Enhancing emergency response capabilities in data center engineering supply chains through government subsidies

Abstract

Emergency capability plays a key role in maintaining the stability of data center engineering supply chains. Government subsidies encourage cooperation among supply chain enterprises by offering financial support, aiding in the improvement of emergency preparedness. In this study, the government subsidy rate is treated as an exogenous variable, and a differential game model involving resource suppliers and service operators is constructed. By comparing different cooperation modes, the study investigates the dynamic mechanisms for enhancing emergency capabilities. The conclusions of this study are contingent upon the assumptions of the proposed model. The findings indicate that collaborative cooperation is the most effective approach for improving emergency capabilities and benefits, as it facilitates resource and information sharing. Resource suppliers align their strategies with the overall supply chain, while service operators need to carefully time contracts to optimize cooperation and maximize benefits. Furthermore, government subsidies and the cost coefficient of emergency efforts exert a stronger influence on emergency capabilities than the level of emergency data utilization. To strengthen emergency capabilities, priority should be placed on increasing government subsidies and optimizing emergency effort costs, which can alleviate financial burdens on enterprises and enhance their emergency responsiveness.

Introduction

In the era of big data, data centers have become indispensable, providing the computing infrastructure needed to transform raw data into valuable information (Lee, 2016). Data center engineering establishes an information infrastructure integrating data storage and computational capabilities to address escalating digitalization demands. The report of the 20th National Congress of the Communist Party of China calls for continued forward-looking planning in developing new infrastructures, such as data centers and computing power, with a focus on improving the resilience and security of industrial and supply chains (Jin, 2024). As the data era advances and digital technologies accelerate, supply chains are increasingly shifting toward global network operations. The growing frequency of unexpected public events has significantly changed the production and operational environments of supply chains, introducing vulnerabilities and risks not commonly seen in traditional scenarios (Zhu et al., 2020). Contemporary supply chains exhibit heightened vulnerability to external disruptions compared to traditional systems (Wang and Chen, 2019). As a key component of industries such as cloud computing, big data, and the Internet of Things, the data center industry plays a vital role. Given dynamic environmental pressures and global uncertainties, there is a pressing need for a robust supply chain to support data center engineering. While the development of data centers encompasses multiple phases, including pre-construction, construction, and post-construction, this study focuses on the construction phase due to its unique challenges and research gaps. The construction phase is a critical period where the foundational infrastructure of data centers is established, involving complex coordination among stakeholders such as equipment suppliers, construction firms, and government entities. This phase is particularly vulnerable to disruptions, such as supply chain delays and environmental uncertainties, which can significantly impact project timelines and costs. By addressing these challenges, our research aims to enhance the resilience and emergency capabilities of Data Center Engineering Supply Chains. Although the pre-construction phase and the post-construction phase are equally important, they have been extensively studied in existing literature. In contrast, the construction phase has received relatively less attention, despite its pivotal role in ensuring the successful deployment of data centers. Therefore, our study fills this research gap by providing targeted insights into the construction phase, while acknowledging the need for future research to explore the interconnectedness of all phases. According to data from the Uptime Institute, the outlook for supply chains in the data center industry remains challenging. The COVID-19 pandemic has significantly disrupted global supply chains, causing delays in the delivery of critical data center equipment, such as servers and cooling systems, which has resulted in project delays and increased costs. Geopolitical tensions have also affected the supply of semiconductors, a key component of data center infrastructure, adding further uncertainty to the industry. Unlike natural disasters or pandemics, the complexity and unpredictability of geopolitical events make modeling and emergency planning more difficult, as any conflict could have widespread implications for the global IT industry. The inherent vulnerability of undersea cable networks exacerbates these systemic risks. These cables carry the majority of intercontinental data traffic globally, but their length and location make them prone to damage from natural or human causes. For instance, the 2006 Luzon Strait earthquake caused multiple cable breaks, leading to widespread internet outages across the Asia-Pacific region and underscoring the security risks associated with undersea cables. Protecting these critical infrastructures presents significant challenges, particularly in international waters, further amplifying the fragility of supply chains.

The emergency preparedness of Data Center Engineering Supply Chains (DCESC) has emerged as a critical research focus, with both government and academia acknowledging the need to enhance it. Extant literature consistently highlights the pivotal role of public-private partnerships in emergency management systems. In emergency organizational systems, improving emergency capabilities requires collaboration among supply chain node enterprises and government support. However, there is no consensus on the most effective organizational cooperation methods in emergency management, which directly impacts the speed of emergency response and the effects of emergencies on projects. As a result, effective coordination decisions are necessary to determine cooperative strategies within organizations. Differential games can effectively model decision interactions among participants in complex environments, particularly under conditions of uncertainty and bounded rationality, enabling optimal responses through dynamic strategy adjustments. However, without profit incentives, enterprises are less likely to actively engage in risk mitigation efforts. Subsidies function as a signaling device. In a business environment marred by information asymmetry, government subsidies play a pivotal role in setting up a signaling mechanism among enterprises. This mechanism facilitates the transformation of operational challenges into measurable performance improvements. (Kleer, 2010) Government subsidies constitute a fundamental policy instrument for enhancing DCESC resilience, as evidenced by their widespread implementation and positive outcomes across various regions. For example, in 2022, the US government introduced the Inflation Reduction Act, which provides tax credits and subsidies to private companies investing in renewable energy and energy-efficient infrastructure, including data centers. Early reports indicate that this policy has encouraged significant investments in sustainable data center infrastructure, improving project timelines and reducing vulnerability to supply chain disruptions. Similarly, the European Union’s Recovery and Resilience Facility, launched in 2021, allocates funds to member states to support digital infrastructure projects, including data centers, with a focus on enhancing their emergency preparedness and sustainability.

The Chinese government, through the National Development and Reform Commission and Ministry of Industry and Information Technology, launched the “East Data West Computing” project in 2022, offering financial incentives such as subsidies and tax reductions to enterprises developing data centers in western regions. This initiative has not only optimized the distribution of computing resources but also strengthened the emergency preparedness of data center projects through targeted subsidies. Additionally, local governments in cities like Beijing and Shanghai have introduced green data center subsidies since 2023, encouraging companies to adopt sustainable practices and improve their emergency preparedness. These examples highlight the importance of government subsidies in addressing the vulnerabilities of DCESC during the construction phase, particularly in the context of increasing global uncertainties and environmental challenges. By integrating these insights, this research contributes to the growing body of literature on the practical benefits of government subsidies in enhancing the resilience and emergency capabilities of critical infrastructure projects.

Current research on DCESC primarily focuses on the operational phase after data centers are deployed, emphasizing technical aspects such as network information security maintenance, renewable energy reliability, and low-energy-consumption studies (Peura and Bunn, 2021). In recent years, the number of data center projects under construction has steadily increased, with over 80% requiring reconstruction based on the latest technologies. This has led to a growing demand for basic equipment. Against the backdrop of integrating new and old technologies, DCESC faces numerous potential risks. Research targeting this phase remains relatively limited, and existing literature has paid little attention to it. This paper focuses on DCESC during the construction phase. Through synthesizing supply chain theory with data center industrial ecosystem frameworks, we define DCESC as a supply chain network formed by multiple node enterprises jointly participating in and cooperating on the construction of data centers. This supply chain encompasses the entire process, from data center infrastructure construction to equipment resource supply, including suppliers providing equipment, technology, and building materials, as well as service operators responsible for designing, constructing, and operating data centers. Its core mission is to ensure the efficient and stable supply of infrastructure, technical equipment, and resources required for data center projects during the construction phase.

In the era of digitalization, data centers serve as critical hubs for information storage and interaction, making the stability and emergency response capabilities of their construction projects vital. This paper innovatively constructs a unified framework that integrates both the government and enterprises related to the Data Center Engineering Supply Chain into the research scope of the construction phase, with government subsidies as a central contextual factor. This framework enables a thorough analysis of the complex relationships among these stakeholders. Building on this foundation, the paper develops an inter-enterprise differential game model to explore optimal strategies for enhancing the emergency response capabilities of DCESC. To address gaps in existing research, this paper focuses on the following key questions:

• 1. How can the resilience and emergency response capabilities of DCESC be effectively improved during the construction phase of data center engineering?

• 2. What role do government subsidies play in motivating enterprises to actively engage in risk mitigation and emergency preparedness?

• 3. How can differential game models be applied to optimize decision-making among DCESC stakeholders in dynamic and uncertain environments?

The structure of this paper is organized as follows: Section 2 systematically reviews the relevant literature in the fields of data center supply chains, government subsidies, and emergency management, clarifying the current research status and developmental context. Section 3 elaborates on the theoretical framework and research methodology, with a focus on the differential game model. Section 4 provides a detailed analysis and comparison of the model results. Finally, Section 5 summarizes the research findings, highlights the core insights, and proposes constructive suggestions for future research directions, aiming to contribute to the sustainable development of this field.

Literature review

DCESC and supply chain emergency capability

President Xi has underscored that global economic digital transformation represents an inevitable trend, explicitly advocating for establishing a national integrated big data center system encompassing multiple national hub nodes and data center clusters (Jin, 2024). The further advancement of initiatives like the “Eastern Data and Western Computing” project has propelled data center infrastructure construction into a phase of rapid development. The increasing scale of data center projects amplifies not only technical and logistical complexity but also exacerbates supply chain vulnerabilities, including critical equipment shortages and prolonged lead times (Robb, 2024). Delays caused by emergencies directly impact the manufacturing, transportation, and delivery of basic hardware and chip products related to the data center engineering supply chain. For instance, a data center in Wuhan planned to add new cloud hosts and bandwidth resources to address an emergency, but basic equipment, personnel, and resources could not be mobilized in time, leading to a temporary supply interruption. This demonstrates that supply shortages or disruptions can destabilize the entire supply chain. These uncertainties represent the primary risk factors affecting DCESC security. The tension between limited emergency capabilities and imminent risks poses a potential threat to the progress of data center construction projects.

Data center resources are characterized by high asset specificity and scarcity (Serror et al., 2021), with limited substitutability and frequent single-source dependencies. Over time, DCESCs have become increasingly distributed across broader geographical areas. This geographical dispersion exposes supply chains to heterogeneous disruption risks, increasing the probability of low-probability but high-consequence disruption events. Additionally, sudden risks caused by disasters can propagate along the supply chain, with adverse effects potentially extending to secondary and tertiary suppliers. As a result, enhancing the emergency capability of the supply chain has become an urgent priority.

Currently, scholars focus on evaluating emergency capabilities and identifying influencing factors. Emergency capability enhancement follows an incremental trajectory, with empirical evidence demonstrating digital transformation’s catalytic effect on organizational responsiveness to disruptions (Cong et al., 2024). Some researchers propose increasing overall supply chain profits by reducing emergency costs or introducing alternative suppliers to achieve emergency management goals. These measures help address data centers’ urgent needs effectively and restore supply chain stability quickly after emergencies. Proactive emergency management during disruptions has been shown to attenuate operational impacts on both supply chain resilience and robustness indices (Shen and Sun, 2023). Crises are often accompanied by uncertainties, making situational awareness and decision support critical for enhancing emergency capability. Emergency response capability assessments are considered essential for improving preemptive control in emergency management. Liao et al. (2018) proposed a hesitant fuzzy linguistic preference utility for selecting the best emergency rescue plan, highlighting that emergency response assessments can also help managers improve their ability to implement optimal plans. Similar to traditional supply chains, data center-related enterprises can enhance emergency capabilities by considering the varied impacts of risks on supply chain members.

The evidence conclusively establishes that multi-stakeholder collaboration constitutes the foundational paradigm for achieving enhanced emergency responsiveness and systemic resilience. Differential game models and cost-sharing contracts offer effective frameworks for understanding and optimizing such collaborations, particularly in the context of government-enterprise partnerships.

Collaboration among emergency entities

During supply chain coordination, cooperation among member enterprises is typically conducive to achieving optimal objectives (Yu et al., 2024). Different stakeholder groups bear distinct social obligations and professional responsibilities while enjoying corresponding rights and interests (Fang et al., 2020). Enterprises are viewed as part of the supply chain, interconnected in complex environments, generating supply chain effects (Blaettchen et al., 2025). It is essential to address production coordination issues among member enterprises and utilize specific intervention methods, such as policies, to coordinate the market. This approach can resolve material supply interruptions, labor shortages, and cost pressures, thereby improving collaboration between upstream and downstream enterprises.

Simatupang et al. (2002) suggested that collaboration among supply chain enterprises can be used to evaluate the risk emergency capabilities and resilience of the supply chain. Due to the mutual influences among entities in complex systems, collaboration and cooperation among supply chain-related entities become critical when facing risks (Zhao and Cheng, 2018). Some scholars have employed differential games to explore the interactive decision-making behaviors among emergency decision entities. Liu and Zhao (2020) utilized a cooperative game model to study the cooperation mode of forming alliances between the government and suppliers.

Given the persistent risks posed by unforeseen public emergencies, establishing regenerative collaborative alliances has been identified as a critical success factor for ensuring uninterrupted implementation of large-scale infrastructure projects. Zhao et al. (2024) explored the enhancement of emergency capabilities in Industrial Internet Engineering Supply Chains, comparing the impacts of random factors and horizontal, vertical, and collaborative cooperation scenarios. Yan et al. (2021) studied the game relationships and interaction mechanisms among emergency entities with bounded rationality. In differential game research, introducing cost-sharing contracts is viewed by some scholars as a means to achieve objectives. Fan et al. (2022) proposed that, alongside government subsidies, enterprises can engage in cost-sharing and introduce a revenue-sharing mechanism to incentivize cooperation. In subsequent research, they incorporated bilateral cost-sharing contracts and found that the improved contract more effectively incentivizes enterprises (Fan et al., 2022).

Collaboration among supply chain entities is crucial for enhancing emergency capabilities and resilience. Differential game models and cost-sharing contracts offer effective frameworks for understanding and optimizing such collaborations, particularly in the context of government-enterprise partnerships.

Government subsidy policies

Government subsidies, as policy instruments, can provide financial support to alleviate the pressures faced by enterprises, reducing their need to acquire resources externally (Yu et al., 2024). Targeted government subsidies can enhance enterprises’ willingness to collaborate. Awareness of government subsidies among upstream and downstream entities in the supply chain can significantly incentivize member collaboration, especially when these entities already intend to establish cooperative relationships. The government can thus enhance the competitiveness and risk resilience of the supply chain. Chen et al. (2019) established a government-led, three-tier supply chain model, studying how two different subsidies influence the cooperative enhancement of innovation capabilities between producers and sellers. Liang and Wei (2020) designed a benefit-sharing and cost-sharing mechanism, discovering that dual government subsidies effectively enhance corporate R&D capabilities, increase sales volume, and raise prices, thereby achieving supply chain coordination. Tao et al. (2022) pointed out that government subsidies can promote market consumption, revitalize the market, and drive the consumption of low-demand goods within the supply chain, fostering its development. Ivanov and Das (2020) discussed the impact of government subsidy strategies on supply chain disruptions during emergencies, considering demand and recovery capabilities, and concluded that subsidies can indirectly enhance supply chain emergency capabilities.

Effective information exchange and knowledge sharing between governmental agencies and enterprises have been empirically demonstrated to significantly improve emergency preparedness (Liu and Zhao, 2020). When government subsidies are coupled with corporate social responsibility, they can improve the overall profitability of the supply chain. For enterprises, government subsidies can optimize internal cost structures related to emergencies. Targeted subsidy policies serve as an effective policy instrument for governments to shape enterprise emergency response strategies and bolster operational resilience.

Government subsidies play a critical role in enhancing supply chain emergency capabilities by incentivizing enterprise collaboration and optimizing cost structures. However, further research is needed to explore the dynamic impacts of subsidies on supply chain resilience, particularly during the construction phase of data center projects.

Currently, research on data center management typically focuses on green energy-saving technologies and network information security. According to the 2023 China Data Center White Paper, the total investment in large-scale infrastructure energy-saving renovations for data centers is expected to reach 7–8 billion yuan between 2023 and 2025. Such substantial capital investment also brings increasing risks. However, few studies address the emergency capabilities of supply chains during the construction phase of data center projects. Most existing research concentrates on evaluating emergency capabilities during the operational phase, primarily within static frameworks, and lacks dynamic quantitative analysis. In supply chain emergency management research, studies on government subsidy measures are relatively scarce. Although Zhao et al. (2024) incorporated emergency capabilities of the industrial internet into a dynamic framework, they primarily considered the impacts of cooperation methods and random factors, without examining the government’s role as a third party in influencing supply chain emergency capabilities. Therefore, this paper adopts the perspective of government subsidies, focusing on different cooperation mechanisms among supply chain node enterprises. By employing a differential game model, we discuss the effects of various factors on enhancing emergency capabilities and validate our findings through numerical simulation methods.

Building upon the identified research gaps, this study systematically investigates the efficacy of government subsidy mechanisms in strengthening emergency capabilities within Data Center Engineering Supply Chains (DCESC), specifically during the construction phase. We investigate the optimal cooperation mechanisms among supply chain node enterprises under government subsidy policies and apply dynamic quantitative analysis to evaluate the impact of various factors on supply chain emergency capabilities. By addressing these issues, our research fills the gap in the existing literature by providing a comprehensive framework that integrates government subsidies, enterprise cooperation mechanisms, and dynamic quantitative analysis. This approach not only enhances the understanding of supply chain emergency capabilities but also offers practical insights for policymakers and industry practitioners.

Model construction

Problem description

This section focuses on cooperative emergency decision-making among DCESC enterprises under government subsidies, with the goal of maximizing the enhancement of DCESC’s emergency capabilities. The emergency system involves multiple entities. Since this paper addresses the infrastructure construction and optimization phase of data centers, the model excludes downstream service experience users and focuses solely on resource suppliers and service operators under government subsidies. Both resource suppliers and service operators aim to enhance DCESC’s emergency capabilities and reduce emergency costs. The purpose of government subsidies is to increase enterprises’ motivation for cooperative emergency responses and assist them in improving emergency capabilities to maintain DCESC’s stability.

The subjects examined in this paper are rational DCESC resource supplier enterprise S and service operator enterprise P, both of which possess symmetric information. The two enterprises mutually constrain and cooperate with each other. When introducing cost-sharing contracts, the service operator is considered the leading enterprise because, compared to dispersed resource suppliers, service operators have more centralized emergency resources and higher emergency awareness. Resource suppliers (enterprise S) include equipment suppliers and engineering service providers, primarily responsible for utilizing advanced technologies to build data center projects, providing intelligent equipment, and offering standardized components for data center core supporting facilities, servers, network equipment, and related renovations. Operators (enterprise P) are primarily responsible for demand transmission and planning. Through close cooperation with resource suppliers, they clarify the equipment and network infrastructure required for data center engineering and establish compliance standards for suppliers’ products and services to ensure they meet the specific operational needs of the project, thus ensuring the overall operational reliability of DCESC. Considering government subsidies, the cooperation model between resource suppliers and IDC service operators is crucial for enhancing DCESC’s emergency capabilities.

To illustrate the cooperative emergency decision-making process among DCESC enterprises under government subsidies, we present a conceptual model in Fig. 1. The model highlights the interactions among the government, infrastructure suppliers, and service operators, as well as the three cooperation strategies that can be adopted to enhance emergency capabilities. This framework provides a clear understanding of how government subsidies and cooperation strategies contribute to improving DCESC’s emergency capabilities during the construction phase of data center projects.

Fig. 1

Analysis of DCESC emergency capability enhancement under government subsidies.

Basic assumptions

Resource suppliers (S) and service operators (P) are rational entities. In the Nash non-cooperative game, the game mode introducing cost-sharing contracts, and the collaborative cooperative game, we discuss the emergency effort costs of resource suppliers and service operators, the optimal trajectory of emergency capabilities, and the final emergency benefits. The enhancement of emergency capabilities depends on the joint efforts of both entities. To incentivize DCESC enterprises to actively cooperate in enhancing emergency capabilities, government subsidies are treated as exogenous variables, with the government providing policy subsidies to both enterprises. Therefore, the following assumptions are proposed:

Assumption 1: In the context of unexpected public events, ${E}_{S}$ and ${E}_{P}$ represent the efforts to enhance emergency capabilities by data center resource suppliers and service operators, respectively (${E}_{S}$, ${E}_{P}\ge 0$). The cost for both parties to enhance emergency capabilities is positively correlated with their level of effort. Combining Chen’s study on setting emergency costs based on emergency capability level incentives (Chen et al. 2023), the cost functions for emergency capability enhancement of data center resource suppliers and operators are set as follows. Here, μ_S and μ_P are the effort cost coefficients for emergency capability enhancement by data center resource suppliers and operators. As an important part of new infrastructure, data centers—compared to other infrastructure projects—can make timely and scientific emergency responses based on scientific data resources and information technology through multiple data engineering systems, such as multi-source data integration and aggregation. This provides emergency scientific data for the prevention and control of sudden disasters. ${\eta }_{S}$ and ${\eta }_{P}$ represent the emergency data utilization capabilities of data center enterprises. High levels of ${\eta }_{S}$ and ${\eta }_{P}$ ensure that data centers can provide necessary data support when facing unexpected events, thereby reducing emergency costs. ${C}_{1}\left(t\right)$ and ${C}_{2}\left(t\right)$ represent the effort costs of data center resource suppliers and operators at time t during the process of enhancing emergency capabilities.

