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D&D 1 Maths Coursework - Apple Pie Investigation.

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Introduction

Aseel Tungekar 02 November 2003 D&D 1 Maths Coursework Apple Pie Investigation Introduction 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. Aim 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. Constraints 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. ...read more.

Middle

Integer Search There are 3 points that are closest to the vertex B at: Vertex p at ( 0 , 3 ) q at ( 0, 2 ) r at ( 1 , 2 ) Substituting these values into x + 2y (the objective function) gives: at point p a profit of: 0 + (3*2) = 6 at vertex q a profit of: 0 + (2*2) = 4 at point r a profit of: 1 + (2*2) = 5 Therefore the most profitable real world solution is at point P, coordinates ( 0 , 3 ) giving a profit of �6. This requires making no sweet pies and three pies. Conclusion From this investigation we can conclude that the optimum integer solution is at ( 0 , 3 ), which creates three fruity pies and no sweet pies for a profit of �6. However the linear programming solution is at ( 1/2 , 3 ) which would create half a sweet pie and three fruity pies giving a profit of �6.50. This solution is not used because this is an integer problem and half is not an integer. Slack Variables We can introduce slack variables to determine exactly how much of each of the key ingredients are wasted when producing three fruity pies. The constraints would now be: 2x + 3y + S1 ? 10 10x + 2y + S2 ? 25 4x + 3y + S3 ? 12 For the integer solution, coordinates ( 0 , 3 ), the slack would be: (2*0) + (3*3) + S1 = 10 S1 = 10 - 9 S1 = 1 (10*0) + (2*3) + S2 = 25 S2 = 25 - 6 S2 = 19 (4*0) + (3*3) + S3 = 12 S3 = 12 -9 S3 = 3 Therefore the full coordinates of the integer solution are ( 0 , 3 , 1 , 19 , 3 ). ...read more.

Conclusion

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 �5.33. 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 �5.00 exactly, by making one sweet pie and two fruity pies. Extensions 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. 1 of 11 ...read more.

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