Maximise $22x_1 + 33x_2 +22x_3$
Subject to $x_1 +2x_2 +x_3 \leq 4$
where $x_1, x_2, x_3 \geq 0$ and are integers
I used the dynamic programming approach to solve this IP problem, i found that there exists no unique solution to this problem and 1 of the solutions to this problem is $(0,0,4)$ would it be possible for someone to verify this?
There's no need for dynamic programming.
I would solve it this way . . .
Firstly, since each coefficient of the objective function is a multiple of $11$, the triples $(x_1,x_2,x_3)$ which maximize $$f = 22x_1 + 33x_2 +22x_3$$ are the same as the triples $(x_1,x_2,x_3)$ which maximize $$g = 2x_1 + 3x_2 +2x_3$$
Also, since all coefficients of $g$ are positive, there's no need to consider triples $(x_1,x_2,x_3)$ for which $x_1 +2x_2 +x_3 < 4$, since for any such triple, if we increase $x_1$ by $1$, the constraint $x_1 +2x_2 +x_3 \le 4$ would still be satisfied, and the value of $g$ would increase.
Thus, we can assume $x_1 +2x_2 +x_3 = 4$.
But then \begin{align*} g &= 2x_1 + 3x_2 +2x_3\\[4pt] &= (x_1+2x_2+x_3) + (x_1+x_2+x_3)\\[4pt] &= 4 + x_1 + x_2 + x_3\\[4pt] \end{align*} hence, the triples $(x_1,x_2,x_3)$ which maximize $g$ subject to the constraints $x_1,x_2,x_3 \ge 0$, and $x_1 +2x_2 +x_3 = 4$, are the same as the triples the triples $(x_1,x_2,x_3)$ which maximize $x_1+x_2+x_3$, subject to the same constraints.
The objective function and the constraints are symmetric with respect to the variables $x_1,x_3$, hence we can assume $x_1 \le x_3$, and then at the end, if any solution $(x_1,x_2,x_3)$ has $x_1 < x_3$, we can include the solution $(x_3,x_2,x_1)$ as well.
But there are only $5$ nonnegative integer triples $(x_1,x_2,x_3)$ with $x_1 \le x_3$ such that $x_1 +2x_2 +x_3 = 4$, namely $$(0,0,4),\;\;(0,1,2),\;\;(0,2,0),\;\;(1,0,3),\;\;(1,1,1)$$ and of those, $x_1+x_2+x_3$ is maximized for the triples $(0,0,4)$ and $(1,0,3)$.
Adjusting for the symmetry of $x_1,x_3$, we have the $4$ solutions triples $$(0,0,4),\;(4,0,0),\;(1,0,3),\;(3,0,1)$$ each yielding the same maximum value, $88$, for the original objective function.