$${C}_{1}\left(t\right)=\frac{{\mu }_{S}{E}_{S}^{2}}{2\left(1+{\eta }_{S}\right)}$$

(1)

$${C}_{2}\left(t\right)=\frac{{\mu }_{P}{E}_{P}^{2}}{2\left(1+{\eta }_{P}\right)}$$

(2)

Assumption 2: When the game subjects are DCESC resource suppliers and service operators, the effort levels of both entities should positively impact the enhancement of emergency capabilities. At time $t\in \left[0,+\propto \right]$, the dynamic change of DCESC’s emergency capability is expressed as $Q\left(t\right)$. Referring to Zhao and Li et al.‘s expression of emergency capability under unexpected events in the industrial internet (Zhao et al., 2024), emergency capability has timeliness and naturally decays over time. The expression of $Q\left(t\right)$ is as follows:

$${Q}^{{\prime} }\left(t\right)=\frac{{dQ}\left(t\right)}{{dt}}={\alpha }_{1}{E}_{S}+{\alpha }_{2}{E}_{P}-\gamma Q\left(t\right)$$

(3)

$$Q\left(0\right)={Q}_{0}\left({Q}_{0}\ge 0\right)$$

(4)

Where ${\alpha }_{1}$ and ${\alpha }_{2}$ represent the influence coefficients of the emergency efforts of DCESC resource suppliers and service operators on emergency capabilities (${\alpha }_{1}\, > \,0,{\alpha }_{2}\, > \,0$). sγ represents the decay coefficient of DCESC’s emergency capability over time when it is not iteratively updated with the alternation of new and old ($\gamma \,> \,0$), ${Q}_{0}$ represents the initial emergency capability of DCESC at time zero.

Assumption 3: The efforts made by DCESC resource suppliers and service operators during emergencies are directly related to the stable development of data center infrastructure projects. Additionally, the improvement of the supply chain’s emergency response level brings gains to the project. Considering these factors, it is assumed that the enhancement benefit $G\left(t\right)$ generated by enterprises jointly enhancing emergency capabilities at time t is:

$$G\left(t\right)={\lambda }_{1}{E}_{S}+{\lambda }_{2}{E}_{P}+\varepsilon Q\left(t\right)+{G}_{0}$$

(5)

Where ${\lambda }_{1}$ and ${\lambda }_{2}$ represent the influence coefficients of DCESC resource suppliers and service operators on project benefits when enhancing emergency capabilities. $\varepsilon$ represents the impact degree of DCESC’s emergency capability on project benefits, and ${G}_{0}$ represents the project’s initial benefit.

Assumption 4: Government subsidies are an important means of reducing enterprises’ cost expenditures. To incentivize DCESC resource suppliers and service operators to actively enhance emergency response capabilities, it is assumed that the government provides subsidies to the relevant enterprises in this supply chain. The government allocates subsidies to resource suppliers and operators at proportions a and b, respectively.

Assumption 5: The enhancement benefit ${\rm{G}}\left({\rm{t}}\right)$ generated by the enhancement of emergency capabilities to data center engineering is distributed between DCESC resource suppliers and service operators according to a certain proportion. The resource supplier obtains a proportion θ, and the service operator obtains a proportion 1−θ, where ${\rm{\theta }}{\rm{\epsilon }}\left(0,1\right)$. The distribution proportion should consider both enterprises’ contributions to emergency inputs and their importance during emergencies. Both enterprises have a positive discount rate ρ and aim to maximize the enhancement of emergency capabilities over an infinite time horizon.

Taking DCESC resource suppliers and service operators as the research objects, the model includes decision variables and state variables. The symbols and their meanings involved in the model are shown in Table 1.

Table 1 Model Symbols and Meanings.

Model analysis

Nash non-cooperative game (non-cooperative mode N without cost-sharing)

In the non-cooperative mode without cost sharing, DCESC resource suppliers and service operators do not consider cooperating to enhance emergency capabilities but instead choose to maximize their own profits. Both parties are independent entities; the service operator does not share costs with the resource supplier. Only the government provides subsidies to each enterprise at certain proportions. The optimal value functions for the DCESC resource supplier and service operator can be expressed as (All variables with the superscript N (e.g., ${E}_{S}^{N}$, ${E}_{P}^{N}$) represent variables under the Nash Non-cooperative Game mode.):

$$\max ({\rm{N}}){J}_{S}={\int }_{0}^{{{\infty }}}{e}^{-\rho t}\left[\theta \left[{\lambda }_{1}{E}_{S}^{N}+{\lambda }_{2}{E}_{P}^{N}+\varepsilon Q\left(t\right)+{G}_{0}\right]-\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{N}}^{2}}{2\left(1+{\eta }_{S}\right)}\right]{dt}$$

(6)

$$\max ({\rm{N}}){J}_{P}={\int }_{0}^{{{\infty }}}{e}^{-\rho t}\left[\left(1-\theta \right)\left[{\lambda }_{1}{E}_{S}^{N}+{\lambda }_{2}{E}_{P}^{N}+\varepsilon Q\left(t\right)+{G}_{0}\right]-\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{N}}^{2}}{2\left(1+{\eta }_{P}\right)}\right]{dt}$$

(7)

Proposition 1: In the non-cooperative mode without cost sharing, the optimal equilibrium strategies for the DCESC resource supplier and service operator are as follows:

$${E}_{S}^{N}=\frac{\theta (1+{\eta }_{S})[{\lambda }_{1}(\rho +\gamma )+\varepsilon {\alpha }_{1}]}{(\rho +\gamma )(1-a){\mu }_{S}}$$

(8)

$${E}_{P}^{N}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{2}\right]}{\left(\rho +\gamma \right)\left(1-b\right){\mu }_{P}}$$

(9)

From the above equations, it is evident that the optimal efforts of the DCESC resource supplier and service operator to enhance emergency capabilities are inversely proportional to the cost coefficients of emergency efforts and the decay coefficient of emergency capabilities over time. They are directly proportional to the enterprise’s emergency data utilization level, the influence coefficients of emergency efforts on emergency capabilities, and the impact degree of DCESC’s emergency capabilities on project benefits.

Proof: To achieve equilibrium in this non-cooperative game, suppose there exist continuous and bounded differential functions ${{{{V}_{S}}^{N}\left(Q\right){andV}}_{P}}^{N}\left(Q\right)$, which are the objective benefit functions of the resource supplier and service operator, respectively. For $Q\ge 0$, the Hamilton-Jacobi-Bellman (HJB) equations are:

$$\rho {{V}_{S}}^{N}\left(Q\right)=\max \left\{\theta \left[{\lambda }_{1}{E}_{S}^{N}+{\lambda }_{2}{E}_{P}^{N}+\varepsilon Q+{G}_{0}\right]-\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{N}}^{2}}{2\left(1+{\eta }_{S}\right)}+\frac{{{{dV}}_{S}}^{N}\left(Q\right)}{{dQ}}\left[{\alpha }_{1}{E}_{S}^{N}+{\alpha }_{2}{E}_{P}^{N}-\gamma Q\right]\right\}$$

(10)

$$\rho {{V}_{P}}^{N}\left(Q\right)=\max \left\{\begin{array}{c}\left(1-\theta \right)\left[{\lambda }_{1}{E}_{S}^{N}+{\lambda }_{2}{E}_{P}^{N}+\varepsilon Q+{G}_{0}\right]-\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{N}}^{2}}{2\left(1+{\eta }_{P}\right)}\\ +\frac{{{{dV}}_{P}}^{N}\left(Q\right)}{{dQ}}\left[{\alpha }_{1}{E}_{S}^{N}+{\alpha }_{2}{E}_{P}^{N}-\gamma Q\right]\end{array}\right\}$$

(11)

Taking partial derivatives of Eqs. (10) and (11) with respect to ${E}_{S}$ and ${E}_{P}$, we obtain:

$$\begin{array}{c}\left(\frac{d\rho {{V}_{S}}^{N}\left(Q\right)}{{{dE}}_{S}^{N}},\frac{d\rho {{V}_{P}}^{N}\left(Q\right)}{d{E}_{P}^{N}}\right)\\ =\left[\theta {\lambda }_{1}-\left(1-a\right)\frac{{\mu }_{S}{E}_{S}^{N}}{1+{\eta }_{S}}+{\alpha }_{1}\frac{{{{dV}}_{S}}^{N}\left(Q\right)}{{dQ}},\left(1-\theta \right){\lambda }_{2}-\left(1-b\right)\frac{{\mu }_{P}{E}_{P}^{N}}{1+{\eta }_{P}}+{\alpha }_{2}\frac{{{{dV}}_{P}}^{N}\left(Q\right)}{{dQ}}\right]\end{array}$$

(12)

Setting the derivatives to zero, we obtain the optimal expressions for the efforts of both parties to enhance emergency management capabilities:

$${E}_{S}^{N}=\frac{\left(1+{\eta }_{S}\right)\left[\theta {\lambda }_{1}+{\alpha }_{1}\frac{{{{dV}}_{S}}^{N}\left(Q\right)}{{dQ}}\right]}{{\mu }_{S}\left(1-a\right)}$$

(13)

$${E}_{P}^{N}=\frac{\left(1+{\eta }_{P}\right)\left[\left(1-\theta \right){\lambda }_{2}+{\alpha }_{2}\frac{{{{dV}}_{P}}^{N}\left(Q\right)}{{dQ}}\right]}{{\mu }_{P}\left(1-b\right)}$$

(14)

Substituting Eqs. (13) and (14) back into (10) and (11), we obtain $\rho {{V}_{S}}^{N}\left(Q\right)\,{and}\,\rho {{V}_{P}}^{N}\left(Q\right)$. Given the characteristics of these equations, we can assume that the solution is the optimal benefit function for enhancing emergency capabilities, expressed as:

$$\left\{\begin{array}{c}{{V}_{S}}^{N}\left(Q\right)={m}_{1}Q+{n}_{1}\\ {{V}_{P}}^{N}\left(Q\right)={m}_{2}Q+{n}_{2}\end{array}\right.$$

(15)

From Eq. (15), we have: $\frac{{{{dV}}_{S}}^{N}\left(Q\right)}{{dQ}}={m}_{1},\frac{{{{dV}}_{P}}^{N}\left(Q\right)}{{dQ}}={m}_{2}$. Using the method of undetermined coefficients, we derive the following expressions:

$${{V}_{S}}^{N}\left(Q\right)=\left\{\begin{array}{c}\frac{\varepsilon \theta }{\rho +\gamma }Q+\frac{{\theta }^{2}\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}\\ +\frac{\theta \left(1-\theta \right){\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}}{\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}+\frac{\theta {G}_{0}}{\rho }\end{array}\right\}$$

(16)

$${{V}_{P}}^{N}\left(Q\right)=\left\{\begin{array}{c}\frac{\varepsilon \left(1-\theta \right)}{\rho +\gamma }Q+\frac{\left(1+{\eta }_{P}\right){\left(1-\theta \right)}^{2}{\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}\\ +\frac{\theta \left(1-\theta \right)\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}^{2}}{\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{\left(1-\theta \right){G}_{0}}{\rho }\end{array}\right\}$$

(17)

Substituting $\frac{{{{dV}}_{S}}^{N}\left(Q\right)}{{dQ}}={m}_{1}$ and $\frac{{{{dV}}_{P}}^{N}\left(Q\right)}{{dQ}}={m}_{2}$ and the results for ${m}_{1}$ and ${m}_{2}$ from Eqs. (16) and (17) into Eqs. (13) and (14), we can prove the existence of Proposition 1:

$${E}_{S}^{N}=\frac{\theta \left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)}$$

(18)

$${E}_{P}^{N}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{2}\right]}{\left(\rho +\gamma \right)\left(1-b\right){\mu }_{P}}$$

(19)

Based on the optimal benefit functions ${{V}_{S}}^{N}\left(Q\right){{{\,and\,V}}_{P}}^{N}\left(Q\right)$ of both parties, and the effort levels ${E}_{S}$ and ${E}_{P}$ of entities in the supply chain, combined with Eq. (3), and considering ${Q}_{0}\ge 0$, setting the derivative of Eq. (3) to zero, we derive the optimal trajectory function of DCESC’s emergency capability:

$${Q}^{N}=\left(\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[\frac{{\alpha }_{1}{E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma }\right]\right]$$

(20)

Therefore, the optimal gain function for enhancing DCESC’s emergency capabilities under the Nash non-cooperative game mode is:

$${V}^{N}\left(Q\right)={{V}_{S}}^{N}\left(Q\right)+{{V}_{P}}^{N}\left(Q\right)=\left\{\begin{array}{c}\frac{\varepsilon }{\rho +\gamma }Q+\frac{\left(1+{\eta }_{P}\right)\left(1-\theta \right)\left(1+\theta \right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}\\ +\frac{\theta \left(2-\theta \right)\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{{G}_{0}}{\rho }\end{array}\right\}$$

(21)

Collaborative cooperative game mode (C)

In this scenario, to further enhance the emergency capabilities of the DCESC and improve emergency resource levels, the resource supplier and the service operator shift from a non-cooperative mode to a collaborative cooperative mode. They aim to jointly determine the optimal strategies by maximizing the shared benefits that their collaboration brings to the project supply chain. The optimal value function is thus (Variables with the superscript C (e.g., ${E}_{S}^{C}$, ${E}_{P}^{C}$) represent variables under the Collaborative Cooperation mode.):

$$\max \left({\rm{C}}\right)J={\int }_{0}^{{{\infty }}}{e}^{-\rho t}\left\{\begin{array}{c}{\lambda }_{1}{E}_{S}^{C}+{\lambda }_{2}{E}_{P}^{C}+\varepsilon Q+{G}_{0}\\ -\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{C}}^{2}}{2\left(1+{\eta }_{S}\right)}-\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{C}}^{2}}{2\left(1+{\eta }_{P}\right)}\end{array}\right\}{dt}$$

(22)

Proposition 2: Under the collaborative cooperative game model, the optimal equilibrium strategies for the resource supplier and the service operator are as follows:

$${E}_{S}^{C}=\frac{\left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)}$$

(23)

$${E}_{P}^{C}=\frac{\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$$

(24)

The optimal effort levels ${E}_{S}^{C}$ and ${E}_{P}^{C}$ under the collaborative cooperation mode highlight the importance of government subsidies (a and b) in enhancing emergency capabilities. The results show that higher subsidies lead to greater effort levels, which in turn improve the resilience of the DCESC. Additionally, the discount rate ρ and decay rate γ reflect the trade-off between immediate and future benefits, as well as the natural decline of emergency capabilities over time.

Proof: To achieve equilibrium in the collaborative cooperative game, suppose there exists a continuous and bounded differential function V^C(Q), which represents the collaborative cooperative objective benefit function of the resource supplier and the service operator. For $Q\ge 0$, the Hamilton-Jacobi-Bellman (HJB) equation is:

$$\rho {V}^{C}\left(Q\right)=\max \left\{\begin{array}{c}{\lambda }_{1}{E}_{S}^{C}+{\lambda }_{2}{E}_{P}^{C}+\varepsilon Q+{G}_{0}-\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{C}}^{2}}{2\left(1+{\eta }_{S}\right)}\\ -\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{C}}^{2}}{2\left(1+{\eta }_{P}\right)}+\frac{d{V}^{C}\left(Q\right)}{{dQ}}\left[{\alpha }_{1}{E}_{S}^{C}+{\alpha }_{2}{E}_{P}^{C}-\gamma Q\right]\end{array}\right\}$$

(25)

According to the HJB equation, taking the first-order partial derivatives of Eq. (25) with respect to ${E}_{S}^{C}$ and ${E}_{P}^{C}$ and setting them to zero, we can derive the optimal strategy expressions for the resource supplier and the service operator. Substituting these back into the HJB equation, we get:

$$\rho {V}^{C}\left(Q\right)=\max \left\{\begin{array}{c}\left[\varepsilon -\gamma \frac{d{V}^{C}\left(Q\right)}{{dQ}}\right]Q+\frac{\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}+{\alpha }_{1}\frac{d{V}^{C}\left(Q\right)}{{dQ}}\right]}^{2}}{2{\mu }_{S}\left(1-a\right)}\\ +\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}+\,\frac{{\alpha }_{2}d{V}^{C}\left(Q\right)}{{\rm{dQ}}}\right]}^{2}}{2{\mu }_{P}\left(1-b\right)}+{G}_{0}\end{array}\right\}$$

(26)

From Eq. (26), we find that $\rho {V}^{C}\left(Q\right)$ is a linear function of Q. Therefore, the solution of ${V}^{C}\left(Q\right)$ can be expressed as:

$${V}^{C}\left(Q\right)={m}_{3}Q+{n}_{3}$$

(27)

Combining Eqs. (26) and (27), and noting that ${m}_{3}$ and ${n}_{3}$ are constants, we obtain:

$${m}_{3}=\frac{\varepsilon -\gamma \frac{d{V}^{C}\left(Q\right)}{{dQ}}}{\rho }=\frac{d{V}^{C}\left(Q\right)}{{dQ}}\to {m}_{3}=\frac{\varepsilon }{\rho +\gamma }$$

(28)

$${n}_{3}=\frac{\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}+\frac{{G}_{0}}{\rho }$$

(29)

Substituting ${m}_{3},{n}_{3}$ into equations, the emergency effort levels are expressed as:

$${E}_{S}^{C}=\frac{\left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)}$$

(30)

$${E}_{P}^{C}=\frac{\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$$

(31)

Thus, Proposition 2 is proved.

Under the collaborative cooperative game mode, the optimal gain function for enhancing the emergency capabilities of the DCESC resource supplier and service operator is:

$${V}^{C}\left(Q\right)=\frac{\varepsilon }{\rho +\gamma }Q+\frac{\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}+\frac{{G}_{0}}{\rho }$$

(32)

Based on the optimal benefit function ${V}^{C}\left(Q\right)$, and the effort levels ${E}_{S}$ and ${E}_{P}$ of the entities in the supply chain, combined with Eq. (3), and knowing that ${Q}_{0}\ge 0$, setting the result of Eq. (3) to zero, we derive the optimal trajectory function of DCESC’s emergency capabilities:

$${Q}^{C}=\left(\frac{{{\alpha }_{1}E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[{\alpha }_{1}\frac{{E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma }\right]\right]$$

(33)

Game mode based on cost-sharing contract (Y)

In reality, participants in game decisions rarely concentrate exclusively on maximizing the overall benefits of collaborative cooperation systems, making such scenarios highly over-idealized and less likely to occur. However, this idealized scenario can function as a standard for strategy coordination in the cost-sharing contract game mode.

This model is based on a cost-sharing contract. Besides government subsidies, internal enterprises within the supply chain also provide incentives. The service operator shares information with the resource supplier and subsidizes its input costs to enhance the emergency capability and responsiveness of the foundational engineering supply chain. During the process of enhancing emergency capabilities, the service operator, as the central enterprise of DCESC, holds advantages in capital and technology. It possesses a higher level and awareness of emergency capability enhancement compared to the resource supplier and can improve emergency management levels more precisely and effectively. Simultaneously, it can reduce part of the emergency costs for the resource supplier by providing guidance on technical solutions. Suppose the service operator shares a proportion $\partial \left(t\right)$ of the effort cost to help the resource supplier enhance emergency capabilities. With this condition introduced, the objective function values of the DCESC resource supplier and service operator are as follows (Variables with the superscript Y (e.g., ${{E}_{S}}^{Y}$, ${{E}_{P}}^{Y}$) represent variables under the Cost-Sharing Contract mode):

$$\max \left({\rm{Y}}\right){J}_{S}={\int }_{0}^{{{\infty }}}{e}^{-\rho t}\left\{\begin{array}{c}\theta \left[{\lambda }_{1}{{E}_{S}}^{Y}+{\lambda }_{2}{{E}_{P}}^{Y}+\varepsilon Q\left(t\right)+{G}_{0}\right]\\ -\left(1-a\right)\left(1-\partial \left(t\right)\right)\frac{{\mu }_{S}{{E}_{S}^{Y}}^{2}}{2\left(1+{\eta }_{S}\right)}\end{array}\right\}{dt}$$

(34)

$$\max \left({\rm{Y}}\right){J}_{P}={\int }_{0}^{{{\infty }}}{e}^{-\rho t}\left\{\begin{array}{c}\left(1-\theta \right)\left[{\lambda }_{1}{{E}_{S}}^{Y}+{\lambda }_{2}{{E}_{P}}^{Y}+\varepsilon Q\left(t\right)+{G}_{0}\right]\\ -\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{Y}}^{2}}{2\left(1+{\eta }_{P}\right)}-\partial \left(t\right)\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{Y}}^{2}}{2\left(1+{\eta }_{S}\right)}\end{array}\right\}{dt}$$

(35)

Proposition 3: Under the condition of a cost-sharing contract, the optimal proportion $\partial \left(t\right)$ for the DCESC service operator to help the resource supplier share effort costs, as well as the optimal equilibrium strategies of the DCESC resource supplier and service operator, are as follows:

$${{E}_{S}}^{Y}=\frac{\theta \left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)\left(1-\partial \left(t\right)\right)}$$

(36)

$${{E}_{P}}^{Y}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$$

(37)

In the cost-sharing contract mode, the optimal cost-sharing proportion $\partial \left(t\right)=1-\theta$ indicates that the service operator bears a significant portion of the resource supplier’s effort costs. This mechanism encourages the resource supplier to increase its effort level ${{E}_{S}}^{Y}$, while the service operator’s effort level ${{E}_{P}}^{Y}$ is influenced by its share of the benefits $\left(1-\theta \right)$.

Proof: To ensure that the game model based on the cost-sharing contract reaches equilibrium, suppose there exist continuous and bounded differential functions ${{{{V}_{S}}^{Y}\left(Q\right){\,and\,V}}_{P}}^{Y}\left(Q\right)$, representing the objective benefit functions of the resource supplier and service operator, respectively. For $Q\ge 0$, the Hamilton-Jacobi-Bellman (HJB) equations are:

$$\rho {V}_{S}^{Y}(Q)=\,\max \left\{\begin{array}{l}\theta \theta [{\lambda }_{1}{E}_{S}^{Y}+{\lambda }_{2}{E}_{P}^{Y}+\varepsilon Q+{G}_{0}]-(1-a)(1-\partial (t))\frac{{\mu }_{S}{E}_{S}^{{Y}^{2}}}{2(1+{\eta }_{S})}\\ \qquad\qquad+\frac{d{V}_{S}^{Y}(Q)}{dQ}[{\alpha }_{1}{E}_{S}^{Y}+{\alpha }_{2}{E}_{P}^{Y}-\gamma Q]\end{array}\right\}$$

(38)

$$\rho {{V}_{P}}^{Y}\left(Q\right)=\max \left\{\begin{array}{c}\left(1-\theta \right)\left[{\lambda }_{1}{{E}_{S}}^{Y}+{\lambda }_{2}{{E}_{P}}^{Y}+\varepsilon Q+{G}_{0}\right]-\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}^{Y}}^{2}}{2\left(1+{\eta }_{P}\right)}\\ +\frac{{{{dV}}_{P}}^{Y}\left(Q\right)}{{dQ}}\left[{\alpha }_{1}{{E}_{S}}^{Y}+{\alpha }_{2}{{E}_{P}}^{Y}-\gamma Q\right]-\partial \left(t\right)\left(1-a\right)\frac{{\mu }_{S}{{E}_{S}^{Y}}^{2}}{2\left(1+{\eta }_{S}\right)}\end{array}\right\}$$

(39)

Taking partial derivatives of Eqs. (38) and (39) with respect to ${E}_{S}$ and ${E}_{P}$, we obtain:

$$\left(\frac{d\rho {{V}_{S}}^{Y}\left(Q\right)}{{{{dE}}_{S}}^{Y}},\frac{d\rho {{V}_{P}}^{Y}\left(Q\right)}{d{{E}_{P}}^{Y}}\right)=\left(\begin{array}{c}{\lambda }_{1}\theta -\left(1-a\right)\left(1-\partial \left(t\right)\right)\frac{{\mu }_{S}{{E}_{S}}^{Y}}{1+{\eta }_{S}}+{\alpha }_{1}\frac{d{{V}_{S}}^{Y}\left(Q\right)}{{dQ}},\\ {\lambda }_{2}\left(1-\theta \right)-\left(1-b\right)\frac{{\mu }_{P}{{E}_{P}}^{Y}}{1+{\eta }_{P}}+{\alpha }_{2}\frac{{{{dV}}_{P}}^{Y}\left(Q\right)}{{dQ}}\end{array}\right)$$

(40)

Setting both partial derivatives to zero yields the optimal expressions for the efforts of the resource supplier and service operator:

$${{E}_{S}}^{Y}=\frac{\left(1+{\eta }_{S}\right)\left[{\alpha }_{1}\frac{{{{dV}}_{S}}^{S}\left(Q\right)}{{dQ}}+\theta {\lambda }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(1-\partial \left(t\right)\right)}$$

(41)

$${{E}_{P}}^{Y}=\frac{\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(1-\theta \right)+{\alpha }_{2}\frac{{{{dV}}_{P}}^{Y}\left(Q\right)}{{dQ}}\right]}{{\mu }_{P}\left(1-b\right)}$$

(42)

In this paper, the collaborative cooperative game mode is considered the standard for strategy coordination in the benefit-sharing and cost-sharing contract game mode. Letting Eqs. (36–37) align with Proposition 2, and using the method of undetermined coefficients, we obtain: $\partial \left(t\right)=1-\frac{\left(\rho +\gamma \right)\left[{\lambda }_{1}\theta +{\alpha }_{1}\frac{{{{dV}}_{S}}^{Y}\left(Q\right)}{{dQ}}\right]}{{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}}$, Substituting the obtained ${{E}_{S}}^{Y},{{E}_{P}}^{Y}$ and $\partial \left(t\right)$ into the HJB equations, and assuming the optimal value functions of the resource supplier and service operator have a linear relationship with the data center’s emergency capability: ${{V}_{S}}^{Y}\left(Q\right)={m}_{4}Q+{n}_{4},{{V}_{P}}^{Y}\left(Q\right)={m}_{5}Q+{n}_{5}$, where ${m}_{i},{n}_{i}$ are constant. We derive the following relationships:

$${m}_{4}=\frac{\varepsilon \theta -\gamma \frac{d{{V}_{S}}^{Y}\left(Q\right)}{{dQ}}}{\rho }=\frac{d{{V}_{S}}^{Y}\left(Q\right)}{{dQ}}=\frac{\varepsilon \theta }{\rho +\gamma }$$

(43)

$${m}_{5}=\frac{\varepsilon \left(1-\theta \right)-\gamma \frac{d{{V}_{P}}^{Y}\left(Q\right)}{{dQ}}}{\rho }=\frac{d{{V}_{P}}^{Y}\left(Q\right)}{{dQ}}=\frac{\varepsilon \left(1-\theta \right)}{\rho +\gamma }$$

(44)

$${n}_{4}=\frac{{\theta }^{2}\left(1+{\eta }_{S}\right){\left[{\alpha }_{1}\varepsilon +{\lambda }_{1}\left(\rho +\gamma \right)\right]}^{2}}{{2\rho \mu }_{S}{\left(\rho +\gamma \right)}^{2}\left(1-a\right)\left(1-\partial \left(t\right)\right)}+\frac{\theta \left(1-\theta \right)\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}}{{\mu }_{P}\left(1-b\right)}+\frac{\theta {G}_{0}}{\rho }$$

(45)

$${n}_{5}=\left\{\begin{array}{c}\frac{{\left(1-\theta \right)}^{2}\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}}{2{\rho \mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}\\ +\frac{\theta \left[1-\partial \left(t\right)-\theta \right]\left(1+{\eta }_{S}\right){\left[{\alpha }_{1}\varepsilon +\left(\rho +\gamma \right){\lambda }_{1}\right]}^{2}}{{2\rho \mu }_{S}\left(1-a\right){\left(1-\partial \left(t\right)\right)}^{2}{\left(\rho +\gamma \right)}^{2}}+\frac{\left(1-\theta \right)}{\rho }{G}_{0}\end{array}\right\}$$

(46)

Substituting ${m}_{4}$ and ${m}_{5}$ into ${{E}_{S}}^{Y}$ and ${{E}_{P}}^{Y}$, we have:

$${{E}_{S}}^{Y}=\frac{\theta \left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)\left(1-\partial \left(t\right)\right)}$$

(47)

$${{E}_{P}}^{Y}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$$

(48)

Thus, Proposition 3 is established, and we find $\partial \left(t\right)=1-\theta$.

Based on the optimal benefit functions and the effort levels ${{E}_{S}}^{Y}$ and ${{E}_{P}}^{Y}$ of the entities in the supply chain, combined with Eq. (3), and knowing ${Q}_{0}\ge 0$, setting the result of Eq. (3) to zero, we derive the optimal trajectory function of DCESC’s emergency capability under the cost-sharing contract game mode:

$${Q}^{Y}=\left(\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }\right]\right]$$

(49)

Using the method of undetermined coefficients, we obtain the optimal gain function for enhancing DCESC’s emergency capability:

$$\begin{array}{c}{V}^{Y}\left(Q\right)={{V}_{S}}^{Y}\left(Q\right)+{{V}_{P}}^{Y}\left(Q\right)\\ =\left\{\begin{array}{c}\frac{\varepsilon }{\rho +\gamma }Q+\frac{\left[\theta \left(1-\partial \left(t\right)\right)-{\theta }^{2}\partial \left(t\right)\right]\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{{2\rho \mu }_{S}\left(1-a\right){\left(1-\partial \left(t\right)\right)}^{2}{\left(\rho +\gamma \right)}^{2}}\\ +\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}\left[{\left(1-\theta \right)}^{2}+2\rho \left(1-{\theta }^{2}\right){\left(\rho +\gamma \right)}^{2}\right]}{2{\rho \mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}+\frac{{G}_{0}}{\rho }\end{array}\right\}\end{array}$$

(50)

Comparative analysis of the models

By comparing the efforts of resource suppliers and service operators to enhance emergency capabilities, the optimal benefit functions, and the emergency levels of the DCESC under the three modes above, we conduct a comparative study based on these results:

Proposition 4: Under different circumstances, the comparison of the optimal effort levels of both parties is as follows: For ${E}_{S}$, since $\theta =1-\partial \left(t\right)$, then ${{E}_{S}}^{C}={{E}_{S}}^{Y}\, > \,{{E}_{S}}^{N}$. For ${E}_{P}$, when $\theta \epsilon \left(0,1\right),{{E}_{P}}^{C}\, > \,{{E}_{P}}^{Y}={{E}_{P}}^{N}$. The effort levels of DCESC entities in enhancing emergency capabilities are negatively correlated with factors such as effort cost coefficients and positively correlated with factors like data center emergency data utilization capabilities, enhancement benefit distribution ratios, and government subsidy ratios.

Proof: from propositions 1, 2, and 3, we have:

$${{E}_{S}}^{N}=\frac{\theta \left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}{\left(\rho +\gamma \right)\left(1-a\right){\mu }_{S}}={{\theta E}_{S}}^{C},{{E}_{S}}^{Y}=\frac{\theta }{1-\partial \left(t\right)}{{E}_{S}}^{C}={{E}_{S}}^{C};$$

$${{E}_{P}}^{N}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{2}\right]}{\left(\rho +\gamma \right)\left(1-b\right){\mu }_{P}}=\left(1-\theta \right){{E}_{P}}^{C},{{E}_{P}}^{Y}=\frac{\left(1-\theta \right)\left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}=\left(1-\theta \right){{E}_{P}}^{C}.$$

Since $\theta =1-\partial \left(t\right)$, then ${{E}_{S}}^{C}={{E}_{S}}^{Y}$, and ${{E}_{S}}^{C}={{E}_{S}}^{Y}\, > \,{{E}_{S}}^{N}$. Similarly, ${{E}_{P}}^{C}\, > \,{{E}_{P}}^{Y}$ = ${{E}_{P}}^{N}$.

In the game cooperation mode, introducing cost-sharing contracts, since $\theta =1-\partial \left(t\right)$, the emergency enhancement efforts of resource suppliers are the same under both collaborative cooperation and the game mode based on cost-sharing contracts, unaffected by other factors. However, under the Nash non-cooperative mode, the emergency effort is less than in the other two cases and is directly proportional to $\theta$. Therefore, it can be inferred that the enhancement benefit distribution ratio $\theta$ is a key factor affecting the emergency enhancement efforts of data center resource suppliers.

Collaborative cooperation can effectively enhance the emergency enhancement efforts of data center service operators because, under system cooperation, enterprises consider maximizing total benefits rather than focusing on individual gains, giving them stronger motivation to improve emergency capabilities.

Under the collaborative cooperation game model, the emergency enhancement efforts of both parties are maximized; in other cases, the impact of the benefit distribution ratio needs to be considered. In summary, DCESC enterprises can choose to make optimal cooperation decisions based on the value of $\theta$ in reality, either through collaborative cooperation or by introducing cost-sharing contracts.

Proposition 5: The optimal comparative result of the project gains from enhancing DCESC emergency capabilities is: ${V}^{C}\left(Q\right)\, > \,{V}^{N}\left(Q\right)$.

Proof: The project gains from enhancing DCESC emergency capabilities under the Nash non-cooperative game, collaborative cooperation, and the cooperation mode based on cost-sharing contracts are as follows:

$$\begin{array}{c}{V}^{N}\left(Q\right)=\frac{\varepsilon }{\rho +\gamma }Q+\frac{{G}_{0}}{\rho }+\frac{\theta \left(2-\theta \right)\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}\\ +\frac{{\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\alpha }_{2}\varepsilon \right]}^{2}\left(1+{\eta }_{P}\right)\left(1-{\theta }^{2}\right)}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}\end{array}$$

(21)

$${V}^{C}\left(Q\right)=\frac{\varepsilon }{\rho +\gamma }Q+\frac{{G}_{0}}{\rho }+\frac{\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}$$

(32)

$$\begin{array}{c}{V}^{Y}\left(Q\right)=\frac{\varepsilon }{\rho +\gamma }Q+\frac{{G}_{0}}{\rho }+\frac{\left[\theta \left(1-\partial \left(t\right)\right)-{\theta }^{2}\partial \left(t\right)\right]\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{{2\rho \mu }_{S}\left(1-a\right){\left(1-\partial \left(t\right)\right)}^{2}{\left(\rho +\gamma \right)}^{2}}\\ +\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}\left[{\left(1-\theta \right)}^{2}+2\rho \left(1-{\theta }^{2}\right){\left(\rho +\gamma \right)}^{2}\right]}{2{\rho \mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}\end{array}$$

(50)

Calculations show that: ${V}^{Y}\left(Q\right)-{V}^{N}\left(Q\right)=\frac{\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}\left[\left({\theta }^{2}-\theta \right)+\left({1-\theta }^{2}\right)\rho {\left(\rho +\gamma \right)}^{2}\right]}{{\rho \mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}+\frac{\theta \left[\left(1-\theta \right)\left[2-3\theta \right]+\left(\theta -2\right)\left(1-\theta \right)\right]\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}^{2}}{{2\rho \mu }_{S}\left(1-a\right){\theta }^{2}{\left(\rho +\gamma \right)}^{2}};{V}^{C}\left(Q\right)-{V}^{N}\left(Q\right)=\frac{{\left(\theta -1\right)}^{2}\left(1+{\eta }_{S}\right){\left[{\lambda }_{1}\left(\rho +\gamma \right)+\varepsilon {\alpha }_{1}\right]}^{2}}{2\rho {\mu }_{S}\left(1-a\right){\left(\rho +\gamma \right)}^{2}}+\frac{{\theta }^{2}\left(1+{\eta }_{P}\right){\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}^{2}}{2\rho {\mu }_{P}\left(1-b\right){\left(\rho +\gamma \right)}^{2}}$> 0, Since $\theta \epsilon \left(0,1\right),$ the value of ${V}^{Y}\left(Q\right)-{V}^{N}\left(Q\right)$ depends on the specific values of other parameters, which will be compared in the numerical simulation later. Since ${V}^{C}\left(Q\right)\, > \,{V}^{N}\left(Q\right)$, Proposition 5 is proven.

From Proposition 5, it can be seen that the project gains from enhancing DCESC emergency capabilities under collaborative cooperation are higher than those under the cost-sharing contract and the Nash non-cooperative mode, and are not affected by other parameters. However, in reality, enterprises as individual entities have their own demands for enhancing emergency capabilities. If the emergency gains allocated to supply chain members under collaborative cooperation are lower than those obtained under non-cooperation or cost-sharing contracts, most enterprises tend to prefer non-cooperation or short-term cooperation. Therefore, in DCESC, to achieve coordinated development of emergency capabilities between resource suppliers and service operators, the distribution of emergency gains under collaborative cooperation needs to ensure that they are all higher than the project emergency gains obtained under non-cooperation or cost-sharing contracts, while reducing unnecessary losses through coordination to achieve Pareto optimality in enhancing emergency capabilities for both parties.

Proposition 6: Under different cooperation modes, the comparison of the optimal trajectory functions of DCESC emergency capabilities is as follows:

$${Q}^{C} \,> \,{Q}^{N}{\rm{and}}{Q}^{Y}\, > \,{Q}^{N}$$

Proof: Simplifying Eqs. (20), (33), and (49), we find that under the three cooperation modes, the optimal trajectory functions of DCESC emergency capabilities are as follows:

$$\left\{{Q}^{N}=\left(\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma }\right]\right]\right.$$

(20)

$$\left\{{Q}^{C}=\left(\frac{{{\alpha }_{1}E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[{\alpha }_{1}\frac{{E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma }\right]\right]\right.$$

(33)

$$\left\{{Q}^{Y}=\left(\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }\right)+{e}^{-\gamma t}\left[{Q}_{0}-\left[\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }\right]\right]\right.$$

(49)

Based on the structure of these formulas, we can compare the sizes of ${Q}^{N},{Q}^{C},{Q}^{Y}$ by comparing the values of $\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma },\frac{{{\alpha }_{1}E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma }$, and $\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }$. Assuming the function $Q=x+{e}^{-\gamma t}\left({Q}_{0}-x\right),$ which is an increasing function, the comparison is as follows:

Let ${x}_{1}=\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma },{x}_{2}=\frac{{{\alpha }_{1}E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma },{x}_{3}=\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }$, ${x}_{2}-{x}_{1}={\alpha }_{1}\frac{\left(1-\theta \right)\left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\gamma \mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)}+{\alpha }_{2}\frac{\theta \left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\gamma \mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$, thus ${x}_{2}-{x}_{1} > 0$, ${Q}^{N} < {Q}^{C};{x}_{3}-{x}_{1}={\alpha }_{1}\frac{\partial \left(t\right)\left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\gamma \mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)\left(1-\partial \left(t\right)\right)} > 0$, thus ${Q}^{N} < {Q}^{Y}$; ${x}_{2}-{x}_{3}={\alpha }_{1}\frac{\left(1-\theta -\partial \left(t\right)\right)\left(1+{\eta }_{S}\right)\left[{\lambda }_{1}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{1}\right]}{{\gamma \mu }_{S}\left(1-a\right)\left(\rho +\gamma \right)\left(1-\partial \left(t\right)\right)}+{\alpha }_{2}\frac{\theta \left(1+{\eta }_{P}\right)\left[{\lambda }_{2}\left(\rho +\gamma \right)+{\varepsilon \alpha }_{2}\right]}{{\gamma \mu }_{P}\left(1-b\right)\left(\rho +\gamma \right)}$, The size relationship depends on the value ranges of $\theta ,\partial \left(t\right)$ which will be compared in the numerical analysis.

Compared with the optimal situation of DCESC emergency capabilities under the Nash non-cooperative mode, the emergency levels under the cost-sharing contract and collaborative cooperation are higher. DCESC enterprises can achieve optimal emergency capabilities through different forms of cooperation. From the perspective of project gains from enhancing DCESC emergency capabilities, collaborative cooperation brings the greatest project emergency gains compared to the other two methods. As DCESC enterprises collaborate or cooperate under cost-sharing contracts to enhance emergency capabilities, their emergency enhancement efforts and the optimal situations of DCESC emergency capabilities are both better than those in the enhancement process under the Nash non-cooperative mode. The choice of specific cooperation methods depends on the distribution ratio of project emergency gains and the cost-sharing ratio.

Simulation analysis

This paper utilizes MATLAB 2016b for simulation to more intuitively analyze the efforts of DCESC enterprises in enhancing emergency capabilities, the optimal situations of DCESC emergency capabilities, and the models of project gains brought by improvements in emergency levels over time. Additionally, we analyze the sensitivity of certain parameters. Based on the subsidy ratios proposed in the “56 Fiscal Policies to Support High-Quality Development of SMEs” issued by the Henan Provincial Finance Department, we assume the government subsidy ratios to be $a=b=0.6$, while satisfying the parameter assumptions, given that not all parameters listed in Table 2 were available in existing literature, we refer to the literature of Fan et al. (2022), Zhang et al. (2021) and Zhao et al. (2024) to assign values to the parameters: $\rho =0.8,{Q}_{0}=5,{\mu }_{S}={\mu }_{P}=2,{\eta }_{P}={\eta }_{S}=0.4,{G}_{0}=1,{\lambda }_{1}={\lambda }_{2}=3,\varepsilon =2,\gamma =0.2,{\alpha }_{1}={\alpha }_{2}=2,\theta =0.6$.

Table 2 Parameter Assignment.

Through calculation, we obtain comparative results for the non-cooperative mode, collaborative cooperation situation, and cost-sharing contract mode. The analysis of Table 3 reveals several key insights. Under both collaborative cooperation and cost-sharing contract modes, the emergency effort levels of resource suppliers remain consistent and are higher than those in the non-cooperative mode. This indicates that collaborative mechanisms, whether through full cooperation or cost-sharing contracts, effectively incentivize resource suppliers to increase their emergency efforts. On the other hand, the emergency effort levels of service operators under the non-cooperative and cost-sharing contract modes are consistent but lower than those under the collaborative cooperation mode. This suggests that service operators are more responsive to full collaborative cooperation, which provides stronger incentives for enhancing their emergency efforts. These findings highlight the importance of cooperation modes in influencing the behavior of different stakeholders within the DCESC. Substituting the parameter values into Propositions 1, 2, and 3, we obtain the emergency efforts of DCESC resource suppliers and service operators under the three modes, as shown in Table 3.

Table 3 Emergency efforts of DCESC resource suppliers and service operators under different modes.

The impact of the three scenarios on DCESC emergency capabilities and emergency benefits

According to Proposition 6, the function $Q={e}^{-\gamma t}\left({Q}_{0}-x\right)+x,$ where: ${x}_{1}=\frac{{{\alpha }_{1}E}_{S}^{N}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{N}}{\gamma },{x}_{2}=\frac{{{\alpha }_{1}E}_{S}^{C}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{C}}{\gamma },{x}_{3}=\frac{{{\alpha }_{1}E}_{S}^{Y}}{\gamma }+\frac{{\alpha }_{2}{E}_{P}^{Y}}{\gamma }$ and $\partial \left(t\right)=1-\theta$. Substituting the parameters, we obtain ${x}_{1}=122.5,{x}_{2}=245,{x}_{3}=171.5$. Substituting into the function, we get: ${Q}_{N}=-{117.5e}^{-0.2t}+122.5;{Q}_{C}=-{240e}^{-0.2t}+245;{Q}_{Y}=-{166.5e}^{-0.2t}+171.5$. Assigning parameters based on formulas (19)-(20), (24), (38), and (59)-(61), we obtain: ${{V}_{S}}^{N}=-{141e}^{-0.2t}+192.76875,{{V}_{S}}^{Y}=-{199.8e}^{-0.2t}+275.364375,{{V}_{P}}^{N}=-{94e}^{-0.2t}+132.8,{{V}_{P}}^{Y}=-{133.2e}^{-0.2t}+146.275,{V}^{C}=-{480e}^{-0.2t}+598.4375,{V}^{N}=-{235e}^{-0.2t}+325.56875,{V}^{Y}=-{333e}^{-0.2t}+421.639375.$ Based on the above, the evolution trends of DCESC emergency capabilities and overall emergency enhancement benefits over time under different modes are shown in Figs. 2 and 3. Additionally, excluding the collaborative cooperation, the evolution of emergency enhancement benefits for DCESC resource suppliers and service operators under the other two game modes is shown in Fig. 4.

Fig. 2

Evolution of DCESC emergency capability over time.

Fig. 3

Comparison of overall emergency capability enhancement benefits of DCESC.

Fig. 4: Changes in emergency capability enhancement benefits under two scenarios.

a Changes in resource suppliers' emergency capability enhancement benefits; b Changes in service operators' emergency capability enhancement benefits.

The peak emergency capability level represents the maximum achievable emergency capability under different cooperation modes. As shown in Fig. 2: (1) The collaborative cooperation mode yields the highest emergency capability, followed by the cost-sharing contract mode, while the non-cooperation mode results in the lowest emergency capability. This finding underscores the importance of cooperation among supply chain enterprises, particularly within a collaborative framework, in significantly enhancing the supply chain’s ability to respond to emergencies. (2) Collaborative cooperation can most effectively enhance DCESC emergency capabilities, and under this cooperation condition, emergency capabilities tend to reach the optimal stable state. This also verifies and complements Proposition 6: ${Q}^{C} \,>\, {Q}^{Y}\, > \,{Q}^{N}$. In Fig. 3, the overall benefits of DCESC emergency capability enhancement increase over time and gradually stabilize. The comparative relationship among the three is: ${V}^{C}\left(Q\right)\, > \,{V}^{Y}\left(Q\right)\, > \,{V}^{N}\left(Q\right)$, consistent with the conclusion of Proposition 5.

Figure 4 illustrates the evolving benefits of emergency capability enhancement for resource suppliers and service operators across collaboration models. Key observations include:(1) Over time, for resource supplier enterprises, the benefits stemming from emergency capability improvements under cost-sharing contracts consistently surpass those in non-cooperative scenarios. Furthermore, the growth differential in non-cooperative mode is substantially larger compared to cost-sharing agreements. (2) The overall trend for service operators is similar to that of resource suppliers. Suppose in Fig. 4b, the emergency benefits under the two modes are equal at the intersection time ${t=T}_{0}$. When $0 \,<\, t \,<\, {T}_{0}$, as shown in the figure, the non-cooperative game is superior to the cost-sharing contract mode. When ${T}_{0}\, < \,t$, the emergency benefits under the cost-sharing contract model exceed those of the non-cooperative game. Specifically, the growth rate of emergency benefits under the cost-sharing contract is higher than that of the non-cooperative model. However, under the cost-sharing contract, where the service operator subsidizes the resource supplier as the central enterprise, the service operator is more likely to opt for the non-cooperative model if the benefits fail to meet the cost-sharing threshold.(3) Comparing the peak values in Fig. 4a, b, the cost-sharing contract has a more significant impact on the emergency capability enhancement benefits for DCESC resource suppliers than for service operators. In practice, factors such as the duration of cooperation should be carefully considered to determine the most suitable approach for improving emergency capabilities.

Sensitivity analysis of relevant parameters

Figure 5 shows the sensitivity analysis of the emergency effort cost coefficient, and Fig. 6 shows the sensitivity analysis of the emergency data utilization level.

Fig. 5

Sensitivity analysis of emergency effort cost coefficient.

Fig. 6

Sensitivity analysis of emergency data utilization level.

The emergency effort cost coefficient generally represents the proportion of additional efforts and resources an enterprise invests to swiftly restore normal operations during emergencies. These costs encompass expenses related to contingency planning, resource allocation, and information communication. From Fig. 5: (1) DCESC emergency capabilities tend to stabilize over time and are not affected by $\mu$, but in terms of the degree of improvement, the reduction of the emergency effort cost coefficient is inversely proportional to the improvement of emergency capability levels. When $\mu =2$, the emergency level under the non-cooperative mode is higher than the emergency level when $\mu =4$ under the cost-sharing contract. (2) When the emergency cost coefficient changes similarly (i.e., $\mu =2\to 4$), the enhancement of emergency capabilities under collaborative cooperation is always higher than that under the cost-sharing contract, which is higher than under the Nash non-cooperative mode. Under collaborative cooperation, the difference in emergency capabilities due to changes in the cost coefficient is the largest, while under the non-cooperative mode, the difference is the smallest, indicating that this mode has the lowest sensitivity to the emergency effort cost coefficient. (3) When the emergency effort cost coefficient increases, enterprises need to provide high funds for emergency response, have weaker enthusiasm, and are prone to losses; when the coefficient decreases, the emergency effort cost provided by enterprises reduces, enhancing their enthusiasm while bringing about growth in emergency capabilities. DCESC can quickly respond to emergencies to reduce losses, ensure the stable operation of data centers, and enhance the resilience and stability of the supply chain.

Emergency data utilization level can more effectively process data flows in emergencies and assist in achieving information sharing and collaboration, thereby enhancing enterprises’ coordination abilities and response capabilities when facing unexpected events. Improving emergency data utilization capabilities can help enterprises quickly identify potential risks, assess impacts, and formulate response measures, while also accumulating more experiences and lessons in daily operations. Figure 6 shows the impact of emergency data utilization level on DCESC emergency capability enhancement at the same time: (1) In all three cooperation modes, the sensitivity of emergency capability to emergency data utilization level is highest under collaborative cooperation, with the largest difference from the starting point. Under the cost-sharing contract, emergency capabilities are always higher than under the non-cooperative mode but always lower than under the collaborative cooperation mode. Collaborative cooperation mode and cost-sharing contract can strengthen the effect of emergency capability enhancement. (2) DCESC emergency capability is directly proportional to the improvement of emergency data utilization level, i.e., at the same time, the more sensitive emergency capability is to the emergency data utilization level, the greater the enhancement effect of emergency capability. (3) Comparing Figs. 5 and 6, we can see that the change ratio of the influencing factors is consistent ($\mu =2\to 4,\eta =0.2\to 0.4$). From the figures, it can be clearly observed that the change in $\mu$ brings a much higher improvement to emergency capabilities than the impact brought by the same level of change in $\eta .$ Therefore, it can be judged that the sensitivity of DCESC emergency capability to the emergency effort cost coefficient is higher than to the emergency data utilization level.

As shown in Fig. 7, the government subsidy ratio plays an important role in enhancing DCESC emergency capabilities. Government subsidy can directly alleviate the financial pressure of enterprises in crisis situations, spur technological innovation, and drive infrastructure enhancement, thereby impacting the response efficiency and bolstering the resilience of data center projects under construction in the face of unexpected events. Increasing the government subsidy ratio can significantly enhance DCESC's emergency capabilities.

Fig. 7

Sensitivity analysis of government subsidy ratio.

Figures 8–10 compare the impact of government subsidy ratio, emergency effort cost coefficient, and emergency data utilization level on DCESC emergency capabilities. According to Fig. 8, when the government subsidy tends to zero, increasing the emergency data utilization level from 0 to 1 only brings a small improvement in the supply chain’s emergency capability; but when the emergency data utilization level tends to zero, changes in government subsidies can still bring a substantial enhancement to emergency capabilities. This situation indicates that under the same conditions, priority should be given to improving the supply chain’s emergency capability by increasing the government subsidy rate. The sensitivity of government subsidy rate to emergency capability improvement is higher compared to the emergency data utilization level.

Fig. 8

Sensitivity analysis of government subsidy ratio and emergency data utilization level.

Fig. 9

Sensitivity analysis of government subsidy ratio and emergency effort cost coefficient.

Fig. 10

Sensitivity analysis of emergency effort cost coefficient and emergency data utilization level.

Figure 9 discusses the impact of government subsidy ratio and emergency effort cost coefficient on emergency capabilities. When the government subsidy ratio is at its maximum and the cost coefficient is at its minimum, the emergency capability reaches its peak. Unlike the emergency data utilization level, when the government subsidy tends to zero and the cost coefficient is at its maximum, the supply chain’s emergency capability is similar to the level when the government subsidy tends to one and the cost coefficient is at its minimum. From this, it is inferred that the sensitivity of government subsidy level and emergency effort cost coefficient to emergency capabilities is similar. In Fig. 10, the impact of emergency effort cost coefficient and emergency data utilization level on DCESC emergency capabilities is compared. Similarly, it can be judged that the sensitivity of the emergency effort cost coefficient to emergency capabilities is higher than that of the emergency data utilization level. Therefore, to enhance emergency capabilities during data center construction, we should first consider reducing the emergency effort cost coefficient and increasing government subsidies, and then consider improving enterprises’ emergency data utilization levels. These influencing factors complement each other and jointly affect the emergency capabilities of data centers. In reality, enterprises should comprehensively consider multiple factors to form a complete emergency response mechanism.

Impact of different subsidy rates implemented by different enterprises on emergency capabilities

The above simulations assume that the government subsidy ratios to both enterprises are the same, i.e., $a=b=0.6$. However, in practice, the government may implement different subsidy policies depending on the type of enterprise. Combining with the actual situation, we adjust the parameters to $a=0.3,b=0.6$, while keeping other parameters unchanged.

As shown in Fig. 11, when the government subsidy ratio to resource supplier enterprises is greater than that to service operators, the effect of emergency capability enhancement is stronger, and this feature is most evident when introducing cost-sharing contracts. When engaging in collaborative cooperation, whether the government subsidizes enterprises differently does not affect DCESC emergency capabilities. Therefore, in actual subsidies, the government should consider providing more subsidies to resource suppliers to more efficiently enhance emergency capabilities.

Fig. 11

Impact of subsidy ratios to both enterprises on DCESC emergency capabilities.

Conclusions and recommendations

Conclusions

This paper investigates supply chain emergency preparedness during the construction phase of data-center infrastructure projects, with specific focus on resource suppliers and service operators within DCESC. We explore the collaborative interactions among enterprises that influence DCESC’s emergency capabilities. We develop a multi-objective, multi-agent dynamic differential game model to analyze the emergency efforts of resource suppliers and service operators, the optimal levels of DCESC’s emergency capability, and the optimal emergency benefits of the supply chain. The dynamic variations in these factors are quantitatively studied, and the model results are analyzed under three scenarios. Key factors influencing the enhancement of emergency capabilities are investigated, and the model’s findings are validated using numerical simulations in MATLAB 2016b. This study contributes to both the literature and practice of DCESC in several important ways. First, it addresses a significant gap in existing research by focusing on the construction phase of data center projects, which has been underexplored despite its critical role. Second, it introduces a multi-objective, multi-agent dynamic differential game model to analyze the collaborative interactions among enterprises and their influence on DCESC’s emergency capabilities. Third, it offers actionable insights for policymakers and managers by identifying optimal cooperation modes and highlighting the role of government subsidies in enhancing emergency preparedness. These findings advance both theoretical understanding of DCESC and offer actionable guidance for improving supply chain resilience in the context of global uncertainties and environmental challenges. Based on the research assumptions of this study, the following conclusions have been drawn:

• (1) The sensitivity analysis of factors influencing the enhancement of DCESC’s emergency capabilities reveals that the government subsidy ratio and the emergency effort cost coefficient have a greater impact than the emergency data utilization level. Therefore, when aiming to improve DCESC’s emergency capabilities, priority should be given to increasing government subsidies and reducing the emergency effort cost coefficient. This approach not only provides effective incentives for enterprises but also lowers their operational costs. By actively leveraging government subsidies to reduce the emergency effort cost coefficient and promoting inter-enterprise collaboration, the vulnerability of the supply chain can be minimized. This enables enterprises to respond more effectively to unexpected events, enhancing the overall recovery speed and risk resistance of the supply chain. DCESC enterprises should fully utilize their strengths and existing policies to make informed decisions and implement scientifically sound emergency strategies.

• (2) From the perspective of enhancing supply chain emergency capabilities and benefits, collaborative cooperation emerges as the optimal mode, followed by cost-sharing contracts, while the non-cooperative mode shows the least improvement. However, the choice of cooperation mode varies depending on the type of enterprise. Resource supplier enterprises generally align with the overall supply chain strategy. For service operators, the non-cooperative mode is more advantageous than cost-sharing contracts when $0\, < \,t \,<\, {T}_{0}$, but cost-sharing contracts become more favorable when ${T}_{0} \,<\, t$. Enterprises should carefully evaluate the timing of introducing contracts and consider their own capacities as well as the willingness of partners to collaborate. This ensures that investments in emergency efforts align with actual needs and yield optimal results.

• (3) Across all three cooperation modes, DCESC’s emergency preparedness metrics and operational benefits demonstrate asymptotic convergence over extended time horizons. Collaborative cooperation significantly enhances the emergency efforts of supply chain enterprises, with gains in emergency capability improvement being substantially higher than those achieved through cost-sharing contracts. Governments should implement incentive measures to encourage DCESC enterprises to adopt collaborative cooperation, thereby increasing its likelihood. Emergency collaboration among enterprises is a key pathway to improving supply chain resilience. Through data exchange and the integration of information flows, enterprises can quickly adjust production and logistics strategies during unexpected events, achieving efficient resource allocation (Sarkis et al., 2020). This collaboration extends beyond emergency capability enhancement to include cross-enterprise data sharing and risk management mechanisms. Improving data utilization levels, particularly through big data analysis and artificial intelligence technologies, can significantly enhance enterprises’ ability to predict and respond to crises (Kache and Seuring, 2017).

• (4) The role of emergency data utilization in enhancing supply chain emergency capabilities cannot be underestimated. Insufficient data capabilities increase the vulnerability of the supply chain and complicate emergency response efforts. In supply chain emergencies, data serves as the foundation for information acquisition, sharing, and decision-making, as well as a critical component of resource allocation. Within complex supply chain networks, data sharing and integration can strengthen the supply chain’s ability to manage uncertain events (Ivanov and Dolgui, 2020). Additionally, the accumulation and analysis of historical data enable enterprises to develop more effective emergency plans, mitigating the impact of unexpected events on the supply chain (Dubey et al., 2018).

From a managerial perspective, this study highlights the critical significance of government subsidies in motivating enterprises to improve their emergency capabilities. Policymakers are advised to implement targeted subsidy policies, such as tax credits or direct financial support, to alleviate cost pressures on enterprises and encourage the adoption of advanced technologies. For managers, fostering collaboration among supply chain partners is essential, as it has been demonstrated to result in the most substantial enhancements in emergency capabilities and overall benefits. Additionally, the findings highlight the value of enhancing data utilization levels and establishing data-driven monitoring systems. These measures are likely to strengthen supply chain resilience by facilitating real-time risk identification and response. Governments and enterprises should collaboratively develop data-sharing platforms that facilitate information exchange and support more informed decision-making during emergencies. These insights offer a practical framework for stakeholders to enhance the ability of Data Center Engineering Supply Chains (DCESC) to respond to unexpected events and maintain project stability.

Recommendations and outlook

The choice of cooperation mode should align with actual conditions, as the required emergency efforts vary across the three modes. When DCESC enterprises possess strong capabilities and a high willingness to collaborate, collaborative cooperation represents the optimal approach. Otherwise, suitable modes should be selected based on specific cooperation contexts. The following recommendations are proposed:

• 1. Optimize Government Subsidy Policies, reduce emergency costs, support the procurement and reserve of key equipment, and encourage enterprises to enhance emergency capabilities.

• 2. Establish Data-Driven Monitoring Systems, implement supply chain monitoring systems to identify risks in advance, and develop precise emergency plans. Governments and enterprises should collaborate to build data-sharing platforms to promote information exchange.

• 3. Promote Supply Chain Collaboration, strengthen relationships with long-term partners to improve emergency capabilities, and foster collaborative cooperation within the supply chain.

• 4. Develop Multi-Level Supply Chain Networks, enhance emergency resource scheduling capabilities through multi-party cooperation, increasing the resilience and flexibility of the supply chain.

By implementing these measures, data center enterprises can effectively strengthen the supply chain’s emergency capabilities, ensure the smooth progression of projects, and maintain supply chain stability. Government subsidy policies play a critical role in enhancing DCESC’s emergency capabilities by incentivizing suppliers to adopt high-quality equipment and supporting service operators in implementing advanced design solutions. This, in turn, improves the sustainability and emergency responsiveness of DCESC.

The Data Center Engineering Supply Chains, encompassing service operators and resource suppliers, stand to gain significant advantages from government subsidies. These subsidies serve as a powerful incentive for the adoption of internationally advanced technologies, ultimately contributing to a notable enhancement in both the quality of data centers and their emergency response capabilities. Within the framework of global cooperation, domestic data centers can undertake international tasks through cross-border collaboration, alleviating global computing power pressures and enhancing their competitiveness. In the context of globalization, domestic data centers not only address local needs but also process international data through cross-border partnerships. By meeting some global data and computing demands, domestic data centers can further strengthen their position and influence within the global supply chain. This collaborative model aligns with the Eastern Data, Western Computing strategy, optimizing the scheduling of data and computing power between Eastern and Western regions to achieve efficient resource utilization. It also fosters cooperation with international data centers, promoting the interconnection of global data center networks.

Limitations and future research directions

While this study provides valuable insights into emergency management within Data Center Engineering Supply Chains, it is important to acknowledge several limitations that warrant attention.

From a modeling perspective, the current framework focuses on a two-tier supply chain system involving resource suppliers and service operators, relying on simplifying assumptions such as symmetric information among participants and linear relationships in benefit and cost functions. Although these assumptions facilitate tractable analysis and yield meaningful insights, they may not fully capture the complexities of real-world DCESC networks. In practice, DCESC networks often involve multiple stakeholders, including governments, multiple suppliers, and end-users, and information asymmetry is common, where one party may possess more or better information than another. Furthermore, nonlinear relationships, such as diminishing marginal returns or increasing marginal costs, are frequently observed in supply chain operations. For instance, in emergency preparedness, additional investments may yield progressively smaller improvements in emergency capabilities, while the costs associated with scaling up emergency responses may rise disproportionately.

These real-world complexities highlight limitations in the model’s ability to fully reflect dynamic supply chain behaviors. Relaxing the assumption of symmetric information, for example, could allow the model to incorporate scenarios where resource suppliers and service operators have varying levels of access to emergency-related information. This could lead to more nuanced decision-making processes, where parties with superior information might gain a strategic advantage, potentially influencing optimal cooperation modes and the overall enhancement of emergency capabilities. Similarly, introducing nonlinear relationships in benefit and cost functions could alter the optimal levels of government subsidies and the effectiveness of different cooperation modes, offering new perspectives on enhancing supply chain emergency capabilities.

To tackle these limitations, future research could investigate several directions. First, the model could be more comprehensively expanded to include additional stakeholders, such as governments, multiple suppliers, and end - users, so as to better reflect the multi-tiered nature of real-world supply chains. This would provide a more comprehensive understanding of how emergency capabilities can be enhanced across diverse supply chain configurations. Second, empirical research could be conducted to gather real-world data, offering stronger evidence-based guidance for improving DCESC emergency capabilities. Finally, future studies could examine the interconnections among all project phases—pre-construction, construction, and post-construction—to develop a more holistic view of DCESC resilience and identify strategies for enhancing emergency capabilities throughout the entire lifecycle of data center projects. By addressing these areas, the robustness of the model could be further improved, contributing to a more comprehensive understanding of supply chain emergency management.