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I have the following linear optimization model:

$$\begin{array}{ll} \text{maximize} & x_1 + 2 x_2\\ \text{subject to} & 3x_1+2x_2 \leq 12\\ & x_1+3x_2 \leq 9\\ &2x_2\leq2\end{array}$$

Could someone please tell me a proper way to determine all solutions (feasible and non feasible ones).

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    $\begingroup$ It's two dimensional problem. Plot the set and do the optimization graphically. $\endgroup$ – A.Γ. Jul 4 '16 at 15:18
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Observe that: $$ x_1+2x_2=\frac{1}{3}(3x_1+2x_2)+\frac{2}{3}(2x_2)\leq 4+\frac{4}{3}=\frac{16}{3}, $$ and the equality occurs if and only if $(x_1,x_2)=(10/3,1)$.

For $(x_1,x_2)=(10/3,1)$ we have $$ 3x_1+2x_2=12\leq 12,\quad x_1+3x_2=\frac{13}{3}\leq 9,\quad 2x_2=2\leq 2. $$ Hence, the optimal value is $\displaystyle\frac{16}{3}$ and the solution set of the problem is $$ S=\left\{\left(\frac{10}{3},1\right)\right\}. $$

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  • $\begingroup$ Thank you very much for your answer! Could you please describe your procedure a bit more. $\endgroup$ – Bastian Jul 4 '16 at 20:09

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