Solution

Introduce slack variables

s_{1}, s_{2}, s_{3}

. In the initial simplex algorithm dictionary,

x_{2}

has the largest positive objective coefficient. The ratio test gives

min {\frac{7}{3}, \frac{5}{2}, \frac{2}{1}} = 2,

x_{2}

enters and

s_{3}

leaves. Solving the third constraint for

x_{2}

gives

x_{2} = 2 - x_{1} - 2 x_{3} - s_{3} .

The objective becomes

z = 3 x_{1} + 6 x_{2} + 4 x_{3} = 12 - 3 x_{1} - 8 x_{3} - 6 s_{3} .

Every reduced cost is now nonpositive, so the simplex optimality criterion gives

(x_{1}, x_{2}, x_{3}) = (0, 2, 0), z_{m a x} = 12 .

The other two slacks both equal one.

minimize subject to 7 y_{1} + 5 y_{2} + 2 y_{3}, 2 y_{1} + 4 y_{2} + y_{3} \geq 3, 3 y_{1} + 2 y_{2} + y_{3} \geq 6, y_{1} + 2 y_{2} + 2 y_{3} \geq 4, y_{1}, y_{2}, y_{3} \geq 0.

The vector

(y_{1}, y_{2}, y_{3}) = (0, 0, 6)

is dual feasible and has objective value

12

. By weak duality, it and the displayed primal point are optimal; they also satisfy complementary slackness.

Solved by gpt-5.6-sol high.

 Ancestors (10)