## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The goal is to minimize the cost of buying pills A and B while ensuring the patient receives at least the required amounts of muscle relaxant, anxiety medication, and pain reliever.

Let's define the decision variables:
- \(x\): the number of pills A to buy
- \(y\): the number of pills B to buy

The objective function to minimize is the total cost:
\[ \text{Minimize:} \quad 3x + 5y \]

Subject to the constraints:
1. Muscle relaxant: \( 2x + 5y \geq 30 \)
2. Anxiety medication: \( 3x + y \geq 15 \)
3. Pain reliever: \( 4x + 3y \geq 20 \)
4. Non-negativity: \( x \geq 0, y \geq 0 \) and \( x, y \) are integers (though for the purpose of linear programming, we'll initially solve it as if \( x \) and \( y \) can be continuous, and then round if necessary).

## Gurobi Code

```python
import gurobi

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

    # Define the variables
    x = model.addVar(name="pill_A", lb=0, vtype=gurobi.GRB.CONTINUOUS)  # Number of pills A
    y = model.addVar(name="pill_B", lb=0, vtype=gurobi.GRB.CONTINUOUS)  # Number of pills B

    # Objective function: Minimize cost
    model.setObjective(3*x + 5*y, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(2*x + 5*y >= 30, name="muscle_relaxant")  # Muscle relaxant constraint
    model.addConstr(3*x + y >= 15, name="anxiety_medication")  # Anxiety medication constraint
    model.addConstr(4*x + 3*y >= 20, name="pain_reliever")  # Pain reliever constraint

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal cost: {model.objVal}")
        print(f"Number of pills A: {x.varValue}")
        print(f"Number of pills B: {y.varValue}")
    else:
        print("The model is infeasible")

solve_pill_problem()
```

This code defines a linear programming problem with Gurobi, solves it, and prints out the optimal cost and the number of each pill to buy. Note that the solution might not be integer, so for practical purposes, you might need to round the solution and check its feasibility. For a more accurate integer solution, consider changing `vtype` to `gurobi.GRB.INTEGER` and solving the model as an integer program, which can be computationally more expensive.