The Development Strategy of Dual-Channel Supply Chain of Smart Elderly Care Service from the Perspective of Time Perception

,


Introduction
China's 14th Five-Year Plan will upgrade the development of smart elderly care industry as a national strategy.Te 20th National Congress of the Communist Party of China in October 2022 proposed to "Implement the national strategy of actively responding to the aging of the population, develop old-age care and elderly care service industry, optimize the services for the elderly who are alone and widowed, and promote the realization of basic elderly services for all the elderly."Today, the aging of the population has become the most distinctive trend of globalization in the current era.Te United Nation's 《World Society Report 2023》 shows that the proportion of the global population over the age of 65 will reach 761 million in 2021.China's aging is characterized by large scale, deep degree, and fast speed.It is expected that by 2035, the elderly population over 60 years old will exceed 400 million and will be entering the severe aging stage.China will also become the country with the most serious aging population in the world.So the aging of population is an important factor afecting China's economic development, and actively responding to the aging of population is an inevitable requirement for the realization of Chinese modernization [1,2].
In order to conform to the development trend of the silver economy era, the smart elderly industry has taken shape under the promotion of modern information technology such as AI.According to the statistics of 《2022-2028 China Smart Pension Industry Development Trend and Investment Decision Proposal Report in 2020》: Te scale of China's smart elderly service industry reached 3.75 trillion RMB in 2020 and is expected to expand to 17.5 trillion RMB in 2040.However, from the current development situation, the development of smart elderly industry in China is still in the primary stage, and does not perfectly ft elderly distribution characteristics as 90% of old people choose to be at home, 7% old people choose community, and 3% old people choose institutional, that means the smart elderly industry development and "9073 endowment service pattern" are deviation, make more than 70% of the elderly in 90% of the home elderly population cannot be efectively met, while elderly care institutions and communities provide services with a high rate of empty beds and serious waste of resources.Investigate its root cause: supply and demand do not match; elderly demand will change with time migration.Afected by time perception, when the elderly to life time perception is not obvious, behavior motivation by capture information to increase knowledge growth, namely, to meet the demand of the future-orientation (Fo-stage), and as the elderly's perception time becomes increasingly strong, old people will pay more attention to emotion related to meet the present-orientation demand (Po-stage) [3].But the current operation mechanism of China's smart elderly care service supply chain does not take into account the migration of elderly demands under time perception, which leads to the imbalance of supply and demand structure.Te specifc manifestations are the construction of elderly care service platforms and elderly care service institutions has weak basic functions, narrow coverage, and single intelligent service content, which do not fully play the "wisdom" effciency [4]; the physical heterogeneity of the elderly is enhanced; and the characteristics of personalized, diferentiated, and sporadic needs are revealed.However, the elderly care services are still mainly to meet the basic life care needs, but the spiritual needs such as humanistic care are not valued [5].Limited by multiple factors such as service scope, technology application, ability, and level, elderly service providers are difcult to provide accurate and efcient elderly care services to the elderly [6].
Based on this, it is the focus of the paper to solve the fragmentation of supply resources, give play to the diferentiated service advantages of the platform and the elderly service center, and coordinate the main relationship of the elderly service supply chain to meet the diversifed needs considering users' time perception of the elderly.Around the above problems, the paper considers the user time perception factors, and from the perspective of meeting the future oriented elderly needs and the current oriented elderly needs, to build the smart elderly service supply chain including elderly service providers and elderly service integrator, to break the fat efciency limit of online service platform and ofine elderly care service center as the means and achieve supply chain coordination as the goal, design smart elderly service dual-channel supply chain include online and ofine service channel, so as to meet the demand of elderly, promote smart elderly industry supply and demand matching, elderly service high quality development have important signifcance.
Te rest of the paper is organized as follows.Section 2 reviews and summarizes the literature related to the smart elderly dual-channel supply chain and user time perception, then puts forward the research questions and summarizes the innovation points according to the research gap.Section 3 provides the problem description and the parameter symbol description.Section 4 designs the smart elderly dualchannel supply chain coordination optimization model under no-contract and beneft-sharing contract.Section 5 analyzes the infuence of each parameter on the coordination efect of supply chain through example analysis.Section 6 is management insights.Section 7 contains conclusions, research limitations, and future prospects.

Literature Review
2.1.Smart Elderly Care Service Supply Chain.Te supply chain of elderly care services has always been a hot topic of academic research.For this research, many scholars have studied the coordination and optimization of supply chains including elderly service providers and elderly service integrators from multiple perspectives.For example, Chen is based on the perspective of information incentive mechanism [7], Ma et al. [8] and Zhao [9] are based on the perspective of contract mechanisms such as revenue sharing and punishment contracts, and Zhang et al. [10] are based on the perspective of government incentives.Among them, service quality [9,11], medical services [12], and nursing staf scheduling [13] are the focus of research on the supply chain of elderly care services.
In the 50s of the twentieth century, the British Trust Fund frst put forward the concept of smart elderly care, and a new type of elderly care model began to enter the public eye.Scholars such as Wang et al. and Song et al. have proposed that the information-based comprehensive service platform for elderly care has achieved breakthrough innovation on the supply side of elderly care services, providing conditions for grasping the demand for elderly care [14,15].However, at present, there are relatively few studies on the use of intelligence advantages to study the smart elderly care service supply chain, and only Chen designed the operation mechanism of the smart elderly care service supply chain with the service platform as the core [16].Furthermore, from the perspective of global aging, it is of great signifcance to study the supply chain of smart elderly care services to alleviate social pressure and promote the development of the silver economy.

