Innovation Prioritization Decisions in the Product–Service Supply Chain: The Impact of Data Mining and Information Sharing Strategies

Abstract

Driven by the servicing and digital transformation of manufacturing enterprises, product and service innovation for manufacturers and service providers to promote integrated solutions collaboratively has become an important way for enterprises to maintain market competitiveness. Building on this foundation, this paper develops an innovation priority decision model for the product–service supply chain, which comprises manufacturers and service providers, considering the data mining and information sharing strategies of service providers. It analyzes the optimal decisions and profits of the members when product innovation is prioritized as well as when service innovation is prioritized, and subsequently explores the selection of innovation strategies for the product–service supply chain under varying conditions. The results of the study show that, firstly, service providers’ data mining and information sharing strategies are not always favorable to the innovation decisions of both parties. Only when data resources can be transformed into real innovation value at a reasonable cost can data mining and information sharing play the role of ‘external incentives’ to promote collaborative innovation between the two parties. Secondly, when service providers do not adopt data mining and information sharing strategies, the efficiency of product and service innovation plays a decisive role in innovation prioritization. The party with high innovation efficiency adopts the sub-priority innovation strategy, which can lead to a larger market share for the innovation results. Finally, under the service provider’s data mining and information sharing strategy, the innovation priority selection of the product–service supply chain depends on the information value transformation ability of the manufacturer and the service provider. Moreover, the profits of manufacturers and service providers under the same innovation priority do not always ‘advance or retreat together’, and there may be cases where one of them suffers a loss of profits. This study provides a theoretical basis for the choice of innovation strategies given to manufacturers and service providers, and promotes the development of collaborative innovation between them.

1. Introduction

Driven by the wave of global digital transformation and smart manufacturing, services carried in the product entity have become a common means for manufacturing enterprises to realize added value during the product technology life cycle [1]. More and more manufacturing enterprises promote the digital transformation of their new product development by ‘leveraging’ digital service enterprises [2], and, through the integration and management of the latter’s information technology, the Internet of Things, big data analysis [3], and other means, they provide customers with integrated product–service customized solutions [4,5], realizing a seamless transition from product manufacturing to service delivery. For example, Haier cooperates with Aliyun to provide a complete set of smart home solutions, through which smart home appliances are connected to the cloud, and users can use mobile phone apps to control and manage home appliances to achieve equipment interconnectivity remotely. Sany Heavy Industries and Baidu Cloud developed the ‘Root Cloud Platform’ to achieve real-time data collection of engineering machinery and equipment worldwide, through remote monitoring of the working status of the equipment, to pre-judge potential problems, and to achieve accurate maintenance and repair. Compared with the independent provision of products or services, hybrid products can achieve greater customer value creation through the optimal matching of products and services [6]. However, under the impetus of servicing and digitizing manufacturing enterprises, customers’ demand for innovation in products and value-added services is increasing, and manufacturing enterprises and service enterprises must maintain their competitiveness in the market through continuous innovation. As a result, hybrid product offerings, driven by both product and service innovation, have become the specific focus of many manufacturing companies’ servicing efforts [7].

The wide application of digital scenarios provides more possibilities to achieve synergy in product and service innovation. The empowerment of digital intelligence technology enables the various experiences, advice, and information that traditional market insights rely on to be stored in the form of massive numbers of data [8]. The collected data can be organized to form an information flow so that scientific and precise actions can be taken under the guidance of data support [9,10]. For example, the Root Cloud Platform not only helps Sany Heavy Industry optimize equipment operation and maintenance, but also analyzes the efficiency of excavators under different working conditions through the data collected from users’ use and feedback. By understanding the different requirements of customers for the performance of equipment under different geological conditions, Sany Heavy Industry can improve product design and launch machinery and equipment that are more suitable for specific working conditions. In the delivery of integrated solutions for innovative products and services, service providers provide last-mile digital value-added services, which also means that service providers are able to achieve access to product operation and customer feedback data over a relatively long period of time through technologies such as the Internet of Things (IoT) and big data [11]. Digital service companies use advanced analytics to mine and analyze these data to gain insights into customer needs [12] as well as correlative information that can improve the quality of their products and services [13]. As a result, manufacturers often require service providers to report the necessary data during the contracting process. The value of these ‘non-monetary transaction’ data becomes the intermediary that drives product innovation. At the same time, the continuous emergence of back-office product innovation triggers the iteration of front-office service innovation [14], and the complementary nature of these two types of innovations drives overall innovation and competitiveness enhancement.

In the traditional product-led logic, enterprises usually regard products as the core carriers of value, and product innovation becomes the main means of enterprise competition, while service innovation enhances the use of products and strengthens customer experience by optimizing services. Under this innovation logic, service innovation is more often used as a support for and an extension of product innovation [7]. As the digitalization process of manufacturing enterprises advances, the higher level of product intelligence supports the development of higher-order services [15]. It creates an interaction and transaction environment where multiple subjects participate and diverse scenarios coexist, and a massive number of data/amount of information is generated by customers in the process of service experience [16]. Deep mining and analysis based on massive data breaks the unidirectional pattern of relying on product innovation to drive service innovation. From only being able to access explicit knowledge, such as knowledge of product usage, functional defects, and improvement opinions, it has evolved to draw on the tacit knowledge hidden in product operation and customer feedback data [1]. This helps decision-makers to identify unmet innovation needs and generate new innovation insights [17,18]. Service innovation is no longer just a passive response to product innovation, but may become the source driving overall innovation. In this new innovation ecosystem, the boundaries between product and service innovation become more blurred, and this change requires us to rethink the roles of manufacturers and service providers in the innovation process. Through rational decision-making with regard to the priority of product and service innovation, we can ensure the internal matching of innovative products and services with the external demand market, and realize greater value creation for both supply and demand.

2. Literature Review

The research area of this paper primarily encompasses two aspects: data resource mining and information sharing among supply chain members and the innovation of the product–service supply chain. Consequently, the related literature review was conducted from these two perspectives.

2.1. Supply Chain Innovation Decisions

Currently, scholars are examining innovation decision-making in product–service supply chains from various perspectives. Several scholars have explored the factors influencing innovation: Du et al. [19] investigated the effects of supply chain information asymmetry and retailer overconfidence on members’ decisions regarding innovation inputs and revenues utilizing a signaling game approach. Liu et al. [20] explores the impact of remanufacturing process innovation on closed-loop supply chains from the perspective of government subsidies. Qu et al. [21] examines joint innovation investment and pricing decisions for revenue-sharing contracts and customer value. Xing et al. [22] constructed a green innovation supply chain consisting of suppliers, manufacturers, and retailers, examining the effects of various government subsidies on this supply chain. Yi et al. [23] employed a differential game approach to analyze the impact of disappointment aversion on supply chain members’ innovation decisions and proposed a coordination mechanism that includes subsidies and revenue matching. Another group of scholars has examined the issue of innovation choice within product–service supply chains: Liu et al. [24] found that retailers provide extended warranty services only when the cost advantage is significant. Furthermore, manufacturers can enhance retailers’ incentives to affiliate through innovation investment, fostering cooperation between both parties. Yu et al. [25] compared and analyzed optimal decision-making under two innovation modes, product innovation alone and retailer equity-financed product innovation, considering suppliers’ varying dominant positions and highlighting the influence of multiple factors on the choice of innovation mode. Li et al. [26] proposed two innovation strategies—closed innovation and leading user innovation—and analyzed the influence of consumer heterogeneity on each strategy’s innovation decisions. Lin et al. [27] investigated the equilibrium results under three innovation strategies: manufacturer’s non-innovation, independent innovation, and technology purchasing in the context of uncertain technical efficiency. Jiang et al. [28] examine the conditions applicable to new energy vehicle enterprises for implementing process or model innovation, discussing the impact of government subsidies on innovation decisions. Current research on innovation primarily emphasizes product innovation; however, the strong correlation between product and service innovation, which jointly influences demand, remains unexamined. Furthermore, there is a paucity of research addressing decision-making issues related to innovation prioritization stemming from the blurred boundaries between product and service innovations. Moreover, the decision-making problem regarding innovation prioritization, resulting from the blurred boundaries between product and service innovation, has received limited scholarly attention.

