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# L19_10 - CHAPTER 9 INTEGER PROGRAMMING Integer Programming...

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CHAPTER 9: INTEGER PROGRAMMING

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Integer Programming definition: Simply stated, an Integer Programming problem (IP) is a Linear Programming (LP) in which some or all the variables are required to be nonnegative integers. An integer programming problem in which all variables are required to be integer is called a pure integer programming problem . If some variables are restricted to be integer and some are not then the problem is a mixed integer programming problem . The case where the integer variables are restricted to be 0 or 1 comes up surprisingly often. Such problems are called pure (mixed) 0-1 programming problems or pure (mixed) binary integer programming problems . Integer programming problems (IPs) are usually much harder to solve than linear programming problems.
Types of Integer Programming Models Models Types of Decision Models All – integer (IP) All are integers Mixed-integer (MIP) Some, but not all, are integers Binary (BIP) All are either 0 or 1 The LP obtained by omitting all integer or 0-1 constraints on variables is called LP relaxation LP relaxation of the IP

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Binary integer variables can be used to model yes/no decisions, such as whether to build a plant or buy a piece of equipment. Integer variables make an optimization problem far more difficult to solve . Memory and solution time may rise exponentially as you add more integer variables. Even with highly sophisticated algorithms and modern supercomputers, there are models of just a few hundred integer variables that have never been solved to optimality. This is because many combinations of specific integer values for the variables must be tested, and each combination requires the solution of a "normal" linear or nonlinear optimization problem . The number of combinations can rise exponentially with the size of the problem. Mixed-Integer Programming (MIP) Problems
Integer Programming Formulations Fixed-Charge Problems Suppose activity i incurs a fixed charge if undertaken at any positive level. Let x i = Level of activity i y i = 1 if activity i is undertaken at positive level ( ) 0 > x i 0 if x i = 0 Then a constraint of the form x i M i y i must be added to the formulation. Here, M i must be large enough to ensure that x i will be less than or equal to M i .

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Integer Programming Formulations (Cont) Either-Or Constraint Suppose we want to ensure that at least one of the following two constraints (and possibly both) are satisfied. () 0 ,..., 2
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L19_10 - CHAPTER 9 INTEGER PROGRAMMING Integer Programming...

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