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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.3

# Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.3

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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 a systematic way of examining the ver- tices of the feasible region to determine the optimal value of the objective function. This method is introduced with the following example. Suppose you want to find the maximum value of where and subject to the following constraints. Because the left-hand side of each inequality is less than or equal to the right-hand 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 , x 1 x 1 2 x 1 x 2 x 2 5 x 2 s 1 s 2 s 3 11 27 90 s 3 s 1 , s 2 , 2 x 1 5 x 2 90 x 1 x 2 27 x 1 x 2 11 x 2 0, x 1 0 z 4 x 1 6 x 2 , 530 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 objec- tive 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 a m 2 x 2 . . . a mn x n s m b m . . . a 21 x 1 a 22 x 2 . . . a 2 n x n s 2 b 2 a 11 x 1 a 12 x 2 . . . a 1 n x n s 1 b 1 b i 0. x i 0 a m 1 x 1 a m 2 x 2 . . . a mn x n b m . . . a 21 x 1 a 22 x 2 . . . a 2 n x n b 2 a 11 x 1 a 12 x 2 . . . a 1 n x n b 1 z c 1 x 1 c 2 x 2 . . . c n x n

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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 called 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. Basic x 1 x 2 s 1 s 2 s 3 b Variables 1 1 0 0 11 s 1 1 1 0 1 0 27 s 2 2 5 0 0 1 90 s 3 0 0 0 0 Current z–value For this initial simplex tableau, the basic variables are and and the nonbasic variables are and The nonbasic variables have a value of zero, yielding a current z -value of zero. From the columns that are farthest to the right, you can see that the basic
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.3

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