Solve the following linear programming problem by simplex method: Maximize

$z= 3x_1+5x_2+4x_3$

subject to

$2x_1+3x_2\leq8 \\ 2x_2+5x_3\leq10 \\ 3x_1 + 2x_2 + 4x_3 \leq 15 \\ x_1,x_2,x_3 \geq 0.$

Adding slack variables, $x_4, x_5, x_6$, we get the simplex table as follows:

Base$C_b$$b$$x_1$$x_2$$x_3$$x_4$$x_5$$x_6$$b/a$
$x_4$08230100$\frac{8}{3}$
$x_5$0100250105
$x_6$015324001$\frac{15}{2}$
Z0-3-5-4000

Taking pivot variable as indicated,

Base$C_b$$b$$x_1$$x_2$$x_3$$x_4$$x_5$$x_6$$b/a$
$x_2$5$\frac{8}{3}$$\frac{2}{3}10\frac{1}{3}00- x_50\frac{14}{3}$$\frac{-4}{3}$05$\frac{-2}{3}$10$\frac{14}{15}$
$x_6$0$\frac{29}{3}$$\frac{5}{3}04\frac{-2}{3}01\frac{29}{12} Z\frac{40}{3}$$\frac{1}{3}$0-4$\frac{5}{3}$00

Again taking pivot variable as indicated,

Base$C_b$$b$$x_1$$x_2$$x_3$$x_4$$x_5$$x_6$$b/a$
$x_2$5$\frac{8}{3}$$\frac{2}{3}10\frac{1}{3}004 x_34\frac{14}{15}$$\frac{-4}{15}$01$\frac{-2}{15}$$\frac{1}{5}0\frac{-7}{2} x_60\frac{89}{15}$$\frac{41}{15}$00$\frac{-2}{15}$$\frac{-4}{15}1\frac{89}{41} Z\frac{256}{15}$$\frac{-11}{15}$00$\frac{17}{15}$$\frac{4}{5}0 Again taking the indicated element as pivot variable, we get the simplex table as: BaseC_b$$b$$x_1$$x_2$$x_3$$x_4$$x_5$$x_6$$b/a x_25\frac{50}{41}010\frac{15}{41}$$\frac{8}{41}$$\frac{-10}{41}- x_34\frac{62}{41}001\frac{-6}{41}$$\frac{5}{41}$$\frac{4}{41}- x_13\frac{89}{41}100\frac{-2}{41}$$\frac{-12}{41}$$\frac{15}{41}- Z\frac{765}{15}000\frac{45}{41}$$\frac{24}{41}$$\frac{11}{41}$

This gives us the optimal solution as

$z = \frac{765}{41}$

at

$x_1 = \frac{89}{41} \\ x_2 = \frac{50}{41} and \\ x_3 = \frac{62}{41}$

answered Dec 21, 2017 by (1,920 points)