By definition, a short food supply chain (SFSC) is a system that emphasizes local production, transparency, and sustainability while minimizing the number of intermediaries between producers and consumers along the supply chain. Based on this, it is proposed to strengthen SFSCs by establishing cooperatives where producers focus on adding value to their products, transitioning cocoa through its different stages to become producers and manufacturers, and ultimately reaching end customers with branded finished products. However, several persistent issues in the field continue to undermine these efforts.

Consequently, the sustainability and competitiveness of the CSC have become central issues in agricultural sustainable research, encompassing environmental management, labor conditions, and fair trade practices. In general, a commercial chocolate bar contains only 25%−35% cocoa mass; cocoa in semisweet chocolate bars range from 45 to 99%, while white chocolate bars only contain cocoa butter, milk, and sugar (Duana-Ávila et al., 2023). In addition, the low application of fertilizers and improper management of shade conditions within plantations limit productivity and quality. Currently, researchers are editing the cocoa genome to produce disease-resistant plants (Fister et al., 2018). As global demand for high-quality cocoa continues to rise, Mexico's cocoa-producing regions will remain critical players in the international market, especially if they continue to prioritize sustainability and quality.

The decision-making problem of the producer and retailer under time delay effects in a two-echelon supply chain was examined by Li et al. (2024). Time delays are intrinsic to many stages of the AFSC and can significantly affect decision-making, efficiency, and overall system performance. In the context of modern supply chains, a dynamic system is one that is continually shaped by various fluctuating factors, such as production rates, consumer demand shifts, transportation delays, and external influences including market conditions, economic trends, and unforeseen weather events. These variables interact in real time. Consequently, businesses must remain agile and responsive, adjusting their strategies and operations as needed to sustain efficiency and mitigate risks.

In the context of cocoa production, for example, delays can arise in areas such as crop growth, harvesting schedules, processing, and market feedback. Yu et al. (2020) addressed optimal control for multiple delays in supply chains. By accounting for time delays, stakeholders can make more informed decisions, improve efficiency, reduce costs, and enhance the resilience of the supply chain. As the global agri-food sector faces growing challenges, such as climate change, market volatility, and shifting consumer preferences, the role of mathematical modeling in managing time delays has become more critical for achieving sustainable and efficient supply chains.

Dynamic pricing models have gained relevance as tools to address volatility and improve value for cocoa farmers. These models, however, face several key limitations: (1) market structure constraints; (2) institutional and socioeconomic issues; (3) environmental concerns; (4) mathematical modeling limitations, such as biological delays in cocoa production (trees take 3–5 years to mature); and (5) positive and negative externalities in production. Therefore, mathematical modeling has emerged as a critical approach for analyzing and optimizing these systems, enabling better decision-making under uncertainty and enhancing supply chain resilience. Techniques such as ordinary differential equations (ODEs) (Davizón et al., 2014), system dynamics, and optimal control theory have been applied to model inventory levels, production rates, and pricing strategies in DSCs (Mehrbanfar et al., 2020). Therefore, dynamic pricing in the CSC faces both practical and structural limitations due to the characteristics of agricultural commodity markets and the socioeconomic context of low-volume cocoa production.

Supply chain ethnography refers to a qualitative analysis, from a sociological perspective, of the practices of the social and cultural environment where the processes of planting, harvesting, and production are carried out in agricultural fields by workers, farmers, and traders. Therefore, it is important to understand the ethnography of cocoa producers in Mexico from a qualitative perspective for supporting industrialization efforts aimed at adding and creating value within the CSC. Furthermore, it is essential to highlight the ethnographic Key Performance Indicators (KPIs) as defined by Rasmussen et al. (2020): the behavioral dimension (socio-cultural factors), the structural dimension (interactions among supply chain participants), and the contextual dimension (market and environmental variables). The working conditions of cocoa producers can be improved through ethnographic analysis, which examines the social, cultural, and economic factors affecting workers. Through the application of surveys and interviews, this approach can identify labor inequalities, leverage practices, and informal contractual trade-offs along the CSC.

Sustainability is the ability to meet present needs without limiting future generations, which is achieved by balancing environmental, social, and economic priorities (Giddings et al., 2010). In addition, an AFSC encompasses the “farm-to-fork” activities from production to marketing. In this context, sustainability involves managing these stages to reduce environmental impact, promote social equity, and ensure economic viability (De Silva et al., 2023). Therefore, the CSC is a complex system where sustainability addresses trade-offs, which requires coordination among farmers, enterprises, and government, adopting a systemic perspective. This perspective incorporates life cycle assessment [a method to validate the environmental impact of a product or service across its entire life cycle (from raw material to end customer)], socioecological analysis (the study of interactions between human societies and the environment, aimed at understanding and managing the sustainability of social and ecological systems), and ethnographic studies (analysis of local cocoa supply chain behaviors, as previously discussed) (Yuan et al., 2024).

