First of all, what do you mean by "random inputs" - do you mean you vary the values of x or starting points? Both of these can cause changes; the starting point value means you just need to do a more complete search, while the x changes are expected.
More importantly, why are you combining your likelihood functions like that? If they are likelihoods, you need to look at the probabilistic model which is generating your data to see how they should be combined; even if they're independent x variables you should be multiplying most likely.