2.2. Supply Chain Information Sharing

A data resource is a kind of resource that contains great value. Through their data mining and sharing strategy, an enterprise can obtain effective data and analyze, store, and use this data resource, which can increase the revenue of the enterprise [17]. Brinch [29] emphasizes that the analysis of data resource value is emerging as a significant research hotspot in the field of supply chain management. A related area of inquiry is the sharing of demand information within the supply chain. Research on demand information sharing in supply chains primarily investigates the effects of various factors on information sharing and its impact on decision-making among stakeholders. For instance, Zhang et al. [30] found that, when both manufacturers and retailers forecast market demand and retailers possess asymmetric information, they will voluntarily share this information when the manufacturer’s innovation efficiency is high. Shi et al. [31] examined product network externalities and found that demand information sharing between retailers and manufacturers correlates with the network externality coefficient, making information sharing profitable when this coefficient is high. Wang et al. [32] investigated dynamic strategies for knowledge sharing and emission reduction benefits in low-carbon technologies within green supply chains using differential game methods, identifying optimal trajectories for suppliers’ and manufacturers’ knowledge stocks and emission reduction benefits under various strategies. Tang et al. [33] analyzed the impact of information prediction and sharing strategies on manufacturing companies’ supply chains under independent and joint innovation models. Nie et al. [34] discovered that incorporating remanufactured products into the product line can create a win–win–win scenario for retailers, manufacturers, and the environment through effective information sharing strategies. Wei et al. [35] revealed that, irrespective of the supplier’s distribution model and the retailer’s information sharing strategy, greater consumer sensitivity to the product’s green level correlates with higher expected profits, product prices, and green levels among channel members. The existing literature primarily considers information sharing as a factor influencing supply chain decision-making, neglecting the value of data resources as economic assets. Data resources hold significant potential value that can be transformed into business profits through effective acquisition, analysis, storage, and utilization. This paper, therefore, discusses the impact of data resource sharing behavior on product and service innovation by incorporating service providers’ data resource sharing into the supply chain and considering the economic value and uncertainty associated with data resources.

To address the issues in the existing research, this study considers two scenarios—information sharing and non-sharing by service providers—and constructs four decision-making models based on product innovation priority strategies and service innovation priority strategies. It analyzes the optimal decision-making, market demand, and profits of each member by solving the different models. This study addresses the following questions: What conditions influence product and service innovation priorities within the product–service innovation supply chain? What impact do service providers’ demand information sharing strategies have on their product and service innovation levels and pricing decisions under the same priority level, and how does this affect profit levels? Further comparisons of market demand for innovative products and services, along with manufacturers’ and service providers’ profits under varying prioritization levels, reveal how different factors influence innovation prioritization choices in product–service supply chains. The research findings aim to provide managerial insights into the collaborative innovation decisions made by manufacturers and service providers.

3. Model Construction and Analysis

3.1. Problem Description and Assumptions

Consider a product–service supply chain consisting of a digital service provider (hereinafter referred to as ‘service provider’, S) and a product manufacturer (hereinafter referred to as ‘manufacturer’, M) which, together, provide an integrated solution, a ‘product + digital service’ (hereinafter referred to as ‘integrated solution’), to a customer. Among them, the service provider provides digital services such as product system modification and additional function expansion of the integrated solution. The manufacturer produces products according to customer and market demand, combines them with the digital services provided by the service provider, and delivers a comprehensive solution integrating products and services to the customer. In order to meet the increasingly individualized needs of customers, both parties must continuously improve and innovate their products and services. Manufacturers, for example, upgrade technology through modular product design, intelligent product upgrades, and ecosystem product design. Service providers, on the other hand, offer innovative services through remote monitoring, troubleshooting for data-driven service optimization, and customized solutions. In the process of implementing product and service innovations, the manufacturer and the service provider first decide on an innovation prioritization strategy and the service provider decides whether to share information. With the service provider adopting an information sharing strategy, if both parties decide to prioritize product innovation, the manufacturer initiates product modification and innovation activities and sets the final sales price of the integrated solution. The service provider then innovates the service and also decides on the unit service price. After both parties have completed their innovations, the manufacturer delivers to the market a comprehensive solution that integrates the innovative products and services. At the end of the sales period, the service provider collects product operation and customer feedback data during the service process, converts them into demand information, and shares them with the manufacturer to guide further innovation. When both parties decide to prioritize service innovation, the service provider takes the lead in initiating service innovation activities and determines the unit service price. At the same time, the service provider shares service innovation information and customer demand information with the manufacturer so that the manufacturer can adapt its product innovation program to the new service characteristics. The manufacturer then innovates the product, sets the selling price, and brings the integrated solution to the market. In cases where the service provider does not share information, the two parties set innovation levels and prices according to different innovation priorities. The product service innovation supply chain structure is shown in Figure 1.

Demand Assumption: Product innovation enhances the performance, stability, and functionality of hardware, software, or systems, making integrated solutions more relevant and flexible, and increasing the attractiveness of the solutions. Service innovation, on the other hand, enhances customer satisfaction and loyalty by providing personalized, continuously optimized services and improving service quality and responsiveness [36]. This has led customers to focus more and more on the overall value provided by integrated solutions, rather than just on single products or services. Therefore, it is further assumed that both the level of product innovation and the level of service innovation have a positive impact on the demand for integrated solutions. With reference to similar studies [37,38], the demand for innovative integrated solutions is portrayed as $\mathrm{D}=\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}$, where $\mathrm{a}$ is the potential market demand for integrated solutions; $\mathrm{p}$ is the selling price of integrated solutions; $\mathrm{b}$ is the price elasticity coefficient of integrated solutions, which measures the sensitivity of the market demand to the selling price; and $\mathsf{\lambda }$ is the product innovation utility and $\mathsf{\beta }$ is the service innovation utility, which represent the degree of customer demand for product and service innovations, respectively.

Innovation Cost Assumptions: Consistent with most innovation cost function assumptions [39,40,41], the innovation input cost functions of manufacturers and service providers are convex functions with respect to the level of innovation. Assuming that the level of service innovation by the service provider is ${\mathrm{e}}_{\mathrm{s}}$ and the level of product innovation by the manufacturer is ${\mathrm{e}}_{\mathrm{m}}$, the cost function of service innovation input by the service provider is $\mathrm{c}({\mathrm{e}}_{\mathrm{s}})={\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$ and the cost function of product innovation input by the manufacturer is $\mathrm{c}({\mathrm{e}}_{\mathrm{m}})={\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$, where ${\mathsf{\mu }}_{\mathrm{s}}$ represents the cost coefficient of service innovation by the service provider and ${\mathsf{\mu }}_{\mathrm{m}}$ is the cost coefficient of product innovation by the manufacturer. For the convenience of the study, the initial production cost of products and services in this study is set to 0, which does not affect the conclusions of the study.

Information Sharing Costs and Benefits Assumptions: According to Ma [42] and Correani [43], the ability of an enterprise to ‘process’ massive data resources into valuable knowledge or information depends largely on the enterprise’s ability to analyze and process massive data and big data. This includes the ability to make data-driven decisions, data productization, and value realization, which is expressed in terms of shorter decision-making cycles, cost reductions as a result of data optimization, and the number of innovative products supported by data. Manufacturers and service providers have the ability to convert data resources into information values, ${\mathrm{k}}_{\mathrm{m}}$ and ${\mathrm{k}}_{\mathrm{s}}$. In this study, integrated solutions are sold to customers as integrated innovative products and services. Therefore, the number of data resources acquired by the service provider is equal to the number of its services under the product innovation priority model [44]. The value of the shared information received by the manufacturer is then quantified as ${\mathrm{k}}_{\mathrm{s}}\mathrm{D}$. In the service innovation priority model, the service provider transforms the collected market demand information into service innovation, and the level of its service innovation represents the data value of the data resources that the service provider converts into information. The manufacturer converts the shared data value into product innovation knowledge, which increases the value of the manufacturer’s product innovation [45] by ${\mathrm{k}}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{s}}$. Additionally, the cost of information sharing by service providers in both models is categorized as ${\mathsf{\eta }}_{\mathrm{s}}\mathrm{D}$ and ${\mathsf{\eta }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}$, including, for example, technical operation and maintenance, data conversion, and communication costs [46]. Among them, ${\mathsf{\eta }}_{\mathrm{s}}$ is the cost coefficient of information sharing for service providers.

The primary parameters utilized in this paper are presented in

Table 1. Description of the primary parameters.

Notation	Description
$\mathrm{a}$	The potential market demand
$\mathrm{b}$	The price elasticity coefficient
$\mathsf{\lambda }$	The product innovation utility
$\mathsf{\beta }$	The service innovation utility
${\mathsf{\mu }}_{\mathrm{m}}$	The cost coefficient of product innovation
${\mathsf{\mu }}_{\mathrm{s}}$	The cost coefficient of service innovation
${\mathsf{\eta }}_{\mathrm{s}}$	The cost coefficient of information sharing
${\mathrm{k}}_{\mathrm{m}}$	Information value transformation coefficient of manufacturer
${\mathrm{k}}_{\mathrm{s}}$	Information value transformation coefficient of service provider
${\mathrm{e}}_{\mathrm{m}}$	The level of product innovation
${\mathrm{e}}_{\mathrm{s}}$	The level of service innovation
$\mathsf{\omega }$	The unit service price
$\mathrm{p}$	The selling price
${\mathsf{\pi }}_{\mathrm{m}}$	The profits of manufacturer
${\mathsf{\pi }}_{\mathrm{s}}$	The profits of service provider

below.

For the sake of clarity, superscripts $\mathrm{S}\mathrm{P}$, $\mathrm{D}\mathrm{P}$, $\mathrm{S}\mathrm{P}\mathrm{I}$, and $\mathrm{D}\mathrm{P}\mathrm{I}$ represent service innovation priority without information sharing, product innovation priority without information sharing, service innovation priority with information sharing, and product innovation priority with information sharing, respectively. The subscripts $\mathrm{m}$ and $\mathrm{s}$ denote manufacturers and service providers, respectively. The symbol ‘*’ indicates the optimal solution.

3.2. Model Construction

This study considers two scenarios—information sharing and non-sharing by service providers—and constructs four decision-making models based on product innovation priority strategies and service innovation priority strategies. Among them, the DP model and SP model do not take into account the information-sharing behavior of service providers, while the DPI model and SPI model do. The utility functions for manufacturers and service providers in the four models are as follows:

(DP Model) In the product innovation priority model, which does not take into account information value conversion and sharing by service providers, the manufacturer takes the lead in completing the product innovation and setting the final sales price for the integrated solution to be released in the market. The service provider determines the level of service innovation and the price per unit of service based on the manufacturer’s decision program.

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}}=\left(\mathrm{p}-\mathsf{\omega }\right)\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$$

(1)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}=\mathsf{\omega }\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$$

(2)

(SP Model) In the service innovation priority model, which does not take into account information value conversion and sharing by service providers, the service provider takes the lead in service innovation and sets the unit price of the innovative service. The manufacturer determines the level of product innovation and the final selling price of the integrated solution released in the market based on the service provider’s decision scenario.

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}=\left(\mathrm{p}-\mathsf{\omega }\right)\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$$

(3)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}}=\mathsf{\omega }\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$$

(4)

In the DP and SP models, $\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)$ is the overall market demand created by the manufacturer and the service provider. Equation (3) is the manufacturer’s utility function, where $\left(\mathrm{p}-\mathsf{\omega }\right)$ and ${\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$ represent the manufacturer’s profit gained from the unit of the integrated solution and the total cost of product innovation, respectively. Equation (4) is the service provider’s utility function, where $\mathsf{\omega }$ and ${\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$ represent the service provider’s profit gained from the unit of the integrated solution and the total cost of service innovation, respectively.

(DPI Model) Similar to the DP model’s decision-making process, the service provider commits to converting the product operation and customer feedback data resources obtained during the delivery of the integrated solution into valuable information and sharing it with the manufacturer. Considering the value of the information shared by the service provider, the manufacturer decides the level of product innovation and sets the selling price of the integrated solution. The service provider decides on the level of service innovation and the price per unit of service based on the manufacturer’s decision scheme.

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}}=\left(\mathrm{p}-\mathsf{\omega }\right)\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)+{\mathrm{k}}_{\mathrm{s}}\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$$

(5)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}}=\mathsf{\omega }(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}})-{\mathsf{\eta }}_{\mathrm{s}}(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}})-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$$

(6)

In the DPI model, the value of the information that the manufacturer receives from the service provider’s data sharing is ${\mathrm{k}}_{\mathrm{s}}\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)$. At the same time, the service provider’s cost of data resource sharing is ${\mathsf{\eta }}_{\mathrm{s}}(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}})$.

(SPI Model) Similar to the SP model’s decision process, the service provider takes the lead in completing the service innovation and determining the unit price of the innovative service. At the same time, market demand and service innovation information is shared with the manufacturer. Based on the service provider’s innovation decisions and the value of the shared information, the manufacturer determines the level of product innovation and the final selling price of the innovative integrated solution released to the market.

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}}=\left(\mathrm{p}-\mathsf{\omega }+{\mathrm{k}}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{s}}\right)\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}^2$$

(7)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}}=\mathsf{\omega }\left(\mathrm{a}-\mathrm{b}\mathrm{p}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}\right)-{\mathsf{\eta }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}-{\displaystyle \frac{1}{2}}{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}^2$$

(8)

In the SPI model, the preferred innovation service enables the manufacturer to obtain an additional value of ${\mathrm{k}}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{s}}$ from the unit integrated solution. Meanwhile, the service provider’s cost of sharing data resources is ${\mathsf{\eta }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}$.

3.3. Model Analysis

By employing backward induction, the equilibrium solutions of the four models can be derived. The derivation of Propositions 1 and 2 is presented below, and the procedure for proving the remaining propositions is analogous. The proofs of the corollaries are included in Appendix A.

Proposition 1 

(DP Model). Manufacturers prioritize product innovation without considering service provider information sharing.

(i) The optimal levels of product innovation (${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$), service innovation (${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$), price per unit of service (${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}$), and selling price (${\mathrm{p}}^{\mathrm{D}\mathrm{P}*}$) for both manufacturers and service providers are defined as follows:

$${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathsf{\lambda }{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(9)

$${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}}{{4\mathrm{b}\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(10)

$${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(11)

$${\mathrm{p}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(3\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2)}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(12)

(ii) The optimal profits for manufacturers (${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$) and service providers (${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$) are defined as follows:

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{{\mathrm{a}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{2(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(13)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{{\mathrm{a}}^2{\mathsf{\mu }}_{\mathrm{s}}{\mathsf{\mu }}_{\mathrm{m}}^2(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}{2{(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}^2}}$$

(14)

Proof. 

The manufacturer first sets the sales premium $\mathrm{r}$ for the innovative integrated solution, and the sales price determined by the manufacturer is the price per unit of service provided by the service provider $\mathsf{\omega }$ plus its sales premium $\mathrm{r}$, which can be expressed as $\mathrm{p}=\mathsf{\omega }+\mathrm{r}$. Substituting into Equations (1) and (2) and applying the inverse induction method of solving for Equation (2) to find the first-order derivatives of ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}$ with respect to $\mathsf{\omega }$ and ${\mathrm{e}}_{\mathrm{s}}$ and making the first-order derivatives equal to 0 yields ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial \mathsf{\omega }}}=\mathrm{a}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}-\mathrm{b}\mathsf{\omega }-\mathrm{b}(\mathrm{r}+\mathsf{\omega })=0$, ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{s}}}}=\mathsf{\beta }\mathsf{\omega }-{\mathsf{\mu }}_{\mathrm{s}}{\mathrm{e}}_{\mathrm{s}}=0$. Therefore, the Hessian matrix of ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}$ is ${\mathrm{H}}_1=\left[\begin{array}{cc}-2\mathrm{b}& \mathsf{\beta }\\ \mathsf{\beta }& -{\mathsf{\mu }}_{\mathrm{s}}\end{array}\right]$. It can be shown that the Hessian matrix $\left|{\mathrm{H}}_1\right|$ is negative definite when $2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2>0$. The service provider profit ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}$ is a joint concave function of $\mathsf{\omega }$ and ${\mathrm{e}}_{\mathrm{s}}$, and there exists an optimal solution with great value. Substituting the result of ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial \mathsf{\omega }}}$ and ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{s}}}}$ being equal to 0 into Equation (1) and taking the first-order derivatives of $\mathrm{r}$ and ${\mathrm{e}}_{\mathrm{m}}$, respectively, the Hessian matrix of ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}}$ is ${\mathrm{H}}_2=\left[\begin{array}{cc}{\displaystyle \frac{2{\mathrm{b}}^2{\mathrm{u}}_{\mathrm{s}}}{{\mathsf{\beta }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}}}& {\displaystyle \frac{\mathrm{b}\mathsf{\lambda }{\mathsf{\mu }}_{\mathrm{s}}}{2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2}}\\ {\displaystyle \frac{\mathrm{b}\mathsf{\lambda }{\mathsf{\mu }}_{\mathrm{s}}}{2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2}}& -{\mathsf{\mu }}_{\mathrm{m}}\end{array}\right]$. When $4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}>0$, the Hessian matrix $\left|{\mathrm{H}}_2\right|$ is negative definite and the profit function ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}}$ is the joint concave function of $\mathrm{r}$ and ${\mathrm{e}}_{\mathrm{m}}$. Seeking its first-order derivatives equal to 0, the joint solution to the manufacturer’s optimal level of innovation and the optimal sales premium, which will be brought into the ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial \mathsf{\omega }}}$, ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{s}}}}$, are solved to obtain the optimal level of innovation of the service provider and the optimal unit price of service results for Formulas (10) and (11). The manufacturer’s optimal sales price $\mathrm{p}$ is equal to the optimal unit service price $\mathsf{\omega }$ plus $\mathrm{r}$. Substituting the above optimal decision variables into the profit function of the manufacturer, the service provider can be obtained. In the product innovation priority, the manufacturer and the service provider profit results in Formulas (13) and (14), and Proposition 1 is proved. □

