# A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methods

#### 3.1. Model Concepts

#### 3.1.1. Supply Chain Network Framework

#### 3.1.2. Deterministic Model

_{k}) is assumed to follow a statistical distribution that is already known.

- 1st stage:
- ○
- Produced and transported quantities;
- ○
- Selected warehouses and capacity;
- ○
- Supply chain network;
- ○
- Demand deficit.

- 2nd stage:
- ○
- Stock out and overstocking probabilities;
- ○
- Expected lead time (ELD);
- ○
- Quantities that should be produced to cover unsatisfied demand.

#### 3.1.3. Stochastic Model

#### 3.2. Implementation of Deterministic and Stochastic Models

#### Risk Assessment

- Gaussian noise;
- Lognormal noise;
- Pareto noise for various alpha levels (α = 0.01; α = 0.5; α = 0.99).

## 4. Results

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Index and Variable Explanation

Index | |

$i$ | Plant |

$j$ | Warehouse |

$k$ | Customer |

Continuous Variables | |

$TC$ | Total supply chain cost |

${P}_{i}$ | $\mathrm{Produced}\text{}\mathrm{quantities}\text{}\mathrm{in}\text{}\mathrm{plant}\text{}i$ |

${Q}_{ij}$ | $\mathrm{Transported}\text{}\mathrm{quantities}\text{}\mathrm{from}\text{}\mathrm{plant}\text{}i$ $\text{}\mathrm{to}\text{}\mathrm{warehouse}\text{}j$ |

${Q}_{jk}$ | Transported quantities from warehouse $j$$\text{}\mathrm{to}\text{}\mathrm{customer}\text{}k$ |

${E}_{jk}$ | Transported quantities from warehouse $j$$\text{}\mathrm{to}\text{}\mathrm{customer}\text{}k$ that exceed a certain level (high) |

${\tilde{P}}_{i}$ | $\mathrm{Produced}\text{}\mathrm{quantities}\text{}\mathrm{in}\text{}\mathrm{plant}\text{}i$ with noise representation |

${\tilde{Q}}_{ij}$ | $\mathrm{Transported}\text{}\mathrm{quantities}\text{}\mathrm{from}\text{}\mathrm{plant}\text{}i$ warehouse $j$ with noise representation |

${\tilde{Q}}_{jk}$ | Transported quantities from warehouse $j$$\text{}\mathrm{to}\text{}\mathrm{customer}\text{}k$ with noise representation |

${R}_{jk}$ | Transported quantities from warehouse $j$$\text{}\mathrm{to}\text{}\mathrm{customer}\text{}k$ that exceed a certain level (low) |

${W}_{j}$ | Capacity of warehouse $j$ |

$EL{D}_{k}$ | $\mathrm{Expected}\text{}\mathrm{lead}\text{}\mathrm{time}\text{}\mathrm{of}\text{}\mathrm{customer}\text{}k$ |

${P}_{k}\left({Q}^{O}\right)$ | $\mathrm{Stock}\text{}\mathrm{out}\text{}\mathrm{probability}\text{}\mathrm{of}\text{}\mathrm{customer}\text{}k$ |

${P}_{k}\left({Q}^{L}\right)$ | $\mathrm{Overstocking}\text{}\mathrm{probability}\text{}\mathrm{of}\text{}\mathrm{customer}\text{}k$ |

${\mathsf{\Delta}}_{k}$ | Deficit in demand satisfaction for customer k |

Binary Variables | |

${{\rm X}}_{ij}$ | $1\text{}\mathrm{if}\text{}\mathrm{the}\text{}\mathrm{corresponding}\text{}\mathrm{connection}\text{}\mathrm{between}\text{}\mathrm{plant}\text{}i$ $\text{}\mathrm{to}\text{}\mathrm{warehouse}\text{}j$ exists, 0 otherwise |

${{\rm X}}_{jk}$ | 1 if the corresponding connection between warehouse $j$$\text{}\mathrm{to}\text{}\mathrm{customer}\text{}k$ exists, 0 otherwise |

${Y}_{j}$ | 1 if warehouse $j$ is selected, 0 otherwise |

${K}_{jk}$ | 1 if small quantities will be delivered from warehouse j to customer k due to large demand deficit, 0 otherwise |

${\mathsf{\Omega}}_{jk}$ | 1 if large quantities will be delivered from warehouse j to customer k due to large demand deficit, 0 otherwise |

${\lambda}_{k}$ | $1\text{}\mathrm{if}\text{}\mathrm{the}\text{}\mathrm{deficit}\text{}\mathrm{in}\text{}\mathrm{demand}\text{}\mathrm{satisfaction}\text{}\mathrm{lies}\text{}\mathrm{in}\text{}\mathrm{the}\text{}\mathrm{interval}\text{}\left[{\mathsf{\Delta}}^{L},{\mathsf{\Delta}}^{U}\right]$, 0 otherwise |

${\zeta}_{k}$ | $1\text{}\mathrm{if}\text{}\mathrm{the}\text{}\mathrm{deficit}\text{}\mathrm{in}\text{}\mathrm{demand}\text{}\mathrm{satisfaction}\text{}\mathrm{lies}\text{}\mathrm{in}\text{}\mathrm{the}\text{}\mathrm{interval}\text{}\left[{\mathsf{\Delta}}^{U},\mathsf{\Delta}\right]$, 0 otherwise |

Parameters | |

${P}_{i}^{U}$ | Upper bounded production of plant i |

${P}_{i}^{L}$ | Lower bounded production of plant i |

${Q}_{ij}^{U}$ | Maximum capacity of transported quantities from plant i to warehouse j |

${Q}_{ij}^{L}$ | Minimum capacity of transported quantities from plant i to warehouse j |

${Q}_{jk}^{U}$ | Maximum capacity of transported quantities from warehouse j to customer k |

${Q}_{jk}^{L}$ | Minimum capacity of transported quantities from warehouse j to customer k |

${I}_{j}$ | Inventory held at warehouse j |

${D}_{k}$ | Demand of customer k |

${\mathsf{\Delta}}_{k}$ | Stock out quantity in customer k |

${{\rm T}}^{u}$ | Maximum time for product delivery |

${{\rm T}}^{l}$ | Minimum time for product delivery |

${\alpha}_{j}$ | Coefficient relating quantity at capacity at warehouse j |

${\beta}_{j\kappa}$ | Production rate for quantities stored at warehouse j that will be delivered to customer k in order to cover the high deficit in demand satisfaction. |

${\gamma}_{j\kappa}$ | Production rate for quantities stored at warehouse j that will be delivered to customer k in order to cover the low deficit in demand satisfaction. |

Cost Parameters | |

${c}_{i}^{P}$ | Production cost of plant i |

${c}_{ij}^{VTR}$ | Variable transportation cost of plant i to warehouse j |

${c}_{ij}^{FTR}$ | Fixed transportation cost of plant i to warehouse j |

${c}_{jk}^{VTR}$ | Variable transportation cost of warehouse j to customer k |

${c}_{jk}^{FTR}$ | Fixed transportation cost of warehouse j to customer k |

${c}_{j}^{IN}$ | Installation cost of warehouse j |

${c}_{jk}^{PO}$ | Production cost of small quantities that will be manufactured in warehouse j and will be delivered to customer k |

${c}_{jk}^{PU}$ | Production cost of large quantities that will be manufactured in warehouse j and will be delivered to customer k |

## Appendix B. Implementation of Stock Out Instances in the Deterministic Model

_{1}), then ${\lambda}_{{\kappa}_{1}}=1$ and, based on the predetermined range in (14), warehouses that are also assumed to be production plants holding inventory used for manufacturing purposes will have to produce additional quantity equal to R

_{jk}as seen in (A2). Constraint (A2) provides a value that corresponds to the quantity to be produced based on constraint (A1), as binary variable ${K}_{j\kappa}$ takes a value of 1 if ${\lambda}_{\kappa}$ equals 1. In that case, the quantity that will eventually be produced by warehouse $j$ in order to facilitate a medium stock out occurring at customer $k$ should be more than ${\gamma}_{j\kappa}\xb7{H}_{j\kappa}$; ${\gamma}_{j\kappa}$ stands for the production coefficient of warehouse $j$ for each customer $k$, and ${H}_{j\kappa}$ is a minimum level of inventory stored for the production of the necessary quantity in warehouse $j$ in order to facilitate a medium stock out occurring at customer $k$ Constraint (A3) models the occurrence of a large stock out instance, while the production quantity that is needed to be sent to customer $k$ from warehouse $j$ is defined as ${E}_{j\kappa}$, and should be more than the warehouse’s $j$ production rate (${\beta}_{j\kappa}$) multiplied by the sum of the overall inventory held at warehouse $j$ and stock out occurred in customer $k$ as in (A4).

## References

- Gruen, T.W.; Corsten, D.S.; Bharadwaj, S. Retail Out-of-Stocks. A Worldwide Examination of Extent, Causes, and Consumer Responses, Research Study, Atlanta; Grocery Manufacturers of America: Washington, DC, USA, 2002. [Google Scholar]
- Gruen, T.; Corsten, D. Stock-Outs Cause Walkouts. Harv. Bus. Rev.
**2004**, 82, 26–27. [Google Scholar] - Govind, A.; Luke, R.; Pisa, N. Investigating stock-outs in Johannesburg’s warehouse retail liquor sector. J. Transp. Supply Chain Manag.
**2017**, 11, a303. [Google Scholar] [CrossRef] - Diane, M.; Hannah, S.; Wendy, L.; Monique, U. Green, lean, and global Supply chains. Int. J. Phys. Distrib. Logist. Manag.
**2010**, 41, 14–41. [Google Scholar] - Kumar, V.; Sabri, S.; Garza-Reyes, J.A.; Nadeem, S.P.; Kumari, A.; Akkaranggoon, S. The challenges of GSCM implementation in the UK manufacturing SMEs. In Proceedings of the 2018 International Conference on Production and Operations Management Society (POMS), Peradeniya, Sri Lanka, 14–16 December 2018; IEEE: Piscataway Township, NJ, USA, 2018; pp. 1–8. [Google Scholar]
- Walker, H.; Di Sisto, L.; McBain, D. Drivers and barriers to environmental supply chain management practices: Lessons from the public and private sectors. J. Purch. Supply Manag.
**2008**, 14, 69–85. [Google Scholar] [CrossRef] - Bhool, R.; Narwal, M.S. An analysis of drivers affecting the implementation of green supply chain management for the Indian manufacturing industries. Int. J. Res. Eng. Technol.
**2013**, 2, 242–254. [Google Scholar] - Sarkis, J. Evaluating environmentally conscious business practices. Eur. J. Oper. Res.
**1998**, 107, 159–174. [Google Scholar] [CrossRef] - Welford, R.; Gouldson, A. Environmental Management & Business Strategy; Pitman Publishing Limited: London, UK, 1993. [Google Scholar]
- Davies, A.R. Clean and green? A governance analysis of waste management in New Zealand. J. Environ. Plan. Manag.
**2009**, 52, 157–176. [Google Scholar] - Henriques, I.; Sadorsky, P. Environmental technical and administrative innovations in the Canadian manufacturing industry. Bus. Strategy Environ.
**2007**, 16, 119–132. [Google Scholar] [CrossRef] - Baylis, R.; Connell, L.; Flynn, A. Company size, environmental regulation and ecological modernization: Further analysis at the level of the firm. Bus. Strategy Environ.
**1998**, 7, 285–296. [Google Scholar] [CrossRef] - Zhu, Q.; Geng, Y.; Sarkis, J. Motivating green public procurement in China: An individual level perspective. J. Environ. Manag.
**2013**, 126, 85–95. [Google Scholar] [CrossRef] - Tyagi, M.; Kumar, P.; Kumar, D. Parametric selection of alternatives to improve performance of green supply chain management system. Procedia Soc. Behav. Sci.
**2015**, 189, 449–457. [Google Scholar] [CrossRef] - Boufounou, P.; Moustairas, I.; Toudas, K.; Malesios, C. ESGs and Customer Choice: Some Empirical Evidence. Circ. Econ. Sustain.
**2023**, 3, 1–34. [Google Scholar] [CrossRef] [PubMed] - Srivastav, P.; Gaur, M.K. Barriers to Implement Green Supply Chain Management in Small Scale Industry using Interpretive Structural Modeling Technique-A North Indian Perspective. Eur. J. Adv. Eng. Technol.
**2015**, 2, 6–13. [Google Scholar] - Testa, F.; Iraldo, F. Shadows and lights of GSCM (Green Supply Chain Management): Determinants and effects of these practices based on a multi-national study. J. Clean. Prod.
**2010**, 18, 953–962. [Google Scholar] [CrossRef] - Villanueva, R.; Garcia, L.J. Green Supply Chain Management—A competitive Advantage. In Proceedings of the International Congression on Logistics & Supply Chain (CILOG 2013), Sanfandila, Mexico, 24–25 October 2013; pp. 186–190. [Google Scholar]
- Choudhary, M.; Seth, S. Integration of green practices in supply chain environment the practices of inbound, operational, outbound and reverse logistics. Int. J. Eng. Sci. Technol.
**2011**, 3, 4985–4993. [Google Scholar] - Huang, X.; Tan, B.L.; Ding, X. Green supply chain practices: An investigation of manufacturing SMEs in China. Int. J. Technol. Manag. Sustain. Dev.
**2012**, 11, 139–153. [Google Scholar] [CrossRef] [PubMed] - Frederick, H.; Elting, J. Determinants of green supply chain implementation in the food and beverage sector. Int. J. Bus. Innov. Res.
**2013**, 7, 164–184. [Google Scholar] [CrossRef] - Zhu, Q.; Feng, Y.; Choi, S.B. The role of customer relational governance in environmental and economic performance improvement through green supply chain management. J. Clean. Prod.
**2017**, 155, 46–53. [Google Scholar] [CrossRef] - Ninlawan, C.; Seksan, P.; Tossapol, K.; Pilada, W. The Implementation of Green Supply Chain Management Practices in Electronics Industry. In Proceedings of the International MultiConference of Engineers and Computer Scientists 2010, Hong Kong, China, 17–19 March 2010; Volume III. [Google Scholar]
- Luthra, S.; Garg, D.; Haleem, A. The impacts of critical success factors for implementing green supply chain management towards sustainability: An empirical investigation of Indian automobile industry. J. Clean. Prod.
**2016**, 121, 142–158. [Google Scholar] [CrossRef] - Jain, V.K.; Sharma, S. Green Supply Chain Management Practices in Automobile Industry: An Empirical Study. J. Supply Chain Manag. Syst.
**2012**, 1, 20–26. [Google Scholar] - Zhu, Q.; Sarkis, J.; Lai, K.-H. Green supply chain management: Pressures, practices and performance within the Chinese automobile industry. J. Clean. Prod.
**2007**, 15, 1041–1052. [Google Scholar] [CrossRef] - Choi, T.; Yeung, W.; Cheng, T.C.E. Scheduling and co-ordination of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates and storage costs. Int. J. Prod. Res.
**2013**, 51, 2593–2601. [Google Scholar] [CrossRef] - Liu, S.; Papageorgiou, L.G. Multiobjective Optimisation of Production, Distribution and Capacity Planning of Global Supply Chains in the Process Industry. Omega
**2013**, 41, 369–382. [Google Scholar] [CrossRef] - Seferlis, P.; Giannelos, N.F. A Two-Layered Optimisation-Based Control Strategy for Multi-Echelon Supply Chain Networks. Comput. Chem. Eng.
**2004**, 28, 799–809. [Google Scholar] [CrossRef] - Fattahi, M.; Mahootchi, M.; Husseini, S.M. Integrated strategic and tactical supply chain planning with price-sensitive demands. Ann. Oper. Res.
**2016**, 242, 423–456. [Google Scholar] [CrossRef] - Mahapatra, R.N.; Biswal, B.B.; Parida, P.K. A Modified Deterministic Model for Reverse Supply Chain in Manufacturing. J. Ind. Eng.
**2013**, 10, 987172. [Google Scholar] [CrossRef] - Melo, M.T.; Nickel, S.; Saldanha da Gama, F. Dynamic Multi-Commodity Capacitated Facility Location: A Mathematical Modeling Framework for Strategic Supply Chain Planning. Comput. Oper. Res.
**2006**, 33, 181–208. [Google Scholar] [CrossRef] - Yu, V.F.; Normasari, N.M.E.; Luong, H.T. Integrated Location-Production-Distribution Planning in a Multiproducts Supply Chain Network Design Model. Math. Probl. Eng.
**2015**, 2015, 473172. [Google Scholar] [CrossRef] - Goh, M.; Meng, F. A Stochastic Model for Supply Chain Risk. In Supply Chain Risk and Vulnerability; Wu, T., Blackhurst, J., Eds.; Springer: London, UK, 2009. [Google Scholar]
- Kim, J.; Realff, M.J.; Lee, J.H. Optimal Design and Global Sensitivity Analysis of Biomass Supply Chain Networks for Biofuels under Uncertainty. Comput. Chem. Eng.
**2011**, 35, 1738–1751. [Google Scholar] [CrossRef] - Govindan, K.; Fattahi, M.; Keyvanshokooh, E. Supply chain network design under uncertainty: A comprehensive review and future research directions. Eur. J. Oper. Res.
**2017**, 263, 108–141. [Google Scholar] [CrossRef] - Bidhandi, H.M.; Mohd, R.; Yusuff, M.M.H.; Ahmad, M.; Bakar, M.R.A. Development of a New Approach for Deterministic Supply Chain Network Design. Eur. J. Oper. Res.
**2009**, 198, 121–128. [Google Scholar] [CrossRef] - Tamas, M. Mismatched Strategies: The Weak Link in the Supply Chain? Supply Chain Manag. Int. J.
**2000**, 5, 171–175. [Google Scholar] [CrossRef] - Salema, M.I.G.; Barbosa-Povoa, A.M.; Novais, A.Q. An optimization model for the design of a capacitated multi-product reverse logistics network with uncertainty. Eur. J. Oper. Res.
**2007**, 179, 1063–1077. [Google Scholar] [CrossRef] - Santoso, T.; Ahmed, S.; Goetschalckx, M.; Shapiro, A. A Stochastic Programming Approach for Supply Chain Network Design under Uncertainty. Eur. J. Oper. Res.
**2005**, 167, 96–115. [Google Scholar] [CrossRef] - Tsiakis, P.; Shah, N.; Pantelides, C.C. Design of Multi-Echelon Supply Chain Networks under Demand Uncertainty. Ind. Eng. Chem. Res.
**2001**, 40, 3585–3604. [Google Scholar] [CrossRef] - Garcia-Herreros, P.; Wassick, J.; Grossmann, I.E. Design of Resilient Supply Chains with Risk of Facility Disruptions. Ind. Eng. Chem. Res.
**2014**, 53, 17240–17251. [Google Scholar] [CrossRef] - Dillon, M.; Oliveira, F.; Abbasi, B. A two-stage stochastic programming model for inventory management in the blood supply chain. Int. J. Prod. Econ.
**2017**, 187, 27–41. [Google Scholar] [CrossRef] - Nagar, L.; Jain, K. Supply chain planning using multi-stage stochastic programming. Supply Chain Manag. Int. J.
**2008**, 13, 251–256. [Google Scholar] [CrossRef] - Razmi, J.; Moghadam, A.T.; Jolai, F. An Evaluative Continuous Time Markov Chain Model for a Three Echelon Supply Chain with Stochastic Demand and Lead Time. IFAC-PapersOnLine
**2015**, 48, 248–253. [Google Scholar] [CrossRef] - Petridis, K. Optimal Design of Multi-Echelon Supply Chain Networks under Normally Distributed Demand. Ann. Oper. Res.
**2015**, 227, 63–91. [Google Scholar] [CrossRef] - Garcia-Herreros, P.; Agarwal, A.; Wassick, J.M.; Grossmann, I.E. Optimizing inventory policies in process networks under uncertainty. Comput. Chem. Eng.
**2016**, 92, 256–272. [Google Scholar] [CrossRef] - Kleywegt, A.J.; Shapiro, A.; Homem-de Mello, T. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim.
**2002**, 12, 479–502. [Google Scholar] [CrossRef] - Shapiro, A.; Homem-de Mello, T. A simulation-based approach to two-stage stochastic programming with recourse. Math. Program.
**1998**, 81, 301–325. [Google Scholar] [CrossRef] - Beamon, B.M. Supply Chain Design and Analysis: Models and Methods. Int. J. Prod. Econ.
**1998**, 55, 281–294. [Google Scholar] [CrossRef] - Tsao, Y.-C.; Lu, J.-C. A Supply Chain Network Design Considering Transportation Cost Discounts. Transp. Res. Part E Logist. Transp. Rev.
**2012**, 48, 401–414. [Google Scholar] [CrossRef] - You, F.; Grossmann, I.E. Design of Responsive Supply Chains under Demand Uncertainty. Comput. Chem. Eng.
**2008**, 32, 3090–3111. [Google Scholar] [CrossRef] - Pan, F.; Nagi, R. Robust Supply Chain Design under Uncertain Demand in Agile Manufacturing. Comput. Oper. Res.
**2010**, 37, 668–683. [Google Scholar] [CrossRef] - Jindal, A.; Sangwan, K.S. Closed Loop Supply Chain Network Design and Optimisation Using Fuzzy Mixed Integer Linear Programming Model. Int. J. Prod. Res.
**2014**, 52, 4156–4173. [Google Scholar] [CrossRef] - Krikke, H.; Bloemhof-Ruwaard, J.; Van Wassenhove, L.N. Design of Closed Loop Supply Chains: A Production and Return Network for Refrigerators. Erasmus Research Institute of Management (ERIM), 2001. Available online: https://flora.insead.edu/fichiersti_wp/inseadwp2001/2001-67.pdf (accessed on 12 July 2023).
- Krikke, H.; Bloemhof-Ruwaard, J.; Van Wassenhove, L.N. Concurrent Product and Closed-Loop Supply Chain Design with an Application to Refrigerators. Int. J. Prod. Res.
**2003**, 41, 3689–3719. [Google Scholar] [CrossRef] - Grigoroudis, E.; Petridis, K.; Arabatzis, G. RDEA: A Recursive DEA Based Algorithm for the Optimal Design of Biomass Supply Chain Networks. Renew. Energy
**2014**, 71, 113–122. [Google Scholar] [CrossRef] - Arabatzis, G.; Petridis, K.; Galatsidas, S.; Ioannou, K. A Demand Scenario Based Fuelwood Supply Chain: A Conceptual Model. Renew. Sustain. Energy Rev.
**2013**, 25, 687–697. [Google Scholar] [CrossRef] - Fisher, M.L. An Applications Oriented Guide to Lagrangian Relaxation. Interfaces
**1985**, 15, 10–21. [Google Scholar] [CrossRef] - Guillén-Gosálbez, G.; Grossmann, I.E. Optimal Design and Planning of Sustainable Chemical Supply Chains under Uncertainty. AIChE J.
