# Reconfiguration point decision method based on dynamic complexity for reconfigurable manufacturing system (RMS)

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## Abstract

To address the problem of how to identify the best time to implement reconfiguration for the reconfigurable manufacturing system (RMS), a dynamic complexity-based RMS reconfiguration point decision method is proposed. This method first identifies factors that affect RMS dynamic complexity (including both positive and negative complexity) at the machine tool and manufacturing cell levels. Next, based on information entropy theory, a quantitative model for RMS dynamic complexity is created, which is solved via state probability analysis for processing capability and the processing function. This model is combined with cusp catastrophe theory to establish an RMS reconfiguration decision model. Both positive and negative complexity are control variables for cusp catastrophe. Cusp catastrophe’s state condition is used to identify RMS state catastrophe at the final stage of production. This catastrophe point is the RMS reconfiguration point. Finally, the case study result shows that this method can effectively identify the RMS state catastrophe moment so that system reconfiguration is implemented promptly to improve RMS’s responsiveness to the market.

## Keywords

Reconfigurable manufacturing system Reconfiguration point decision Information entropy Cusp catastrophe Dynamic complexity## Introduction

In 2010, the German government proposed the concept of Industry 4.0 in its report “High-Tech Strategy (2010)”. That report marked the prologue to the fourth Industrial Revolution, which is dominated by intelligent manufacturing. Lee Jie, Professor and Distinguished Chair at The University of Cincinnati (US) and director of the National Science Foundation’s Intelligent Maintenance System Industry/University Cooperative Center, believes that there are three pillars in the development of Industry 4.0. One of them is that “manufacturing systems, including the machine tool itself, are able to make adjustments automatically and promptly according to differences in the processed product and changes in processing condition, and possess the so-called ‘self-reflection’ capability” (Lee et al. 2015). The RMS reconfiguration capability satisfies the requirement of intelligent manufacturing and is a critical component in future intelligent manufacturing environments. Mehrabi et al. (2000a, 2002) provide an outlook for RMS research focus and future development and suggest that RMS is a desirable next step in the evolution of production systems. The reconfiguration moment decision is essential for RMS to be the backbone of intelligent manufacturing; it is also a critical and fundamental problem waiting for a solution on the road to intelligent manufacturing.

RMS is a manufacturing system with fast response capability. Its architecture, hardware and software can be reassembled and adjusted to rapidly change the system function to respond to impacts from market fluctuation, technology innovation and policy change (Koren et al. 1999; Koren 2013; Mehrabi et al. 2000b). Because of its huge market potential, researchers have studied various aspects of RMS, including reconfigurability (Wang 2000), system layout planning (Goyal et al. 2012; Wu et al. 2007), system performance analysis (Cai 2004), part-family formation (Wang et al. 2016a; Goyal et al. 2013; Gupta et al. 2013; Hasan et al. 2014) and reconfiguration schemes evaluation (Wang et al. 2016b). Zhao et al. (2000a, b, 2001a, b) provide a stochastic RMS model that includes a framework, optimal configuration, optimal selection policy and performance measurement. Bi et al. (2008) provides an overview of RMS. Renzi et al. (2014) elaborate the advantages of an RMS in terms of cost and efficiency and conduct an in-depth analysis of key technologies in RMS. Although RMS has excellent architecture that can respond to rapid changes in market demand (Wang 2000), the issue of when to implement such a change is a complex and difficult decision problem. The decision about when to implement RMS reconfiguration should consider various production factors such as cost, time and order; a conclusion based on the analysis of any single factor is not convincing. Therefore, this paper has analyzed the RMS reconfiguration point from the system’s perspective. Researchers have conducted comprehensive analyzing manufacturing systems via system state. Rao (2006) provide a quantitative description of manufacturing system state complexity from the perspective of information theory; they analyze the manufacturing system’s static complexity and dynamic complexity and create an information entropy-based static and dynamic complexity measurement model. To address deficiencies in the manufacturing system architecture complexity modeling method, Duan (2012) analyze the effect of the intermediate buffer zone state on the manufacturing system state; they also employ the universal generating function and information entropy theory to create and evaluate a model for non-serial manufacturing system complexity. Smart et al. (2013) study the manufacturing system’s dynamic complexity based on improved information entropy theory; their main focus is an analysis of equipment and queue state in the production process and an elaboration of the relationship between system complexity and system operation. Zhang (2011) analyze the manufacturing system dissipation structure; create a manufacturing system entropy variation model based on information entropy theory; analyze the relationship between the manufacturing system control variable and the state variable based on catastrophe theory; and perform a quantitative analysis of system dissipation structure formation. The aforementioned manufacturing state analysis is only based on a part state or the machine tool state, whereas the manufacturing system state is determined by interaction between parts and machine tools. The part type demands a specific processing function from the manufacturing system; the part quantity demands a corresponding processing capability from the manufacturing system. As provider of processing function and processing capability, the machine tool provides one or more processing functions and processing capabilities of specific quantities. To summarize, the analysis of manufacturing system complexity in terms of processing function and processing capability fully describes the interaction between parts and machine tools, reflects the nature of the manufacturing system state, and thus provides evidence for the RMS reconfiguration decision.

