We study an integrated multi-product production and distribution problem considering a network of multiple plants and customers, who are geographically dispersed, with direct shipment from the plants to the costumers. In addition to the decisions on production and distribution, a decision needs to be taken on the level of process flexibility in the network, i.e., which products can be produced in which plants. There is a clear trade-off between these decisions. On one hand, a network with total flexibility where each plant can produce all types of products allows for lower transportation costs, but requires large investments in flexibility and frequent setups. On the other hand, a network with a limited amount of flexibility where each plant produce only few products, will increase the transportation costs, but requires a lower investment in flexibility. We model this problem as an extension of the capacitated lot-sizing problem. We limit the investment in flexibility by a budget constraint and minimize the operational costs. Varying this budget allows us to analyze different levels of flexibility. In this paper, we propose mathematical models and a hybrid solution method that combines a mixed integer programming-based approach and a kernel search heuristic. Our computational results using data sets from the literature show that the proposed hybrid method produces on average better solutions with significantly lower computational times when compared with the results produced by a state-of-the-art optimization software. Additional computational results are presented by varying key parameters and analyzing their impact on the value of flexibility. These computational experiments indicate that some of the main managerial insights which were derived in the literature for the case without transportation costs are no longer valid when we consider transportation costs.