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Chapter 5: Problem 24

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

Short Answer

Expert verified
Linear programming is used to solve problems that involve optimizing an objective given several constraints, and where all relationships are linear. This method is applicable in multiple domains such as business, economics, manufacturing, and others which need the optimal allocation of scarce resources.

Step by step solution

01

Define Linear Programming

Linear programming is a method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It's an optimization technique.
02

Identify Characteristics of Problems

Linear programming is typically used for problems that have the following characteristics: \n 1. Optimization: Linear programming is used when we have certain resources and we want to allocate them in a way that maximizes profit or minimizes cost.\n 2. Constraints: These are problems where there are restrictions on the use of resources.\n 3. Linearity: The relationships of any constraints and the optimization objective should be linear in nature.
03

Application Areas

The applications of linear programming span many areas including business, economics, transportation, energy, telecommunications, manufacturing, and many more. Examples of specific problems could include: maximizing revenue in a business given constraints on labor and materials or minimizing waste in a manufacturing process with given input constraints.

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Most popular questions from this chapter

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x=3 y+7\\\ &x=2 y-1 \end{aligned} $$

Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. A company that manufactures bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) One pan pizza and two beef burritos provide 1980 calories. Two pan pizzas and one beef burrito provide 2670 calories. Find the caloric content of each item.

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\frac{x}{6}-\frac{y}{2}=\frac{1}{3}\) \(x+2 y=-3\)

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=-16\) \(5 x-10 y=30\)
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