# A Dynamic Analysis for Mitigating Disaster Effects in Closed Loop Supply Chains

^{*}

## Abstract

**:**

## 1. Introduction

- Demand Pattern 1: Demand does not vary as a result of the phenomenon (for instance, food products during COVID-19 effect).
- Demand Pattern 2: Demand marks a sudden increase when disaster materializes. This increase is maintained for as long as the phenomenon last (disaster-period) and thereafter (post-disaster period) it returns to its pre-disaster level. An indicative example for this case is the sudden increase in demand for medical masks owing to the outbreak of COVID-19 [24].
- Demand Pattern 3: Demand marks a sudden decrease during the disaster period. During the post-disaster period, it increases to reach higher levels compared with pre-disaster conditions and after a certain amount of time, it returns to its pre-disaster level (for instance, the automotive industry during COVID-19 [24]).

## 2. Literature Review

## 3. System and Problem Description

#### 3.1. System Description

#### 3.2. Problem Description

^{D}is the unknown time of occurrence of a disaster event, while h

^{D}is the unknown duration of disaster event, and h

^{rec}is the recovery period during which each mitigation policy is implemented by the manufacturer. This approach is based on the disaster resilience triangle concept originally introduced by Bruneau [74] and thereafter extended by Zobel and Khansa [75].

^{D}), two options are considered regarding the disruptions of the manufacturer’s production rate due to the disaster effect: a 50% reduction of the production rate and a 20% reduction of the production rate.

## 4. The SD Model

^{®}, Vensim

^{®}, i-think

^{®}and Stella

^{®}[77].

#### 4.1. Generic Stock and Flow Structure

#### 4.2. Mitigation Policies at the Manufacturer Level

_{i}= 1) or deactivating (MP

_{i}= 0) the corresponding MP

_{i}. The sets of control parameters that fully describe MP1 and MP2 are the following:

- MP1: [MP
_{1,}cover time of MI, cover time of MI due to event, adjust time of MI, adjust time of MI due to event, h^{rec}]. - MP2: [MP
_{2}, cover time of MI, cover time of MI due to event, adjust time of MI, adjust time of MI due to event percentage offered by contracted manufacturer, h^{rec}].

_{1}and MP

_{2}is given by the following equations; where equations are differentiated, the values of parameters MP

_{i}are given to indicate the active mechanism.

#### 4.3. Stock Equations

#### 4.4. Flow Equation

#### 4.5. Auxiliary Equations

^{rec}as the combination of the expected wholesaler’s orders with an adjustment that brings the MI in line with its desired value (desired MI). The production rate is subject to the availability of actual stock in parts. The percentage offered by contracted manufacturer in Equation (5) is defined by Equation (10) under MP

_{2}, while the expected wholesaler’s orders is a forecasted value for wholesaler’s orders (see Figure 3 and Equation (15)) calculated from a first-order exponential smoothing. The adjustment of MI is based on a proportional rule that controls the discrepancy (discrepancy of MI) between desired MI and actual MI (Equation (11)).

^{rec}. Especially during the disaster period and during the recovery period h

^{rec}, the value of adjust time of MI is decreased so as for the system to be able to respond to the changes caused by the disaster event (see Equation (13)). Cover time of MI (see Equation (14)) is the base stock expressed in time units.

#### 4.6. Profit Equations

_{2}. The detailed income/cost equations are based on standard calculations.

## 5. Numerical Experimentation and Discussion

#### 5.1. Validation of the SD Model

#### 5.2. Settings

^{D}= 12 [week], while the duration of the disaster period (h

^{D}) was examined under three sets of value: 2 weeks (small-scale); 6 weeks (medium-scale); 10 weeks (large-scale). The alternative demand patterns (Demand Pattern 1, Demand Pattern 2 and Demand Pattern 3) due to the disaster event (see explanation in Section 1) are given in Figure 6, Figure 7 and Figure 8.

- MP1: [MP
_{1,}cover time of MI, cover time of MI due to event, adjust time of MI, adjust time of MI due to event, h^{rec}]. - MP2: [MP
_{2}, cover time of MI, cover time of MI due to event, time of MI, adjust time of MI due to event, percentage offered by contracted manufacturer, h^{rec}].

- BS: [Demand ~N(10,000 items/week, 1000 items/week, stock management: cover time of MI = 4 weeks; adjust time of MI = 6 weeks].

#### 5.3. Recommendations for Mitigating Disaster Effects

^{rec}(4, 8 and 12 weeks).

#### 5.3.1. Profitability of the CLSC System

^{rec}.

^{rec}). In addition, it is preferable for the duration of h

^{rec}to be long (8 or 12 weeks). For example, in the case of h

^{D}= 6 weeks and reduction of production rate = 20%, from Table 2, we found that the best policy recommendations were: MP2, h

^{rec}= 12 weeks, adjust time due to event = 2 weeks and cover time due to the event = 2 weeks for all demand patterns.

^{D}) or the reduction of production rate fluctuate among small values. In the opposite cases, MP1 with the appropriate adjustments to the rest of the parameters is preferable. Finally, we observe that, if the best mitigation policies are implemented, the system returns to the same or an even better condition compared with the one in the pre-disaster period.

