In this paper we study optimal scheduling for two part-type, symmetric manufacturing systems subject to non negligible setup times, and characterized by finite buffer capacities. The modelling approach proposed is based on a two step transformation: the original fluid approximation is converted to a sampled model, according to the necessary condition of optimality of the transient policies, and then such a two-dimensional sampled model is mapped onto an equivalent scalar one. Based on the scalar model, the optimal solution of the original scheduling problem is proved, whenever a threshold condition holds for demand loss cost parameter.

http://ieeexplore.ieee.org/iel5/8969/28459/01272508.pdf

A modelling approach for the dynamic scheduling problem of manufacturing systems with non negligible setup times and finite buffers

Over the last decades complexity of products and production processes has increased tremendously, moving towards high-tech production systems: manufacturing expensive products with even more expensive resources. Failures or mistakes have therefore become expensive too and need to be avoided to ful??ll the manufacturers' targets: generating products while maximizing the pro??t. In general, this is to be achieved by keeping vast control over the manufacturing processes, resources, stocks and labour. Easier said than done, since controlling all phenomena that occur in a manufacturing system is very expensive, if possible at all. Therefore, speci??c parts of a manufacturing facility are modelled to reduce its complexity. With the models, predictions of future behavior of the system can be made. Moreover, different control strategies can be tested of??ine at low risks, before implementing them on the real production system. In this dissertation, manufacturing ??ow lines are modelled using several modelling paradigms. These are divided into three groups: discrete event models, continuous models and hybrid models. The presented modelling methods are used throughout the remainder of the thesis and therefore the survey is by no means an attempt to give a complete overview of the whole area of modelling manufacturing systems. A state space representation of a manufacturing workstation is introduced, which is ??nite dimensional, can be measured instantaneously and does not contain any information about production or control policy. This state space representation is used in the coupling of different model paradigms, facilitating the use of analysis techniques in both time domain and event domain. In addition, the introduced state space representation is used in the development of a continuous time receding horizon state feedback controller. For a ??ow line of multiple workstations, each with its own buffer capacity and process time, and for an event based control horizon, an optimal production schedule controller is developed. This state feedback controller provides optimal schedules, even when unexpected disturbances occur. Switching servers are found in a wide variety, like in manufacturing industries, traf??c networks and call-centers. Switching servers process multiple types of jobs, with a switchover time involved. A hybrid ??uid model is used to describe the dynamics of switching servers. Continuous dynamics is used to describe the evolution of buffer levels. The discrete event part of the model describes the switches between the product types. For a workstation processing two product types that arrive at constant rates, optimal switching policies with respect to minimal time averaged weighted work in process levels are de??ned for situations with and without maximum buffer level capacities. An important insight is the possible appearance of a slow-mode in optimal process cycles. During a slow-mode, a buffer is empty and products are served at their arrival rate, instead of switching to the other product type. The slow-mode represents a tradeoff between losing capacity due to serving products at a lower rate than the maximum rate and losing capacity due to relatively often switching in time. Conditions on the appearance of a slow-mode in optimal process cycles are derived explicitly. In a manufacturing network, arrivals of products at a workstation are in general not at a constant rate. For a switching server processing two product types and with piecewise constant periodic arrival rates (on/off), optimal system behavior with respect to work in process levels is de??ned. The total optimization problem then splits into several subproblems, which need to be solved separately. The optimal mean work in process level for a single switching server is also a lower bound on the mean work in process level that can be realized for a switching server ??ow line in which that server resides. For a ??ow line consisting of two switching servers, each processing two product types, it is investigated under which conditions this lower bound can actually be achieved. An important conclusion is that, in certain cases, workstations in a ??ow line need a synchronization mechanism to get to the desired process cycles. For the single switching server and the switching server ??ow lines with constant arrival rates of products, state feedback controllers are proposed to steer the trajectory of the switching server (??ow line) to the determined optimal process cycle from arbitrary feasible starting point. Contrary to many methods proposed in literature, in this research ??rst the desired (optimal) system behavior is de??ned, regardless of any control policy. Then a control policy is formulated to achieve this desired behavior. It is questionable whether optimal system behavior should be looked for. The studies in this thesis show that optimal process cycles can be determined for a rather small class of workstations. Taking more than two product types into account or more than two workstations leads to very complex optimization problems. Apart from the complexity, it is not even known whether optimal cyclic behavior for these larger systems exists. Manufacturers might not be interested in the theoretically optimal solution. Often, a better solution than the current solution will do. Moreover, a manufacturer might prefer suboptimal solutions over optimal solutions when the suboptimal solution handles disturbances or uncertainties better. This dissertation can serve as a starting point for further research on modelling and control of manufacturing networks. The introduced state space representation topic can be extended for larger networks and other manufacturing resources. In addition, other research areas can be linked to this research, including stochastic behavior and effective process times.

