Question 1194453
**a. Formulate this problem as a Linear Programming Problem (LPP)**

**Decision Variables:**

* Let x represent the number of bottles of Punch A to produce.
* Let y represent the number of bottles of Punch B to produce.

**Objective Function:**

* Maximize Profit: 
   * Profit = 4x + 3y 

**Constraints:**

* **Orange Juice Constraint:** 
    * 20x + 10y ≤ 230 
* **Grape Juice Constraint:**
    * 5x + 15y ≤ 120
* **Non-negativity Constraints:**
    * x ≥ 0
    * y ≥ 0

**b. & c. Determine the Optimal Solution (Graphical Method)**

1. **Graph the Constraints:**
   - Plot the lines: 
      - 20x + 10y = 230 
      - 5x + 15y = 120
   - Shade the feasible region (the area that satisfies all constraints).

2. **Find the Corner Points of the Feasible Region:**
   - Determine the coordinates of the vertices of the feasible region.

3. **Evaluate the Objective Function at Each Corner Point:**
   - Calculate the profit (4x + 3y) at each corner point.

4. **Identify the Optimal Solution:**
   - The corner point that yields the highest profit is the optimal solution.

**b. & c. Determine the Optimal Solution (Simplex Method)**

1. **Standard Form:**
   - Convert the inequalities to equalities by introducing slack variables:
      - 20x + 10y + s1 = 230
      - 5x + 15y + s2 = 120
   - Where s1 and s2 are slack variables.

2. **Initial Simplex Tableau:**
   - Create the initial simplex tableau.

3. **Perform Iterations:**
   - Apply the simplex method rules (finding the pivot column, pivot row, and performing row operations) to improve the objective function value in each iteration.

4. **Optimal Solution:**
   - The final simplex tableau will indicate the optimal values of x and y, as well as the maximum profit.

**Note:**

* The graphical method is suitable for problems with two decision variables.
* The simplex method is a more general algorithm that can be used for problems with any number of variables.

**By performing the graphical method or the simplex method, you will find the optimal number of bottles of Punch A and Punch B to produce in order to maximize profit, along with the maximum profit value.**

**Disclaimer:** This solution provides a general framework. The specific calculations and graphical representation will need to be carried out to obtain the exact numerical results.