title: How to implement shutdown, start-up, ramp-up and ramp-down in a MILP if capacity is optimized as well and minimum power exists
body: <p>I am currently trying to solve a problem with Mixed-Integer linear problems. I have a component, which capacity and operation I want to optimize. This component has a lower limit of operation, p_low. The optimization can either run the component between its upper limit (the capacity) or p_low, but it should also be able to shut-down the component.</p>
  <p>Here starts the issue: I have no idea how to implement both possibilities, as a shut-down would violate the minimal power constraint.</p>
  <p>I thought about using binary variables for decision (b_status), but I have a problem with the the capacity of the component, because it is optimized as well and combining the capacity variable with the binary variable:</p>
  <p>p &gt;= p_low * capacity * b_status -&gt; if b_status is 0 -&gt; p &gt;= 0; else: p &gt;= p_low * capacity</p>
  <p>... would result in a non-linear problem as the combination of capacity and b_status is not allowed in linear models (both variables).</p>
  <p>Does someone know a way to implement this? I know this is not the main topic here, but it might be worth the try asking.</p>