Inequality Constraint Linearization of a product of an integer and a binary variable

Question

I have thought I had found the answer here: How to linearize the multiplication of an integer and a binary integer variable?

But the answers to that questions didn't help me find a solution for my problem.

I have the following constraints

\begin{align}\sum_{i=1}^{N}\sum_{k=1}^{M}{{(x_{i,k,t}\cdot r_i)}}&\le C_t\\ r_i&\le R_i \end{align} where $r_i\ge 0$ is an integer variable, $x_{i, k, t}\in\{0,1\}$ is a binary variable, $C_t$ and $R_i$ are known numbers.

Can I linearize this?

Some background: At period $t$ I have a capacity of $C_t$ for the resources. A certain number of resources $r_i$ perform job $i$. $x_{i,k,t}$ tells us, if job $i$ is performed at time $t$ with a different type of resource $k$.

Answer 1

The linked answer depends on an equality constraint that doesn’t appear here. You can still use the same idea of introducing a nonnegative variable, say $y_{i,k,t}$, to represent the product. Because your constraint is $\le$, you need to enforce only that $$x_{i,k,t}=1\implies y_{i,k,t}\ge r_i,$$ which you can do with a big-M constraint: $$r_i-y_{i,k,t}\le R_i (1-x_{i,k,t})$$