SOLUTION: 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 pun
Algebra.Com
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.
RELATED QUESTIONS
CalJuice Company has decided to introduce three fruit juices made from blending two or... (answered by Theo,ikleyn)
The Juice Company offers three kinds of smoothies: Midnight Mango, Tropical Torrent, and... (answered by ikleyn)
The Juice Company offers three kinds of smoothies: Midnight Mango, Tropical Torrent, and... (answered by mananth)
Maggie makes some fruit punch. She mixes 2 1/2 quarts of grape juice with 1 1/2 quarts of (answered by josgarithmetic,josmiceli)
How much of two grape drinks, one 10% grape juice and the other 40% grape juice should be (answered by ikleyn)
A fruit juice company makes two special drinks by blending apple and pineapple juices.... (answered by amoresroy,ikleyn)
two kinds of orange juice one with 20% and the other with 5%, How can a mixture be make... (answered by josgarithmetic)
A pure fruit juice is made by mixing the juices of oranges and mangoes in the ratio of... (answered by Cromlix)
A beverage company is planning to make a fruit juice blend using grape juice and... (answered by josgarithmetic)