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Chapter 8: 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 where the best outcome needs to be achieved subject to constraints. This spans across a wide range of domains including business, economics, operations management etc. where the aim could be to minimize resources or cost, or maximize profits, reach, efficiency etc.

Step by step solution

01

Understanding Linear Programming

Linear programming method is a technique where we depict complex relationships through linear functions and then find the optimum points. It is used to solve problems that seek the minimum or maximum value of a linear function, subject to a set of constraints.
02

Identifying typical problems solved using Linear Programming

It's widely used in business and economics, and also in various industries. Examples could be maximizing profits or minimizing costs given several constraints in sectors like manufacturing industries, in operations management to minimize the cost of resources, in marketing to maximize reach, in logistics to maximize efficiency of transport and supply chain, in telecommunications for data routing, etc. The number of applications is vast and spans across various domains.
03

Summarization

So, the types of problems solved by linear programming are mostly optimization problems where some sort of restriction is imposed and the goal is to either minimize or maximize a particular criterion.

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Most popular questions from this chapter

Sketch the graph of the solution set for the following system Of inequalities: $$ \left\\{\begin{array}{l} {y \geq n x+b(n<0, b>0)} \\ {y \leq m x+b(m>0, b>0)} \end{array}\right. $$

Consider the objective function \(z=A x+B y(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9\) \(x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the objective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A=\frac{2}{3} B\).

For the linear function \(f(x)=m x+b, f(-2)=11\) and \(f(3)=-9 .\) Find \(m\) and \(b\).

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=10 x+12 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {x+y \leq 7} \\ {2 x+y \leq 10} \\ {2 x+3 y \leq 18} \end{array}\right. \end{aligned} $$
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