Operational risk is the risk of loss because of ineffective or failed internal processes, people, systems, or external events, which can disrupt the flow of business operations. An inviolable aspect of a business organization is the distribution of supply lines, both on the input side of the business as well as in relation to its output deliverables, together with the supply chain management (SCM) of its overall throughput. Due to rapid economic globalization, the majority of operations, ranging from manufacturing to transportation sectors and from warehousing to the customer base, are conducted by supply chain contractors or third-party logistics (3PL) companies. Recent research in logistics developments predicts that, in the foreseeable future, approximately 80% of economic transactions will be based on services. Thus, the better the design of supply chain operations, the better the service level the customers will experience. Currently, for the majority of products transported and sold throughout the world, customers rarely subscribe to brand loyalty; thus, any stock clearance may result in a reduction in sales and future loss of income for firms ^[1][2][3][1,2,3].

Furthermore, entering the 21st century, the business environment is becoming more and more challenging because of the worldwide effort to meet the SDGs’ challenges [4]. Green supply chain management (GSCM) integrates environmental thinking into supply chain management, creating a sustainable supply chain [5]. As noted in the relevant literature (see, e.g., ^{[6][7][8][9][10][11][12][13][14]}[6,7,8,9,10,11,12,13,14]) companies are under immense pressure to adopt GSCM practices that are driven towards the environment by a combination of external factors (government rules and legislation; environmental concerns and regulation; social and environmental responsibility; customer awareness, pressure, and support; supplier pressure and willingness; global climate pressure) and internal factors (green image; global marketing; competitiveness; economic beliefs or cost reduction benefits; investor and shareholder pressure; employee motivation; health and safety issues; waste management issues) towards meeting SDGs, as they directly affect customer choice [15]. Furthermore, many studies are currently focusing on discussing the implementation of GSCM in different sectors of the economy and/or specific countries (see, e.g., ^{[16][17][18][19][20][21][22][23][24][25][26]}[16,17,18,19,20,21,22,23,24,25,26]).

More specifically, [51] proposed a nonlinear programming (NLP) model that integrated a complex multidimensional framework comprising the facility location and inventory allocation problem with cost discounts. A two-phase approximation approach was deployed as a solution to provide numerical results that could demonstrate the impact of different simulated data to the supply chain decisions and cost. ^[27][33] propose a multi-echelon supply chain model that includes suppliers, plants, and distribution centers and aims at minimizing the total cost of the supply chain. The proposed methodology involves sensitivity analysis to show that the customer demand parameter has the greatest impact on the optimal solution. ^[28][31] propose a deterministic model for the supply chain uncertainty in the demand. The suggested model assumes that returned items from the customers can be remanufactured at a fixed rate.

Choi et al. ^[29][27] study the supply chain scheduling and co-ordination problem comprising multiple suppliers, a single warehouse operator, a single manufacturer, and multiple retailers. Fattahi et al. [30] investigate the supply chain network design and planning for a multi-commodity and multi-layer network over a planning horizon with multiple periods, in which the demands of customer zones are considered price dependent through the development of a mixed-integer linear programming (MILP) model.

Similarly, in another development directed towards incorporation of systemic uncertainties as random fluctuations, ref. ^[31][41] used a mixed-integer linear programming model wherein both binary and continuous variables are considered with the objective of assigning uncertainty in the structure of the hierarchical variables, e.g., demand as deterministic uncertainty in their respective numbers, without explicit incorporation of statistical stochastic terms. The first are used for network representation, while the latter for facility capacity and flows of goods throughout the channels of the supply chain network [32]. Similar models have been proposed, considering the demand uncertainty and measuring the customers’ service level through the calculation of lead time and normally distributed demand (see, e.g., ^[33][34][46,52]). The formulation of an agile or flexible supply chain network with the use of a heuristic algorithm as a solution procedure has been also proposed by ^[35][53] as a means of incorporating certain non-deterministic fluctuations perpetrating changed functionalities in the supply chain.

Closed-loop supply chains (CLSC) are generally used to model the reusability and recycling of products (ICT, food, etc.) In a more recent work, ref. ^[36][54] proposed a fuzzy MILP model to capture the uncertainty in demand, cost, and other parameters. Similar modeling approaches have been proposed in the literature using mathematical programming techniques for the optimal closed-loop supply chain network design (CLSCND) ^[37][38][55,56].

Recent works focus mostly on biomass-based supply chain networks due to a global turn towards bioenergy production. In their work, ref. ^[39][57] proposed a data envelopment analysis (DEA) based algorithm for optimal biomass supply chain network design. An optimal design of a forest supply chain network has been proposed by ^[40][58]. In this work, the authors employed a Lagrangian relaxation algorithm ^[41][59] to design the fuel–wood supply chain, considering demand uncertainty. The optimal design of a biofuel supply chain network has been also examined using a Monte Carlo simulation approach to provide a sensitivity analysis for various parameters ^[42][35].

