## Problem Description and Formulation

The problem is a classic example of a linear programming problem. We have two types of capsules, A and B, with different compositions of targeted medicine, pain reliever, and filler. The goal is to minimize the cost of creating a final product that meets certain minimum requirements of these components.

Let's define the decision variables:
- \(x\): The number of capsules A to use.
- \(y\): The number of capsules B to use.

The objective function to minimize is the total cost:
\[ \text{Minimize:} \quad 2x + 3y \]

Subject to the constraints based on the minimum requirements:
1. Targeted medicine: \(2x + 3y \geq 20\)
2. Pain reliever: \(3x + y \geq 20\)
3. Filler: \(x + 3y \geq 15\)
4. Non-negativity: \(x \geq 0, y \geq 0\)

## Gurobi Code

To solve this problem using Gurobi in Python, we will use the Gurobi Python interface. First, ensure you have Gurobi installed and a valid license.

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the decision variables
x = model.addVar(lb=0, name="x")  # Number of capsules A
y = model.addVar(lb=0, name="y")  # Number of capsules B

# Objective function: Minimize 2x + 3y
model.setObjective(2*x + 3*y, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(2*x + 3*y >= 20, name="medicine_constraint")
model.addConstr(3*x + y >= 20, name="pain_reliever_constraint")
model.addConstr(x + 3*y >= 15, name="filler_constraint")

# Optimize the model
model.optimize()

# Print the results
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of capsules A: {x.varValue}")
    print(f"Number of capsules B: {y.varValue}")
    print(f"Optimal cost: {model.objVal}")
else:
    print("No optimal solution found.")
```

This code defines the problem in Gurobi, solves it, and then prints out the optimal values for \(x\) and \(y\) (the number of capsules A and B to use) and the minimum cost. If no optimal solution is found, it indicates that the problem is infeasible or unbounded.