The chilling adventures of Sabrina¶
Problem definition¶
Spellman´s Ltd is a company that manufactures chilling soft drinks. They want to manufacture two types of drinks A, and B. Both beverages use a semi-elaborate C, another expensive ingredient D and other ingredients that are not relevant for production planning. Sabrina is a young student of engineering and management doing an internship at Spellman´s. She needs to formulate a Continuous Linear Program to configure the optimal daily production plan for the company.
The selling price of drink A is 3€/liter and the selling price of drink B is 2€/liter.
1 liter of drink A uses 3 grams of ingredient D. A liter of drink B uses 1 gram of ingredient D. There are only 3 grams of ingredient D available per day.
The factory only has one mixer to elaborate both drink types and the semi-elaborate. It takes 1 hour to process a liter of drink A, 1 hour to process 1 liter of drink B, and 1 hour to process 1cl of semi-elaborate C. The mixer is available 6 hours per day.
Drink A uses 2cl of semi-elaborate C and drink B uses 1cl of semi-elaborate C. The company has 3cl of semi-elaborate C plus the amount they decide to produce available per day.
1. Write a Continuous Linear Problem to help Sabrina design the optimal production plan that maximises revenues for the company.
2. Write the dual problem
3. Given the following solution:
0 Liters of drink A
3 Liters of drink B
3 cl of semi-elaborate C
Verify the solution. Is the solution feasible? What are the values of the slack variables?
4. Use complementary slackness to find the dual solution corresponding to this vertex. Is the dual solution feasible? Is the solution optimal? Motivate your response