Program Arrives Home Smoothly: Uncertainty-Based Routing Scheduling of Home-Based Elderly Care Programs

Abstract

In China, the home-based elderly care program system plays an important role in meeting the needs of elderly individuals. The routing scheduling of home-based elderly care (RSHEC) programs is closely related to the quality of the home-based elderly care programs. The structural supply problem faced by home-based elderly care programs is a prominent problem, and the RSHEC programs are an important aspect that has rarely been studied. This paper explores RSHEC programs under uncertainty by comprehensively considering the costs of home-based elderly care, such as the fixed costs, time, and transportation costs. First, a deterministic mixed integer programming (MIP) model was constructed to solve the general routing scheduling problem. In addition, to effectively cope with the uncertainty and risk of the modern market, the robust optimization theory and algorithm model are introduced, namely, the mixed integer box set robust optimization (MIBRO) model and the mixed integer ellipsoid set robust optimization (MIERO) model. Finally, MATLAB and the Gurobi package are applied to obtain the solutions of the models. The case verification shows that the MIP model has the lowest total cost under deterministic conditions. However, the MIERO and MIBRO models can achieve more robust RSHEC programs under uncertain conditions. The results prove the effectiveness and feasibility of the optimization model and algorithm, which provides reference value for management decisions regarding home-based elderly care programs.

1. Introduction

Since 2000, the aging population has accelerated significantly in China. In 2021, 267 million people aged 60 accounted for 18.9% of the national population, and the population aged 65 exceeded 200 million, accounting for 14.2% of the national population [1]. The increased aging of the population has led to real-life problems. Firstly, the traditional family structure has disintegrated, as small nuclear families have become the norm. Due to these changes, traditional family functions have weakened, and intergenerational support and care are reduced at the same time. However, due to the influence of Confucian tradition, Chinese seniors prefer to spend their remaining years at home rather than in nursing homes [2,3]. Secondly, the social and industrial structure has transformed, which increases the pressure on social retirement. The decline of the labor force not only increases the burden on social pension pillars but also drives the labor demand adjustments for new industries and economic structures. Thirdly, the diversified demands for senior care programs are growing [4] in various fields, such as basic medical, cultural, and spiritual care programs, forcing the program standards of the senior care industry to be overhauled with model innovation and level improvement. China’s main elderly care programs are divided into the following three categories: family elderly care, institutional elderly care, and (community) home-based elderly care. Family elderly care, which is tied to blood relations, is weakening due to the transformation of social development. In addition, market-oriented institutional elderly care is facing multiple difficulties in practice, such as uneven standards, high program costs, and single-program content. Among them, home-based elderly care is community-based, solving the problems of daily care, life care, and social programs through door-to-door visits from professional nursing staff and volunteers such as community workers and social workers [5,6]. The advantages of home-based elderly care programs are related to several issues. For example, they fit the traditional concepts that the elderly believe in and provide links with social elderly resources while allowing the elderly individuals to leave home as little as possible. Additionally, the multilevel needs for the elderly with professional skills and professional ethics can be met without inconveniencing elderly individuals [7].

In fact, China has a long history of providing home-based elderly care programs [8]. The concept of home-based elderly care in China can be traced back to the 1980s. In 2000, the policy “The Decision to Strengthen the Work of the Elderly” was issued, which first proposed to “establish an elderly care mechanism based on family elderly care, relying on community programs and supplemented by social elderly care” [9]. In 2008, the policy, “The Opinions on Comprehensively Promoting Home-based Elderly Care Programs” clearly stated that, “home-based care programs refer to a program form in which the government and social forces rely on the community to provide programs such as life care, home management, rehabilitation and spiritual comfort to the elderly at home” [10]. To meet the growing demand for diversified elderly care, in 2019, the decision to “improve the home and community-based elderly care policy” was made [11]. With the promotion of strong national policy promulgation and policy resources, China’s home-based care programs have developed rapidly. However, although RSHEC programs are effective measures to cope with increased aging of the population, they have not gained much attention.

Specifically, RSHEC programs face two operational planning concerns, namely, the locations of program stations and the vehicle routing schedule. Moreover, moderate vehicle routing scheduling can greatly reduce logistics costs [12]. Although home-based elderly care program centers seek to maximize efficiency, how to effectively plan the use of limited resources is still one of the greatest challenges [13,14]. Therefore, reducing the operational costs of RSHEC programs and improving program quality have become important topics of current research. In studies related to routing scheduling, most scholars treat the parameters in the RSHEC programs as deterministic values [15]. However, in practice, where fluctuations and market risks exist, even relying on modern technologies such as big data, the Internet of Things, and smart tools, some parameters remain in an unknown state. Uncertainty includes the immediate skills and statuses of drivers, waiting time of elderly program clients, unexpected extreme weather, and unstable traffic conditions, which can affect the elderly program supply and routing scheduling system. Therefore, if the demand and supply data rely merely on historical data and algorithms, there may be non-exact distribution conditions that affect the supply and evaluation of senior care programs. As a result, the concepts of stochastic programming and robust optimization are introduced to solve problems involving uncertainty. Robust optimization theory is used to construct uncertain sets, such as box sets, ellipsoids, or polyhedral, to describe problems involving uncertainty, and these uncertain sets are used to reflect risk and robustness [16,17]. Robust optimization theory and models can, to some extent, consider the risk preferences and conservativeness of decision makers, and have received considerable attention in recent years [18,19].

Combining RO theory with home-based elderly care offers the possibility of developing sustainable strategies. The RSHEC programs under the RO model explore the comprehensive costs of programs; moreover, minimizing the costs of RSHEC programs is an important manifestation of achieving economic sustainability. The development of health care and the sustainable development of society interact in both directions [20]. Responding to the aging population and meeting the demand for home-based care programs for the elderly is a significant initiative for attaining sustainable social development. An important feature of sustainable social development is the provision of the problems and protection needed by members, providing health support and functional support [21]. With the deepening population aging, RSHEC programs should also be guaranteed to a certain extent, to ensure that all elderly people have equal and timely access to elderly programs, thus ensuring the sustainability of elderly programs from both social and economic aspects. In addition, this paper contributes to previous research and reality. At the research level, this paper enriches the existing research. Firstly, a stochastic model is constructed to solve the RSHEC program problem considering the integrated costs, and the uncertainty is measured by setting uncertainty parameters. Furthermore, we optimize the mixed integer programming (MIP) model into a mixed integer box set robust optimization (MIBRO) model and mixed integer ellipsoid robust optimization (MIERO) model. Secondly, the proposed models have strong risk response capabilities, and decision makers can choose the appropriate models according to their risk preferences or cost strategies. At the practical level, we validate and evolve the algorithms with the example of city H in China and show the applicability of the three models through examples to provide references and aid in decision-making regarding program support of RSHEC programs.

2. Literature Review

China’s home-based elderly care has gone through the following four stages: planned economy, initial awareness of aging basic system construction, aging before preparation, and active development. Its development concept has undergone a gradual optimization process from focusing on macro reform to fine development [22]. Currently, program delivery is one of the current research perspectives [23], that is, the routing scheduling. Problem innovation in programs is one of the focuses of this research [24], which indicates the improvement of the model approach regarding the programs. For the RSHEC, the optimization of route scheduling is also the innovation of home-based elderly care, so this paper is an echo of the real issue and a supplement to the research.

In terms of specific content, there are practical dilemmas regarding insufficient integration of multiple resources and incomplete multifunctional configurations in home-based elderly care programs [25], and there are also problems regarding non precision in terms of program content supply and targeting of program guarantee support [26]. However, routing scheduling plays a certain role in both resource allocation optimization and guaranteeing program supply. Moreover, the routing scheduling model of home-based elderly care is an extension of the smart home-based elderly care approach. Smart aging-at-home is the use of modern technology to provide elderly individuals with online and offline integrated programs [27], but most regions currently have difficulty integrating information technology into aging-at-home programs [28]. In addition, the use of a platform to design a scheduling solution based on the balance of cost and program efficiency is important, which simplifies the combination of information technology and aging, programs, and manages with the aging population problem and the development of intelligent technology. Finally, research on RSHEC programs is not only one of the initiatives to respond to the national strategy but is also aimed at promoting the synergistic development of the elderly program business and the elderly program industry [29].

The current considerable quantitative research on home-based elderly care programs also provides some basis for this paper. Firstly, scholars explored the selection of supplying subjects, and a two-tier planning model considering cost and program quality was constructed [30]. Among them, quality games for each subject in the home-based elderly care supply chain under different situations are also considered [31]. Secondly, regarding caregiver scheduling, some scholars consider multiple types of caregivers and build a mixed integer planning model [32], and some scholars construct mathematical models based on constraints such as program time windows and program demand [33]. Similarly, some scholars integrate prospect theory and fuzzy theory to consider maximizing the comprehensive perceived satisfaction of the elderly by building a mixed integer nonlinear planning mathematical model [34]. Thirdly, in terms of the optimization of routing scheduling, some scholars study the scheduling of community home-based elderly care appointments through improved functional algorithms [35], and some explore vehicle routing scheduling under stochastic conditions [36]. Fourthly, with the increase in RO theory and model algorithms, the models are mainly applied to enterprise logistics and emergency logistics. For example, Leung et al. constructed an RO model to solve cross-border logistics problems under uncertain environments [37]; Baron et al. applied RO theory to locate equipment facing uncertain demands in multiple periods [38]; Sun et al. addressed the uncertainty regarding the number of casualties in post-disaster rescue workers casualties and established an RO model combining facility location and casualty transportation [39]; and Marques et al. applied the RO model to base location and ambulance allocation problems for fire departments [40].

