ABC is a national company that produces trains and trucks to meet the demand of its customers. The company has grown its capital from $500,000 to $1,000,000 and its labour force from 40 to 60 personnel. Due to the growth, the company decided to produce both trains and trucks for the international market subject to its resources. However, the company aims to maximise profit by entering the global market. Use the additional information in the table to determine how many trains and trucks the company will produce to maximise profits.

Resources	Trains	Trucks
Labour	0.2	0.1
Production cost	$1,500	$4,000
Profit	$5,000	$3,000


To determine how many trains and trucks the company should produce to maximize profits, we need to approach this as an optimization problem using the given resources. We'll use a linear programming model (LPM), which is suitable for maximizing or minimizing a linear objective function, subject to linear equality and inequality constraints.

### Let’s define the variables:
- Let $ x $ represent the number of trains produced.
- Let $ y $ represent the number of trucks produced.

### Step 1: Objective Function
The company wants to maximize its profit. The profit for each train is $5,000, and the profit for each truck is $3,000.

The objective function (which we want to maximize) is:
\[
\text{Maximize } P = 5000x + 3000y
\]

### Step 2: Constraints
#### 1. Labour Constraint
- Each train requires 0.2 labour units, and each truck requires 0.1 labour units.
- The company has a total of 60 labour units available.

The labour constraint is:
\[
0.2x + 0.1y \leq 60
\]

#### 2. Production Cost Constraint
- The production cost for each train is $1,500, and for each truck is $4,000.
- The company has a total capital of $1,000,000.

The production cost constraint is:
\[
1500x + 4000y \leq 1,000,000
\]

#### 3. Non-Negativity Constraints
Since the number of trains and trucks cannot be negative:
\[
x \geq 0, \quad y \geq 0
\]

### Step 3: Solving the Linear Programming Problem
Now, we'll solve this system of inequalities to maximize the profit function $ P = 5000x + 3000y $, subject to the constraints.

Let me compute this for you.from scipy.optimize import linprog

# Coefficients of the objective function (negative for maximization)
c = [-5000, -3000]

# Coefficients matrix for the inequality constraints (Ax <= b)
A = [
    [0.2, 0.1],   # Labour constraint
    [1500, 4000]  # Production cost constraint
]

# Right-hand side values for the inequality constraints
b = [60, 1000000]

# Bounds for each variable (x and y must be non-negative)
x_bounds = (0, None)
y_bounds = (0, None)

# Solving the linear program
result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')

resultThe company should produce:
- **215 trains** and
- **169 trucks** 

to maximize its profit, subject to the available resources. The maximum possible profit will be approximately **$1,584,615.38**.

Would you like further details on how these calculations were derived or any related questions?

Here are 8 questions to extend your understanding:

1. How would the solution change if the available labor were reduced by 10%?
2. What happens to the optimal production if the capital increases to $1.5 million?
3. How do changing the production costs affect the outcome?
4. How would a reduction in the profit margin per truck impact the solution?
5. What other optimization techniques could be used besides linear programming?
6. How can sensitivity analysis be applied to this solution?
7. What role do shadow prices play in this optimization?
8. How could this model be adapted for future resource growth?

**Tip**: Always verify if assumptions such as linearity in constraints and objective functions hold when using linear programming for real-world applications.

Learn how to maximize profits by optimizing the production of trains and trucks. Explore the use of linear programming to solve resource allocation problems efficiently. Find out the optimal quantities of trains and trucks to produce for maximum profitability.

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