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Chapter 8: Problem 26

What is a constraint in a linear programming problem? How is a constraint represented?

Short Answer

Expert verified
In linear programming, a constraint is a condition that any solution to the problem must satisfy, often representing the limitations or requirements. Constraints are usually represented as linear inequalities that define feasible regions within a Cartesian plane.

Step by step solution

01

Understanding Constraints

In the context of linear programming, a constraint is a condition that the solution to an optimization problem must satisfy. It's the specific limitations or requirements that a problem's solutions need to meet. For example, if an optimization problem aims to maximize profit in a factory, constraints may set limits on the available resources, like raw materials or labour hours.
02

Representation of Constraints

Constraints in a linear programming problem are typically represented as linear inequalities. If the problem is represented in two dimensions (two variables), these inequalities define regions in a Cartesian plane; a point in this region satisfies all constraints and is a feasible solution to the problem. For instance, if labor hours can't exceed a certain number, it may be represented as a constraint like \( x \leq 40 \), where x is the number of labour hours.

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Most popular questions from this chapter

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$\begin{array}{lll} {} & {\text { Model } A} & {\text { Model } B} \\ {\text { Assembling }} & {5} & {4} \\ {\text { Painting }} & {2} & {3} \end{array}$$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are 25 for model A and 15 for model B . How many of each type should be produced to maximize profit?

\(g(x)=\frac{x-6}{x^{2}-36}\) (Section 3.5, Example 1)

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.

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

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A theater is presenting a program for students and their parents on drinking and driving. The proceeds will be donated to a local alcohol information center. Admission is 2.00 for parents and 1.00 for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?
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