In summary, the research on home-based elderly care programs has some deep excavation value, especially due to the lack of research on routing schedules under uncertainty. In the global context in which the population is aging at an accelerated and increased level, the diversified demand for elderly programs is growing. However, in practice, greater challenges and requirements for elderly programs exist according to the heterogeneity of elderly individuals in various regions. Meanwhile, both traditional traffic conditions and emerging COVID-19 problems show that uncertain factors affect the immediate and efficient supply in practice during the process of the program. However, the RO model has the natural advantage of coping with uncertain conditions embodied in theory and practice, which can be applied to the optimization strategy of RSHEC programs. Through this combination, the two-way longitudinal expansion of home-based elderly care programs and RO theory can be realized.

3. Problem Description

The RSHEC program problem is assumed to be divided into two stages. The first stage involves the selection of the center location and planning of the home-based elderly care center. In this stage, the center location is selected based on the radiated demand of the elderly in the community, based on the principle of proximity, as well as optimality. Subsequently, the home-based elderly care center responds rapidly, including the determination of resources and manpower. The centers are supposed to make decisions on the suitable allocation of home-based elderly care that is needed within a short period of time. The second stage is the initial RSHEC programs. After receiving the demand target, the program center integrates all costs, caregivers, and materials from the program center into home-based elderly care in the community. The community’s demand for an aging-in-place program is influenced not only by the needs of the elderly but also by the comprehensive evaluation of the program center, the caregivers, and the resources available, as well as many other aspects. In the closed-loop program feedback process, limits are placed on the needs of participants in the programs. Each patient can make single or multiple requests on the same day during the program. However, it is important to note that the continuity of requests between different elderly individuals and the interruption of the program are not considered in the MIP model but can be explained by incurring high costs in the MIERO and MIBRO models. For the two stages of home-based elderly care requests, on the one hand, it is necessary to coordinate planning to ensure that the demand is met to the maximum extent. On the other hand, on the basis of meeting the program demand, it is also necessary to adopt reasonable routing scheduling by minimizing operational management costs and achieving dual social and economic benefits.

In the home-based elderly care program network, there are $m$ home-based elderly care program centers and $n$ communities (as shown in Figure 1). Firstly, the paper establishes a MIP model to explore the RSHEC under deterministic conditions. The purpose of this model is to achieve the lowest total cost under the satisfaction of the program demand while maximizing the storage capacity, time constraints, and road constraints to the greatest extent possible.

The following scheduling problems are assumed for the RSHEC program.

• Assume that the program center can satisfy the basic needs of multiple communities.
• The program demands of all nodes can be satisfied.
• Each program demand node corresponds to at least one program center.
• The vehicles used in the same phase are of the same type, as well as having the same fuel consumption and load capacity.
• The geographic location and time window of the program centers are known.
• The average speed of vehicles varies in different areas and inconsistently depends on the road conditions and time periods.

Furthermore, the descriptions of the corresponding parameters and variables are shown in

Table 1. Descriptions of specific parameters and variables.

Note	Description
$D_n$	Demand of the home-based elderly care community
$c_f$	Fixed costs
$H_m^{Max}$	Maximum inventory of home-based elderly care program centers
$v$	The set of vehicles
$h_v$	The maximum load capacity of large transport vehicles
$c_v$	Fuel consumption cost per unit of vehicle transportation
$c_t$	The unit delay penalty cost
$d_{mn}$	Distance between home-based elderly care program centers and home-based elderly care community
${\overline{v}}_n$	Average vehicle speed
$t$	Base arrival time
$T_n^{Max}$	Maximum arrival time
$M$	A collection of home-based elderly care programs centers, consisting of $m$
$N$	A collection of home-based elderly care communities, consisting of $n$
$x_m$	$x_m\in [0,1]$, binary variable, if $x_m$ = 1, the program center is selected for demand community, 0 otherwise
$y_{mn}$	$y_{mn}\in [0,1]$, continuous variable, if $y_{mn}\ne 0$, select the routing schedule $mn$, the larger the value of $y_{mn}$, the higher the proportion

.

4. Model Construction

In the study of the RSHEC program, the costs are incorporated into the model operations. Among them, the specific costs include fixed costs, transportation costs, and time waiting costs. Each cost is explained in detail below.

(1)

Fixed Costs

Fixed costs are infrastructure investment costs, which are divided into office equipment wear and tear costs, basic utilities costs for the centers, and management program staff salaries. Fixed costs are not related to vehicle transportation costs. The formula is as follows:

$$C_f={\displaystyle \sum _{m=1}^m{c}_f}[x_m],x_m\in [0,1],$$

(2)

Vehicle Transportation Costs

To ensure the smooth arrival of the care program, transportation costs are as follows:

$$C_v=c_v{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\frac{D_n}{h_v}}}{y}_{mn}{d}_{mn}$$

In the equation, $c_v$ is the cost per unit of fuel consumed during vehicle transportation, $h_v$ is the maximum capacity of the center, where $v$ refers to the set of vehicles, and $d_{mn}$ is the distance from the center to the demand community.

(3)

Time Waiting Costs

This cost represents the reward (or penalty) due to time advancement (or delay). The formula is as follows:

$$C_t=c_t{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n[y_{mn}]\frac{d_{mn}}{{\overline{v}}_n}}}-t$$

where $c_t$ represents the time-related unit reward (or penalty) cost, $t$ represents the basic program door-to-door time, and $\overline{v}$ represents the average vehicle speed in the area where the program center is located [41].

After considering the various cost conditions, an MIP model is first constructed to plan the RSHEC program scheme under deterministic conditions, and then the model is extended into MIBRO and MIERO models by introducing RO theory to address the uncertain factors in the market and society.

4.1. MIP Model

This paper constructs an MIP model, that aims to minimize the total cost on the basis of satisfying the demands of elderly individuals. The specific model is as follows:

$$\min C_f+C_v+C_t$$

(1)

$$\left\{{\mathit{X}}^{*},{\mathit{Y}}^{*}\right\}=\arg \min {\displaystyle \sum _{m=1}^m{c}_f\lceil x_m\rceil }+c_v{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\frac{D_n}{h_v}{y}_{mn}{d}_{mn}}}+c_t{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\lceil y_{mn}\rceil (\frac{d_{mn}}{{\overline{v}}_n}}}-t)$$

(2)

$$s.t.{\displaystyle \sum _{m=1}^m{y}_{mn}=1},\forall n\in N$$

(3)

$$y_{mn}\le x_m,\forall m\in M$$

(4)

$${\displaystyle {\sum }_{n=1}^n{D}_n{y}_{mn}}\le H_m^{Max},\forall m\in M$$

(5)

$$\lceil y_{mn}\rceil \cdot \frac{d_{mn}}{{\overline{v}}_n}\le T_t^{Max},\forall m\in M,\forall n\in N$$

(6)

$$0\le x_m\le 1,\forall m\in M$$

(7)

$$0\le y_{mn}\le 1,\forall m\in M,\forall n\in N$$

(8)

The specific objective function and constraints conditions of the basic MIP model are as follows: objective function (1) is the synthesis of the cost analysis, which can be converted to constraints (2) in the model. The first item of the objective function is the fixed cost $c_f$, which includes the cost of office equipment loss, basic water and electricity costs of the program centers, and the salary of the management program staff. Meanwhile, fixed costs are independent from vehicle routing scheduling. Among them, $x_m$ is a binary variable, if $x_m$ = 1, the program center $m$ is selected for demand community, 0 otherwise. The second item of the objective function is the vehicle transportation cost, which aims to ensure that home-based elderly care providers arrive smoothly. Among them, $c_v$ is the unit fuel consumption cost during vehicle transportation, which represents the maximum passenger capacity of the home-based elderly care vehicle. The third item of the objective function is the time cost $C_t$, which is the reward (or penalty) caused by the advance (or delay) of time. Among them, $c_t$ is the time-related unit reward (or punishment) cost, $t$ is the basic time to reach the program center, and ${\overline{v}}_m$ is the average vehicle speed in the area where the program center is located. Constraints (3) means that all care programs under the regional platform are provided by program centers, while there is no other outflow, and the needs of any community must be met. Constraints (4) means that only selected program centers can participate in staff scheduling jobs. Constraints (5) shows that the care program capacity provided by program centers $m$ is less than or equal to its maximum program dispatch capacity $H_m^{Max}$. Constraints (6) is time constraints, that is, the time consumed by any route is less than or equal to its maximum routing period $T_t^{Max}$. Constraints (7) and (8) describe the domain of the variable, $x_m$ is an integer, and they are in the range of collection $M$ and $N$.

4.2. MIBRO Model

In reality, the external market environment is sophisticated. Thus, it is difficult to obtain the distribution of key parameters, especially the accurate value or probability of the program. This may cause the ideal model to be less feasible or even nonexistent in the real world. That is, the routing scheduling scheme of the basic model has low anti-interference ability under uncertainty. Existing studies have shown that the RO model can effectively resist the interference of uncertain parameters, so the application of RO is naturally more attractive. In this paper, the theory of RO is used to transform the abovementioned deterministic model into an RO model so that the routing scheduling does not depend on the probability distribution of the parameters. The greater the customer demand volatility, the greater the uncontrollability. Define a random demand ${\tilde{D}}_n=D_n^0+\epsilon D_n^0$, where $D_n^0$ is the nominal demand and $\epsilon$ is the disturbance ratio. On this basis, the MIBRO model is established. In the MIBRO model, the uncertain demand is ${\tilde{D}}_n$, and the uncertain set is Box. According to RO theory, the robust equivalent model is (9)–(15), $Dom{U}^B=\{\epsilon :{\Vert \epsilon \Vert }_{\infty }\le \mathsf{\Psi }\}=\{\epsilon :|{\epsilon }_n|\le \mathsf{\Psi }\}$, $\mathsf{\Psi }$ is the uncertainty level parameter (that is, the safety parameter), and the uncertainty level parameter indicates that at most one parameter deviates from the nominal value. In the following constraints, the objective of Constraints 9 is to minimize the total cost. Constraints (9)–(15) have the same meaning as the constraints in the MIP model above.

$$\left\{{\mathit{X}}^{*},{\mathit{Y}}^{*}\right\}=\underset{X,Y}{\arg \min }{Z}_B$$

