# Resilience Regulation Strategy for Container Port Supply Chain under Disruptive Events

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Supply Chain Risks and Dynamics

_{2}emissions of food supply chains. They developed a spatially and temporally disaggregated price equilibrium mathematical model and compared three emission reduction interventions (carbon tax, technology innovation, and land sparing). Papanagnou [24] implemented a stochastic state-space model to capture the dynamics of a new four-echelon closed-loop supply chain model, and introduced an optimization method to study the impact of the Internet of Things on inventory variance and the bullwhip effect. Cuong et al. [25] proposed a fractional-order sliding mode control algorithm based on the adaptive mechanism, which ensured the robust stability of the goods flow in the supply chain network. Alkaabneh et al. [26] developed a dynamic programming model to optimize resource allocation by food banks among the agencies they serve. Khamseh et al. [27] established a general model based on bounded optimal control theory to optimize supply chain recovery and cost. Fu et al. [28] developed a distributed model predictive control approach to handle supply chain operations and achieve effective supply chain management with minimal information exchange and communication. Alkaabneh and Diabat [29] proposed and compared two different algorithms (branch-and-price and a two-stage meta-heuristic) to solve the multi-objective home healthcare delivery problem. The aim is to minimize the service and routing costs while maximizing compatibility of nurses and patients. Yan et al. [30] designed a stabilizing linear feedback controller to stabilize the supply chain mathematical model with a computer aided digital manufacturing process. Based on system dynamics, Xu et al. [31] established a four-dimensional differential equation with chaotic behavior to describe the multi-level supply chain, and combined this with modern control theory to implement a novel fractional-order adaptive sliding mode control algorithm to achieve efficient management of the supply chain.

#### 2.2. Resilience of Supply Chain

## 3. Modified Two-Stage CPSC System

#### 3.1. Container Handling System

_{adj}), and the adjustment of container handling in process (CHIP

_{adj}). Specifically, there are three feedforward links in the CHS. The AVCHR is obtained by the first-order forecasting link defined by the smoothing constant T

_{A}, and AVCHR can obtain PCHR. Simultaneously, AVCHR reacts with T

_{Q}to obtain the ECHIP level, and reacts with T

_{H}to obtain the EFCHR. T

_{P}represents the delay in the particular container handling process. T

_{CHIP}and T

_{UCHR}are used as the error adjustment time, adjusting the error between ECHIP and CHIP and the error between EFCHR and UCHR, respectively.

#### 3.2. Container Pretreatment System

_{WAIT}represents the container waiting time constant. After the effect of the first-order delay set by T

_{WAIT}, the average container arrive rate (AVRATE) can be estimated from the actual container arrive rate (CARATE). The estimated value from CARATE (AVRATE) can further set the expected FCPR (EFCPR), where T

_{E}is the expected lead-time of the pretreatment system. The adjustment of FCPR (FCPR

_{adj}) can be obtained by the difference between the expected FCPR and the FCPR after the adjustment time T

_{FCPR}, and T

_{FCPR}can be used as a proportional control. The sum of the average container arrive rate (AVRATE) and the FCPR adjustment is taken as the expected allocation completion rate (ECOM). U

_{A}represents the unit allocation rate, and the gross allocation completion rate (COMRATE

_{1}) can be obtained by the ECOM. The FCPR is derived from the difference between the net allocation completion rate (COMRATE

_{2}) and the CARATE.

#### 3.3. Two-Stage Container Port Supply Chain System

_{1}(ACHR) as the system coupling point to construct the two-stage container port supply chain system. Based on the elaboration of the system in Section 3.1 and Section 3.2, the specific formula description of the two-stage container port supply chain system can be obtained. For the CPS, the only input is the CARATE, and the output COMRATE

_{1}is taken as the input of the next stage. The CARATE can be deduced as

_{WAIT}, and $\Delta T$ represents the interval between two adjacent sampling times [42]. CARATE

_{1}is obtained by ECOM through the link $1/{U}_{A}$, that is,

_{1}is further used as the input of the CHS, that is, $CARAT{E}_{1}(t)=ACHR(t)$. As described in Section 3.1, the PCHR equals to the AVCHR plus the UCHR

_{adj}and the CHIP

_{adj}:

_{FCPR}, T

_{CHIP}and T

_{UCHR}in the formulas can be used as the proportional control of the system, while T

_{P}and T

_{P}

_{1}in the established system are regarded as the inherent properties of the system and are generally not adjusted.

