Case 2: Red Brand Canners (RBC)
Red Brand Canners would like to determine production levels for its tomato products this season in order to maximize profits. It is my understanding that each product is sold by the case, and that a case of canned whole tomatoes takes 18 pounds of tomatoes, a case of tomato juice takes 20 pounds, and a case of tomato paste takes 25 pounds. Each product is made up of Grade A tomatoes (Average Quality: 9) and/or Grade B tomatoes (Average Quality: 5).
Physical production limitations include the supply of Grade A tomatoes (600,000) and the supply of Grade B tomatoes (2,400,000). According to Myers’ demand forecasts, production should ...view middle of the document...
The objective function serves to maximize the profit from the sales of the three tomato products. The contribution margin from selling a pound of each product can be seen in Exhibit A. So, for each product type, the total income is given by the contribution margin multiplied by the number of pounds used. The objective function is therefore 0.0822AC + 0.0822BC + 0.066AJ + 0.066BJ + 0.074AP + 0.074BP. The first set of constraints is necessary to keep production of each tomato product below its respective demand. Additionally, the number of tomatoes used must be kept under the amount available for each grade. The final type of constraint ensures that the mixture of tomato grades used for the canned tomato and tomato juice products meets the required quality score (this is not needed for the tomato paste, as any grade will meet the requirements). For the canned whole tomatoes, the quality score (9AC + 5BC) should be greater than or equal to 8. For the tomato juice, the quality score (9AJ + 5BJ) should be greater than or equal to 6.
2. AC = Pounds of Grade A tomato allocated to produce canned whole tomatoes.
BC = Pounds of Grade B tomato allocated to produce canned whole tomatoes.
AJ = Pounds of Grade A tomato allocated to produce tomato juice.
BJ = Pounds of Grade B tomato allocated to produce tomato juice.
AP = Pounds of Grade A tomato allocated to produce tomato paste.
BP = Pounds of Grade B tomato allocated to produce tomato paste.
Maximize 0.0822AC + 0.0822BC + 0.066AJ + 0.066BJ + 0.074AP + 0.074BP
1AC + 1BC 14,400,000 (Canned Whole Tomatoes Demand)
1AJ + 1BJ 1,000,000 (Tomato Juice Demand)
1AP + 1BP 2,000,000 (Tomato Paste Demand)
1AC + 1AJ + 1AP 600,000 (Grade A Tomato Supply)
1BC + 1BJ + 1BP 2,400,000 (Grade B Tomato Supply)
1AC – 3BC 0 (Canned Whole Tomatoes Quality)
3AJ – 1BJ 0 (Tomato Juice Quality)
3. Based on my analysis, the optimal production allocation plan would consist of 525,000 lbs of Grade A &175,000 lbs of Grade B for whole tomatoes, 75,000 lbs of Grade A & 225,000 lbs of Grade B for tomato juice, and 200,000 lbs of Grade B for tomato paste. This would allow the company to maximize profits and reach an optimal objective function value of $225,340. With this optimal solution, none of the forecasted demands are fully satisfied. There is a shortage of 13,700,000 lbs for whole tomatoes, 700,000 lbs for tomato juice and 1.4 x 106 lbs for tomato paste. If it is possible to acquire a larger supply of Grade A and/or Grade B tomatoes, the company should be able to fill more demand and increase profit (Exhibit B).
4. The only binding constraints are the quality constraints for the whole tomatoes and tomato juice. The shadow price for whole tomato and juice demand is 0, while the shadow price for tomato paste demand is 0.0161. For Grade A supply it is 0.0903 and for Grade B supply it is 0.0579. The shadow price for both whole tomato and tomato juice quality is -0.0081....