mthsc810-lecture01

mthsc810-lecture01 - Excerpt from the syllabus Homework...

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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 Modelinglanguages : 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
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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 Ifweknew 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 0 1 2 3 4 5 6 0 50 100 150 200 250 Q ( r )[ in 2 ] r [in] minimum r = 1.471in h = V π ( 1 . 471 ) 2 = 2 . 942 in
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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 > 0 r > 0 Optimization Models, in general, have: Variables (e.g. height and radius, number of trucks, . . . ): the unknown (and desired) part of the problem (one thing your boss cares about).
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