#### 3.4. State Space Description

_{1}reflects the status of the actual container arrive rate, and x

_{2}indicates the status of the FCPR. Furthermore, the derivative state of each state variable can be obtained as follows:

_{1}= ACHR, Equation (10) can be rewritten as

_{1}, x

_{2}, and x

_{3}can be obtained as

#### 3.5. Dynamic Characteristic Analysis

_{A}, T

_{UCHR}, T

_{CHIP}, T

_{P}, T

_{Q}and T

_{H}. When T

_{P}= T

_{Q}, U

_{A}= 1 and T

_{H}= 0, uchr (∞) = 0.

_{WAIT}and T

_{FCPR}. When the values of T

_{WAIT}and T

_{FCPR}increase, ${\omega}_{n}$ decreases accordingly, causing the response to take longer to return to the steady state. For $\zeta $, since both a and b are positive, it is easy to obtain ${T}_{WAIT}{}^{2}+{T}_{FCPR}{}^{2}\ge 2{T}_{WAIT}{T}_{FCPR}$, thus making $\zeta \ge 1$. The system is always in a critically damped or overdamped state, which means that the CPS can maintain the stability under any positive control parameters.

_{1}, and p

_{2}, it is easy to identify one of the poles as $p=-1/{T}_{A}$. Then,

_{WAIT}is taken as the main independent variable, the states involved are shown in Figure 6.

_{1}, PCHR, and UCHR show an initial rise, while the FCPR shows a rapid initial drop. The reason lies in the transient response caused by the mutated input unit step signal to the responses. Under the influence of the increasing T

_{WAIT}, the setting time of each response increases correspondingly, at the cost of the decreasing oscillation level in varying degrees. The simulation results preliminarily prove the correctness of the calculation and analysis.

## 4. Two-Dimensional Resilience Index

_{UCHR}and ITAE

_{FCPR}, respectively. When the values of $ITA{E}_{UCHR}$ and $ITA{E}_{FCPR}$ are smaller, the response and recovery are better. α and β are the proportional coefficients, which is to coordinate the order of magnitude of each calculation link. Therefore, for the novel two-dimensional index R, smaller R represents better resilience. In Equation (41), ${E}_{UCHR}=uchr(t)-uchr(\infty )$, ${E}_{FCPR}=fcpr(t)-fcpr(\infty )$. Furthermore, it has been obtained from Equation (33) that $uchr(\infty )={T}_{UCHR}\left({T}_{P}-{T}_{Q}\right)/{T}_{CHIP}$, $fcpr(\infty )=1$.

## 5. Adaptive Fuzzy Double-Feedback Adjustment Strategy

#### 5.1. Overall Strategy Design

_{WAIT}is the main independent variable, while T

_{P}and TP

_{1}are intrinsic properties. The first-level fuzzy logic system is used to realize the adaptive optimization of the control parameters T

_{FCPR}and T

_{UCHR}, and the second-level adaptive fuzzy adjustment system is used to synchronously update the adjustment factors in the fuzzy control process to further optimize the modified two-stage CPSC system control effect. The overall design is shown in Figure 7.

_{1}), and its change rate (ec

_{1}) as the inputs of first-level fuzzy logic subsystem 1. The smoothing coefficient ${\alpha}_{1}$ is regulated by the set fuzzy logic, and the updated FCPR obtained by the updated control parameter T

_{FCPR}is fed back to the modified two-stage CPSC system. Meanwhile, K

_{1}and K

_{2}are the adjustment factors of the deviation e

_{1}and deviation change rate ec

_{1}respectively, and the control ratio of e

_{1}and ec

_{1}can be adjusted in real time, so as to better adapt to changes of external environment. The specific fuzzy design of the first-level fuzzy logic system and the second-level adaptive fuzzy adjustment system will be detailed in Section 5.2 and Section 5.3. For the first-level fuzzy logic subsystem 2, the principle is roughly similar to that of subsystem 1; the main difference is that it receives the deviation between the unfinished container handling requirement UCHR and the CARATE (e

_{2}) and the change rate of deviation (ec

_{2}) as input. The updated T

_{UCHR}is obtained by the smoothing coefficient ${\alpha}_{2}$. The updated UCHR obtained by T

_{UCHR}is fed back to the modified two-stage CPSC system. K

_{3}and K

_{4}also adjust the corresponding control ratio to better optimize their control effect.

