Vaccine supply chain coordination using blockchain and artificial intelligence technologies

Abstract

Currently, the global spread of COVID-19 is taking a heavy toll on the lives of the global population. There is an urgent need to improve and strengthen the coordination of vaccine supply chains in response to this severe pandemic. In this study, we consider a vaccine supply chain based on a combination of artificial intelligence and blockchain technologies and model the supply chain as a two-player dynamic game with inventory level as the dynamic equation of the system. The study focuses on the applicability and effectiveness of the two technologies in the vaccine supply chain and provides management insights. The impact of the application of the technologies on environmental performance is also considered in the model. We also examine factors such as the number of people vaccinated, positive and side effects of vaccines, vaccine decay rate, revenue-sharing/cost-sharing ratio, and commission ratio. The results are as follows: the correlation between the difficulty in obtaining certified vaccines and the profit of a vaccine manufacturer is not monotonous; the vaccine manufacturer is more sensitive to changes in the vaccine attenuation rate. The study’s major conclusions are as follows: First, the vaccine supply chain should estimate the level of consumers’ difficulty in obtaining a certified vaccine source and the magnitude of the production planning and demand forecasting error terms before adopting the two technologies. Second, the application of artificial intelligence (AI) technology is meaningful in the vaccine supply chain when the error terms satisfy a particular interval condition.

1. Introduction

1.1. Background and motivation

According to World Health Organization statistics, from the first outbreak at the end of 2019 to December 2021, the COVID-19 virus infected 464,809,377 people and caused 6,062,536 deaths. The virus continues to wreak havoc worldwide (World-Health-Organization, 2022). A study published in Science showed that the coronavirus is responsible for a monthly average of hundreds of thousands of deaths, reducing the global gross domestic product by hundreds of billions of dollars per month and causing ongoing and cumulative damage to human health and education (Castillo et al., 2021). People dread the virus, and effective solutions are urgently needed to drastically improve the situation.

Research published in Science by Knipe et al. (2020) showed that vaccination is one of the most successful medical and public health measures implemented to date. Vaccination is now recognized as a cost-effective method to suppress or even eradicate infectious diseases, and vaccination has effectively controlled many infectious diseases (Yong et al., 2020). Vaccination is a long-term solution for the global response to the COVID-19 pandemic (Schaffer et al., 2020). However, Dai and Song (2021) show that as of mid-2021, several countries, including Canada, Israel, the United Kingdom, and the United States (US), have vaccinated over 40% of their populations; however, there are still 90 countries with fewer than 10 vaccine doses per 100 people. The global average vaccination coverage has remained as low as 10.91% as of June 1, 2021, especially in low- and middle-income countries (Johns Hopkins Coronavirus Resource Center, 2021).

However, the endgame of the COVID-19 pandemic is not the successful development of a vaccine but its sustained availability and effective vaccination (Dai & Song, 2021). The COVID-19 vaccine scandal in Peru in August 2020 reinforced the need to address the shortage of vaccine resources and provide timely and effective vaccination in the face of outbreaks worldwide (Estivariz et al., 2017).

A survey by Edwards and Hackell (2016) noted that the issue of vaccine reliability and accessibility is the most critical concern about vaccination, reflecting concerns about future vaccine supply chain issues. Historically, vaccine safety and supply incidents have had severe negative consequences. In 1955, two batches of polio vaccines from Cutter Laboratories were contaminated with live poliovirus because of supply chain mismanagement, resulting in 51 cases of permanent paralysis, 5 deaths, and transmission to families and community members (Nathanson & Langmuir, 1995). Estivariz et al. (2017) investigated the shortage of inactivated polio vaccines in Bangladesh between August and October 2015 and showed that it could be attributed to the vaccine supply–demand mismatch, which caused massive localized wastage, resulting in an overall vaccine shortage and a severe governmental breach of trust.

Since the publication of the genetic sequence of SARS-CoV-2 of COVID-19 on January 11, 2020, several vaccines have been developed by national research teams, with 115 vaccine candidates worldwide as of April 2020 (Thanh Le et al., 2020). As of June 1, 2021, the WHO has approved 8 vaccines for full use, 7 for restricted use, 30 for phase III trials, 34 for phase II trials, and 51 for phase I trials (Zimmer et al., 2022). Requirements for vaccine storage temperature and vaccination timeliness are stringent, posing new challenges to the vaccine supply chain. Let us consider the Pfizer-BioNTech vaccine as an example. Its long-term storage temperature must be between −80 °C and $-$60 °C. Tham (2022) showed that as of December 2020, the Pfizer-BioNTech vaccine is only usable for five days given standard refrigerator temperatures of 2 °C to 8 °C (i.e., 35.6°F and 46.4 °F) and that the interval between vaccinations is four weeks. Schifflfling and Breen (2022) showed that vaccine waste rates may be as high as 30% in some developed countries owing to poor vaccine cold chain logistics or vaccine inventory management.

In summary, strengthening vaccine supply chain management and ensuring its reliability and accessibility is urgent.

1.2. Literature review

Since the birth of blockchain technology in 2008, it has received much attention from business, academia, and the government, with increasingly widespread applications. Blockchain technology is an emerging technology platform that tracks and manages shipment activities in the supply chain using peer-to-peer, secure, and distributed ledgers without the involvement of intermediaries or trusted third parties (Hasan et al., 2019). Blockchain technology has emerged as an effective solution for drug supply chain management and related security issues, including vaccines (Haq & Muselemu, 2018). In March 2017, the US Food and Drug Administration began to apply blockchain technology to trace and track prescription drug use. In July 2018, China’s Henan Ziyun Cloud Computing Co. officially released a vaccine traceability platform based on blockchain technology. When used for vaccines, the technology ensures that the vaccine information in circulation is valid and not tampered with, thus maximizing the traceability and reliability of vaccines. From the consumers’ perspective, a vaccine information traceability platform built on blockchain technology can make it easier to find certified vaccine sources and increase the credibility of vaccine information, thus increasing vaccine reliability. Several scholars have investigated vaccine supply chains based on blockchain technology. Yong et al. (2020) developed a “blockchain vaccine” system based on blockchain and machine learning technology to support vaccine traceability and smart contract functionality and address vaccine expiration and vaccine record fraud. Chauhan et al. (2021) proposed a blockchain-based solution to improve the security and transparency of COVID-19 vaccine traceability and monitor vaccine supply and distribution through intelligent contracts. Shah et al. (2021) provided a comprehensive review of blockchain technology application in the COVID-19 pandemic, presented a case study of a digital vaccine passport based on this technology, and analyzed its complexity. Bamakan et al. (2021) proposed a variety of solutions for blockchain technology in the medical cold chain.

Artificial intelligence (AI) is the core technology of Industry 4.0. It creates new economic and business value by analyzing large amounts of data to identify relevant content, improve operations, and provide actionable recommendations (Richardson, 2021). Several scholars have studied AI-based vaccine supply chains. Arora et al. (2021) focused on the potential application of AI technologies to COVID-19 surveillance, diagnosis, outcome prediction, drug discovery, and vaccine development, discussing the clinical utility of AI-based models and the limitations and challenges faced by AI systems. Golan et al. (2021) proposed that AI technology helps supply chain managers better quantify efficiency/resilience trade-offs across all associated networks/domains and support optimal system performance post disruption.

The vaccine supply chain is a particular class of physical supply chains for perishable products, and dynamic changes in vaccine inventory level will seriously affect the coordination of the vaccine supply chain. Physical supply chains based on AI or Vendor Managed Inventory (VMI) model have been amply researched. Naz et al. (2021) proposed a framework and urgent recommendations for improving supply chain resilience based on AI technology in response to frequent supply chain disruptions caused by outbreaks, which can help researchers and practitioners improve supply chain management. De Giovanni (2021) modeled a dynamic game of a physical supply chain with VMI model to assess the benefits of utilizing AI technology to improve supply chain coordination. Riewpaiboon et al. (2015) compared and performed economic analyses of the traditional and VMI models in the Thai vaccine supply chain system, concluding that the VMI model would improve vaccine supply chain efficiency. VMI model means the vaccine manufacturer located upstream directly controls the downstream vaccination unit’s inventory to ensure better vaccine supply. In VMI model, the reliability and accessibility of the vaccine supply chain is enhanced because the vaccine manufacturer directly manages the inventory. In addition, VMI model has strong applicability to the vaccine supply chain for the following reasons.

• •Since the vaccine manufacturer directly manages the inventory of the vaccination unit in the VMI model, the vaccine manufacturer has the quickest access to all information about the inventory of the vaccination unit, which not only creates the conditions for the overall vaccine supply chain synergy, but also makes the necessary preparations for the addition of new technologies.
• •VMI is a typical physical supply chain management model, and the physical attributes of the vaccine supply chain confirm its applicability.
• •As the shelf life of vaccines is minimal, the VMI model can enable vaccines to reach vaccinators faster and in time, thereby reducing vaccine losses.
• •The VMI model can effectively avoid duplicate inventory, reduce inventory costs, lower inventory risks, ensure vaccine inventory accuracy, and enhance vaccine supply chain coordination.

Since the pandemic, many studies have been done on vaccine supply chain management. Niu et al. (2020) established a static game model to explain the preference of overseas vaccine suppliers for exclusive and Competitive Retailing strategies, aiming at the profit and social responsibility of overseas vaccine suppliers. Liu et al. (2021) applied blockchain technology to vaccine supply chain management and constructed a static game model to reveal the impact of blockchain on the vaccine supply chain. Sun et al. (2021) analyzed the risk of vaccine production management and established the integration mode and outsourcing mode of vaccine supply chain by using the static game model. Xie et al. (2021) analyzed subsidy selection in a vaccine supply chain with risk-averse buyers using evolutionary game methods. Shamsi et al. (2018) used optimal control theory, Stackelberg game model, and a nonlinear programming method to construct SIR epidemic model to minimize procurement and social costs in the vaccine supply chain. Pan et al. (2022) discussed the impact of public and private hospital supply on the vaccine supply chain.

1.3. Research gap and objective

At present, the studies of scholars do not involve the temporal impact of industry 4.0 technology on the dynamic game results of all parties in modern vaccine supply chain management, the details are shown in Table 1.

Table 1.

Summary of different existing content/model in previous literature.

Literature category	Authors	Dynamic game model	Vaccine supply chain	AI/Blockchain technology
	Yong et al. (2020)	–	✓	AI and Blockchain
	De Giovanni (2021)	✓	–	AI
	Liu et al. (2021)	–	✓	Blockchain
	Chauhan et al. (2021)	–	✓	Blockchain
New technology	Shah et al. (2021)	–	✓	Blockchain
	Bamakan et al. (2021)	–	✓	Blockchain
	Arora et al. (2021)	–	✓	AI
	Naz et al. (2021)	–	✓	AI
	Golan et al. (2021)	–	✓	AI
	Niu et al. (2020)	–	✓	–
	Sun et al. (2021)	–	✓	–
Supply chain management	Xie et al. (2021)	✓	✓	–
	Shamsi et al. (2018)	✓	✓	–
	Pan et al. (2022)	–	✓	–
	This paper	✓	✓	AI and Blockchain

Based on the robust results of existing research, this study considers a vaccine supply chain based on a combination of AI and blockchain technologies and models the supply chain as a two-player dynamic game with inventory level as the dynamic equation of the system. The application of blockchain technology is likely to improve the security of the vaccine supply chain and the accessibility of vaccine consumers. The application of AI technology is expected to enhance the resilience of the vaccine supply chain and build a flexible vaccine supply chain to coordinate the production plans and demand forecasts of vaccines and avoid redundancies and shortages. As far as we know, this paper is the only one that uses the differential game method to build a dynamic game model and considers the combination of AI technology and blockchain technology in the model to improve vaccine supply chain coordination.

The rest of the paper is organized as follows: Section 2 introduces the two dynamic game models proposed in this study; Section 3 presents the equilibrium results of the models; Section 4 compares these equilibrium results and offers management insights; and Section 5 presents the study’s conclusions and outlook.

2. Model

2.1. The basic assumptions

According to Davido et al. (2021), Given that antibody levels are declining over time and that the danger of contracting the infection often reappears after a few months, Covid booster shots should be suggested to and advised to all fully immunized individuals. This category of vaccine supply chain has the following characteristics:

• •The demand for a continuous supply of vaccines.
• •The wide area covered by vaccines.
• •The huge business volume.
• •The high vaccine penetration rate (especially under government leadership).
• •The short response time required.

In this study, we assumed that the supply of vaccine is a continuous process in an infinite time interval. The vaccine supply chain in this paper represents a vaccine supply chain with a continuous supply of vaccines. We assume a two-player supply chain consisting of a vaccine manufacturer (the leader M, A vaccine manufacturer, such as Pfizer.) and a vaccination unit (the follower U, A facility that provides vaccination services, such as a hospital or a government-run vaccination site (Colombo et al., 2006).) and model this as a dynamic game in a time interval $t\in \left[0,\infty \right)$. M is located upstream of the vaccine supply chain and decides the optimal production efficiency strategy $u\left(t\right)$. U is located downstream of the vaccine supply chain and decides the optimal sales price strategy $p\left(t\right)$. M needs to pay the production cost $c$ per vaccine. The players share revenues according to the exogenous revenue-sharing/cost-sharing ratio $\varphi$ ($\varphi \in \left(0,1\right)$) agreed upon in the sharing contract signed before the collaboration. It is assumed that the state variable is the inventory level $Y\left(t\right)$. The two players develop their respective strategies based on the current inventory level $Y\left(t\right)$. This approach to the inventory level as the state variable is necessary and it sets the stage for the important role of blockchain technology to be reflected later. This setup method is similar to De Giovanni (2021) and He et al. (2020). Therefore, strategies $u\left(t\right)$ and $p\left(t\right)$, on both sides, are functions of the state variable.

M and U partner strategically with two technology service providers, BVP and G. BVP provides blockchain technology services to the vaccine supply chain to enhance the traceability of vaccines and increase the transparency of vaccine products in circulation, thereby reducing the consumers’ effort in locating certified vaccine sources. G provides AI technology services to the vaccine supply chain. When M implements a production plan, it generates a vaccine production error term ${ϵ}_U$, which is the difference in quantity between the actual and planned production. Similarly, when U forecasts the market for the vaccine, it generates a vaccine demand error term ${ϵ}_D$, the quantitative difference between the actual and projected vaccine demand. G eliminates these errors (${ϵ}_U,{ϵ}_D$) by building an AI system for the vaccine supply chain to accurately match production efficiency and demand forecasts. We use superscripts $B$ and $AI$ to identify the basic dynamic game model that applies only blockchain technology and the AI dynamic game model that applies both blockchain technology and AI technology, respectively. A similar assumption approach can be seen in De Giovanni (2021).

2.2. Basic dynamic game model with blockchain technology application only (B-model)

First, we assume that the vaccine demand is $D^B\left(t\right)$. The vaccine price is $p^B\left(t\right)$ ($p^B\left(t\right)>0$), which is negatively correlated with the utility of vaccination $Z$, i.e., the utility of vaccination decreases as the price of vaccine $p^B\left(t\right)$ increases, and its utility conversion coefficient is $\beta$ ($\beta >0$). The inventory level is $Y^B\left(t\right)$ ($Y^B\left(t\right)>0$), which is positively correlated with the utility of vaccination $Z$, that is, the utility of vaccination $Z$ increases as the inventory level $Y^B\left(t\right)$ increases. The more inventory at a vaccination unit, the more consumers are willing to go to that unit for vaccination; its utility conversion coefficient is $\alpha$ ($\alpha >0$). Under the same conditions, consumers are more likely to spend in shopping places that offer a greater quantity of goods and better quality and service. This setup method is similar to Liu et al. (2021).