#### 5.3.2. Demand Backlog and Manufacturer Inventory

_{index}for the case of medium scale (h

^{D}= 6 weeks) disaster events and 20% reduction of production rate, for the three demand patterns described above. From the results, we found that the system entered into different equilibrium state for each demand pattern compared to BS (in all cases equilibrium state is differential to 1).

_{index}, is independent from adjust time of MI due to event. As far as the safety stock is concerned, it reaches high levels in this case. For example, in the case of h

^{D}= 6 weeks and reduction of production rate = 50%, from Table 3, we found that the best policy recommendations were: MP2, h

^{rec}=4 weeks, cover time due to event = 6 weeks or 8 weeks and is independent from adjust time of MI due to event for Demand Pattern 3. In most cases, MP2 is recognized as the best mitigation policy. Nevertheless, there are cases where MP1 is preferable, mainly when Demand Pattern 1 applies (for example when h

^{D}= 2 weeks, reduction of production rate = 50% for Demand Pattern 1 MP1 is preferable). Finally, as far as h

^{rec}is concerned, there are cases where the best policy is indicated for short time lapses, for long time lapses or is independent from time lapse.

## 6. Summary, Limitations and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Likelihood pattern for the production rate and level of inventory under disaster effects at the manufacturer.

**Figure 5.**Dynamic behavior of Inventory level at the manufacturer. (

**a**) This study (MI, disaster-free model); (

**b**) Yadav et al. [80]. Copyright Year: 2022, Copyright Owner’s Name: Elsevier.

**Figure 9.**DBMI

_{index}for medium scale (h

^{D}= 6 weeks) disaster event and reduction of production rate 20%.

Approaches | Reference |
---|---|

Complex Adaptive Systems Theory | [35,36,37] |

Chaos Theory | [38,39,40] |

Catastrophe Theory | [41] |

Catastrophe-Risk Approaches | [42] |

Disaster Preparedness | [43] |

System Dynamics | - |

Non-Linear Dynamic Approaches | [40,44,45] |

h^{D}(Weeks) | Reduction of Production Rate | Demand Pattern | Best/Worst Cases | MP * | h^{rec}(Weeks) | Adjust Time due to Event (Weeks) | Cover Time due to Event (Weeks) | $\mathbf{Total}\mathbf{Profit}\mathbf{Index}\phantom{\rule{0ex}{0ex}}\left[\frac{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{f}\mathbf{i}\mathbf{t}{}_{\mathbf{t}}}{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{f}\mathbf{i}\mathbf{t}{}_{\mathbf{t}}^{\left(\mathbf{B}\mathbf{S}\right)}}\ast 100\right]$ | ||
---|---|---|---|---|---|---|---|---|---|---|

