# A Predictive Control Model of Bernoulli Production Line with Rework Loop for Real-Time WIP Optimization in Permutation Flowshop

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## Abstract

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## 1. Introduction

- (1)
- Most current modeling efforts mainly focus on static production processes, and steady-state analysis is effective in the case of a single large-scale production. In complex and changeable streamlined production processes, compared with transient analysis, the results obtained by steady-state analysis are not accurate enough in some cases. However, the current transient modeling work on production systems is still insufficient.
- (2)
- The rework and reuse of defective products is of great significance to reducing costs, improving manufacturing efficiency, and realizing green manufacturing. Currently, the existing transient modeling work mainly focuses on continuous manufacturing systems, and it is difficult to consider complex re-entrant systems. There is still insufficient research on this link.
- (3)
- In the actual production process, human factors play a large role in the safety, risk management, and quality control of the production system. However, current research on the replacement process usually only considers automatic machines and does not consider manual machines that are affected by the human actions of the machine operator.

- (1)
- This paper presents an analysis of automated and manually operated semi-automated machines and their integration into a displacement flowshop with a rework loop.
- (2)
- This study establishes an instantaneous productivity model suitable for arranging flow operations with rework loops and human factors, and measures basic production performance indicators through a recursive method.
- (3)
- To address the challenges of intelligent control in permutation flowshops and to furnish comprehensive, real-time production insights, a model predictive control system based on discrete event-driven feedback is employed. As a result of these research outcomes, there is a discernible enhancement in the ability to perceive and predict work-in-progress, leading to significant savings in human resources.

## 2. Problem Description and Model Assumptions

- (1)
- There are M machines and M-1 buffers on the main production line. Among them, there are two rework production lines, on which is an inspection machine which can check whether the products are qualified. Qualified products are recorded as class A products, and unqualified products are recorded as class B products. Qualified products are directly transferred to the buffer, and unqualified products are sent for or wait for reprocessing through the rework production line [31].
- (2)
- At least one machine has sufficient raw materials, and the last machine on the production line of Line 3 will not be blocked [32].
- (3)
- The start time of each production determines the working status of the machine, and the end time of each production determines the status of the buffer [33].
- (4)
- The entire replacement process with re-entry satisfies the assumptions of “time dependent failure” and “pre-processing blocking” [34].

## 3. Production System Modeling and Predictive Model Control

#### 3.1. Production System Modeling

#### 3.1.1. Machine Reliability Model for Both Manual and Automatic Machines

#### 3.1.2. Transient Transition Modeling

- (1)
- Operator P describes the probability of occurrence of event E (i.e., P[E]).
- (2)
- The operator Φ describes the event that the object O is in state S at time t, respectively, Φ (O, S, t).
- (3)
- The operator H describes multiple objects (O
_{1}, O_{2}, O_{3}, …) At time t, they are in states (S_{1}, S_{2}, S_{3}, …), respectively, H (O_{1}, O_{2}, O_{3}, …/S_{1}, S_{2}, S_{3}, …, t) - (4)
- The operator T describes the probability of a particular object O going from state S
_{1}to S_{2}in time t, respectively, ${T}^{{S}_{2},{S}_{1}}(t)$.

#### 3.1.3. Transient Mapping Analysis

#### 3.1.4. Reverse Modeling of Machine Transient Behavior

#### 3.2. Model Predictive Control

#### 3.2.1. r-WIP Optimization Problem Formulation

_{p}step having a re-entry link. The goal is to reduce the cost and the rate of rework with the least WIP cost. The product quality information will feed back to the dynamic permutation flowshop performance check.

- (1)
- To represent the dynamic behavior of a replacement process model with re-entry links, a mathematical model can be established.
- (2)
- To determine when there are non-conforming parts, an event-driven production performance identification method based on the model can be proposed.
- (3)
- To produce the best release time of r-WIP optimization jobs, a discrete event-driven model predictive control is proposed. The following assumptions will be defined so that the dynamic behavior of permutation flow stores with retransmission links can be modeled.
- a.
- SM* defines the last and slowest machine closest to the end of the line, assuming that one or more machines are likely to be hungry or blocked in the re-entry link.
- b.
- When a disturbing event happens, the processing time of the k
_{i}-th part at the i, j-th machine is ${\sigma}_{i,j}^{\prime}({k}_{i})={\sigma}_{i,j}({k}_{i})+{d}_{i}$. - c.
- There is a finite capacity for each buffer ${b}_{i,j}$.
- d.
- The interference event depends on the operation and can be detected in real time.
- e.
- If the customer’s demand exceeds the production capacity of the replacement process, the replacement process should be run at maximum production capacity.
- f.
- The transportation time between the machine and the buffer can be ignored [35].