Second, we assume that the number of potential consumers is $n$. The perceived value of the vaccine to the consumers is $v$, and its distribution function $f\left(v\right)$ follows a uniform distribution on the interval $\left(0,1\right)$. The positive effect of vaccination on the consumers is $s$ ($s>0$), the consumers’ difficulty in obtaining a certified vaccine source (the acquisition effort) before vaccination is $W$ ($W>0$), and the increase in difficulty has a negative utility on the vaccination utility of the vaccine. The negative utility conversion coefficient is $\gamma$ ($\gamma >0$), and the side effects $\theta$ of vaccination, such as discomfort and other adverse effects, also have a negative utility of vaccination $Z$. The assumptions here are referenced to Liu et al. (2021). Therefore, we express the utility of vaccination by the following equation:

$$Z=v+\alpha Y^B\left(t\right)-\beta p^B\left(t\right)-\gamma W-\theta +s,$$ (1)

when $Z>0$, consumers receive the vaccine. Therefore, the number of people who eventually choose to be vaccinated, that is, the demand for vaccines, is

$$\begin{array}{ccc}\hfill D^B\left(t\right)\hfill & \hfill =\hfill & \hfill {\int }_{-\alpha Y^B\left(t\right)+\beta p^B\left(t\right)+\gamma W+\theta -s}^{1}f\left(v\right)dv\hfill \\ \hfill \hfill & \hfill =\hfill & \hfill n\left(1+\alpha Y^B\left(t\right)-\beta p^B\left(t\right)-\gamma W-\theta +s\right).\hfill \end{array}$$ (2)

$Y^B\left(t\right)$ ($Y^B\ge 0$) represents the inventory accumulated over time according to the following dynamic equation:

$${\dot{Y}}^B\left(t\right)=u^B\left(t\right)\left(1+{ϵ}_U\right)-D^B\left(t\right)\left(1+{ϵ}_D\right).$$ (3)

This dynamic equation expresses inventory accumulation per unit time as the difference between vaccine production per unit time (productivity $u^B\left(t\right)$) and vaccine demand per unit time ($D^B\left(t\right)$), where ${ϵ}_U,{ϵ}_D$ are the production efficiency and demand forecast error terms, respectively. Notably, the state variable $Y\left(t\right)$ represents the change in the physical vaccine inventory, which is a physical quantity change. $Y\left(t\right)$ differs from state variables like brand value and goodwill, whose error terms can continuously and systematically affect the inventory state variable. Unlike the traditional error term, the error term in this paper is independent of time, state and policy, which is the same as the second method in Sethi (1983).

Based on the current extensive application and practice of AI technology in industry 4.0 manufacturing and supply systems, Section 2.3 will consider eliminating these error terms from the model, which will become feasible with the help of modern AI technology. Tao et al. (2018) pointed out that AI technology has the characteristic of proactively adapting to productivity amplitude, thus helping M eliminate random errors in production efficiency. Frank et al. (2019) pointed out that artificial intelligence technology, by its big data analysis ability, could realize amplitude instability and the last kilometer problem of long-term demand prediction, thus helping U eliminate demand prediction error. Gilchrist (2016) described and discussed the application and practice of artificial intelligence technology in modern industry in detail, which further supported our view.

In the B-model, M and U enter into a strategic partnership through a contract. Both players’ revenues as per the contract are distributed according to a revenue-sharing/cost-sharing ratio $\varphi$. The marginal profit of M is ${\pi }_M^B=p^B\left(t\right)\varphi -c$ and the marginal profit of U is ${\pi }_U^B=p^B\left(t\right)\left(1-\varphi \right)$, where $c$ is the vaccine’s marginal cost of production.

Referring to Erickson (2011), let the vaccine production cost $C_u^B$ be

$$C_u^B\left(u^B\left(t\right),t\right)=\frac{h}{2}{\left(u^B\left(t\right)\right)}^2,$$ (4)

where $h$ is the production cost conversion factor.

Referring to Ben-Daya et al. (2013), let the inventory cost $C_h^B$ of U located downstream be:

$$C_h^B=c_h{\left(Y^B\left(t\right)\right)}^2,$$ (5)

where $c_h$ denotes the inventory cost conversion factor. After negotiations between M and U, both players decide to share the inventory cost using the revenue-sharing/cost-sharing ratio $\varphi$. The players’ shares can be expressed as $\varphi c_h{\left(Y^B\left(t\right)\right)}^2$ and $(1-\varphi )c_h{\left(Y^B\left(t\right)\right)}^2$, respectively.

In addition, we also consider the environmental and profit impacts of vaccine production emissions. Assume that $\omega$ is the emissions per vaccine, and $\overline{e}$ is the upper limit of emissions set by the local government. Then, the profit of M affected by emissions $E^B$ is

$$E^B=c_e\left(\omega u^B\left(t\right)-\overline{e}\right).$$ (6)

When $\omega u^B\left(t\right)>\overline{e}$, M pays a penalty to the government; when $\omega u^B\left(t\right)<\overline{e}$, the government pays a subsidy to M. The conversion factor used is $c_e$.

Assume that the number of vaccine losses because of mishandling in vaccine transportation, storage, warehousing, and network operations, as a percentage of vaccine demand ($D^B\left(t\right)$), is $\lambda$ ($\lambda >0$); the total vaccine loss is $\lambda D^B\left(t\right)$, and the vaccine loss causes lowers profits for U, which is expressed as

$$L^B=\lambda D^B\left(t\right){\pi }_U^B=\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi ).$$ (7)

After negotiations between M and U, both players decide to share the decrease in profit $L^B$ using the contracted revenue-sharing/cost-sharing ratio $\varphi$. The two players’ shares can be expressed as $\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi )\varphi$ and $\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi ){}^2$, respectively.

BVP is committed to applying blockchain technology to build a product traceability service platform, improve product information transparency and the difficulty of information tampering, and develop blockchain applications to control consumers’ difficulty in obtaining certified vaccine sources. After negotiations, BVP’s fees are paid from the total revenues of both M and U, at a commission ratio $\eta$. The blockchain fee to be paid by M is

$$F_M^B=\eta D^B\left(t\right)\left(1+\lambda \right)\left(p^B\left(t\right)\varphi -c\right).$$ (8)

The blockchain fees to be paid by U is

$$F_U^B=\eta D^B\left(t\right)p^B\left(t\right)(1-\varphi ).$$ (9)

The BVP company removes its own costs after collecting the blockchain service fees paid by M and U, which is the profit of BVP. According to the results of Eqs. (8), (9), BVP’s profit is obtained as

$${\Pi }_{BVP}^B=F_M^B+F_U^B-c_{BVP}{D}^B\left(t\right)\left(1+\lambda \right),$$ (10)

where $c_{BVP}$ ($c_{BVP}>0$) is the retrospective marginal cost per vaccine borne by the vaccine traceability service platform.

Finally, considering the above components affecting the expected profits and benefits, M’s total profit ${\Pi }_M^B$ is

$${\Pi }_M^B=\underset{u^B\left(t\right)}{max}{\int }_0^{+\infty }{e}^{-\rho t}\left\{\begin{array}{c}{D}^B\left(t\right)\left(1+\lambda \right)\left(p^B\left(t\right)\varphi -c\right)\hfill \\ -F_M^B-\varphi L^B-\varphi C_h^B-C_u^B-E^B\hfill \end{array}\right\}dt$$ (11)

U’s total profit ${\Pi }_U^B$ is

$${\Pi }_U^B=\underset{p^B\left(t\right)}{max}{\int }_0^{+\infty }{e}^{-\rho t}\left\{\begin{array}{c}{D}^B\left(t\right)p^B\left(t\right)(1-\varphi )-F_U^B\hfill \\ -L^B(1-\varphi )-(1-\varphi )C_h^B\hfill \end{array}\right\}dt$$ (12)

M and U have the same discount factor $\rho$ ($\rho >0$).

Referring to Liu et al. (2021), this study uses consumer surplus (CS) to denote the impact on the utility of consumers and social welfare (SW) to denote the impact on the sum of the utilities of all vaccine supply chain participants. In the B-model, the $C{S}^B$ is

$$\begin{array}{ccc}\hfill C{S}^B\hfill & \hfill =\hfill & \hfill n{\int }_{\genfrac{}{}{0ex}{}{-\alpha Y^B\left(t\right)+\beta p^B\left(t\right)}{+\gamma W+\theta -s}}^1\left(\begin{array}{c}v+\alpha Y^B\left(t\right)-\beta p^B\left(t\right)\hfill \\ -\gamma W-\theta +s\hfill \end{array}\right)f\left(v\right)dv\hfill \\ \hfill \hfill & \hfill =\hfill & \hfill \frac{n}{2}(1+\alpha Y^B\left(t\right)-\beta p^B\left(t\right)-\gamma W-\theta +s){}^2.\hfill \end{array}$$ (13)

The $S{W}^B$ is

$$S{W}^B={\Pi }_M^B+{\Pi }_U^B+{\Pi }_{BVP}^B+C{S}^B.$$ (14)

2.3. Dynamic game model applying AI technology and blockchain technology (AI-model)

In the AI-model, M and U decide to adopt AI technology to aid vaccine supply chain collaboration and approach G, which specializes in AI technology solutions for manufacturing companies. Owing to the adoption of AI technology, the marginal manufacturing profit of M is ${\pi }_M^{AI}=p^{AI}\left(t\right)\varphi -c-c_{AI}$, where $c_{AI}$ represents the impact of AI technology adoption on M’s marginal cost, and U’s marginal revenue is ${\pi }_U^{AI}=p^{AI}\left(t\right)\left(1-\varphi \right)$.

The benefit of using AI technology is the elimination of the errors ${ϵ}_D,{ϵ}_U$ that appear in Eq. (3) in the B-model and their negative impact on the operational decisions of the vaccine supply chain, thereby making the dynamic supply–demand coordination more intelligent. The AI-model inventory-level dynamic equation is as follows:

$${\dot{Y}}^{AI}\left(t\right)=u^{AI}\left(t\right)-D^{AI}\left(t\right),$$ (15)

where $u^{AI}\left(t\right),D^{AI}\left(t\right)$ represent the production efficiency and demand, respectively. The functional forms of the demand ($D^{AI}\left(t\right)$), production cost ($C_u^{AI}$), and inventory cost ($C_h^{AI}$) in this model are the same as those in Eqs. (2), (4), (5) in the B-model, respectively. The total cost paid by the vaccine supply chain for G’s AI services is:

$${\Pi }_G^{AI}=A\left(A+\delta \left({\left({ϵ}_u\right)}^2+{\left({ϵ}_D\right)}^2\right)+k{Y}^{AI}\left(t\right)\right),$$ (16)

where $A$ is the cost to G to allow the vaccine supply chain access to the “AI Cloud” (Intelligent Cloud Service); $\delta$ is the conversion factor for the fee charged by G based on the magnitude of the error term eliminated from the vaccine supply chain; and $k$ is the conversion factor for the fee charged by G based on the vaccine supply chain’s inventory level. According to the revenue-sharing/cost-sharing ratio $\varphi$, the AI service costs must be shared by M and U. Therefore, the shares paid by M and U are $\varphi {\Pi }_G^{AI}$ and $\left(1-\varphi \right){\Pi }_G^{AI}$, respectively.

In addition, the adoption of AI technology changes M’s emissions. In the AI-model, M’s profit, affected by emissions $E^{AI}$, is

$$E^{AI}=c_e\left(\left(\omega +{\omega }_{AI}\right)u^{AI}\left(t\right)-\overline{e}\right),$$ (17)

where ${\omega }_{AI}$ is the increase in emissions per vaccine owing to AI technology adoption. This setting enables the vaccine supply chain to make the right decision regarding AI technology adoption.

In this model, the functional forms of the lost profit because of vaccine loss $L^{AI}$, the blockchain cost to be paid by M $F_M^{AI}$, the blockchain cost to be paid by U $F_U^{AI}$, and BVP’s profit ${\Pi }_{BVP}^{AI}$ are consistent with the functional forms in Eqs. (7), (8), (9), and (10) in the B-model, respectively.

Finally, considering the above components affecting the expected profits and benefits, M’s total profit ${\Pi }_M^{AI}$ in the AI-model is:

$${\Pi }_M^{AI}=\underset{u^{AI}\left(t\right)}{max}{\int }_0^{+\infty }{e}^{-\rho t}\left\{\begin{array}{c}\hfill D^{AI}\left(t\right)\left(1+\lambda \right)\left(p^{AI}\left(t\right)\varphi -c-c_{AI}\right)\hfill \\ \hfill \begin{array}{c}-F_M^{AI}-\varphi L^{AI}-C_u^{AI}\hfill \\ -E^{AI}-\varphi C_h^{AI}-\varphi {\Pi }_G^{AI}\hfill \end{array}\hfill \end{array}\right\}dt$$ (18)

U’s total profit ${\Pi }_U^{AI}$ is

$${\Pi }_U^{AI}={\begin{array}{c}\hfill max\hfill \end{array}}_{p^{AI}\left(t\right)}{\int }_0^{+\infty }{e}^{-\rho t}\left\{\begin{array}{c}{D}^{AI}\left(t\right)p^{AI}\left(t\right)\left(1-\varphi \right)\hfill \\ -F_U^{AI}-\left(1-\varphi \right)L^{AI}\hfill \\ -\left(1-\varphi \right)C_h^{AI}-\left(1-\varphi \right){\Pi }_G^{AI}\hfill \end{array}\right\}dt$$ (19)

M and U have the same discount factor $\rho$ ($\rho >0$). In the AI-model, the $C{S}^{AI}$ is

$$\begin{array}{ccc}\hfill C{S}^{AI}\hfill & \hfill =\hfill & \hfill n{\int }_{\genfrac{}{}{0ex}{}{-\alpha Y^{AI}\left(t\right)+\beta p^{AI}\left(t\right)}{+\gamma W+\theta -s}}^1\left(\begin{array}{c}v+\alpha Y^{AI}\left(t\right)-\beta p^{AI}\left(t\right)\hfill \\ -\gamma W-\theta +s\hfill \end{array}\right)f\left(v\right)dv\hfill \\ \hfill \hfill & \hfill =\hfill & \hfill \frac{n}{2}(1+\alpha Y^{AI}\left(t\right)-\beta p^{AI}\left(t\right)-\gamma W-\theta +s){}^2.\hfill \end{array}$$ (20)

The $S{W}^{AI}$ is

$$S{W}^{AI}={\Pi }_M^{AI}+{\Pi }_U^{AI}+{\Pi }_{BVP}^{AI}+{\Pi }_G^{AI}+C{S}^{AI}.$$ (21)

All notations and definitions have been listed in Appendix C, as shown in Table C.5.

3. Equilibria

Drawing on Fershtman and Kamien (1987), this section analyzes the results of Nash equilibrium calculations for the B-model and the AI-model. We propose the perfect closed-loop Nash equilibrium strategy for the subgame in both models $u_{SS},p_{SS}$. Furthermore, we derive the equilibrium trajectory $Y\left(t\right)$ of the inventory level and specify its stability condition and calculate the equilibrium solutions for the other indicators (denoted by the subscript $SS$). For brevity, the time variable $t$ is omitted in the following calculations.

3.1. B-model calculation results

In the B-model, ${ϵ}_U,{ϵ}_D$ are not effectively eliminated because of a lack of AI technology, resulting in a mismatch between production planning and actual demand that makes supply chain collaboration challenging and highly detrimental to consumers.

