Question

Two statements are provided: an Assertion (A) and a Reason (R). Select the correct answer from the codes (A), (B), (C), and (D) given below: Assertion (A): Consider a Linear Programming Problem with minimise $Z = x + 2y$ subject to constraints $2x + y \geq 3$, $x + 2y \geq 6$, $x, y \geq 0$ which gives minimum $Z$ at infinitely many points. The corner points of feasible region are $(0, 3)$ and $(6, 0)$.
Reason (R): If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

When the coefficients of the variables in the objective function $Z = ax + by$ are perfectly proportional to the coefficients of a bounding constraint line $ax + by = c$, the optimal solution will always occur along the entire corresponding line segment, generating infinitely many solutions.

Answer 1

The Correct Option is A

Concept: In Linear Programming Problems (LPP), the feasible region is determined by the intersection of all given linear constraints, including non-negativity restrictions. According to the Fundamental Theorem of Linear Programming, if an optimal value (maximum or minimum) of an objective function exists, it must occur at one of the extreme boundary points, known as the corner points or vertices of the feasible region. Furthermore, if the optimal value is attained at two distinct corner points, then the objective function will yield that identical optimal value at any and every point lying along the line segment connecting those two specific corner points, creating infinitely many optimal solutions.

Step 1: Determining the boundary lines and finding the exact corner points of the feasible region.
To locate the corner points of the region defined by the system of linear inequalities, let us convert the inequality constraints into standard linear equations to locate their geometric intercepts:
• Constraint 1: $2x + y \geq 3$. The corresponding boundary line equation is $2x + y = 3$.
• To find the $y$-intercept, set $x = 0$: $2(0) + y = 3 \implies y = 3$. Point is $A(0, 3)$.
• To find the $x$-intercept, set $y = 0$: $2x + 0 = 3 \implies x = \frac{3}{2} = 1.5$. Point is $B(1.5, 0)$.
• Constraint 2: $x + 2y \geq 6$. The corresponding boundary line equation is $x + 2y = 6$.
• To find the $y$-intercept, set $x = 0$: $0 + 2y = 6 \implies y = 3$. Point is $A(0, 3)$.
• To find the $x$-intercept, set $y = 0$: $x + 2(0) = 6 \implies x = 6$. Point is $C(6, 0)$. Now, let us consider the inequalities $2x + y \geq 3$ and $x + 2y \geq 6$ along with the non-negativity constraints $x \geq 0, y \geq 0$. Testing the origin $(0,0)$ in both inequalities gives $0 \geq 3$ (False) and $0 \geq 6$ (False), indicating the feasible region lies away from the origin in the first quadrant. The boundary vertices that bound this open, unbounded region on the low side are exactly the points $(0, 3)$ and $(6, 0)$. The point $(1.5, 0)$ is inside the non-feasible zone since it does not satisfy $x + 2y \geq 6$ ($1.5 + 0 = 1.5 \nsim \geq 6$). Thus, the actual corner points of the feasible region are indeed $(0, 3)$ and $(6, 0)$.

Step 2: Evaluating the objective function $Z = x + 2y$ at each valid corner point.
Let us calculate the exact quantitative value of our objective function $Z$ at the two identified vertices:
• At the first corner point $(0, 3)$: \[ Z_{(0,3)} = 0 + 2(3) = 6 \]
• At the second corner point $(6, 0)$: \[ Z_{(6,0)} = 6 + 2(0) = 6 \] We observe that the objective function values at both independent boundary coordinates are precisely equal to $6$.

Step 3: Assessing the validity of Assertion (A) and Reason (R).
Since both distinct corner points yield the exact same minimum value ($Z = 6$), any linear convex combination of these two points will yield the exact same value. Since a line segment contains an infinite number of points, the LPP possesses infinitely many points that minimize $Z$. Thus, Assertion (A) is completely True. Now looking at Reason (R): It states that if two corner points produce the same minimum value, then every point on the line segment connecting them will also give that minimum value. This statement is a standard, mathematically sound theorem of optimization theory. Therefore, Reason (R) is True. Moreover, Assertion (A) concludes that there are infinitely many minimizing points precisely because the two corner points yield identical values, which is the exact behavior described by Reason (R). Hence, Reason (R) is the direct and correct explanation for Assertion (A).