Optimization of an objective where cost parameter takes step value

Question

I have the following simplified optimization problem:

min $C*x$

s.t, some constraints

I want the following to be enforced as additional constraints:

$C=1$, when $0<x<=40$

$C=2$, when $40<x<=140$

$C=3$, when $140<x<=400$

How do i implement that as an MIP as I don't think this can be converted to a LP problem.

Answer 1

Introduce three binary variables $d_i$ which sum to 1 and a variable $f$ to represent objective, and then use big-M to enforce that $d_1$ implies $0\leq x \leq 40, f = x $ etc.

To model $d_i$ implies $g(x)\leq 0$, you use $g(x) \leq M(1-d_i)$.