Using PDE-models for a unified theory of production lines

В статье обсуждается введенный автором (2003 г.) [1] класс моделей производственных систем с поточным способом организации производства, широко используемый в настоящее время для построения эффективных систем управления производственными линиями [2,3]. Модели класса, определяющие поведение параметров производственной линии с помощью уравнений в частных производных, получили название PDE-моделей производственных систем [2–4], в последнее десятилетие успешно использованы для описания квазистатических, так и некоторых неустановившихся переходных процессов [5].This paper discusses a class of models of production systems [1]. Models are widely used today for building effective systems of production lines [2,3]. Class model parameters determine the behavior of the production line with the help of partial differential equations (PDE-model) [2-4], in the last decade successfully used to describe the quasi-static and transient transients [5]

PDE–Based Modelling and Control Strategies for Manufacturing Processes

This work aims to design boundary control strategies to solve demand tracking and backlog problems for manufacturing systems in terms of conservation laws coupled with ODEs in different network topologies. The OCPs are investigated in the dispersing and the merging networks. The problems are optimized utilizing open-loop optimal control based on the direct and the indirect approaches. The proposed approaches enable the solution of the OCPs. All of the approaches, in general, reach a local minima with similar behaviour that leads to the steady-state. The results analysis reveals that each method has its own distinct characteristics. The indirect methodology is characterized by excellent accuracy and minimal processing burden; yet, due to the information necessary to compute the gradient, it is a sensitive method. The ease of use and flexibility to any problem distinguishes the direct method. However, this approach takes substantially longer to achieve a solution when compared to the indirect method. Also, the AMPC was introduced to investigate demand tracking and backlog problems in the context of the complex network of production systems. The addressed network includes structures that are dispersing and merging. Furthermore, the appropriate way to handle the parameters of the AMPC for both control and prediction horizons is addressed. Moreover, the proposed AMPC provides for the solutions of demand tracking and backlog problems. In general, AMPC and traditional MPC attain local minima with similar behaviour that leads to steady-state convergence. When compared to a typical MPC, the AMPC's performance shows a considerable reduction in computational time. Additionally, because it provides a mathematical insight into the method's structure, the AMPC allows for great accuracy of optimal solutions. Finally, the AMPC is characterized by its robustness according to perturbation effects

Controlling the Bullwhip Effect in a Supply Chain Network with an Inventory Replenishment Policy by a Robust Control Method

This paper develops a mathematical model using differential equations and considers a bullwhip effect in a supply chain network with multiple retailers and distributors. To ensure the stability of the entire system and reduce the bullwhip effect, a robust control method and an inventory replenishment policy are proposed. This shows that the choice of the output matrix may reduce the bullwhip effect. It has also observed in the inventory replenishment mechanism may be a negative impact on the robustness of the bullwhip effect. However, the inventory replenishment behavior may lead to the bullwhip effect on the presented model. This means that the complex supply relationships may have a significant role in controlling or reducing the bullwhip effect of fluctuations

Использование PDE-моделей для построения единой теории производственных линий

В статье обсуждается введенный автором (2003 г.) [1] класс моделей производственных систем с поточным способом организации производства, широко используемый в настоящее время для построения эффективных систем управления производственными линиями [2,3]. Модели класса, определяющие поведение параметров производственной линии с помощью уравнений в частных производных, получили название PDE-моделей производственных систем [2–4], в последнее десятилетие успешно использованы для описания квазистатических, так и некоторых неустановившихся переходных процессов [5]. This paper discusses a class of models of production systems [1]. Models are widely used today for building effective systems of production lines [2,3]. Class model parameters determine the behaviour of the production line with the help of partial differential equations (PDE-model) [2-4], in the last decade successfully used to describe the quasi-static and transient processes [5]

Основы статистической теории построения континуальных моделей производственных линий

