The corner points of the bounded feasible region of a linear programming problem are \((0, 0)\), \((8, 0)\), \((6, 4)\) and \((0, 7)\). Let \(Z = px + qy\), where \(p, q \gt 0\). Consider the following statements:
(A) If \(p = 2q\), the maximum of Z occurs at both \((8, 0)\) and \((6, 4)\).
(B) If \(p = 2q\), the maximum of Z occurs at both \((6, 4)\) and \((0, 7)\).
(C) If \(p = q\), the maximum of Z occurs only at \((6, 4)\).
(D) If \(q = 3p\), the maximum of Z occurs only at \((0, 7)\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
Values of \(Z = px + qy\) at the corner points (other than the origin, where \(Z = 0\)):
\(Z(8, 0) = 8p\)
\(Z(6, 4) = 6p + 4q\)
\(Z(0, 7) = 7q\)
The maximum occurs at two corners together only when Z has equal values there and no other corner is larger.
At (8, 0) and (6, 4):
\(8p = 6p + 4q\)
\(p = 2q\)
At (6, 4) and (0, 7):
\(6p + 4q = 7q\)
\(q = 2p\)
So the maximum is at both (8, 0) and (6, 4) when \(p = 2q\), and at both (6, 4) and (0, 7) when \(q = 2p\).
Checking the statements
(A) If \(p = 2q\), the maximum of Z occurs at both \((8, 0)\) and \((6, 4)\). — Correct. Values \(16q, 16q, 7q\): the largest value is at both points.
(B) If \(p = 2q\), the maximum of Z occurs at both \((6, 4)\) and \((0, 7)\). — Incorrect. \(Z(0, 7) = 7q\) is less than \(16q\); the condition needed is \(q = 2p\).
(C) If \(p = q\), the maximum of Z occurs only at \((6, 4)\). — Correct. Values \(8p, 10p, 7p\): the largest is \(10p\) at (6, 4).
(D) If \(q = 3p\), the maximum of Z occurs only at \((0, 7)\). — Correct. Values \(8p, 18p, 21p\): the largest is \(21p\) at (0, 7).