I have $n$ objects $O_i$, each of them having $3$ values, $O_i = (A_i, B_i, C_i)$. I am trying to group them into $k$ groups $P_u$ such as $P_u =(A_u, B_u, C_u)$ such that

$$\text{minimize} \quad \sum_u^k M_k \left( a A_u + b B_u + c C_u \right)$$

for all $O_i \in P_u$, $A_i \leq A_u$, $B_i \leq B_u$, $C_i \leq C_u$. Here, $M_k$ is the numbers of objects $O_i$ in each category ($\sum M_k =n $). $a$, $b$ and $c$ are constants, as $A$, $B$ and $C$ do not have the same weight.

Can anyone link me to a theorem or some literature that would give me an idea of how to solve my problem?

I am not a mathematician or anything alike. I am an engineer with what I assume is a trivial optimization problem. Let me clarify what I'm trying to achieve.

I am a structural engineer and I have to design beam bearing plates, each bearing plates having $3$ dimensions (length, width and thickness). In a given building it is not uncommon to need $100$ or more bearing plates, each with their own dimensions. However for the sake of practicality we oversize some and group them so that at the end we only have a few different bearing plates ($k$ groups of plates, $k$ being a number chosen arbitrarily by the engineer and not a variable to optimize). We cannot undersize any plates.

I am trying to find a way to optimize the design of those groups so that the act of grouping the plates waste as little money as possible, so the values to be optimized are the dimensions of each group of plates.

Factors $a$, $b$ and $c$ comes into play because bumping up a plate thickness is more costly than bumping up its length or width.

I hope this clears up any misunderstanding, if not just let me know. Thanks all!

**Fedja's possible clarification** (I'm not sure if this is what OP had in mind)

There are $n$ triples $(A_i,B_i,C_i)$ of real numbers ($i=1,\dots,n$). We want to partition the index set $\{1,\dots,n\}$ into $k$ subsets $P_u$ so that the sum $$ \sum_u |P_u|[a\max_{i\in P_u}A_i+b\max_{i\in P_u}B_i+c\max_{i\in P_u}C_i] $$ is as small as possible, where $|P_u|$ is the cardinality of $P_u$.

What is an efficient algorithm for doing that?

I would also risk to assume that $a,b,c>0$ (which actually makes them redundant in the mathematical formulation of the problem: just replace $A_i$ by $aA_i$ and so on).