(448c) Optimization of Discrete Time Supply Chain Models with Guaranteed Robust Stability and Feasibility

We present a method for the simultaneous economic optimization and optimal design for disturbance rejection of discrete time systems. The method is an extension of the normal vector methods for the design of nonlinear dynamical systems with guaranteed robust stability and feasibility [6-10]. The normal vector method was originally developed for parametric uncertainty with respect to stability properties in optimization problems. However, it naturally applies to the robust treatment of more general dynamical properties as well as robust feasibility. In this contribution we extend the method to a new set of constraints on the location of poles of discrete time systems. This development was motivated by the need to optimize discrete time systems while simultaneously imposing constraints that ensure robust disturbance rejection. The method is an extension of a previously presented approach to the optimization of discrete time systems with constraints on robust stability [9].

Essentially, the normal vector method is based on the fact that equilibrium solutions of dynamical systems can be characterized by their parametric distance to the manifolds of critical points [6]. Typical critical points of interest are bifurcation points or points at which state variable constraints are violated. Once the locations of the critical points of a system are known, normal vectors on the critical manifolds can be used to measure the distance from the nominal point of operation to stability and feasibility boundaries in the space of the system design parameters. By staying sufficiently far away from all critical manifolds we can guarantee robust stability and feasibility of the system.

In this contribution we present the extension to bounds for robust disturbance rejection. In order to ensure robust disturbance rejection we introduce a simple but new type of critical point that is an extension and generalization of fold, Neimark-Sacker and flip bifurcations. Applying the normal vector method to this type of critical point amounts to forcing all eigenvalues of the linearized discrete time system into a circle of radius r<1 in the complex plane. By specifying the radius r, the user of the method can specify the desired decay rate. We stress that while the normal vector method makes use of the linearized system and its eigenvalues, the decay rate constraint holds for the nonlinear system in a finite ball around the nominal optimal system parameters [4]. By specifying the size of this ball, the user of the method can specify the desired parametric robustness.

The proposed approach is applied to the optimization of two supply chain models. The first model is the generic model of an automatic pipeline feedback compensated inventory and order-based production control system (APIOBPCS) [1, 5]. The second model is the model of APIOBPCS embedded within a vendor managed inventory supply chain where the demand profile is deemed to change significantly over time (VMI-APIOBPCS) [2, 3]. We present and discuss the optimal points for APIOBPCS and VMI-APIOBPCS that result if constraints on decay rates for robust disturbance rejection are imposed.

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