Didactisch materiaal bij de cursus Academiejaar Tel: 09/ Fax: 09/ Prof. dr. ir. W. Philips Optimalisatietechnieken

© W. Philips, Universiteit Gent, versie: 21/3/ a. 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, ” 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

Verzwaarde lineaire programma’s Herhaling

© W. Philips, Universiteit Gent, versie: 21/3/ a. 4...Voorbeeld 1: “Swedish Steel” koolstof: 0.65 T  0.8 x x x x 4  0.75 T nikkel: 3 T  1.8 x x x 5  3.5 T … Noem x j de hoeveelheid van bestanddeel j (schroot of additief) Beperkingen: gewenste hoeveelheid: Dit is een grotendeels lineair programma alle x j niet-negatief Te minimaliseren: de kostprijs 16 x x x x x x x 7 En nu nog de alles-of-niets voorwaarden: x 1 = 75 y 1 en x 2 = 250 y 2 met y 1, y 2 = 0,1 Besluit: lineair programma met twee binaire variabelen

© W. Philips, Universiteit Gent, versie: 21/3/ a. 5 …Voorbeeld 3: Modellering... Modellering: definieer binaire beslissingsvariabelen x j : x j = 1 als project j wordt gekozen Projecten 1 …n mutueel exclusief  project j vereist i  x j  x i Besluit: ingewikkelde als-dan-beperkingen kunnen weer worden vervangen door eenvoudige lineaire beperkingen resultaat: lineair programma met enkele binaire variabelen De lineaire modellering is niet uniek: b.v. 1,2,3 mutueel exclusief  x 1 +x 2 +x 3  1  x 1 +x 2  1 en x 1 +x 3  1 en x 2 +x 3  1 Een goede keuze kan de oplossing sterk vereenvoudigen let op: enkel voor alle x j binair!

© W. Philips, Universiteit Gent, versie: 21/3/ a. 6 Nasa: modellering Maximaliseer 200 x x x x x x x x x x x x x x 14 mits 6 x x x 3 +x x x 12  10 (2000 tot 2004) 3 x x x x x x x 12 +x x 14  12 (2005 tot 2009) 8 x 5 +x x x x x 14  14 (2010 tot 2014) 8 x x x 11 + x x 14  14 (2015 tot 2019) 10 x x 6 +x x 14  14 (2020 tot 2024) x 4 +x 5  1 x 8 +x 11  1 x 9 +x 14  1(mutueel exclusief) x 11  x 2 x 4  x 3 x 5  x 3 x 6  x 3 x 7  x 3 (afhankelijkheden) alle x j = 0 of 1  zero-one programma