# Solve optimization problem with constant cost

Given that

$$x_1 + 2x_2 + x_3 + 2x_4 =C$$

with constraints

$$x_1+2x_2\leq C_1$$

$$x_3+2x_4\leq C_2$$

Find $$x_1$$, $$x_2$$, $$x_3$$ and $$x_4$$

My doubt is that can we have a solution for this type of problems?

• It depends on the values of $C,C_1,C_2$. It may or may not be feasible. Commented Oct 3, 2018 at 15:27

A $$4$$-tuple $$(x_1,x_2,x_3,x_4)$$ satisfying the conditions exists if and only if $$C\le C_1+C_2$$.
Assuming $$C\le C_1+C_2$$, and assuming no further constraints, the feasible region is infinite.
Explicitly, assume $$C\le C_1+C_2$$, and let $$w=(C_1+C_2)-C$$.
Let $$x_2,x_4$$ be arbitrary real numbers, and let \begin{align*} x_1=C_1-2x_2-{\small{\frac{w}{2}}}\\[4pt] x_3=C_2-2x_4-{\small{\frac{w}{2}}}\\[4pt] \end{align*} Then all the specified conditions are satisfied.
On the other hand if $$C > C_1+C_2$$, summing the inequality constraints yields $$x_1+2x_2+x_3+2x_4 \le C_1+C_2 < C$$ contradicting the equality constraint.