Dual-Channel Supply
Chain.Te research of distribution channels has always been a very important concept, and the study of dual-channel supply chains has matured [17].In recent years, scholars such as Jaśkowski et al. [18], He et al. [19], and Ha et al. [20] have focused on the two directions of supply chain channel confict and coordination strategy.How to efectively deal with channel conficts caused by the loss of self-interest and other reasons, and improve the eforts of supply chain members to achieve the overall coordination of direct sales and retail channels through the design of contract coordination mechanisms has become an important area of concern [21,22].With the rise of ecommerce platforms, such as Zhang et al. [23], Barman 2 Complexity et al. [24], and Kazancoglu et al. [25], they have promoted the online agency and resale model and the online and ofine dual-channel model to become the research hotspots.Te Hotelling and Stackelberg game methods have gradually become the main means to achieve dual-channel supply chain coordination.Among them, Hotelling proposed that under the premise of considering the behavior of other competitors, the utility model should be constructed through the Hotelling game method, and the implementation of product diferentiation would be more conducive to maximizing profts [26].Also, Wang and Chen used Hotelling to build a consumer utility model to achieve dual-channel supply chain coordination [27].Stackelberg is the most important method to achieve dual-channel supply chain optimization as a sequential game [28,29].Tere are relatively few studies on the issue of dual-channel supply chain for elderly care services, and only one scholar, Zhao has constructed a dual-channel supply chain for elderly medical services, arguing that the community service model of O2O is more conducive to the efective coordination of the elderly care service supply chain [30].Terefore, there is still a lot of research space for using the Hotelling and Stackelberg game methods to solve the problem of elderly care service supply chain coordination.

Smart Elderly Care Service Supply Chain considering the
Needs of the Elderly.Regarding the research on the smart elderly care service supply chain considering the needs of the elderly, we found that only Cai et al. designed the servicesensitive demand index [31] and Zhang et al. analyzed the quality decision-making problem of the elderly care service supply chain from the perspective of the scale of elderly care demand [32].But nobody has yet considered the condition that the demand for elderly care will change over time when studying the supply chain of smart elderly care services.Carstensen et al. proposed that the behavioral motivation of the elderly will be afected by the user's time perception, and over time, the elderly will transition from the future-orientation needs that focus on information growth to the presentorientation needs that focus on emotional companionship [3].At present, user time perception is mostly used to study the emotional changes, judgment and attention, intertemporal choice, and other felds of the elderly [33][34][35].If the user's time perception factor can be added to the research scope of the dual-channel supply chain development strategy of smart elderly care services, it will make an important breakthrough for realizing the balance of supply and demand and supply chain coordination of the smart elderly care industry.
To sum up, there are three limitations on the current research about the dual-channel development strategy of smart elderly service supply chain: (1) Most of the existing literature focuses on the pricing or the improvement of service quality level of the elderly care service supply chain, and there is a lot of research space for how to give full play to the "smart" efciency of the elderly care service.(2) Te research on the selection of dual-channel development strategy of supply chain using Stackelberg or other game methods is mostly focused on the manufacturing feld.Nevertheless, research on the dual-channel supply chain coordination of the smart elderly care industry relying on the smart elderly service platform is worth more indepth exploration.(3) Although a few scholars have considered factors such as uncertain elderly demand or demand preference when studying the elderly service supply chain, no literature has studied the coordination and optimization of the dual-channel supply chain of smart elderly service on the premise of subdividing the elderly demand based on the user time perception factors of elderly behavior psychology.
In order to further explore the smart elderly service dualchannel supply chain development strategy, the paper starts from the unique perspective of user time perception, divided the elderly demand into the present-orientation demand and the future-orientation demand two patterns, for discussing how to choose the service channel to bring the elderly better service experience, so as to realize the smart elderly service supply chain efcient coordination.At length to alleviate the mismatch between supply and demand in the smart elderly industry from the root causes.Specifcally, the paper mainly focuses on the following three research issues: (1) How does the user time perception time afect the elderly needs?(2) Under the no-contract and beneft-sharing contract, what is the optimal decision and proft of the members of the smart elderly care service supply chain considering the elderly demand?(3) What is the optimal channel choice strategy for the whole supply chain and old people?
Te main contributions of the paper are as follows: (1) Combined with the theory of geriatric behavioral psychology, this paper analyzes the time perception of the elderly to identify the future-orientation and the present-orientation of the elderly care needs, and increases the parameters of the level of elderly needs.(2) Proposing the "online + ofine" dual-channel parallel mode relying on the smart elderly service care platform, to better elaborating the "smart" effciency.(3) Putting forward the dual-channel diferentiated service strategy, using the Hotelling-Stackelberg coordination of the smart elderly care service dual-channel supply chain, and optimizing the supply and demand structure of the smart elderly care industry.

