Optimization options to select multiple items with different features and values

Question

I'm trying to identify which approach would work best to select a set of elements that have different features that minimise a certain value. To be more specific, I might have a group of elements with Feature 1, 2, 3, 4 and another group with Feature 2, 3, 4, 5.

I'm trying to minimise the overall value of Feature 2 and 3, and I also need to pick a certain number of elements of each group (for instance 3 from the first group and 1 from the second).

From the research I did it seems that combinatorial optimization and integer programming are the best suited for the job. Is there any other option I should consider? How should I set up the problem in terms of cost function, constraints, etc.?

Edit: I've simplified the problem a bit, but I still can't setup the IP problem. Let's say I have three options:

$A$ with $x_1 = 1000$ and $x_2 = -400$

$B$ with $x_1 = 800$ and $x_2 = -500$

$C$ with $x_1 = 1200$ and $x_2 = -600$

I want to pick three elements that minimise for $x_1$ and $x_2$ (it's probably the wrong notation, but those are the values for the features for each elements).

Answer 1

I think linear integer programming is a good idea. Based on your specifications it would look like the following program.

Udate

I specify your model. You have 3 different features in each group. The costs of picking a element of feature j from group 1 are: $(c_{11},c_{12},c_{13})=(2,4,3)$. The costs of picking a element of feature j from group 2 are: $(c_{21},c_{22},c_{23})=(5,2,1)$.

$x_{ij}$: Amount of elements of feature j from group i // The values are not given.

You have to pick $(b_1,b_2,b_3)=(3,5,4)$ elements of feature j

$c_{ij}$: Costs of picking one feature j of group i. The model for 3 features and 2 groups is:

$\text{min} \ c_{11}\cdot x_{11}+c_{21}\cdot x_{21}+c_{12}\cdot x_{12}+c_{22}\cdot x_{22}+c_{13}\cdot x_{13}+c_{23}\cdot x_{23}$

$x_{11}+x_{21}=3$

$x_{12}+x_{22}=5$

$x_{13}+x_{23}=4$

$x_{ij} \in \mathbb N $

$c_{ij} >0 $

This is the model i would suggest. You have to insert the values for $c_{ij}$ And keep in mind, that the values of $x_{ij}$ are not given. They are the solution of the model. And all values are greater or equal to zero and whole numbers.