Single-item lot sizing problem with carbon emission under the cap-and-trade policy

We study the integration of the carbon emission constraint into the single item uncapacitated lot sizing problem (ULSP) under the cap-and-trade policy. Besides a limitation on the total carbon emitted through the production and storage activities over the entire hoziron, the cap-and-trade policy allows the firm to buy and to sell carbon units in case of need or surplus. In addition to the classical lot sizing costs (setup cost, unit production and unit holding costs) we take into account a cost of buying and a cost of selling carbon units. The speculative trades are not allowed by assuming stationary prices both for selling and buying activities. With an unlimited budget assumption, we show that the problem is equivalent to the classical ULSP which is polynomially solvable. We study two budget constraints and we show the problem to be NP-hard in the ordinary sense under a limited budget assumption using a recent result in Helmrich et al., 2012. We also show that both problems under different budget constraints and carbon trade costs can be reduced to the problem studied in Helmrich et al., 2012.