constraintsschedulingnonlinear-optimization

Formulating an optimization problem with non linear constraints


I am trying to solve a dynamic machine scheduling problem using what I think qualifies as a non linear programming constraints.

I have academic knowledge of linear programming techniques and appreciate the opportunity to try my hands at non linear programming.

However, before anything I am struggling to write down a well forumulated problem. Here is the situation:

Can you please help me properly formulate the problem? Here is how I started putting it:

objective function: Find optimal X_mt such that task are started on time (limit penalty)

Objective function

constraints:

balanced duration

balanced number

specialization

makespan

singleton

I am also looking for an algorithm to find some epsilon-optimal solution based on how I relax the workload balancing parameter.

But before I can try to look for an algorithm, I need help to properly formulate the problem.

I've seen example where variance constraints are formulated as the minimization of the difference of upper and lower boundaries but I don't know how to apply it in this case since I have multiple variance constraints.

If this is too much, how would you translate this problem into a Reinforcement Learning one?


Solution

  • Let's look at one of the variance constraints:

    enter image description here

    is wrong IMHO. There should be no "for all m".

    I rewrite this as:

    v[m] = sum(t,d[m,t])  for all m 
    variance(v) <= eps    (this is a single, scalar constraint)
       
    <==>
    
    v[m] = sum(t,d[m,t])  for all m
    minv <= v[m]          for all m 
    maxv >= v[m]          for all m
    maxv - minv <= eps    (different eps. now to limit the bandwidth)