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simplexIII

# simplexIII - OR320/520 Prof Bland The Simplex Method Parts...

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OR320/520 9/18/07 Prof. Bland The Simplex Method: Parts III and IV Part III: Tableaus In our illustration of the simplex method in Part II on the example from Part I we used the term Tableau to describe a system of equations that gets updated at each iteration. To simplify the presentation of the tableau we usually write down in each row the coefficients of the variables without repeating the variable names. Return to the example from Parts I and II. The linear programming problem we were solving there has a vector x = ( x 1 , x 2 , x 3 , x 4 , x 5 ) T of five nonnegative variables, an objective function cx with coefficients ( c 1 , c 2 , c 3 , c 4 , c 5 ) = (20 , 30 , 0 , 0 , 0) , and a 3 × 5 system of equations Ax = b with: A = 2 2 1 0 0 4 2 0 1 0 3 6 0 0 1 b = 80 120 210 The initial tableau in this example is particularly simple: three of its four rows are just the system Ax = b , but we present it in tabular form, leaving out the x j ’s and simply writing down the entries from A and b . These three rows comprise what is called the body of the tableau. The other row of the tableau is its objective function row . First we add a variable z ; for any choice of the vector x the value of z is the objective function value cx . So we think of an equation like z = cx being appended to the Ax = b system. However, z is a variable, and we like to keep all of our variables on the left-hand-side of our equations, so we re-write this as - z + cx = 0 and append it to the Ax = b system to get an extended system with one additional row, the objective function row, and one additional column, the ( - z ) column. (It turns out that it will be convenient if we think of ( - z ) as the added variable, rather than z ).

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simplexIII - OR320/520 Prof Bland The Simplex Method Parts...

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