(9)

$$s.t.{\displaystyle \sum _{m=1}^m{c}_f\lceil x_m\rceil }+c_t{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\lceil y_{mn}\rceil (\frac{d_{mn}}{{\overline{v}}_n}}}-t)+\underset{\epsilon \in \mathbb{U}{U}^{{}_B}}{\sup }\mathsf{\Psi }\left\{c_v{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\frac{{\tilde{D}}_n}{h_v}{y}_{mn}{d}_{mn}}}\right\}\le Z_B$$

(10)

$${\displaystyle {\sum }_{m=1}^m{y}_{mn}}=1,\forall n\in N$$

(11)

$${\displaystyle \sum _1^n{D}_n^0{y}_{mn}}+\mathsf{\Psi }{\displaystyle \sum _1^n\epsilon D_n^0{y}_{mn}}\le H_m^{Max},\forall m\in M$$

(12)

$$\lceil y_{mn}\rceil \cdot \frac{d_{mn}}{{\overline{v}}_n}\le T_t^{Max},\forall m\in M,\forall n\in N$$

(13)

$$0\le x_m\le 1,\forall m\in M$$

(14)

$$0\le y_{mn}\le 1,\forall m\in M,\forall n\in N$$

(15)

4.3. MIERO Model

In the MIERO model (16)–(24), the ellipsoid set $Dom{U}^E=\{\varsigma :{\Vert \epsilon \Vert }_2\le \mathsf{\Omega }\}=\{\epsilon \sqrt{{\displaystyle {\sum }_N{\epsilon }_n^2}}\le \mathsf{\Omega }\}$, among them, where $\mathsf{\Omega }$ is an adjustable safety parameter, and the spherical diameter of the uncertain set. The ellipsoidal uncertainty set is $U^E=\{\tilde{D}\in R,{\displaystyle {\sum }_{n=1}^n[{({\tilde{D}}_n-D_n^0)/{\widehat{D}}_n]}^2}\le {\mathsf{\Omega }}^2\}$. Since this model is a nonlinear constraints problem, the set $U^E$ is equivalent to $U^E\iff \{\tilde{D}\in R,{({\tilde{D}}_n-D_n)}^T{C}^{-1}({\tilde{D}}_n-D_n)\le {\mathsf{\Omega }}^2\}$, where C is an n-th order diagonal matrix with ${\widehat{D}}^2{}_n$ (nonzero) elements. Then, set $\mathit{P}\ge \sqrt{{\displaystyle \sum _1^n{({\tilde{D}}_n)}^2}{\mathit{R}}^2}$, where $\mathit{R}\ge \sqrt{{\displaystyle \sum _1^m{(\frac{c_v{d}_{mn}}{h_v})}^2}}$, which increases the slack constraint. Then, the upper formula can be transformed into the following formula: ${\displaystyle \sum _{m=1}^m{c}_f\lceil x_m\rceil }+c_t{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\lceil y_{mn}\rceil (\frac{d_{mn}}{{\overline{v}}_n}}}-t)+\underset{\epsilon \in U^{E}U}{\sup }\mathsf{\Omega }\mathit{P}\le Z_E$, whose goal is to minimize the total cost. Constraints (17) shows that the combination of fixed and stochastic costs is less than the minimized cost. Constraints (18) and (19) are slack constraints to facilitate solving over a larger feasible domain. Constraints (20) indicates that program center $n$ provides less than or equal to its maximum capacity of care. The other constraints are defined as above.

$$\left\{{\mathit{X}}^{*},{\mathit{Y}}^{*}\right\}=\underset{X,Y}{\arg \min }{Z}_E$$

(16)

$$s.t.{\displaystyle \sum _{m=1}^m{c}_f\lceil x_m\rceil }+c_t{\displaystyle \sum _{m=1}^m{\displaystyle \sum _{n=1}^n\lceil y_{mn}\rceil (\frac{d_{mn}}{{\overline{v}}_n}}}-t)+\underset{\epsilon \in U^E\mathbb{U}}{\sup }\mathsf{\Omega }\mathit{P}\le Z_E$$

(17)

$$\mathit{P}\ge \sqrt{{\displaystyle \sum _1^n{({\tilde{D}}_n)}^2}{\mathit{R}}^2},\forall n\in N$$

(18)

$$\mathit{R}\ge \sqrt{{\displaystyle \sum _1^m{(\frac{c_v{d}_{mn}}{h_v})}^2}},\forall m\in M$$

(19)

$${\displaystyle {\sum }_1^m{y}_{mn}}=1,\forall n\in N$$

(20)

$${\displaystyle \sum _1^n{D}_n^0{y}_{mn}}+\mathsf{\Omega }\mathit{P}\le H_m^w,\forall m\in M$$

(21)

$$\lceil y_{mn}\rceil \cdot \frac{d_{mn}}{{\overline{v}}_n}\le T_t^{Max},\forall m\in M,\forall n\in N$$

(22)

$$0\le x_m\le 1,\forall m\in M$$

(23)

$$0\le y_{mn}\le 1,\forall m\in M,\forall n\in N$$

(24)

5. Numerical Algorithm

Through the three model constructions, Y district of H City, Zhejiang Province in China is selected as an example to verify the effectiveness and applicability of the RO model to the RSHEC program. The reason for choosing district Y of city H is that it is typical of district Y. On the one hand, in 2020, district Y of city H is seriously aging, with 161,000 people over 65 years old, accounting for 13.86% of the total population. In addition, there is an increase in the number of senior citizens and a strong demand for home-based elderly care programs. The life expectancy of the household population in city H has reached the leading level in developed countries and regions in the world. At the same time, the economy of city H is more developed, and the development of home-based elderly care programs is more developed. The case study of district Y of city H is selected not only to provide suggestions for the optimization of home-based elderly care programs in China, but also to provide references for other countries and regions. On the other hand, compared with existing studies, this case study focuses on district Y, which further relaxes the demand constraint of home-based elderly care programs and makes its application broader. In the actual program, the decision-makers of the home-based elderly care program face two stages of vehicle routing scheduling. The first stage is to determine the location of the home-based elderly care program center. The determination of the center has the following functions: one is to provide resources and programs for the corresponding demands from elderly groups, and the other is to determine the radiation range of the center. Based on the comprehensive consideration of various location factors, the home-based senior program center selects five program center sites in the area, namely, Hongyuan Center, Colorful Light Center, Social Center, Mengchuang Center, and Yijia Center. There are 20 demand nodes, as shown in Figure 2 below, and the distance between each demand node and the program center point is based on the official GIS data.

In the RSHEC program system, there are five program centers and 20 community demand nodes, and any alternative routes correspond to different cost schemes. Among them, the transportation cost is set according to the real-time fuel price. The model is performed on the basis of comprehensive consideration of relevant costs, taking the actual distance, traffic congestion and calculated time constraints into account. According to the geographical data, the map annotations of buildings, lakes, and mountains are hidden, and the relative location map is obtained as shown in Figure 3.

5.1. Basic Data

The basic data include the nominal demands of the demand nodes, the maximum program supply of the program center, the regional traffic conditions, and the fixed operation costs, as shown in

Table 2. Basic data of the demand community.

Demand Area	Nominal Demand	Demand Area	Nominal Demand
$D_1$	145	$D_{11}$	300
$D_2$	150	$D_{12}$	180
$D_3$	140	$D_{13}$	105
$D_4$	210	$D_{14}$	105
$D_5$	290	$D_{15}$	280
$D_6$	120	$D_{16}$	120
$D_7$	215	$D_{17}$	170
$D_8$	115	$D_{18}$	92
$D_9$	100	$D_{19}$	95
$D_{10}$	190	$D_{20}$	90
Program Center	Maximum Program Supply	Average Speed	Fixed Cost
$N_1$	590	40	2700
$N_2$	610	45	3100
$N_3$	650	50	3500
$N_4$	640	45	4800
$N_5$	660	40	5100

. The actual distances between nodes are shown in

Table 3. Actual geographic location distance.

${\mathit{d}}_{\mathit{i}\mathit{j}}\cdot {\mathbf{km}}^{-1}$	${\mathit{N}}_1$	${\mathit{N}}_2$	${\mathit{N}}_3$	${\mathit{N}}_4$	${\mathit{N}}_5$
$D_1$	37	16	37	21	55
$D_2$	42	8.7	50	29	59
$D_3$	29	14	42	10	46
$D_4$	25	18	39	14	33
$D_5$	29	28	21	8.6	27
$D_6$	9.5	45	8.4	30	21
$D_7$	11	49	2.2	22	16
$D_8$	6	39	7.9	15	21
$D_9$	15	52	4.3	26	19
$D_{10}$	39	9.9	48	29	57
$D_{11}$	37	7	51	26	55
$D_{12}$	40	13	46	27	61
$D_{13}$	32	11	39	20	54
$D_{14}$	3.2	38	13	16	35
$D_{15}$	35	8.4	49	21	53
$D_{16}$	29	51	17	30	5.7
$D_{17}$	22	41	14	17	21
$D_{18}$	40	58	27	37	16
$D_{19}$	43	61	34	41	20
$D_{20}$	51	70	42	49	28

.

The actual distance between nodes can be directly obtained, as shown in Table 3. In addition, the time of the basic door-to-door program is 20 min, the time delay cost is $10 per hour, and the maximum delay time is 60 min. If the maximum time is exceeded, the demand point will cancel the demand program order.

5.2. Algorithm Design and Routing Scheduling Scheme

In this paper, the algorithm framework is designed based on the MATLAB (R2016a) platform, and the solution is solved by Gurobi (9-1-0). To ensure the scientific nature of the experiment, all the model algorithms are run on the same computer and in the same operating environment (Windows 10, Intel(R) Core(TM) i5-8300H CPU @2.3 GHZ, RAM8GB, 512G SSD). The specific algorithm is shown in

Table 4. Specific steps of algorithm design.

