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Chapter 5: Problem 27

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

Short Answer

Expert verified
To solve a linear programming problem, first understand the problem and identify the decision variables, constraints, and the objective function. Then, formulate these into mathematical expressions. Finally, use the graphical or Simplex method to find where the objective function is optimized subject to the constraints. Interpret the results in terms of the original problem situation.

Step by step solution

01

Understand the Problem

The first thing to be done is identify the decision variables, limitations or constraints and the objective function from our problem statement. Decision variables are those quantities that we have control over and want to choose in a way that achieves our goal. Constraints are the set of limitations or conditions that are imposed on us and the decision variables. The objective function is the function that we wish to maximize or minimize.
02

Formulate the Mathematical Model

After identifying the key parts, these need to be translated into mathematical expressions. Decision variables are usually denoted by symbols like \(x\), \(y\), etc. Constraints are expressed as linear inequalities in terms of the decision variables. The objective function is expressed as a linear equation in terms of the decision variables and represents the quantity to be minimized or maximized.
03

Apply Linear Programming Method

Finally, once we have a proper represented mathematical model, we are now ready to solve the problem. If our problem involves two variables, we can use graphical methods: plot the feasible region which is determined by the constraints and then find the point in this region where the objective function is maximized or minimized. If our problem involves more than two variables, numerical methods such as the Simplex method can be used to find the solution.
04

Interpretation of Results

After solving the problem, the obtained solution must be interpreted in terms of the original problem, checking whether it's feasible or not and it satisfies the posed conditions.

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Most popular questions from this chapter

The figure shows the healthy weight region for various heights for people ages 35 and older. GRAPH CAN'T COPY If \(x\) represents height, in inches, and \(y\) represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$\left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right.$$ Use this information to solve Exercises 77-80. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

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?

On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. "The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 \(\cdot\) The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000 .\) Total weekly costs could not exceed \(\$ 300,000\) Find the number of American and British planes that were used to maximize cargo capacity.

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.

Without graphing, in Exercises 73鈥�76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} 3 x+y<9 \\ 3 x+y>9 \end{array}\right.$$
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