3.1.1. Proposition 1

Assuming an interior solution, the equilibrium productivity and price strategies in the B-model are given by:

$$u_{SS}^B=\frac{\left(Y_{SS}^B{M}_1^B+M_2^B\right)\left(1+{ϵ}_U\right)-\omega c_e}{h},$$ (22)

$$p_{SS}^B=\frac{1}{2}\left(\frac{1+s+Y_{SS}^B\alpha -\gamma W-\theta }{\beta }+\frac{\left(Y_{SS}^B{U}_1^B+U_2^B\right)\left(1+{ϵ}_D\right)}{\left(-1+\varphi \right)\left(-1+\eta +\lambda -\lambda \varphi \right)}\right).$$ (23)

The equilibrium profits of M and U are

$$\begin{array}{c}{\Pi }_{MSS}^B=\frac{M_1^B}{2}\left(Y_{\mathrm{SS}}^B\right){}^2+M_2^B{Y}_{\mathrm{SS}}^B+M_3^B,\hfill \\ {\Pi }_{USS}^B=\frac{U_1^B}{2}\left(Y_{\mathrm{SS}}^B\right){}^2+U_2^B{Y}_{\mathrm{SS}}^B+U_3^B,\hfill \end{array}$$ (24)

where $M_i^B,U_i^B$ ($i=1...3$) are the Riccati coefficients. The equilibrium trajectory of the inventory level is:

$$\begin{array}{cc}\hfill Y^B\left(t\right)=\hfill & \hfill \left(1-e^{t\left\{\begin{array}{c}(1+{ϵ}_D)(\beta U_1^B(1+{ϵ}_D)\hfill \\ -\alpha (1-\varphi )(1-\eta -\lambda \left(1-\varphi \right)))\hfill \\ +2(1-\varphi )(1-\eta \hfill \\ -\lambda \left(1-\varphi \right))M_1^B(1+{ϵ}_U){}^2\hfill \end{array}\right\}}\right)Y_{SS}^B\hfill \\ \hfill \hfill & \hfill +e^{t\left\{\begin{array}{c}(1+{ϵ}_D)(\beta U_1^B(1+{ϵ}_D)\hfill \\ -\alpha (1-\varphi )(1-\eta -\lambda \left(1-\varphi \right)))\hfill \\ +2(1-\varphi )(1-\eta \hfill \\ -\lambda \left(1-\varphi \right))M_1^B(1+{ϵ}_U){}^2\hfill \end{array}\right\}}{Y}_0^B,\hfill \end{array}$$ (25)

where

$$Y_{SS}^B=-\frac{\left(\begin{array}{c}hn(1+{ϵ}_D)(-(1+s-\gamma W-\theta )(-1+\varphi )(-1+\eta \hfill \\ +\lambda -\lambda \varphi )+\beta U_2^B(1+{ϵ}_D))-2(-1+\varphi )(-1+\eta \hfill \\ +\lambda -\lambda \varphi )\omega c_e(1+{ϵ}_U)+2(-1+\varphi )(-1+\eta +\lambda \hfill \\ -\lambda \varphi )M_2^B(1+{ϵ}_U){}^2\hfill \end{array}\right)}{\left(\begin{array}{c}hn(1+{ϵ}_D)(-\alpha (-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )\hfill \\ +\beta U_1^B(1+{ϵ}_D))+2(-1+\varphi )(-1+\eta \hfill \\ +\lambda -\lambda \varphi )M_1^B(1+{ϵ}_U){}^2\hfill \end{array}\right)},$$ (26)

is the equilibrium inventory level. This equilibrium is globally asymptotically stable if and only if:

$$\left(\begin{array}{c}hn(1+{ϵ}_D)(\alpha (1-\varphi )(1-\eta -\lambda \left(1-\varphi \right))-\beta U_1^B(1+{ϵ}_D))\hfill \\ -2(1-\varphi )(1-\eta -\lambda \left(1-\varphi \right))M_1^B(1+{ϵ}_U){}^2\hfill \end{array}\right)<0.$$

The equilibrium demand can be calculated as

$$D_{SS}^B=\frac{1}{2}n(1+s+Y_{SS}^B\alpha -\gamma W-\theta -\frac{\beta (Y_{SS}^B{U}_1^B+U_2^B)(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}),$$ (27)

The equilibrium BVP profit is

$${\Pi }_{BVPSS}^B=\eta D_{SS}^B(1+\lambda )(p_{SS}^B\varphi -c)+\eta D_{SS}^B{p}_{SS}^B(1-\varphi )-c_{BVP}{D}_{SS}^B(1+\lambda ).$$ (28)

The equilibrium CS is

$$C{S}_{SS}^B=\frac{1}{8}n{\left(\begin{array}{c}-1-s-Y_{SS}^B\alpha +\gamma W+\theta \hfill \\ +\frac{\beta (Y_{SS}^B{U}_1^B+U_2^B)(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}\hfill \end{array}\right)}^2.$$ (29)

The equilibrium SW is

$$S{W}_{SS}^B=\left(\begin{array}{c}\hfill D_{SS}^B{p}_{SS}^B\eta (1-\varphi )+D_{SS}^B\eta (1+\lambda )(-c+p_{SS}^B\varphi )\hfill \\ \hfill -D_{SS}^B(1+\lambda )c_{BVP}+\frac{1}{2}\left(Y_{SS}^B\right){}^2{M}_1^B+Y_{SS}^B{M}_2^B+M_3^B\hfill \\ \hfill +\frac{1}{2}\left(Y_{SS}^B\right){}^2{U}_1^B+Y_{SS}^B{U}_2^B+U_3^B\hfill \\ \hfill +\frac{1}{8}n(-1-s-Y_{SS}^B\alpha +W\gamma +\theta +\frac{\beta (Y_{SS}^B{U}_1^B+U_2^B)(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}){}^2\hfill \end{array}\right).$$ (30)

From Proposition 1, it follows that $u_{SS}^B,p_{SS}^B,D_{SS}^B,C{S}_{SS}^B,S{W}_{SS}^B,{\Pi }_{MSS}^B,{\Pi }_{USS}^B$ depend on the state of the inventory ($Y_{SS}^B$) and are related to the value of $M_i^B,U_i^B$ ($i=1...3$). As the inventory-level dynamic equation is a linear quadratic equation with fully coupled coefficients $M_1^B,U_1^B$, the benchmark parameter values must be determined to solve the Riccati system before analyzing the individual parameters and obtaining management insights. Referring to El Ouardighi et al. (2008) and Prasad and Sethi (2004), the benchmark parameter values are set as listed in Table C.6.

The values of ${ϵ}_D$ and ${ϵ}_U$ in Table C.6 are fixed and set assuming the following: On the one hand, ${ϵ}_D$ and ${ϵ}_U$ constantly and systematically affect the entire vaccine supply chain and lead to a logical solution; on the other hand, these can help enterprises better determine whether the error term lies within the valid region, providing a basis for deciding upon AI technology adoption. The case for non-fixed values of ${ϵ}_D$ and ${ϵ}_U$ is discussed in Section 4.

To determine the value of $M_i^B,U_i^B$ ($i=1...3$), we substitute the values in Table C.6 into the Riccati system to obtain two sets of real roots. Please refer to Appendix C for details.

3.1.2. Corollary 1

• •The trajectory of the equilibrium inventory level $Y_{SS}^B$ is monotonic. When $Y_0^B<Y_{SS}^B$, $Y^B\in \left[Y_0^B,Y_{SS}^B\right]$; when $Y_0^B>Y_{SS}^B$, $Y^B\in \left[Y_{SS}^B,Y_0^B\right]$.
• •The trajectory of $u_{SS}^B\left(Y_{SS}^B\right),p_{SS}^B\left(Y_{SS}^B\right),D_{SS}^B\left(Y_{SS}^B\right),{\Pi }_{MSS}^B\left(Y_{SS}^B\right),{\Pi }_{USS}^B\left(Y_{SS}^B\right)$ decreases monotonically.

(Proof: omitted)

3.2. The AI-model calculation results

In the AI-model, the production planning error ${ϵ}_U$ and demand forecasting error ${ϵ}_D$ in the production and supply processes of the vaccine supply chain, respectively, are effectively eliminated using AI technology. This helps in coordinating production planning and actual demand.

The vaccine supply chain pays G to access AI cloud services, eliminate errors, and effectively control inventory levels. M and U share costs according to the revenue-sharing/cost-sharing ratio $\varphi$.

3.2.1. Proposition 2

Assuming an interior solution, the equilibrium productivity and price strategies in the AI-model are given by:

$$u_{\mathrm{SS}}^{\mathrm{AI}}=\frac{(Y_{\mathrm{SS}}^{\mathrm{AI}}{M}_1^{\mathrm{AI}}+M_2^{\mathrm{AI}})-c_e(\omega +{\omega }_{\mathrm{AI}})}{h},$$ (31)

$$p_{\mathrm{SS}}^{\mathrm{AI}}=\frac{1}{2}(\frac{1+s+Y_{\mathrm{SS}}^{\mathrm{AI}}\alpha -W\gamma -\theta }{\beta }+\frac{Y_{\mathrm{SS}}^{\mathrm{AI}}{U}_1^{\mathrm{AI}}+U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}).$$ (32)

M and U’s equilibrium profits are

$$\begin{array}{ccc}\hfill {\Pi }_{MSS}^{AI}\hfill & \hfill =\hfill & \hfill \frac{M_1^{AI}}{2}\left(Y_{\mathrm{SS}}^{AI}\right){}^2+M_2^{AI}{Y}_{\mathrm{SS}}^{AI}+M_3^{AI},\hfill \\ \hfill {\Pi }_{USS}^{AI}\hfill & \hfill =\hfill & \hfill \frac{U_1^{AI}}{2}\left(Y_{\mathrm{SS}}^{AI}\right){}^2+U_2^{AI}{Y}_{\mathrm{SS}}^{AI}+U_3^{AI}.\hfill \end{array}$$ (33)

where $M_i^{AI},U_i^{AI}$ ($i=1...3$) are the Riccati coefficients. The equilibrium trajectory of the inventory level is:

$$\begin{array}{cc}\hfill Y^{AI}\left(t\right)=\hfill & \hfill \left(1-e^{t\left\{\begin{array}{c}\left(2M_1^{AI}-hn\alpha \right)(1-\varphi )(1-\eta \hfill \\ -\lambda \left(1-\varphi \right))+hn\beta U_1^{AI}\hfill \end{array}\right\}}\right)Y_{SS}^{AI}\hfill \\ \hfill \hfill & \hfill +e^t\left\{\begin{array}{c}\left(2M_1^{AI}-hn\alpha \right)(1-\varphi )(1-\eta \hfill \\ -\lambda \left(1-\varphi \right))+hn\beta U_1^{AI}\hfill \end{array}\right\}{Y}_0^{AI},\hfill \end{array}$$ (34)

where

$$Y_{SS}^{AI}=\frac{\left(\begin{array}{c}-(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )(hn(1+s\hfill \\ -W\gamma -\theta )-2M_2^{AI})+hn\beta U_2^{AI}\hfill \\ -2(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )c_e(\omega +{\omega }_{AI})\hfill \end{array}\right)}{\left(\begin{array}{c}(-1+\varphi )(-1+\eta +\lambda \hfill \\ -\lambda \varphi )(hn\alpha -2M_1^{AI})-hn\beta U_1^{AI}\hfill \end{array}\right)},$$ (35)

is the equilibrium inventory level. This equilibrium is globally asymptotically stable if and only if:

$$\left(hn\alpha -2M_1^{AI}\right)(1-\varphi )(1-\eta -\lambda \left(1-\varphi \right))-hn\beta U_1^{AI}<0$$

The equilibrium demand can be calculated as

$$D_{SS}^{AI}=\frac{1}{2}n(1+s+Y_{SS}^{AI}\alpha -\gamma W-\theta -\frac{\beta (Y_{SS}^{AI}{U}_1^{AI}+U_2^{AI})}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}),$$ (36)

The equilibrium BVP profit is

$${\Pi }_{BVPSS}^{AI}=\eta D_{SS}^{AI}(1+\lambda )(p_{SS}^{AI}\varphi -c)+\eta D_{SS}^{AI}{p}_{SS}^{AI}(1-\varphi )-c_{BVP}{D}_{SS}^{AI}(1+\lambda ).$$ (37)

The equilibrium G profit is

$${\Pi }_{GSS}^{AI}=A(A+\delta ({ϵ}_U^2+{ϵ}_D^2)+k{Y}_{SS}^{AI}).$$ (38)

The equilibrium CS is

$$C{S}_{SS}^{AI}=\frac{1}{8}n{\left(\begin{array}{c}-1-s-Y_{SS}^{AI}\alpha +\gamma W+\theta \hfill \\ +\frac{\beta (Y_{SS}^{AI}{U}_1^{\mathrm{AI}}+U_2^{\mathrm{AI}})}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}\hfill \end{array}\right)}^2.$$ (39)

The equilibrium SW is

$$\begin{array}{cc}\hfill S{W}_{SS}^{AI}=\hfill & \hfill -D_{SS}^{AI}{p}_{SS}^{AI}\eta (-1+\varphi )-D_{SS}^{AI}\eta (1+\lambda )(c-p_{SS}^{AI}\varphi +c_{AI})\hfill \\ \hfill \hfill & \hfill -D_{SS}^{AI}(1+\lambda )c_{BVP}+\frac{{\left(Y_{SS}^{AI}\right)}^2{M}_1^{AI}}{2}+Y_{SS}^{AI}{M}_2^{AI}\hfill \\ \hfill \hfill & \hfill +M_3^{AI}+\frac{{\left(Y_{SS}^{AI}\right)}^2{U}_1^{AI}}{2}+Y_{SS}^{AI}{U}_2^{AI}\hfill \\ \hfill \hfill & \hfill +\frac{1}{8}n{\left(\begin{array}{c}-1-s-Y_{SS}^{AI}\alpha +\gamma W+\theta \hfill \\ +\frac{\beta (Y_{SS}^{AI}{U}_1^{AI}+U_2^{AI})}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}\hfill \end{array}\right)}^2\hfill \\ \hfill \hfill & \hfill +U_3^{AI}+A(A+k{Y}_{SS}^{AI}+\delta ({ϵ}_D^2+{ϵ}_U^2))\hfill \end{array}$$ (40)

(Proof: see Appendix A)

From Proposition 2, it follows that the equilibrium strategy ($u_{SS}^{AI}$, $p_{SS}^{AI}$), demand ($D_{SS}^{AI}$), CS ($C{S}_{SS}^{AI}$), SW ($S{W}_{SS}^{AI}$), and profit (${\Pi }_{MSS}^{AI},{\Pi }_{USS}^{AI}$) depend on the state of inventory ($Y_{SS}^{AI}$) and are related to the value of $M_i^{AI},U_i^{AI}$ ($i=1...3$).

To determine the value of $M_i^{AI},U_i^{AI}$ ($i=1...3$), we substitute the values in Table C.6 into the Riccati system to obtain two sets of real roots. Please refer to Appendix C for details.

3.2.2. Corollary 2

• •The trajectory of the equilibrium inventory level $Y_{SS}^{AI}$ is monotonic. When $Y_0^{AI}<Y_{SS}^{AI}$, $Y^{AI}\in \left[Y_0^{AI},Y_{SS}^{AI}\right]$; when $Y_0^{AI}>Y_{SS}^{AI}$, $Y^{AI}\in \left[Y_{SS}^{AI},Y_0^{AI}\right]$.
• •The trajectory of $u_{SS}^{AI}\left(Y_{SS}^{AI}\right),p_{SS}^{AI}\left(Y_{SS}^{AI}\right),D_{SS}^{AI}\left(Y_{SS}^{AI}\right),{\Pi }_{MSS}^{AI}\left(Y_{SS}^{AI}\right),{\Pi }_{USS}^{AI}\left(Y_{SS}^{AI}\right)$ decreases monotonically.

(Proof: omitted)

3.2.3. Lemma 1

Table C.1 shows the sensitivity analysis of each exogenous variable in the B-model to $M_i^B,U_i^B$ ($i=1...3$) and Table C.2 shows the sensitivity analysis of each exogenous variable in the B-model for each performance indicator. Please refer to Appendix C for details. For details on the data in Table C.2, please refer to Table B.2 in Appendix B. The results in Table C.1, Table C.2 can be used to analyze the focus covariate for our research.

From Table C.1, we get ${\omega }_{AI}$, $\overline{e}$, $c_{AI}$, $A$, $k$, $\delta$, $c_{BVP}$ have almost no effect on $M_i^B,U_i^B\left(i=1...3\right)$, Therefore, they cannot be the focus variable of our analysis. $c$, $c_e$, $\omega$, $\gamma$, $W$, $\theta$, $s$ have no effect on the linear quadratic coefficient $M_1^B$, $U_1^B$, but $\gamma$, $W$, $\theta$, $s$ are effect on the linear primary term coefficients and the constant terms, this can be verified by their impact on the performance indicator in Table C.2. Combined with the theme of this paper, we finally selected ${ϵ}_U,{ϵ}_D$, $W,\lambda ,\theta ,s,n$, $\varphi ,\eta$ as the focus covariate for our research.

Similar analytical results we can derive from Table C.3, Table C.4, but note that since the AI model eliminates ${ϵ}_D$ and ${ϵ}_U$, ${ϵ}_D$ and ${ϵ}_U$ in Table C.4 have no effect on performance indicators and $M_i^B,U_i^B\left(i=1....3\right)$ has no effect, but its research value cannot be negated. ${ϵ}_D$ and ${ϵ}_U$ are the key variables for our study of the application of AI techniques.

