## Problem Description and Formulation

The problem is a linear programming (LP) optimization problem. The goal is to maximize the profit of an electronics factory that produces DVD players and projectors under certain constraints.

### Decision Variables

- \(x\): Number of DVD players produced per week
- \(y\): Number of projectors produced per week

### Objective Function

The profit per DVD player is $250, and the profit per projector is $200. The objective is to maximize the total profit:

\[ \text{Maximize:} \quad 250x + 200y \]

### Constraints

1. **Silicon Chips**: Each DVD player requires 5 silicon chips, and each projector requires 3 silicon chips. The company can buy 250 silicon chips per week.

\[ 5x + 3y \leq 250 \]

2. **Engineering Time**: Each DVD player requires 6 hours of engineering time, and each projector requires 5 hours. There are 200 hours of engineering time available per week.

\[ 6x + 5y \leq 200 \]

3. **Assembly Time**: Each DVD player requires 2.5 hours of assembly time, and each projector requires 2 hours. There are 240 hours of assembly time available per week.

\[ 2.5x + 2y \leq 240 \]

4. **Minimum Production Requirements**: The company wants to produce at least 30 units of DVD players and at least 30 units of projectors each week.

\[ x \geq 30 \]
\[ y \geq 30 \]

5. **Non-Negativity Constraints**: The number of DVD players and projectors produced cannot be negative.

\[ x \geq 0 \]
\[ y \geq 0 \]

However, since \(x \geq 30\) and \(y \geq 30\) are specified, the non-negativity constraints are implicitly satisfied.

## Gurobi Code

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the decision variables
    x = model.addVar(lb=30, name="DVD_players")  # At least 30 DVD players
    y = model.addVar(lb=30, name="projectors")  # At least 30 projectors

    # Objective function: Maximize profit
    model.setObjective(250 * x + 200 * y, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5 * x + 3 * y <= 250, name="silicon_chips")
    model.addConstr(6 * x + 5 * y <= 200, name="engineering_time")
    model.addConstr(2.5 * x + 2 * y <= 240, name="assembly_time")

    # Optimize the model
    model.optimize()

    # Print the results
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"DVD Players: {x.varValue}")
        print(f"Projectors: {y.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible.")

solve_optimization_problem()
```