## RMS reconfiguration mechanism and problem analysis

After RMS construction is completed, a running-in period is required; that period is called the ramp-up time (Rösiö and Säfsten 2013). During this period, system stability is relatively low, and problems such as machine failure and substandard products are likely to emerge. After a period of tuning and running-in, problems in the new system are gradually fixed; the system operates in a highly efficient and stable state and is capable of high-quality, high-yield and low-cost production. This period is called the stable production period. As RMS continues its operation, because of random internal and external factors such as machine tool failure and new production order queue-jumping, the system fails to provide sufficient processing functions and capability. The leads to delayed order delivery, more difficult scheduling, declined productivity and chaos in production, which prompts system reconfiguration. At this moment, the system runs into the end stage of production. RMS stops production at the right moment and enters the reconfiguration period. Machine tools and components are added, rearranged, removed and adjusted to reconfigure the RMS processing function and processing capability. When reconfiguration is completed, the RMS processing function and processing capability are updated; system capability is able to meet new market demand; and RMS starts a new round of production cycles. In this cyclic process, the quantitative accumulation of machine tool blocks and failures leads to a qualitative change in the system state, or state catastrophe. After a catastrophe, system-level reconfiguration is required and scheduling alone is unable to solve system problems. The key to this process is to identify the catastrophe moment, which is also this paper’s focus. Every RMS reconfiguration updates system capability and thus always maintains a relatively high market response capability and productivity. Figure 1 shows the RMS reconfiguration mechanism and implementation process. That diagram shows that after a temporary ramp-up time, RMS can provide processing function and processing capability for current production requirements, and the system is experiencing stable production. At the end stage of production, because of the combined effect of system internal and external factors, three problems arise in the RMS processing function and processing capability system: (1) the processing-function requirement is met, but the processing capability is inadequate; (2) processing capability is adequate, but the processing function is inadequate; and (3) both processing function and processing capability are inadequate. At this moment, system operation efficiency declines and system production capacity gradually declines, leading to delayed order delivery. At a certain moment, processing function and processing capability deficiency leads to manufacturing system state catastrophe; the system enters a reconfiguration period to rebuild processing function and processing capability and then moves to the next production cycle.

When the RMS processing function or processing capability fails to meet order demand, if reconfiguration is implemented immediately, the normal result is a relatively high cost attributable to production down time. When there are a large number of delayed orders, system reconfiguration is more difficult and costly. The decision about the reconfiguration moment becomes a critical step in RMS implementation. In other words, at the end stage of production, system complexity variation is analyzed to identify the catastrophe point of the system stable state. When system stable state catastrophe occurs, continued production normally results in a dramatic increase in cost. Moreover, if production continues until the system is in an extremely unstable state, system reconfiguration requires significantly higher cost and a much longer time. Therefore, when a system stable state catastrophe occurs, it is the best time to implement system reconfiguration. This reconfiguration moment is the RMS reconfiguration point. In this paper, based on information entropy, system complexity variation is analyzed quantitatively and is combined with cusp catastrophe to complete the decision about the RMS reconfiguration point.

## Information entropy-based quantitative analysis of RMS complexity

### RMS complexity analysis

Manufacturing system complexity variation should be considered in the RMS reconfiguration decision process. There are two categories of manufacturing system complexities: static complexity and dynamic complexity. Static complexity is primarily concerned with the production state of the manufacturing system according to its scheduling plan. However, in actual production, because of uncertain factors such as market fluctuation and machine failure, production normally deviates from the scheduling plan. Dynamic complexity describes a manufacturing system’s actual operational state, which represents system state variation in actual production. Therefore, in this paper, system complexity is simplified as dynamic complexity and RMS state analysis is based on dynamic complexity.

