# Towards Digital Twinning of Fabrication Lines in Shipyards

^{*}

## Abstract

**:**

## 1. Introduction

#### Brief Literature Review

## 2. Mathematical Modeling

#### 2.1. The Finite-State Method

^{n}is the total number of split flows of the nth order and $\sum _{i=1}^{{K}^{n}}{r}_{i}^{n}}=1$ (Figure 1). The primary flow is composed of S machines and S−1 buffers, where S stands for a machine that performs the flow splitting, while the total numbers of machines and buffers in a line equal M and M−1, respectively. Each machine (m

_{i}, I = 1, 2, …, M) is of the Bernoulli reliability (${p}_{i}\sim Bernoulli\left({p}_{i}\right)$), and it is thus in the state {up} with the probability p

_{i}and in the state {down} with the probability 1−p

_{i}. Each buffer (b

_{i}, i = 1, 2, …, M−1) is of the capacity N

_{i}, where N

_{i}∈ N. The system at hand obeys the mass conservation law; thus, material can enter it only at the first machine, while it exits the system only at the last machine of a particular branch. In addition, the usual assumptions on the infinite capacities of supply and delivery storage, as well as the homogeneity of the machine cycles, hold.

_{m}) of the primary flow and the weakest machine ${\left({\mathrm{m}}_{i}^{n}\right)}_{m}$ associated with the ith secondary flow of the nth order. Hence, its reliability (${\left({p}_{i}^{n}\right)}_{m}$) equals

^{n}and ${p}_{m}^{0}=\mathrm{min}\left({p}_{1},{p}_{2},...,{p}_{S}\right)$, while p

_{ζ}stands for a set of machines that form the ith secondary flow of the nth order. Once the weakest machines have been identified, M−1 finite-state elements can be established. An example of the finite-state element arrangement along the hypothetical splitting line concerning the weakest machines is presented in Figure 2, where their general classification into upstream and downstream elements can be noted.

^{e}is the total capacity of buffer b

_{e}, while p

_{L}and p

_{R}stand for the reliabilities of the left-hand-side and right-hand-side machines of the two-machine–one-buffer problem. Thus, in the case of upstream elements, p

_{L}always equals p

_{e}, while p

_{R}amounts to ${\left({p}_{i}^{n}\right)}_{m}$. Conversely, in the case of downstream elements, p

_{L}equals ${\left({p}_{i}^{n}\right)}_{m}$, and p

_{R}amounts to p

_{e}. Once the distributions $\u2329{P}^{e}\u232a$ are known, the system-level steady-state probability distribution ($\u2329P\u232a$) can be determined by assuming the independence of events at each buffer. Therefore, the probability that the production system at hand is in the state h, h = 1, 2, …, D, where $D={\displaystyle \prod _{i=1}^{M-1}\left({N}_{i}+1\right)}$, equals

_{i}, i = 1, 2, …, M−1, is the state of the buffer b

_{i}. Equation (4) is also known as the system-level steady-state probability distribution.

#### 2.2. Key Performance Indicators

_{ξ}), as an expected number of finished products per cycle time, equals

_{i}) can be determined as follows:

_{i}) of machines other than those performing the splitting of the material flows is equal to

_{M}= 0. However, the probability of blockage is more complex in the case of splitting machines, as it is influenced by events originating from different secondary flows. Thus, BL

_{ψ}, where ψ = {S, S + R

_{1}, S + R

_{1}+ R

_{2}, …}, equals

_{i}, i = 1, 2, …, K

^{n}, is a set of buffers corresponding to the ith secondary flow and placed immediately after the splitting point (χ) in the system. Finally, the probability of starvation (ST

_{i}) of the ith machine in the system (except the first one that is, by definition, never starved) takes a simpler form:

#### 2.3. Digital Threading

## 3. Application of the Developed Theory

_{6}and m

_{21}) and cutting (machines m

_{7}, m

_{9}, m

_{13}, m

_{15}, m

_{22}, and m

_{24}) operations, respectively. Thus, in total, four separate models can be formulated, two of them corresponding to plate fabrication (models A1 and A2), and two in the case of profile processing (models B1 and B2) (Figure 5). A detailed list of the operations attributed to each machine is presented in Table 1, along with the pertaining reliability data. Moreover, a breakdown of the buffering capacities is listed in Table 2. The machine reliability data were determined on the shipyard’s floor using a digital thread in such a way that each working station was related to a dedicated database storing operative logs and indicating the time that each machine spent in the states {up} and {down}. Recall that the state {down} stands for the portion of time spent in breakdown, repair, or maintenance. Thus, the probability that the machine (m

_{i}) is in the state {up} (p

_{i}) equals ${T}_{\left\{\mathrm{up}\right\}}/{T}_{\mathrm{Total}}$, where T

_{{up}}is the time spent in the state {up}, and T

_{Total}is the time elapsed since the beginning of data collection.