Problem Description and Symbol Description
3.1.Problem Description.Tis paper constructs a dualchannel supply chain system for smart elderly care services, including elderly care service providers, elderly care service integrators (ofine service channels and online service channels) and elderly care users (shown as Figure 1).Among them, according to the user's time perception coefcient, the elderly service provider divides the level of efort to meet the needs of the elderly into N Fo for the information service efort level to meet the needs of the future-orientation and N Po for the emotional service efort level to meet the needs of the presentorientation, and provides personalized services to the elderly users through online service channels and ofine service channels, respectively.Elderly users in the Po-stage will spend the market service price p r to enjoy the elderly care service with the service level of s r through ofine service channels, and the elderly users in the Fo-stage will spend the market service price p e to enjoy the elderly care service with the service level of s e .

Complexity
Combined with the Hotelling model, this paper implements a diferentiated service strategy by expanding the supply channels of elderly care service integrators.Ofine elderly care center A and online elderly care service platform B are located at both ends of a linear market.Te elderly user O is distributed between AB, and the line OA is the elderly user who chooses the ofine service channel (x), and the OB represents the elderly user who chooses the online service channel (1-x).Among them, referring to the research design of Gu et al. [36], Wei and Lu [37], ofine service channels and online service channels will generate comprehensive costs such as transportation (t) and search (c), respectively, and the hidden loss cost of the smart elderly care market is divided into the loss cost of ofine channels based on transportation costs (t * x) and the loss cost of online channels based on search costs (c * (1-x)), as shown in Figure 2.
In response to the above problems, the paper puts forward the following hypotheses: (1) Elderly users obey an even distribution in the range of the pension market (0, 1).( 2) In order to meet the strategic conditions of service diferentiation, it is assumed that there are certain competitive behaviors between online service channels and ofine service channels.(3) In order to meet the laws of reality, set: 0 < c, t < p r , p e ; 0 < w < p r , p e .(4) Tis research considers the sensitivity of the price of one channel service to the price change of another channel service, and sets the cross-sensitivity coefcient of dual-channel price α and β of the elderly service integrator.Also, 0 < α < 1, 0 < β < 1. (5) Considering the heterogeneity of dual-channel services, this paper assumes that the service improvement cost impact coefcients of ofine service channels and online service channels are θ and φ, respectively, 0 < θ < 1 and 0 < φ < 1. Te cost of service improvement is 1/2s 2 r and 1/2s 2 e .Te research objectives of this paper are to use Hotelling to construct a utility function frstly, then with the online service channel as the leader and the ofine service channel as the follower to carry out the Stackelberg game, so as to analyze the impact of the service efort level of the elderly service provider to meet the present and future orientation needs and the service quality of elderly service integrator on the coordination of the smart elderly care service dualchannel supply chain under the no-contract model and the beneft-sharing contract and at last, obtain the optimal decision.1.

Coordination and Optimization Model of the
Dual-Channel Supply Chain of Smart Elderly Care Service

Hotelling Model and Proft Function.
According to the Hotelling model, the utility functions of ofine service channels and online service channels can be expressed in the following formulas: For elderly at the point of no diference, the total utility obtained by choosing online service channels and choosing ofine service channels is equal, which can be expressed in the following formulas: Combined with Figure 2 and formulas (3) and ( 4) you can see, the elderly who in the left range (0, x) belong to choose ofine service channels, and in the right (x, 1) range belong to choose online service channels, respectively, corresponding to ofine elderly care service demand Q r and online elderly care service demand Q e , can be expressed in the following formulas: Table 1: Model parameters and variables.

Parameters α Ofine service channel price cross-coefcient [38] β
Online service channel price cross-coefcient [38] θ Te infuence coefcient of the service improvement cost of the ofine service channel [39] φ Te infuence coefcient of the service improvement cost of the online service channel [39] μ Te coefcient of efort paid by elderly care service providers to content the current oriented needs of the elderly [40] (0 < μ < 1) λ Beneft sharing coefcient of elderly care service integrators [8]

Variables w
Te price of basic service from elderly care service providers to elderly care service integrators x No diference point [36,37] t Te sum of transportation costs such as unit service costs to choose ofine service channels [36,37] N Po Te level of service eforts paid by elderly care service providers to content the present-orientation needs of the elderly [40] N Fo Te level of service eforts paid by elderly care service providers to content the future-orientation needs of the elderly [ Service level of online service channels

Complexity
Because the market share is set to 1, the proft function of the elderly care service providers are expressed as: M s � w − c s .Te ofine service channels proft functions M r and online service channels proft functions M e can be expressed in the following formulas:

No-Contract Decision Model.
In the no-contract decision model, the Stackelberg decision order dominated by the online service channels of the elderly care service integrator is: First, the online service channels determine their service price p e and service level s e according to the elderly market; the ofine service channels determine their service price p r and service level s r according to the online service channels.Te online and ofine channels of elderly care service integrator meet the master and slave game relationship.In order to simplify the calculation, the service cost is only expressed as the basic elderly care service price w paid to elderly care service providers.Te no-contract decision model is represented by superscript "d," and the optimal solution is represented by superscript " * ." In the no-contract decision model, we count the frst partial derivatives of p r , s r , p e , and s e as formulas (7) and (8) according to the backward induction method.Combined with (zM r /zp r ) � 0, (zM r /zs r ) � 0, (zM e /zp e ) � 0, (zM e /zs e ) � 0, the optimal service price and the optimal service level of the online service channels and the ofine service channels of elderly care service integrator are obtained, as shown in the following formulas: In the no-contract decision model, the following conclusions are drawn: Theorem 1. Tere must be p d * r and s d * r to maximize the proft of ofine service channels.
Prove: Using formula (8) to count the frst partial derivative of p r and s r : (zM r /zp r ) and (zM r /zs r ), obtain the following formulas: Calculate the second partial derivatives of p r and s r , we obtain 6 Complexity According to formula (15), if 2(α + β) > 0 and (c + t) > 0, z 2 M r /zp r 2 < 0 is established.According to formula (16), − 1 < 0 and z 2 M r /zp r 2 < 0 is established.On this basis, make formulas ( 13) and ( 14) equal to 0, can obtain Solve the optimal ofine service price p d * r and the optimal ofine service level s d * r of the ofine service channels, as shown in the following formulas.
Terefore, the optimal ofine service price p d * r and the optimal ofine service level s d * r can maximize the ofine optimal proft M d * r , Teorem 1 proof complete.
Theorem .Tere may be p d * e and s d * e to maximize the proft of online service channels M d * e .Te restriction was: Prove: Substitute the optimal service price p d * r and the optimal service level s d * r of ofine service channels into formula (8), and get the reaction function M e (p d * r , s d * r ) d of the proft function of the online service channels of the elderly service integrator, as shown in the following formula: Calculate the second partial derivative of p e and s e , we can obtain According to formula (25), because 2(α Based on this, it can be determined that optimal service price of Online service channels has an equilibrium solution in 2αc + 2βc + 2αt + 2βt − θ 2 > 0, at this time, exist p d * e and s d * e , can make M d * e maximize.Teorem 2 has been proven.Finally, the optimal proft of ofine service channels M d * r and the optimal proft of online service channel M d * e of elderly care service integrator under no-contract decision are obtained as shown in the following formulas: Because the diversity and quality services provided by elderly care service providers are closely related to the satisfaction of geriatric, moreover the satisfaction of geriatric has a direct impact on the service price and service level.Based on this, under the no-contract decision model, we further analyze the relationship between the optimal service price (p d * r and p d * e ), the optimal service level (s d * r and s d * e ) and the present-orientation and Future-orientation service efort level (N Po and N Fo ), get Inferences 3 and 4.
Inference 3.Under certain conditions, p d * r is the increasing function of N Po and the minus function of N Fo , the limit condition is 4(α + β)(c + t) > 2θ 2 + φ 2 ; at the same time, there may be the increasing function of p d * e is N Fo and the minus function of N Po , the limit condition is 0 Prove: From formulas ( 12) and ( 14), respectively, count the frst partial derivatives for N Po and N Fo , obtained: is the increasing function of N Po and the minus function of N Fo ; when 4(α Prove: According to formula (11), count the frst partial derivatives for N Po and N Fo , (zs d * r /zN Po ) � (μθ)/(4 . Te proof process is similar to Inference 3, and will not be repeated. Under the no-contract model, the service price (p) and the service level (s) of dual-channel elderly care service are closely related to the elderly care service providers' service efort level (N) to meet the present-orientation and futureorientation needs of elderly.Generally, the elderly care service providers eforts to satisfy the demand of presentorientation (N Po ) will directly improve the price (p r ) and service level (s r ) of ofine service channels, it means that as the geriatric for emotional company of humanistic care demand increase, can promote the service level of ofine service channels, and also will bring more profts for ofine service channels.Similarly, when the elderly care service providers to meet the demand of future-orientation (N Fo ) will directly improve the price (p e ) and service level (s e ) of online service channels, it means that when the elderly demand related to learning and growth increase, online service channels need to provide more quality service.At the same time, with the improvement of market service prices will make it get more profts.
However, under certain conditions, the level of eforts made by elderly care service providers for diferent elderly needs will also be negatively correlated with the service price and service quality of the dual-channels.Infuenced by the price cross-coefcient, the more the value of α and β approaches to 1, the stronger substitution of online service channels and ofine service channels becomes.At this time, no matter how high the level of efort paid by N Po and N Fo is, it cannot attract geriatric.Under this condition, take ofine service channels as an example: if the geriatric pay ofine transportation costs (t) increase, the demand of ofine service channels will decrease, and the demand of online service channels will increase, will inevitably afect the elderly users of ofine service channels, cause ofine service price (p r ) and service level reduction (s r ).Ofine service channels need to pay more service improvement costs to maintain the service level.

Beneft-Sharing Contract Decision Model.
In the dualchannel supply chain of smart elderly care service, the development of online service channels has seized the market share of some ofine service channels.In this scenario, designing a beneft-sharing contract can help the whole dualchannel supply chain of smart elderly care service to be greater than that of the no-contract decision model.Further realize the coordination and optimization of the smart elderly care service supply chain.Tus, elderly care service providers still provide elderly care services to elderly care service integrators at the reasonable price w, and the online and ofine channels of elderly care service integrators take the form of cooperation.Elderly care service integrators recommend corresponding service channels to elderly users according to the needs of diferent stages, and a certain proportion will income subsidies from online service channels to ofine service channels.In the context of beneft-sharing contract, the needs of elderly users in diferent stages can match different service channels, which can not only increase the satisfaction of elderly users, but also bring more benefts to the smart elderly care service supply chain.Assuming that the beneft-sharing coefcient of the online service channel of the elderly care service integrator is λ, and the beneft-sharing coefcient of the ofine service channel is (1-λ).Te decision result is represented by the superscript "c." At this point, the proft function of the ofine service channel and the ofine service channel can be expressed in the following formulas: M c e � λp e Q e − wQ e − In the case of beneft-sharing contract, according to the backward induction method, count the frst partial derivative of p r , s r , p e , and s e in formulas ( 29) and (30), combined (zM c r /zp r ) � 0, (zM c r /zs r ) � 0, (zM c e /zp e ) � 0, (zM c e /zs e ) � 0, and obtain the optimal service price and the optimal service level of the elderly care service integrator, as shown in the following formulas: Complexity Under the beneft-sharing contract decision model, the following conclusions are drawn: Theorem 5.In the context of beneft-sharing contract, there must be p c * r and s c * r to maximize the proft M c * r of the ofine service channel of the elderly care service integrators.
Inference 8 has been proven.
Under the beneft-sharing contract mechanism, the decision-making status of ofine service channel has changed, and the optimal service price (p c * r and p c * e ) and the optimal service level (s c * r and s c * e ) maximize the proft of elderly care service integrators (M c * At the same time, carding Teorems 1, 2, 5, and 6, found: At the same time, comparing Teorems 1, 2, 5, and 6, it is found that in the no-contract decision mode, the secondorder partial derivatives of the optimal price p d * r and optimal service level s d * r of the ofine service channel in Teorem 1z 2 M r /zp r 2 < 0 and z 2 M r /zs r 2 < 0 are constant establishment.Also, second partial derivative of the optimal price p d * e and optimal service level s d * e in Teorem 2: z 2 M e /zp e 2 < 0 and z 2 M e /zs e 2 < 0 are only at 2(α + β)(c + t) − θ 2 > 0 can be established.After joining the beneft-sharing contract, it is not difcult to fnd through the proof results of Teorems 5 and 6 that the second derivative of the optimal price (p c * r and p c * e ) and the optimal service level (s c * r and s c * e ) of dualchannels are constant establishment.Tis means that after adding the beneft-sharing coefcient λ, the proft optimization of online service channels breaks through the limit of 2(α + β)(c + t) − θ 2 > 0. Tis is due to the fact that the dualchannels of elderly service integrators realize the efcient matching of channels and elderly demands in the context of beneft-sharing, so as to further optimize the conditions for proft maximization.So that there must be the optimal service price and the optimal service level, make the proft optimal.
Finally, according to the results of Inferences 3, 4, 7, and 8, we found that after adding parameters about meeting the elderly demands of Future-orientation and present-orientation of service providers' service efort level (N Fo and N Po ), which can optimize the price and level of elderly services.So as to promote the coordination of the smart elderly services dual-channel supply chain, whether it is the no-contract decision-making model (Inferences 3 and 4) or the beneft-sharing contract decision-making model (Inferences 7 and 8).Next, the research will further demonstrate the feasibility of the conclusion through example analysis.

Sensitivity Analysis of the Decision Model in No-Contract.
In this section, MATLAB software is used to verify the efectiveness of beneft-sharing contract for coordinating the smart elderly care service supply chain through numerical examples, and the relevant parameters are set as: w � 20; c s � 10; c � 12; α � 0.6; β � 0.4; θ � 0.4; φ � 0.3; t � 18.

No-Contract Decision Model.
In the no-contract situation, the impact of the level of efort paid to satisfy the two-stage needs of the elderly on the optimal service price and the optimal service level was further analyzed.Figures 3(a) and 3(b), respectively, show the sensitivity analysis of the efort level (N Po and N Fo ) paid by elderly care service providers to satisfy the present-orientation needs and future-orientation needs to the optimal market price (p d * r and p d * e ) of elderly care service integrators.In general, under the infuence of China 9073 pension model, 90% of elderly users choose the home care model.Terefore, after opening online service channels, home care is more convenient, and various functional services tend to be more convenient and intelligent.As can be clearly seen from Figures 3(a) and 3(b), the optimal market price (p d * e ) of online service channels is always higher than the optimal market price (p d * r ) of ofine service channels, that is, p d * e > p d * r .From the changing trend: in Figure 3(a), as elderly care service providers make increasing eforts of (N Po ) to satisfy the present-orientation needs of the elderly, the elderly are increasingly satisfed with the present-orientation of humanistic care such as emotional companionship, and will stimulate the market share (Q r ) of elderly care service integrators.At this time, the service price (p d * r ) of ofine service channels will lead to an upward trend.Te service price (p d * e ) of the online service channel has an obvious declining trend; opposite, when N Fo increasing, the trend of p d * e is rising.However, the p d * r showed a decreasing trend.Te outcome is consistent with the Inference 3.
Figures 4(a) and 4(b), respectively, show the sensitivity analysis of the efort level (N po and N Fo ) paid by elderly care service providers to satisfy the present-orientation needs and future-orientation needs to the optimal service level (s d * r and s d * e ) of elderly care service integrators.
In general, the experience brought by ofine service channels to elderly users is more intuitive and specifc, so the service level of ofine service channels is often higher than that of online service channels, that is, s d * e > s d * r .Te changing trend of s d * r and s d * e shown in Figures 4(a) and 4(b) is consistent with Figure 3. Te larger N Po is the enthusiasm of ofine service channels and s d * r will be increasing; the larger N Fo is, the increasing trend of s d * e is more obviously, in stent with Inferences 3 and 4.

Complexity
Continue to discuss the impact of the changes in the parameters such as the price cross-coefcient (α, β) related to the elderly care service price (p) and the level of elderly care service (s) with the cost coefcient of service improvement (θ, φ) on the supply chain of smart elderly care service as shown in Figures 5 and 6.
Figures 5(a) and 5(b) on the price cross-coefcient (α and β) analyzes the infuence of elderly care service price efect, you can see whether ofine service channel price cross-coefcient α or online service channel price cross-coefcient of β, also means that as α and β infnite approach to 1, alternative online service channels and ofine service channels will be stronger, both channels cannot give full play to their own advantages, will inevitably lead to the price (p d * r and p d * e ), hope by reducing its price to get more market share.
Figures 6(a) and 6(b) analyze the efect of the service improvement coefcient (θ and φ) on the service level.It can be seen that with the service improvement of Ofine service channels, the greater the cost impact coefcient θ, the service level of Ofine service channels is getting higher and higher.Te dual channels achieve equilibrium at θ � 0.2.Similarly, with the greater the service improvement coefcient of the online service channel, the online service level of the service channel φ is also on the rise, and the dual channels achieve equilibrium at φ � 0.59.At the same time, Figure 6 shows that the infuence coefcient of the cost of service improvement of 12 Complexity the dual channels only has an impact on their own service level, which means that the elderly care service integrator can only adjust the service level of ofine service channels or online service channels by controlling the size of θ or φ.

Sensitivity Analysis of Each Parameter of the Beneft-Sharing Contract.
After adding the beneft-sharing contract, the efect of the efort level to meet the elderly on the optimal service price and the optimal service level are as follows: Figures 7(a) and 7(b), respectively, show the sensitivity analysis of the efort level (N Po and N Fo ) of the elderly care service providers to the ofine and online optimal market price (p c * r and p c * e ) after joining the beneft-sharing contract.
Figures 7(a) and 7(b) can clearly see, to join the beneftsharing contract, the future-orientation demand and presentorientation demand of elderly users have achieved a more accurate match with service supply dual channels.While the demand for the elderly is efectively satisfed, makes the dual channels of the optimal service price (p c * r and p c * e ) are signifcantly improved.Also, it is still in line with the development trend of China's 9073 elderly care model, p c * r > p c * e .From the perspective of the growth trend, with the eforts (N Po ) made by elderly service providers to meet the presentorientation needs of the elderly are creasing, the price of elderly service in ofine service channels p d * r is still on the upward trend, while the price of elderly service in online service channels p d * e shows a downward trend; on the other Complexity hand, elderly care service providers to meet the needs of the future-orientation of elderly care eforts (N Fo ) is more and more bigger, ofine service channels of elderly care service price p d * r is still declining, and online service channels of elderly care service prices are on the rise, and the growth of online service channels compared with ofine service channels is more obvious.Te outcome is consistent with the Inference 7.
Figures 8(a) and 8(b), respectively, show the sensitivity analysis of the efort level (N Po and N Fo ) on the ofine and online optimal service level (s c * r and s c * e ) of the elderly care service integrators after joining the beneft-sharing contract.
From the level of service (s) change, online service channels and ofine service channel docking elderly user demand more clearly, make service improvement cost decreased, and ofine service level (s d * r ) is still rising, even the rising trend is more obvious, and it representative the beneft-sharing contract has a signifcant efect on reducing service improvement costs and improving service level s d * r efect is signifcant.Te infuence route of N fo change on the optimal service price and the optimal service level is basically consistent with N po , which will not be repeated here.
Continue to discuss the impact of the changes in the price cross-coefcient (α, β) and the service price (p) and the service level (s) of the service on the supply chain of the smart elderly care service under the beneft-sharing contract as shown in Figures 9 and 10.
Te efect of the price cross-coefcient (α and β) on the price of elderly care services in Figures 9(a) and 9(b) show that: After the original loss of ofine service channels is subsidized by online service channel, price advantage increasing obviously, although there is still a price war between dual channels, but the price gap narrowed under the beneftsharing contract.Proving that the beneft-sharing contract not only can improve the profts but also reduce the damage brought about by the price war.In Figures 10(a) and 10(b), the impact of service improvement coefcient (θ and φ) on service level is analyzed.It can be seen that θ and φ balance online and ofine service channels at 0.16 and 0.74, respectively, and the service improvement cost is reduced.

