Problem 30 - Finite Mathematics and Applied Calculus

Question

Music CD Sales Your music store's main competitor, Nuttal Hip Hop Classic Store, also wishes to stock at most 20,000 CDs, with at least half as many rap CDs as rock CDs and at least 2,000 classical CDs. It anticipates an average sale price of \(\$ 15 /\) rock CD, \$10/rap CD and \$10/classical CD. How many of each type of CD should it stock to get the maximum retail value, and what is the maximum retail value?

Step-by-Step Solution

Answer

To maximize the retail value, Nuttal Hip Hop Classic Store should stock \(12,000\) rock CDs, \(6,000\) rap CDs, and \(2,000\) classical CDs. The maximum retail value with this stock is \(\$270,000\).

Step 1: (Step 1: Define the variables)

Let's define the variables for the different types of CDs the store wishes to stock: x1: number of rock CDs x2: number of rap CDs x3: number of classical CDs

Step 2: (Step 2: Formulate the objective function)

We want to maximize the total sale price, so the objective function will be the sum of the sale prices for each type of CD. \(Z = 15x1 + 10x2 + 10x3\)

Step 3: (Step 3: Formulate the constraints)

We have the following constraints due to the store's preferences: 1. The total number of CDs (rock, rap, and classical) should be at most 20,000: \(x1 + x2 + x3 \leq 20000\) 2. At least half as many rap CDs as rock CDs: \(x2 \geq \frac{1}{2}x1\), which can also be expressed as: \(x1 - 2x2 \leq 0\) 3. At least 2,000 classical CDs: \(x3 \geq 2000\) 4. All variables must be non-negative: \(x1, x2, x3 \geq 0\)

Step 4: (Step 4: Graph the constraints)

To graph the constraints, express each of them in terms of two variables (fix one variable) and find the feasible region. First, fix x3 = 2000 (minimum classical CDs). The constraints become: \(x1 + x2 \leq 18000\) \(x1 - 2x2 \leq 0\) Plot these (along with the non-negativity constraints) on an x1-x2 plane and shade the feasible region.

Step 5: (Step 5: Identify the corner points of the feasible region)

Evaluate the corner points generated by the intersection of these lines. Label them as A, B, C, and D, and note their coordinates.

Step 6: (Step 6: Solve the linear programming problem)

Substitute the coordinates of each of these corner points (A, B, C, and D) into the objective function Z to find the optimal solution that maximizes Z. The maximum value is the largest value of Z from these evaluations.

Step 7: (Step 7: Find the optimal solution)

Identify the corner point corresponding to the maximum value of Z. This point will provide the optimal solution, i.e., the number of each type of CD the store should stock for maximum retail value. Calculate this retail value using the optimal solution.

Key Concepts

Objective Function

In the realm of linear programming, the objective function is a crucial component. It is the mathematical expression that represents a goal that needs to be optimized, either maximized or minimized.
In this exercise, our goal is to maximize the total revenue generated from selling different types of CDs. Therefore, the objective function is formulated based on the expected sale price of each CD type:

• For rock CDs, the price is \(15 each, represented as \(15x_1\).
• For rap CDs, the price is \)10 each, contributing as \(10x_2\).
• For classical CDs, again priced at $10 each, forming \(10x_3\).

By adding these together, we obtain the objective function: \(Z = 15x_1 + 10x_2 + 10x_3\). Maximizing \(Z\) means we look for the combination of rock, rap, and classical CDs that results in the highest total revenue.

Constraints

Constraints in linear programming define the limitations or restrictions within which the solution must comply. They are expressed as mathematical inequalities or equations.
In this problem, several constraints set the boundaries for the decision variables:\(x_1\),\(x_2\), and \(x_3\). They are listed as follows:

• The store can stock a maximum of 20,000 CDs, forming the constraint: \(x_1 + x_2 + x_3 \leq 20000\).
• The number of rap CDs must be at least half the number of rock CDs, which translates to: \(x_2 \geq \frac{1}{2}x_1\) or \(x_1 - 2x_2 \leq 0\).
• The store aims to keep at least 2,000 classical CDs: \(x_3 \geq 2000\).
• All the variables must be non-negative, meaning \(x_1, x_2, x_3 \geq 0\).

Constraints like these ensure the solution remains realistic and aligned with the physical and business limits of the scenario.

Feasible Region

The feasible region is a set of all possible points that satisfy the constraints of a linear programming problem. This region defines where the solution could potentially be found.
For our CD stocking problem, identifying the feasible region involves plotting the constraints on a graph. Typically, this is done:

• By representing constraints as lines or curves in a coordinate system where each axis represents a decision variable, such as \(x_1\) and \(x_2\).
• Shading or highlighting the area where all constraints intersect or overlap.

In this given exercise, constraints must be plotted for \(x_1\) and \(x_2\), with \(x_3\) fixed at its minimum of 2000 CDs. The feasible region is the area where these constraints all hold true. Any points within this region represent a viable solution set for the linear programming problem.

Optimization Problem

An optimization problem in linear programming is all about finding the best solution from a set of feasible solutions. This could involve maximizing or minimizing the objective function. Steps to solve an optimization problem include:

• Formulating the objective function, which specifies what is being optimized.
• Identifying the constraints that form the boundaries of the problem.
• Graphing the constraints to determine the feasible region.
• Finding the optimal point, usually located at the intersection (corner) of the constraints.

In our CD store example, we aim to maximize the retail value by adjusting how many CDs of each type are stocked. Solving the optimization problem involves checking all corner points of the feasible region and assessing which yields the highest value of the objective function \(Z = 15x_1 + 10x_2 + 10x_3\). This step is fundamental to ensure decisions lead to the best possible outcome for the given constraints.