#### 3.2.2. Event-Based Time-Varying Model Predictive Control

_{p}step (if customer demand is deterministic, it is necessary to update the mechanism for releasing the plan every N

_{p}step).

_{p}[35].

## 4. Solution of the Established Model

## 5. Case Study

_{max}of the system is 0.8959 by simulation. For the last remaining flowchart layout, the average stock level, the productivity (PR), the rework rate (RR), and the power decrease ratio (ECRA) are shown in Table 4.

_{in}, but also the tracking error J

_{out}of the reference signal. In Figure 8a, the tardiness values for layouts 1 to 7 are Y (700)-r (700) = 4261 s, 4725 s, 5125 s, 5944 s, 4958 s, 8027 s, and 10,669 s, respectively. Layout 1 can also achieve maximum throughput in the presence of non-conforming parts, but this can only be achieved by maintaining a high level of work-in-progress (WIP) buffering. In addition, the flexibility of event model predictive control (WIP) and discrete model predictive control (DMPC) in work-in-progress (WIP) is limited when machines are lacking or blocked. In Figure 8b, the circle represents the delay gap Jout (k + 1) − Jout (k) for different layouts. It can be seen from the results that when additional work is actively added in the unpredictable event, it is possible to increase the throughput of the sorting process by the event model prediction control and the discrete model prediction of the WIP. While DPM has similar output performance as WIP, it is not highly stable, as illustrated in Figure 8a of the drawing. In particular, r-WIP in discrete model predictive control of WIP is highly volatile, as illustrated in Figure 9b, which can add complexity to process management. On the other hand, WIP, an event model predictive control, is able to keep a constant WIP level through real-time adjustment of production schedule. Specifically, the r-WIP is very volatile in discrete model predictive control; as illustrated in Figure 9b, the complexity of process management may increase. On the contrary, if the production plan is adjusted in real time, event model predictive control of work-in-progress can maintain a constant level of it.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Permutation flowshop throughput with various control mechanisms. (

**a**) Y(700)-r(700), (

**b**) J1 (k + 1)-J1 (k).

**Figure 9.**r-WIP with different control mechanisms. (

**a**) The WIP in layout 1, (

**b**) The WIP in layout 2-layout 7.

**Figure 10.**The calculation results of the proposed method and the classical steady-state performance algorithm under different parameter settings.

**Figure 11.**Comparison of calculation results between the proposed method and the traditional method under different parameter settings.

Notation | Interpretation |
---|---|

$U{P}_{i,j}(t)$ | The machine is in a state of non-fault operation in unit time t |

$D{N}_{i,j}(t)$ | A machine or buffer that is out of order within a time unit t |

$P{R}_{i,j}(t)$ | The operation of a machine within a time unit t |

$NP{R}_{i,j}(t)$ | The state in which a machine operates non-productively within a time unit t |

$S{T}_{i,j}(t)$ | The machine or buffer operates in a hungry state for a unit time t |

$NS{T}_{i,j}(t)$ | The machine or buffer unit is not hungry within unit time t |

$B{L}_{i,j}(t)$ | The operation of a machine or buffer that is blocked for a unit of time t |

$NB{L}_{i,j}(t)$ | The state of unblocked operation of a machine or buffer for a unit of time t |

$\theta $ | The complete set of all possible states of a machine |

${X}_{i,j}(t)$ | Time instant at which the machine ${m}_{i,j}$ starts to work on the t-th part |

$y(t)$ | Time instant at which the t-th part leaves the permutation flowshop |

${\sigma}_{i,j}(t)$ | Processing time of the t-th part at ${m}_{i,j}$ |

${N}_{i}^{-}\left(t\right)$ | The buffer level after the t-th part’s entrance into ${b}_{i,j}$ |

${N}_{i}^{+}\left(t\right)$ | The buffer level of ${b}_{i,j}$ just after the t-th part leaves ${b}_{i,j}$ |

${N}_{c}$ | The control range of discrete event model predictive control |

${N}_{p}$ | The prediction horizon of the discrete event model predictive control, N_{c} ≤ N_{p} |

${N}_{i}$ | The capacity of the buffer ${b}_{i,j}$ |

$r(t)$ | The expiration date of the finished product |

$u(t)$ | The time when the t-th component is fed to the system |

${\overrightarrow{\mathrm{e}}}_{i}=(\mathrm{j},{t}_{i},{\mathrm{d}}_{i})$ | A disturbing event that lasts d_{i} time when the machine ${m}_{i,j}$ processes the t-th part |

Type | ${\mathit{m}}_{1,1}$ | ${\mathit{m}}_{2,1}$ | ${\mathit{m}}_{3,1}$ |
---|---|---|---|

A | × | × | × |

B | × | × | √ |

C | × | √ | × |

D | × | √ | √ |

E | √ | × | × |

F | √ | × | √ |

G | √ | √ | × |

H | √ | √ | √ |

Layout Code | Machines on the Main Production Line | Rework Line Machine | Machine Separation | ||
---|---|---|---|---|---|

P_{i} | C | P_{i} | C | h | |

1 | {0.9, 0.8, 0.8, 0.7} | 8 | {0.9, 0.8} | 4 | 1 |

2 | {0.9, 0.7, 0.85, 0.8, 0.9} | 10 | {0.9, 0.8} | 4 | 1 |

3 | {0.9, 0.7, 0.85, 0.8, 0.9} | 10 | {0.9} | 2 | 1 |

4 | {0.9, 0.7, 0.85, 0.8, 0.9} | 10 | {0.9, 0.8} | 4 | 1 |

5 | {0.9, 0.7, 0.85, 0.8, 0.9} | 10 | {0.9, 0.8, 0.7} | 6 | 1 |

6 | {0.9, 0.7, 0.85, 0.8, 0.9, 0.7} | 12 | {0.9} | 2 | 1 |

7 | {0.9, 0.7, 0.8, 0.8, 0.9, 0.7, 0.8, 0.7, 0.9, 0.9} | 20 | {0.9, 0.8, 0.9, 0.7, 0.85} | 10 | 1 |

Layout Code | Average Inventory Level (${\mathit{Q}}_{\mathit{i},\mathit{j}}$) | Production Rate (PR) | Rework Rate (RR) | Energy-Consuming Reduction Rate (ECRA) |
---|---|---|---|---|

1 | 0.53 | 0.8959 | 0.31 | 0.11 |

2 | 0.47 | 0.7853 | 0.26 | 0.12 |

3 | 0.43 | 0.6792 | 0.22 | 0.14 |

4 | 0.38 | 0.5878 | 0.18 | 0.18 |

5 | 0.34 | 0.6862 | 0.26 | 0.23 |

6 | 0.27 | 0.4865 | 0.12 | 0.27 |

7 | 0.23 | 0.4842 | 0.09 | 0.31 |

Processing Times | Average Inventory (${\mathit{Q}}_{\mathit{i},\mathit{j}}$) | Production Rate (PR) | Rework Rate (RR) | Energy-Consuming Reduction Rate (ECRA) |
---|---|---|---|---|

10 | 5 | 0.8863 | 0.25 | 0.206 |

20 | 8 | 0.8886 | 0.20 | 0.235 |

50 | 10 | 0.8873 | 0.15 | 0.259 |

80 | 10 | 0.8896 | 0.10 | 0.293 |

100 | 10 | 0.8912 | 0.10 | 0.326 |

Processing Times | Average Inventory (${\mathit{Q}}_{\mathit{i},\mathit{j}}$) | Production Rate (PR) | Rework Rate (RR) | Energy-Consuming Reduction Rate (ECRA) |
---|---|---|---|---|

10 | 20 | 0.8852 | 0.15 | 0.157 |

20 | 25 | 0.8867 | 0.17 | 0.209 |

50 | 35 | 0.8886 | 0.20 | 0.248 |

80 | 40 | 0.8898 | 0.25 | 0.316 |

100 | 60 | 0.8926 | 0.25 | 0.324 |

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## Share and Cite

**MDPI and ACS Style**

Gu, W.; Guo, Z.; Wang, X.; Yang, Y.; Yuan, M.
A Predictive Control Model of Bernoulli Production Line with Rework Loop for Real-Time WIP Optimization in Permutation Flowshop. *Machines* **2024**, *12*, 20.
https://doi.org/10.3390/machines12010020

**AMA Style**

Gu W, Guo Z, Wang X, Yang Y, Yuan M.
A Predictive Control Model of Bernoulli Production Line with Rework Loop for Real-Time WIP Optimization in Permutation Flowshop. *Machines*. 2024; 12(1):20.
https://doi.org/10.3390/machines12010020

**Chicago/Turabian Style**

Gu, Wenbin, Zhenyang Guo, Xianliang Wang, Yiran Yang, and Minghai Yuan.
2024. "A Predictive Control Model of Bernoulli Production Line with Rework Loop for Real-Time WIP Optimization in Permutation Flowshop" *Machines* 12, no. 1: 20.
https://doi.org/10.3390/machines12010020