Problem 105 - A Graphical Approach to Precalculus with Limits

Question

Shipment Costs A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(A\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

Step-by-Step Solution

Answer

Ship 20 refrigerators to warehouse A and 80 to warehouse B; minimum cost is $1040.

Step 1: Define Variables

Let's define variables for the number of refrigerators shipped to each warehouse. Let \( x \) be the number of additional refrigerators shipped to warehouse A, and \( y \) be the number of refrigerators shipped to warehouse B.

Step 2: Create the Objective Function

The cost function we need to minimize is: \( C = 12x + 10y \), where 12 and 10 are the costs per refrigerator for each respective warehouse.

Step 3: Set Up Constraints

Given constraints include: \( x + y \geq 100 \) (at least 100 refrigerators shipped), \( x \leq 75 \) (warehouse A max capacity is 100 minus the 25 it already has), and \( y \leq 80 \) (warehouse B max capacity is 100 minus the 20 on hand).

Step 4: Analyze the Constraints

Graph the inequalities on the Cartesian plane or solve manually to determine the feasible region. Our constraints are: \( x + y \geq 100 \), \( x \leq 75 \), \( y \leq 80 \), \( x \geq 0 \), and \( y \geq 0 \).

Step 5: Identify Potential Solutions

Look at the vertices of the feasible region (points satisfying the constraints) which could be potential solutions: (20, 80), (75, 25), and (75, 25) being repeated. Compute cost at these points.

Step 6: Calculate Costs at Vertices

Calculate the cost at each vertex:- \( (20, 80) \): \( C = 12(20) + 10(80) = 240 + 800 = 1040 \)- \( (75, 25) \): \( C = 12(75) + 10(25) = 900 + 250 = 1150 \)

Step 7: Find the Minimum Cost

Compare the costs: The minimum cost is at \( (20, 80) \), with a cost of \( 1040 \). Thus, ship 20 refrigerators to warehouse A and 80 to warehouse B to minimize cost.

Key Concepts

Objective Function

The objective function in linear programming is a mathematical expression used to describe a goal or outcome that needs to be optimized. In our exercise, the goal is cost minimization, which means we are trying to reduce the shipping cost as much as possible.
We define the cost function as the objective function:

• The formula is: \( C = 12x + 10y \).
• Here, \( C \) is the total cost that we want to minimize.
• \( x \) is the number of refrigerators shipped to warehouse A.
• \( y \) is the number of refrigerators shipped to warehouse B.
• The coefficients 12 and 10 represent the cost per refrigerator for warehouses A and B, respectively.

Understanding the objective function is crucial because it is the expression we manipulate to find the most cost-effective way to ship the refrigerators.

Inequality Constraints

Inequality constraints are essential in linear programming because they limit the possible values that a solution can take. They act like boundaries that define feasible solutions. In the refrigerator shipping problem, several constraints need to be considered:

• \( x + y \geq 100 \): This constraint ensures that at least 100 refrigerators are shipped in total to both warehouses.
• \( x \leq 75 \): Limits the number of refrigerators to warehouse A to not exceed its capacity after accounting for the 25 refrigerators already there.
• \( y \leq 80 \): Similarly ensures that warehouse B does not receive more than it can handle considering its 20 already stored refrigerators.
• \( x \geq 0 \) and \( y \geq 0 \): These constraints guarantee that we do not ship a negative number of refrigerators.

Properly setting up and understanding these constraints is vital as they shape the overall solution space.

Feasible Region

The feasible region in linear programming represents all the possible solutions that satisfy the inequality constraints. It is typically depicted graphically as an area on a graph where all the constraints overlap.

• For the refrigerator problem, the feasible region is the space in which all conditions described by our constraints (i.e., \( x + y \geq 100 \), \( x \leq 75 \), and \( y \leq 80 \)) are true simultaneously.
• This region is often a polygon, and it contains all the potential shipping combinations we can consider without violating any constraints.
• Feasible regions are the playground where we search for the best possible solution to our objective function.

Understanding feasible regions allows us to pinpoint where on the graph we should focus our optimization efforts.

Cost Minimization

Cost minimization, the primary goal of the refrigerator transport problem, involves finding the shipping combination with the least cost within the feasible region. Here’s how it’s done:

• First, identify the vertices (corners) of the feasible region. These points often represent potential solutions because, according to linear programming theory, the optimal solution will be at one of these vertices.
• In our problem, the vertices were points like \( (20, 80) \), and \( (75, 25) \).
• Next, calculate the total shipping cost at each vertex using the objective function \( C = 12x + 10y \).
• Finally, compare these costs to find the minimum. In this example, shipping 20 refrigerators to warehouse A and 80 to warehouse B incurred the lowest cost \( 1040 \).

The focus of cost minimization is to strategically utilize available resources to achieve the lowest possible spending.