Proposition 2 

(SP Model). Service providers prioritize service innovation without considering service provider information sharing.

(i) The optimal levels of product innovation (${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$), service innovation (${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$), price per unit of service (${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}$), and selling price (${\mathrm{p}}^{\mathrm{S}\mathrm{P}*}$) for both manufacturers and service providers are defined as follows:

$${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathsf{\lambda }{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(15)

$${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(16)

$${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(17)

$${\mathrm{p}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}}(3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(18)

(ii) The optimal profits for manufacturers (${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$) and service providers (${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$) are defined as follows:

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{{\mathrm{a}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}^2(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}{2{(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}^2}}$$

(19)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{{\mathrm{a}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{2(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(20)

Proof. 

The inverse induction method is used to solve Equation (3) to find the first-order derivative of ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}$ with respect to $\mathrm{p}$, ${\mathrm{e}}_{\mathrm{m}}$ and make it equal to 0 to obtain ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial \mathrm{p}}}=\mathrm{a}+\mathsf{\lambda }{\mathrm{e}}_{\mathrm{m}}+\mathsf{\beta }{\mathrm{e}}_{\mathrm{s}}-\mathrm{b}\mathrm{p}-\mathrm{b}(\mathrm{p}-\mathsf{\omega })=0$, ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{m}}}}=\mathsf{\lambda }(\mathrm{p}-\mathsf{\omega })-{\mathsf{\mu }}_{\mathrm{m}}{\mathrm{e}}_{\mathrm{m}}=0$. Then, the Hessian matrix of ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}$ is ${\mathrm{H}}_1=\left[\begin{array}{cc}-2\mathrm{b}& \mathsf{\lambda }\\ \mathsf{\lambda }& -{\mathsf{\mu }}_{\mathrm{m}}\end{array}\right]$. It can be seen that the Hessian matrix $\left|{\mathrm{H}}_1\right|$ is negatively definite when $2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2>0$. The manufacturer profit ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}$ is a joint concave function of $\mathrm{p}$ and ${\mathrm{e}}_{\mathrm{m}}$, and there exists an optimal solution with a great value. Substituting the result of ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial \mathrm{p}}}$ and ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{m}}}}$ equal to 0 into Equation (4) and taking the first-order derivatives of $\mathsf{\omega }$ and ${\mathrm{e}}_{\mathrm{s}}$, respectively, it can be seen that the Hessian matrix of ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}}$ is ${\mathrm{H}}_2=\left[\begin{array}{cc}{\displaystyle \frac{2{\mathrm{b}}^2{\mathrm{u}}_{\mathrm{m}}}{{\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}}}& {\displaystyle \frac{\mathrm{b}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}}{2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2}}\\ {\displaystyle \frac{\mathrm{b}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}}{2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2}}& -{\mathsf{\mu }}_{\mathrm{s}}\end{array}\right]$. When $4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}>0$, the Hessian matrix $\left|{\mathrm{H}}_2\right|$ is negative definite and the profit function ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}}$ is the joint concave function of $\mathsf{\omega }$ and ${\mathrm{e}}_{\mathrm{s}}$. Seeking its first-order derivatives equal to 0, the joint solution to the optimal level of innovation of the service provider and the optimal price of services, respectively, can be found for the results of Equations (16) and (17), with ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial \mathrm{p}}}$ and ${\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}}}{\partial {\mathrm{e}}_{\mathrm{m}}}}$ being used to solve for the optimal level of innovation of the manufacturer and the optimal price of the results of the sales from Equations (15)–(18). Substituting the above optimal decision variables into the profit function of the manufacturer, the service provider can be obtained. When the service innovation is prioritized, the manufacturer and the service provider profits result in Equations (19) and (20), respectively, and Proposition 2 is proved. □

Corollary 1. 

The optimal decision, when product innovation is prioritized, increases with the product innovation utility $\mathsf{\lambda }$ and decreases with the product innovation cost coefficient ${\mathsf{\mu }}_{\mathrm{m}}$, without considering service provider information sharing.

$$\begin{gathered} {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0, {\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0, {\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}}{\partial \mathsf{\lambda }}}>0; \\ {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0, {\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0, {\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{m}}}}<0; \end{gathered}$$

Corollary 2. 

The optimal decision, when service innovation is prioritized, increases with the service innovation utility $\mathsf{\beta }$ and decreases with the service innovation cost coefficient ${\mathsf{\mu }}_{\mathrm{s}}$, without considering service provider information sharing.

$$\begin{gathered} {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0, {\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0, {\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}}{\partial \mathsf{\beta }}}>0; \\ {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}}{\partial {\mathsf{\mu }}_{\mathrm{s}}}}<0; \end{gathered}$$

Corollaries 1 and 2 suggest that both manufacturers and service providers benefit from increased service and product innovation effects. Greater product and service innovation utility implies greater customer acceptance of the innovative content and model. When service providers introduce more flexible and personalized services to give new functions and usage scenarios to their products, this expands the scope of application of the products, brings new market opportunities for manufacturers, and encourages manufacturers to innovate their products to meet different customer needs. At the same time, product innovation by manufacturers usually introduces new technologies, which change the operation mode and service process of existing service providers, prompting service providers to adjust and innovate their existing services to adapt to new products. Both parties cooperate in technology development and the production process, and the innovation effects promote each other to form a spiral development mode. On the contrary, the increase in the innovation cost coefficient implies that manufacturers and service providers have insufficient innovation capability and cannot meet the market demand despite investing more in innovation costs. The uncertainty of innovation leads both parties to be more cautious when investing in innovation, or even to reduce their investment in innovation, which leads to a decrease in the level of innovation.

Proposition 3 

(DPI model). Considering the scenario of information sharing by service providers, product innovation is prioritized.

(i) The optimal levels of product innovation (${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$), service innovation (${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$), price per unit of service (${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$), and selling price (${\mathrm{p}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$) for both manufacturers and service providers are defined as follows:

$${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{(\mathrm{a}+\mathrm{b}({\mathrm{k}}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}))\mathsf{\lambda }{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(21)

$${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{(\mathrm{a}+\mathrm{b}({\mathrm{k}}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}))\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(22)

$${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{(\mathrm{a}+\mathrm{b}(3{\mathsf{\eta }}_{\mathrm{s}}+{\mathrm{k}}_{\mathrm{s}})){\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}(2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(23)

$${\mathrm{p}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(3\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2)+\mathrm{b}({\mathsf{\eta }}_{\mathrm{s}}-{\mathrm{k}}_{\mathrm{s}})(-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}+\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(24)

(ii) The optimal profits for manufacturers (${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$) and service providers (${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$) are defined as follows:

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{{(\mathrm{a}+\mathrm{b}({\mathrm{k}}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}))}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{8\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-4{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(25)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{{(\mathrm{a}+\mathrm{b}({\mathrm{k}}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}))}^2{\mathsf{\mu }}_{\mathrm{m}}^2{\mathsf{\mu }}_{\mathrm{s}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2)}{2{(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}^2}}$$

(26)

To ensure the existence and practical significance of the equilibrium solution, the following conditions must be satisfied: $2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2>0$, $4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}>0$, $0<{\mathsf{\eta }}_{\mathrm{s}}<{\displaystyle \frac{(-\mathrm{a}+\mathrm{b}{\mathrm{k}}_{\mathrm{s}}){\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}(2\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}+{\mathrm{k}}_{\mathrm{s}}({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}))}{\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2)}}$.

