viewing it, presenting it to an audience in a lecture, distributing it to students or employees for self-teaching purposes,... Use, modification, copying and distribution for commercial purposes or by commercial organisations is not covered by this licence and is not permitted without the author’s consent. A fee may be charged for such use. Disclaimer: Note that no warrantee is offered, neither for the correctness of the contents of this presentation, nor to the safety of its use. Electronic documents such as this one are inherently unsafe because they may become infected by macro viruses. The programs used to view and modify this software are also inherently unsafe and may contain bugs that might corrupt the data or the operating system on your computer. If you use this presentation, I would appreciate being notified of this by . I would also like to be informed of any errors or omissions that you discover. Finally, if you have developed similar presentations I would be grateful if you allow me to use these in my course lectures. Prof. dr. ir. W. Philips Department of Telecommunications and Information ProcessingFax: University of GentTel: St.-Pietersnieuwstraat 41, B9000 Gent, Belgium

Sensitiviteitsanalyse

© W. Philips, Universiteit Gent, versie: 14/3/ c. 4 Overzicht Primaal/duaal probleem in formulevorm Te bestuderen: hoe verandert het optimum van een lineair programma bij wijzigen van de winstfunctie of de (rechterleden van) de beperkingen De eigenlijke sensitiviteitsanalyse kwalitatief kwantitatief voor kleine veranderingen: berekenen van sensitiviteit kwantitatieve parametrische studies: winst als functie van de wijzigende coëfficiënt Toepassing: de parametrische zelf-duale simplexmethode

© W. Philips, Universiteit Gent, versie: 14/3/ c. 5 Notaties Primaal probleem in standaardvorm maximaliseer c t x over x en w mits x  0, w  0 en Ax + w = b met originele veranderlijken x en reserveveranderlijken w Duaal probleem in standaardvorm maximaliseer - b t y over y en z mits y  0, z  0 en -A t y + z = - c met originele veranderlijken y en reserveveranderlijken z Compactere notaties: primaal probleem: maximaliseermits  Door herdefiniëren van A, c en x : maximaliseermits ct xct x herdefinitie duale veranderlijken: z Ax= bAx= b

© W. Philips, Universiteit Gent, versie: 14/3/ c. 6 …Notaties… Opmerkingen onze voornaamste bedoeling is het invoeren van een uniforme notatie voor de veranderlijken omdat simplexstappen de variabelen mengt In de simplextableaus blijft er wel negatieve transponeringssymmetrie, maar de namen van de veranderlijken zijn nu anders P2P2 D2D xxx xxx xxx xx    ...max zzzz zzzz zzz   ... max door de keuze om z uit te breiden i.p.v. y, hebben duale variabelen dezelfde index

© W. Philips, Universiteit Gent, versie: 14/3/ c. 7 xNxN cNcN N …Notaties Net zoals vroeger maken we onderscheid tussen de B en NB- veranderlijken in een bepaalde fase: basiskolommen van A worden verzameld in B, de andere in N idem voor de basisrijen van x en c, en z B xBxB cBcB Maximaliseer c B x B + c N x N mits B x B +Nx N =b

© W. Philips, Universiteit Gent, versie: 14/3/ c. 8 Duaal/primaal in formulevorm… Winst: Primaal probleem: formules in simplextableau Stelsel: Winst: Duaal probleem: formules in simplextableau (via negatieve-transponering bovenstaande formules) Let op: z N bevat de rijen van z corresponderend met x N ; in het duale simplextableau zijn dit de B-variabelen (niet de NB-variabelen)

© W. Philips, Universiteit Gent, versie: 14/3/ c. 9 …Duaal/primaal in formulevorm Winst Primaal: In een specifieke primale en corresponderende duale basisoplossing: x N *=0  z B *=0  Duaal: Stelsel Winst Primaal: Duaal: Stelsel Vereenvoudige formules: in termen van de waarden van de huidige B- variabelen in de huidige basisoplossing