*X*represents a system;

*E*(

*X*) represents information contained in system

*X*or the information entropy of system

*X*; \(p_{i }(i=1,2,\ldots ,\hbox {n})\) represents the probability of system

*X*in the

*i*-th symbol; and lb represents logarithmic to the base 2.

Based on the RMS definition, RMS’s rapid-response capability is determined by each manufacturing cell’s ability to adjust processing capability and processing function; i.e., RMS reconfiguration is driven by adjusting the processing capability and processing function. Therefore, the RMS processing capability and processing function are chosen as indexes to describe system complexity. At end stage of each production period, RMS processing capability and processing function state probability are analyzed to create the RMS complexity information-entropy model. In the analysis of the system processing capability state, machine tool productivity and the buffer zone state are chosen as analysis indexes; in the analysis of the system processing function state, the machine tool processing state is chosen as the analysis index.

### Information entropy-based RMS dynamic complexity model

RMS is composed of several manufacturing cells. Each manufacturing cell contains numerous machine tools. Each type of machine tool is classified as the same type of processing function. The processing route establishes the relationship between processing functions. Part quantity is a test of a machine tool’s processing capability; i.e., that affects a machine tool’s productivity. When RMS is in operation, the machine tool is in specific state; machine tool quantity and machine tool state in the cellt directly determine cell state. Similarly, overall states of cells in the system determine the system state. Therefore, the machine tool state in the cell should be identified.

*N*represents the quantity of cells in RMS; \(G_{i}\) represents types of parts in the

*i*-th cell; and \(S_{j}\) represents the processing route for the

*j*-th type of part in the

*i*-th cell. For instance, when a manufacturing cell contains 2 types of parts, processing routes for parts 1 and 2 are \(\{a \; b \; c\}\) and \(\{a \; c \; d\}\), respectively; next, there is \(S_{1}=\{a \; b \; c\}\) and \(S_{2}=\{a \; c \; d\}\), where

*a*,

*b*,

*c*and

*d*represent processing functions. Each type of processing function can contain multiple machine tools. \({\vert }S_{j}{\vert }\) represents processing route length; \(p^{1}_{\textit{ijk}}\) represents the probability of the operative/non-blocked state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}_{j}\); \(p^{2}_{\textit{ijk}}\) represents the probability of the operative/blocked state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}_{j}\); \(p^{3}_{\textit{ijk}}\) represents the probability of the inoperative state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}_{j}\).

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}j\); \(p_{\textit{ijk}}^C\) represents the probability of the non-blocked state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}j\); \(p_{\textit{ijk}}^D \) represents the probability of the blocked state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}j\); and \(p_{\textit{ijk}}^B \) represents the probability of the inoperative state when the

*j*-th type of part in the

*i*-th cell calls for the

*k*-th processing function in processing route \(\hbox {S}j\).

### Analysis of processing function probability

*i*-th manufacturing cell in RMS;

*n*represents the types of processing functions in the manufacturing cell; and \(D_{ij}\) represents the quantity of machine tools with the

*j*-th type of processing function in the

*i*-th manufacturing cell.

*m*-th machine tool with the

*k*-th type of processing function for the

*j*-th type of part in the

*i*-th cell; \(p_{\textit{ijkm}}^{\bar{{J}}} \) represents the probability of the

*m*-th machine tool with the

*k*type of processing function for the

*j*-th type of part in the

*i*-th cell that is in the operative state but fails to provide the processing function required by this type of part; and \(M_{k }\)represents the quantity of machine tools with the

*k*-th processing function. Both \(P_{\textit{ijk}}^G \) and \(P_{\textit{ijk}}^{\bar{{J}}} \) are represented as state time frequency. Machine tool state duration in a specific period is collected, and the frequency of each state is calculated as ratio of state duration to overall time.

To summarize, the probability analysis for the processing function is completed, which includes the probability of the processing function in the operative state and the probability of the processing function in the inoperative state.

### Probability analysis of processing capability

Buffer zone probability is analyzed further. Assume that the buffer zone capacity is *h*. Then, from empty to full stock, there are \(h+1\) states. When the machine tool before the buffer zone completes the processing of a part, the buffer zone state moves backward one cell; i.e., the previous machine tool’s productivity is the buffer zone’s input transition rate. When the machine tool after the buffer zone starts to process a part, the buffer zone state moves forward one cell; i.e., the subsequent machine tool’s productivity is the buffer zone’s output transition rate. The buffer zone state transition process is shown in Fig. 3.

