How to model manufacturing shift data with irregular production times?
From the information at hand, you could break this down into two problems - 


Predicting the production for the shift, and 
Finding the probability of breakdown during the shift


For (1) you could either go down the time-series route (ARIMA, Box-Jenkins, Exponential smoothening) or the regression route (provided you have good features)

For (2) you could build a logistic regression model that gives you a probability score

Possible features you could look into are 


Time of day, week, month and year
Time between breakdowns
Available labor
Electricity Board data?