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notes_9 - Lecture Notes For Optimization Theory &...

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Lecture Notes For Optimization Theory &Ana lys is (ESE 304) Michael A. Carchidi November 24, 2010 Chapter 9 - Integer Programming The following notes are based on the text entitled: Introduction to Mathemat- ical Programming by Wayne L. Winston and Munirpallam Venkataramanan (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. 9.1 Introduction to Integer Programming A Linear Programming (LP) problem in which at least one the decision vari- ables is required to be integer is called an Integer Programming (IP) problem. If all of the decision variables are required to be integer, then the IP problem is called a Pure Integer Programming (PIP) problem, otherwise it is called a Mixed (MIP) problem. Example : Revised Giapetto’s Woodcarving Giapetto’s Woodcarving Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14 .At ra ins e l l sfo r $21 and uses $9 worth of raw materials. Each train that is manufactured increases Giapetto’s variable labor and overhead costs by $10 .The manufacture of wooden soldiers and trains requires two types of labor: f nishing
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and carpentry. A soldier requires 4 hours of f nishing labor (instead of 2 as in the original statement of the problem in Chapter 3 )and 1 hour of carpentry labor. A train requires 1 hour of f nishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw materials but only 100 f nishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly pro f ts (revenues minus costs). We had shown in chapter 3 that a statement of this problem is as follows max z =3 x 1 +2 x 2 (Objective Function) s.t. 4 x 1 + x 2 100 (Finishing Constraint) x 1 + x 2 80 (Carpentry Constraint) x 1 40 (Constraint on Demand for Soldiers) x 1 ,x 2 0 (Sign Restrictions) x 1 2 are integer (Integer Restrictions) where x 1 = the number of soldiers produced each week, and x 2 = thenumbero ftra insproducedeachweek . With the minor change of a soldier requiring 4 hours of f nishing labor instead of 2 hours, the optimal solution to the problem without the integer restrictions is x max = 6 2 3 73 1 3 with z max =$166 2 3 Although we have been ignoring the integer restrictions in the past chapters, we must technically include these restrictions in the full development of this problem since we cannot make a fractional number of soldiers or trains. We see then that the Giapetto problem is really a Pure Integer Programming problem. Using LINDO (which we shall discuss later) we f nd that the solution to the pure IP problem is given by x max = 6 74 ¸ with z max .
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notes_9 - Lecture Notes For Optimization Theory &...

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