Question

Let $R$ be the feasible region (convex polygon) for a linear programming problem and $Z=a x+b y$ be the objective function. Then, which of the following statements is false?

• A. When $Z$ has an optimal value, where the variables $x$ and $y$ are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region
• B. If $R$ is bounded, then the objective function $Z$ has both a maximum and a minimum value on $R$ and each of these occurs at a corner point of $R$
• C. If $R$ is unbounded, then a maximum or a minimum value of the objective function may not exist
• D. If $R$ is unbounded and a maximum or a minimum value of the objective function, then it does not occur at a corner point

Answer 1

Answer: (D)

(a) Let $R$ be the feasible region (convex polygon) for a linear programming problem and let $Z=a x+b y$ be the objective function. When $Z$ has an optimal value (maximum or minimum), where the variables $x$ and $y$ are subject to constraints described by linear inequalities, this optimal value must occur at a comer point (vertex) of the feasible region.
(b) Let $R$ be the feasible region for a linear programming problem and let $Z=a x+b y$ be the objective function. If $R$ is bounded, then the objective function $Z$ has both a maximum and a minimum value on $R$ and each of these occurs at a corner point (vertex) of $R$.
(c) If $R$ is unbounded, then a maximum or a minimum value of the objective function may not exist.
(d) If the maximum or a minimum value of $Z$ exists, it must occur at a corner point of $R$.