*l*-th state; when \(l=0,p_{\textit{ijk}}^0 =p_{\textit{ijk}}^Q \); i.e., the probability of a buffer zone with no stock; when \(l=h\), \(p_{\textit{ijk}}^h =p_{\textit{ijk}}^H \); i.e., the probability of a buffer zone with no vacancy; \(w_{\textit{ijk}}^R \)represents the productivity of the machine tool before the buffer zone; and \(w_{\textit{ijk}}^C \)represents the productivity of the machine tool after the buffer zone; i.e., the productivity of the

*k*-th processing function for the

*j*-th part in the

*i*-th cell.

## Cusp catastrophe model-based RMS reconfiguration point identification

Various complexity components control system operation and determine system stability. When positive complexity of a maintenance system’s stability is predominant, the system has relatively high stability and productivity. When the negative complexity component that leads to system instability is predominant, system stability significantly declines, which could result in system state catastrophe and trigger system reconfiguration. Analysis shows that during operation, RMS will experience state catastrophe because of internal and external factors such as new orders and machine tool failure. Catastrophe theory, which was proposed by French mathematician THOM in 1972, is a universal method used to investigate transition, discontinuity and catastrophe. In catastrophe theory, the system potential function is the study object, which comprises the state variable and external control parameter; the critical point of the system balance state is calculated from the potential function (Barunik and Vosvrda 2009; Chow et al. 2012; Dou and Ghose 2006; Hu and Xia 2015; Sethi and King 1998; Thom et al. 1975; Saunders 1980). Therefore, in this paper, RMS dynamic complexity is analyzed and combined with catastrophe theory to calculate and analyze the reconfiguration point.

*S*represents the RMS state;

*M*represents the machine tool state; and \(a_{i}(i= 0, 1, {\ldots }, 5)\) are a set of data related to RMS state complexity.

*V*represents the standard format of cusp catastrophe function; \(b_{i}\) are the matrix transformation result of \(a_{i}\; (i = 0, 1, \ldots , 5)\).

*V*, as shown in formula (18).

*u*and

*v*are two control variables in RMS. Based on the previous analysis, the value of the control variable is determined by positive complexity and negative complexity. The value of the control variable is calculated from the ratio of the complexity component with a different effect to overall system complexity.

*v*represents the positive complexity control variable;

*u*represents the negative complexity control variable. Because negative complexity has a negative effect on system stability, the value for the negative complexity control variable is negative, i.e., \(u<0.\; b_{0}\) is a constant, which does not change the potential function’s catastrophe characteristic. Generally, the origin of the potential function is changed to eliminate constant item \(b_{0}\). Therefore, the potential function is expressed as formula (19) (Saunders 1980).

*V*represents RMS’s standard format potential function; u and v are the control variable of RMS based on information entropy theory.

*V*in formula (19). The derived function is set to 0 to obtain a balanced surface equation, as shown in formula (20).

## Test verification

Processing task

Part no. | Processing route | Quantity |
---|---|---|

01 | 1, 3 | 1000 |

02 | 1, 4 | 2000 |

03 | 6, 7, 9 | 2000 |

04 | 3, 5, 6, 7, 9 | 1000 |

05 | 1, 5 | 1000 |

06 | 2, 8 | 1000 |

Single piece processing time for various parts

Part no. | Processing time (min) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cell 1 | Cell 2 | Cell 3 | ||||||||||

\(\text {M}_{11}\)/ \(\text {M}_{12}\) | \(\text {M}_{3}\) | \(\text {M}_{4}\) | \(\text {M}_{3}\) | \(\text {M}_{5}\) | \(\text {M}_{6}\) | \(\text {M}_{7}\) | \(\text {M}_{9}\) | \(\text {M}_{1}\) | \(\text {M}_{2}\) | \(\text {M}_{5}\) | \(\text {M}_{8}\) | |

01 | 2 | 4 | ||||||||||

02 | 3 | 3 | ||||||||||

03 | 2 | 2 | 5 | |||||||||

04 | 3 | 5 | 1 | 2 | 1 | |||||||

05 | 4 | 4 | ||||||||||

06 | 3 | 2 |

Probability analysis for machine tool M1/2

Monitoring time | Machine tool state probability | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Processing function | \(p^{K}_{\textit{ijk}}\) | \(p^{B}_{\textit{ijk}}\) | Processing capability | \(p^{C}_{\textit{ijk}}\) | \(p^{D}_{\textit{ijk}}\) | ||||||

