In order to obtain the better analysis of the multiple reentrant manufacturing systems (MRMSs), their modeling and analysis from both micro- and macroperspectives are considered. First, this paper presents the discrete event simulation models for MRMS and the corresponding algorithms are developed. In order to describe MRMS more accurately, then a modified continuum model is proposed. This continuum model takes into account the re-entrant degree of products, and its effectiveness is verified through numerical experiments. Finally, based on the discrete event simulation and the modified continuum models, a numerical example is used to analyze the MRMS. The changes in the WIP levels and outflux are also analyzed in details for multiple re-entrant supply chain networks. Meanwhile, some interesting observations are discussed.
In recent years, factories and production systems have become larger and more complicated. The reentrant manufacturing system is a typical example, in which the work in process (WIP) repeatedly passes through the same workstation at different stages of the process routes. Figure
The structure of a reentrant manufacturing system.
Currently, most of the semiconductor manufacturing systems are described by the discrete models, the advantages of the discrete models are (1) some of the actual static problems can be directly reflected in the discrete models. (2) In reality, the methods of data collection are discrete. (3) There are many existing methods to solve the discrete models. Based on the theory of the discrete event dynamic systems, there are several discrete methods used to model the multiple reentrant semiconductor production flows.
First, queuing networks can intuitively describe the discrete production process of the wafer production line and obtain the analytical expression of the performance evaluation. A methodology for supply chain inventory analysis and optimization was presented by linking production authorization (PA) strategy to queuing models [
Second, Petri net (PN) model has been widely used in modeling, simulating, analyzing, and controlling the discrete event dynamic systems. Compared with some other description tools, PN model is especially easy to describe concurrent phenomena and simulate the parallel systems. However, with the increasing complexity and size of the manufacturing systems, the complexity of the PN model analysis is also a corresponding increase. Lin et al. [
Third, fluid networks model comes from traffic theory and was introduced by Newell [
On the other hand, the continuous models also have several advantages in description of the production lines. That is, they are scalable, more detailed results that can be found as compared to fluid models, and more important, they are amenable to optimization and control. Recently, the continuous models have been applied to many fields and have achieved some significant research results. Anderson [
The continuous models can more accurately reflect the actual situation, while the corresponding discrete models need to be discretized in time or space, then takes a constant value in each discrete node to represent the state of a period of time. This method is the approximation of the real situations and can not fully reflect the actual situations. Using the common ground and similarities of the semiconductor manufacturing systems, the large-scale complex network can be decomposed into relatively small-scale simple problems. The multiscale methods can be more accurate modeling and analysis of the manufacturing systems. Armbruster and Ringhofer [
In this paper, in order for better analysis of the multiple reentrant manufacturing systems, their modeling and analysis from both micro- and macroperspectives are considered. First, the discrete event simulation models and their basic algorithm are proposed. In order to describe the multiple reentrant semiconductor manufacturing systems more precisely, a modified continuous model is proposed, which can reflect how the reentrant degree of a product impacts on the system performance. Finally, the changes of the WIP levels and outflux are analyzed based on the discrete event simulation models and the modified continuous models.
The structure of this paper is organized as follows: Section
In the discrete event simulation (DES) models, each item to be produced is treated as individual. The production process consists of
In engineering practices, the most significant influence on the form of this distribution comes from the total number of items in progress, that is, the work in progress (WIP). So the TPT distribution
For a high-volume multiple reentrant manufacturing system, the time interval between jobs becomes less important, and then the jobs can be seen as a continuous way. In these continuous production models, the WIP
For a large number of lots, the individual lots are replaced by a continuous quantity, and the fluxes
So the question can be used to solve outflux
However, if supplier
In phase models, the evolution of a large ensemble of lots is modeled by describing the trajectories of each individual lot in phase space based on Newton equations. The trajectories
A random variable
Set the initial phase to 0 as the item enters the factory, increase the WIP
Compute
Compute the TPT
Update the distribution TPT
The function
Partial differential equation (PDE) models are actually continuum approximation of fluid models. Recently, PDE models for large-scale multiple reentrant production systems have become an important research topic. PDE models do have several advantages, that is, they might be preferred when evaluating the overall performance of large-scale manufacturing systems or solving the optimization problem for a longer time plan. Currently, the continuous models have been applied to many fields and have achieved some significant research results.
