1 Introduction

The proportion of individuals aged 65 and above in China’s total population is steadily rising. As of 2023, this age group accounts for 15.4% of the population. With the aging trend accelerating, it is projected that by approximately 2031, the proportion of individuals aged 65 and above will exceed 20%, marking the transition into a super-aging society (1). The issue of aging has emerged as a significant concern in the rejuvenation of the Chinese people. Focusing on this matter will undoubtedly facilitate comprehensive economic and social transformation. Home care service is a type of assistance society offers to older individuals who live at home to address the challenges they face in their everyday lives. Based on survey data from the China Health and Wellness Commission, China’s older adults care pattern can be summarized as “9,073.” This means that around 90% of the older adult population live in their own homes, 7% depend on community-based care, and 3% reside in older adults care institutions. Aging in place is the predominant form of aging in China. Consequently, there is a substantial need for home care services for older people at home (2, 3). In China, home care facilities’ operational modes are primarily categorized into four types: government-funded operation, privately funded operation, government-funded private operation, and civilian-operated public assistance (4). Government-funded operation denotes an establishment funded, managed and administered by the government. Civilian-operated public assistance entails the government’s selection of appropriate service providers via open tender, wherein the government subsidizes but does not engage in the operational management of these providers. These two models are non-profit and primarily focused on low-income older individuals in poor physical condition with limited flexibility, predominantly providing low-cost or complimentary aged care services, lacking universality. The government-funded private operation model denotes a government-funded establishment operated by the private sector for profit. However, it struggles to deliver the leisure-oriented aged care services sought by seniors, resulting in incomplete coverage. The model is appropriate for the initial phase of aging development, characterized by market-oriented self-financing. However, as the government supplies hardware facilities, investors may refrain from augmenting their investment during operations, potentially compromising service quality. The privately funded operation model denotes the private sector financing a venture’s initiation and operation. The model offers substantial growth potential, is capable of addressing the comprehensive needs of the older adult population, exhibits high flexibility, and guarantees that the service retains the structural integrity of the product while complying with national requirements to accommodate the increasing demands of aging individuals. Consequently, the privately established and managed home care model possesses more significant potential for advancement.

In actuality, the majority of privately funded operation older adults care institutions function independently (2, 4). This study seeks to delineate the operational framework of privately funded operation home care businesses, which often cater to a defined demographic within a particular locale. Provision of services in accordance with ‘the Basic Norms for Home-Based Older adults Services’ (GB/T 43153–2023). It should be noted that if an organization provides home care services for the older adults in the form of a single service, a lot of problems may arise in the course of implementation. From the customer’s perspective, when multiple services are required simultaneously, they must sequentially order each individual service. Additionally, for customers with consistent and regular care requirements, repeated bookings of the same services are necessary. If they neglect to place an order, they forfeit access to the services, resulting in significant inconvenience and diminishing their overall experience. From the enterprise perspective, consistent long-term orders will substantially reduce the variability and unpredictability of orders, facilitating more precise allocation of service resources, effectively lowering operational costs, and enhancing the enterprise’s capacity to address spontaneous orders, thereby improving customer satisfaction. Consequently, for service programs characterized by elevated demand and increased frequency, it is essential to concurrently offer a composite service product comprising a range of individual services, specifically the older adult care package. To adequately meet the varied needs of senior citizens, different service packages might be concurrently launched, termed the older adults care service product family. It is often necessary to adjust the older adults care service product family offerings based on prevailing demand. The creation of older adults care service product families tailored to the specific regional context is a research area warranting significant attention.