4. Comparison of results and discussion

This section analyzes the equilibrium results of the B-model and AI-model to investigate the necessity of adopting AI technology. Propositions 1 and 2 present the numerical results of the B-model and AI-model equilibrium strategies and performance indicators when in equilibrium, respectively, as shown in Table Table 2.

Table 2.

Comparison of B-Model and AI-model equilibrium numerical results.

	B-model	AI-model
$Y_{SS}$	2.44426	5.53872
$u_{SS}$	0.30211	0.882744
$p_{SS}$	2.73512	6.17528
$D_{SS}$	0.369246	0.882744
${\Pi }_{MSS}^{}$	1.06091	1.56877
${\Pi }_{USS}^{}$	0.596422	2.35682
${\Pi }_{BVPSS}^{}$	0.043772	0.258913
${\Pi }_{GSS}^{}$	0	0.603898
$C{S}_{SS}^{}$	0.0454476	0.259746
$S{W}_{SS}^{}$	1.74656	5.04365

Table 2 shows that ${\Pi }_{MSS}^{},{\Pi }_{USS}^{},{\Pi }_{BVPSS}^{},{\Pi }_{GSS}^{},C{S}_{SS},S{W}_{SS}$ significantly improve with AI technology adoption. Notably, the increase in inventory levels $Y_{SS}$ and the resulting price increase $p_{SS}$ are potentially beneficial for vaccine supply chain enterprises. The increase in vaccine inventory levels $Y_{SS}$ increases the consumers’ willingness to go to U. Furthermore, the increase in vaccine inventory levels $Y_{SS}$ reduces the risk of vaccine stock-outs and increases the consumers’ willingness to go to U even if the vaccine price $p_{SS}$ increases.

Referring to the sensitivity analysis results, this section analyzes the effects of the three sets of critical variables on the vaccine supply chain system equilibrium. First, we investigate the applicability of AI technology to the system by analyzing the impact of ${ϵ}_U,{ϵ}_D$ on the system equilibrium. Second, we examine the applicability of blockchain technology to the system by analyzing the impact of $W,\lambda ,\theta ,s,n$ variables of the two models on the system equilibrium and the impact of the vaccine product characteristics on the vaccine supply chain. Finally, we study the impact of the revenue-sharing/cost-sharing ratio and the commission ratio on the system by analyzing the impact of the variables $\varphi ,\eta$ on the system equilibrium.

4.1. Effect of ${ϵ}_D$ and ${ϵ}_U$ on the system equilibrium

In Lemmas 1 and 2, the effects of ${ϵ}_D$ and ${ϵ}_U$ on the Riccati system in two different models are analyzed, showing that the vaccine supply chain, as a physical supply chain, has a change in ${ϵ}_U$ and ${ϵ}_D$, which affects the system equilibrium.

4.1.1. Corollary 3

In the space corresponding to the couple $\left({ϵ}_D,{ϵ}_U\right)$ and $Y$, a subspace $\Omega$ includes all realizations of the couple $\left({ϵ}_D,{ϵ}_U\right)$, such that $Y^B>Y^{AI}$.

Proof: From Lemmas 2 and 4, we obtain the results of the sensitivity analysis of ${ϵ}_D,{ϵ}_U$ with respect to the inventory level $Y$ in the two models, that is, $\frac{\partial Y_{SS}^B}{\partial {ϵ}_D}>0,\frac{\partial Y_{SS}^B}{\partial {ϵ}_U}<0,\frac{\partial Y_{SS}^{AI}}{\partial {ϵ}_D}=0,\frac{\partial Y_{SS}^{AI}}{\partial {ϵ}_U}=0$. Therefore, there exists a subspace $\Omega$ that includes all realizations of the couple $\left({ϵ}_D,{ϵ}_U\right)$, such that $Y^B>Y^{AI}$.

In the vaccine supply chain, when ${ϵ}_D>0$ and ${ϵ}_U<0$, supply and demand are always satisfied at equilibrium, and M and U can control the supply chain by adjusting their strategies; AI technology adoption is thus not necessary to eliminate errors. Instead, when using ${ϵ}_D<0$ or ${ϵ}_U>0$, M and U should apply AI technology to eliminate the errors.

The effects of ${ϵ}_D$ and ${ϵ}_U$ on the equilibrium inventory level are shown in Fig. 1. The B-model and AI-model are presented in orange and blue, respectively.

Fig. 1.

Effect of ${ϵ}_D$ and ${ϵ}_U$ on the system equilibrium.

Fig. 1 shows the expected results of applying AI technology to the vaccine supply chain when ${ϵ}_D<0$ or ${ϵ}_U>0$. An increase in inventory level $Y$ guarantees the tapping of higher levels of market potential and improves each party’s profits. There is an increase in $u_{SS},p_{SS},D_{SS},{\Pi }_{MSS}^{},{\Pi }_{USS}^{},{\Pi }_{BVPSS}^{},{\Pi }_{GSS}^{},C{S}_{SS},S{W}_{SS}$ as well.

4.2. Effect of $W,\lambda ,\theta ,s,n$ on the system equilibrium

This section focuses on the effect of the variables $W,\lambda ,\theta ,s,n$ on the equilibrium indicator $Y_{SS},u_{SS},p_{SS},D_{SS},{\Pi }_{MSS}^{},{\Pi }_{USS}^{},{\Pi }_{BVPSS}^{},{\Pi }_{GSS}^{},C{S}_{SS},S{W}_{SS}$. Fig. 2 shows the effect of a consumer’s effort to obtain a certified vaccine $W$ on the system equilibrium.

Fig. 2.

Effect of $W$ on the system equilibrium.

As shown in Fig. 2, all equilibrium performance indicators in the AI-model are generally higher than those in the B-model. The valid interval for M as the supply chain leader $W$ is $\left(1.5,7.1\right)$, as shown in Fig. 2(e).

When $W\in \left(1.5,5.5\right)$, AI technology application significantly affects the improvement in M’s profit, which peaks in each of the two models. This is because, when the value of $W$ is small, consumers can obtain certified vaccines more efficiently, thereby lowering the vaccine inventory level $Y$, the demand for vaccines, and M’s profit. Similarly, when the value of $W$ is higher, the stock of vaccines and M’s inventory cost increases, thus making M less profitable.

However, when $W\in \left(5.5,7.1\right)$ is used, M’s profit is lower in the AI-model than that in the B-model. This is because an increase in $W$ increases the inventory level and reduces M’s profit, as the AI service fee charged by G is positively correlated with the inventory level.

As shown in Fig. 2(e)–(h), the values of $W$ corresponding to M and U’s optimal profits, optimal SW, and optimal CS are different; therefore, the vaccine supply chain should reasonably coordinate all parties’ profit shares when applying blockchain technology. When the value of $W$ is larger, it means that the vaccine supply chain is not cooperating with BVP at this time, which we observe leads to a decrease in M’s profitability, which is worse in the B model. Currently, most of the pharmaceutical sector has covered blockchain technology, especially in the vaccine sector, such as he US Food and Drug Administration and China’s Henan Ziyun Cloud Computing Co. The above results also illustrate this trend.

Fig. 3 shows the effect of the decay rate $\lambda$ during vaccine transportation or storage on the system equilibrium.

Fig. 3.

The effect of $\lambda$ on the system equilibrium.

As shown in Fig. 3, the value of each equilibrium performance indicator decreases as the decay rate $\lambda$ during vaccine transportation or storage increases, with each indicator generally higher for the AI-model than for the B-model. Notably, as shown in Fig. 3(e) and (f), M is more sensitive to the increase in $\lambda$. There is a critical decay rate $\lambda \ast$ above which vaccine suppliers are more inclined to discontinue using AI technology; this critical decay rate is calculated to be $\lambda \ast =0.034$.

Fig. 4 shows the effect of the vaccine’s side effect $\theta$ on the system equilibrium.

Fig. 4.

The effect of $\theta$ on the system equilibrium.

As shown in Fig. 4, the indicators of the side effects $\theta$ are generally higher in the AI-model than in the B-model within a specific interval. As shown in Fig. 4(f)–(j), the increase in the vaccine’s side effects $\theta$ in the AI-model partially lowers U’s profit, CS, SW, and BVP and G’s profits.

Fig. D1 shows the vaccine’s positive effect $s$ on the system equilibrium. As shown in Fig. 5, the vaccine’s positive effect $s$ is generally higher in the AI-model than in the B-model for a specific interval. As shown in Fig. D1(f)–(j), an increase in the vaccine’s positive effect $s$ in the AI-model partially lowers U’s profit, CS, SW, and BVP and G’s profits.

Fig. D1.

The effect of $s$ on the system equilibrium.

Fig. 5.

The effect of $n$ on the system equilibrium.

Fig. 5 shows the effect of the number of vaccinators $n$ on the system equilibrium.

As shown in Fig. 5(e), as the number of vaccinators in the distribution $n$ increases, the indicators are generally higher in the AI-model than in the B-model. When $n\in \left(0,1\right)$, a smaller number of vaccinators nevertheless increases M and U’s profits. This is because supply and demand are satisfied at the system equilibrium, which raises the vaccine price and profits. However, the positive utility of vaccinators with higher stockpiles and fewer vaccinations offsets the negative utility of higher prices.

4.3. Effect of $\varphi ,\eta$ on the system equilibrium

This section focuses on the effect of the variables $\varphi ,\eta$ on $Y_{SS},u_{SS},p_{SS},D_{SS},{\Pi }_{MSS}^{},{\Pi }_{USS}^{},{\Pi }_{BVPSS}^{},{\Pi }_{GSS}^{},C{S}_{SS},S{W}_{SS}$ in the equilibrium. Fig. 6 shows the effect of the revenue-sharing/cost-sharing ratio on the system equilibrium.

Fig. 6.

The effect of $\varphi$ on the system equilibrium.

The revenue-sharing/cost-sharing ratio $\varphi$ in the sharing contract between M and U is the primary parameter variable for revenue allocation between them and the basis for determining cost sharing for inventory, vaccine decay, and AI technology between the two players. Its value affects the system equilibrium. The rationale for setting the revenue-sharing ratio is the same as that for setting the cost-sharing ratio: In a realistic supply chain sharing contract, when either player’s revenue-sharing ratio is lower than their cost-sharing ratio, they are prone to opportunism by exchanging a lower level of effort for a higher revenue. As shown in Fig. 6(e), M’s profit reaches the maximum value when $\varphi \in \left[0.4,0.5\right]$ in the AI-model, which results from setting $\varphi$ as the revenue-sharing/cost-sharing ratio. A reasonable value of $\varphi$ should be ensured before the players sign the cooperation contract.

Fig. D2 shows the effect of the commission ratio $\eta$ on the system equilibrium. As shown in Fig. D2, the indicators in the AI-model are generally higher than those in the B-model when the commission ratio $\eta$ varies within a specific range; BVP, M, and U prefer to enter into cooperation contracts with a commission ratio of $\eta$ in the interval $\eta \in \left(0.25,0.3\right)$ in the AI-model.

Fig. D2.

The effect of $\eta$ on the system equilibrium.

4.4. Summary of the managerial insights

This section analyze the equilibrium results of the B-model and AI-model, and the management insights obtained are as follows:

• •The increase in vaccine inventory levels $Y_{SS}$ reduces the risk of vaccine stock-outs and increases the consumers’ willingness to go to U even if the vaccine price $p_{SS}$ increases. The benefits of increased inventory level outweigh the drawbacks for the vaccine supply chain as a whole. Therefore, companies should pay more attention to avoiding the risk of out-of-stock as opposed to the increase in inventory costs.
• •AI technology adoption is thus not necessary to eliminate errors. The companies in the vaccine supply chain should consider the application of AI technology when the production schedule for vaccines is lower than expected and the demand for vaccines is higher than expected.
• •The vaccine manufacturer and the vaccination unit need to be concerned about the peak of $W$. Excessive demands on the blockchain service company to raise $W$ will result in lower profits.
• •There is no need to consider the application of AI technology when the vaccine loss rate is too high and has a greater impact on the vaccine manufacturer.
• •In a realistic supply chain sharing contract, when either player’s revenue-sharing ratio is lower than their cost-sharing ratio, they are prone to opportunism by exchanging a lower level of effort for a higher revenue.
• •M’s profit reaches the maximum value when the revenue-sharing/ cost-sharing ratio $\varphi \in \left[0.4,0.5\right]$ in the AI-model.
• •M, and U prefer to enter into cooperation contracts with a commission ratio of $\eta$ in the interval $\eta \in \left(0.25,0.3\right)$ in the AI-model.

5. Conclusions

Against the background of the highly developed modern Industry 4.0, this study focuses on the impact of two widely-used modern digital industrial technologies on collaborative vaccine supply chain management: The first, blockchain technology, enhances transparency in vaccine regulation by recording each vaccine product’s unique label on the blockchain, reduces the possibility of tampering with label information through various blockchain nodes, and improves the traceability of vaccines, making it more convenient for consumers to obtain certified vaccine sources and enhancing the need for vaccinators; The second, AI technology, obtains the datasets of vaccine supply chain node enterprises through numerous data interactions among vaccine supply chain enterprises using machine learning technology to eliminate the production planning and demand forecasting errors attributable to unexpected epidemic events. This helps match production planning and demand forecast in the vaccine supply chain and enables accurate collaborations, thus reducing the stock-out risk and inventory risk caused by overcapacity.

To better analyze the benefits of the two technologies for the vaccine supply chain and lay the foundation for their application, we first introduce the VMI model, controlled by a dynamic inventory-level equation. Cooperation between two companies with a shared contract allows better coordination of profitability and revenue/cost-sharing between the two companies while improving the availability of vaccines. In this study, we use blockchain technology to establish the demand function and apply the idea of AI technology to eliminate the errors and establish a dynamic equation. Based on the two-player dynamic game modeled, we evaluate the improvements in the vaccine supply chain because of the application of the two technologies.

The results show that the application of blockchain and AI technologies enhances vaccine supply chain efficiency and improves vaccine supply chain coordination. However, while the application of blockchain technology depends on the level of $W$, that is, consumers’ difficulty in obtaining a certified vaccine source, the application of AI technology depends on the magnitude of the error terms ${ϵ}_D,{ϵ}_U$. Therefore, the vaccine supply chain should estimate the level of $W$ and magnitude of the error terms ${ϵ}_D,{ϵ}_U$ before adopting the two technologies. In addition to analyzing the above critical variables, this study also analyzes the impact of other critical variables on the system, such as the vaccine’s positive effects, vaccine’s side effects, revenue-sharing/cost-sharing ratio, and commission percentage.

The sensitivity analysis section allows us to better examine the impact of these variables on system equilibrium in the B-model and the AI-model, respectively, and the comparative analysis section indicates that M’s performance indicators are more sensitive to changes in the critical variables that make them more willing to abandon AI technology adoption. AI technology costs, production costs, and emissions have a more significant impact on M than on U and are more closely related to M.

Owing to the gradual normalization of the COVID-19 pandemic, collaborative research on the direction of the normalization of the pandemic and the environment for an emergency response can be performed in the future.