В статье обсуждается введенный автором в опубликованных ранее работах (2003 г.) класс моделей производственных систем с поточным способом организации производства, широко используемый в настоящее время для построения эффективных систем управления производственными линиями. Модели класса, определяющие поведение параметров производственной линии с помощью уравнений в частных производных, получили название PDE-моделей производственных систем This paper discusses a class of PDE-models of production systems [1]. PDE-Models are widely used today for building effective systems of production lines [2, 3]. Class model parameters determine the behaviour of the production line with the help of partial differential equations (PDE-model) [2–4], in the last decade successfully used to describe the quasi-static and transients

Основы статистической теории построения континуальных моделей производственных линий

В статье обсуждается введенный автором в опубликованных ранее работах (2003 г.) класс моделей производственных систем с поточным способом организации производства, широко используемый в настоящее время для построения эффективных систем управления производственными линиями. Модели класса, определяющие поведение параметров производственной линии с помощью уравнений в частных производных, получили название PDE-моделей производственных систем This paper discusses a class of PDE-models of production systems [1]. PDE-Models are widely used today for building effective systems of production lines [2, 3]. Class model parameters determine the behaviour of the production line with the help of partial differential equations (PDE-model) [2–4], in the last decade successfully used to describe the quasi-static and transients

Performance analysis of manufacturing networks : surplus-based control

In the modern market, keeping high competition in brands and varieties in type of products is the way for survival of manufacturing industries. Therefore production control methods with capabilities of quick responses to rapid changes in the demand and efficient distribution of the raw material throughout the network are of importance among leading manufacturers. Nowadays, the production control problem has been widely studied and a lot of valuable approaches including queuing theory, Petri nets, dynamic programming, linear programming, hybrid systems were proposed and some of them are implemented. Though up to this moment many methods have been developed, the factory performance remains a challenging problem for further research. Motivated by this problem we study the performance of several manufacturing networks operated by surplus-based control. In the surplus-based control, decisions are made based on the demand tracking error, which is the difference between the cumulative demand and the cumulative output of the network. The studied networks are a single machine, a manufacturing line, a multi-product manufacturing line, a re-entrant machine and a re-entrant line. The performance analysis is based on the performance factors such as demand tracking errors and inventory levels. Specifically, given the presence of unknown but bounded production speed perturbations as well as demand rate fluctuations, we investigate how close the cumulative production output of a manufacturing network follows its cumulative production demand under a surplus-based control policy. The research is subdivided into theoretical analysis, simulation-based analysis and experimental analysis parts. Theoretical analysis By means of analytical tools, the relation between the production demand tracking accuracy and the inventory levels of the networks is investigated. In order to find this relation, classical tools from control theory are used. Models of production flow processes are formulated by means of difference as well as differential equations. In order to analyze their performance, optimal control theory and Lyapunov theory approaches are exploited. Simulation based analysis By means of simulation tools, the theoretical results on performance are evaluated by time-based simulation models. Thus, all theoretical results are illustrated and confirmed by computer simulation. Also two comparative studies are conducted. The first comparative study is realized in order to test the theoretical results on more accurate models, which are event-based. The results are shown to be in agreement with the theory. The second comparative study is on time-based models, where the behavior of a line, a single re-entrant machine and a re-entrant line is tested under three commonly used surplus-based production policies. The performance of each network is evaluated and the results are presented. Experimental analysis An experimental prototype is invented, designed and developed for education and research purposes. The prototype is a hardware tool that serves as a liquid-based emulator of manufacturing network processes. In its core, the liquid-based emulator consists of several electrical pumps and liquid reservoirs. The electrical pumps emulate manufacturing machine behavior, while the liquid reservoirs serve as the intermediate product storages, also called buffers. In the platform, pumps and tanks can be interconnected in a flexible manner. In that way the prototype permits an easy and intuitive way of studying manufacturing control techniques and performance of several network topologies. A detailed system description is provided. Several network configurations and experimentations are presented and discussed

Stability and performance for multi-class queueing networks with infinite virtual queues

We generalize the standard multi-class queueing network model by allowing both standard queues and in??nite virtual queues which have infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Towards this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. We further establish simple diffusion limits for the departure and work allocation processes. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem