A Saddle-Point Variational Model for Dynamic Production Planning Under Uncertainty

Abstract

For Uncertain dynamic production planning problems, there must be efficient optimization methods that can deal with the changing state of the system, uncertain disturbances, and equilibrium relations. This paper presents a saddle-point variational framework to optimize dynamic production plans based on minimax optimization in combination with variational inequality formulations. In the saddle-point problem formulation, the uncertain production process becomes a convex–concave problem with production strategies minimizing costs and disturbance uncertainties. Equilibrium relations are then converted into the variational inequalities problem using the monotone operator approach. Existence of saddle-point solutions is proved using the minimax theorem, uniqueness follows from strong monotonicity properties, and convergence of the projected iterative algorithm is established via proper operator properties. An example of a dynamic production planning problem with multiple production lines, changing demands, and restrictions is taken into account in order to perform numerical experiments. Numerical results show that the saddle-point variational iteration method converges faster with only 41 iterations and a lower cost value of 104.2 than the gradient descent-ascent method and penalty method.