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Chapter 7: Problem 27

In your own words, describe how to solve a linear programming problem.

Short Answer

Expert verified
To solve a linear programming problem, one would understand the problem, define the variables, formulate the objective function and constraints, represent the problem graphically to find the feasible region, and finally interpret the solution obtained.

Step by step solution

01

- Understand the Problem

Before attempting to solve the problem, it's crucial to understand it thoroughly. Analyze the real world scenario that the problem is presenting. Identify what the problem is asking and what you need to solve.
02

- Define the Variables

Clarify the unknowns that you are trying to solve. These are typically the quantities that you wish to optimize. Define them with clear and concise variable names.
03

- Formulate the Objective Function

Determine the objective of the problem and formulate it as a mathematical expression. Typically, a linear programming problem is about maximizing or minimizing something. This something is described by the objective function, which is a linear expression in terms of the variables.
04

- Set up the Constraints

Identify the limitations or restrictions of the problem and turn them into mathematical inequalities. The solution to the problem must respect these constraints. All constraints form a system of inequalities.
05

- Solve the Problem Graphically

Plot the constraints on a graph to form a feasible region. The solutions of the linear programming lie in this feasible region. Then, determine the point in the feasible region at which the objective function is optimized (maximum or minimum). Evaluate the objective function at the corner points of the feasible region to find this point.
06

- Interpret the Results

Once you've acquired the solutions to the variables, ensure to interpret your results. This would typically involve going back to the original problem's context and explaining what the number values obtained for your variables signify in the real-world scenario.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every linear system has infinitely many ordered-pair solutions.

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Between 1990 and 2013 , there was a drop in violent crime and a spike in the prison population in the United States. The bar graph shows the number of violent crimes per \(100,000\) people and the number of imprisonments per \(100,000\) people for six selected years from 1990 through 2013. (GRAPH CAN NOT COPY) a. Based on the information in the graph, it appears that there was a year when the number of violent crimes per \(100,000\) Americans was the same as the number of imprisonments per \(100,000\) Americans. According to the graph, between which two years did this occur? b. The data can be modeled by quadratic and linear functions. Violent erime rate \(\quad y=0.6 x^{2}-28 x+730\) Imprisonment tate \(-15 x+y=300\) In each function, \(x\) represents the number of years after 1990 and \(y\) represents the number per \(100,000\) Americans. Solve a nonlinear system to determine the year described in part (a). Round to the nearest year. How many violent crimes per \(100,000\) Americans and how many imprisonments per \(100,000\) Americans were there in that year?
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