Consider the integer knapsack set as:
$$ K_{I} = \{ x \in Z^{N}_{+} : ax \leq b, x \leq u\} $$
Let $C \subseteq N$ be a cover if $\lambda = \sum_{i \in C} u_ia_i − b \gt 0$ and consider the restriction $K_I (N \text{\\} C, \varnothing)$ obtained by fixing all $x_i, i \in N \text{\\} C$ to zero. Extending the combinatorial argument on covers for the $0$-$1$ knapsack set, since not all variables $x_i, i \in C$ can be at their upper bound simultaneously, inequality:
$$ \sum_{i \in C} x_i \leq u(C)-1 \quad (1)$$
is valid for $K_I (N \text{\\} C, \varnothing)$. Another way to write the above inequality is:
$$ \sum_{i \in C} (u_{i} - x_{i}) \geq 1 \quad (2)$$
which states that at least one $x_{i}$ must be less than its upper bound. However, observe that if $a_i \lt \lambda$ for all $i \in C$ the left-hand side of (2) is greater than $1$ for all feasible solutions. This suggests the strengthening:
$$ \sum_{i \in C} (u_{i} - x_{i}) \geq \lceil \lambda/ \bar{a} \rceil $$
where $\bar{a} = \text{max}_{i \in C} a_{i}$. Suppose the integer knapsack is given as:
$$ K_{I}=\{ x \in \text{Z}^2: 5x_{1} + 9x_{2} \leq 45, x_{1} \leq 6, x_{2} \leq 4\}$$
Let $C = \{1,2\}$ be an integer cover with $\lambda = 30 + 36 − 45 = 21$. The following inequality:
$$ (6-x_{1}) + (4-x_{2}) \geq 1 $$
does not support $K_{I}$ . However, the strengthened inequality:
$$ (6-x_{1}) + (4-x_{2}) \geq \lceil 21/9 \rceil $$
defines a facet of $K_{I}$ for this example. For more details please, see the following paper.
Also, it seems the mentioned integer knapsack set can be supposed as a capacity constraint in the planning models that other valid inequalities like Gomory cuts, MIR cuts, etc. can be applied on.
Atamtürk, A. Cover and Pack Inequalities for (Mixed) Integer Programming. Ann Oper Res 139, 21–38 (2005). https://doi.org/10.1007/s10479-005-3442-1