CRediT authorship contribution statement

Ye Gao: Writing – original draft, Formal analysis, Methodology, Software, Validation, Writing – review & editing. Hongwei Gao: Conceptualization, Funding acquisition, Project administration, Supervision, Writing – review & editing. Han Xiao: Validation, Writing – review & editing. Fanjun Yao: Software, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Proof of Proposition 1

According to the Bellman continuum dynamic planning theory, for any state variable $Y^B$, there exists a continuously differentiable optimal value function ${\Pi }_{MSS}^B\left(Y_{SS}^B\right)$ and ${\Pi }_{USS}^B\left(Y_{SS}^B\right)$, which satisfies the following Hamilton–Jacobi–Bellman (hereafter abbreviated as the HJB) equation:

$$\begin{array}{cc}\hfill \rho \left({\Pi }_{MSS}^B\right)=\hfill & \hfill \left(1-\eta \right)D^B\left(t\right)\left(1+\lambda \right)\left(p^B\left(t\right)\varphi -c\right)\hfill \\ \hfill \hfill & \hfill -\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi )\varphi -\frac{h}{2}{\left(u^B\left(t\right)\right)}^2\hfill \\ \hfill \hfill & \hfill -c_e\left(\omega u^B\left(t\right)-\overline{e}\right)-\varphi c_h{\left(Y^B\right)}^2\hfill \\ \hfill \hfill & \hfill +{\Pi }_{MSS}^B\prime (u^B\left(t\right)\left(1+{ϵ}_u\right)-D^B\left(t\right)\left(1+{ϵ}_D\right)),\hfill \end{array}$$ (A.1)

$$\begin{array}{cc}\hfill \rho \left({\Pi }_{USS}^B\right)=\hfill & \hfill \left(1-\eta \right)D^B\left(t\right)p^B\left(t\right)(1-\varphi )\hfill \\ \hfill \hfill & \hfill -\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi ){}^2-c_h(1-\varphi ){\left(Y^B\right)}^2\hfill \\ \hfill \hfill & \hfill +{\Pi }_{USS}^B\prime (u^B\left(t\right)\left(1+{ϵ}_u\right)-D^B\left(t\right)\left(1+{ϵ}_D\right)).\hfill \end{array}$$ (A.2)

${\Pi }_{MSS}^B\prime$ and ${\Pi }_{USS}^B\prime$ are the first-order partial derivatives of the optimal value function with respect to the inventory level state variable $Y^B$, respectively, and are the marginal contributions of the inventory level to the value function. According to the structure of Eqs. (A.1), (A.2), let ${\Pi }_{MSS}^B,{\Pi }_{USS}^B,{\Pi }_{MSS}^B\prime ,{\Pi }_{USS}^B\prime$ be.

$$\begin{array}{ccc}\hfill {\Pi }_{MSS}^B\hfill & \hfill =\hfill & \hfill \frac{M_1^B}{2}\left(Y_{\mathrm{SS}}^B\right){}^2+M_2^B{Y}_{\mathrm{SS}}^B+M_3^B,\hfill \\ \hfill {\Pi }_{USS}^B\hfill & \hfill =\hfill & \hfill \frac{U_1^B}{2}\left(Y_{\mathrm{SS}}^B\right){}^2+U_2^B{Y}_{\mathrm{SS}}^B+U_3^B,\hfill \end{array}$$ (A.3)

$$\begin{array}{c}{\Pi }_{MSS}^B\prime =Y_{\mathrm{SS}}^B{M}_1^B+M_2^B,\hfill \\ {\Pi }_{USS}^B\prime =Y_{\mathrm{SS}}^B{U}_1^B+U_2^B,\hfill \end{array}$$ (A.4)

where $M_1^B,M_2^B,M_3^B,U_1^B,U_2^B,U_3^B$ are constants. Substituting Eqs. (A.3), (A.4) into Eqs. (A.1), (A.2) respectively gives

$$\begin{array}{cc}\hfill \rho \left(\begin{array}{c}\frac{{\left(Y_{\mathrm{SS}}^B\right)}^2{M}_1^B}{2}\hfill \\ +Y_{\mathrm{SS}}^B{M}_2^B+M_3^B\hfill \end{array}\right)=\hfill & \hfill \left(1-\eta \right)D^B\left(t\right)\left(1+\lambda \right)\left(p^B\left(t\right)\varphi -c\right)\hfill \\ \hfill \hfill & \hfill -\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi )\varphi -\frac{h}{2}{\left(u^B\left(t\right)\right)}^2\hfill \\ \hfill \hfill & \hfill -c_e\left(\omega u^B\left(t\right)-\overline{e}\right)-\varphi c_h{\left(Y^B\right)}^2\hfill \\ \hfill \hfill & \hfill +(Y_{\mathrm{SS}}^B{M}_1^B+M_2^B)(u^B\left(t\right)\left(1+{ϵ}_u\right)\hfill \\ \hfill \hfill & \hfill -D^B\left(t\right)\left(1+{ϵ}_D\right)),\hfill \end{array}$$ (A.5)

$$\begin{array}{cc}\hfill \rho \left(\begin{array}{c}\frac{{\left(Y_{\mathrm{SS}}^B\right)}^2{U}_1^B}{2}\hfill \\ +Y_{\mathrm{SS}}^B{U}_2^B+U_3^B\hfill \end{array}\right)=\hfill & \hfill \left(1-\eta \right)D^B\left(t\right)p^B\left(t\right)(1-\varphi )\hfill \\ \hfill \hfill & \hfill -\lambda D^B\left(t\right)p^B\left(t\right)(1-\varphi ){}^2\hfill \\ \hfill \hfill & \hfill -c_h(1-\varphi ){\left(Y^B\right)}^2\hfill \\ \hfill \hfill & \hfill +(Y_{\mathrm{SS}}^B{U}_1^B+U_2^B)(u^B\left(t\right)\left(1+{ϵ}_u\right)\hfill \\ \hfill \hfill & \hfill -D^B\left(t\right)\left(1+{ϵ}_D\right)).\hfill \end{array}$$ (A.6)

In a dynamic decision-making environment, the vaccine supply chain need to consider the impact of changes in inventory levels on future profits and make optimal decisions to maximize long-term profits. According to the first-order optimization conditions, it is obtained that

$$u_{SS}^B=\frac{(Y_{SS}^B{M}_1^B+M_2^B)(1+{ϵ}_U)}{h},$$ (A.7)

$$p_{SS}^B=\frac{1}{2}(\frac{1+s+Y_{SS}^B\alpha -W\gamma -\theta }{\beta }+\frac{(Y_{SS}^B{U}_1^B+U_2^B)(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}).$$ (A.8)

Substituting Eqs. (A.5), (A.6), the Riccati system is obtained after collation as follows.

$$\begin{array}{c}\frac{1}{4}(\frac{n{\alpha }^2\varphi (1-\eta (1+\lambda )+\lambda \varphi )}{\beta }-4\varphi c_h+\frac{n\beta \varphi (-1+\eta +\eta \lambda -\lambda \varphi ){\left(U_1^B\right)}^2{(1+{ϵ}_D)}^2}{{(-1+\varphi )}^2{(-1+\eta +\lambda -\lambda \varphi )}^2}\hfill \\ +2M_1^B(-n\alpha -\rho -n\alpha {ϵ}_D+\frac{n\beta U_1^B{(1+{ϵ}_D)}^2}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})+\frac{2{\left(M_1^B\right)}^2{(1+{ϵ}_U)}^2}{h})=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{2}\frac{n\alpha (c\beta (-1+\eta )(1+\lambda )+(1+s-W\gamma -\theta )\varphi (1-\eta (1+\lambda )+\lambda \varphi ))}{\beta }\hfill \\ +\frac{(n\beta U_1^B(1+{ϵ}_D)(-c(-1+\eta )(1+\lambda )(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )+\varphi (-1+\eta +\eta \lambda -\lambda \varphi )U_2^B(1+{ϵ}_D))}{\left((-1+\varphi ){}^2{(-1+\eta +\lambda -\lambda \varphi )}^2\right)}\hfill \\ +M_2^B(-n\alpha -2\rho -n\alpha {ϵ}_D+\frac{n\beta U_1^B{(1+{ϵ}_D)}^2}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ +M_1^B(n(1+{ϵ}_D)(-1-s+W\gamma +\theta \hfill \\ +\frac{\beta U_2^B(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})+\frac{2(1+{ϵ}_U)(-\omega c_e+M_2^B(1+{ϵ}_U))}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(\frac{2{\omega }^2{c}_e^2}{h}-4\rho M_3^B+2n{M}_2^B(1+{ϵ}_D)-1-s+W\gamma +\theta +\frac{\beta U_2^B(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ +(n(-(1+s-T\gamma -\theta )(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )\hfill \\ +\beta U_2^B(1+{ϵ}_D))(-(-1+\varphi )\hfill \\ (-1+\eta +\lambda -\lambda \varphi )(2c\beta (-1+\eta )(1+\lambda )\hfill \\ +(1+s-W\gamma -\theta )\varphi (1-\eta (1+\lambda )+\lambda \varphi ))\hfill \\ +\frac{\beta \varphi (-1+\eta +\eta \lambda -\lambda \varphi )U_2^B(1+{ϵ}_D))}{\left(\beta {(-1+\varphi )}^2{(-1+\eta +\lambda -\lambda \varphi )}^2\right)}+\frac{2{\left(M_2^B\right)}^2{(1+{ϵ}_U)}^2}{h}+\frac{4c_e(eh-\omega M_2^B(1+{ϵ}_U))}{h})=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(\frac{n{\alpha }^2(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}{\beta }+4(-1+\varphi )c_h\hfill \\ +U_1^B(\frac{n\beta U_1^B{(1+{ϵ}_D)}^2}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}-2(n\alpha +\rho +n\alpha {ϵ}_D)+\frac{4M_1^B{(1+{ϵ}_U)}^2}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{2}(\frac{n\alpha (1+s-W\gamma -\theta )(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}{\beta }-U_2^B(n\alpha +2\rho +n\alpha {ϵ}_D)+\frac{2M_1^B{U}_2^B{(1+{ϵ}_U)}^2}{h}\hfill \\ +U_1^B(\frac{n(1+{ϵ}_D)(-(1+s-W\gamma -\theta )(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )+\beta U_2^B(1+{ϵ}_D))}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}\hfill \\ +\frac{2(1+{ϵ}_U)(-\omega c_e+M_2^B(1+{ϵ}_U))}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(\frac{n{(-1-s+W\gamma +\theta )}^2(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}{\beta }\hfill \\ -4\rho U_3^B+U_2^B(n(1+{ϵ}_D)(2(-1-s+W\gamma +\theta )+\frac{\beta U_2^B(1+{ϵ}_D)}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ +\frac{4(1+{ϵ}_U)(-\omega c_e+M_2^B(1+{ϵ}_U))}{h}))=0.\hfill \end{array}$$

Table B.1.

Results of sensitivity analysis of B-Model performance.

Variables	Value	$Y_{SS}^B$	$u_{SS}^B$	$p_{SS}^B$	$D_{SS}^B$	$C{S}_{SS}^B$	$S{W}_{SS}^B$	${\Pi }_{MSS}^B$	${\Pi }_{USS}^B$	${\Pi }_{BVPSS}^B$	${\Pi }_{GSS}^B$
	0.9	3.66395	0.490596	3.60976	0.599617	0.119847	3.01826	1.6589	1.14179	0.097723	0
$\alpha$	0.95	2.93653	0.377055	3.0906	0.460845	0.0707927	2.20329	1.27191	0.797638	0.0629525	0
	1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.7	1.71254	0.193138	2.20738	0.236057	0.0185744	1.19927	0.818393	0.340649	0.0216547	0
$\beta$	0.75	2.04222	0.241314	2.44745	0.294939	0.0289963	1.42366	0.915815	0.448197	0.0306532	0
	0.8	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$c$	0.25	2.61397	0.32306	2.92592	0.394851	0.0519691	1.8427	1.05849	0.682607	0.0496276	0
	0.3	2.78368	0.344009	3.11672	0.420456	0.0589277	1.94099	1.05161	0.774607	0.0558489	0
	0.8	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\varphi$	0.85	2.22696	0.275065	2.49104	0.33619	0.0376746	1.42911	0.984389	0.371324	0.0357267	0
	0.9	2.04437	0.252343	2.28594	0.30842	0.0317076	1.19503	0.925097	0.20863	0.0295943	0
	1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$h$	1.05	2.50924	0.310147	2.80816	0.379069	0.0478978	1.83008	1.10702	0.628824	0.046343	0
	1.1	2.57736	0.318573	2.88473	0.389367	0.0505356	1.92046	1.15709	0.663717	0.0491164	0
	0.11	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$c_h$	0.115	2.60741	0.332692	2.90791	0.406623	0.0551142	1.91444	1.13408	0.673471	0.0517721	0
	0.12	2.78535	0.366606	3.09579	0.448074	0.0669234	2.11148	1.22072	0.762514	0.0613262	0
	0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\rho$	0.105	2.39378	0.300462	2.6737	0.367231	0.044953	1.67784	1.02188	0.568619	0.0423873	0
	0.11	2.34926	0.299344	2.61919	0.365864	0.0446189	1.6175	0.987252	0.544416	0.0412164	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$c_e$	0.25	2.87973	0.355867	3.22471	0.434948	0.06306	2.42427	1.46957	0.829254	0.0623782	0
	0.3	3.31521	0.409623	3.7143	0.50065	0.0835502	3.19977	1.9316	1.10037	0.0842527	0
	1.5	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\omega$	1.55	2.50232	0.309278	2.8004	0.378006	0.0476296	1.81127	1.09232	0.625255	0.046064	0
	1.6	2.56039	0.316445	2.86568	0.386767	0.0498628	1.87772	1.12467	0.654768	0.0484141	0
	0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
${\omega }_{AI}$	0.15	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.3	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\overline{e}$	0.35	2.44426	0.30211	2.73512	0.369246	0.0454476	1.84656	1.16091	0.596422	0.043772	0
	0.4	2.44426	0.30211	2.73512	0.369246	0.0454476	1.94656	1.26091	0.596422	0.043772	0
	0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$c_{AI}$	0.15	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.4	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$A$	0.45	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.5	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$k$	0.25	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.3	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\delta$	0.15	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.2	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	−0.15	2.16676	0.257552	2.4182	0.333303	0.0370303	1.50377	0.940527	0.49207	0.0341451	0
${ϵ}_D$	−0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	−0.05	2.7651	0.354806	3.10152	0.410827	0.0562597	2.05972	1.21842	0.728695	0.0563479	0
	0.05	2.66107	0.34457	2.97884	0.401998	0.0538674	2.0209	1.20664	0.70776	0.0526317	0
${ϵ}_U$	0.1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.15	2.26771	0.268108	2.53665	0.342583	0.039121	1.54482	0.955717	0.512822	0.0371571	0
	0.02	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\lambda$	0.025	2.3926	0.295941	2.67683	0.361706	0.0436104	1.69212	1.036	0.570548	0.0419645	0
	0.03	2.34282	0.289998	2.62066	0.354442	0.0418764	1.64064	1.01233	0.546171	0.0402568	0
	1.5	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$n$	1.55	2.40942	0.297617	2.70593	0.363754	0.0426828	1.71974	1.04937	0.585109	0.0425815	0
	1.6	2.37766	0.293522	2.6793	0.358749	0.040219	1.69556	1.03897	0.574866	0.0415102	0
	0.49	2.35013	0.28896	2.68085	0.353173	0.0415771	1.73892	1.08018	0.57627	0.040893	0
$\gamma$	0.495	2.3972	0.295535	2.70799	0.36121	0.0434908	1.74326	1.07103	0.586421	0.0423214	0
	0.5	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	3.9	2.3266	0.285672	2.66729	0.349155	0.0406364	1.73635	1.08439	0.571139	0.0401871	0
$W$	3.95	2.38543	0.293891	2.7012	0.359201	0.0430083	1.74227	1.07341	0.583897	0.0419622	0
	4	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
	0.01	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\theta$	0.015	2.45603	0.303754	2.74191	0.371255	0.0459435	1.74722	1.05824	0.598899	0.0441381	0
	0.02	2.46779	0.305398	2.74869	0.373264	0.0464421	1.74781	1.0555	0.601367	0.0445056	0
	1	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$s$	1.05	2.3266	0.285672	2.66729	0.349155	0.0406364	1.73635	1.08439	0.571139	0.0401871	0
	1.1	2.20894	0.269234	2.59945	0.329064	0.0360944	1.7196	1.10184	0.544921	0.0367407	0
	0.01	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$c_{BVP}$	0.015	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74467	1.06091	0.596422	0.0418888	0
	0.02	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74279	1.06091	0.596422	0.0400057	0
	0.05	2.44426	0.30211	2.73512	0.369246	0.0454476	1.74656	1.06091	0.596422	0.043772	0
$\eta$	0.055	2.47886	0.30754	2.77284	0.375882	0.0470957	1.78317	1.0762	0.609689	0.0501901	0
	0.06	2.51446	0.313139	2.81164	0.382726	0.0488263	1.82147	1.09215	0.623481	0.0570098	0

Table B.2.

Results of sensitivity analysis of AI-Model performance.

