How to solve mixed-integer problem?

Question

I don't know how to solve this equation,

$$\left\lceil\frac{x-A}{B}\right\rceil C + D x < E, \quad x\in \mathbb{Z}$$

In this equation, only $x$ is unknown and $x$ is integer, but $A,B,C,D,E$ are maybe decimals and they are greater than zero. How can I get the upper bound of $x$?

Answer 1

Introduce an integer variable $y$ and a continuous variable $s$, and write: $$\begin{align} & y C + Dx \le E\\ &y = \frac{x-A}{B} + s \\ &s \in [0,0.999] \\ &y \in \{...,-3,-2,-1,0,1,2,3,...\} \end{align}$$ Of course a $<$ constraint is no good, so I made it a $\le$ constraint.