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Problem 10

# Solve the given LP problem. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. $\begin{array}{ll}\text { Minimize } & c=0.4 x+0.1 y \\ \text { subject to } & 30 x+20 y \geq 600 \\ & 0.1 x+0.4 y \geq 4 \\ & 0.2 x+0.3 y \geq 4.5 \\ & x \geq 0, y \geq 0\end{array}$

Expert verified
The optimal solution to the given LP problem exists at vertex A, $$(x, y) = (15, 6.25)$$, with a minimum cost of $$c = 6.625$$. The feasible region is not empty, and the objective function is not unbounded.
See the step by step solution

## Step 1: Identify the feasible region

First, let's write down the inequality constraints: 1. $$30x + 20y \geq 600$$ 2. $$0.1x + 0.4y \geq 4$$ 3. $$0.2x + 0.3y \geq 4.5$$ 4. $$x \geq 0$$, $$y \geq 0$$ The feasible region is the area in the $$xy$$-plane that satisfies all of these inequalities.

## Step 2: Find the vertices of the feasible region

We can find the vertices of the feasible region by graphing the inequality constraints and identifying the points of intersection of the boundary lines. 1. For constraint 1: At the boundary line, $$30x + 20y = 600$$. Re-write it as $$y = -\frac{3}{2}x + 30$$. 2. For constraint 2: At the boundary line, $$0.1x + 0.4y = 4$$. Re-write it as $$y = -\frac{1}{4}x + 10$$. 3. For constraint 3: At the boundary line, $$0.2x + 0.3y = 4.5$$. Re-write it as $$y = -\frac{2}{3}x + 15$$. Graph these lines and identify the vertices of the feasible region, which are the points of intersection of the boundary lines that satisfy all constraints. We have 3 vertices: 1. The intersection of constraints 1 and 2: $$(A) (15, 6.25)$$ 2. The intersection of constraints 1 and 3: $$(B) (20, 5)$$ 3. The intersection of constraints 2 and 3: $$(C) (18.75, 3.75)$$

## Step 3: Evaluate the objective function at each vertex

Now let's evaluate the objective function $$c = 0.4x + 0.1y$$ at each vertex: 1. For vertex A: $$c_A = 0.4(15) + 0.1(6.25) = 6 + 0.625 = 6.625$$ 2. For vertex B: $$c_B = 0.4(20) + 0.1(5) = 8 + 0.5 = 8.5$$ 3. For vertex C: $$c_C = 0.4(18.75) + 0.1(3.75) = 7.5 + 0.375 = 7.875$$

## Step 4: Determine the optimal solution

To minimize the objective function, we choose the vertex with the lowest value. Comparing the values of $$c_A$$, $$c_B$$, and $$c_C$$, we see that the minimum is achieved at vertex A, with a cost of 6.625. Thus, the optimal solution exists and is $$(x, y) = (15, 6.25)$$ with a minimum cost of $$c = 6.625$$. The feasible region is not empty and the objective function is not unbounded.

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