Variables	Value	$Y_{SS}^{AI}$	$u_{SS}^{AI}$	$p_{SS}^{AI}$	$D_{SS}^{AI}$	$C{S}_{SS}^{AI}$	$S{W}_{SS}^{AI}$	${\Pi }_{MSS}^{AI}$	${\Pi }_{USS}^{AI}$	${\Pi }_{BVPSS}^{AI}$	${\Pi }_{GSS}^{AI}$
	0.9	10.2086	1.70716	10.0495	1.70716	0.971463	18.8726	8.51854	7.5771	0.836705	0.977486
$\alpha$	0.95	7.22312	1.177	7.58412	1.177	0.461779	9.13376	3.57634	3.93354	0.429457	0.73865
	1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	0.7	3.44007	0.51984	4.40501	0.51984	0.0900779	1.33633	−0.14981	0.856988	0.105722	0.436005
$\beta$	0.75	4.33094	0.67198	5.16394	0.67198	0.150519	2.70413	0.462928	1.42426	0.162571	0.507275
	0.8	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	0.2	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$c$	0.25	5.78792	0.919178	6.45642	0.919178	0.281629	5.45294	1.6612	2.61059	0.280383	0.623834
	0.3	6.03713	0.955612	6.73757	0.955612	0.304398	5.87569	1.7539	2.87578	0.302708	0.64377
	0.8	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\varphi$	0.85	4.92487	0.792248	5.48338	0.792248	0.209219	3.20164	0.900145	1.33678	0.204741	0.55479
	0.9	4.43136	0.7195	4.92711	0.7195	0.17256	1.92722	0.39191	0.685343	0.165766	0.515308
	1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$h$	1.05	5.8551	0.929113	6.53211	0.929113	0.28775	5.92882	2.04507	2.68218	0.289355	0.629208
	1.1	6.20135	0.979874	6.92262	0.979874	0.320051	6.96123	2.60496	3.05971	0.324602	0.656908
	0.11	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$c_h$	0.115	6.08221	0.992541	6.76315	0.992541	0.328379	6.41073	2.22206	2.89722	0.320758	0.647377
	0.12	6.71112	1.12127	7.442	1.12127	0.419085	8.16194	3.06678	3.58308	0.401028	0.697689
	0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\rho$	0.105	5.4237	0.877351	6.03601	0.877351	0.256581	4.89822	1.55418	2.24611	0.251123	0.594696
	0.11	5.32413	0.873709	5.91457	0.873709	0.254456	4.78164	1.54797	2.15224	0.244691	0.58673
	0.2	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$c_e$	0.25	6.31261	0.995888	7.04835	0.995888	0.330598	7.21823	2.70838	3.18225	0.336268	0.665808
	0.3	7.08649	1.10903	7.92142	1.10903	0.409985	9.68376	4.01009	4.11796	0.423659	0.727719
	1.5	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\omega$	1.55	5.63546	0.896887	6.28442	0.896887	0.268135	5.28081	1.68361	2.45397	0.268033	0.611637
	1.6	5.73219	0.91103	6.39355	0.91103	0.276658	5.52252	1.80098	2.55284	0.277311	0.619375
	0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
${\omega }_{AI}$	0.15	5.63546	0.896887	6.28442	0.896887	0.268135	5.28081	1.68361	2.45397	0.268033	0.611637
	0.2	5.73219	0.91103	6.39355	0.91103	0.276658	5.52252	1.80098	2.55284	0.277311	0.619375
	0.3	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\overline{e}$	0.35	5.53872	0.882744	6.17528	0.882744	0.259746	5.14365	1.66877	2.35682	0.258913	0.603898
	0.4	5.53872	0.882744	6.17528	0.882744	0.259746	5.24365	1.76877	2.35682	0.258913	0.603898
	0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$c_{AI}$	0.15	5.78792	0.919178	6.45642	0.919178	0.281629	5.45294	1.6612	2.61059	0.282727	0.623834
	0.2	6.03713	0.955612	6.73757	0.955612	0.304398	5.87569	1.7539	2.87578	0.307581	0.64377
	0.4	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$A$	0.45	5.65283	0.908481	6.29647	0.908481	0.275113	4.89121	1.26815	2.36837	0.272054	0.712155
	0.5	5.76694	0.934218	6.41766	0.934218	0.290921	4.69867	0.927892	2.37142	0.285513	0.827694
	0.2	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$k$	0.25	5.76694	0.934218	6.41766	0.934218	0.290921	5.51047	1.64949	2.55182	0.285513	0.737494
	0.3	5.99516	0.985693	6.66004	0.985693	0.323863	5.99693	1.73165	2.75285	0.313381	0.880219
	0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\delta$	0.15	5.53872	0.882744	6.17528	0.882744	0.259746	5.04005	1.56557	2.35602	0.258913	0.604298
	0.2	5.53872	0.882744	6.17528	0.882744	0.259746	5.03645	1.56237	2.35522	0.258913	0.604698
	−0.15	5.53872	0.882744	6.17528	0.882744	0.259746	5.03915	1.56477	2.35582	0.258913	0.604398
${ϵ}_D$	−0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	−0.05	5.53872	0.882744	6.17528	0.882744	0.259746	5.04635	1.57117	2.35742	0.258913	0.603598
	0.05	5.53872	0.882744	6.17528	0.882744	0.259746	5.04635	1.57117	2.35742	0.258913	0.603598
${ϵ}_U$	0.1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	0.15	5.53872	0.882744	6.17528	0.882744	0.259746	5.03915	1.56477	2.35582	0.258913	0.604398
	0.02	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\lambda$	0.025	5.37053	0.858782	5.98501	0.858782	0.245835	4.64863	1.38423	2.188	0.244525	0.590442
	0.03	5.21044	0.83598	5.80391	0.83598	0.232954	4.28268	1.21272	2.03248	0.231199	0.577636
	1.5	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$n$	1.55	5.42689	0.865507	6.07312	0.865507	0.241646	4.82847	1.4711	2.27583	0.249365	0.594951
	1.6	5.32576	0.849928	5.9807	0.849928	0.225743	4.63738	1.38463	2.2036	0.240886	0.586861
	0.49	5.39831	0.860095	6.06864	0.860095	0.246588	4.91937	1.55791	2.27899	0.24761	0.592665
$\gamma$	0.495	5.46851	0.871419	6.12196	0.871419	0.253124	4.98169	1.56366	2.31785	0.253231	0.598281
	0.5	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	3.9	5.3632	0.854433	6.04198	0.854433	0.243352	4.88806	1.55479	2.2596	0.244823	0.589856
$W$	3.95	5.45096	0.868588	6.10863	0.868588	0.251482	4.96615	1.56228	2.30812	0.25182	0.596877
	4	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
	0.01	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\theta$	0.015	5.55627	0.885575	6.18861	0.885575	0.261414	5.05908	1.56995	2.36659	0.260343	0.605302
	0.02	5.57383	0.888406	6.20194	0.888406	0.263088	5.07449	1.57109	2.37635	0.261777	0.606706
	1	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$s$	1.05	5.3632	0.854433	6.04198	0.854433	0.243352	4.88806	1.55479	2.2596	0.244823	0.589856
	1.1	5.18768	0.826122	5.90867	0.826122	0.227492	4.73016	1.53685	2.1631	0.231116	0.575815
	0.01	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$c_{BVP}$	0.015	5.53872	0.882744	6.17528	0.882744	0.259746	5.03915	1.56877	2.35682	0.254411	0.603898
	0.02	5.53872	0.882744	6.17528	0.882744	0.259746	5.03465	1.56877	2.35682	0.249909	0.603898
	0.05	5.53872	0.882744	6.17528	0.882744	0.259746	5.04365	1.56877	2.35682	0.258913	0.603898
$\eta$	0.055	5.65308	0.903148	6.30122	0.903148	0.271892	5.32715	1.69497	2.45364	0.298664	0.613046
	0.06	5.77212	0.924418	6.4323	0.924418	0.284849	5.62894	1.82924	2.5562	0.341732	0.622569

Table B.3.

B-model $M_i^B,U_i^B\left(i=1...3\right)$ sensitivity analysis test results.

Variables	Value	$M_1^B$	$U_1^B$	$M_2^B$	$U_2^B$	$M_3^B$	$U_3^B$
	0.9	0.125098	0.192497	0.260371	−0.0514928	−0.134775	0.0383666
$\alpha$	0.95	0.119926	0.208927	0.263337	−0.0445435	−0.0184649	0.0276265
	1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.7	0.105545	0.263462	0.267557	−0.0344089	0.205421	0.0132369
$\beta$	0.75	0.110447	0.242918	0.266548	−0.0366957	0.14115	0.0165753
	0.8	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$c$	0.25	0.115393	0.224834	0.264784	−0.041788	−0.0278781	0.0237123
	0.3	0.115393	0.224834	0.264246	−0.0443198	−0.131051	0.0268734
	0.8	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\varphi$	0.85	0.115316	0.16903	0.265982	−0.0274728	0.106112	0.0133657
	0.9	0.115238	0.11294	0.266542	−0.0171778	0.139372	0.00773423
	1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$h$	1.05	0.121156	0.224615	0.264768	−0.0396823	0.0612326	0.0212746
	1.1	0.126917	0.224392	0.264189	−0.0401173	0.0546416	0.021819
	0.11	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$c_h$	0.115	0.119026	0.222864	0.264826	−0.041024	0.0389674	0.0228571
	0.12	0.122685	0.220862	0.264286	−0.0428705	0.00868454	0.0251787
	0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\rho$	0.105	0.117455	0.224684	0.264714	−0.0401077	0.0516979	0.0208866
	0.11	0.119516	0.224531	0.264083	−0.0409653	0.0370447	0.021056
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$c_e$	0.25	0.115393	0.224834	0.332123	−0.0457528	0.0346778	0.0287501
	0.3	0.115393	0.224834	0.398924	−0.0522494	−0.0250345	0.0380527
	1.5	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\omega$	1.55	0.115393	0.224834	0.274229	−0.0401224	0.0448346	0.0217405
	1.6	0.115393	0.224834	0.283136	−0.0409886	0.0215002	0.0227552
	0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
${\omega }_{AI}$	0.15	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.3	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\overline{e}$	0.35	0.115393	0.224834	0.265322	−0.0392562	0.167694	0.0207488
	0.4	0.115393	0.224834	0.265322	−0.0392562	0.267694	0.0207488
	0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$c_{AI}$	0.15	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.4	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$A$	0.45	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.5	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$k$	0.25	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.3	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\delta$	0.15	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.2	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	−0.15	0.111286	0.238387	0.265735	−0.0400885	0.103505	0.0193357
${ϵ}_D$	−0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	−0.05	0.119497	0.212689	0.264855	−0.0385961	0.0292438	0.0223324
	0.05	0.12663	0.224403	0.276904	−0.0413591	0.0214209	0.023284
${ϵ}_U$	0.1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.15	0.105586	0.225199	0.254569	−0.0374415	0.106942	0.018686
	0.02	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\lambda$	0.025	0.115484	0.224778	0.265459	−0.0389682	0.0703198	0.0204092
	0.03	0.115574	0.22472	0.265593	−0.0386836	0.0729148	0.0200769
	1.5	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$n$	1.55	0.115213	0.226172	0.26569	−0.0377668	0.0747802	0.0196031
	1.6	0.115046	0.227419	0.266024	−0.0363948	0.0812611	0.0185725
	0.49	0.115393	0.224834	0.264229	−0.0266853	0.140539	0.0180919
$\gamma$	0.495	0.115393	0.224834	0.264776	−0.0329708	0.104752	0.0194483
	0.5	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	3.9	0.115393	0.224834	0.263956	−0.0235426	0.157955	0.0173927
$W$	3.95	0.115393	0.224834	0.264639	−0.0313994	0.113818	0.0191144
	4	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.01	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\theta$	0.015	0.115393	0.224834	0.265459	−0.0408276	0.0582314	0.0210652
	0.02	0.115393	0.224834	0.265596	−0.042399	0.0486889	0.0213781
	1	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$s$	1.05	0.115393	0.224834	0.263956	−0.0235426	0.157955	0.0173927
	1.1	0.115393	0.224834	0.26259	−0.00782893	0.240272	0.0136873
	0.01	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$c_{BVP}$	0.015	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.02	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
	0.05	0.115393	0.224834	0.265322	−0.0392562	0.0676944	0.0207488
$\eta$	0.055	0.115816	0.223395	0.265218	−0.0394339	0.0629324	0.0210877
	0.06	0.116243	0.221955	0.265111	−0.0396154	0.0580684	0.0214363

Table B.4.

AI-model $M_i^{AI},U_i^{AI}\left(i=1...3\right)$ sensitivity analysis test results.

Variables	Value	$M_1^{A}I$	$U_1^{A}I$	$M_2^{A}I$	$U_2^{A}I$	$M_3^{A}I$	$U_3^{A}I$
	0.9	0.162375	0.170674	0.369542	−0.110132	−3.71491	−0.192013
$\alpha$	0.95	0.155529	0.186002	0.373601	−0.0941706	−3.17946	−0.238436
	1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.7	0.136306	0.235902	0.370939	−0.0717604	−2.23239	−0.291989
$\beta$	0.75	0.142874	0.21721	0.373203	−0.0767181	−2.49334	−0.280584
	0.8	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.2	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$c$	0.25	0.149491	0.200696	0.373938	−0.0849842	−3.0071	−0.259205
	0.3	0.149491	0.200696	0.373119	−0.0875508	−3.2229	−0.253043
	0.8	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\varphi$	0.85	0.149402	0.15099	0.37646	−0.0584755	−2.76571	−0.206319
	0.9	0.149309	0.100947	0.377857	−0.0370823	−2.7485	−0.141476
	1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$h$	1.05	0.156941	0.20038	0.376663	−0.0838597	−2.85047	−0.261547
	1.1	0.164387	0.200056	0.378444	−0.0853501	−2.90278	−0.257753
	0.11	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$c_h$	0.115	0.154355	0.19876	0.373721	−0.0858041	−2.90604	−0.257303
	0.12	0.159251	0.196782	0.372522	−0.0893884	−3.01952	−0.248468
	0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\rho$	0.105	0.151977	0.200502	0.373074	−0.0837783	−2.70458	−0.248541
	0.11	0.154462	0.200302	0.371336	−0.0851527	−2.61827	−0.233307
	0.2	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$c_e$	0.25	0.149491	0.200696	0.452213	−0.090388	−3.12479	−0.24594
	0.3	0.149491	0.200696	0.52967	−0.0983585	−3.49699	−0.224343
	1.5	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\omega$	1.55	0.149491	0.200696	0.384439	−0.0834139	−2.85667	−0.262851
	1.6	0.149491	0.200696	0.394121	−0.0844102	−2.91418	−0.260549
	0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
${\omega }_{AI}$	0.15	0.149491	0.200696	0.384439	−0.0834139	−2.85667	−0.262851
	0.2	0.149491	0.200696	0.394121	−0.0844102	−2.91418	−0.260549
	0.3	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\overline{e}$	0.35	0.149491	0.200696	0.374757	−0.0824176	−2.6999	−0.265116
	0.4	0.149491	0.200696	0.374757	−0.0824176	−2.5999	−0.265116
	0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$c_{AI}$	0.15	0.149491	0.200696	0.373938	−0.0849842	−3.0071	−0.259205
	0.2	0.149491	0.200696	0.373119	−0.0875508	−3.2229	−0.253043
	0.4	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$A$	0.45	0.149491	0.200696	0.383436	−0.0864479	−3.28779	−0.349537
	0.5	0.149491	0.200696	0.392115	−0.0904782	−3.81926	−0.444142
	0.2	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$k$	0.25	0.149491	0.200696	0.392115	−0.0904782	−3.09766	−0.263742
	0.3	0.149491	0.200696	0.409473	−0.0985388	−3.40969	−0.263105
	0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\delta$	0.15	0.149491	0.200696	0.374757	−0.0824176	−2.8031	−0.265916
	0.2	0.149491	0.200696	0.374757	−0.0824176	−2.8063	−0.266716
	−0.15	0.149491	0.200696	0.374757	−0.0824176	−2.8039	−0.266116
${ϵ}_D$	−0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	−0.05	0.149491	0.200696	0.374757	−0.0824176	−2.7975	−0.264516
	0.05	0.149491	0.200696	0.374757	−0.0824176	−2.7975	−0.264516
${ϵ}_U$	0.1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.15	0.149491	0.200696	0.374757	−0.0824176	−2.8039	−0.266116
	0.02	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\lambda$	0.025	0.149619	0.200668	0.375249	−0.081783	−2.78875	−0.266663
	0.03	0.149747	0.200637	0.375732	−0.081156	−2.77773	−0.268172
	1.5	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$n$	1.55	0.149255	0.201974	0.375518	−0.079115	−2.76466	−0.269008
	1.6	0.149035	0.203163	0.376201	−0.0760775	−2.73253	−0.272457
	0.49	0.149491	0.200696	0.373099	−0.070843	−2.6344	−0.262901
$\gamma$	0.495	0.149491	0.200696	0.373928	−0.0766303	−2.71641	−0.263978
	0.5	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	3.9	0.149491	0.200696	0.372685	−0.0679494	−2.59396	−0.262385
$W$	3.95	0.149491	0.200696	0.373721	−0.0751835	−2.69577	−0.263703
	4	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.01	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\theta$	0.015	0.149491	0.200696	0.374964	−0.0838644	−2.821	−0.26541
	0.02	0.149491	0.200696	0.375172	−0.0853112	−2.8422	−0.265708
	1	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$s$	1.05	0.149491	0.200696	0.372685	−0.0679494	−2.59396	−0.262385
	1.1	0.149491	0.200696	0.370612	−0.0534813	−2.39732	−0.260032
	0.01	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$c_{BVP}$	0.015	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.02	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
	0.05	0.149491	0.200696	0.374757	−0.0824176	−2.7999	−0.265116
$\eta$	0.055	0.150055	0.199379	0.374874	−0.0828592	−2.8219	−0.263753
	0.06	0.150626	0.198061	0.374986	−0.0833115	−2.84446	−0.26234

Table C.1.

