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notes_4 - Notes For Optimization Theory & Analysis (ESE...

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Notes For Optimization Theory & Analysis (ESE 304) Michael A. Carchidi September 23, 2010 Chapter 4 - The Simplex Algorithm and Goal Programming The following notes are based on the text entitled: Operations Research by Wayne L. Winston (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. 4.1 How to Convert An LP Problem To Standard Form The general form for an LP problem involving n decision variables and m constra intsisasfo l lows : max or min z = c 1 x 1 + c 2 x 2 + ··· + c n x n (Objective Function) s.t. a 11 x 1 + a 12 x 2 + + a 1 n x n ( )(=)( ) b 1 (Constraint #1 ) a 21 x 1 + a 22 x 2 + + a 2 n x n ( )(=)( ) b 2 (Constraint #2 ) a 31 x 1 + a 32 x 2 + + a 3 n x n ( )(=)( ) b 3 (Constraint #3 ) . . . . . . a m 1 x 1 + a m 2 x 2 + + a mn x n ( )(=)( ) b m (Constraint # m ) x 1 ,x 2 3 ,...,x n 0 (Sign Restrictions) where all parameters c i , a ij and b i are known constants where the b i ’s can (and should) always be choosen to be greater than or equal to zero .Theno ta t ion ( )(=
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)( ) is used to indicate either: ( ) , (=) ,or ( ) as the appropriate sign for that constraint. An LP in which: (1) all the constraints involve the equality sign (=) , (2) all decision variables are non-negative and (3) the objective is to maximize z , is said to be in standard form . All LP problems that already have the sign restrictions can be put in standard form. To see this, we f rst note that any minimization problem of the form min w = c 1 x 1 + c 2 x 2 + ··· + c n x n can be expressed as a maximization problem of the equivalent form max z = w =( c 1 ) x 1 +( c 2 ) x 2 + c n ) x n . Next, any inequality of the form a k 1 x 1 + a k 2 x 2 + + a kn x n b k can be expressed as an equality a k 1 x 1 + a k 2 x 2 + + a kn x n + s k = b k by introducing the non-negative slack variable s k , and any inequality of the form a k 1 x 1 + a k 2 x 2 + + a kn x n b k can be expressed as an equality a k 1 x 1 + a k 2 x 2 + + a kn x n e k = b k by introducing the non-negative excess variable e k . Therefore, by including the necessary number of slack variables and excess variables, we can write the above LP problem as max or min z = c 1 x 1 + c 2 x 2 + + c N x N (Objective Function) s.t. a 11 x 1 + a 12 x 2 + + a 1 N x N = b 1 (Constraint #1 ) a 21 x 1 + a 22 x 2 + + a 2 N x N = b 2 (Constraint #2 ) a 31 x 1 + a 32 x 2 + + a 3 N x N = b 3 (Constraint #3 ) . . . . . . a m 1 x 1 + a m 2 x 2 + + a mN x N = b m (Constraint # m ) x 1 ,x 2 3 ,...,x N 0 (Sign Restrictions) 2
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wh ichisanLPprob lemhav ing N = n +(# of Slack Variables )+(# of Excess Variables ) decision variables and m equality constraints. In addition, we may always formu- late the problem so that b j 0 for j =1 , 2 , 3 ,...,m . For example, suppose we have the LP problem max z =2 x 1 +3 x 2 +4 x 3 + x 4 (Objective Function) s.t. x 1 + x 2 + x 3 + x 4 10 (Constraint #1 ) x 1 +2 x 2 + x 3 +5 x 4 30 (Constraint #2 ) 2 x 1 + x 2 x 3 + x 4 20 (Constraint #3 ) 5 x 1 x 2 x 3 + x 4 =5 (Constraint #4 ) x 1 ,x 2 3 4 0 (Sign Restrictions) then we may introduce two slack variables s 1 and s 2 and one excess variable e 1 , and write this as max z x 1 x 2 x 3 + x 4 +0 s 1 s 2 e 1 (Objective Function) s.t.
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notes_4 - Notes For Optimization Theory & Analysis (ESE...

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