The corner points of the feasible region determined by: x + y ≤ 8, 2x + y ≥ 8, x ≥ 0, y ≥ 0 are A(0, 8), B(4, 0), and C(8, 0). If the objective function Z = ax + by has its maximum value on the line segment AB, then the relation between a and b is:
a = 2b
Analyzing the Objective Function Maximum in Linear Programming
The problem involves finding the relationship between the coefficients 'a' and 'b' of an objective function \(Z = ax + by\), given that its maximum value occurs on a specific edge of the feasible region defined by a set of linear inequalities. The feasible region is determined by the constraints: \(x + y \le 8\), \(2x + y \ge 8\), \(x \ge 0\), and \(y \ge 0\). The corner points of this feasible region are given as A(0, 8), B(4, 0), and C(8, 0).
Understanding Maximum Value Location
In Linear Programming, the optimal (maximum or minimum) value of the objective function always occurs at one of the corner points of the feasible region. However, if the optimal value occurs at two adjacent corner points, it means that the entire line segment connecting these two points also yields the same optimal value. In this problem, the maximum value of the objective function \(Z = ax + by\) is stated to occur on the line segment AB.
Calculating Objective Function Value at Corner Points
Since the maximum value occurs on the line segment AB, the value of the objective function \(Z\) must be the same at both corner point A(0, 8) and corner point B(4, 0). Let's calculate the value of Z at these two points.
At point A(0, 8):
Substitute \(x = 0\) and \(y = 8\) into the objective function \(Z = ax + by\).
\[Z_A = a(0) + b(8) = 0 + 8b = 8b\]
At point B(4, 0):
Substitute \(x = 4\) and \(y = 0\) into the objective function \(Z = ax + by\).
\[Z_B = a(4) + b(0) = 4a + 0 = 4a\]
Finding the Relation Between a and b
As the maximum value occurs on the line segment AB, the value of Z at A must be equal to the value of Z at B.
\[Z_A = Z_B\]
\[8b = 4a\]
To find the relation between a and b, we can simplify this equation by dividing both sides by 4:
\[\frac{8b}{4} = \frac{4a}{4}\]
\[2b = a\]
So, the relation between a and b is \(a = 2b\).
This relationship indicates that for the objective function \(Z = ax + by\) to have its maximum along the edge connecting (0,8) and (4,0), the coefficient 'a' must be twice the coefficient 'b'.
Let's verify this against the given options:
- Option 1: \(8a + 4 = b\) (Incorrect)
- Option 2: \(a = 2b\) (Correct)
- Option 3: \(b = 2a\) (Incorrect, this is the reverse relation)
- Option 4: \(8b + 4 = a\) (Incorrect)
Therefore, the correct relation between a and b is \(a = 2b\).
Revision Table: Linear Programming Concepts
| Concept | Description |
|---|---|
| Objective Function | The function (e.g., \(Z = ax + by\)) that is to be maximized or minimized. |
| Feasible Region | The set of all points \((x, y)\) that satisfy all the given constraints (inequalities). It is usually a convex polygon. |
| Corner Points (Vertices) | The points where the boundary lines of the feasible region intersect. Optimal solutions often occur at these points. |
| Corner Point Theorem | States that if a linear programming problem has an optimal solution, it must occur at a corner point of the feasible region. If it occurs at two corner points, it occurs at every point on the line segment connecting them. |
Additional Information: Gradient and Optimal Edge
The objective function \(Z = ax + by\) can be represented by a family of parallel lines \(ax + by = Z\). The value of Z increases as the line moves in a certain direction determined by the coefficients 'a' and 'b'. The direction of maximum increase is given by the vector \((a, b)\).
When the maximum value of Z occurs along an edge of the feasible region (like line segment AB), it means that the objective function line \(ax + by = Z_{max}\) is parallel to that edge AB. The edge AB connects A(0, 8) and B(4, 0).
The slope of the line segment AB is given by:
\[m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{0 - 8}{4 - 0} = \frac{-8}{4} = -2\]
The slope of the objective function line \(ax + by = Z\) (or \(by = -ax + Z\), so \(y = -\frac{a}{b}x + \frac{Z}{b}\)) is \(-\frac{a}{b}\), assuming \(b \neq 0\).
If the objective function line is parallel to edge AB, their slopes must be equal:
\[-\frac{a}{b} = -2\]
\[\frac{a}{b} = 2\]
\[a = 2b\]
This confirms the relation found by evaluating the objective function at the endpoints. This method provides an alternative way to think about why the maximum occurs on the edge – it's because the slope of the objective function's level curves matches the slope of the feasible region's boundary edge.
Important Questions from Linear Programming
An objective function Z = ax + by is maximum at points (8, 2) and (4, 6). If a ≥ 0, b ≥ 0, and ab = 25, then the maximum value of the function is:
Which of the following cannot be the direction ratios of the straight line:
\( \frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1} \)
Corner points of a feasible bounded region are (0,10), (4,2), (3,7) and (10,6). Maximum value 50 of objective function \( z = ax + by \) occurs at two points (0,10) and (10,6). The value of \( a \) and \( b \) are:
The corner points of the feasible region determined by system of linear constraints are (60,0), (120,0), (40,20), and (60,30). Let \( z = ax + by \), where \( a, b > 0 \) be the objective function. Find condition on \( a \) and \( b \) so that the maximum of \( z \) occurs at (120,0) and (60,30).
The feasible region corresponding to an LPP represented by the constraints \( x \geq 7, y \geq 4, x + 2y \geq 8 \) is:
