Question

Find the maximum value of the objective function \(Z=5x+10y\) by graphical method under the following constraints: \(x+2y\le120\), \(x+y\ge60\), \(x-2y\ge0\), \(x\ge0\), \(y\ge0\).

Hint:

Find corner points by solving the boundary lines in pairs, then evaluate Z at each one.

Answer 1

Step 1: Understanding the Concept:
We must find the feasible region formed by all constraints, locate its corner points, and evaluate Z at each corner point.
The graphical method says the maximum of a linear objective function over a bounded feasible region occurs at a corner point.

Step 2: Key Formula or Approach:
Plot the boundary lines \(x+2y=120\), \(x+y=60\), \(x-2y=0\), find their pairwise intersections that lie inside all constraints, then evaluate \(Z=5x+10y\) at every corner.

Step 3: Detailed Explanation, find the corner points:
Solve \(x+2y=120\) and \(x-2y=0\) together: adding gives \(2x=120\), so \(x=60\), then \(y=30\). Point \((60,30)\).
Solve \(x+y=60\) and \(x-2y=0\) together: substituting \(x=2y\) gives \(2y+y=60\), so \(y=20\), \(x=40\). Point \((40,20)\).
Solve \(x+y=60\) with \(y=0\) (the x axis): gives \(x=60\). Point \((60,0)\).
Solve \(x+2y=120\) with \(y=0\): gives \(x=120\). Point \((120,0)\).
Check that \(x+2y=120\) and \(x+y=60\) meet at \((0,60)\), but this fails \(x-2y\ge0\) since \(0-120<0\), so it is rejected as not feasible.
So the feasible region is a quadrilateral with corners \((60,0)\), \((120,0)\), \((60,30)\), \((40,20)\).

Step 4: Evaluate Z at each corner point:

Corner Point	Z = 5x + 10y
(60, 0)	300
(120, 0)	600
(60, 30)	600
(40, 20)	400

The largest value, 600, is achieved at two corner points, \((120,0)\) and \((60,30)\).
Since both lie on the line \(x+2y=120\), and \(Z=5(x+2y)\) on this whole line, Z equals 600 at every point of the segment joining them, not just at the two corners.

Final Answer:
The maximum value of Z is 600, attained along the whole edge from (60,30) to (120,0), giving infinitely many optimal solutions. \[ \boxed{Z_{max} = 600 \text{ at all points on the segment joining } (60,30) \text{ and } (120,0)} \]