Proposition 4 

(SPI model). Considering the scenario of information sharing by service providers, service innovation is prioritized.

(i) The optimal levels of product innovation (${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$), service innovation (${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$), price per unit of service (${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$), and selling price (${\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$) for both manufacturers and service providers are defined as follows:

$${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{\mathsf{\lambda }(\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }))}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(27)

$${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{\mathrm{a}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }){\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\eta }}_{\mathrm{s}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(28)

$${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)(\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }))}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(29)

$${\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{{\mathrm{b}}^2{\mathrm{k}}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{m}}({\mathsf{\eta }}_{\mathrm{s}}-\mathrm{a}{\mathrm{k}}_{\mathrm{m}})+{\mathsf{\lambda }}^2({\mathsf{\eta }}_{\mathrm{s}}\mathsf{\beta }-\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}})-\mathrm{b}({\mathsf{\eta }}_{\mathrm{s}}{\mathrm{k}}_{\mathrm{m}}{\mathsf{\lambda }}^2+3{\mathsf{\eta }}_{\mathrm{s}}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}+\mathrm{a}{\mathrm{k}}_{\mathrm{m}}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}-3\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(30)

(ii) The optimal profits for manufacturers (${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$) and service providers (${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$) are defined as follows:

$${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{m}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2){(\mathrm{b}{\mathsf{\eta }}_{\mathrm{s}}{\mathrm{k}}_{\mathrm{m}}+{\mathsf{\eta }}_{\mathrm{s}}\mathsf{\beta }-\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}})}^2}{2{(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}^2}}$$

(31)

$${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{{\mathrm{a}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\eta }}_{\mathrm{s}}(\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })-{\mathsf{\eta }}_{\mathrm{s}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2))}{2(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$$

(32)

To ensure the existence and practical significance of the equilibrium solution, the following conditions must be satisfied: $2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2>0$, $4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}>0$, $0<{\mathsf{\eta }}_{\mathrm{s}}<{\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}{2(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}}$.

Corollary 3. 

(i) In the service provider information sharing scenario, the optimal decision ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$, ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$, ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$, ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ for the product innovation prioritization decision increases with the increase in the service provider information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$ and decreases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$.

$$\begin{gathered} {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{s}}}}>0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{s}}}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{s}}}}>0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{s}}}}>0; \\ {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0, {\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0 \end{gathered}$$

(ii) When ${\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}>0$, the optimal decision ${\mathrm{p}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ increases with the increase in the service provider information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$ and decreases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$. When ${\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}<0$, the optimal decision ${\mathrm{p}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ decreases with the increase in the service provider’s information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$ and increases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$.

(iii) The optimal decision ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ increases with the increase in the service provider’s information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$. When $3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}>0$, the optimal decision ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ increases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$. When $3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}<0$, the optimal decision ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ decreases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$.

Corollary 4. 

In the service provider information sharing scenario, the optimal decision to prioritize service innovation ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$, ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$, ${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$, ${\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$, ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$, ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}$ increases with the increase in the value transformation coefficient of the manufacturer’s information ${\mathrm{k}}_{\mathrm{m}}$ and decreases with the increase in the information sharing cost coefficient ${\mathsf{\eta }}_{\mathrm{s}}$.

$$\begin{gathered} {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0,{\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0,{\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0,{\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0,{\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0,{\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathrm{k}}_{\mathrm{m}}}}>0 \\ {\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0,{\displaystyle \frac{\partial {\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0,{\displaystyle \frac{\partial {\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0,{\displaystyle \frac{\partial {\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0,{\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0,{\displaystyle \frac{\partial {\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}}{\partial {\mathsf{\eta }}_{\mathrm{s}}}}<0 \end{gathered}$$

It follows from Corollaries 3 and 4 that, on the one hand, information sharing by service providers allows manufacturers to capture economic value beyond the benefits of the innovation itself. For example, by obtaining more innovation information based on product operation data and customer feedback data, the cost of innovation for the manufacturer can be reduced through information sharing. On the other hand, enterprises process and analyze product operation data, customer feedback, and market demand information to identify potential market opportunities and changes in consumer demand, improve R&D efficiency through optimal allocation of resources, and reduce the risk of innovation due to market uncertainty. In this case, manufacturers and service providers will tend to increase the level of innovation of their products and services to meet specific market demands. Customers are willing to pay higher prices to try and accept innovative products when the manufacturer’s and service provider’s innovations greatly satisfy their needs. In this case, both parties can set higher prices in order to make higher profits. On the contrary, higher information sharing costs mean that the service provider needs to invest more resources in information collection, processing, and delivery. The increase in information sharing costs means service providers have to undermine their investment in service innovation, leading to a decrease in the level of service innovation. The decrease in the level of service innovation directly affects the value of innovation information and indirectly increases the cost of product innovation for manufacturers, which leads to a decrease in the level of product innovation. In the case of a decline in the level of product and service innovation, service providers and manufacturers maintain market competitiveness and market share by lowering wholesale and selling prices.

Corollary 5 

(DPI Model vs. DP Model).

(i) There exists $\overline{{\mathrm{k}}_{\mathrm{s}1}}$. When $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}1}}$, if $0<{\mathsf{\eta }}_{\mathrm{s}}<{\mathrm{k}}_{\mathrm{s}}$, the level of product innovation, the level of service innovation, the selling price, the manufacturer’s profit, and the service provider’s profit under the product innovation priority strategy satisfy ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}>0$, ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}>0$, ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$, ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$; if ${\mathrm{k}}_{\mathrm{s}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{d}}}$, the level of product innovation, the level of service innovation, the selling price, the manufacturer’s profit, and the service provider’s profit satisfy $0<{\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}<{\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$, $0<{\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}<{\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$, ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}<{\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$, ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}<{\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$. When ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}1}}$, if $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{d}}}$, the level of product innovation, the level of service innovation, the selling price, the manufacturer’s profit, and the service provider’s profit satisfy ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}>0$, ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}>0$, ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{D}\mathrm{P}*}$, ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{D}\mathrm{P}*}$. The comparison between ${\mathrm{p}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$ and ${\mathrm{p}}^{\mathrm{D}\mathrm{P}*}$ has the same threshold, but the magnitude of both depends on ${\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}$.

Where  $\overline{{\mathrm{k}}_{\mathrm{s}1}}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}({\mathsf{\beta }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}})}{\mathrm{b}(2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+({\mathsf{\lambda }}^2-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}){\mathsf{\mu }}_{\mathrm{s}})}}$, $\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{d}}}={\displaystyle \frac{(-\mathrm{a}+\mathrm{b}{\mathrm{k}}_{\mathrm{s}}){\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}(2\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}+{\mathrm{k}}_{\mathrm{s}}({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}))}{\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2)}}$.

(ii) There exists  $\overline{{\mathrm{k}}_{\mathrm{s}2}}$. When  $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}2}}$, if  $0<{\mathsf{\eta }}_{\mathrm{s}}<{\displaystyle \frac{\mathrm{b}{\mathrm{k}}_{\mathrm{s}}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+({\mathsf{\lambda }}^2-3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}){\mathsf{\mu }}_{\mathrm{s}}}}$, the price of the service satisfies  ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}$; if  ${\displaystyle \frac{\mathrm{b}{\mathrm{k}}_{\mathrm{s}}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+({\mathsf{\lambda }}^2-3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}){\mathsf{\mu }}_{\mathrm{s}}}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{d}}}$, the price of the service satisfies  ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}<{\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}$. When  ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}2}}$, if  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{d}}}$, the price of the service satisfies  ${\mathsf{\omega }}^{\mathrm{D}\mathrm{P}\mathrm{I}*}>{\mathsf{\omega }}^{\mathrm{D}\mathrm{P}*}$, where  $\overline{{\mathrm{k}}_{\mathrm{s}2}}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}({\mathsf{\beta }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}})(2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-3\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}{\mathrm{b}(\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$.

