Formulating Solution for Branch and Bound I have a linear programming question which I am setting up for a branch and bound solution.
I am having issues with where to begin. The question is asking to find the minimum operating cost to produce a minimum of 225,000 cars from multiple plants. Each plant can run in 3 different modes, each mode with a different cost / car production. Not all of the plants need to be used. 
The data is as follows
Plant     Mode         Operating Cost          Cars Produced
1         1            40,000                  80,000
1         2            70,000                  130,000
1         3            110,000                 190,000
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2         1            63,000                  95,000
2         2            98,000                  165,000
2         3            150,000                 230,000
------------------------------------------------------------
3         1            83,000                  110,000
3         2            108,000                 175,000
3         3            125,000                 210,000

What I'm having troubles with is how to handle the fact that each plant can only be used once, and the fact that it isn't necessary for a plant to be used. How should I show this in my objective function and constraints?
Thanks

I would take the following variable:
$y_{ij}=\begin{cases} 1, \ \text{if plant i in mode j is used} \\ 0, \ \text{else} \end{cases}$
At most one mode for each plant
$\sum_{j=1}^3 y_{ij} \leq 1$
Equal or more 225,000 cars have to be produced
$80,000\cdot y_{11}+130,000\cdot y_{12}+190,000\cdot y_{13}+95,000\cdot y_{21}+160,000\cdot y_{22}$ $+230,000\cdot y_{23}+110,000\cdot y_{31}+ 175,000\cdot y_{32}+210,000\cdot y_{33}\geq 225,000$
