## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the return on investment by allocating $1,000,000 across four industries: biotech, food, finance, and health care, with specific constraints.

### Variables

Let's denote the amount invested in each industry as:
- \(B\): biotech industry
- \(F\): food industry
- \(FN\): finance industry
- \(H\): health care industry

### Objective Function

The return on investment for each industry is given as:
- Biotech: 4% or 0.04
- Food: 6% or 0.06
- Finance: 8% or 0.08
- Health Care: 10% or 0.10

The objective function to maximize the total return on investment (\(R\)) is:
\[ R = 0.04B + 0.06F + 0.08FN + 0.10H \]

### Constraints

1. **Total Investment Constraint**: The total investment across all industries cannot exceed $1,000,000.
\[ B + F + FN + H \leq 1,000,000 \]

2. **Health Care vs. Biotech Constraint**: The amount invested in the health care industry cannot exceed the amount invested in the biotech industry.
\[ H \leq B \]

3. **Food vs. Finance Constraint**: The amount invested in the food industry cannot exceed the amount invested in the finance industry.
\[ F \leq FN \]

4. **Health Care Investment Limit**: At most 30% of the investment can be in the health care industry.
\[ H \leq 0.30 \times 1,000,000 \]
\[ H \leq 300,000 \]

5. **Non-Negativity Constraints**: The investment in each industry must be non-negative.
\[ B \geq 0, F \geq 0, FN \geq 0, H \geq 0 \]

## Gurobi Code

```python
import gurobi

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

    # Define variables
    B = model.addVar(name="Biotech", lb=0)
    F = model.addVar(name="Food", lb=0)
    FN = model.addVar(name="Finance", lb=0)
    H = model.addVar(name="HealthCare", lb=0)

    # Objective function: maximize return on investment
    model.setObjective(0.04 * B + 0.06 * F + 0.08 * FN + 0.10 * H, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(B + F + FN + H <= 1000000, name="Total_Investment")
    model.addConstr(H <= B, name="HealthCare_vs_Biotech")
    model.addConstr(F <= FN, name="Food_vs_Finance")
    model.addConstr(H <= 300000, name="HealthCare_Limit")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Biotech: ${B.varValue}")
        print(f"Food: ${F.varValue}")
        print(f"Finance: ${FN.varValue}")
        print(f"Health Care: ${H.varValue}")
        print(f"Max Return: ${model.objVal}")
    else:
        print("No optimal solution found.")

solve_investment_problem()
```

This Gurobi code defines the optimization problem as described, with the objective to maximize the return on investment under the given constraints. It then solves the problem and prints out the optimal investment amounts for each industry if a solution exists.