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Chapter 7: Problem 19

What kinds of problems are solved using the linear programming method?

Short Answer

Expert verified
Linear Programming is useful in solving different types of optimization problems - typically that involve finding the best way to use available resources, such as maximizing profit or output, minimizing cost or waste, and similar. Examples include production planning, diet planning, transportation problems, and task assignment problems.

Step by step solution

01

Definition of Linear Programming

Linear Programming is a mathematical method used for optimization of a linear objective function, subject to linear equality and linear inequality constraints. It's used to find the best (optimal) way to use available resources.
02

Main Usage of Linear Programming

The main usage of Linear Programming is in maximization or minimization problems, like maximizing profit, revenue, output, efficiency and similar, or minimizing cost, waste, loss, risk and similar. These sorts of problems can be found across a multitude of sectors (business, economics, engineering, etc).
03

Example of Linear Programming Problems

Here are some traditional types of problems solved using linear programming: 1) Production problems where the manufacturer wishes to generate a production plan to maximize profit or minimize cost while respecting constraints of raw material availability and production capacity. 2) Diet problems where one wants to minimize the cost of a certain nutrition plan, while still maintaining a minimum intake of certain necessary nutrients. 3) Transportation problems where routes or supply chains are optimized to minimize cost and time. 4) Assignment problems where tasks are assigned to resources in an optimal way.

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Most popular questions from this chapter

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x} \text {, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { [adadReg }\\\ &\begin{aligned} &y=3 \times 2+b x+c \\ &\bar{y}=.8 \\ &b=2.4 \\ &c=6 \end{aligned} \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs ____ feet from where it was thrown and the maximum height is ____ feet.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 2 \\ y \leq 3\end{array}\right.\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline 4 & 4 \\ \hline \end{array} $$

In Exercises 29-30, find the vertex for the parabola whose equation is given by writing the equation in the form \(y=a x^{2}+b x+c\). \(y=(x-3)^{2}+2\)
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