A food factory is making a beverage for a customer from mixing two different existing
products A and B. The compositions of A and B and prices ($/L) are given as follows,
Amount.(L) in/100L of A and B
{Lime,Orange,Mango,Cost($/L)}; {A,2,6,4,5}; {B,7,4,8,15}. The customer requires that there must be atleast 5 Litres(L)Orange and atleast 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage.The customer needs atleast 150 Litres of the beverage per week.

We are tasked with formulating a beverage blend from two existing products, A and B, which contain lime, orange, and mango concentrates. The goal is to meet specific customer requirements for minimum orange and mango concentrate levels, maximum lime concentrate levels, and an overall quantity of at least 150 liters of beverage per week. Additionally, we want to minimize the cost of producing this blend.

Let’s break the problem down into variables and constraints.

### Variables:
- Let $ x $ represent the amount of product A (in liters) used per 100 liters of beverage.
- Let $ y $ represent the amount of product B (in liters) used per 100 liters of beverage.

### Compositions of A and B:
- Product A: 
  - Lime: 2 L
  - Orange: 6 L
  - Mango: 4 L
  - Cost: $5/L
- Product B: 
  - Lime: 7 L
  - Orange: 4 L
  - Mango: 8 L
  - Cost: $15/L

### Customer Constraints:
1. **Minimum Orange Concentrate**: 
   The beverage must contain at least 5 liters of orange concentrate per 100 liters.
   \[
   6x + 4y \geq 5
   \]
2. **Minimum Mango Concentrate**: 
   The beverage must contain at least 5 liters of mango concentrate per 100 liters.
   \[
   4x + 8y \geq 5
   \]
3. **Maximum Lime Concentrate**: 
   The beverage must contain no more than 6 liters of lime concentrate per 100 liters.
   \[
   2x + 7y \leq 6
   \]
4. **Total Amount of Beverage**: 
   The customer needs at least 150 liters of beverage per week. For 100 liters, this constraint doesn't apply directly, but later it will help us scale the production.

### Objective:
Minimize the cost of producing the beverage, which is given by:
\[
\text{Cost} = 5x + 15y
\]

### Problem Setup:
We need to solve this as a linear programming problem, subject to the constraints mentioned above.

### Constraints in summary:
\[
\begin{align*}
6x + 4y &\geq 5 & \text{(Minimum Orange)} \\
4x + 8y &\geq 5 & \text{(Minimum Mango)} \\
2x + 7y &\leq 6 & \text{(Maximum Lime)} \\
x + y &= 1 & \text{(Total Proportion per 100 L)}
\end{align*}
\]
Additionally, $x, y \geq 0$ to ensure non-negative amounts of A and B.

### Next Steps:
1. Solve the linear programming problem to minimize the cost.
2. Ensure that the solution meets all constraints for orange, mango, lime, and total beverage amount.

Do you want me to proceed with solving this linear programming problem, or would you prefer further clarification on any of the constraints?

### Relative Questions:
1. What would happen if the cost of product B was reduced?
2. How would a different ratio of minimum orange to mango concentrate change the solution?
3. What if the customer requested a different maximum lime limit (e.g., 7 L)?
4. How would increasing the weekly beverage requirement to 200 liters impact the solution?
5. How does changing the cost structure (e.g., increasing the cost of A) affect the optimal solution?

### Tip:
Understanding how to convert word problems into mathematical models is key to solving real-world optimization problems like this one.

A food factory is making a beverage for a customer from mixing two different existing products A and B. The compositions of A and B and prices ($/L) are given as follows, Amount.(L) in/100L of A and B {Lime,Orange,Mango,Cost($/L)}; {A,2,6,4,5}; {B,7,4,8,15}. The customer requires that there must be at least 5 Litres(L) Orange and at least 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at least 150 Litres of the beverage per week.

This page presents a linear programming problem where a food factory blends two products, A and B, to meet customer constraints on orange, mango, and lime concentrates, while minimizing costs. It covers key mathematical concepts, including optimization, linear inequalities, and the simplex method, ideal for undergraduate students.

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