mthsc810-lecture01

mthsc810-lecture01 - MthSc 810 – Mathematical Programming...

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Unformatted text preview: MthSc 810 – Mathematical Programming instructor: Pietro Belotti Martin Hall, O- 321 phone: (864) 656 6765 office hrs: Tue/Thu, 3:30pm – 5pm (or by appt.) email: [email protected] web page: http://myweb.clemson.edu/˜pbelott Excerpt from the syllabus Homework: 25% (approx. one every week) Midterm I: 25% (end of September) Midterm II: 25% (end of October) Final exam: 25% Learning and exercising material: ◮ Textbook : D. Bertsimas, J.N. Tsitsiklis, Introduction to Linear Optimization , Athena Scientific. ◮ Extra material: handouts and the online material by Bob Fourer: www.4er.org/CourseNotes ◮ Modeling languages : for modeling optimization problem. Many have a limited version available to students. AMPL: preferred. No Graphical User Interface (GUI), but I know it better (read: I can help) – www.ampl.com Mosel: very nice GUI – google “xpress mosel” GAMS: Has version with even nicer GUI (Aimms) – www.gams.com , www.aimms.com Lecture plan ◮ Relaxations, lower/upper bounds, convexity, complexity ◮ Linear Optimization problems ◮ The simplex method ◮ Duality theory ◮ Sensitivity analysis ◮ Large scale problems: decomposition methods ◮ Network flow models Optimization models ◮ are used to find the best configuration of processes, systems, products, etc. ◮ rely on a theory developed mostly in the past 50 years ◮ when applied in an industrial, financial, military context, they can yield a better use of budget/resources Success stories Source: http://www.informs.com (see also http://www.ScienceOfBetter.org ) yr company result 86 Eletrobras (hydroelectric energy) 43M$ saved 90 Taco Bell (human resources) 7.6M$ saved 92 Harris semicond. prod. planning 50% → 95% orders “on time” 95 GM – Car Rental +50M$ 96 HP printers — re-designed prod. 2x production 99 IBM — supply chain 750M$ saved 00 Syngenta — corn production 5M$ saved An example You are charged with designing a can , a cylinder made of tin (a can is obtained by cutting and soldering tin foil). ◮ The can must contain V = 20 cu.in. (11 fl.oz., 33 cl) ◮ Tin (foil) is expensive, use as little as possible ⇒ Design a cylinder with volume V using as little tin (i.e., total area) as possible. Example r h If we knew radius r and height h , ◮ the volume would be π r 2 h ◮ qty of tin would be 2 π r 2 + 2 π rh π r 2 h must be V = 20in 3 ⇒ h = V π r 2 Rewrite the quantity of tin as Q ( r ) = 2 π r 2 + 2 π r V π r 2 , or Q ( r ) = 2 π r 2 + 2 V r ⇒ Find the minimum of Q ( r ) ! Minimize the quantity of tin 1 2 3 4 5 6 5 0 1 0 0 1 5 0 2 0 0 2 5 0 Q ( r )[ in 2 ] r [in] minimum r = 1.471 in h = V π ( 1 . 471 ) 2 = 2 . 942 in Your first Optimization model Variables r : radius of the can’s base h : height of the can Objective 2 π rh + 2 π r 2 ( minimize ) Constraints π r 2 h = V h > r > Optimization Models, in general, have: Variables (e.g. height and radius, number of trucks, . . . ): the unknown (and desired) part of the problem (one thing...
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This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

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mthsc810-lecture01 - MthSc 810 – Mathematical Programming...

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