#### 5.2. First-Level Fuzzy Logic System

_{1}and the error change rate signal ec

_{1}between the FCPR and CARATE are received, T

_{FCPR}is adjusted by the smoothing coefficient ${\alpha}_{1}$, and the calculation principle can be expressed as

_{1}are relatively uniform, the membership function which can be set as the uniformly distributed trigonometric functions, has good accuracy and is convenient to calculate. Specific settings can be found in Appendix A. The variation range of the trigonometric membership function is consistent with the domain corresponding to the fuzzy subset.

_{UCHR}is regulated by the smoothing coefficient ${\alpha}_{2}$. According to the input–output logic relationship, the rule table of fuzzy control of the two fuzzy logic subsystems is shown in Table 2.

_{1}, ec

_{2}) and opposite to error (e

_{1}, e

_{2}). Therefore, the basic control principle of the first-level fuzzy logic system can be summarized as follows: when the change rate is relatively large, the smoothing coefficient should be increased to alleviate the rapid change of the modified two-stage CPSC system under disruptive events, and when the error is relatively large, it is necessary to reduce the smoothing coefficient to try to restore the error to ideal states.

#### 5.3. Second-Level Adaptive Fuzzy Adjustment System

## 6. Simulations and Analysis

#### 6.1. The Effect of CPS on the Modified Two-Stage CPSC

_{WAIT}, can effectively reflect the adverse influences of disruptive events, and the preparations for the follow-up regulation research can be made.

_{A}= 6 days. T

_{A}can be given different values according to the specific prediction mechanism, but T

_{A}does not change the overall response trend of the system to a large extent; T

_{Q}and T

_{P}are set to be the same value, that is, T

_{Q}= T

_{P}= 4 days.

_{WAIT}on the dynamic behaviors of responses in the CPS. It should be emphasized that, compared with other three performance indicators, it is interesting to see that the OSC shows an opposite change trend, because when the waiting time increases, at the cost of longer setting time, the fluctuation becomes less. Furthermore, the responses always experience an initial rise, which is particularly evident in the UCHR response. The reason for this is that the mutated step demand is transmitted to the CPSC through the container preprocessing system, and the responses has to satisfy the output signal transmitted by the CPS. Then, with the continuous port productivity, the UCHR level is gradually reduced to the required level. Consistent with the analysis in Section 3.4, when T

_{WAIT}increases, the oscillation levels of the UCHR and PCHR decrease, which are achieved at the expense of longer setting time. In comparison, CHIP is less sensitive to the change of T

_{WAIT}. When T

_{WAIT}is small, the oscillation peak of CHIP does not change significantly, which proves that when the CPSC is less affected (small T

_{WAIT}), a certain balance can still be maintained between the port productivity and the planned container handling requirement.

_{WAIT}can make the resilience response have a faster response speed. When T

_{WAIT}increases, the response speed of R becomes slower and the final stable value is obviously larger, which means that when the modified two-stage CPSC system is more seriously affected (larger T

_{WAIT}), the resilience will be significantly reduced. Compared with the initial system, CPS has weakened the resilience to a certain extent under larger values of T

_{WAIT}, and the resilience is reduced by 66.95% in the worst case.

#### 6.2. Adaptive Fuzzy Double-Feedback Adjustment Strategy for Modified Two-Stage CPSC System

_{s}can be shortened by up to 35.94%. When the container waiting time delay is (3, 6, 12, 24) days, AFDA can ultimately shorten the stability time by (28.25%, 28.62%, 35.44%, 35.94%), respectively.

_{WAIT}significantly decreases the resilience. However, using the AFDA strategy, the resilience is enhanced. It is worth noting that R under the AFDA strategy has an obvious initial rapid rise. The reason is that when the feedback signal is used for adjustment, the initial rise of response affects the measurement of R. Eventually, both the response and R can be restored to the stable states.

_{WAIT}is small, R is significantly affected by FL more than RE. This trend begins to change as T

_{WAIT}grows larger. When the value of T

_{WAIT}becomes large, R is obviously dependent on the change of RE, while the change of FL is no longer obvious.

## 7. Managerial Findings and Conclusions

#### 7.1. Managerial Findings

_{WAIT}= 24, T

_{FCPR}= 4, the resilience can be reduced to as little as 1/3 of that of the original system. (3) Since resilience reflects the inherent property of the CPSC, it is more difficult to measure resilience than other performance indicators. The proposed two-dimensional resilience index can integrate recovery and affordability. Under the same T

_{WAIT}, the resilience can be improved by (44.95%, 36.88%, 25.39%, 12.90%), respectively. (4) When T

_{WAIT}is small, R is mainly affected by the affordability. When T

_{WAIT}becomes larger, the CPSC needs more recovery, and R begins to mainly depend on the recovery.

