Question 1194453: 1. Calculate a Juice Company has available two kinds of food Juices: Orange Juice and Grape Juice. The company produces two types of punches: Punch A and Punch B. One bottle of punch A requires 20 liters of Orange Juice and 5 liters of Grape Juice.1 Bottle of punch B requires 10 liters of Orange Juice and 15 liters of Grape Juice.
From each of bottle of Punch A a profit of $4 is made and from each bottle of Punch B a profit of $3 is made .Suppose that the company has 230 liters of Orange Juice and 120 liters of Grape Juice available.
Required:
a. Formulate this problem as a LPP
b. How many bottles of Punch A and Punch B the company should produce in order to maximize profit?
c. What is this maximum profit? (solve it using both the graphical and Simplex method)
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website! **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.
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