For the objective function $Z=ax+by$, if the optimal value occurs at two distinct corner points, then the number of optimal solutions is:

Solution

1. Let the optimal value be $k$ at points $P_1$ and $P_2$.
2. Any point on the line segment $P_1{P}_2$ can be written as $P=\lambda P_1+(1-\lambda )P_2$ for $0\le \lambda \le 1$.
3. Since the objective function is linear, $$\begin{array}{rl}Z(P)& =\lambda Z(P_1)+(1-\lambda )Z(P_2)\\ & =\lambda k+(1-\lambda )k\\ & =k\end{array}$$.
4. Thus, every point on the segment is an optimal solution.
5. Hence the answer is (D).