LU Decomposition problem

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jeffo

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Apr 1, 2022
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The question is a bit long, I have simplified it:
Contractor can provide 80 hours of planning, 120 hours of cutting and 160 hours of fitting per day.
There are three types of Gardens the contractors work on, each require the following:
Garden 1: 2 hours planning, 3 hours cutting, 4 hours fitting
Garden 2: 1 hours planning, 1 hours cutting, 3 hours fitting
Garden 3: 5 hours planning, 2 hours cutting, 6 hours fitting
Using LU decomposition, find the best balance of work for each garden to make use of all work power/hours.

I am pretty stuck on this, I am sure the square 3x3 matrix will be from the hours each garden requires, this multiplied by some unknown should equal total hours for each planning cutting and fitting. But I am not sure what the unknowns I am solving for represents.
Also what is the benefit of using LU decomposition over inverse matrix?

Any help appreciated.
 
jeffo said:
The question is a bit long, I have simplified it:
Contractor can provide 80 hours of planning, 120 hours of cutting and 160 hours of fitting per day.
There are three types of Gardens the contractors work on, each require the following:
Garden 1: 2 hours planning, 3 hours cutting, 4 hours fitting
Garden 2: 1 hours planning, 1 hours cutting, 3 hours fitting
Garden 3: 5 hours planning, 2 hours cutting, 6 hours fitting
Using LU decomposition, find the best balance of work for each garden to make use of all work power/hours.

I am pretty stuck on this, I am sure the square 3x3 matrix will be from the hours each garden requires, this multiplied by some unknown should equal total hours for each planning cutting and fitting. But I am not sure what the unknowns I am solving for represents.
Also what is the benefit of using LU decomposition over inverse matrix?

Any help appreciated.
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Apologies,
So far I have this:
[math]\begin{bmatrix} 2 & 3 & 4\\ 1 & 1 & 3\\ 5&2&6 \end{bmatrix} * \begin{bmatrix} x1\\ x2\\ x3\\ \end{bmatrix} = \begin{bmatrix} 80\\ 120\\ 160\\ \end{bmatrix}[/math]
I can apply the LU decomposition method to solve for the unknowns. Where I am stuck is with what the unknowns actually represent/mean since the question, i believe, is asking to optimize the number of gardens that can be made? Is this asking what proportion of the total hours for each action (planning, cutting fitting) should be for each garden every day?
 
Can you please define x1, x2, x3, do the multiplication and see if you should be getting the results on the rhs of the equal sign?
 
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