A dynamic lot sizing model with carbon emission constraint and multi-mode production: A general property and a polynomially solvable case

This paper extends the classical dynamic lot sizing model to the case with carbon emission and multi-mode production. Production activities generate emission and an emission cap is given for each period. The manufacturer is equipped with more than one mode for production. If a mode is less costly for producing one unit of product, it however yields more emission. As a consequence, the maximum production quantity in each period depends on the production mode selected and the emission cap. The model proposed in this paper represents applications in which the production is emission-dependent and constrained by the government-imposed emission regulation. Furthermore, with multi-mode production and emission regulation, emission abatement may be prompted through technological innovation and optimization of production planning. Compared with classical production planning problems, the problem considered here is challenging due to the emission constraint (therefore cumulative production capacity) and the multiple production modes (therefore multiple setup cost structures). In this paper, we present a general property of optimal solutions. Then, we consider a special but practical case involving two production modes. The practicalness stems from the fact that few manufacturers are equipped with more than two technologies, because of the heavy investment required for a new technology. We first prove some properties of optimal solutions. As a consequence, we can focus on solutions with these properties. This enables us to develop a polynomial dynamic programming algorithm to solve the model into optimality.