In another setting, the use of multiple objective functions may be seen as providing a more realistic approach to real-world problems. In such domains, multi-objective programming (MOP) models have been traditionally employed, including the optimal design of chemical supply chains ^[43][60], biofuel/biomass supply chains ^[44][45][28,61], in forest supply chains ^[46][62], and considering green supply chains with environmental factors ^[47][63].

The introduction of noise realization has been examined in many production–allocation systems (including the supply chain network design problem). The main modeling method for noise representation is optimal control. In these lines, ^[48][64] have proposed a multi-echelon control model to describe a production–allocation supply chain network. In their work, the authors assumed that noise corrupted demand and system delays. A popular approach is based on the value of stochastic solution (VSS) ^[49][65] to compare relative contributions between deterministic and stochastic amplitudes within the remit of the same model.

A game theoretical model is proposed by ^[50][66], where through a collaborative approach, a noise (read fluctuation) reduction scheme was propounded. Noise, in terms of uncertainty, has also been modeled through different demand and supply scenarios identifying disruptions to the production process ^[51][67]. A decision support system is proposed by ^[52][68], where the performance of service level or customer satisfaction was examined through a simulation study. Uncertainty has been modeled by adding noise to the demand parameter or by sampling from statistical distributions.

One recent article titled “Stochastic Inventory Control in a Multi-Echelon Supply Chain: A Review” ^[53][69] examines the existing literature on stochastic inventory control in multi-echelon supply chains. It delves into various mathematical models, optimization techniques, and decision-making approaches employed to manage uncertainty in inventory levels across different stages of the supply chain. The review emphasizes the need for robust inventory policies and coordination mechanisms to mitigate the impact of stochasticity.

Johnson et al. ^[54][70] provide a comprehensive analysis of supply chain risk management, encompassing stochastic events. It discusses the identification, assessment, and mitigation of risks associated with stochasticity, such as demand volatility, supplier disruptions, and natural disasters. The article emphasizes the importance of building resilient supply chains through effective risk management strategies.

Liu et al. ^[55][71] focus on managing disruptions caused by stochastic events in supply chains. They explore strategies such as redundancy, flexibility, and collaboration that can help to mitigate the impact of disruptions and improve supply chain robustness. The article also discusses the role of technology, such as real-time monitoring and predictive analytics, in enhancing supply chain resilience.

In the realm of supply chain management, various studies have been conducted to explore different aspects and challenges. Ref. ^[56][72] conducted a study to analyze the impact of financial risk on the manufacturer–supplier relationship in a two-echelon supply chain. They developed a multi-objective decision model for supplier selection and order allocation, aiming to maximize the manufacturer’s total profit while minimizing the financial risk faced by selected suppliers. The study considered foreign exchange risk, default risk, market risk, and price fluctuation risk, and explored three case scenarios to understand the behavior of suppliers in response to different financial risks, both in the short and long term.

Building on the concept of risk aversion in supply chains, ref. ^[57][73] examine a two-echelon supply chain with two competing manufacturers and one retailer. One manufacturer adopted sustainable technology to reduce carbon emissions under cap-and-trade regulations, while the other followed traditional business practices. The study considered two configurations involving risk-neutral and risk-averse agents and analyzed operational decisions using a retailer–leader game optimization approach under the mean variance framework. The results showed that risk-averse agents benefited from low-scale risk aversion, and low carbon emissions were attainable when the underlying manufacturer had small or moderate risk aversion.

In a different approach, ref. ^[58][74] explore the application of thermodynamics in describing the behavior of economic and financial systems. They discuss the first and second laws of thermodynamics and construct a mathematical model for a constant price process. The focus is on examining the dynamics of economic processes using thermodynamics principles. However, more specific findings and conclusions from the research were not provided in the summary.

Additionally, ref. ^[59][75] conduct a literature review on risk and disruption management in production–inventory and supply chain systems. They reviewed works that considered real-life risk factors, such as imperfect production processes, disruptions in production, supply, demand, and transportation. The review emphasized the mathematical models and solution approaches used to address these problems, both in hypothetical and real-world scenarios. The review concluded by discussing future research directions in this area.

Furthermore, ref. ^[60][76] proposed a nonlinear programming (NLP) model, providing an integrating framework for the facility location and inventory allocation problems with cost discounts. They deployed a two-phase approximation approach as a solution to provide numerical results that demonstrate the impact of different simulated data on supply chain decisions and cost.

More realistic, explicit incorporations of multiplicative noise routines have rarely come across in the relevant literature. This is partly due to computational difficulty, and to the minimalistic nature of most problems considered.

This study, emphasizes the significance of subjective and objective uncertainty in achieving optimized decisions by incorporating stochastic fluctuations into the supply chain structure. We focus on a processing production plant as a model for a chain of operations and supply chain actions. Through stochastic optimization, it is demonstrated that the plant producer can benefit from improved financial outcomes by setting higher sale prices while simultaneously lowering optimized production costs. This can be accomplished by selectively choosing producers whose production cost probability density function follows a Pareto distribution.