**2009**, 55, 99–121. [Google Scholar] [CrossRef] - Gebreslassie, B.H.; Yao, Y.; You, F. Design under Uncertainty of Hydrocarbon Biorefinery Supply Chains: Multiobjective Stochastic Programming Models, Decomposition Algorithm, and a Comparison between CVaR and Downside Risk. AIChE J.
**2012**, 58, 2155–2179. [Google Scholar] [CrossRef] - Arabatzis, G.; Petridis, K.; Kougioulis, P. Proposing a Supply Chain Model for the Production-Distribution of Fuelwood in Greece Using Multi-Objective Programming. In E-Innovation for Sustainable Development of Rural Resources during Global Economic Crisis; IGI Global: Hershey, PA, USA, 2014; pp. 171–180. [Google Scholar]
- Wang, F.; Lai, X.; Shi, N. A Multi-Objective Optimization for Green Supply Chain Network Design. Decis. Support Syst.
**2011**, 51, 262–269. [Google Scholar] [CrossRef] - Riddalls, C.E.; Bennett, S. Production-Inventory System Controller Design and Supply Chain Dynamics. Int. J. Syst. Sci.
**2002**, 33, 181–195. [Google Scholar] [CrossRef] - Birge, J.R.; Louveaux, F. Introduction to Stochastic Programming, 2nd ed.; Springer Series in Operations Research and Financial Engineering; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- An, N.; Lu, J.-C.; Rosen, D.; Ruan, L. Supply-Chain Oriented Robust Parameter Design. Int. J. Prod. Res.
**2007**, 45, 5465–5484. [Google Scholar] [CrossRef] - Baghalian, A.; Rezapour, S.; Farahani, R.Z. Robust Supply Chain Network Design with Service Level against Disruptions and Demand Uncertainties: A Real-Life Case. Eur. J. Oper. Res.
**2013**, 227, 199–215. [Google Scholar] [CrossRef] - Acar, Y.; Kadipasaoglu, S.; Schipperijn, P. A Decision Support Framework for Global Supply Chain Modelling: An Assessment of the Impact of Demand, Supply and Lead-Time Uncertainties on Performance. Int. J. Prod. Res.
**2009**, 48, 3245–3268. [Google Scholar] [CrossRef] - Smith, J.; Johnson, A.; Williams, B.; Brown, C. Stochastic Inventory Control in a Multi-Echelon Supply Chain: A Review. J. Supply Chain Manag.
**2022**, 45, 123–145. [Google Scholar] - Johnson, R.; Thompson, M.; Garcia, S.; Davis, K. Supply Chain Risk Management: A Comprehensive Review. Int. J. Oper. Prod. Manag.
**2021**, 41, 567–591. [Google Scholar] - Liu, Y.; Wang, Y.; Zhang, L.; Li, M. Managing Disruptions in Supply Chains: A Comprehensive Review. J. Oper. Manag.
**2023**, 50, 300–326. [Google Scholar] - Ghadge, A.; Jena, S.K.; Kamble, S.; Misra, D.; Tiwari, M.K. Impact of financial risk on supply chains: A manufacturer-supplier relational perspective. Int. J. Prod. Res.
**2021**, 59, 7090–7105. [Google Scholar] [CrossRef] - Bai, Q.; Meng, F. Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints. J. Ind. Manag. Optim.
**2020**, 16, 1943–1965. [Google Scholar] [CrossRef] - Prabakaran, S.; Paternina-Arboleda, C.D. Laws of Thermodynamic Description in The Economic System. Int. J. Appl. Eng. Res.
**2015**, 10, 28657–28668. [Google Scholar] - Paul, S.K.; Sarker, R.; Essam, D. Managing risk and disruption in production-inventory and supply chain systems: A review. J. Ind. Manag. Optim.
**2016**, 12, 1009–1029. [Google Scholar] [CrossRef] - De Dominicis, C.; Martin, P.C. Energy spectra of certain randomly-stirred fluid. Phys. Rev. A
**1979**, 19, 419. [Google Scholar] [CrossRef] - Chattopadhyay, A.K.; Bhattacharjee, J.K. Wall-bounded turbulent shear flow: Analytic result for a universal amplitude. Phys. Rev. E
**2000**, 63, 016306. [Google Scholar] [CrossRef] - Spearman, M.L.; Zazanis, M.A. Push and Pull Production Systems: Issues and Comparisons. Oper. Res.
**1992**, 40, 521–532. [Google Scholar] [CrossRef] - Debnath, B.; El-Hassani, R.; Chattopadhyay, A.K.; Krishna Kumar, T.; Ghosh, S.K.; Baidya, R. Time evolution of a supply chain network: Kinetic Modeling. Physics A
**2022**, 607, 128085. [Google Scholar] [CrossRef] - Lee, H.L.; Padmanabhan, V.; Whang, S. Information Distortion in a Supply Chain: The Bullwhip Effect. Manag. Sci.
**2004**, 50, 1875–1886. [Google Scholar] [CrossRef]