Lower | Upper | Equilibrium State | ||||||||

2 | 0% | 1 | Basic Scenario (BS) | 100 | 100 | 100 | ||||

20% | 1 | Best | MP2 | 12 | 1 | 2 | 100 | 106.58 | 100.88 | |

Worst | MP2 | 12 | 2 | 8 | 80.41 | 99.93 | 95.90 | |||

20% | 2 | Best | MP2 | 12 | 4 | 2 | 99.81 | 142.83 | 105.60 | |

Worst | MP2 | 12 | 2 | 8 | 95.67 | 119.84 | 103.60 | |||

20% | 3 | Best | MP2 | 12 | 2 | 2 | 82.59 | 160.12 | 137.00 | |

Worst | ΜP1 | 12 | 4 | 8 | 73.29 | 154.10 | 134.00 | |||

50% | 1 | Best | MP1 | 12 | 2 | 2 | 100 | 109.66 | 100 | |

Worst | MP2 | 12 | 4 | 8 | 85.73 | 99.30 | 96.60 | |||

50% | 2 | Best | MP1 | 12 | 4 | 2 | 99.87 | 145.95 | 106.80 | |

Worst | MP1 | 12 | 3 | 6 | 96.61 | 130.04 | 105.45 | |||

50% | 3 | Best | MP1 | 12 | 3 | 2 | 85.49 | 162.30 | 137.40 | |

Worst | MP1 | 8 | 6 | 8 | 76.92 | 148.91 | 123.90 | |||

6 | 20% | 1 | Best | MP2 | 12 | 2 | 2 | 98.93 | 108.97 | 99.00 |

Worst | MP2 | 12 | 2 | 8 | 82.13 | 97.99 | 94.58 | |||

20% | 2 | Best | MP2 | 12 | 2 | 2 | 99.26 | 156.45 | 108.20 | |

Worst | MP1 | 12 | 4 | 8 | 95.90 | 132.55 | 107.50 | |||

20% | 3 | Best | MP2 | 12 | 2 | 2 | 65.08 | 151.73 | 133.86 | |

Worst | MP1 | 12 | 4 | 8 | 45.07 | 145.35 | 131.60 | |||

50% | 1 | Best | MP1 | 8 | 4 | 2 | 100.33 | 114.48 | 100.53 | |

Worst | MP2 | 12 | 1 | 8 | 91.84 | 99.17 | 97.79 | |||

50% | 2 | Best | MP1 | 12 | 3 | 2 | 101.13 | 163.40 | 109.82 | |

Worst | MP2 | 8 | 3 | 6 | 97.82 | 147.99 | 107.97 | |||

50% | 3 | Best | MP1 | 12 | 4 | 2 | 67.64 | 155.12 | 135.10 | |

Worst | MP1 | 12 | 3 | 8 | 53.80 | 147.33 | 132.97 | |||

10 | 20% | 1 | Best | MP2 | 8 | 6 | 2 | 98.11 | 110.12 | 98.15 |

Worst | MP2 | 8 | 3 | 8 | 83.04 | 98.43 | 95.35 | |||

20% | 2 | Best | MP2 | 12 | 1 | 2 | 100.13 | 166.10 | 111.00 | |

Worst | MP1 | 12 | 4 | 8 | 96.17 | 143.76 | 110.90 | |||

20% | 3 | Best | MP1 | 12 | 3 | 2 | 50.33 | 144.16 | 131.52 | |

Worst | MP2 | 12 | 2 | 8 | 30.72 | 135.70 | 125.90 | |||

50% | 1 | Best | MP1 | 12 | 4 | 2 | 99.94 | 119.43 | 100.90 | |

Worst | MP1 | 4 | 6 | 8 | 93.41 | 102.90 | 99.13 | |||

50% | 2 | Best | MP1 | 8 | 2 | 2 | 100.88 | 178.86 | 113.40 | |

Worst | MP1 | 8 | 4 | 8 | 97.66 | 158.60 | 112.70 | |||

50% | 3 | Best | MP1 | 12 | 4 | 2 | 58.92 | 148.35 | 133.80 | |

Worst | MP1 | 12 | 2 | 8 | 42.75 | 137.77 | 127.70 |

h^{D}(Week) | Reduction of Production Rate | Demand Pattern | MP | h^{rec}[Weeks] | Adjust Time Due to Event (Week) | Cover Time Due to Event (Week) | $\mathbf{D}\mathbf{B}\mathbf{M}{\mathbf{I}}_{\mathbf{i}\mathbf{n}\mathbf{d}\mathbf{e}\mathbf{x}}$ | |
---|---|---|---|---|---|---|---|---|

Max | Equilibrium State (Value/Time (Week) ^{1}) | |||||||

2 | 0% | 1 | Basic Scenario (BS) | 1 | - | |||

20% | 1 | MP2 | 12 | All values | 6 or 8 | 1.01 | 0.91/7 | |

20% | 2 | MP2 | 8 or 12 | All values | 6 or 8 | 1.52 | 0.93/36 | |

20% | 3 | MP1 or MP2 | 4 | All values | 6 or 8 | 1.54 | 1.02/60 | |

50% | 1 | MP1 | 12 | All values | 6 or 8 | 1.35 | 0.84/20 | |

50% | 2 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 2.05 | 1.06/50 | |

50% | 3 | MP1 or MP2 | 4 | All values | 6 or 8 | 2.05 | 1.11/74 | |

6 | 20% | 1 | MP1 | 12 | All values | 8 | 1.07 | 0.72/20 |

20% | 2 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 2.09 | 1.06/45 | |

20% | 3 | MP1 or MP2 | 4 | 1 | 8 | 1.23 | 0.96/50 | |

50% | 1 | MP1 | 12 | All values | 6 or 8 | 2.80 | 1.09/45 | |

50% | 2 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 5.50 | 1.11/80 | |

50% | 3 | MP2 | 4 | All values | 6 or 8 | 2.40 | 1.11/70 | |

10 | 20% | 1 | MP1 | 12 | All values | 8 | 1.14 | 0.72/25 |

20% | 2 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 4.85 | 1.12/70 | |

20% | 3 | MP1 or MP2 | 4 | 1 | 8 | 1.48 | 1.06/80 | |

50% | 1 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 7.41 | 1.09/45 | |

50% | 2 | MP1 or MP2 | All values | All values | 4 or 6 or 8 | 11.09 | 1.14/75 | |

50% | 3 | MP1 or MP2 | 4 | 1 or 2 or 3 or 4 | 6 or 8 | 3.67 | 1.12/70 |

^{1}in number of weeks (t − t

^{D}).

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**MDPI and ACS Style**

Katsoras, E.; Georgiadis, P.
A Dynamic Analysis for Mitigating Disaster Effects in Closed Loop Supply Chains. *Sustainability* **2022**, *14*, 4948.
https://doi.org/10.3390/su14094948

**AMA Style**

Katsoras E, Georgiadis P.
A Dynamic Analysis for Mitigating Disaster Effects in Closed Loop Supply Chains. *Sustainability*. 2022; 14(9):4948.
https://doi.org/10.3390/su14094948

**Chicago/Turabian Style**

Katsoras, Efthymios, and Patroklos Georgiadis.
2022. "A Dynamic Analysis for Mitigating Disaster Effects in Closed Loop Supply Chains" *Sustainability* 14, no. 9: 4948.
https://doi.org/10.3390/su14094948