_{i}) attributed to each split flow based on the final product that will be assembled out of the fabricated structural elements. Thus, the rates (r

_{i}) are highly sensitive to the final properties of the product, including its scale, shape, material, geometrical properties, and assembly sequence. In the case considered in this study, two typical cruise ship sections were evaluated: the ship fore and midship sections (Figure 6). Therefore, the rates (r

_{i}) were determined according to data and technical documentation provided by the shipyard (Table 3). It can be seen that in the present case, the shipyard is employing only the plasma cutting of plates and the manual cutting of stiffeners, as r

_{3}, r

_{4}, r

_{8}, r

_{9}, r

_{10}, r

_{12}, r

_{15}, and r

_{16}take zero values.

#### Results and Discussion

_{6}and PR

_{21}) of 0.78 plates and 0.8 stiffeners per cycle, with rather low levels of blockage and starvation probabilities, except in the case of the drying and abrasive cleaning of plates, where some blockage may be expected. Moreover, the capacity of the buffers in the case of model A1 is quite balanced, as the values of the work-in-process occupy almost all of the available storage space. However, this is not true in the case of buffers b

_{19}and b

_{20}(model B1), which are of significantly larger capacity compared to the expected number of stiffeners lined for the coat-drying and marking operations. This fact indicates that model B1 has a capacity for improvements toward leaner solutions. Conversely, models A2 and B2 remain quite sensitive to the splitting rates, yielding lower production rates, taking values of 0.64 (PR

_{8}, plasma cutting), 0.04 (PR

_{11}and PR

_{12}, plate-forming operations), 0.54 (PR

_{10r3}, plasma cutting), 0.15 (PR

_{22r14}, stiffener forming), and 0.62 (PR

_{23}, stiffener cutting). In addition, they demonstrate more pronounced starvation probabilities, especially in the case of the plate- and stiffener-forming operations, reaching 0.81 and 0.78, respectively. However, this can be expected, as a rather low number of structural elements, as compared to the total number of structural elements, is expected to be formed using roller-bending machines or a hydraulic press. Similar to the case of model B1, it can be seen that production lines A2 and B2 demonstrate an almost adequate balancing of the buffering capacity. However, some improvement capacity towards leaner solutions may be identified in the cases of buffers b

_{10}, b

_{12}, and b

_{22}.

_{4}, m

_{7}, m

_{19}, and m

_{22}are improved through an improved maintenance strategy, investments, and better working conditions. This will also positively affect the complete production system through a decrease in blockages BL

_{1}, BL

_{2}, BL

_{3}, BL

_{17}, and BL

_{18}. Conversely, a slight increase in the probabilities of starvation may be expected. However, generally, they are at a rather low level (except for the specific cases of ST

_{11}, ST

_{12}, and ST

_{23}), and no significant impact on the complete production system can be expected.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Mathematical models (digital twins) of the considered fabrication lines: models A1 and A2—fabrication of plate elements; models B1 and B2—fabrication of stiffeners.

**Figure 7.**A numerical model of the fabrication line in a shipyard: (1) prefabrication of steel plates; (2) prefabrication of profiles; (3) plasma cutting of plates; (4) manual cutting and forming of profiles; (5) robotic cutting of profiles; (6) oxy-fuel cutting of plates; (7) plate forming.

**Figure 8.**A comparison of KPIs obtained using the mathematical model (MM) and numerical model (NM): (

**a**) production rate (PR); (

**b**) work-in-process (WIP); (

**c**) probability of blockage (BL); and (

**d**) probability of starvation (ST).

i | m_{i} | Model | Operation | p_{i} |
---|---|---|---|---|

1 | m_{1} | A1 | Plate straightening | 0.90 |

2 | m_{2} | Drying | 0.91 | |

3 | m_{3} | Abrasive cleaning | 0.98 | |

4 | m_{4} | Shop priming | 0.80 | |

5 | m_{5} | Coat drying | 0.91 | |

6 | m_{6} | Marking | 0.96 | |

7 | m_{7} | A2 | Plasma tracing and cutting | 0.70 |

8 | m_{8} | Plasma marking | 0.77 | |

9 | m_{9} | Plasma tracing and cutting | 0.71 | |

10 | m_{10} | Plasma marking | 0.77 | |

11 | m_{11} | Plate-forming–roller-bending machine | 0.85 | |

12 | m_{12} | Plate forming–hydraulic press | 0.85 | |

13 | m_{13} | Oxy-fuel tracing and cutting | 0.70 | |

14 | m_{14} | Oxy-fuel marking | 0.77 | |

15 | m_{15} | Oxy-fuel tracing and cutting | 0.71 | |

16 | m_{16} | Oxy-fuel marking | 0.77 | |

17 | m_{17} | B1 | Drying | 0.91 |

18 | m_{18} | Abrasive cleaning | 0.98 | |

19 | m_{19} | Shop priming | 0.80 | |

20 | m_{20} | Coat drying | 0.91 | |

21 | m_{21} | Marking | 0.96 | |

22 | m_{22} | B2 | Oxy-fuel manual cutting | 0.77 |

23 | m_{23} | Stiffener-forming–roller-bending machine | 0.93 | |

24 | m_{24} | Oxy-fuel robotic cutting | 0.81 |

BC * | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{7} | b_{9} | b_{10} | b_{11} | b_{13} | b_{15} | b_{17} | b_{18} | b_{19} | b_{20} | b_{22} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

