91Ó°ÊÓ

Purchasing You are in charge of purchases at the student-run used-book supply program at your college, and you must decide how many introductory calculus, history, and marketing texts should be purchased from students for resale. Due to budget limitations, you cannot purchase more than 650 of these textbooks each semester. There are also shelf-space limitations: Calculus texts occupy 2 units of shelf space each, history books 1 unit each, and marketing texts 3 units each, and you can spare at most 1,000 units of shelf space for the texts. If the used book program makes a profit of \(\$ 10\) on each calculus text, \(\$ 4\) on each history text, and \(\$ 8\) on each marketing text, how many of each type of text should you purchase to maximize profit? What is the maximum profit the program can make in a semester? HINT [See Example 3.]

Step by step solution

01

Define the variables

To begin, let's define our variables: \(x_1\) = number of calculus texts \(x_2\) = number of history texts \(x_3\) = number of marketing texts.

02

Set up the objective function

The objective is to maximize the profit from purchasing the textbooks. The profit is given by: \(P = 10x_1 + 4x_2 + 8x_3\)

03

Set up the constraints

Now, we need to set up the constraints due to budget limitations and shelf space: Budget constraint: The total number of textbooks cannot exceed 650. \(x_1 + x_2 + x_3 \leq 650\) Shelf space constraint: The total shelf space occupied by the textbooks cannot exceed 1,000 units. \(2x_1 + x_2 + 3x_3 \leq 1000\) Non-negative constraint: The number of textbooks purchased cannot be negative. \(x_1, x_2, x_3 \geq 0\)

04

Solve the linear programming problem

With the objective function and constraints set up, we can now use a graphical method or a linear programming solver to find the optimal solution for purchasing the textbooks: Objective Function: \(P = 10x_1 + 4x_2 + 8x_3\) Constraints: \(x_1 + x_2 + x_3 \leq 650\) \(2x_1 + x_2 + 3x_3 \leq 1000\) \(x_1, x_2, x_3 \geq 0\) The optimal solution is found at the point where all constraints are met, and the objective function is maximized. Using a linear programming solver or a graphical method, we find that: \(x_1 = 225\) \(x_2 = 0\) \(x_3 = 425\)

05

Calculate the maximum profit

Finally, substitute the optimal textbook purchase numbers back into the profit function to find the maximum profit: \(P = 10(225) + 4(0) + 8(425) = 2250 + 0 + 3400 = \(\$ 5650\)\) The maximum profit that the program can make in one semester is \$5650.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Optimization

Optimization in the context of linear programming is the process of finding the best solution within a defined set of constraints. This is typically accomplished by maximizing or minimizing an objective function. In the exercise, the objective is to maximize the profit generated from purchasing and reselling textbooks.

Every optimization problem has an objective function, which in this case is represented by the equation for profit, expressed as \(P = 10x_1 + 4x_2 + 8x_3\). The variables \(x_1\), \(x_2\), and \(x_3\) represent the number of calculus, history, and marketing texts to purchase, respectively. Optimization seeks the combination of \(x_1\), \(x_2\), and \(x_3\) that yields the highest possible profit while satisfying all constraints.

Constraints

Constraints are limitations or requirements that must be met in any optimization problem. In the textbook purchasing example, there are two primary constraints: the budget limit and the shelf space limit.

The budget constraint is defined by the total number of textbooks, \(x_1 + x_2 + x_3 \leq 650\), and inherently restricts the amount of each textbook type that can be purchased. Similarly, the shelf space constraint, defined as \(2x_1 + x_2 + 3x_3 \leq 1000\), accounts for the physical space each type of textbook occupies. These constraints create a feasible region within which the optimal solution must lie. It's critical to correctly identify and incorporate all constraints to accurately represent the problem in a linear programming model.

Objective Function

The objective function in a linear programming problem is a mathematical representation of the goal you're trying to achieve. It quantifies what you're optimizing for—in this scenario, it's the profit from selling books.

The objective function is \(P = 10x_1 + 4x_2 + 8x_3\), where \(P\) represents the total profit, and the variables represent the number of textbooks purchased. Each textbook type contributes a different amount to the profit, reflected by the coefficients 10, 4, and 8 in the function. Understanding and correctly setting up the objective function is vital as it drives the direction of the optimization process in seeking the highest (maximization) or lowest (minimization) value possible.

Non-Negative Variables

Non-negative variables are a key component of linear programming problems, ensuring that solutions make sense in real-world contexts. In our exercise, the variables \(x_1\), \(x_2\), and \(x_3\) represent the numbers of calculus, history, and marketing textbooks to be purchased. These values cannot be negative, as you cannot purchase a negative number of books.

Therefore, the non-negativity constraint is expressed as \(x_1, x_2, x_3 \geq 0\). Ensuring that all decision variables meet this non-negativity condition is crucial for finding a practical and realistic solution to the optimization problem.