Dealing with demand uncertainty in multi-item lot sizing problems poses huge challenges due to the inherent complexity. The resulting stochastic formulations typically determine production plans which minimize the expected total operating cost while ensuring that a predefined service level constraint for each product is satisfied. We extend these stochastic programming models to a more general setting where, in addition to the individual service level constraints, an aggregate service level constraint is also imposed. Such a situation is relevant in practical applications where the service level aggregated from a variety of products or components must be collectively satisfied. These extended models allow the decision maker to flexibly assign different individual service levels to different products while ensuring that the overall aggregate service level is satisfied and these aggregated service level measures can be used in conjunction with the commonly adopted individual service levels. Different mathematical models are proposed for this problem with different types of service levels. These models are a piece-wise linear approximation for the \(\beta\), \(\gamma\), and \(\delta\) service levels and a quantile-based model for the \(\alpha_{c}\) service level. Computational experiments are conducted to analyze the impact of aggregate service levels and demonstrate the value of the proposed models as opposed to standard service levels imposed on individual items.