It is important to recognize that the design of older adults care product families diverges from conventional product family design. Conventional product family design is an economical approach to assessing market demand, categorizing product components into modular types to create product family module libraries, configuring various series of segmented products from a shared platform in a market-driven manner, and producing a final product family to meet diverse customer segments (5). Traditional product family design contrasts with older adults care product family design in the following aspects. Primarily, the traditional approach is oriented toward the manufacturing sector, where products possess predetermined components and modules. Consequently, the final product family is generated solely through the combination of these fixed components and modules, which fails to address the needs of individual customers. The older adults care service project can be offered as an individual service addressing individual needs, while the older adults care service product family design caters to the requirements of diverse client segments. Secondly, the prior product family design just needs to account for the influence of external competition on the demand for the final product during product configuration; individual components do not possess market share and are not required to consider the competition among internal components. But older adults care services are presently experiencing an oversupply, with a restricted number of organizations offering such services in a specific area, and individual services can be delivered independently, necessitating consideration of the competitive influence of these individual services on the demand for service packages during the product family design for older adults care services. Third, prior product family design commenced with the identification of consumer perceptions and rival performance, quantitatively integrating these factors into product configuration design to link the demand and product configuration design stages for determining the product configuration aspect (6, 7). In the context of aging, the risk of market entry is unpredictable due to the specificity of the target demographic, and customer needs and preferences can only be anticipated through market research, which is constrained by technology and the particularities of the customer group in directly gathering expectation data. Compelling companies to tailor services to meet the needs of customers in particular regions. This approach mitigates resource wastage and excessive costs, necessitating more intricate design decisions for the development of products aimed at older adult services. In summary, the conventional product family design methodology is not entirely applicable to the design of product families for older adults care services. This research suggests a novel method to product family design for older adults care services.

2 Related work 2.1 Designing services for the older adults

The increasing population of older adult folks and the evolution of the ailments they encounter have necessitated enhanced social assistance and extended care services. Contemporary society emphasizes personalized care for older adult individuals, considering their distinct health problems, financial circumstances, and familial structures (8–10). Home older adults care services offer personalized care to address the varied needs of older persons, making the design of such services essential. Zhou et al. (11) propose that service design focused on older adults care can optimize the utilization of social resources to deliver precise and high-quality home care services for seniors in their residences. Consequently, home medical care, rehabilitation services, day care, in-home assistance, meal provision, bathing support, and other older adults care services are progressively becoming accessible to seniors, making a practical comprehension of their actual needs essential for addressing their requirements (12). Wei et al. (13) establish service design metrics for the rehabilitation requirements of the long-term care system, addressing the needs of long-term care stakeholders while also taking into account the emotional experiences of direct stakeholders, thereby enhancing the quality of care. Hu et al. (14) propose the reconstruction of functional service modules for semi-disabled older adult individuals within the community, establishing a health food system tailored to their needs, optimizing community resources, addressing the diverse healthcare requirements of the older adults, and offering an innovative perspective on the integration of medical care and older adults care within familial support systems (15). Chen et al. (16) used a service design methodology for an older adults demographic with mild cognitive impairment to summarize the hierarchy of high-level needs and guide design practice by observationally mapping user journeys, summarizing the service design strategies for mild cognitive impairment, and exploring more possibilities for design interventions for mild cognitive impairment. Thapaliya et al. (17) examine access to additional care services and the consumption of hospital and ambulance services by those utilizing the aging in place package, concluding that services must be tailored for this demographic to ensure sufficient support. The establishment of the home care service system constitutes a social system initiative. The persistent increase in demand for social pensions, coupled with constrained social resources, significantly hampers the advancement of social pension initiatives. Therefore, directing social organizations to offer home care services can effectively mitigate the pressures of social pensions while addressing the actual needs of the older adults, thereby promoting a diversified supply of older adult services (18–21). The design of older adult services in the process of continuous optimization of the home-based older adults service system can integrate older adults service resources from the needs of the older adults and improve the quality of life of the older adults.

2.2 Product family design

Product Family Design (PFD) is a strategy for expanding product offerings to satisfy a varied marketplace (22). Many businesses utilize product families and platform-based product development to address a broad spectrum of client requirements (23, 24). Liu et al. (25) consider the relative importance of components in responding to customer needs and the interrelationships between components to achieve optimal product architecture. To maintain market competitiveness, companies expand their product lines by launching product families; however, diverse customer needs cannot be sufficiently met through mass marketing strategies, highlighting the importance of product family positioning based on customer purchasing behaviors. Zhang et al. (26) proposed a fuzzy clustering-based market segmentation method that helps to effectively and efficiently plan the right product. The scope of the product family design issue was broadened to encompass the determination of suitable market positioning for each product within the family. Kumar et al. (27) proposed a new market-driven product family design (MPFD) approach to study the impact of increased variety in product offerings in different market segments and to explore the cost savings associated with communal decision-making. Xu et al. (28) proposed an information integration modeling architecture for the full life cycle of product families. Geng et al. (29) employed Quality Function Deployment (QFD) to translate demand attributes into engineering characteristics, thereby offering sustainable functional solutions for product design aligned with customer needs. In response to varied client expectations, abbreviated product development cycles, and cost constraints compelling manufacturing firms to adopt mass customization, product family design emerges as an effective strategy (30). The subsequent rise in the number of companies vying to provide a wide array of tailored services alongside customized products to enhance revenues and customer satisfaction. The application of new concepts such as service families and service platforms to the service industry through the use of the product family design methodology, the increasing diversity of service offerings leading to complexity and difficulty in estimating service costs. Tay et al. (31) proposed a service family cost estimation method based on service modularization and the job costing approach. Traditional product family design is a single-objective optimization problem. Du et al. (32) present a complex type of leader-following-joint optimization problem involving multiple decision makers encompassing different levels of decision hierarchies, consisting of many conflicting objectives competing to reach an equilibrium solution. Wu et al. (33) proposed an evolutionary planning model that has a stronger response to product evolution and can maximize firm performance in effective time. In response to escalating demand for personalization, a growing number of service businesses must provide customized service offerings while improving customer satisfaction and service quality in a cost-effective manner. The open product design allows for the incorporation of personalization modules into the product structure, thus meeting the unique requirements and preferences of individual clients. Tan et al. (34) proposed an optimization method that integrates individual consumer preferences via a genetic algorithm for individualized module allocation. Zhou et al. (35) proposed a hierarchical joint optimization model for the design of personalized service product families based on service resource families considering crowdsourcing of service operations, which was solved using a nested genetic algorithm to obtain a scheme for personalized service products. Feldman et al. (36) examined a classification optimization problem aimed at selecting the sequential presentation of products to maximize anticipated revenues. This approach breaks with the conventional product portfolio framework by focusing on the sequence of product portfolios presented in a customer-centric manner. Zhang et al. (37) consider uncertainty in service utility and customer behavior to maximize expected customer satisfaction as well as sales profit. Therefore, in the design of the service product family, we should focus on considering the customer’s preference, the probability of the customer’s choice of the products provided by the enterprise, and the related costs into the architecture system to ensure that the designed product architecture can be loved by the customer at the same time to achieve the goal of the enterprise to obtain profits.

A review of the literature on older adults care services and product family design indicates a significant rise in demand for services catering to the aging population, with home older adults care services demonstrating efficacy in mitigating the challenges faced by the older adults. To accommodate consumer preferences while minimizing operational expenses, product family design effectively addresses the elevated costs faced by companies due to the diverse customer requirements and the extensive range of home care services available in the market. This paper addresses the scarcity of studies on product family design for older adults care services by proposing a research methodology that prioritizes customer preferences for individual services. It emphasizes demand as a driving force, integrating the likelihood of customer selection of service products offered by the enterprise and associated costs into the design architecture. The approach aims to create a subset of service packages that maximizes profit while fulfilling the needs of the customer groups, concurrently managing individual services to address specific requirements, thereby enhancing the quality of life for the older adults and alleviating their challenges.

3 Principles and methods 3.1 Problem description

This study addresses the optimization of the design of a range of older adults care service goods within a category of privately funded operation models. In this type of problem, the agency selects to offer in-home care services in a designated region with a known clientele size. After identifying the individual services in alignment with national standards, it amalgamates those with high demand and frequent delivery. Additionally, it evaluates the optimal array of older adults care service products from various options, constrained by limited resources. This article addresses the issue of the product family design program shown in Figure 1. This paper proposes a product family design for older adults care services that addresses the selection of an optimal package subset. The market offers a variety of service items, which can be categorized into distinct modules based on their functional characteristics, thereby organizing the service items systematically and enabling the segmentation of customer demand within the target market. According to the Kano model, the corresponding services under the service module can be categorized into attractive attributes A, one-dimensional attributes O, must-be attributes M and irrelevant attributes I. Demand characteristics for individual services vary across distinct market segments. By combining service modules with corresponding service items, a package collection is created. Subsequently, packages are categorized based on price, allowing for the estimation of demand at each level within various market segments. The firm’s operational constraints are integrated by considering customer preferences, while essential configuration factors, including module type, customer demand, market competition for a singular service, selection probability, and variable versus fixed costs, are included to maintain the subset of packages that optimize the firm’s profitability during standard operations of a single service.

