find the optimal value of the cost function?

Question

Use duality to find the optimal value of the cost function in the following linear programming problem:

Max. $x + y + z$

such that $3x + 2y + 2z = 1,$

$x ≥ 0, y ≥ 0, z ≥ 0$

One of my friend said me that used Largangian and dual function but im not getting

Im completely struck Pliz help me

Siong Thye Goh · Accepted Answer · 2020-03-10T06:22:19.407

3

The dual is

$$\min p$$

subject to $$3p \ge 1, 2p \ge 1, 2p \ge 1$$

Solve this optimization problem and use strong duality to make conclusion.

edited Mar 10 '20 at 06:22

answered Mar 10 '20 at 06:11

Siong Thye Goh

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thanks u @siong is $p= 1/2$ ?? – jasmine Mar 10 '20 at 06:14
1

yes, optimal $p$ is $\frac12$. Practise writing down dual. :) Also you might like to visualize what is the set of the optimal solution in the original optimization problem. – Siong Thye Goh Mar 10 '20 at 06:20
okss thanks u so much @Siong Thye Goh – jasmine Mar 10 '20 at 06:22

score 1 · Answer 2 · answered Mar 10 '20 at 06:36

Alternatively, you can use Simplex method: $$\begin{array}{rrrr|rr} & x_1 & x_2 & s_1 & C & \text{ratio} \\ \hline s & 3 & 2 & 2 & 1 & 1/2 \\\hdashline & -1 & -1 & -1 & 0 & \\ \hline x_2 \text{ or } x_3 & 3/2 & 1 & 1 & \color{red}{1/2} & \\\hdashline & 1/2 & 0 & 0 & \color{blue}{1/2} & \end{array}$$ So, $u(0,\color{red}{1/2},0)=u(0,0,\color{red}{1/2})=\color{blue}{1/2}$ is maximum.

find the optimal value of the cost function?

2 Answers2