1. **State the problem:** We want to maximize the expression $a_1 + t_1, t_4$ subject to the constraints: $$ \begin{cases} y_1 - a_1 \leq r, \\ y_1 \leq r, \\ y_1 \geq a_1, \\ y_1 \geq r. \end{cases} $$ 2. **Understand the constraints:** - $y_1 - a_1 \leq r$ means $y_1 \leq a_1 + r$. - $y_1 \leq r$ restricts $y_1$ to be at most $r$. - $y_1 \geq a_1$ means $y_1$ is at least $a_1$. - $y_1 \geq r$ means $y_1$ is at least $r$. 3. **Analyze the feasible region:** From $y_1 \geq r$ and $y_1 \leq r$, we get $y_1 = r$. Also, $y_1 \geq a_1$ and $y_1 = r$ imply $a_1 \leq r$. From $y_1 \leq a_1 + r$ and $y_1 = r$, we get $r \leq a_1 + r$ which simplifies to $0 \leq a_1$. 4. **Summarize feasible values:** - $y_1 = r$ - $0 \leq a_1 \leq r$ 5. **Maximize the objective:** The objective is $a_1 + t_1, t_4$. Since $t_1$ and $t_4$ are not constrained here, we focus on $a_1$. Maximizing $a_1$ over $[0, r]$ gives $a_1 = r$. 6. **Final answer:** The maximum value of $a_1$ under the constraints is $r$. Hence, the solution is: $$ \max a_1 = r $$ with $$ y_1 = r, \quad 0 \leq a_1 \leq r. $$