N_{i} | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 |

Splitting Rate | r_{1} | r_{2} | r_{3} | r_{4} | r_{5} | r_{6} | r_{7} | r_{8} | r_{9} | r_{10} | r_{11} | r_{12} | r_{13} | r_{14} | r_{15} | r_{16} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r_{i} | 0.5 | 0.5 | 0 | 0 | 0.05 | 0.05 | 0.9 | 0 | 0 | 0 | 1 | 0 | 0.2 | 0.8 | 0 | 0 |

**Table 4.**KPIs of production processes: production rate (PR

_{i}(pieces/cycle)), work in process (WIP

_{i}(pieces)), residence time (RT

_{i}= WIP/PR

_{i}(cycle)).

A1 | PR_{6} | WIP_{1} | WIP_{2} | WIP_{3} | WIP_{4} | WIP_{5} | RT_{6} | |||||

MM | 0.78 | 3.35 | 0.93 | 0.98 | 0.81 | 0.81 | 8.82 | |||||

NM | 0.80 | 4.00 | 1.00 | 1.00 | 0.00 | 0.00 | 7.50 | |||||

A2 | PR_{8} | WIP_{7} | RT_{8} | |||||||||

MM | 0.64 | 1.18 | 1.84 | |||||||||

NM | 0.70 | 0.00 | n/a | |||||||||

PR_{11} | PR_{12} | PR_{10r3} | WIP_{9} | WIP_{10} | WIP_{11} | RT_{11} | RT_{12} | RT_{10r3} | ||||

MM | 0.04 | 0.04 | 0.64 | 1.20 | 0.04 | 0.04 | 31 | 31 | 1.87 | |||

NM | 0.04 | 0.04 | 0.63 | 0.00 | 0.00 | 0.00 | n/a | n/a | n/a | |||

B1 | PR_{21} | WIP_{17} | WIP_{18} | WIP_{19} | WIP_{20} | RT_{21} | ||||||

MM | 0.80 | 7.35 | 7.91 | 1.45 | 0.99 | 22.12 | ||||||

NM | 0.80 | 7.98 | 9.99 | 0.00 | 0.00 | 22.46 | ||||||

B2 | PR_{22r14} | PR_{23} | WIP_{22} | RT_{22r14} | RT_{23} | |||||||

MM | 0.15 | 0.62 | 0.17 | / | 0.27 | |||||||

NM | 0.15 | 0.62 | 0.00 | / | n/a |

**Table 5.**KPIs of production processes: the probability of blockage (BL

_{i}) and the probability of starvation (ST

_{i}).

A1 | BL_{1} | BL_{2} | BL_{3} | BL_{4} | BL_{5} | ST_{2} | ST_{3} | ST_{4} | ST_{5} | ST_{6} | |

MM | 0.17 | 0.24 | 0.27 | 0.08 | 0.03 | 0.00 | 0.07 | 0.01 | 0.17 | 0.19 | |

NM | 0.10 | 0.11 | 0.18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.16 | |

A2 | BL_{7} | ST_{8} | |||||||||

MM | 0.06 | 0.13 | |||||||||

NM | 0.00 | 0.07 | |||||||||

BL_{9} | BL_{10} | ST_{10} | ST_{11} | ST_{12} | |||||||

MM | 0.06 | 0.00 | 0.12 | 0.81 | 0.81 | ||||||

NM | 0.00 | 0.00 | 0.07 | 0.81 | 0.81 | ||||||

B1 | BL_{17} | BL_{18} | BL_{19} | BL_{20} | ST_{18} | ST_{19} | ST_{20} | ST_{21} | |||

MM | 0.11 | 0.18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.16 | |||

NM | 0.11 | 0.18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.16 | |||

B2 | BL_{22} | BL_{23} | ST_{23} | ||||||||

MM | 0.00 | 0.00 | 0.78 | ||||||||

NM | 0.00 | 0.00 | 0.78 |

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**MDPI and ACS Style**

Hadžić, N.; Ložar, V.; Opetuk, T.; Keser, R.
Towards Digital Twinning of Fabrication Lines in Shipyards. *J. Mar. Sci. Eng.* **2023**, *11*, 1053.
https://doi.org/10.3390/jmse11051053

**AMA Style**

Hadžić N, Ložar V, Opetuk T, Keser R.
Towards Digital Twinning of Fabrication Lines in Shipyards. *Journal of Marine Science and Engineering*. 2023; 11(5):1053.
https://doi.org/10.3390/jmse11051053

**Chicago/Turabian Style**

Hadžić, Neven, Viktor Ložar, Tihomir Opetuk, and Robert Keser.
2023. "Towards Digital Twinning of Fabrication Lines in Shipyards" *Journal of Marine Science and Engineering* 11, no. 5: 1053.
https://doi.org/10.3390/jmse11051053