#### 7.2. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Fuzzy Subsets | Input x (e, ec) | Abscissas of Trigonometric Function (a, b, c) |
---|---|---|

VS | $0\le x\le 5\%$ | (a, b, c) = (0, 0, 0.1667) |

S | $5\%\le x\le 16.67\%$ | (a, b, c) = (0, 0.1667, 0.3333) |

RS | $16.67\%\le x\le 33.33\%$ | (a, b, c) = (0.1667, 0.3333, 0.5) |

M | $33.33\%\le x\le 50\%$ | (a, b, c) = (0.3333, 0.5, 0.6667) |

RB | $50\%\le x\le 66.67\%$ | (a, b, c) = (0.5, 0.6667, 0.8333) |

B | $66.67\%\le x<83.33\%$ | (a, b, c) = (0.6667, 0.8333, 1) |

VB | $83.33\%\le x\le 100\%$ | (a, b, c) = (0.8333, 1, 1) |

Fuzzy Subsets | Coordinate Range x (e, ec) | Abscissas of Trapezoidal Function (a, b, c, d) |
---|---|---|

VS | $0\le x\le 25\%$ | (a, b, c, d) = (0, 0, 0.05, 0.1) |

S | $25\%\le x\le 37.5\%$ | (a, b, c, d) = (0.05, 0.1, 0.15, 0.3) |

M | $37.5\%\le x\le 62.5\%$ | (a, b, c, d) = (0.1, 0.2, 0.35, 0.65) |

B | $62.5\%\le x\le 75\%$ | (a, b, c, d) = (0.55, 0.8, 0.9, 1) |

VB | $75\%\le x\le 100\%$ | (a, b, c, d) = (0.9, 0.95, 1, 1) |

**Table A3.**The initial settings of the main variables and parameters before the fuzzification procedure.

Symbols | Values |
---|---|

T_{UCHR} | 6 |

T_{FCPR} | 4 |

${\alpha}_{1}$ | 1/5 |

${\alpha}_{2}$ | 1/7 |

K_{1} = K_{2} | 0.5 |

K_{3} = K_{4} | 0.5 |

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**Figure 6.**Changes of each state in the CPS with different control parameters. (

**a**) COMRATE

_{1}under the change of T

_{WAIT}; (

**b**) FCPR under the change of T

_{WAIT}; (

**c**) PCHR under the change of T

_{WAIT}; (

**d**) UCHR under the change of T

_{WAIT}.

**Figure 9.**Resilience of original system and the system under the influence of T

_{WAIT}. (

**a**) R of original system; (

**b**) R under the change of T

_{UCHR}.

**Figure 10.**The 3D mesh, contour map and trajectory of decomposed R. (

**a**) 3D mesh of decomposition; (

**b**) contour map; and (

**c**) the trajectories of FL and RE.

**Figure 13.**The decomposition of R under AFDA. (

**a**) T

_{WAIT}= 3; (

**b**) T

_{WAIT}= 6; (

**c**) T

_{WAIT}= 12; (

**d**) T

_{WAIT}= 24.

