810-notes-1x2

810-notes-1x2 - MthSc810 – Mathematical Programming...

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Unformatted text preview: 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 > r > 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 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 . . . the number of employees starting on a certain day ◮ 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 ≥...
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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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810-notes-1x2 - MthSc810 – Mathematical Programming...

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