Apple Pie Investigation
I know two recipes of apple pie, which are famously known and are assumed to sell well, with the key ingredients of one of the pies being apples, honey and pastry. The other pie contains the same key ingredients but in different portions. The fruits of this labour will be sold in a fete to the general public and the proceeds will go to charity. It is desired to have as much money as possible so that the charity will benefit the most. Because half pies are unlikely to be sold, this creates an integer problem. Other ingredients in the pies are considered plentiful and are therefore taken out of the equation. Additionally there is only enough room in the car for at most three pies.
The objective of this project is to make as much money as possible, to donate as much as you can to the charity. This profit can only be calculated once the costs of buying the ingredients are taken into account. The selling price will therefore be a larger value than the cost of making the pie. This particular problem is suitable to linear programming because the various limits such as quantity of ingredients constrain the maximum number of pies that can be produced, affecting the profit.
Profit can be expressed as:
Profit margin = selling price – cost of ingredients
The time taken and the cost of labour are irrelevant because the project takes up your own time and you work for free as this project benefits charity.
The first pie recipe will be known as a “sweet” pie as it contains a higher proportion of honey than the second pie. The variant of this recipe will be known as the “fruity” pie as it contains more apples than the first recipe. The fruity pies tend to create larger pies than the sweet pies, therefore at most only three fruity pies will be able to fit into the car as stated previously.Upon investigation of the original solutions of the investigation, it was found that the apple constraint was tight at the optimum solution with no apples left over, with a little slack on the integer solution. After refinements, the transportation constraint was no longer used, as it had become redundant. In addition it was found that the original solutions were no longer viable as they had burst the apple constraint in both optimum and integer solutions.
The new optimum solution was found to be at ( 0 , 2.33 , 0 , 17.16 , 2.20 ) with no slack for apples, a lot of slack for honey and a little slack in terms of pastry. This optimum solution gave a profit of ï¿½5.33. However, as stated previously, this cannot be the real world solution as this is an integer problem. After an integer search, the final solution was found to be at ( 1 , 2 , 1 , 6.5 , 0.2 ). This gave some slack in terms of honey, much less than the optimum solution, and very little slack in terms of apples or pastry. This solution gave a profit of ï¿½5.00 exactly, by making one sweet pie and two fruity pies.
This project could be extended by creating a production line in which the optimum solution would represent the ratio of sweet pies to fruity pies. Currently this project is intended as a one off, to benefit a charity fete. The ratio could be scaled up to create integer values, using the values obtained would create eight fruity pies and zero sweet pies. However the amount of ingredients would also have to be scaled up accordingly. In addition, this would require the additional constraints of time and labour costs.
There are other uses for linear programming, such as organising a production line effectively. For example, if there were a set number of parts available for different type cars, linear programming could help to decide the ratio of cars to be made. Another example would be in the production of two different types of lemonade produced using the same key ingredients, which could provide the best number of each type of drink to be produced.