Unbounded Feasible Region

Question

I know that "Unbounded feasible regions may or may not have an optimal solution." I know the example where unbounded feasible region has an optimal solution but I need two examples where a maximum and in another a minimum don't exist.

Answer 1

1. a feasible LPP where a maximum doesn't exist \begin{alignat}{3} \max\quad & z = & 3x & + & 2y \\ \text{s.t.} & & x & - & y & \le -3 \\ & & 5x & - & 7y & \le 10 \\ & & \rlap{x,y \ge 0} \end{alignat} \begin{cases} y &\ge x - 3 \\ y &\ge \dfrac{5x - 10}{7} \\ x,y &\ge 0 \\ z &= 3x + 2y \ge C_1 x + M \end{cases} for some constants $C_1> 0$ and $M \in \Bbb{R}$ independent of $x$ and $y$.
2. a feasible LPP where a minimum doesn't exists \begin{alignat}{3} \min\quad & z = & & 4x & - & 3y \\ \text{s.t.} & & & 3x & - & 2y & \ge -8 \\ & & - & 2x & + & 7y & \ge 10 \\ & & \rlap{x,y \ge 0} \end{alignat} \begin{cases} y &\le \dfrac{3x + 8}{2} \\ y &\ge \dfrac{2x + 10}{7} \\ x,y &\ge 0 \\ z &= 4x - 3y \end{cases}

As $x$ get larger, it is possible to choose some $y$ so that \begin{cases} \dfrac32 x &\le y \le \dfrac{3x + 8}{2} \\ y &\ge \dfrac{2x + 10}{7} \\ x,y &\ge 0 \\ z &= 4x - 3y \le 4x - 3\left( \dfrac32 x \right) = -\dfrac12 x \end{cases} Hence this LPP is unbounded and it has no minimum.