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Chapter 7: Problem 10

Optimize \(6 x+4 y\) subject to $$ \begin{array}{r} x+y \geq 6 \\ 2 x+y \geq 9 \end{array} $$

Short Answer

Expert verified
Maximum value is 36 at \((6,0)\) and \((0,9)\).

Step by step solution

01

Identify the Objective Function and Constraints

The goal is to optimize the objective function, which is given as \(6x + 4y\). The constraints are \(x + y \geq 6\) and \(2x + y \geq 9\).
02

Graph the Inequalities

Plot the lines \(x + y = 6\) and \(2x + y = 9\) on a coordinate grid. For \(x + y = 6,\) the line has intercepts (6,0) and (0,6). For \(2x + y = 9,\) the line has intercepts (4.5,0) and (0,9). To shade the feasible region, choose points that satisfy both \(x + y \geq 6\) and \(2x + y \geq 9\). The feasible region will be above these lines.
03

Determine the Feasible Region

The feasible region is the intersection of the shaded areas from the inequalities. Check corners to find the vertices, which are points where the bounding lines intersect each other. The region is unbounded towards the positive x and y direction, but has vertices to test.
04

Find Intersection Points

Calculate where the lines intersect by solving the system of equations: \(x + y = 6\) and \(2x + y = 9\). Subtract one equation from the other: \((2x+y)-(x+y)=9-6\), which simplifies to \(x = 3\). Substitute \(x = 3\) back into one of the original equations, e.g., \(3 + y = 6\), giving \(y = 3\). Thus, the intersection point is \((3,3)\).
05

Evaluate the Objective Function at Vertex

Check the value of the objective function \(6x + 4y\) at the point \((3,3)\). Substitute these values into the function: \(6(3) + 4(3) = 18 + 12 = 30\).
06

Check for Other Vertices

Since both inequalities intersect and the region is unbounded to the top and right, check points along the boundary closest to the origin, like \((6,0)\) and \((0,9)\). Substitute into the objective function: \(6(6) + 4(0) = 36\) and \(6(0) + 4(9) = 36\).
07

Compare Values

Evaluate all feasible points. The objective function reached a maximum value of 36 at both \((6,0)\) and \((0,9)\). The point \((3,3)\) yielded 30, which is less than the maximum found at the endpoints.
08

Conclusion

The maximum value of \(6x + 4y\) occurs along points of the feasible region on the boundary, specifically points \((6,0)\) and \((0,9)\), with objective value being 36.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Objective Function
In linear optimization, the objective function is what you're trying to optimize, either maximizing or minimizing. Here, the given objective function is \(6x + 4y\). This means for every unit of \(x\), the result is scaled by 6, and for every unit of \(y\), it is scaled by 4.
To optimize this function, we determine how the values of \(x\) and \(y\) can be chosen to yield the highest or lowest value, depending on whether it is a maximization or minimization problem.
The objective function plays the role of guiding us in choosing the best combination of variables that satisfies certain conditions.
Constraints
The constraints in a linear optimization problem set the limitations or boundaries within which the solution must lie. In our problem, they are: \(x + y \geq 6\) and \(2x + y \geq 9\).
These constraints can be visualized as lines when graphed on a coordinate plane. They create a border, shaping the space within which the potential solutions exist.
  • Constraints not only shape the feasible region but also influence which solutions are permissible.

  • Effective problem-solving requires checking each constraint to ensure any solution lies within all bounds.
The constraints define a system of inequalities that bound the solution space, often resulting in a region that needs to be explored for possible optimal solutions.
Feasible Region
The feasible region is a crucial concept in linear optimization, consisting of all possible points that satisfy the problem's constraints. For this exercise, the region is formed by the intersection of the shaded areas that meet the conditions \(x + y \geq 6\) and \(2x + y \geq 9\).
  • By graphing these inequalities, we visually locate the area that meets both requirements.

  • The feasible region may be bounded or unbounded, depending on the constraints.
In this case, the feasible region is unbounded towards the positive x and y directions, however, it still possesses certain key vertices that provide critical points for calculation.
Understanding the feasible region is key to finding potential solutions for the objective function, as it limits where we can look for the best values of \(x\) and \(y\).
Intersection Points
Intersection points are where the lines representing constraints meet on the graph. Calculating these points helps in determining the vertices of the feasible region, which are potential candidates for optimizing the objective function.
In our setup, the lines are \(x + y = 6\) and \(2x + y = 9\). Solving these equations simultaneously, we determined the intersection point as \((3,3)\).
  • Intersection points are crucial because they often hold optimal solutions, especially in bounded regions.

  • They are calculated through systems of equations, which can be solved either by substitution or elimination methods.
By evaluating the objective function at these intersection points, we can check if they provide optimal or extreme values within the feasible region.

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Most popular questions from this chapter

Use the Dichotomous Search Method with a tolerance of \(t=0.2\) and \(\varepsilon=0.01\). a. Minimize \(f(x)=x^{2}+2 x,-3 \leq x \leq 6\) b. Maximize \(f(x)=-4 x^{2}+3.2 x+3,-2 \leq x \leq 2\)

For Problems 8-12, find both the maximum solution and the minimum solution using graphical analysis. Assume \(x \geq 0\) and \(y \geq 0\) for each problem. Fit the model to the data using Chebyshev's criterion to minimize the largest deviation. a. \(y=c x\) \begin{tabular}{l|cccc} \(y\) & 11 & 25 & 54 & 90 \\ \hline\(x\) & 5 & 10 & 20 & 30 \end{tabular} b. \(y=c x^{2}\) \begin{tabular}{l|llll} \(y\) & 10 & 90 & 250 & 495 \\ \hline\(x\) & 1 & 3 & 5 & 7 \end{tabular}

Producing Electronic Equipment - An electronics firm is producing three lines of products for sale to the government: transistors, micromodules, and circuit assemblies. The firm has four physical processing areas designated as follows: transistor production, circuit printing and assembly, transistor and module quality control, and circuit assembly test and packing. The various production requirements are as follows: Production of one transistor requires \(0.1\) standard hour of transistor production area capacity, \(0.5\) standard hour of transistor quality control area capacity, and \(\$ 0.70\) in direct costs. Production of micromodules requires \(0.4\) standard hour of the quality control area capacity, three transistors, and \(\$ 0.50\) in direct costs. Production of one circuit assembly requires \(0.1\) standard hour of the capacity of the circuit printing area, \(0.5\) standard hour of the capacity of the test and packing area, one transistor, three micromodules, and \(\$ 2.00\) in direct costs. Suppose that the three products (transistors, micromodules, and circuit assemblies) may be sold in unlimited quantities at prices of \(\$ 2, \$ 8\), and \(\$ 25\) each, respectively. There are 200 hours of production time open in each of the four process areas in the coming month. Formulate a mathematical model to help determine the production that will produce the highest revenue for the firm.

Optimize \(x-y\) subject to $$ \begin{array}{r} x+y \geq 6 \\ 2 x+y \geq 9 \end{array} $$

Optimize \(6 x+4 y\) subject to $$ \begin{array}{r} -x+y \leq 12 \\ x+y \leq 24 \\ 2 x+5 y \leq 80 \end{array} $$
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