Hi,
I'm struggling to formulate an LP for this problem..even just trying to formulate the decisions variable is proving to be hard. Any chance you could help would be great. The question is:
A manufacturer of chicken feed has available four grains: Sunflower, Sorghum, Maize, Wheat. The nutritional composition (per kg) and price of these grains is as follows:
Sunflower Sorghum Maize Wheat
Energy (kJ) 10000 7000 9000 8000
Protein (g) 140 100 50 80
Vitamin A (mg) 12 2 3 6
Price ($/kg) $2.40 $1.00 $0.50 $1.20
The Australian Chook Farmers Association specifies that 1 kg in the diet of a standard chicken should contain at least the following levels:
Component Min. requirement
Energy 6400 kJ
Protein 96 g
Vitamin A 6.4 mg
What blend of ingredients should be used in 1 kg of feed to provide an adequate diet for a chicken at least cost?
What is an LP?
linear program...so ive got to formulate the problem as a linear program in symbols! struggling..
It's going to be something like this (but do check my algebra)...
You've been told what the Energy, Protein, Vitamin and Price are for unit quantities of Sunflower, Sorghum, Maize and Wheat. This gives you four equations - one in each of Energy, Protein, Vitamin A and Price as follows:
Suppose you've got F kgs of Sunflower, G kgs of Sorghum, M kgs of Maize and W kgs of Wheat. Then you know that:
Energy (kJ) = (10000 x F) + (7000 x G) + (9000 x M) + (8000 x W)
Protein (g) = (140 x F) + (100 x G) + (50 x M) + (80 x W)
Vitamin A (mg) = (12 x F) + (2 x G) + (3 x M) + (6 x W)
Price ($/kg) = (2.4 x F) + (1 x G) + (0.5 x M) + (1.2 x W)
Now - you have to ensure that one kg of chicken feed contains at least:
Energy 6400 kJ
Protein 96 g
Vitamin A 6.4 mg
That is...
Energy (kJ) = (10000 x F) + (7000 x G) + (9000 x M) + (8000 x W) >= 6400
Protein (g) = (140 x F) + (100 x G) + (50 x M) + (80 x W) >= 96
Vitamin A (mg) = (12 x F) + (2 x G) + (3 x M) + (6 x W) >= 6.4
and the price of that will be:
Price ($/kg) = (2.4 x F) + (1 x G) + (0.5 x M) + (1.2 x W)
So you now have to minimize Price subject to the three constraints given above. In passing, don't forget that there are four other constraints that are intuitively obvious, but which nevertheless have to be satisfied by the solution, namely F >= 0, G >= 0, M >=0 and W >= 0.
Does that help?
Oh - and one more constraint, namely that the total quantity of Sunflower, Sorghum, Maize and Wheat equals one kilogram, since you're trying to find the optimal mix in that amount of feed.
This isn't actually part of the problem you've been set, but if you're interested I've managed to find the solution to it using Excel Solver, which is an optimization application add-in for MS Excel.
The optimal solution appears to be 0.4kg of Sunflower, 0.2kg of Sorghum, 0.4kg of Maize and no Wheat. This produces 9000 kJ of energy (more than the minimum of 6400 kJ); 96g of protein (the minimum requirement) and 6.4mg of Vitamin A (the minimum requirement). The price of this mixture is $1.36/kg.
Hi David,
Thankyou so much..that helped me alot! I had managed to form the equations but I really wasn't sure if they were correct but your's match up with mine! Thanks again =)