Comparative Analysis of the Optimal Proft.
In the case of the beneft-sharing contract, as the income-sharing coefcient λ changes within the (0, 1) interval, the equilibrium solution of each variable is shown in Table 2.
Trough Table 1 can clearly see: frst of all, as λ gradually smaller, the beneft ratio (1-λ) from online service channels share to ofine service channel gradually bigger, means that the elderly demand will match to more suitable supply channels, so that the equilibrium solution of optimal service price p c r and p c e is higher and higher.In λ � 0.1, ofine service channel optimal service price p c r is 50.95, and the online service channels of optimal service price p c e reached 50.47.Second, as λ gradually decrease, both ofine optimal service level s c r or online optimal service level s c e basically presents a downward trend, this is due to the infuence of beneft-sharing contract, channel recommended behavior between dual channels have a clear demand for geriatric, under the beneft-sharing contract mechanism, dual channels can have more energy to satisfy the specifc needs of the geriatric, which spend less service improvement cost for higher elderly satisfaction.Finally, because the beneftsharing contract mechanism set for the online service channel beneft in proportion to ofine service channels, so the ofine service channels of optimal proft M c r more and more increasing, and the ofine service channels of optimal proft M c e has a slight downward trend, but for the supply chain total proft M c is better than no-contract mode of supply chain total proft M d .
Table 3 shows the analysis of the infuence of N Po and N Fo on the decision variables paid by elderly care service providers to satisfy the present-orientation and futureorientation demands.
Trough the change of time perception by elderly users, when the elderly demand for information increment is greater than the demand for emotional increment, often need to pay more related to the future-orientation service level N Fo , otherwise need to elderly care service providers to pay more related to the present-orientation service efort level N Po .As can be seen from Table 2 (top half ), as N Fo gradually increasing, whether there is a contract, ofine service channel proft increased.Moreover, under the beneft-sharing contract, the ofine service channel of optimal proft is signifcantly higher than that of ofine service channel under the no-contract mode, namely, M d r > M c r .At the same time, the supply chain total proft is higher under the beneft-sharing contract, M c > M d .As can be seen from Table 2 (bottom half), as N Po gradually increasing, whether there is a contract, online service channel proft also increased.It is diferent from the left side, because of the beneft-sharing contract, the online service channel revenue is distributed to ofine service channels, so the optimal proft is slightly lower than the online service channel in nocontract mode on ofine service channel of optimal proft, namely, M c e < M d e .But join the beneft-sharing optimal service level in benefit-sharing contract  16 Complexity contract, supply chain profts compared with no-contract mode of total proft is higher, M c > M d .To sum up, after adding the beneft-sharing contract, no matter how N Fo and N Po change, the total proft of the supply chain is higher than the total proft of the supply chain under the no-contract, that is, M c > M d is constant established, which verifes the efectiveness of the revenue-sharing contract for maximizing the proft of the supply chain.
In general, compared with the noncontract model, the beneft-sharing contract has a more obvious efect on the optimal service price p * r and p * e , but the impact on the optimal service level s * r and s * e is relatively weak.Te optimal proft under the beneft-sharing contract M c is higher than that under the noncontract optimal proft M d .However, regardless of whether there is a contract or not, the larger the price cross-coefcient α and β, the smaller the channel characteristics and diferences, the higher the substitution, and the lower the satisfaction of diferentiated elderly services.From the perspective of elderly demands, the service efort level of elderly service providers to meet the presentorientation needs of elderly users N Po is higher, the market share of ofine service channels Q r is greater, and then the service satisfaction of elderly users in the Po-stage is higher.In the same way, if the level of service efort made by the elderly care service providers to meet the future-orientation needs of the elderly care users N Fo is higher, the market share of the online service channel Q e is greater, and the service experience of the elderly care users in the Fo-stage is stronger.It can be seen that the service quality of the smart elderly service supply chain has an increasingly important impact on the sales Q, and the improvement rate of service improvement cost 1/2s 2 r and 1/2s 2 e is less than the growth rate of sales volume and supply chain profts, and the smart elderly service dual-channel supply chain, considering the needs of the elderly can obtain more profts.

