integer programming - linear program with zeros

Hi, I want to be able to solve a linear program that has constraints that are either zero or a range. An example below in LP_Solve-like syntax shows what I want to do. This doesnt work. In general all the decision variables Qx can be 0 or a <= Qx <= b where a > 0 and b > a. All decision variables must be integers.


max: 0.2 Q1 + 0.4 Q2 + 0.6 Q3 + 0.8 Q4 + 0.4 Q5 + 0.6 Q6 + 0.6 Q7 + 0.7 Q8;


Q1 = 0 or Q1 >= 5;
Q2 = 0 or Q2 >= 1;
Q3 = 0 or Q3 >= 1;
Q4 = 0 or Q4 >= 1;
Q5 = 0 or Q5 >= 1;
Q6 = 0 or Q6 >= 1;
Q7 = 0 or Q7 >= 9;
Q8 = 0 or Q8 >= 1;
Q4 <= 30;
Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + Q7 + Q8 <= 50;

How would I rewrite this to work? Is there a particular solver that should be used for this kind of task?