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Chapter 7: Problem 1

Consider a company that carves wooden soldiers. The company specializes in two main types: Confederate and Union soldiers. The profit for each is \(\$ 28\) and \(\$ 30\), respectively. It requires 2 units of lumber, \(4 \mathrm{hr}\) of carpentry, and \(2 \mathrm{hr}\) of finishing to complete a Confederate soldier. It requires 3 units of lumber, \(3.5 \mathrm{hr}\) of carpentry, and \(3 \mathrm{hr}\) of finishing to complete a Union soldier. Each week the company has 100 units of lumber delivered. There are \(120 \mathrm{hr}\) of carpenter machine time available and \(90 \mathrm{hr}\) of finishing time available. Determine the number of each wooden soldier to produce to maximize weekly profits.

Short Answer

Expert verified
Produce the number of soldiers corresponding to the corner point with highest profit from Step 7.

Step by step solution

01

Define Variables

Let's define the variables for the problem: \( x \) will be the number of Confederate soldiers produced and \( y \) will be the number of Union soldiers produced. Our goal is to determine the values of \( x \) and \( y \) that maximize the company's weekly profit.
02

Formulate the Objective Function

The profit from producing a Confederate soldier is \( \(28 \), and for a Union soldier, it's \( \)30 \). The objective function, representing the total profit \( P \), can be expressed as follows:\[ P = 28x + 30y \]
03

Establish Constraints

Use the given resources to establish constraints:- Lumber: Confederate soldiers require 2 units, and Union soldiers require 3 units. The constraint is: \[ 2x + 3y \leq 100 \]- Carpentry: Confederate soldiers require 4 hours, and Union soldiers require 3.5 hours. The constraint is: \[ 4x + 3.5y \leq 120 \]- Finishing: Confederate soldiers require 2 hours, and Union soldiers require 3 hours. The constraint is: \[ 2x + 3y \leq 90 \]Additionally, both \( x \) and \( y \) must be non-negative: \[ x \geq 0, \quad y \geq 0 \]
04

Graph Constraints

Graph the inequalities from the constraints on a coordinate plane to find the feasible region. The feasible region is bound by where these lines intersect or meet the axes:1. Plot \( 2x + 3y = 100 \).2. Plot \( 4x + 3.5y = 120 \).3. Plot \( 2x + 3y = 90 \).4. Identify the region where all the conditions are satisfied.
05

Identify Corner Points

Find the points of intersection of the constraint lines, as they form the vertices (corners) of the feasible region. Calculate each intersection to get these points:- Intersection of \( 2x + 3y = 100 \) and \( 4x + 3.5y = 120 \).- Intersection of \( 2x + 3y = 100 \) and \( 2x + 3y = 90 \).- Intersection of \( 4x + 3.5y = 120 \) and \( 2x + 3y = 90 \).- Additionally, check boundaries at axis intercepts.
06

Calculate Profit at Each Corner Point

For each corner point found, substitute the \( x \) and \( y \) values into the objective function \( P = 28x + 30y \) to determine the profit at each point. This will show which combination of \( x \) and \( y \) provides the maximum profit.
07

Determine Maximum Profit

Compare all the profit calculations from Step 6. The \((x, y)\) combination with the highest value of \( P \) will indicate the optimal production strategy for maximum profit.

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Key Concepts

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

Objective Function
The objective function is the heart of any linear programming problem, defining what needs to be maximized or minimized. For this exercise, we are dealing with a profit maximization problem. The company crafts two types of wooden soldiers: Confederate and Union soldiers. Each type of soldier provides a different profit: \(28 for a Confederate soldier and \)30 for a Union soldier.

The objective function here is a mathematical expression that represents the total profit. Written mathematically, if we denote the number of Confederate soldiers as \( x \) and Union soldiers as \( y \), the objective function is:
  • \( P = 28x + 30y \)
This equation helps us to determine how many of each type of soldier should be produced to maximize profit. The objective function sets the stage for everything else in this linear programming problem as it directly ties production numbers to profit.
Constraints
Constraints define the limits within which the company must operate. They provide restrictions based on the available resources like materials and labor. For the wooden soldier company, constraints arise from the limited quantities of resources needed to produce each type of soldier, such as lumber and carpentry hours.

These are the constraints based on the resource availability:
  • Lumber: \( 2x + 3y \leq 100 \)
  • Carpentry: \( 4x + 3.5y \leq 120 \)
  • Finishing: \( 2x + 3y \leq 90 \)
Additionally, both \( x \) and \( y \) must be non-negative, as negative production doesn't make sense:
  • \( x \geq 0 \)
  • \( y \geq 0 \)
These constraints ensure that the production strategy is both feasible and practical, keeping within the limits of what resources the company has each week.
Feasible Region
The feasible region is a visual representation of all possible solutions that satisfy the constraints. It is found by graphing the inequalities of the constraints on a coordinate plane. Each constraint line divides the plane into two halves: one that satisfies the inequality and one that does not.

In our exercise, the feasible region is the area where all these constraint inequalities overlap. To determine this region:
  • Plot the lines for each inequality: \( 2x + 3y = 100 \), \( 4x + 3.5y = 120 \), and \( 2x + 3y = 90 \).
  • Identify where these lines intersect or meet the axes.
The feasible region represents all potential production combinations of Confederate and Union soldiers that respect the resource limits. It is the area where our optimal solution鈥攎aximum profit鈥攚ill be found.
Profit Maximization
Profit maximization is the end-goal of the linear programming problem, aiming to identify the combination of production levels of Confederate and Union soldiers that results in the highest possible profit.

Steps to achieve this include:
  • Calculate the intersection points of the boundary lines of the feasible region (corner points).
  • Substitute these corner points into the objective function \( P = 28x + 30y \) to calculate profit.
  • Compare profits for all corner points.
The solution with the highest calculated profit gives the optimal production strategy. Thus, identifying which combination of \( x \) and \( y \) maximizes the company's profits while staying within the resource constraints. This careful analysis links production decisions to financial outcomes, ensuring resources are used efficiently.

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Most popular questions from this chapter

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