Step 1	Construct an algorithm framework, input initial values, set uncertain parameter value boundaries and safety parameters.
Step 2	#1 The mixed integer programming model defines the variable types. #2 MIBRO and MIERO models define variable types and the system randomly generates random numbers.
Step 3	If the random number meets the conditions: $Dom{U}^B=\{\epsilon :{\Vert \epsilon \Vert }_{\infty }\le \mathsf{\Psi }\}=\{\epsilon :|{\epsilon }_n|\le \mathsf{\Psi }\}$$,Dom{U}^E=\{\varsigma :{\Vert \epsilon \Vert }_2\le \mathsf{\Omega }\}=\{\epsilon \sqrt{{\displaystyle {\sum }_N{\epsilon }_n^2}}\le \mathsf{\Omega }\}$, execute step 4; otherwise, return to step 2.
Step 4	Input parameter variable constraints, maximum path constraints, load constraints, time window constraints, and other constraints. #1 MIP model and MIBRO model directly input constraints. #2 MIERO model then input.
Step 5	If step 4 is satisfied, terminate; otherwise, execute step 2.
Step 6	Set the solution environment and solve it through Gurobi.
Step 7	Output the optimal solution and count the total running time.

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From the information above, we can see that $N_1,N_2,N_3,N_4,N_5$ is the home-based elderly care program center, and $D_1,D_2,D_3\dots ,D_{19},D_{20}$ is the community demand node. In the model, the effect of $\mathsf{\Psi }$ on the total cost is constantly changing. When $\mathsf{\Psi }$ = 0, the MIBRO model and the MIERO model are equivalent to the MIP model. With all parameter conditions known, that is, when $\mathsf{\Psi }$ = 0, the routing scheduling scheme of the MIP model is shown in Figure 4. When $\mathsf{\Psi }$ is set to 1 through 20, the MIBRO model and MIERO model are shown in Figure 5 and Figure 6, respectively. Since the MIP model can only solve the problem where all parameter information and data are known, it has a limited application without applying the uncertainty problem. If all information and data are transparent and easily available, the results of the MIP model are undoubtedly more desirable. In Figure 5 and Figure 6, the route scheduling scheme in the model calculation results is complicated slightly due to the influence of uncertain parameters. The original route scheduling scheme fluctuates due to the influence of uncertainty and has to be moderately adjusted. At the same time, compared with the MIBRO model (Figure 5), the MIERO model has significantly fewer long-haul transport strategies; therefore, the route scheduling scheme is simpler. Based on the above analysis, the following preliminary conclusions can be drawn, which also provide recommendations to industry managers and program decision-makers. That is, the MIP model is the most appropriate when all parameter data are transparent and known. Then, the MIERO model provides better route scheduling than the MIBRO model when uncertainties arise, such as additional demand parameters and known parameters that are uncertain.

6. Model Sensitivity Analysis

The model results are derived to compare and analyze the performances of the three models, including run efficiency, uncertainty, and degree of demand volatility.

6.1. Model Runtime Comparison

The running efficiency of each model was compared to that of the others by observing the running times of the models. To ensure the validity of the results, the following analysis was conducted: the comparison was run in the same computer environment and the safety parameters were set as the only variables. The results are shown in Figure 7, from which the running efficiency of the two RO models and the MIP model can be observed. The MIP model has the highest running efficiency, and the overall running efficiency is much higher than that of the RO model (5.2666 s). Between the MIBRO model and MIERO model, the MIERO model (blue solid line) is less efficient than the MIBRO model (red dashed line). With the improvement of the safety parameters, overall, the MIP model is not affected, and the running times of the MIBRO and MIERO models show increasing trends. Among them, the MIBRO model runs more erratically, and the MIERO model is less efficient but more stable with fewer ups and downs. Due to the small size of the routing scheduling problem in this paper, the computation times do not differ much. However, when the constraints and variables in the model increase to thousands or even tens of thousands, there is a significant difference in the running efficiency of the model solutions [42]. Therefore, to solve the actual problem, we can build a matching model according to the size of the dataset.

6.2. Impact of Demand Fluctuations on Total Costs

In this section, the impacts of stochastic demand fluctuations on total cost for the three models are comparatively analyzed. The control variable method, that is, a fixed safety parameter ($\mathsf{\Psi },\mathsf{\Omega }=10$), is used to explore the impact of the fluctuation of the stochastic parameter on the total cost, and the results are shown in Figure 8. By comparing the trends in the graphs, the following conclusions can be drawn: first, the program level of the MIP model is not affected by the stochastic parameters, which means that it cannot solve the routing scheduling problem under uncertainty. Second, although the total costs of the MIBRO and MIERO models are higher than that of the MIP model, they still provide a more suitable route scheduling solution under demand uncertainty. Finally, the total cost of route scheduling tends to increase with the fluctuation of demand parameters, which means that program managers and decision-makers need to incur at higher costs to counteract the uncertainty as the risk factor increases.