This is appropriate to describe a semiconductor manufacturing fab involving a large number of items in many stages:
Assuming that there is a unique entry and exit for the system and the yield is 100%, PDE models can be given bellow according to the conservation law,
An arbitrary initial condition for the density of the products can be expressed as
The production process of the supply chain network can be described as an equivalent
Equation (
Hence, assume that an initial WIP distribution
If the influx
Since the Courant-Friedrich-Levy (CFL) condition is necessary for stability, the time step
Based on the above formula, the density distribution of each moment
Furthermore, the total WIP
Mini-Fab is a simplified model of the semiconductor production lines, which has all the important features of the reentrant semiconductor manufacturing systems, such as reentrant, different processing time, and batch production. Currently, many scholars have done a lot of research work based on the Mini-Fab. The Mini-Fab contains 5 machines grouped into 3 work centers, the product comprises of 6 processing steps and each work center has a reentrant step, which is shown in Figure
Process flow diagram of the Mini-Fab.
For convenience, we make the following basic assumptions on the model. The product yield rate is 100%, namely, there is no rework problem. The system is a continuous production process for 24 hours a day. The system does not take into account the time of carrying, loading and discharging, adjusting equipment, premaintaining equipment, and downtime.
Now assuming that there is one product
Processing time of the product
Machining centers | Processing time (hours) | |
---|---|---|
Machines A & B | Step 1: 1.5 | Step 5: 1.5 |
Machines C & D | Step 2: 0.5 | Step 4: 1 |
Machine E | Step 3: 1 | Step 6: 0.5 |
From the above table, we can see that the total processing time is
Throughput as a function of time for the PDE simulation.
When the throughput of the basic continuous models is obtained, the corresponding Mini-Fab simulation model can be built using simulation package ExtendSim [
Throughput as a function of time for the ExtendSim simulation.
Comparing Figure
It is worth noting that the state equations could reflect the characteristics of systems—any changes of the multiple reentrant production systems may lead to a different state equation. Equation (
Let
Reentrant factor
In reality, the velocity of products in the system is not only related with the WIP level, but also with the reentrant factor. Let
According to the queuing theory, the processing cycle time of the reentrant process
As for the nonreentrant processes, assuming the total number of workstations is
In order to avoid computational complexity of
The corresponding new state equation can be obtained as follows:
Therefore, the resulting modified whole PDE model is given below:
Once the new state equation is obtained, the validity of the modified models can be verified by the same example described in the previous section. According to Figure
Throughput as a function of time for the modified PDE model.
It can be seen that the throughput is initially zero for the reentrant manufacturing systems. This is due to the time delay and the system initialization (the system starts up with an empty factory), then the system begins to have throughput at about 0.25 days and increases drastically until the throughput reaches a stable value of 5 (units/day) at about 1.5 days. Comparing Figure
Once the modified continuum models are obtained, in order to describe the multiple reentrant manufacturing systems from the micro- and macroperspectives, based on the DES and modified PDE models, a numerical experiment is carried out to demonstrate the consistency of the models for the Mini-Fab case. The WIP levels and outflux of the DES and the modified continuum models for the multiple reentrant supply chain systems are presented in this section.
In this section, a mean field assumption is made first, namely, the TPT distribution
Probability distribution of the TPT as a function of WIP.
Same as the previous section, taking the multiple reentrant production system as an example, the system begins to run from an empty factory. The DES model is executed by generating a total of 200 lots. In order to facilitate the computation, we assume that the influx is defined as a constant value in the experiment first. The trajectories are computed by the random phase model, and the lots’ arrival time
WIP levels of the DES and modified PDE models as a function of time.
Outflux of the DES and the modified PDE models as a function of time.
Comparing the WIP levels and outflux computed from the DES models to the solution of the modified PDE models, it can be observed that, although the WIP levels and outflux values of the two models are not exactly equal, the results are close enough to each other to substantiate the consistency of two models. The same results of the DES and modified PDE models are given for the steady state. The WIP levels of the two models show gradual increase, while the outflux of the two models first shows an increasing trend gradually to reach the maximum and then reaches a stable value.
In this paper, better analysis of the multiple reentrant manufacturing systems can be obtained if both micro- and macroperspectives are adopted. The discrete event simulation models are first proposed and their basic algorithm is also presented in detail. The low accuracy of the basic PDE models for the multiple reentrant supply chain networks is explained by a numerical experiment. In order to model such complex systems more precisely, a modified PDE model that takes into account the reentrant degree of the product is presented, while the validity of the modified PDE model is also illustrated through a numerical experiment for multiple reentrant supply chain systems. Then, based on the DES and modified PDE models, a numerical experiment is provided to compare the WIP levels and outflux changes. Meanwhile, some interesting observations are discussed. Once the results of the micro- and macro simulations are obtained, some analysis for the multiple reentrant manufacturing systems based on the multiscale methods becomes possible.
The work presented in this paper has been supported by a Grant from the National High-Tech Research and Development Program (863 Program) of China (2008AA04Z104) and a Grant from National Natural Science Foundation of China (70871077).