Question

Maximize Z = x + y, subject to constraints are x - y $\le$ - 1, - x + y $\le$ 0 and x, y $\ge$ 0.

Answer 1

Our problem is to maximize Z = x + y ........(i)

Subject to constraints are x - y $\le$ -1 .......(ii)

- x + y $\le$ 0 ...........(iii)

x $\ge$ 0, y $\ge$ 0 ........(iv)

Firstly, draw the graph of the line x - y = -1

$\begin{array}{ccc}x& 0& -1\\ y& 1& 0\end{array}$

Putting (0, 0) in the inequality x - y $\le$ - 1, we have

$0-0\le -1\Rightarrow 0\le -1$ (which is false)

So, the half plane is away form the origin.

Secondly, draw the graph of the line - x + y = 0

$\begin{array}{ccc}x& 0& 1\\ y& 0& 1\end{array}$

Putting (2, 0) in the inequality - x + y $\le$ 0, we have - 2 + 0 $-2+0\le 0$

$\Rightarrow -2\le 0$ (which is true)

So, the half plane is towards the X - axis.

Since, x, y $\ge$ 0

So, the feasible region lies in the first quadrant. From the above graph, it is clearly shown that there is no common region. Hence , there is no feasible region and thus Z has no maximum value.

Note If there is no common region, then we do not determine the minimum / maximum value.