Corollary 5 shows that the choice of information sharing strategy under the priority of product innovation is closely related to the information value conversion coefficient and information sharing cost coefficient of the service provider. Only when the service provider’s information value conversion coefficient and information sharing cost coefficient satisfy certain conditions will the information sharing strategy increase the innovation level of both parties as well as bring excess profits. The market demand information acquired by the service provider is not always directly transformed into product innovation knowledge. Only if the service provider transforms it into knowledge that is highly compatible with the manufacturer’s product innovation needs will information sharing be of sufficient economic value to the manufacturer. In addition, service providers incurring additional information sharing costs will demand higher service prices, which will directly affect manufacturers’ product pricing and profit allocation. In conjunction with Corollary 3, the information sharing strategy benefits both parties more when the service provider’s ability to transform the value of information is higher and the cost of information sharing is lower.

Corollary 6 

(SPI model vs. SP model). 

(i) There exists  $\overline{{\mathsf{\eta }}_{\mathrm{s}1}}$. When  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}1}}$, the level of product innovation, the price per unit of service, and the manufacturer’s profit under the service innovation priority decision satisfy  ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$,  ${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}$,  ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$. When  $\overline{{\mathsf{\eta }}_{\mathrm{s}1}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{s}}}$, the level of product innovation, the price per unit of service, and the manufacturer’s profit satisfy  ${\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{e}}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$,  ${\mathsf{\omega }}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathsf{\omega }}^{\mathrm{S}\mathrm{P}*}$,  ${\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathsf{\pi }}_{\mathrm{m}}^{\mathrm{S}\mathrm{P}*}$.

(ii) There exists  $\overline{{\mathsf{\eta }}_{\mathrm{s}2}}$. When  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}2}}$, the level of service innovation satisfies  ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$. When  $\overline{{\mathsf{\eta }}_{\mathrm{s}2}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{s}}}$, the level of service innovation satisfies  ${\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{e}}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$.

(iii) There exists  $\overline{{\mathsf{\eta }}_{\mathrm{s}3}}$. When  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}3}}$, the sales price satisfies  ${\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{p}}^{\mathrm{S}\mathrm{P}*}$. When  $\overline{{\mathsf{\eta }}_{\mathrm{s}3}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{s}}}$, the sales price satisfies  ${\mathrm{p}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{p}}^{\mathrm{S}\mathrm{P}*}$.

(iv) There exists  $\overline{{\mathsf{\eta }}_{\mathrm{s}4}}$. When  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}4}}$, service provider profits satisfy  ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$. When  $\overline{{\mathsf{\eta }}_{\mathrm{s}4}}<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{s}}}$, service provider profits satisfy  ${\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathsf{\pi }}_{\mathrm{s}}^{\mathrm{S}\mathrm{P}*}$.

Where  $\overline{{\mathsf{\eta }}_{\mathrm{s}\mathrm{s}}}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}{2(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2)}}$,  $\overline{{\mathsf{\eta }}_{\mathrm{s}1}}={\displaystyle \frac{\mathrm{a}\mathrm{b}{\mathrm{k}}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+2\mathsf{\beta }){\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })(-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}+4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}}$;  $\overline{{\mathsf{\eta }}_{\mathrm{s}2}}={\displaystyle \frac{\mathrm{a}\mathrm{b}{\mathrm{k}}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{m}}(-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}+{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}\mathsf{\beta }+{\mathsf{\beta }}^2+4\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}))}{2({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}})({\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}}$;  $\overline{{\mathsf{\eta }}_{\mathrm{s}3}}={\displaystyle \frac{\mathrm{a}\mathrm{b}{\mathrm{k}}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{m}}({\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })+\mathrm{b}{\mathsf{\mu }}_{\mathrm{s}}(2\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}+{\mathrm{k}}_{\mathrm{m}}({\mathsf{\lambda }}^2-\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}})))}{({\mathsf{\lambda }}^2\mathsf{\beta }+{\mathrm{b}}^2{\mathrm{k}}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{m}}-\mathrm{b}({\mathrm{k}}_{\mathrm{m}}{\mathsf{\lambda }}^2+3\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}))({\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}}$;  $\overline{{\mathsf{\eta }}_{\mathrm{s}4}}={\displaystyle \frac{\mathrm{a}{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })(-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}+4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})-\sqrt{{\mathrm{a}}^2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}^2({\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})({(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}+2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}}{2({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}})({\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}})}}$.

Corollary 6 shows that the choice of information sharing strategy under service innovation prioritization is not only closely related to the information sharing cost coefficient of the service provider, but also significantly associated with the information value conversion coefficient of the manufacturer. If the manufacturer can effectively use the service innovation information and demand information shared by the service provider to accurately adjust the product design, improve the production process, and quickly launch innovative products in response to the market demand, this efficient information conversion can significantly increase the added value of service innovation to product innovation. Innovative products and services occupy a larger market share, which, in turn, will lead to the growth of service providers’ profits and make up for the economic losses caused by service providers in the process of information sharing. However, the value of information sharing cannot be maximized if the manufacturer’s ability to convert the value of information is weak, even if the service provider provides detailed information on market demand. In this case, there is limited improvement in the service provider’s level of innovation and profitability, and the high cost of information sharing may even cause the service provider’s profit to be lower than it would have been without information sharing.

The previous section analyzed the correlation between the optimal decision and each parameter as well as the impact of the service provider’s information sharing strategy on the optimal decision and profit by solving different models. Comprehensively comparing the four models constructed, it is found that the key factors affecting the product–service innovation supply chain are the information value conversion coefficient of the manufacturer and the service provider and the information sharing cost coefficient of the service provider. Next, we further compare the market demand under different innovation priorities and analyze the impact of these parameters on the choice of supply chain innovation priorities.

Proposition 7. 

(i) Market demand under product innovation prioritization and service innovation prioritization decisions in the scenario of no information sharing by the service provider:

$${\mathrm{D}}^{\mathrm{D}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(33)

$${\mathrm{D}}^{\mathrm{S}\mathrm{P}*}={\displaystyle \frac{\mathrm{a}\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(34)

(ii) Market demand under product innovation prioritization and service innovation prioritization decisions in the scenario of information sharing by the service provider:

$${\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{\mathrm{b}(\mathrm{a}+\mathrm{b}({\mathrm{k}}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}})){\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(35)

$${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}={\displaystyle \frac{\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}(\mathrm{a}{\mathsf{\mu }}_{\mathrm{s}}-{\mathsf{\eta }}_{\mathrm{s}}(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }))}{4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}}}$$

(36)

Corollary 7 

(SP Model vs. DP Model). Introduces two parameters, product innovation efficiency and service innovation efficiency, ${\phi }_{\mathrm{m}}$, ${\phi }_{\mathrm{s}}$. When the party with high innovation efficiency carries out innovation activities as a sub-priority, higher market demand can be obtained.

A party with high innovation efficiency usually has better resource optimization and cost control capabilities, and can further reduce innovation costs on the basis of previous innovations, which helps it to be more competitive in the market. At the same time, by drawing on the technological foundation and market experience of the prior innovation process, it avoids innovation risks, makes use of existing market data and technological advances to carry out more optimized innovation, integrates the latest technology with market demand, and provides more mature and efficient products or services. This precise alignment enables them to quickly meet demand and gain market share when entering the market.

Corollary 8 

(SPI Model vs. DPI Model). 

Case 1: If  ${\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-\sqrt{{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}(2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}<0$, then  $\overline{{\mathrm{k}}_{\mathrm{m}1}}<\overline{{\mathrm{k}}_{\mathrm{m}2}}$.

There exist  $\overline{{\mathrm{k}}_{\mathrm{m}1}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}2}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}3}}$. When  $0<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}1}}$, if  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand under the product innovation priority and service innovation model satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$; if  ${\mathsf{\eta }}_{\mathrm{s}}>\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. When  $\overline{{\mathrm{k}}_{\mathrm{m}1}}<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}2}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. When  $\overline{{\mathrm{k}}_{\mathrm{m}2}}<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}3}}$, if  $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}3}}$,  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$; if  $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}3}}$,  ${\mathsf{\eta }}_{\mathrm{s}}>\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$; if  ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}3}}$, market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$.

Case 2: If  ${\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-\sqrt{{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}(2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}>0$, then  $\overline{{\mathrm{k}}_{\mathrm{m}1}}>\overline{{\mathrm{k}}_{\mathrm{m}2}}$.

