https://file.aiquickdraw.com/mathbot/file/17244686792585tl8gr4w.jpg

Formulate the problem as a linear programming problem to decide how many cars of each design should be produced to maximise the profit.

To formulate the problem as a linear programming problem (LP) based on the given data, we need to define the objective function and the constraints.

### Step 1: Define Variables
Let:
- \( x_1 \) = Number of two-door cars to produce
- \( x_2 \) = Number of four-door cars to produce

### Step 2: Define the Objective Function
The objective is to maximize the total profit. Given the profits from the table:
- Profit from two-door cars: \( 13,500x_1 \)
- Profit from four-door cars: \( 15,600x_2 \)

The objective function is:
\[
\text{Maximize } Z = 13,500x_1 + 15,600x_2
\]

### Step 3: Define the Constraints

#### 1. **Production Time Constraint**:
- Two-door cars require 12 hours per car.
- Four-door cars require 20 hours per car.
- Total available production time is 8,000 hours.

The constraint is:
\[
12x_1 + 20x_2 \leq 8,000
\]

#### 2. **Storage Constraint**:
- Storage available for two-door cars: 400 cars.
- Storage available for four-door cars: 350 cars.

The constraints are:
\[
x_1 \leq 400
\]
\[
x_2 \leq 350
\]

#### 3. **Consumer Demand Constraint**:
- Consumer demand for two-door cars: 700 cars.
- Consumer demand for four-door cars: 500 cars.
- Total consumer demand: 240,000 cars.

These constraints are:
\[
x_1 \leq 700
\]
\[
x_2 \leq 500
\]
\[
x_1 + x_2 \leq 240,000
\]

#### 4. **Non-negativity Constraint**:
Both \( x_1 \) and \( x_2 \) must be non-negative:
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Step 4: Linear Programming Problem Formulation

The linear programming problem can be written as:

\[
\text{Maximize } Z = 13,500x_1 + 15,600x_2
\]

Subject to:
\[
12x_1 + 20x_2 \leq 8,000
\]
\[
x_1 \leq 400
\]
\[
x_2 \leq 350
\]
\[
x_1 \leq 700
\]
\[
x_2 \leq 500
\]
\[
x_1 + x_2 \leq 240,000
\]
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

This LP formulation can be solved using methods such as the Simplex algorithm or using software tools like MATLAB, Python (SciPy), or Excel Solver.

Would you like details on solving this linear programming problem or have any other questions?

---

Here are 5 related questions you might find interesting:
1. How does the Simplex method work for solving linear programming problems?
2. What are the advantages of using linear programming in production planning?
3. How would the solution change if the profit per car increased for one of the car models?
4. What if additional resources, such as production time, were added—how would the constraints change?
5. Can linear programming handle multiple objective functions simultaneously?

**Tip:** When dealing with multiple constraints, always check if any of them are redundant before solving the linear programming problem.

Learn how to formulate a linear programming problem to maximize profit from producing two-door and four-door cars. This problem involves defining objective functions and constraints based on production time, storage, and consumer demand. Suitable for advanced college-level studies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation