# Linear programming

• Jan 10th 2009, 02:35 PM
figo
Linear programming
Hi
I have one problem (Worried)
Trucks production company must produce 1000 trucks. Company has four factories. The cost of truck production (raw materials, work) in each separate factory is shown in table:
[HTML]
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|Factory |Cost x 10^3 |Work |Raw materials |
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|1 |15 |2 |3 |
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|2 |10 |3 |4 |
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|3 |9 |4 |5 |
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|4 |7 |5 |6 |
------------------------------------------------[/HTML]

Automaker trade union requires that at least 400 trucks should be made in 3rd factory. It was given 3300 work hours and 4000 pcs. raw materials to each factory for order lead time. Make linear programming problem which minimize the cost of 1000 trucks production.

Can anyone help me to express the objective function and the constraints??
• Jan 10th 2009, 03:47 PM
Last_Singularity
Quote:

Originally Posted by figo
Hi
I have one problem (Worried)
Trucks production company must produce 1000 trucks. Company has four factories. The cost of truck production (raw materials, work) in each separate factory is shown in table:
[html]
-------------------------------------------------
|Factory |Cost x 10^3 |Work |Raw materials |
-------------------------------------------------
|1 |15 |2 |3 |
-------------------------------------------------
|2 |10 |3 |4 |
------------------------------------------------
|3 |9 |4 |5 |
------------------------------------------------
|4 |7 |5 |6 |
------------------------------------------------[/html]Automaker trade union requires that at least 400 trucks should be made in 3rd factory. It was given 3300 work hours and 4000 pcs. raw materials to each factory for order lead time. Make linear programming problem which minimize the cost of 1000 trucks production.

Can anyone help me to express the objective function and the constraints??

Denote $x_1, x_2, x_3, x_4$ as your variables, representing the amount that each factory produces. So your objective is:
$z = 15x_1 + 10x_2 + 9x_3 + 7x_4$

Because it needs to produce 1,000 trucks, one constraint is
$x_1 + x_2 + x_3 + x_4 = 1000$

At least 400 trucks need to be made in the 3rd factory, so
$x_3 \geq 400$

Subject to work hours, we have:
$2 x_1 + 3 x_2 + 4 x_3 +5 x_4 \leq 3300$

Subject to raw materials, we have:
$3 x_1 + 4 x_2 + 5 x_3 +6 x_4 \leq 4000$

However, I am unsure about whether the previous two constraints are correct or not. They would be if the integers that you provided me with were in units of work hours per truck made and raw materials per truck made. If it is the other way around, you need to flip the numbers around.

Don't forget to implement:
$x_1,x_2,x_3,x_4 \geq 0$ - this is NOT a trivial constraint because if you solve it with linear programming software, it could give you negative production.
• Jan 11th 2009, 06:09 AM
DeMath
MS Excel has shown such a result:
$z_{\min}=11600$ if $x_1=400$, $x_2=200$, $x_3=400$, $x_4=0$.