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Unformatted text preview: 9.3 THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9.2 is convenient. However, for problems involving more than two variables or problems involving a large number of constraints, it is better to use solution methods that are adaptable to computers. One such method is called the simplex method, developed by George Dantzig in 1946. It provides us with a systematic way of examining the vertices of the feasible region to determine the optimal value of the objective function. We introduce this method with an example. Suppose we want to find the maximum value of where and subject to the following constraints. Since the lefthand side of each inequality is less than or equal to the righthand side, there must exist nonnegative numbers and that can be added to the left side of each equation to produce the following system of linear equations . The numbers and are called slack variables because they take up the “slack” in each inequality. s 3 s 1 , s 2 2 x 1 1 5 x 2 1 s 1 1 s 2 1 s 3 5 90 2 x 1 1 5 x 2 1 s 1 1 s 2 1 s 3 5 27 2 x 1 1 5 x 2 1 s 1 1 s 2 1 s 3 5 11 s 3 s 1 , s 2 2 x 1 1 5 x 2 # 90 2 x 1 1 5 x 2 # 27 2 x 1 1 5 x 2 # 11 x 2 $ 0, x 1 $ z 5 4 x 1 1 6 x 2 , 494 CHAPTER 9 LINEAR PROGRAMMING Standard Form of a Linear Programming Problem A linear programming problem is in standard form if it seeks to maximize the objective function subject to the constraints . . . where and After adding slack variables, the corresponding system of constraint equations is . . . where s i $ 0. a m 1 x 1 1 a m 2 x 2 1 . . . 1 a mn x n 1 s m 5 b m a 21 x 1 1 a 22 x 2 1 . . . 1 a 2 n x n 1 s 2 5 b 2 a 11 x 1 1 a 12 x 2 1 . . . 1 a 1 n x n 1 s 1 5 b 1 b i $ 0. x i $ a m 1 x 1 1 a m 2 x 2 1 . . . 1 a mn x n # b m a 21 x 1 1 a 22 x 2 1 . . . 1 a 2 n x n # b 2 a 11 x 1 1 a 12 x 2 1 . . . 1 a 1 n x n # b 1 z 5 c 1 x 1 1 c 2 x 2 1 . . . 1 c n x n R E M A R K : Note that for a linear programming problem in standard form, the objective function is to be maximized, not minimized. (Minimization problems will be discussed in Sections 9.4 and 9.5.) A basic solution of a linear programming problem in standard form is a solution of the constraint equations in which at most m variables are nonzero––the variables that are nonzero are called basic variables. A basic solution for which all variables are nonnegative is called a basic feasible solution. The Simplex Tableau The simplex method is carried out by performing elementary row operations on a matrix that we call the simplex tableau. This tableau consists of the augmented matrix corresponding to the constraint equations together with the coefficients of the objective function written in the form In the tableau, it is customary to omit the coefficient of z . For instance, the simplex tableau for the linear programming problem Objective function is as follows....
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 Spring '10
 asda

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