/pdf/modelling-and-control-of-discrete-event-manufacturing-flow-2660b5g66c.pdf

Modelling and control of discrete event manufacturing flow lines

The paper considers standard fluid models of multi-product multiple-server production systems where setup times are incurred whenever a server changes product. We consider a general approach to the problem of optimizing the long-run average cost per unit time that consists of first determining an optimal steady state (periodic) behavior and then to design a feedback scheduling protocol ensuring convergence to this behavior as time progresses. In this paper, we focus on the latter part and introduce a systematic approach. This approach gives rise to protocols that are cyclic and distributed: the servers do not need information about the entire system state. Each of them proceeds basically from the local data concerning only the currently served queue, although a fixed finite number of one-bit notification signals should be exchanged between the servers during every cycle. The approach is illustrated by simple instructive examples concerning polling systems, single server systems with processor sharing scheme, and the re-entrant two-server manufacturing network with non-negligible setup times introduced by Kumar and Seidman. For the last network considered in the analytical form, some cases of optimal steady-state (periodic) behavior are first recalled. For all examples, based on the desired steady state behavior and using the presented theory, we designed simple distributed feedback switching control laws. These laws not only give rise to the required behaviors but also make them globally attractive, irrespective of the system parameters and initial state.

/pdf/designs-of-optimal-switching-feedback-decentralized-control-5dsxn2c9e5.pdf

Designs of optimal switching feedback decentralized control policies for fluid queueing networks

In this paper we study optimal scheduling for a two part-type, symmetric manufacturing systems subject to non negligible setup times, and characterized by finite buffer capacity. The solution method relies on 1) restricting the possible control policies to those which respect some general necessary conditions of optimality, and on 2) the exploitation of such properties to introduce a two-dimensional sampled version of the original system. The symmetric assumptions allows to map the two-dimensional sampled model onto an equivalent scalar one. By means of such a scalar sampled model we give the analytical solution of the optimal control problem addressed.

Optimal control of two symmetric competing queues with finite capacity and non negligible setup times

We consider a generic deterministic time-invariant fluid model of a polling system, where several buffers (queues) of infinite sizes receive constant rate inflows of jobs from outside the system and share a common source of service (a finite capacity server). The server can serve at most one buffer at a time and has to switch among buffers from time to time; any switch consumes a nonzero switch-over period. With respect to the long-run maximal scaled work in progress (wip) performance metric, near optimality of periodic scheduling and service protocols is established: The optimum can be furnished by such a protocol up to as small error as desired. To prove this, a special class of protocols is introduced, which prescribe to serve any buffer at the maximal rate until its size reduces to a pre-specified percent of its size at the beginning of the visit. It is also shown that the exhaustive policy is optimal for any buffer whose service at the maximal rate implies reduction in the scaled wip.

On Global Near Optimality of Special Periodic Protocols for Fluid Polling Systems with Setups

We consider a setup scheduling problem for a manufacturing system with finite buffers. Some analytical results for the infinite capacity case are also included. The system comprises a non-perfect, reliable single machine processing two part types, modeled by means of a fluid approximation. The objective of the control is to minimize an index cost under steady and transient conditions. Dynamic programming is used to derive the structure of the optimal policy and some initial analytical results are given. A symmetry assumption on part types is considered to simplify the problem.

http://ieeexplore.ieee.org/iel5/7709/21122/00980613.pdf

M. Del Gaudio

Papers

A scheduling problem for two competing queues with finite capacity and non-negligible setup times