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

Abstract

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.

KEYWORDS:

• Inventory control
• multi-modal transport
• uncertain demand
• discrete-time systems
• time-delay systems

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work has been performed in the framework of a project “Robust control solutions for multi-channel networked flows” no. 2021/41/B/ST7/00108 financed by the National Science Centre, Poland.

Notes on contributors

Przemysław Ignaciuk

Przemysław Ignaciuk received the M.Sc. degree in telecommunications and computer science and the Ph.D. degree in control engineering and robotics from Lodz University of Technology, Lodz, Poland, both with honours, in 2005 and 2008, respectively. In January 2014 he obtained the Habilitation degree in computer science. P. Ignaciuk worked for three years at a full-time position as analyst and IT system designer in the telecommunications industry. Next, he joined Institute of Automatic Control, and since 2011 he is with Institute of Information Technology, Lodz University of Technology, where he occupies the position of Assistant Professor and is the head of a research group dedicated to complex system analysis and design. He has authored or coauthored one book, 15 monograph chapters and over 140 journal and conference papers, mainly in the area of logistic and communication networks. His research interests encompass robust control, dynamical optimisation, networked system control, and time-delay systems.