# Supply Chain Risk of Obsolescence at Simultaneous Robust Perturbations

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Perturbed Flows in Global Supply Chains

#### 1.2. Supply Chain Finances and Recommendations of the World Economic Forum

## 2. Literature Review

#### Economic and Technological Components of Sustainable Development

## 3. Methods

#### 3.1. Disruption of Supply Chains on the Infinite Horizon

- What are potential sources of disruption; and
- How can this disruption be avoided or mitigated?

- Specifying sources of risk and vulnerabilities in a chain;
- Risk assessment; and

#### 3.2. Extended MRP Theory as a Framework for Risk Analysis in an SC

**H**and

**G**, respectively. Different rows correspond to different items, and different columns correspond to various activities that transform the input items to the new things. The basic notation is given in Table 1.

**HP**, and the whole outputs are assembled into the column vector

**GP**, from which the net production is determined as (

**G**–

**H**)

**P**.

**P**(and thereby net production) will be a time-varying, vector-valued function. For the details, see Grubbström’s papers and the overview of Bogataj and Bogataj (2019). For the sake of simplicity, we often assume

**G = I.**

**R**(0) collects the initially available inventory levels. $\tilde{F}(s)$ represents exports from the system. If demand is reduced, delivery is also reduced. Substantially reduced demand could influence disruption risk, while high-frequency, small-severity changes of order here are supposed to influence operational risk. For the plan $\tilde{P}(s)$ to be feasible, we must always have ${\pounds}^{-1}\left\{\tilde{R}(s)\right\}\ge 0$.

#### 3.3. The Analysis of the Simultaneously Realised Risks in a Global Supply Chain

**p**should be described as a row vector:

#### 3.4. The Infinite Time Horizon and the NPV of the Costs as Presented in Bogataj et al. (2016)

#### 3.5. The Risk Realisation on the Finite Horizon as a New Approach—The Network Simulation Model

## 4. Results—Mitigating the Supply Chain Risk of Obsolescence at the Simultaneous Robust Perturbations on the Finite Horizon

#### 4.1. The SC without Perturbations of Lead Times on the Infinite Time Horizon—Numerical Example

#### 4.2. The SC with Perturbations of the Lead Times and Expected Obsolescence on the Finite-Time Horizon

## 5. Discussion

## 6. Conclusions and Further Research

- (a)
- To study the impact of obsolescence, together with environmental issues, the extension of the presented network is needed, as explained by Bogataj and Grubbström in the article “On the representation of timing for different structures within MRP theory”, where the close loop analysis is given on the infinite horizon [2]. In such cases, some considerations by Kwak and Kim, regarding green profit maximisation through integrated pricing and production planning [48], would also be further studied and evaluated with the NPV criterion, and environmental constraints and penalties.
- (b)
- Among the social sustainability issues, the main challenges are rising because of the ageing of the European workforce. Therefore, the trade-off between investments in ergonomics and pension schemes for earlier retirement, as presented by Bogataj et al. [49], and other ergonomic improvements should be studied on the finite horizon, which might give different optimal policies for the cases with a smaller ratio between the obsolescence horizon and the value of the length of a cycle. In this case, as well as some other solutions, listed in the articles of Calzavara et al. [50], Battini et al. [51], and Andriolo et al. [52], would be included in the model.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Example of the production in a global supply chain: product structure tree with lead-times shown in the activity cells and on the transportation lines (average lead time $\overline{\tau}$. The longest path is denoted by the bold lines).

**Figure 2.**Net present value (NPV) on the time horizon to the time of obsolescence when the interest rate takes a value on the interval [0.01, 0.1].

**Figure 3.**NPV in details on the time horizon lower than 100. The interest rate takes a value on the interval [0.01, 0.1].

**Table 1.**The basic notation in extended material requirements planning (EMRP). NPV, net present value.

H;${h}_{i,j}\in \text{}$H | Input matrix (bill-of-materials) describing how many items (${h}_{i,j}$) from activity cell i are needed to produce or to package one item in activity cell j |

$i,j$ | Index of fixed activity cell in a supply chain |

G;${g}_{i,j}\in $G | Output matrix. In our model, we shall assume that G is the identity matrix I |

P_{j} ∈ P | The intensity of activities in j-th activity cell as a component of production vector P |

${h}_{i,j}$P_{j} | The intensity of flow from $i$ to $j$ |

${\tau}_{j}$ | Lead time for the manipulation of the item inside activity cell $j$ |

${\tau}_{i,j}$ | Lead time for transportation of items from $i$ to $j\text{}$ |

$\mathit{\tau}$ | Diagonal lead time matrix with element ${e}^{s{\tau}_{j}}$ in its $j$-th diagonal position |

${\tilde{\tau}}^{d}(s)$ | Perturbed diagonal lead time matrix |

$s$ | The Laplace (complex) frequency in the Laplace transformed frequency domain |

${\tau}_{j}^{d}$,${\tau}_{ij}^{d}$ | Additional delays inside the activity cell and transportation line, respectively |

$\stackrel{\u2323}{H}(s);\tilde{H}{(s)}^{\mathsf{\omega}d}$ | Generalised input matrix and the generalised perturbed input matrix |

