Question

The linear programming problem:
Maximize : \(Z=x+y\)
Subject to :
\(x+2y\leq 200,x+y\leq 150,0\leq y\leq 60,x\geq 0\) has:

• A. a finite number of optimal solutions
• B. a unique optimal solution
• C. infinite number of optimal solutions
• D. no feasible solution

Hint:

Find the corner points of the feasible region and compare Z. If two corners give the same maximum, the whole segment between them is optimal.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The maximum of a linear function occurs at corner points of the feasible region.

Step 2: Corner Points
The constraints are \(x+2y\leq200\), \(x+y\leq150\), \(0\leq y\leq60\), \(x\geq0\).
Corners: \((0,0)\), \((150,0)\), \((100,50)\) (from \(x+2y=200\) and \(x+y=150\)), \((80,60)\) and \((0,60)\).

Step 3: Evaluate Z
Z at these points: \(0,\ 150,\ 150,\ 140,\ 60\).

Step 4: Conclusion
The maximum \(150\) is attained at two adjacent corners \((150,0)\) and \((100,50)\).
Every point on the segment between them (on \(x+y=150\)) is also optimal.

Final Answer:
The problem has infinitely many optimal solutions. \[ \boxed{\text{(C) }\text{infinite number of optimal solutions}} \]