The sensitivity analysis of each exogenous variable in the B-model to $M_i^B,U_i^B$ ($i=1...3$).

	$M_1^B$	$U_1^B$	$M_2^B$	$U_2^B$	$M_3^B$	$U_3^B$
$\alpha$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$\beta$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$c$	–	–	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\varphi$	$\downarrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$h$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$c_h$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\rho$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$c_e$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\omega$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
${\omega }_{AI}$	–	–	–	–	–	–
$\overline{e}$	–	–	–	–	$\uparrow$	–
$c_{AI}$	–	–	–	–	–	–
$A$	–	–	–	–	–	–
$k$	–	–	–	–	–	–
$\delta$	–	–	–	–	–	–
${ϵ}_D$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\uparrow$
${ϵ}_U$	$\downarrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$
$\lambda$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$n$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$\gamma$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$W$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\theta$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$s$	–	–	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$
$c_{BVP}$	–	–	–	–	–	–
$\eta$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$

“$\uparrow$”, “$\downarrow$” and “–” indicate that the performance indicators are positively correlated, negatively correlated, and not correlated with the exogenous variables, respectively.

Table C.2.

The sensitivity analysis of each exogenous variable in B-model for each performance indicator.

	$Y_{SS}^B$	$u_{SS}^B$	$p_{SS}^B$	$D_{SS}^B$	$C{S}_{SS}^B$	$S{W}_{SS}^B$	${\Pi }_{MSS}^B$	${\Pi }_{USS}^B$	${\Pi }_{BVPSS}^B$	${\Pi }_{GSS}^B$
$\alpha$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$\beta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
$c$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	–
$\varphi$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$h$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
$c_h$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
$\rho$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$c_e$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
$\omega$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
${\omega }_{AI}$	–	–	–	–	–	–	–	–	–	–
$\overline{e}$	–	–	–	–	–	–	–	–	–	–
$c_{AI}$	–	–	–	–	–	–	–	–	–	–
$A$	–	–	–	–	–	–	–	–	–	–
$k$	–	–	–	–	–	–	–	–	–	–
$\delta$	–	–	–	–	–	–	–	–	–	–
${ϵ}_D$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–
${ϵ}_U$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$\lambda$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$n$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	–
$\gamma$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	–
$W$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	–
$\theta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	–
$s$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	–
$c_{BVP}$	–	–	–	–	–	$\downarrow$	–	–	$\downarrow$	–
$\eta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	–

“$\uparrow$”, “$\downarrow$” and “–” indicate that the performance indicators are positively correlated, negatively correlated, and not correlated with the exogenous variables, respectively.

Table C.3.

The sensitivity analysis of each exogenous variable in the AI-model for $M_i^{AI},U_i^{AI}$ ($i=1...3$).

	$M_1^{AI}$	$U_1^{AI}$	$M_2^{AI}$	$U_2^{AI}$	$M_3^{AI}$	$U_3^{AI}$
$\alpha$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$\beta$	$\uparrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$c$	–	–	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\varphi$	$\downarrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$h$	$\uparrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$c_h$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\rho$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\uparrow$
$c_e$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\omega$	–	–	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$
${\omega }_{AI}$	–	–	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$
$\overline{e}$	–	–	–	–	$\uparrow$	–
$c_{AI}$	–	–	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$A$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
$k$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\delta$	–	–	–	–	$\downarrow$	$\downarrow$
${ϵ}_D$	–	–	–	–	$\uparrow$	$\uparrow$
${ϵ}_U$	–	–	–	–	$\downarrow$	$\downarrow$
$\lambda$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$n$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$
$\gamma$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
$W$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
$\theta$	–	–	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
$s$	–	–	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$
$c_{BVP}$	–	–	–	–	–	–
$\eta$	$\uparrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$

“$\uparrow$”, “$\downarrow$” and “–” indicate that the performance indicators are positively correlated, negatively correlated, and not correlated with the exogenous variables, respectively.

Table C.4.

The sensitivity analysis of each exogenous variable in AI-model for each performance indicator.

	$Y_{SS}^{AI}$	$u_{SS}^{AI}$	$p_{SS}^{AI}$	$D_{SS}^{AI}$	$C{S}_{SS}^{AI}$	$S{W}_{SS}^{AI}$	${\Pi }_{MSS}^{AI}$	${\Pi }_{USS}^{AI}$	${\Pi }_{BVPSS}^{AI}$	${\Pi }_{GSS}^{AI}$
$\alpha$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
$\beta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$c$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\varphi$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
$h$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$c_h$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\rho$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
$c_e$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\omega$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
${\omega }_{AI}$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\overline{e}$	–	–	–	–	–	$\uparrow$	$\uparrow$	–	–	–
$c_{AI}$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$A$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$
$k$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\delta$	–	–	–	–	–	$\downarrow$	$\downarrow$	$\downarrow$	–	$\uparrow$
${ϵ}_D$	–	–	–	–	–	$\uparrow$	$\uparrow$	$\uparrow$	–	$\downarrow$
${ϵ}_U$	–	–	–	–	–	$\downarrow$	$\downarrow$	$\downarrow$	–	$\uparrow$
$\lambda$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
$n$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
$\gamma$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$W$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$\theta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$
$s$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
$c_{BVP}$	–	–	–	–	–	$\downarrow$	–	–	$\downarrow$	–
$\eta$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$

“$\uparrow$”, “$\downarrow$” and “–” indicate that the performance indicators are positively correlated, negatively correlated, and not correlated with the exogenous variables, respectively.

Table C.5.

Notations and definitions.

Notations	Definitions	Unit
$\alpha$	Coefficient of influence of the inventory level on demand	–
$\beta$	Coefficient of influence of the price on demand	–
$c$	The production cost per vaccine	Dollar
$\varphi$	The revenue-sharing/cost-sharing ratio	–
$h$	Coefficient of influence of the production efficiency on the costs of vaccine producers	–
$c_h$	Coefficient of the impact of the inventory level on vaccine production cost	Dollar
$\rho$	Discount rate	–
$c_e$	Coefficient of the impact of emissions on producer costs	Dollar
$\omega$	Emissions per vaccine	Cubic meter
${\omega }_{AI}$	The increase in emissions per vaccine because of AI technology adoption	Cubic meter
$\overline{e}$	The upper limit of total emissions	Cubic meter
$c_{AI}$	Increase in marginal production costs from AI technology adoption	Dollar
$A$	Cost of accessing cloud services in G’s profit	Dollar
$k$	Coefficient of influence of the controlled inventory level on G’s profit	–
$\delta$	The marginal cost of eliminating the error from G’s profit	Dollar
${ϵ}_D$	The demand forecasting error	–
${ϵ}_U$	The production planning error	–
$\lambda$	Proportional coefficient of vaccine loss during transportation or preservation	–
$n$	Number of vaccines under each vaccination facility	Person
$\gamma$	Coefficient of influence of the difficulty in obtaining certified vaccines on demand for	–
$W$	The consumers’ difficulty in obtaining certified vaccine sources	–
$\theta$	Coefficient of the impact of the vaccine’s side effects on demand	–
$s$	Coefficient of the impact of the vaccine’s positive effects on demand	–
$c_{BVP}$	Marginal retrospective cost per vaccine borne by BVP	Dollar
$\eta$	BVP’s commission as a percentage of revenue	–
$M$	A vaccine manufacturer, such as Pfizer	–
$U$	A facility that provides vaccination services, such as a hospital or a government-run vaccination site	–
$u\left(t\right)$	The optimal production efficiency strategy	Piece
$p\left(t\right)$	The optimal sales price strategy	Dollar
$Y\left(t\right)$	The inventory level at time t	Piece
$BVP$	The company that provides blockchain services to the vaccine supply chain	–
$G$	The company that provides AI technology services to the vaccine supply chain	–
$D\left(t\right)$	The vaccine demand	10 000 Person
$Z$	The utility of vaccination	–
$v$	The perceived value of the vaccine to the consumers	–
$f\left(v\right)$	The distribution function of the perceived value $v$	–
${\pi }_M$	The marginal profit of M	Dollar
${\pi }_U$	The marginal profit of U	Dollar
$C_u$	The vaccine production cost	Dollar
$C_h$	The vaccine inventory cost	Dollar
$E$	The profit of M affected by emissions	Dollar
$L$	The vaccine loss causes lowers profits for U	Dollar
$F_M$	The blockchain fee to be paid by M	10 000 Dollar
$F_U$	The blockchain fee to be paid by U	10 000 Dollar
${\Pi }_{BVP}$	The BVP’s profit	10 000 Dollar
${\Pi }_M$	M’s total profit	10 000 Dollar
${\Pi }_U$	U’s total profit	10 000 Dollar
CS	The consumer surplus	–
SW	The social welfare	–
$B$	The superscript of B-model	–
$AI$	The superscript of AI-model	–
${\Pi }_G^{AI}$	The total cost paid by the vaccine supply chain for G’s AI services	10 000 Dollar
$SS$	The subscript of equilibrium solutions	–
$M_i,U_i$ ($i=1...3$)	The Riccati coefficients	–

Table C.6.

Symbols, meanings, and the benchmark parameter values.

Symbols	Meaning	Values
$\alpha$	Coefficient of influence of the inventory level on demand	1
$\beta$	Coefficient of influence of the price on demand	0.8
$c$	The production cost per vaccine	0.2
$\varphi$	The revenue-sharing/cost-sharing ratio	0.8
$h$	Coefficient of influence of the production efficiency on the costs of vaccine producers	1
$c_h$	Coefficient of the impact of the inventory level on vaccine production cost	0.11
$\rho$	Discount rate	0.1
$c_e$	Coefficient of the impact of emissions on producer costs	0.2
$\omega$	Emissions per vaccine	1.5
${\omega }_{AI}$	The increase in emissions per vaccine because of AI technology adoption	0.1
$\overline{e}$	The upper limit of total emissions	0.3
$c_{AI}$	Increase in marginal production costs from AI technology adoption	0.1
$A$	Cost of accessing cloud services in G’s profit	0.4
$k$	Coefficient of influence of the controlled inventory level on G’s profit	0.2
$\delta$	The marginal cost of eliminating the error from G’s profit	0.1
${ϵ}_D$	The demand forecasting error	−0.1
${ϵ}_U$	The production planning error	0.1
$\lambda$	Proportional coefficient of vaccine loss during transportation or preservation	0.02
$n$	Number of vaccines under each vaccination facility	1.5
$\gamma$	Coefficient of influence of the difficulty in obtaining certified vaccines on demand for	0.5
$W$	The consumers’ difficulty in obtaining certified vaccine sources	4
$\theta$	Coefficient of the impact of the vaccine’s side effects on demand	0.01
$s$	Coefficient of the impact of the vaccine’s positive effects on demand	1
$c_{BVP}$	Marginal retrospective cost per vaccine borne by BVP	0.1
$\eta$	BVP’s commission as a percentage of revenue	0.05

Proof of Proposition 2

According to the Bellman continuum dynamic planning theory, for any state variable $Y^{AI}$, there exists a continuously differentiable optimal value function ${\Pi }_{MSS}^{AI}\left(Y_{SS}^{AI}\right)$ and ${\Pi }_{USS}^{AI}\left(Y_{SS}^{AI}\right)$, which satisfies the following Hamilton–Jacobi–Bellman (hereafter abbreviated as the HJB) equation:

$$\begin{array}{cc}\hfill \rho \left({\Pi }_{MSS}^{AI}\right)=\hfill & \hfill \left(1-\eta \right)D^{AI}\left(t\right)\left(1+\lambda \right)\left(p^{AI}\left(t\right)\varphi -c-c_{AI}\right)\hfill \\ \hfill \hfill & \hfill -\lambda D^{AI}\left(t\right)p^{AI}\left(t\right)\left(1-\varphi \right)\varphi -\frac{h}{2}{\left(u^{AI}\left(t\right)\right)}^2\hfill \\ \hfill \hfill & \hfill -c_e\left(\left(\omega +{\omega }_{AI}\right)u^{AI}\left(t\right)-\overline{e}\right)-c_h\varphi {\left(Y^{AI}\left(t\right)\right)}^2\hfill \\ \hfill \hfill & \hfill -\varphi {\Pi }_G^{AI}+({\Pi }_{MSS}^{AI}\prime )(u^{AI}\left(t\right)-D^{AI}\left(t\right)),\hfill \end{array}$$ (A.9)

$$\begin{array}{cc}\hfill \rho \left({\Pi }_{USS}^{AI}\right)=\hfill & \hfill \left(1-\eta \right)D^{AI}\left(t\right)p^{AI}\left(t\right)\left(1-\varphi \right)\hfill \\ \hfill \hfill & \hfill -\lambda D^{AI}\left(t\right)p^{AI}\left(t\right){\left(1-\varphi \right)}^2\hfill \\ \hfill \hfill & \hfill -c_h\left(1-\varphi \right){\left(Y^{AI}\left(t\right)\right)}^2-\left(1-\varphi \right){\Pi }_G^{AI}\hfill \\ \hfill \hfill & \hfill +({\Pi }_{USS}^{AI}\prime )(u^{AI}\left(t\right)-D^{AI}\left(t\right)).\hfill \end{array}$$ (A.10)

${\Pi }_{MSS}^{AI}\prime$ and ${\Pi }_{USS}^{AI}\prime$ are the first-order partial derivatives of the optimal value function with respect to the inventory level state variable $Y^{AI}$, respectively, and are the marginal contributions of the inventory level to the value function. According to the structure of Eqs. (A.9), (A.10), let ${\Pi }_{MSS}^{AI},{\Pi }_{USS}^{AI},{\Pi }_{MSS}^{AI}\prime ,{\Pi }_{USS}^{AI}\prime$ be.

