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Chapter 5: 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 restriction on the decision variables. They are represented mathematically as linear equations or inequalities.

Step by step solution

01

Define Constraint in Linear Programming

A constraint in a linear programming problem is a restriction or limitation imposed on a decision. In mathematical terms, it's an equation or inequality that define the conditions that the solution must satisfy. For instance, if we want to maximize profit in a production process, constraints could be the availability of raw material, labor hours etc. In simple words, constraints define the feasible region within which we search for the optimal solution.
02

Representation of Constraint

Constraints are represented mathematically as linear equations or inequalities depending on the problem statement. For example, if we have a constraint that the total hours worked by an employee must be less than or equal to 40 hours per week, this can be represented as \(x \leq 40\), where \(x\) represents the total hours worked by an employee in a week.

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Most popular questions from this chapter

Without graphing, in Exercises 73鈥�76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} 6 x-y \leq 24 \\ 6 x-y>24 \end{array}\right.$$

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 $_____.

What is an objective function in a linear programming problem?

Consider the objective function \(z-A x+B y \quad(A>0\) 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 \(A-\frac{2}{3} B\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are a circle and a line can have four real ordered-pair solutions
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