However, supply chain systems usually contain large uncertainties, such as costumer demands, production time and transportation time, that can severely degrade the performance of conventional nominal MPC. Robust MPC, which is an enhanced MPC algorithm that includes the uncertainties explicitly in control calculation, can substantially improve the performance of controlling highly uncertain systems, but its application to robust operation of supply chain systems has been rarely seen.
In this paper, a new robust MPC method developed by Li and Marlin (2007) will be applied to a real operational industrial SCO problem. The robust MPC predicts the bounds on the future system behaviors by approximating the uncertain closed-loop system. The approach can handle the future input saturation and avoid unnecessary conservative control. The resulting bi-level stochastic optimization problem is transformed into a single-level, convex Second Order Cone Programs (SOCP, Ben-Tal and Nemirovski, 1999), which can be solved efficiently and reliably with the state-of-the-art interior point optimization solver, e.g. IPOPT (Wächter and Biegler, 2006).
The industrial supply chain system in this paper includes a material production unit, a product processing uniting, a plant distribution center and 5 regional distribution centers (for end customers). The goal of operating this supply chain system is to minimize the total inventory (working capital) and transportation cost while satisfying customer demands (if possible) by deciding the material and product production rates and transportation rates once a day. With proper assumptions, we build a state-space dynamic model with continuous variables for the nominal system. This model is then updated into the canonical form (that is appropriate for the MPC calculation) by the following two steps:
1) Handling the transportation time (which is the time delay between input and output in the state-space model) by introducing additional state variables that denote the products in the transportation. These variables can be directly used for the working capital associated to the inventories in the transit.
2) Handling the inconsistency between the control execution periods and the input implementation periods. If the implementation period of an input is less than the controller execution period, decisions for that input will be implemented as calculated from the most recent controller calculation. If the implementation period is larger than the controller execution period, the decisions will only be changed when possible in the plant, i.e., when a production run has been completed.
The major uncertainties in the system are the costumer demand, product production and transportation time. We assume the costumer demand is normally distributed, and its variance can be obtained from the historical data. The variability of the product production and transportation time is estimated by the industrial collaborator, which is characterized approximately by linear disjunctive models with continuous parameters. In the closed-loop prediction model, the future control policies are approximated by the nominal MPC that drive the output trajectory to a reference trajectory.
The case study compares the performance of nominal and robust MPC for the SCO problem. The simulation results show that undesirable back orders occur in a regional center if the supply chain is operated with nominal MPC, which is due to the inaccurate nominal prediction. But no back orders occur if the supply chain is operated with robust MPC, which is because robust MPC predicts the uncertain closed-loop behavior explicitly (according to the known uncertainties) and is able to keep safety stock ahead of time. This indicates that robust MPC can determine the optimal safety stock to satisfy customer demands and save working capital by avoiding unnecessary storage.
This paper will conclude with a discussion of open issues and the future research.
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