this paper, we study the single item capacitated multi-mode lot sizing problem with periodic carbon emission constraints where the carbon emission constraints define an upper bound for the average emission per product produced in any period. The uncapacitated version of this problem was introduced in Absi et al. (2013) and solved in polynomial time. We show that this generalization of the problem is NP-Hard, in general, and present important properties for the optimal solutions of the problem. We propose algorithms to construct the piecewise linear total production cost functions for each period when the number of modes is fixed. This enables us to solve the problem using the dynamic programming algorithms developed for the lot sizing problem with piecewise concave production cost functions. We also consider an extension of the problem where at most two resources can be used at any period, and develop a polynomial time algorithm to solve it when the number of resources, the cost and emission parameters, and the capacities of the resources are time-invariant.
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