${T}_{j}$ | Length of the j-th cycle |

$p$, $K$, $\Pi $ | Price vector, setup cost vector, and transportation cost vector, respectively |

$\tilde{F}(s)$, D,$\Delta $ | Vector of delivery, the vector of demand and reduction of demand respectively |

R(0) $\tilde{R}(s)$ | Initial available inventory level and the available interest rate at s |

$\rho ,r$ | Continuous and yearly interest rate, respectively |

λ | The correction factor for the NPV for various time to obsolescence and various values of the interest rate |

ϑ | The correction factor for the NPV at different perturbations of lead times and increasing time to the obsolescence |

Interest Rate | Time to Obsolescence [Time Units] | |||||
---|---|---|---|---|---|---|

[%] | 100 | 200 | 300 | 400 | 500 | 1000 |

1 | 39.5 | 53.8 | 59.2 | 61.3 | 62.0 | 62.5 |

2 | 26.2 | 29.7 | 30.2 | 30.3 | 30.3 | 30.3 |

3 | 18.6 | 19.5 | 19.6 | 19.6 | 19.6 | 19.6 |

4 | 14.0 | 14.2 | 14.3 | 14.3 | 14.3 | 14.3 |

5 | 11.0 | 11.1 | 11.1 | 11.1 | 11.1 | 11.1 |

6 | 8.95 | 8.97 | 8.97 | 8.97 | 8.97 | 8.97 |

7 | 7.48 | 7.49 | 7.49 | 7.49 | 7.49 | 7.49 |

8 | 6.38 | 6.38 | 6.38 | 6.38 | 6.38 | 6.38 |

9 | 5.53 | 5.53 | 5.53 | 5.53 | 5.53 | 5.53 |

10 | 4.86 | 4.86 | 4.86 | 4.86 | 4.86 | 4.86 |

**Table 3.**Correction factor λ for calculation of NPV for a different time to obsolescence and various values of the interest rate.

Interest | Time to Obsolescence [Time Units] | |||||
---|---|---|---|---|---|---|

Rate | 100 | 200 | 300 | 400 | 500 | 1000 |

0.01 | 0.63 | 0.86 | 0.95 | 0.98 | 0.99 | 1.00 |

0.02 | 0.86 | 0.98 | 1.00 | 1.00 | 1.00 | 1.00 |

0.03 | 0.95 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 |

0.04 | 0.98 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 |

0.05 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.06 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.07 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.08 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.09 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.10 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

**Table 4.**NPV in 100,000 € for different perturbed lead times of D and increasing time to the obsolescence.

Lead Time | Time to the Obsolescence TTO [Time Units] | |||||
---|---|---|---|---|---|---|

[Time Units] | 100 | 200 | 300 | 400 | 500 | 1000 |

3.00 | 3.95 | 5.38 | 5.92 | 6.13 | 6.20 | 6.25 |

3.60 | 3.63 | 5.12 | 5.67 | 5.86 | 5.94 | 5.98 |

4.20 | 3.52 | 4.95 | 5.43 | 5.62 | 5.6 | 5.73 |

4.80 | 3.42 | 4.71 | 5.22 | 5.40 | 5.46 | 5.50 |

5.40 | 3.35 | 4.57 | 5.02 | 5.19 | 5.25 | 5.29 |

**Table 5.**The correction factor ϑ for the NPV at different perturbations of lead times and increasing time to the obsolescence.

Lead Time | Time to the Obsolescence TTO [Time Units] | |||||
---|---|---|---|---|---|---|

[Time Units] | 100 | 200 | 300 | 400 | 500 | 1000 |

3.00 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

3.60 | 0.919 | 0.952 | 0.958 | 0.956 | 0.958 | 0.957 |

4.20 | 0.891 | 0.920 | 0.917 | 0.917 | 0.918 | 0.917 |

4.80 | 0.866 | 0.875 | 0.882 | 0.881 | 0.881 | 0.880 |

5.40 | 0.848 | 0.849 | 0.848 | 0.847 | 0.847 | 0.846 |

6.00 | 0.777 | 0.809 | 0.814 | 0.814 | 0.815 | 0.814 |

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## Share and Cite

**MDPI and ACS Style**

Campuzano-Bolarín, F.; Marín-García, F.; Moreno-Nicolás, J.A.; Bogataj, M.; Bogataj, D.
Supply Chain Risk of Obsolescence at Simultaneous Robust Perturbations. *Sustainability* **2019**, *11*, 5484.
https://doi.org/10.3390/su11195484

**AMA Style**

Campuzano-Bolarín F, Marín-García F, Moreno-Nicolás JA, Bogataj M, Bogataj D.
Supply Chain Risk of Obsolescence at Simultaneous Robust Perturbations. *Sustainability*. 2019; 11(19):5484.
https://doi.org/10.3390/su11195484

**Chicago/Turabian Style**

Campuzano-Bolarín, Francisco, Fulgencio Marín-García, José Andrés Moreno-Nicolás, Marija Bogataj, and David Bogataj.
2019. "Supply Chain Risk of Obsolescence at Simultaneous Robust Perturbations" *Sustainability* 11, no. 19: 5484.
https://doi.org/10.3390/su11195484