810-notes-2x2 - An example MthSc810 Mathematical...

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MthSc810 – Mathematical Programming Pietro Belotti January 23-27, 2012 An example You work at a company that sells food in tin cans, and are charged with designing the next generation can, which is a cylinder made of tin The can must contain V = 20 cu.in. (11 fl.oz., 33 cl) Cut and solder tin foil to produce cans 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. 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 Optimal solution (computed with any nonlinear solver): r = 1 . 471 in, h = V π ( 1 . 471 ) 2 = 2 . 942 in Optimization Models, in general, have: Variables : The unknown (and sought for) part of the problem Constraints : They define all and only values of the variables that give possible solutions. Objective function : A function of the variables
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Another example The manager of a new hospital division is hiring nurses. Required # of employees for each day of the week: day S M T W Th F Sa # empl. 11 17 13 15 19 14 16 (State regulations impose) that a nurse works five days in a row and then receives two days off The number of nurses is minimum What are the variables of the problem? The number of nurses working each day? The total number of nurses to hire? What to we want to know? If an nurse works on Thu, her work days can be Thu , Fri, Sat, Sun, Mon, or Wed, Thu , Fri, Sat, Sun, or Tue, Wed, Thu , Fri, Sat, or Mon, Tue, Wed, Thu , Fri, or Sun, Mon, Tue, Wed, Thu . We don’t know when she started his working shift. It is the variable we are looking for! Actually, we are only interested in . . . thenumberofemployees starting onacertainday Define it as variable x i , with i ∈ { Sun , Mon , Tue , Wed , Thu , Fri , Sat } . Now that we know what we are looking for . . . We have variables. We can write constraints & objective f. constraint #1: there must be 19 employees on Thursdays. x Thu + x Wed + x Tue + x Mon + x Sun 19 constraint #2: an employee works five consecutive days and then receives two days off. This is already included in the definition of our variables and in the above constraint. objective function: the total number of employees (to be minimized). number of employees starting on Monday, plus those starting on Tuesday, etc. we can sum them up because they define disjoint sets of employees: if one starts working on Thursday, he doesn’t start on Friday . . . The model min x Sun + x Mon + x Tue + x Wed + x Thu + x Fri + x Sat ( Sun ) x Sun + x Wed + x Thu + x Fri + x Sat 11 ( Mon ) x Sun + x Mon + x Thu + x Fri + x Sat 17 ( Tue ) x Sun + x Mon + x Tue + x Fri + x Sat 13 ( Wed ) x Sun + x Mon + x Tue + x Wed + x Sat 15 ( Thu ) x Sun + x Mon + x Tue + x Wed + x Thu 19 ( Fri ) x Mon + x Tue + x Wed + x Thu + x Fri 14 ( Sat ) x Tue + x Wed + x Thu + x Fri + x Sat 16 x Sun , x Mon , x Tue , x Wed , x Thu , x Fri , x Sat 0 x Sun , x Mon , x Tue , x Wed , x Thu , x Fri , x Sat Z
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The solution LP: Linear Programming relaxation – ignore integrality constraints IP: Integer Programming – real problem Sun Mon Tue Wed Thu Fri Sat Total LP 5
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