Lot sizing with capacity adjustment using on-site green and grid electricity

This paper investigates from a theoretical point of view how on-site generation of renewable energy can be incorporated in the optimization of a mid-term production and capacity planning problem. Specifically, we consider the generic case of a manufacturer using two energy sources to supply the electricity demand of its plant: an on-site renewable energy source and the electricity grid. The renewable energy source is considered to be free of use, but its available amount of energy fluctuates over time, whereas the grid power is virtually unlimited but incurs a cost per kWh purchased from the external provider. The objective is to satisfy a time-varying demand at a minimal cost over a mid-term horizon. The plant has a stationary nominal production capacity. To deal with the fluctuation of both the demand and the amount of energy supplied by the on-site source, the production capacity can be temporally increased by installing additional capacity, typically by changing the shift pattern or opening more production lines. Increasing the capacity allows to respond to peak demand and to build stock in periods where the energy is cheap but incurs a fixed cost. We study if an optimal solution of this integrated production, capacity, and energy planning problem can be computed efficiently to provide the best compromise. Our objective is to classify the complexity of the deterministic version of the problem. We establish its NP-hardness under mild assumptions and propose three polynomial time algorithms for special cases, according to the amount of energy provided by the renewable source. These algorithms rely on dominance structural properties which allow us to reduce the problem to well-studied lot-sizing problems with capacity or full batch delivery.