6.3. Impact of Security Parameters on Total Cost and Program Level

This section provides a comparative analysis of the performances of the models, including the effect of the security parameters on total cost and program level. From Figure 9, we analyzed the effect of the variation of the security parameters on the total cost (with a fixed volatility of 0.10). In addition, the MIP model is not affected by the security parameters, while the rate of cost increase varies for the different RO models. With the increase in safety parameters, the overall logistics cost tends to increase, and the MIERO model has the highest growth rate (23.21% increase in cost ratio). The trend indicates that this model has the largest robustness cost to increase the safety level, while the MIBRO model is relatively stable. With the increase in safety parameters, the robustness cost is lower, and the cost increase is significantly different from that of the other two models. The total cost of the MIBRO model is higher than that of the MIERO model when the safety parameters range are from 0 to 4, while the total cost of the MIERO model is higher than that of the MIBRO model when the safety parameters are in the interval 4–20.

In addition, due to the requirement of timeliness, the time difference is used to compare the program levels of the models by analyzing the advantages and disadvantages of the models. The program level is calculated by the formula $SL=[1-{\displaystyle {\sum }_N(y_{mn}){\tilde{D}}_n{d}_{mn}{\overline{v}}^{-1}}-t_n{\displaystyle {\sum }_N{\tilde{D}}_n}/t_n{\displaystyle {\sum }_N{\tilde{D}}_n}]\times 100\%$, where $M,N$ is the indicator parameter in the model. Figure 10 shows the effect of the security parameters on the program level, which is limited with a fixed degree of random fluctuations in demand (fixed volatility of 0.10). In summary, the logistics program level tends to increase as the security level increases, which helps to compensate for the disadvantage of increased total cost. Moreover, it can mitigate the loss of a program level decrease due to random demand fluctuations. In addition, the three models show that equilibrium points of the program level turn out under the condition that the security parameters reach a certain threshold. From the results, the program decision-makers can choose the routing and scheduling scheme flexibly in practice.

Based on Figure 10, Figure 11 further analyzes the responsiveness of logistics program level fluctuations to changes in security parameters. The following conclusions can be drawn from Figure 11.

(1) The program level remains constant because the data are all deterministic and known ($SL=94.99\%$), referred to as the program level of the MIP model. (2) The program level of the MIBRO model tends to increase with the increase in the security parameters, while the MIERO model has strong robustness. Although the MIERO cost increases as the safety parameters increase, the logistics service level increases significantly. Thus, the MIERO model is very robust and can provide a higher quality of service. When the security parameter increases from 1 to 20, first, the logistics program level increases from 94.99% to 96.08% in the routing scheduling stage. Second, the performance of the MIBRO model improves, and the logistics program level increases from 94.99% to 95.46%. (3) In the program level improvement comparison, it can be observed that the MIERO model performs better and maintains a relatively stable growth trend overall, with a maximum increase of 11.24‰. In the process of home-based elderly care programs, managers must make effective decisions in a short period of time. With full consideration of uncertainty, the MIP model only provides routing scheduling within a defined range. Meanwhile, the performance and applicability of the MIBRO model is different from those of the MIERO model. This means that decision-makers need to weigh the objectives, program levels, and total costs according to the actual situation and changing trends. They can choose the most ideal solution among the alternatives to achieve multiple objectives, such as high efficiency of home-based elderly care program levels, program cost savings, and optimal resource allocation.

7. Conclusions

The structural drawbacks of the home-based elderly care program system and the modern risk market have formed a real challenge to home-based elderly care programs. Thus, the optimization of its routing scheduling decisions urgently needs to be explored in depth. The robust optimization model, as an emerging theoretical model that effectively resists uncertain conditions, should be applied to home-based elderly care programs. It is an inherent requirement for theoretical development and an inevitable choice for a practical response.

In this paper, we studied the RSHEC program by introducing robust optimization methods. Firstly, we established the MIP model under the condition of deterministic parameters for solving the routing scheduling problem. In addition, we advanced the model into the MIBRO model and MIERO model based on risk and market uncertainty, and the RO model has high practical application value. It can effectively address the problem of parameter value uncertainty, and the optimization model combined with uncertain parameters is robust and adjustable [43]. Finally, the reliability and validity of the model were verified by actual case operations, and the route scheduling optimization scheme was proposed according to the model results. The decision maker at the senior program center can choose the applicable model scheme according to the actual cost adjustment and logistics program level. This paper shows that the MIP model can achieve better decision benefits when all conditions of the home care program are known and certain, while the MIBRO model and MERO model can obtain more robust routing and scheduling solutions when uncertainties and risks exist and increase. The specific choice of the MIBRO model or MIERO model can be determined according to the goal pursued by the minimum cost or higher logistics level. The robust optimization model constructed in this paper can provide a decision reference for RSHEC programs under uncertain environments to achieve sustainable economic development. It can also provide a theoretical basis for the optimization of market programs to meet the program demands of home-based elderly individuals and achieve sustainable social development. However, there are still some limitations in this paper. For example, uncertainties such as carbon emission costs and traffic node congestion can be further considered in the model, and other uncertainty sets can be expanded. In addition, the advantages of the MIERO model have not been fully explored. More optimal algorithms can be used to study them and limitations can be further solved in depth in the future.