There exist  $\overline{{\mathrm{k}}_{\mathrm{m}1}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}2}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}3}}$. When  $0<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}2}}$, if  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand under the product innovation priority and service innovation model satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$; if  ${\mathsf{\eta }}_{\mathrm{s}}>\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. When  $\overline{{\mathrm{k}}_{\mathrm{m}2}}<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}1}}$, if  $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}3}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. If  ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}3}}$,  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$; if  ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}3}}$,  ${\mathsf{\eta }}_{\mathrm{s}}>\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. When  $\overline{{\mathrm{k}}_{\mathrm{m}1}}<{\mathrm{k}}_{\mathrm{m}}<\overline{{\mathrm{k}}_{\mathrm{m}3}}$, if  $0<{\mathrm{k}}_{\mathrm{s}}<\overline{{\mathrm{k}}_{\mathrm{s}3}}$,  $0<{\mathsf{\eta }}_{\mathrm{s}}<\overline{{\mathsf{\eta }}_{\mathrm{s}5}}$, market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}>{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$. If  ${\mathrm{k}}_{\mathrm{s}}>\overline{{\mathrm{k}}_{\mathrm{s}3}}$, the market demand satisfies  ${\mathrm{D}}^{\mathrm{S}\mathrm{P}\mathrm{I}*}<{\mathrm{D}}^{\mathrm{D}\mathrm{P}\mathrm{I}*}$.

Where  $\overline{{\mathrm{k}}_{\mathrm{s}3}}={\displaystyle \frac{\mathrm{a}({\mathrm{b}}^2{\mathrm{k}}_{\mathrm{m}}^2{\mathsf{\mu }}_{\mathrm{m}}+2\mathrm{b}{\mathrm{k}}_{\mathrm{m}}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}{\mathrm{b}(4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}-{(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}-2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})}}$,  $\overline{{\mathsf{\eta }}_{\mathrm{s}5}}=-{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{s}}(\mathrm{a}({\mathrm{b}}^2{\mathrm{k}}_{\mathrm{m}}^2{\mathsf{\mu }}_{\mathrm{m}}+2\mathrm{b}{\mathrm{k}}_{\mathrm{m}}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}+{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}})+\mathrm{b}{\mathrm{k}}_{\mathrm{s}}({(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })}^2{\mathsf{\mu }}_{\mathrm{m}}+2({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}){\mathsf{\mu }}_{\mathrm{s}}))}{2{\mathsf{\beta }}^2(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }){\mathsf{\mu }}_{\mathrm{m}}-(\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta })(-{\mathsf{\lambda }}^2+\mathrm{b}(4+\mathrm{b}{\mathrm{k}}_{\mathrm{m}}+\mathsf{\beta }){\mathsf{\mu }}_{\mathrm{m}}){\mathsf{\mu }}_{\mathrm{s}}+2\mathrm{b}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2){\mathsf{\mu }}_{\mathrm{s}}^2}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}1}}={\displaystyle \frac{{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{s}}+2{\mathsf{\mu }}_{\mathrm{s}}({\mathsf{\beta }}^2-\mathrm{b}(2+\mathsf{\beta }){\mathsf{\mu }}_{\mathrm{s}})+\sqrt{4{\mathsf{\beta }}^4{\mathsf{\mu }}_{\mathrm{m}}^4+4{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}({\mathsf{\lambda }}^2-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}})+{\mathsf{\mu }}_{\mathrm{s}}^2({({\mathsf{\lambda }}^2-4\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}})}^2-8{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}({\mathsf{\lambda }}^2-2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}))}}{2{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}2}}={\displaystyle \frac{-\mathrm{b}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}+\sqrt{2{\mathrm{b}}^2{\mathsf{\beta }}^2{\mathsf{\mu }}_{\mathrm{m}}^2-{\mathrm{b}}^2{\mathsf{\lambda }}^2{\mathsf{\mu }}_{\mathrm{m}}{\mathsf{\mu }}_{\mathrm{s}}}}{{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}}}$,  $\overline{{\mathrm{k}}_{\mathrm{m}3}}={\displaystyle \frac{-\mathrm{b}\mathsf{\beta }{\mathsf{\mu }}_{\mathrm{m}}+\sqrt{2}\sqrt{{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}(2\mathrm{b}{\mathsf{\mu }}_{\mathrm{m}}-{\mathsf{\lambda }}^2){\mathsf{\mu }}_{\mathrm{s}}}}{{\mathrm{b}}^2{\mathsf{\mu }}_{\mathrm{m}}}}$.

Corollary 8 shows that, in the case of service provider information sharing, the choice of innovation priority for integrated solutions is closely related to the information value conversion coefficients of the manufacturer and the service provider. When the cost coefficient of data resource sharing and the information conversion coefficients of both parties satisfy a certain magnitude relationship, it is possible to decide the priority of product innovation according to the market demand. This means that the delivery and use of integrated solutions are not only expressed in the delivery of physical products and value-added services, but also include regular services, optimization services, and upgrading services in the process of product use and operation. The value of data resources such as customer needs and services for product innovation provides ‘external incentives’ for manufacturers and service providers to innovate.

4. Numerical Simulation

4.1. Numerical Calculation of the Model

Building upon the theoretical analysis, we establish the model parameters and compute the decision-making outcomes for various models. Taking into account the constraints necessary for the existence of equilibrium solutions, and drawing from the existing literature [11,47,48,49], the parameters are set as follows: $\mathrm{a}=5$, $\mathrm{b}=1.2$, $\mathsf{\lambda }=0.5$, $\mathsf{\beta }=0.6$, ${\mathrm{k}}_{\mathrm{m}}=0.7$, ${\mathrm{k}}_{\mathrm{s}}=1$, ${\mathsf{\mu }}_{\mathrm{m}}=0.8$, ${\mathsf{\mu }}_{\mathrm{s}}=1.5$, ${\mathsf{\eta }}_{\mathrm{s}}=0.8$. By substituting the values of each parameter into the analytical solutions of the decision variables across the four innovation strategies, the results are presented in

Table 2. Decision-making results under different decision-making models.

Decision Models	${\mathbf{e}}_{\mathbf{m}}$	${\mathbf{e}}_{\mathbf{s}}$	$\mathsf{\omega }$	$\mathbf{p}$	$\mathbf{D}$	${\mathbf{\pi }}_{\mathbf{m}}$	${\mathbf{\pi }}_{\mathbf{s}}$
DP	0.7798	0.4991	1.2477	3.4934	1.4972	3.1192	1.6812
SP	0.7942	0.5083	2.2104	3.4810	1.5248	1.6852	3.1766
DPI	0.8172	0.5230	2.1075	3.4611	1.5691	3.4258	1.8465
SPI	0.9471	0.9215	2.6362	3.5066	1.8185	2.3970	3.4200

.

As can be seen from Table 2, when service providers do not share information, the innovation level and market demand of both parties do not change significantly under different innovation priorities, and the party that prioritizes innovation is able to obtain higher profits. When service providers share information, the innovation level, market demand, and profits of both manufacturers and service providers increase. This also reflects that the service provider’s data mining and information sharing strategy is conducive to improving the innovation performance of the supply chain.

4.2. Impact Analysis of Information Sharing Cost Coefficients

This section analyzes the impact of variations in information sharing costs on the levels of innovation, unit service pricing, sales prices, and profits of both parties, as illustrated in Figure 2. To satisfy the constraints derived from Propositions 3 and 4, the range of values for the information sharing cost coefficients is set to ${\mathsf{\eta }}_{\mathrm{s}}\in \left(0, 1.6\right]$. The remaining parameters are maintained using the values specified in Section 4.1 of this paper.

It can be seen that the optimal level of innovation and profits for both manufacturers and service providers decreases with increasing information sharing cost coefficients, irrespective of the innovation priority adopted. When service innovation is prioritized, the unit service price and selling price decrease with an increasing information sharing cost coefficient. When product innovation is prioritized, the price per unit of service and selling price increase with the information sharing cost coefficient. Combined with the analyses of Corollaries 3 and 7, the changes in unit service prices and selling prices with the information sharing cost coefficient depend on the innovation efficiency of products and services. Low innovation efficiency implies that the competitiveness enhancement of products and services brought about by innovation has relatively little impact on market demand, and firms are unable to strive for higher premiums through differentiated innovation. It also reflects the passive position of firms in the market, with their pricing strategies relying more on cost orientation. In the face of rising costs, firms often choose to shift cost pressures by raising prices in order to maintain their profitability.

