Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

A coffee company purchases mixed lots of coffee beans and then grades them into premium, regular, and unusable beans. The company needs at least 280 tons of premium-grade and 200 tons of regular-grade coffee beans. The company can purchase ungraded coffee from two suppliers in any amount desired. Samples from the two suppliers contain the following percentages of premium, regular, and unusable beans: $$ \begin{array}{|c|c|c|c|} \hline \text { Supplier } & \text { Premium } & \text { Regular } & \text { Unusable } \\ \hline \text { A } & 20 \% & 50 \% & 30 \% \\ \text { B } & 40 \% & 20 \% & 40 \% \\ \hline \end{array} $$ If supplier A charges $$\$ 900$$ per ton and B charges $$\$ 1200$$ per ton, how much should the company purchase from each supplier to fulfill its needs at minimum cost?

Step-by-Step Solution

Answer

Purchase 400 tons from Supplier A and 100 tons from Supplier B for minimum cost.

Step 1: Define Variables

Let's denote \( x \) as the amount (in tons) of coffee the company purchases from Supplier A, and \( y \) as the amount (in tons) it purchases from Supplier B. Our goal is to find the values of \( x \) and \( y \) that minimize the cost.

Step 2: Understand the Composition Requirements

Supplier A's coffee consists of 20% premium and 50% regular beans. Supplier B's coffee consists of 40% premium and 20% regular beans. The company needs at least 280 tons of premium and 200 tons of regular coffee.

Step 3: Formulate the Linear System

Using the percentages, we can write two inequality equations to represent the minimum requirements: \( 0.2x + 0.4y \geq 280 \) for premium beans and \( 0.5x + 0.2y \geq 200 \) for regular beans.

Step 4: Cost Function

The cost function to minimize is \( 900x + 1200y \), which represents purchasing costs from Suppliers A and B, respectively.

Step 5: Solve the Inequalities

Solve the inequalities for \( y \): 1. From \( 0.2x + 0.4y \geq 280 \), we have \( y \geq 700 - 0.5x \).2. From \( 0.5x + 0.2y \geq 200 \), we have \( y \geq 1000 - 2.5x \).

Step 6: Graphical Representation

Graph these inequalities on a coordinate plane with \( x \) and \( y \). The feasible region will be the overlapping area that satisfies both inequalities.

Step 7: Evaluate the Cost at Corners

Find the cost at each corner of the feasible region. The corners are where the system of inequalities intersect with each other and the axes.

Step 8: Optimal Solution

Identify the corner point corresponding to the minimum cost from Step 7. This will give the optimal amounts of \( x \) and \( y \) to purchase.

Key Concepts

Systems of Inequalities

A system of inequalities involves more than one inequality in the same problem. In a linear programming context, these inequalities help define the constraints that a solution must satisfy. For the coffee company, the key lies in ensuring that certain conditions are met in terms of the types of coffee they purchase.
For instance, the company has specific needs for premium and regular coffee. Supplier A provides 20\% premium and 50\% regular beans, while Supplier B offers 40\% premium and 20\% regular beans.
With these percentages, the company forms inequalities to represent their needs:\( 0.2x + 0.4y \geq 280 \) for premium beans and \( 0.5x + 0.2y \geq 200 \) for regular beans. Here, \( x \) and \( y \) are the variable amounts of coffee purchased from Suppliers A and B, respectively. These inequalities form a system that defines the company's required coffee composition.
By graphing these inequalities on a coordinate plane, the feasible solutions are represented by the region where the conditions of both inequalities are met. This overlapping area is essential when optimizing choices within set constraints.

Cost Function Optimization

The objective of cost function optimization is to determine how to purchase coffee from suppliers at the lowest possible cost, considering the constraints. The key here is the formulation of an equation that represents this cost, which, in mathematical terms, is known as the cost function.
For our coffee company problem, the cost function is expressed as \( 900x + 1200y \). This reflects the cost incurred based on buying \( x \) tons of coffee from Supplier A at \\(900 per ton, and \( y \) tons from Supplier B at \\)1200 per ton.
Optimization involves finding the values of \( x \) and \( y \) that minimize this cost function while still falling within the acceptable limits set by the system of inequalities. In practice, this often means evaluating the cost function at the boundary points or corners of the feasible region.
Essentially, the solver is looking for the optimal combination of purchases that ensure minimum expenditure without violating any of the problem's constraints.

Feasible Region

The feasible region is a core concept in solving linear programming problems. It is the set of all possible points that satisfy all of the problem's inequalities. This region can be visualized by plotting the inequalities on a coordinate plane, where each point within the space represents a permissible solution that meets all constraints.
In the coffee company scenario, the feasible region is determined by the intersection of the areas satisfying both \( 0.2x + 0.4y \geq 280 \) and \( 0.5x + 0.2y \geq 200 \).
Finding the feasible region algebraically involves solving the inequalities for \( y \) in terms of \( x \), and then graphing these lines to see where they intersect and overlap. Graphical representation helps identify where these inequalities are simultaneously true.
A critical element in linear programming is to analyze the vertices or corner points of the feasible region, as these points often contain the optimal solution. By inspecting these points, one can find the particular combination of supplier purchases that achieves the program's goal of minimizing costs while meeting quality requirements.