Optimization Problem

[mercuryv8]

A coworker of mine phoned with this problem.

 

You have been asked to create at least 1000g of a mixture of sunflower seeds and raisins. The mixture must contain at least 75g of dietary fibre, but no more than 40mg of iron. Use the chart below to determine how many units of sunflower seeds (1 unit is 100g) and how many units of raisins (1 unit is 100g) should be used to minimize the cost of producing this mixture.

 

Dietary fibre (g) 10 g / unit sunflower seeds

Dietary fibre (g) 5 g / unit of raisins

 

Iron (mg) 4 mg / unit Sunflower seeds

Iron 2mg / unit raisins

 

I’ve worked out a system of e equations, and have tried to graph the problem. But I don’t know what the best strategy is.

 

Thanks for any input .

 

Nic

[DrP]

Shall we assume that sunflour seeds or rasins are cheaper? or have I missed that one somewhere?

 

It takes about 3 mins by trial and error I think. You can only use 10 units anyway to make up your 1000g. Therefore tryout 10:0 / 9:1 / 8:2 / 7:3 / 6:4 / 50:50 / 4:6 etc... back to 0:10 units of each. (or have I missed the point here?)

 

 

PS - If raisins are cheaper, use 50:50(75g fibre + 30g iron). If sunflour seeds are cheaper then use 100% sunflour seeds (100g fibre + 40g iron).

[mercuryv8]

Shall we assume that sunflour seeds or rasins are cheaper? or have I missed that one somewhere?

 

It takes about 3 mins by trial and error I think. You can only use 10 units anyway to make up your 1000g. Therefore tryout 10:0 / 9:1 / 8:2 / 7:3 / 6:4 / 50:50 / 4:6 etc... back to 0:10 units of each. (or have I missed the point here?)

 

 

PS - If raisins are cheaper, use 50:50(75g fibre + 30g iron). If sunflour seeds are cheaper then use 100% sunflour seeds (100g fibre + 40g iron).

 

 

I see what you are saying. It would make more sense if they specify what the costs of each are.

 

Opps!! sometimes I sould spend more time reading. The Cost of sunflower seeds is .12$ per unit and raisins are .10 $ per unit

Nic

 

So 5:5 must be the optimal mixture.

 

You get 75g of Fiber

 

30mg of Iron

 

and the cost is 1.10$ / 1000g

[Country Boy]

A coworker of mine phoned with this problem.

 

You have been asked to create at least 1000g of a mixture of sunflower seeds and raisins. The mixture must contain at least 75g of dietary fibre, but no more than 40mg of iron. Use the chart below to determine how many units of sunflower seeds (1 unit is 100g) and how many units of raisins (1 unit is 100g) should be used to minimize the cost of producing this mixture.

 

Dietary fibre (g) 10 g / unit sunflower seeds

Dietary fibre (g) 5 g / unit of raisins

 

Iron (mg) 4 mg / unit Sunflower seeds

Iron 2mg / unit raisins

 

I’ve worked out a system of e equations, and have tried to graph the problem. But I don’t know what the best strategy is.

 

Thanks for any input .

 

Nic

Your job involves solving classical high school algebra problems?

[DrDNA]

Your job involves solving classical high school algebra problems?

 

He/she could be a theoretical nutritionist/food scientist.

[MrMongoose]

I wish my job was that easy!

[insane_alien]

He/she could be a theoretical nutritionist/food scientist.

 

well, in food process engineering(i took a class on it just there) there are quite a few things requiring more complex maths than algebra. freezing requires a bit of calculus. granted its still not all that complicated compared to some other problems.

[mercuryv8]

I'm a teacher.

 

Thanks for the discussion, The solution to the problem is the easy part. The person that asked the question was looking for more of a theory behind solving the problem answer. I've worked out a system of equations to help.

 

Nic

[D H]

I’ve worked out a system of e equations, and have tried to graph the problem. But I don’t know what the best strategy is.

This is a simple example of the class of problems known as "linear programming". To graph this problem, put number of units of sunflowers on one axis and number of units of raisins on another. I will assume the sunflower axis is the horizontal axis. You have three constraints: Fiber >= 75 grams, iron <= 40 mg, and total units >= 10 (you have to produce at least 1000 grams of product and 1 unit=100 grams). These constraints generate a feasibility region. In this case, the feasibility region is the area between two parallel lines with slope = -2 that intercept the horizontal axis at 7.5 units (fiber >= 75 grams) and 10 units (iron <= 40mg) and the region above the line with slope = -1 that intercepts the horizontal axis at 10 units (total units >= 10). Now plot some constant cost curves. These are parallel lines with slope = -6/5.

 

Since everything here is linear, the solution must lie somewhere on the boundary of the feasibility region. In fact, the solution must be at one of the vertices of the feasibility region, with one exception. The exception occurs when some edge of the feasibility region has the same slope as the constant cost curves, in which case a vertex is still optimal. The standard technique for solving these problems, the simplex algorithm, takes advantage of the fact that the solution lies on a vertex.

 

For more info, I suggest looking at the Wikipedia articles on linear programming, simplex algorithm, and interior point methods.

[mercuryv8]

This is a simple example of the class of problems known as "linear programming". To graph this problem, put number of units of sunflowers on one axis and number of units of raisins on another. I will assume the sunflower axis is the horizontal axis. You have three constraints: Fiber >= 75 grams, iron <= 40 mg, and total units >= 10 (you have to produce at least 1000 grams of product and 1 unit=100 grams). These constraints generate a feasibility region. In this case, the feasibility region is the area between two parallel lines with slope = -2 that intercept the horizontal axis at 7.5 units (fiber >= 75 grams) and 10 units (iron <= 40mg) and the region above the line with slope = -1 that intercepts the horizontal axis at 10 units (total units >= 10). Now plot some constant cost curves. These are parallel lines with slope = -6/5.

 

Since everything here is linear, the solution must lie somewhere on the boundary of the feasibility region. In fact, the solution must be at one of the vertices of the feasibility region, with one exception. The exception occurs when some edge of the feasibility region has the same slope as the constant cost curves, in which case a vertex is still optimal. The standard technique for solving these problems, the simplex algorithm, takes advantage of the fact that the solution lies on a vertex.

 

For more info, I suggest looking at the Wikipedia articles on linear programming, simplex algorithm, and interior point methods.

 

That's what I was looking for.

 

Thanks

 

Nic

[D H]

That's what I was looking for.

Thanks

 

You're welcome.