\(p^{G}_{\textit{ijkm}}\) | \(p_{\textit{ijkm}}^{\bar{{J}}} \) | \(p^{Q}_{\textit{ijk}}\) | \(p^{H}_{\textit{ijk}}\) | ||||||||

10 | 0.1 | 0.500 | 0.64 | 0.36 | 0.2 | 0.202 | 0.638 | 0.362 | 0.409 | 0.231 | 0.36 |

20 | 0.15 | 0.500 | 0.578 | 0.423 | 0.25 | 0.21 | 0.593 | 0.408 | 0.342 | 0.235 | 0.423 |

30 | 0.2 | 0.500 | 0.51 | 0.49 | 0.298 | 0.25 | 0.527 | 0.474 | 0.269 | 0.241 | 0.49 |

40 | 0.21 | 0.500 | 0.496 | 0.504 | 0.3 | 0.271 | 0.510 | 0.49 | 0.253 | 0.243 | 0.504 |

50 | 0.22 | 0.500 | 0.482 | 0.518 | 0.55 | 0.39 | 0.275 | 0.726 | 0.132 | 0.349 | 0.518 |

60 | 0.23 | 0.500 | 0.467 | 0.533 | 0.595 | 0.41 | 0.239 | 0.761 | 0.112 | 0.355 | 0.533 |

Probability analysis for machine tool in Cell 1

Monitoring time | \(\text {M}_{11}\)/ \(\text {M}_{12}\) | \(\text {M}_{3}\) | \(\text {M}_{4}\) | ||||||
---|---|---|---|---|---|---|---|---|---|

\(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | |

10 | 0.380 | 0.23 | 0.39 | 0.381 | 0.23 | 0.36 | 0.380 | 0.23 | 0.35 |

20 | 0.355 | 0.215 | 0.43 | 0.349 | 0.221 | 0.41 | 0.354 | 0.217 | 0.40 |

30 | 0.320 | 0.24 | 0.44 | 0.308 | 0.252 | 0.44 | 0.338 | 0.222 | 0.44 |

40 | 0.229 | 0.171 | 0.6 | 0.22 | 0.180 | 0.6 | 0.182 | 0.208 | 0.61 |

50 | 0.127 | 0.213 | 0.6 | 0.091 | 0.249 | 0.66 | 0.113 | 0.227 | 0.64 |

60 | 0.112 | 0.228 | 0.66 | 0.087 | 0.253 | 0.66 | 0.079 | 0.261 | 0.66 |

Probability analysis for machine tool in Cell 2

Monitoring time | \(\text {M}_{3}\) | \(\text {M}_{5}\) | \(\text {M}_{6}\) | \(\text {M}_{7}\) | \(\text {M}_{9}\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | |

10 | 0.403 | 0.297 | 0.31 | 0.429 | 0.27 | 0.3 | 0.379 | 0.23 | 0.39 | 0.504 | 0.196 | 0.33 | 0.384 | 0.225 | 0.38 |

20 | 0.331 | 0.269 | 0.4 | 0.328 | 0.271 | 0.41 | 0.347 | 0.223 | 0.43 | 0.422 | 0.177 | 0.4 | 0.41 | 0.159 | 0.43 |

30 | 0.316 | 0.274 | 0.41 | 0.296 | 0.293 | 0.41 | 0.288 | 0.271 | 0.44 | 0.351 | 0.238 | 0.41 | 0.323 | 0.236 | 0.44 |

40 | 0.185 | 0.215 | 0.6 | 0.174 | 0.225 | 0.6 | 0.232 | 0.167 | 0.6 | 0.235 | 0.164 | 0.6 | 0.238 | 0.162 | 0.6 |

50 | 0.119 | 0.221 | 0.66 | 0.114 | 0.225 | 0.66 | 0.108 | 0.231 | 0.66 | 0.115 | 0.244 | 0.64 | 0.19 | 0.149 | 0.65 |

60 | 0.088 | 0.252 | 0.66 | 0.09 | 0.249 | 0.66 | 0.092 | 0.247 | 0.66 | 0.081 | 0.258 | 0.66 | 0.158 | 0.171 | 0.67 |

Probability analysis for machine tool in Cell 3

Monitoring time | \(\text {M}_{1}\) | \(\text {M}_{2}\) | \(\text {M}_{5}\) | \(\text {M}_{8}\) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

