Linear programming

figo · #1 $Old$ January 10th, 2009, 04:35 PM

Hi
I have one problem

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 Code:

-------------------------------------------------
|Factory |Cost x 10^3    |Work   |Raw materials  |
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|1          |15                 |2        |3                  |
-------------------------------------------------
|2          |10                 |3        |4                  |
 ------------------------------------------------
|3          |9                   |4        |5                  |
 ------------------------------------------------
|4          |7                   |5        |6                  |
 ------------------------------------------------

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??

Last_Singularity · #2 $Old$ January 10th, 2009, 05:47 PM

Quote:

Originally Posted by figo $View Post$

Hi
I have one problem

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 Code:

-------------------------------------------------
|Factory |Cost x 10^3    |Work   |Raw materials  |
-------------------------------------------------
|1          |15                 |2        |3                  |
-------------------------------------------------
|2          |10                 |3        |4                  |
 ------------------------------------------------
|3          |9                   |4        |5                  |
 ------------------------------------------------
|4          |7                   |5        |6                  |
 ------------------------------------------------

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.

DeMath · #3 $Old$ January 11th, 2009, 08:09 AM

MS Excel has shown such a result:
$z_{\min}=11600$ if $x_1=400$ , $x_2=200$ , $x_3=400$ , $x_4=0$ .