4.3. Impact Analysis of the Information Value Transformation Coefficient

The impact of the information value transformation coefficients of both the manufacturer and the service provider on the levels of innovation, unit service pricing, selling price, and profits for both parties is analyzed in Figure 3. In this analysis, based on the constraints derived from Propositions 3 and 4, the ranges for the information value transformation coefficients of the manufacturer and the service provider are set to ${\mathrm{k}}_{\mathrm{m}}\in \left(0 , 0.9\right]$ and ${\mathrm{k}}_{\mathrm{s}}\in \left(0 , 2\right]$, respectively. The remaining parameters continue to be selected according to the values specified in Section 4.1 of this paper.

It can be seen that the optimal level of innovation and profits for both manufacturers and service providers increase with the value of the information transformation coefficient, irrespective of the innovation priority adopted. When service innovation is prioritized, the selling price increases with the information value transformation coefficient. However, when product innovation is prioritized, the selling price decreases with the increase in the information value transformation coefficient. The main reason for this phenomenon is the different ways in which manufacturers obtain the value of information under the two innovation priorities. Under the product innovation priority, the manufacturer directly captures the economic value of the information shared by the service provider, and the value of the information increases with the increase in the service provider’s information value transformation coefficient, which enlarges the manufacturer’s profit margin. With low innovation efficiency, manufacturers attempt to capture a larger market share by lowering the selling price. In contrast, the value of information acquired by the manufacturer under the service innovation priority is reflected in the creation of differentiated products to meet market demand based on service innovation, and this differentiation allows for greater freedom in pricing, enabling the manufacturer to earn higher profits by raising prices.

4.4. Impact Analysis of Information Sharing and Non-Sharing

This section compares the levels of innovation, unit service pricing, sales prices, and profits for both parties when service providers adopt information sharing and non-sharing strategies under the same innovation priority. Furthermore, it identifies the relationship governed by the service provider’s information sharing cost coefficient and the conversion coefficients of both the manufacturer and service provider’s information value when the optimal decisions differ between the two strategies.

As illustrated in Figure 4, under the product innovation priority strategy, when the service provider’s information value transformation coefficient exceeds the information sharing cost coefficient, the levels of innovation and profit for both parties under the service provider’s information sharing strategy surpass those when they do not share. Under the service innovation priority strategy, when the service provider’s information sharing cost coefficient and the manufacturer’s information value transformation coefficient fulfill a specific relationship, the innovation decisions of both parties under the service provider’s information sharing strategy exceed those made when not sharing.

4.5. Impact Analysis of Innovation Priorities

In order to show more intuitively the difference between market demand and profit when the information value conversion coefficients of the two parties are at different levels under different innovation priorities, we make a difference between the market demand and the profit values of the two parties under the priority of product innovation and the priority of service innovation, and compare the size of the market demand and profit under the two scenarios by the positive and negative values of the difference.

Figure 5a shows how the difference in market demand under the product innovation priority decision and the service innovation priority decision varies with the information value transformation coefficient ${\mathrm{k}}_{\mathrm{m}}$, ${\mathrm{k}}_{\mathrm{s}}$. From Figure 5a, it can be seen that there exists a critical range for the relationship between the size of the information value transformation coefficient ${\mathrm{k}}_{\mathrm{m}}$ and the information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$. When ${\mathrm{k}}_{\mathrm{s}}$ is greater than this critical range, the market demand under the service innovation priority is smaller than that under the product innovation priority, and the opposite is true if it is less than this critical value. At the same time, it can also be found from the scope covered by the difference of market demand greater than zero that the enhanced information value transformation capability of the manufacturer under the service innovation priority strategy is more likely to increase the market demand. For this reason, measuring the information value transformation capability of manufacturers is a key indicator when deciding on innovation prioritization based on market demand.

Figure 5b shows how the difference between the manufacturer’s and the service provider’s profits under different innovation prioritization decisions varies with the information value transformation coefficient ${\mathrm{k}}_{\mathrm{m}}$ and the information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$. From Figure 5b, it can be seen that, when the manufacturer’s information value transformation coefficient ${\mathrm{k}}_{\mathrm{m}}$ and the service provider’s information value transformation coefficient ${\mathrm{k}}_{\mathrm{s}}$ satisfy a certain size relationship, either product innovation priority or service innovation priority leads to an increase in the profits of both parties compared to the other strategy. It can also be found that, when both information transformation coefficients are low, the service provider’s profit under the service innovation priority strategy is higher than that of the product innovation priority strategy, but the product innovation priority leads to a higher profit for the manufacturer. This suggests that the profits of manufacturers and service providers under the same innovation prioritization do not always ‘advance or retreat together’, and there are cases where one of them loses profits. In this case, the development of an appropriate profit-sharing mechanism between the manufacturer and the service provider plays an important role in fostering collaborative innovation between the two parties.

5. Conclusions and Insights

In the context of the digital economy, data, recognized as a new type of production factor, have become a key resource for enterprises aiming to achieve economic benefits and sustainable competitive advantages. For manufacturers and service providers collaborating to deliver integrated solutions comprising ‘products + digital services’, data mining and analysis empower them to swiftly identify market trends and customer demands, thereby facilitating innovative decision-making. Within the framework of collaborative innovation between manufacturers and service providers, it is essential to examine the impact of service providers’ data mining and information sharing strategies on the innovation decisions and priorities of both parties. Framing this as the research problem, this paper constructs and analyzes the optimal decision-making results of the DP, SP, DPI, and SPI models, examines the influence of various factors on decision-making, and subsequently explores the issue of innovation priority selection in the product–service innovation supply chain. The main findings are summarized below. Firstly, service providers’ data resource mining and information sharing strategies are not always favorable to both parties’ innovative decisions. It is certain that the optimal innovation decisions of manufacturers and service providers increase with the level of information value transformation of both parties and decrease with the increase in the information sharing cost coefficient. Therefore, only when data resources can be transformed into real innovation value at a reasonable cost can data mining and sharing really play the role of ‘external incentives’ to promote collaborative innovation between the two parties. Secondly, product and service innovation efficiency plays a decisive role in innovation prioritization choices when information is not shared, and the party with high innovation efficiency adopts sub-priority innovations, which can lead to a larger market share for the innovative integrated solution. Under the information sharing strategy, product and service innovation efficiency affects the pricing decision under the product innovation prioritization strategy. This dominates whether the supply chain gains higher profits by increasing pricing or captures a greater market share by lowering prices. Finally, the choice of innovation priority under the information sharing strategy depends on the ability of manufacturers and service providers to transform information value. When selecting innovation priorities based on market demand, service innovation is prioritized when the manufacturer’s ability to transform information value is high. Product innovation is prioritized when the service provider’s information value transformation capability is high. When selecting innovation priorities based on the profits of both parties, it is possible that one party’s profits may be compromised.

Based on the research conclusions, the following managerial recommendations are derived: Firstly, enhance data resource sharing and optimize sharing costs. Data resource sharing can improve levels of service and product innovation. Manufacturers should actively establish cooperative relationships with service providers for data mining and information sharing, creating efficient mechanisms for information exchange. Service providers should optimize existing sharing processes through technological improvements and cost reductions, thereby enhancing the overall benefits of information sharing. Secondly, emphasize value transformation of information to improve innovation decision quality. Manufacturers and service providers should introduce advanced data analysis tools and techniques to strengthen their data analysis and information management capabilities, enhancing the value derived from transforming data resources into actionable information. Managers should accurately assess their capabilities to set reasonable innovation priority decisions and promote trust and support among collaborative innovation partners through transparent decision-making processes. Thirdly, establish effective incentive mechanisms to promote supply chain coordination. To address costs arising from information sharing and collaborative innovation, consider the economic capabilities and contributions of all parties. By designing a reasonable cost-sharing mechanism, the financial burden on supply chain members can be alleviated, thereby increasing their willingness to share data resources and engage in collaborative innovation.

Current research focuses on the universal issue of the impact of data resource sharing on innovation prioritization, but different industries and different data mining approaches may have unique characteristics that affect innovation prioritization and data sharing capabilities. Meanwhile, the risk aversion of supply chain members, the long-term dynamic impact of data resource sharing, and competitor pricing decisions can be further discussed in the future.