Managerial Insights
According to the research results, this paper puts forward the following insights for improving the service level of the smart elderly care services dual-channel supply chain and realizing the efcient coordination of the whole supply chain.
First, the beneft-sharing contract mechanism has a limited coordination efect for the whole supply chain, and mainly improves the proft of the supply chain by afecting the service price.In the application of the contract, more reasonable restrictions should be designed by combining with the actual situation of the smart elderly care service supply chain.When necessary, it can be integrated into other contracts to achieve the coordination efect of the supply chain by improving the service level, so as to continue to optimize the supply chain system of smart elderly care service and achieve the goal of proft maximization.
Second, although the elderly care service mainly supplied by elderly care service providers, elderly care service integrators is the main elderly access to service, so in addition to the root of rich elderly care services and improve the quality of service at the same time, also should pay attention to the construction of online and ofine service channels, give full play to the ofine service channel experience and convenience of online service channels, take advantage of diferentiated service to expand dual-channel market share, in order to realize the purpose of maximizing proft of supply chain.
Again, under the action of time perception, according to the change of the elderly needs in diferent times, further refne elderly care service providers provided by multiple elderly care services, such as the elderly daily life care preferences, through online service channels providing door-to-door companion service or accompanying to the hospital services, and ofine service channels can mainly for disability or half disabled elderly provide 24 hours care and company services.
Finally, with the improvement of material living standards, the price is no longer the main factors for the elderly; therefore, when improving service quality need to pay service improvement cost and should consider the comprehensive benefts of service quality improvement to ensure smart elderly care service supply chain to achieve efective coordination, promote the smart elderly care service industry good sequence development, comprehensively improve the elderly's life happiness.

Conclusion and Future Directions
Tis paper combines the user's time perception to depict the present-orientation demands and future-orientation demands and further analyzes the coordinated efect of elderly service providers to meet the present-orientation demands and future-orientation demands of elderly users on the whole dual-channel supply chain of smart elderly care service.We found that: Compared with the no-contract model, the beneft-sharing contract is more conducive to the smart elderly care service supply chain to achieve winwin benefts through cooperation.At the same time, the elderly care service providers should choose diferent channels to provide personalized services to the elderly users according to the degree of time perception, which can give full play to the diferentiated advantages of the two channels.In addition, improving the service level, strengthening the channel construction, and implementing the information and resource sharing between channels are of great significance for realizing the balance between supply and demand in the smart elderly care service supply chain, maximizing the benefts, and improving the happiness of the elderly.
However, there are still some limitations in our study.For example, when we build the utility model, we only subdivide the elderly demands into two deterministic needs: future-orientation demand and present-orientation demand according to the user's time perception, and do not consider other infuencing factors such as region, economic development level, and education level.Nevertheless, these questions also provide research directions for more in-depth research in the future, such as the coordination of the smart elderly services supply chain considering regional, economic, and other factors, and the coordination of smart elderly supply chain considering smart elderly services and products at the same time.

Figure 1 :Figure 2 :
Figure 1: Dual-channel supply chain system of smart elderly care service.
there is (zp d * r /zN Po ) < 0 and (zp d * r /zN Fo ) > 0, p d * e is the increasing function of N Fo , and the minus function of N Po .Inference 3 has been proven.

Inference 4 .
Under certain conditions, s d * r is the increasing function of N Po and the minus function of N Fo , s d * e is the increasing function of N Fo and the minus function of N Po , and the limit condition is 4

r 2 <Theorem 6 .Inference 7 .
0 is established.According to formula (38), if − 1 < 0,z 2 M c r /zs r 2 < 0 is established.Teorem 5 has been proven.In the context of beneft-sharing contract, there must be p c * e and s c * e to maximize the proft M c * e of the online service channel of the elderly care service integrator.Prove: consistent with the above steps, further calculate the second partial derivative of p e and s e , get: z 2 M c e /zp e 2 � − (2λ(α + β))/(c + t) < z 2 M c e /zp e 2 � − 1 < 0. Based on this, there must be p c * e and s c * e that maximize the M c * e .Teorem 6 has been proven.Similarly, under the proft-sharing contract scenario, to further analyze the relationship between the level of elderly demand efort (N Po and N Fo ) and the price of elderly care services (p c * r and p c * e ) and the level of elderly care services (s c * r and s c * e ), we can draw the following inferences: Under certain conditions, p c * r is the increasing function of N Po and the subtraction function of N Fo ; p c * e is the increasing function of N Fo and the subtraction function of N Po .Te limit condition is 2 , there are (zp c * r /zN Po ) > 0, (zp c * r / zN Fo ) < 0, (zp c * e /zN Po ) < 0, (zp c * e /zN Fo ) > 0. Inference 7 has been proven.Inference 8.Under certain conditions, s c * r is an increasing function of N Po and a decreasing function of N Fo ; s c * e is a decreasing function of N Po and an increasing function of N Fo .Te limit condition is r and M c * e ) simultaneously.M c * r and M c * e are shown in the following formulas : 10 Complexity

Figure 3 :Figure 4 :
Figure 3: (a) N Po sensitivity analysis to service price of dual-channels.(b) N Fo sensitivity analysis to service price of dual-channels.

Figure 5 :Figure 6 :
Figure 5: (a) α sensitivity analysis to service price of dual-channels.(b) β sensitivity analysis to service price of dual-channels.

Figure 7 :
Figure 7: (a) N Po sensitivity analysis to service price of dual-channels with beneft-sharing contract.(b) N Fo sensitivity analysis to service price of dual-channels with beneft-sharing contract.

Figure 8 :Figure 9 :Figure 10 :
Figure 8: (a) N Po sensitivity analysis to service level of dual-channels with beneft-sharing contract.(b) N Fo sensitivity analysis to service level of dual-channels with beneft-sharing contract.

Table 3 :
Optimal decision when the elderly demand changes.