BISCO MISR is a food industries Co. produces two types of chocolates; dark chocolate and white chocolate. Ingredients for both chocolates are sugar, nuts, milk and chocolate. Currently, the company has 100 kg of sugar, 20 kg of nuts, 15 kg of milk and 30 kg of chocolate in stock. The mixture of dark chocolate contains at least 20% nuts, no more than 10% milk and at least 50% chocolate. The mixture of white chocolate contains at least 10% nuts, at least 40% milk and no more than 15% chocolate. Assume that one kg of each ingredient contributes one kg to the final chocolates product. Each kg of dark chocolate can be sold at EGP 25 and each kg of white chocolate can be sold at EGP 20. Formulate an LP model to maximize the revenue
To formulate an LP model to maximize the revenue for BISCO MISR, we need to define the decision variables and the objective function, as well as the constraints.
Decision Variables:
Let's define the decision variables as:
- x1: the amount of dark chocolate (in kg) produced
- x2: the amount of white chocolate (in kg) produced
Objective Function:
The objective is to maximize the revenue, which can be calculated by multiplying the amount of each chocolate produced by its selling price. Therefore, the objective function is:
Maximize: 25x1 + 20x2
Constraints:
There are several constraints based on the available ingredients and the mixture requirements for each type of chocolate.
Ingredients Constraints:
- Sugar Constraint: Sugar used in dark chocolate + sugar used in white chocolate <= 100 kg
This can be represented as: x1 + x2 <= 100
- Nuts Constraint: Nuts used in dark chocolate + nuts used in white chocolate <= 20 kg
This can be represented as: 0.2x1 + 0.1x2 <= 20
Note that the percentage requirement is translated into decimal form (20% becomes 0.2, and 10% becomes 0.1).
- Milk Constraint: Milk used in dark chocolate + milk used in white chocolate <= 15 kg
This can be represented as: 0.1x1 + 0.4x2 <= 15
- Chocolate Constraint: Chocolate used in dark chocolate + chocolate used in white chocolate <= 30 kg
This can be represented as: 0.5x1 + 0.15x2 <= 30
Non-negativity Constraints:
The production of chocolates cannot be negative, so we have the following constraints:
x1 >= 0
x2 >= 0
Putting it all together, the LP model to maximize the revenue for BISCO MISR can be summarized as follows:
Maximize: 25x1 + 20x2
Subject to:
x1 + x2 <= 100 (Sugar constraint)
0.2x1 + 0.1x2 <= 20 (Nuts constraint)
0.1x1 + 0.4x2 <= 15 (Milk constraint)
0.5x1 + 0.15x2 <= 30 (Chocolate constraint)
x1 >= 0
x2 >= 0
This LP model can be solved using linear programming techniques to determine the optimal amounts of dark and white chocolate to produce in order to maximize the revenue.