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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 optimization problems. It is used in various fields like operations management, logistics, transportation, economics, diet planning, manufacturing, advertising budget allocation, and many more situations where the most efficient allocation of resources is required considering certain constraints.

Step by step solution

01

Understand Linear Programming

Linear programming is a method used in applied mathematics and computer science to find the best possible outcome in a given mathematical model. Usually, this method is employed to optimize resource allocation with the constraints at hand.
02

Identify Areas of Applications

The areas where linear programming is applied are broad. They are typically problems of optimization where one has a linear objective function and a system of linear inequalities or equations as constraints.
03

Examples

For instance, linear programming is used in operations management for scheduling, planning, and optimizing production. In logistics, it aids in managing and planning the flow of goods in a network. In transportation, it is used to find the optimal route. It is also commonly used in economics to solve profit maximization or cost minimization problems with given constraints. Moreover, linear programming is useful in problems related to diet planning, manufacturing, advertising budget allocation, and many more industries.

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Most popular questions from this chapter

What does a dashed line mean in the graph of an inequality?

Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\)-pound weight restriction.

a. A student earns \(\$ 10\) per hour for tutoring and \(\$ 7\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17), \text { and }(8,12) .]\) Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for hours _____ per week and working as a teacher鈥檚 aide for _____ hours per week. The maximum amount that the student can earn each week is $_____.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{l} 4 x+3 y=4 \\ y=2 x-7 \end{array}\right.$$
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