We consider a two-stage supply chain in which two production plants are collaborating in an industrial symbiosis to satisfy their respective stochastic demands. We formulate the production planning problems of these two plants as an integrated capacitated lot-sizing problem, in which the second production plant uses as an alternative raw material a by-product obtained as a residue from the production of the first plant. The goal is to minimize the overall total cost in the supply chain, including production and inventory of the final product and by-product transfer costs, while meeting the stochastic demands. First, a natural formulation of the problem is proposed, and is solved using the Sample Average Approximation (SAA) method. The analysis of the gaps exhibits however quite large optimality gaps. To improve these optimality gaps, a plant location like reformulation for this integrated lot-sizing problem is developed. The analysis has been carried out again to evaluate both formulations’ performances in terms of the optimality gaps and computational times, both when items demands follow Gamma and Normal distributions. The analysis indicates that despite having a computational time of on average 1.7 times higher than the main formulation, the plant location reformulation provides better optimality gaps on average 22% improved and better ranges for upper and lower bounds under stochastic demands.