\(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | \(p^{1}_{\textit{ijk}}\) | \(p^{2}_{\textit{ijk}}\) | \(p^{3}_{\textit{ijk}}\) | |

10 | 0.409 | 0.231 | 0.36 | 0.409 | 0.231 | 0.36 | 0.393 | 0.346 | 0.261 | 0.204 | 0.2 | 0.36 |

20 | 0.342 | 0.235 | 0.423 | 0.342 | 0.235 | 0.423 | 0.48 | 0.290 | 0.23 | 0.336 | 0.264 | 0.4 |

30 | 0.269 | 0.241 | 0.49 | 0.269 | 0.241 | 0.49 | 0.430 | 0.33 | 0.24 | 0.330 | 0.26 | 0.41 |

40 | 0.253 | 0.243 | 0.504 | 0.253 | 0.243 | 0.504 | 0.448 | 0.352 | 0.2 | 0.392 | 0.308 | 0.5 |

50 | 0.132 | 0.349 | 0.518 | 0.132 | 0.349 | 0.518 | 0.215 | 0.495 | 0.29 | 0.12 | 0.22 | 0.6 |

60 | 0.112 | 0.355 | 0.533 | 0.112 | 0.355 | 0.533 | 0.175 | 0.525 | 0.3 | 0.081 | 0.259 | 0.66 |

Complexity and reconfiguration decision data

Monitoring time | Complexity | Reconfiguration decision | ||||
---|---|---|---|---|---|---|

\(E_{X}\) | \(E_{p}\) | \(E_{n}\) | \(\Delta \) | | | |

10 | 15.34 | 5.28 | 10.06 | 0.942 | 0.344 | \(-\)0.656 |

20 | 16.86 | 5.79 | 11.07 | 0.920 | 0.343 | \(-\)0.657 |

30 | 16.98 | 5.78 | 11.20 | 0.833 | 0.340 | \(-\)0.660 |

40 | 17.00 | 5.75 | 11.25 | 0.770 | 0.338 | \(-\)0.662 |

50 | 14.09 | 4.17 | 9.92 | \(-\)0.427 | 0.296 | \(-\)0.704 |

60 | 13.71 | 3.70 | 10.01 | \(-\)1.147 | 0.270 | \(-\)0.730 |

The system state trend diagram in Fig. 7 shows the trend of system state detection index \(\Delta \): after a relatively stable production period, the system experiences a state catastrophe, which triggers system reconfiguration. In the first 40 working days, when \(\Delta \) is above 0 and stays around 0.8, fluctuation is relatively small. This shows that the system is in a stable operation period and reconfiguration is not needed. During the \(40{\mathrm{th}}\sim 50{\mathrm{th}}\) working days, \(\Delta \) decreases dramatically from 0.77 to \(-0.427\). When \(\Delta \) is <0, the critical point is crossed and the cusp catastrophe condition is met. This shows that system state catastrophe occurs and immediate RMS reconfiguration is required. RMS monitoring is continued. Figure 5 shows that on the 60th working day, the system state continues the downward trend. This further confirms the necessity of reconfiguration.

## Conclusions

In this paper, a dynamic complexity-based RMS reconfiguration point decision method is proposed. First, the effect of processing function and processing capability on RMS is analyzed and the relationship between RMS complexity and RMS state is elaborated. This shows that the moment of an RMS state catastrophe is the best time to implement RMS reconfiguration. Next, based on information entropy theory, a system complexity quantitative model is created to analyze the effect of the system complexity component on the system state. That is then combined with the cusp catastrophe theory to analyze system state variation under the effect of various complexity components. The cusp catastrophe determinant condition helps identify the state catastrophe point and decide the RMS reconfiguration point. The test shows that information entropy-based system complexity analysis provides a quantitative description of system complexity. System complexity variation is analyzed in terms of processing function and processing capability to reveal the nature of the system state. Application of the cusp catastrophe helps identify the system state catastrophe moment. This provides evidence for the decision maker to decide the reconfiguration point, promptly implement system reconfiguration and maintain RMS vitality. Subsequent work focuses on an in-depth investigation of the RMS state catastrophe mechanism, which includes an analysis of the production factors that lead to system state catastrophe.

## Notes

### Acknowledgements

The authors are grateful to the anonymous reviewers for their comments, which have helped to improve this paper.

**Funding** The National Natural Science Foundation, China (Project No. 51105039).

## Compliance with ethical standards

## Conflict of interest

The authors declare that they have no conflict of interest.

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