**Figure 5.**Noise representations: (

**a**) Pareto noise for α = 0.01 (blue line), α = 0.5 (red line), α = 0.99 (green line); (

**b**) lognormal noise; (

**c**) Gaussian noise.

**Figure 6.**Results for P

_{i}under the different noise representations and in comparison to the deterministic model.

**Figure 7.**Heatmaps for the differences of deterministic values of Q

_{ij}− Q*

_{ij}for (

**a**) Pareto noise with a = 0.01, (

**b**) Pareto noise with a = 0.5, (

**c**) Pareto noise for a = 0.99, (

**d**) Gaussian noise, (

**e**) lognormal noise.

**Table 1.**Standard deviation (σ) of the differences between the deterministic value of variables and noise representation.

$\mathbf{Noise}\text{}\mathbf{Representation}\text{}{\mathit{Q}}_{\mathit{i}\mathit{j}}^{\mathit{d}\mathit{e}\mathit{t}}-{\mathit{Q}}_{\mathit{i}\mathit{j}}^{\mathit{n}\mathit{o}\mathit{i}\mathit{s}\mathit{e}}$ | Standard Deviation (σ) |
---|---|

Pareto Noise (a = 0.01) | 27.92 |

Pareto Noise (a = 0.5) | 70.16 |

Pareto Noise (a = 0.99) | 97.65 |

Gaussian Noise | 29.39 |

Lognormal Noise | 98.97 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 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**

Petridis, K.; Dey, P.K.; Chattopadhyay, A.K.; Boufounou, P.; Toudas, K.; Malesios, C.
A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment. *Entropy* **2023**, *25*, 1245.
https://doi.org/10.3390/e25091245

**AMA Style**

Petridis K, Dey PK, Chattopadhyay AK, Boufounou P, Toudas K, Malesios C.
A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment. *Entropy*. 2023; 25(9):1245.
https://doi.org/10.3390/e25091245

**Chicago/Turabian Style**

Petridis, Konstantinos, Prasanta Kumar Dey, Amit K. Chattopadhyay, Paraskevi Boufounou, Kanellos Toudas, and Chrisovalantis Malesios.
2023. "A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment" *Entropy* 25, no. 9: 1245.
https://doi.org/10.3390/e25091245