To minimize the objective function [tex]\( P = 5x - 27y + 1 \)[/tex] subject to the given constraints [tex]\( y \leq 3 \)[/tex], [tex]\( y + x \geq 0 \)[/tex], and [tex]\( y \geq x - 2 \)[/tex], we follow these steps:
1. Rewrite the constraints in a standard form:
[tex]\[
\begin{array}{l}
y \leq 3 \\
y + x \geq 0 \quad \text{or} \quad x \geq -y \\
y \geq x - 2 \quad \text{or} \quad y - x \geq -2
\end{array}
\][/tex]
2. Identify the feasible region: The constraints define a region in the (x, y) coordinate system where all these inequalities are satisfied. This region is a polygonal area on the graph.
3. Find the vertices of the feasible region: The vertices are the points of intersection of the boundary lines of the constraints. To find these points, solve the equations of intersecting lines:
- Intersection of [tex]\( y \leq 3 \)[/tex] and [tex]\( y + x \geq 0 \)[/tex]:
[tex]\[
y = 3,\; x = -3
\][/tex]
The point is [tex]\((-3, 3)\)[/tex].
- Intersection of [tex]\( y \leq 3 \)[/tex] and [tex]\( y \geq x - 2 \)[/tex]:
[tex]\[
y = 3,\; y = x - 2 \implies 3 = x - 2 \implies x = 5
\][/tex]
The point is [tex]\((5, 3)\)[/tex].
- Intersection of [tex]\( y + x \geq 0 \)[/tex] and [tex]\( y \geq x - 2 \)[/tex]:
[tex]\[
y = -x,\; y = x - 2 \implies -x = x - 2 \implies 2x = 2 \implies x = 1,\; y = -1
\][/tex]
The point is [tex]\((1, -1)\)[/tex].
4. Evaluate the objective function [tex]\( P(x, y) = 5x - 27y + 1 \)[/tex] at these vertices:
- At [tex]\((-3, 3)\)[/tex]:
[tex]\[
P(-3, 3) = 5(-3) - 27(3) + 1 = -15 - 81 + 1 = -95
\][/tex]
- At [tex]\((5, 3)\)[/tex]:
[tex]\[
P(5, 3) = 5(5) - 27(3) + 1 = 25 - 81 + 1 = -55
\][/tex]
- At [tex]\((1, -1)\)[/tex]:
[tex]\[
P(1, -1) = 5(1) - 27(-1) + 1 = 5 + 27 + 1 = 33
\][/tex]
Therefore, the minimum value of [tex]\( P \)[/tex] over the feasible region is [tex]\( -95 \)[/tex] at the point [tex]\((-3, 3)\)[/tex].
The minimum is: [tex]\(-95\)[/tex] at [tex]\((-3, 3)\)[/tex].