ACHR | Actual container handling requirement |

AVCHR | Average container handling requirement |

AVRATE | Actual container average arrive rate |

CARATE | Container arrive rate |

CHIP | Container handling in progress |

CHIP_{adj} | CHIP adjustment |

COMRATE | Handling completion rate |

COMRATE_{1} | Gross allocation completion rate |

COMRATE_{2} | Net allocation completion rate |

ECHIP | Expected container handling in progress |

ECOM | Expected allocation completion rate |

EFCPR | Expected finished container pretreatment requirement |

EFCHR | Expected finished container handling requirement |

FCPR | Finished container pretreatment requirement |

FCPR_{adj} | FCPR adjustment |

OSC | Oscillation level |

PCHR | Planned container handling requirement |

R | Resilience index |

T_{A} | Forecasting smoothing constant |

T_{CHIP} | Time to adjust CHIP discrepancy |

T_{E} | Expected lead-time of pretreatment |

T_{FCPR} | Time to adjust FCPR discrepancy |

T_{H} | Expected handling time |

t_{p} | Peak time |

T_{P} | Container handling delay time |

T_{P1} | Allocation time delay |

T_{Q} | Expected lead-time of container handling |

t_{s} | Setting time |

T_{UCHR} | Adjustment time of UCHR |

T_{WAIT} | Waiting time delay |

U_{A} | Unit allocation rate |

UCHR | Unfinished container handling requirement |

UCHR_{adj} | UCHR adjustment |

ec | e | ||||||
---|---|---|---|---|---|---|---|

VS | S | RS | M | RB | B | VB | |

VS | M | RS | RS | S | S | VS | VS |

S | RB | M | RS | RS | S | S | VS |

RS | RB | RB | M | RS | RS | S | S |

M | B | RB | RB | M | RS | RS | S |

RB | B | B | RB | RB | M | RS | RS |

B | VB | B | B | RB | RB | M | RS |

VB | VB | VB | B | B | RB | RB | M |

ec | e | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | Z | PS | PM | PB | |

NB | B | B | M | VS | S | M | B |

NM | B | B | M | S | S | B | B |

NS | VB | B | M | M | M | B | VB |

Z | VB | B | B | M | B | B | VB |

PS | VB | B | M | M | M | B | VB |

PM | B | B | M | S | M | B | B |

PB | B | M | S | VS | S | M | B |

Control Parameters | UCHR | R | |||
---|---|---|---|---|---|

OSC | t_{s} (d) | t_{p} (d) | |||

T_{FCPR} = 4 | T_{WAIT} = 3 | 5.44 | 40.60 | 10.00 | 5.94 |

T_{WAIT} = 6 | 5.33 | 57.30 | 11.20 | 7.24 | |

T_{WAIT} = 12 | 5.03 | 72.80 | 12.30 | 10.16 | |

T_{WAIT} = 24 | 4.70 | 87.10 | 12.90 | 15.82 | |

Ideal situation | 4.70 | 36.40 | 9.10 | 5.19 |

Control Parameters | UCHR | R | ||
---|---|---|---|---|

OSC | t_{s} | t_{p} | ||

T_{WAIT} = 3 | 15.74% | 11.54% | 9.89% | 14.45% |

T_{WAIT} = 6 | 13.40% | 57.42% | 23.08% | 39.50% |

T_{WAIT} = 12 | 7.02% | 100.00% | 35.16% | 95.76% |

T_{WAIT} = 24 | 0 | 139.29% | 41.76% | 204.48% |

UCHR | R | |||
---|---|---|---|---|

OSC | t_{s} | t_{p} | ||

Increase in T_{WAIT} | 2.80 | 35.40 | 5.70 | 3.27 |

2.69 | 40.90 | 6.60 | 4.57 | |

2.60 | 47.00 | 7.60 | 7.58 | |

2.51 | 55.80 | 9.00 | 13.78 |

UCHR | R | |||
---|---|---|---|---|

OSC | t_{s} | t_{p} | ||

Increase in T_{WAIT} | 2.86 | 82.50 | 6.60 | 7.58 |

2.63 | 86.50 | 7.30 | 8.78 | |

2.36 | 94.00 | 7.80 | 11.05 | |

2.14 | 102.00 | 8.10 | 15.40 |

UCHR | R | |||
---|---|---|---|---|

OSC | t_{s} | t_{p} | ||

Increase in T_{WAIT} | 5.44 | 40.70 | 10.00 | 5.87 |

5.33 | 57.30 | 11.20 | 6.84 | |

5.03 | 72.80 | 12.30 | 9.89 | |

4.70 | 87.20 | 12.90 | 15.64 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, B.; Liu, W.; Li, J. Resilience Regulation Strategy for Container Port Supply Chain under Disruptive Events. *J. Mar. Sci. Eng.* **2023**, *11*, 732.
https://doi.org/10.3390/jmse11040732

**AMA Style**

Xu B, Liu W, Li J. Resilience Regulation Strategy for Container Port Supply Chain under Disruptive Events. *Journal of Marine Science and Engineering*. 2023; 11(4):732.
https://doi.org/10.3390/jmse11040732

**Chicago/Turabian Style**

Xu, Bowei, Weiting Liu, and Junjun Li. 2023. "Resilience Regulation Strategy for Container Port Supply Chain under Disruptive Events" *Journal of Marine Science and Engineering* 11, no. 4: 732.
https://doi.org/10.3390/jmse11040732