notes_9 - Lecture Notes For Optimization Theory...

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Lecture Notes For Optimization Theory & Analysis (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 Integer Programming (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 . A train sells for $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: fi nishing
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and carpentry. A soldier requires 4 hours of fi 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 fi nishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw materials but only 100 fi 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 fi 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 , x 2 are integer (Integer Restrictions) where x 1 = the number of soldiers produced each week, and x 2 = the number of trains produced each week. With the minor change of a soldier requiring 4 hours of fi 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 fi nd that the solution to the pure IP problem is given by x max = 6 74 ¸ with z max = $166 .
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