Linear-quadratic optimal control of multi-modal distribution systems with imperfect channels

The paper examines the perspectives of linear-quadratic (LQ) optimal control in steering the process of goods distribution in logistic systems with multiple transportation options. In the considered class, the distribution centre governs the stock replenishment process of subordinate depots, from which un uncertain market demand is served. The centre is linked with the depots via shared supply channels with different characteristics regarding delay, reliability, and capacity, e.g. train vs truck delivery. The design objective is a rule of dynamical channel allocation – how many goods to send in a period using a given mode – so that balanced, cost-efficient system performance and high customer service rate are achieved. The received goods are inspected for quality defects and rejected when faulty. Thus, one needs to cope with two major sources of uncertainty: unpredictable demand variations and channel imperfections. A multi-variable LQ optimal controller is designed and presented in closed form for detailed analytical and numerical treatment. It is formally shown that despite perturbations, the controller always establishes a non-negative and upper-bounded replenishment signal, and the stock level does not cross the reference value. Conditions for warehouse space selection and obtaining full demand satisfaction at the depots are specified and formally proved.