A two-echelon dynamic supply chain was analyzed by Tshinangi et al. (2025), with similar approaches applied to two-echelon cold supply chains (Acerce and Denizhan, 2025; Sebatjane, 2025), using the following ordinary differential equation, which considers both producers and consumers in a DSC:

Equation 1 represents a differential equation that relates the inventory level (I), production rate S(t), and demand rate d(t). A proper lead time (θ) is considered because CSCs are related to time delay approaches for perishable inventory systems.

In general, the cocoa dynamic supply chain structure can be expressed as:

To present a model reduction for the system, in general, we have the following equation:

Applying the Taylor series to the time delay term, we have (Insperger, 2015):

By applying Equations 4–6, we have:

Assuming an equilibrium condition where S₁(t) = d₂(t ).

The final system is as follows:

By adding Equations 9, 10, we obtain:

In general, we have:

Assuming a decreasing production rate of the form S₁(t) = −kt, and applying in Equation 12 for k > 0, we obtain:

Equation 13 applies the derivative of both sides of the equation:

Equation 14 implies that the rate of change (the sum of variations in both inventories) is constant and equal to the net production with lead times (time delay). This reflects a steady-state scenario where the supply chain is in equilibrium. Under this condition, the production rate, adjusted for lead times, matches the demand and consumption rate, ensuring that inventory levels fluctuate predictably. After analyzing the two-tier cocoa supply chain, further applications for improving the efficiency, sustainability, and equity of the entire DSC are summarized in Table 2.

In previous studies, the application of OC in DSCs (Kappelman and Sinha, 2021) has been addressed. For the system dynamics described in Equations 9, 10, we have:

Therefore, the optimal control (OC) problem for this system is:

s.t.

With the following initial and final conditions in a time horizon of T = 100 days, MT refers to metric tons. Assuming an artisanal production rate in rural Mexico, the average inventory level is 200 kilograms (Kgs).

Applying PMP to solve the OC problem in Equation 17, we define the Hamiltonian as:

Here, λ₁ and λ₂ are Lagrange multipliers for the costates.

Applying the Hamilton equations, we have:

From Equation 19, we have:

In addition, the second condition follows:

Therefore, we have:

By integrating Equation 22 with respect to time, we have:

By substituting u(t) into Equation 20, we have:

Considering that:

Finally, we have:

Applying the initial condition I₂(0) = 200, where W = 200.

In a similar manner, we can calculate I₁(t) to achieve:

After integrating Equation 27 with respect to time and applying the initial condition I₁(0) = 100, we obtain:

We can apply the final conditions. After applying algebra, we obtain:

Adding Equations 29, 30, we have:

By substituting λ1=6θ2 into Equation 30, we have:

Finally, we have:

Thus, the optimal solution and the evolution of the inventory levels depend on time delay θ.

The stability analysis of supply chains as dynamical systems relies on distinct mathematical approaches with proper system architecture. The Routh–Hurwitz stability criterion provides a necessary and sufficient condition for the stability of linear, continuous-time, single-input and single-output (SISO) systems. However, to address the inherent complexities of non-linear supply chain dynamics, Lyapunov's direct and indirect methods are employed, offering a generalized framework for evaluating equilibrium stability without explicitly solving the system's differential equations. Within this framework, to evaluate system stability under time delays and quantify inventory fluctuations, we propose a model that accounts for losses during both the production and consumption phases of the supply chain.

Theorem 1.

The system described in Equations 35, 36 is asymptotically stable if the conditions α₁ > 0 and α₂ > 0 are satisfied.

Proof.

We have a system of the form in the Laplace transform:

Presenting Equations 35, 36 in transfer function form, we have:

For the transfer function Equation 39, the system is asymptotically stable if α₁ > 0, which is based that the pole in the left is on the complex plane s = −α₁. In addition, for the transfer function Equation 40, the system is asymptotically stable if α₂ > 0, which is based that the pole in the left complex plane is on s = −α₂.

To present a sensitivity analysis of the system dynamics of the DSC, we consider the following:

Assuming the operating point:

The partial derivatives evaluated at the operating point: u=1,000, u˙o=-100, θ=20, and k= 100

Partial derivatives with respect to k:

Partial derivatives with respect to u(t):

Partial derivatives with respect to u˙: