  In this paper, we propose the optimal production transport model, which is an
extension of the classical optimal transport model. We observe in economics,
the production of the factories can always be adjusted within a certain range,
while the classical optimal transport does not take this situation into
account. Therefore, differing from the classical optimal transport, one of the
marginals is allowed to vary within a certain range in our proposed model. To
address this, we introduce a multiple relaxation optimal production transport
model and propose the generalized alternating Sinkhorn algorithms, inspired by
the Sinkhorn algorithm and the double regularization method. By incorporating
multiple relaxation variables and multiple regularization terms, the inequality
and capacity constraints in the optimal production transport model are
naturally satisfied. Alternating iteration algorithms are derived based on the
duality of the regularized model. We also provide a theoretical analysis to
guarantee the convergence of our proposed algorithms. Numerical results
indicate significant advantages in terms of accuracy and efficiency.
Furthermore, we apply the optimal production transport model to the coal
production and transport problem. Numerical simulation demonstrates that our
proposed model can save the production and transport cost by 13.17%.
