This study considers a non-homogeneous discrete grey forecasting model with arbitrary accumulation operators, abbreviated as NDGM(1,1,k,c), to discuss the gross regional product of Sichuan. Firstly, we provide the definition of an arbitrary accumulation operator, and discuss several particular cases including the integer-order accumulation operator, the new information priority accumulation operator, the fractional-order accumulation operator, the conformable fractional-order accumulation operator, the Hausdorff fractional-order accumulation operator, the damping accumulation operator, and the probabilistic accumulation operator. Secondly, the NDGM(1,1,k,c) model with arbitrary accumulation operators is systematically studied with the grey modelling technique, the least squares estimation method and the salp swarm optimization algorithm. The perturbation bounds of the new model are discussed by the matrix perturbation theory. Finally, the NDGM(1,1,k,c) model is applied to the gross regional product of counties (cities, districts) in Sichuan. With raw data from 2011 to 2022, we develop different grey models under 20 subcases for each county/city/district which show that the NDGM(1,1,k,c) model can obtain competitive results and be suitable for studying and analyzing Sichuan county economy.