$$\begin{array}{ccc}\hfill {\Pi }_{MSS}^{AI}\hfill & \hfill =\hfill & \hfill \frac{M_1^{AI}}{2}\left(Y_{\mathrm{SS}}^{AI}\right){}^2+M_2^{AI}{Y}_{\mathrm{SS}}^{AI}+M_3^{AI},\hfill \\ \hfill {\Pi }_{USS}^{AI}\hfill & \hfill =\hfill & \hfill \frac{U_1^{AI}}{2}\left(Y_{\mathrm{SS}}^{AI}\right){}^2+U_2^{AI}{Y}_{\mathrm{SS}}^{AI}+U_3^{AI},\hfill \end{array}$$ (A.11)

$$\begin{array}{ccc}\hfill {\Pi }_{MSS}^{AI}\prime \hfill & \hfill =\hfill & \hfill Y_{\mathrm{SS}}^{AI}{M}_1^{AI}+M_2^{AI},\hfill \\ \hfill {\Pi }_{USS}^{AI}\prime \hfill & \hfill =\hfill & \hfill Y_{\mathrm{SS}}^{AI}{U}_1^{AI}+U_2^{AI}.\hfill \end{array}$$ (A.12)

where $M_1^{AI},M_2^{AI},M_3^{AI},U_1^{AI},U_2^{AI},U_3^{AI}$ are constants. Substituting Eqs. (A.11), (A.12) into Eqs. (A.9), (A.10) respectively give

$$\begin{array}{cc}\hfill \rho \left(\begin{array}{c}\frac{{\left(Y_{\mathrm{SS}}^{\mathrm{AI}}\right)}^2{M}_1^{\mathrm{AI}}}{2}\hfill \\ +Y_{\mathrm{SS}}^{\mathrm{AI}}{M}_2^{\mathrm{AI}}\hfill \\ +M_3^{\mathrm{AI}}\hfill \end{array}\right)=\hfill & \hfill \left(1-\eta \right)D^{AI}\left(t\right)\left(1+\lambda \right)\left(p^{AI}\left(t\right)\varphi -c-c_{AI}\right)\hfill \\ \hfill \hfill & \hfill -\lambda D^{AI}\left(t\right)p^{AI}\left(t\right)\left(1-\varphi \right)\varphi \hfill \\ \hfill \hfill & \hfill -\frac{h}{2}{\left(u^{AI}\left(t\right)\right)}^2-c_e\left(\left(\omega +{\omega }_{AI}\right)u^{AI}\left(t\right)-\overline{e}\right)\hfill \\ \hfill \hfill & \hfill -c_h\varphi {\left(Y^{AI}\left(t\right)\right)}^2-\varphi {\Pi }_G^{AI}\hfill \\ \hfill \hfill & \hfill +(Y_{\mathrm{SS}}^{\mathrm{AI}}{M}_1^{\mathrm{AI}}+M_2^{\mathrm{AI}})(u^{AI}\left(t\right)-D^{AI}\left(t\right)),\hfill \end{array}$$ (A.13)

$$\begin{array}{cc}\hfill \rho \left(\begin{array}{c}\frac{{\left(Y_{\mathrm{SS}}^{\mathrm{AI}}\right)}^2{U}_1^{\mathrm{AI}}}{2}\hfill \\ +Y_{\mathrm{SS}}^{\mathrm{AI}}{U}_2^{\mathrm{AI}}\hfill \\ +U_3^{\mathrm{AI}}\hfill \end{array}\right)=\hfill & \hfill \left(1-\eta \right)D^{AI}\left(t\right)p^{AI}\left(t\right)\left(1-\varphi \right)\hfill \\ \hfill \hfill & \hfill -\lambda D^{AI}\left(t\right)p^{AI}\left(t\right){\left(1-\varphi \right)}^2\hfill \\ \hfill \hfill & \hfill -c_h\left(1-\varphi \right){\left(Y^{AI}\left(t\right)\right)}^2-\left(1-\varphi \right){\Pi }_G^{AI}\hfill \\ \hfill \hfill & \hfill +(Y_{\mathrm{SS}}^{\mathrm{AI}}{U}_1^{\mathrm{AI}}+U_2^{\mathrm{AI}})(u^{AI}\left(t\right)-D^{AI}\left(t\right)).\hfill \end{array}$$ (A.14)

In a dynamic decision-making environment, the vaccine supply chain need to consider the impact of changes in inventory levels on future profits and make optimal decisions to maximize long-term profits. According to the first-order optimization conditions, it is obtained that

$$u_{SS}^{AI}=\frac{(Y_{SS}^{AI}{M}_1^{AI}+M_2^{AI})-c_e(\omega +{\omega }_{AI})}{h},$$ (A.15)

$$p_{SS}^{AI}=\frac{1}{2}(\frac{1+s+Y_{SS}^{AI}\alpha -W\gamma -\theta }{\beta }+\frac{Y_{SS}^{AI}{U}_1^{AI}+U_2^{AI}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}).$$ (A.16)

Substituting Eqs. (A.13), (A.16), the Riccati system is obtained after collation as follows.

$$\begin{array}{c}\frac{1}{4}(-4\varphi c_h+\frac{2{\left(M_1^{\mathrm{AI}}\right)}^2}{h}+M_1^{\mathrm{AI}}(-2(n\alpha +\rho )+\frac{2n\beta U_1^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ +\frac{n\varphi (1-\eta (1+\lambda )+\lambda \varphi )({\alpha }^2-\frac{{\beta }^2{\left(U_1^{\mathrm{AI}}\right)}^2}{{(-1+\varphi )}^2{(-1+\eta +\lambda -\lambda \varphi )}^2})}{\beta })=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{2}(cn\alpha (-1+\eta )(1+\lambda )-2Ak\varphi +\frac{n\alpha (1+s-W\gamma -\theta )\varphi (1-\eta (1+\lambda )+\lambda \varphi )}{\beta }\hfill \\ +n(-1+\eta )(1+\lambda )c_{\mathrm{AI}}(\alpha -\frac{\beta U_1^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ +(-(1-\eta +\lambda (-1+\varphi ))(-1+\varphi )(-(n\alpha +2\rho )(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )M_2^{\mathrm{AI}}\hfill \\ +n\beta (-c(-1+\eta )(1+\lambda )+M_2^{\mathrm{AI}})U_1^{\mathrm{AI}})\hfill \\ +n\beta \varphi (-1+\eta +\eta \lambda -\lambda \varphi )U_1^{\mathrm{AI}}{U}_2^{\mathrm{AI}})/\left((-1+\varphi ){}^2(-1+\eta +\lambda -\lambda \varphi ){}^2\right)\hfill \\ +M_1^{\mathrm{AI}}(n(-1-s+W\gamma +\theta +\frac{\beta U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})+\frac{2M_2^{\mathrm{AI}}-2c_e(\omega +{\omega }_{\mathrm{AI}})}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(2cn(-1+\eta )(1+s-W\gamma -\theta )(1+\lambda )-4A^2\varphi +\hfill \\ \frac{n{(-1-s+W\gamma +\theta )}^2\varphi (1-\eta (1+\lambda )+\lambda \varphi )}{\beta }+4e{c}_e+\frac{2{\omega }^2{c}_e^2}{h}+\frac{2{\left(M_2^{\mathrm{AI}}\right)}^2}{h}\hfill \\ -4\rho M_3^{\mathrm{AI}}-\frac{2cn\beta (-1+\eta )(1+\lambda )U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}+\frac{n\beta \varphi (-1+\eta +\eta \lambda -\lambda \varphi ){\left(U_2^{\mathrm{AI}}\right)}^2}{{(-1+\varphi )}^2{(-1+\eta +\lambda -\lambda \varphi )}^2}\hfill \end{array}$$

$$\begin{array}{c}+2n(-1+\eta )(1+\lambda )c_{\mathrm{AI}}(1+s-W\gamma -\theta -\frac{\beta U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})\hfill \\ -4A\delta \varphi ({ϵ}_D^2+{ϵ}_U^2)+\frac{2c_e^2{\omega }_{\mathrm{AI}}(2\omega +{\omega }_{\mathrm{AI}})}{h}\hfill \\ +2M_2^{\mathrm{AI}}(n(-1-s+W\gamma +\theta +\frac{\beta U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})-\frac{2c_e(\omega +{\omega }_{\mathrm{AI}})}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(\frac{n{\alpha }^2(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}{\beta }+4(-1+\varphi )c_h\hfill \\ +U_1^{\mathrm{AI}}(-2(n\alpha +\rho )+\frac{4M_1^{\mathrm{AI}}}{h}+\frac{n\beta U_1^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{2}(\frac{(-1+\varphi )(2Ak\beta +n\alpha (1+s-W\gamma -\theta )(-1+\eta +\lambda -\lambda \varphi ))}{\beta }+(-n\alpha -2\rho +\frac{2M_1^{\mathrm{AI}}}{h})U_2^{\mathrm{AI}}\hfill \\ +U_1^{\mathrm{AI}}(n(-1-s+W\gamma +\theta +\frac{\beta U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})+\frac{2M_2^{\mathrm{AI}}-2c_e(\omega +{\omega }_{\mathrm{AI}})}{h}))=0,\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{4}(-4\rho U_3^{\mathrm{AI}}+\frac{(-1+\varphi )(4A^2\beta +n{(-1-s+W\gamma +\theta )}^2(-1+\eta +\lambda -\lambda \varphi )+4A\beta \delta ({ϵ}_D^2+{ϵ}_U^2))}{\beta }\hfill \\ +U_2^{\mathrm{AI}}(n(2(-1-s+W\gamma +\theta )+\frac{\beta U_2^{\mathrm{AI}}}{(-1+\varphi )(-1+\eta +\lambda -\lambda \varphi )})+\frac{4M_2^{\mathrm{AI}}-4c_e(\omega +{\omega }_{\mathrm{AI}})}{h}))=0.\hfill \end{array}$$

Appendix B.

In Appendix B, we will analyze the sensitivity of each performance indicator in the model to each variable to test the robustness of the results obtained in this paper. The baseline values of the variables for the sensitivity analysis are

$$\begin{array}{c}\alpha =1,\beta =0.8,c=0.2,\varphi =0.8,h=1,c_h=0.11,\rho =0.1,\hfill \\ c_e=0.2,\omega =1.5,{\omega }_{\mathrm{AI}}=0.1,\overline{e}=0.3,c_{\mathrm{AI}}=0.1,A=0.4,\hfill \\ k=0.2,\delta =0.1,{ϵ}_D=-0.1,{ϵ}_U=0.1,\lambda =0.02,n=1.5,\hfill \\ \gamma =0.5,W=4,\theta =0.01,s=1,c_{\mathrm{BV\; P}}=0.01,\eta =0.05.\hfill \end{array}$$

The results of the benchmark tests for each performance indicator derived from the baseline values of the variables are

$$\begin{array}{c}{Y}_{SS}^B=2.444,u_{SS}^B=0.302,p_{SS}^B=2.735,D_{SS}^B=0.369,C{S}_{SS}^B=0.0454,\hfill \\ S{W}_{SS}^B=1.747,{\Pi }_{MSS}^B=1.061,{\Pi }_{USS}^B=0.596,{\Pi }_{BVPSS}^B=0.0438,\hfill \\ {\Pi }_{GSS}^B=0;\hfill \end{array}$$

$$\begin{array}{c}{Y}_{SS}^{AI}=5.539,u_{SS}^{AI}=0.883,p_{SS}^{AI}=6.175,D_{SS}^{AI}=0.883,C{S}_{SS}^{AI}=0.260,\hfill \\ S{W}_{SS}^{AI}=5.0437,{\Pi }_{MSS}^{AI}=1.569,{\Pi }_{USS}^{AI}=2.357,\hfill \\ {\Pi }_{BVPSS}^{AI}=0.259,{\Pi }_{GSS}^{AI}=0.604.\hfill \end{array}$$

The $M_i^B,U_i^B,M_i^{AI},U_i^{AI}$ ($i=1...3$) benchmark test results derived from the baseline values of the variables are

$$\begin{array}{c}{M}_1^B=0.115,U_1^B=0.225,M_2^B=0.265,U_2^B=-0.039,\hfill \\ M_3^B=0.068,U_3^B=0.021;M_1^{AI}=0.149,U_1^{AI}=0.201,\hfill \\ M_2^{AI}=0.375,U_2^{AI}=-0.082,M_3^{AI}=-2.800,U_3^{AI}=-0.265.\hfill \end{array}$$

Based on the baseline test results, we varied the values of each variable within a specific range and observed the consequent changes in each performance indicator and $M_i^B,U_i^B,M_i^{AI},U_i^{AI}$ ($i=1...3$). The results of the performance indicator tests are detailed in Table B.1, Table B.2. The results of the $M_i^B,U_i^B,M_i^{AI},U_i^{AI}$ ($i=1...3$) tests are detailed in Table B.3, Table B.4.

Appendix C.

To determine the value of $M_i^B,U_i^B$ ($i=1...3$), we substitute the values in Table C.6 into the Riccati system to obtain two sets of real roots.

$$\begin{array}{c}\hfill root1:\hfill \\ \hfill M_1^B=1.2960704908814287,U_1^B=-0.09196658004726502,\hfill \\ \hfill M_2^B=0.5922616130080464,U_2^B=0.06813554569491495,\hfill \\ \hfill M_3^B=2.4427568427601654,U_3^B=0.3277503414865632;\hfill \\ \hfill root2:\hfill \\ \hfill M_1^B=0.11539303236839006,U_1^B=0.22483403189608073,\hfill \\ \hfill M_2^B=0.2653223752286094,U_2^B=-0.039256229180258374,\hfill \\ \hfill M_3^B=0.06769443130977659,U_3^B=0.02074879624526238.\hfill \end{array}$$

Substituting the above two sets of roots and the values in Table C.6 into Eq. (26), we obtain

$$Y_{SS1}^B=-0.8651424149046247,Y_{SS2}^B=2.4442615495361406,$$

where $Y_{SS2}^B$ satisfies the assumption of a positive inventory level. Substituting $Y_{SS}^B=2.4442615495361406$ into Eqs. (22)–(24), we obtain

$$\begin{array}{c}\hfill u_{SS}^B=0.3021104400641489,p_{SS}^B=2.735121859077061,\hfill \\ \hfill D_{SS}^B=0.36924609341173775,C{S}_{SS}^B=0.04544755916660992,\hfill \\ \hfill S{W}_{SS}^B=1.746556311228034,{\Pi }_{MSS}^B=1.0609146155035325,\hfill \\ \hfill {\Pi }_{USS}^B=0.5964221573408529,{\Pi }_{BVPSS}^B=0.043771979217038674,\hfill \end{array}$$

To determine the value of $M_i^{AI},U_i^{AI}$ ($i=1...3$), we substitute the values in Table C.6 into the Riccati system to obtain two sets of real roots, as follows:

$$\begin{array}{c}\hfill root1:\hfill \\ \hfill M_1^{AI}=1.8112093987987976,U_1^{AI}=-0.07469752136786388,\hfill \\ \hfill M_2^{AI}=0.6889018498476401,U_2^{AI}=0.06335557940748664,\hfill \\ \hfill M_3^{AI}=1.776357377333373,U_3^{AI}=-0.01939392395107195;\hfill \\ \hfill root2:\hfill \\ \hfill M_1^{AI}=0.14949053965698567,U_1^{AI}=0.2006964431004427,\hfill \\ \hfill M_2^{AI}=0.37475715621200184,U_2^{AI}=-0.08241756711145327,\hfill \\ \hfill M_3^{AI}=-2.799895848958785,U_3^{AI}=-0.2651161940697662.\hfill \end{array}$$

Substituting the above two sets of roots and the values in Table C.6 into Eq. (35), we obtain

$$Y_{SS1}^{AI}=-0.7003524174197305,Y_{SS2}^{AI}=5.538722258883102$$

where $Y_{SS2}^{AI}$ satisfies the assumption of a positive inventory level. Substituting $Y_{SS}^{AI}=5.538722258883102$ into Eqs. (31)–(33), we obtain:

$$\begin{array}{c}\hfill u_{SS}^{AI}=0.8827437357025953,p_{SS}^{AI}=6.1752830438517154,\hfill \\ \hfill D_{SS}^{AI}=0.8827437357025962,C{S}_{SS}^{AI}=0.25974550097405835,\hfill \\ \hfill S{W}_{SS}^{AI}=5.043650460038378,{\Pi }_{MSS}^{AI}=1.5687738027688787,\hfill \\ \hfill {\Pi }_{USS}^{AI}=2.35682276574916,{\Pi }_{BVPSS}^{AI}=0.25891260288771567,\hfill \\ \hfill {\Pi }_{GSS}^{AI}=0.6038977807106483\hfill \end{array}$$

Appendix D.

The effect of $s$ on the system equilibrium and the effect of $\eta$ on the system equilibrium are in Fig. D1, Fig. D2.

Data availability

Data will be made available on request.