Figure 1. Design of a product family for older adult services.

The symbolic description used to represent the sets and quantities involved in the design problem for this type of older adults care service product family is shown in Table 1.

Table 1. Symbolic description.

3.2 Gathering customer requirements

In reality, the service coverage area of an in-home older adults care business is established. Due to the variability in the physical conditions and service requirements of the older adults across different regions, it is imperative for each older adults care service organization to conduct a survey within its jurisdiction to assess the physical conditions and service needs of the older adult population, subsequently enabling the design of tailored older adults care service product families. A questionnaire survey was undertaken specifically for the older population in a designated area, employing a stratified random sampling method to guarantee that each eligible older adult individual had an equal opportunity to be picked for participation. The questionnaire was developed through interviews with pertinent service-oriented enterprises, examining the preliminary classification and statistical methodologies of service demand in the home care sector. The questionnaire was formulated following an extensive analysis of interview results, national standards for home care services, and variables such as the physical health, economic capacity, and age demographics of the older adults in the targeted operational region of the company. Due to the primary demographic of the questionnaire being the older adults, the execution of an online survey poses significant challenges. Therefore, a hybrid approach of both online and offline methods is employed. Additionally, considering the varying physical conditions of the older adults, those capable of completing the questionnaire independently may do so, while those facing difficulties may utilize either a question-and-answer format or have family members complete the questionnaire on their behalf.

The data collected from the questionnaire was utilized to classify the characteristics of each service item using the Kano model. The must-be attributes of a service are the minimum requirements that must be met to provide basic life care. One-dimensional attributes are additional aspects of the service that add value and enhance the competitiveness of similar products after the must-be attributes have been satisfied. Attractive attributes are extra elements that aim to surprise and attract users. Irrelevant attributes are optional and do not significantly impact needs or preferences. Figure 2 displays the attributes of each category of demand in the Kano model. The horizontal coordinate is the level of sufficient, and the vertical coordinate is the level of satisfaction.

Figure 2. Kano model.

The questionnaire survey was conducted to assess the expectations of the senior population regarding different home services for the older adults. Let η be the size of the sample (38). The η calculation formula is shown in Equation 1.

η=μ∂/22σ2φ2    (1)

μ∂/2 is the standardized score. The value reflects the confidence level. φis the absolute error, the overall variance is σ2=ρ1−ρ, ρ is the proportion of indicators. Upon determining the sample size, the questionnaire was disseminated to gather data. The collected data were then analyzed using SPSS 25 software to ensure that the questionnaire structure was reliable and valid. Descriptive statistical analysis was conducted to determine the proportion of relevant parameters.

3.3 Computation of utility functions

Based on the pre-processing of the questionnaires obtained from the acquisition of the needs of the older adults, the physical condition of the older adults in the region was analyzed, and the market was subdivided based on self-care into MS=MS1,⋯,MSf. If the traditional Kano model is followed, the attributes of the service line are determined based on the attribute category that accounts for the most; many in-home senior care services have attractive attributes. This is not beneficial for evaluating the market and providing customers with tailored service programs that meet their actual needs. Therefore, it is important to introduce specific satisfaction and dissatisfaction indices for each service program in different market segments to assist in categorizing the types of demands. The satisfaction index of the jth service under the ith module under the rth market segment is represented by the variable csijr, whereas the discontent index is represented by the variable cdijr. The csijr and cdijr calculation formulas are shown in Equation 2 and Equation 3 (35).

csijr=Aijr+OijrAijr+Mijr+Iijr+Oijr    (2) cdijr=−1Mijr+OijrAijr+Mijr+Iijr+Oijr    (3)

Aijr,Oijr,Mijr,Iijr represent the number of services classifying the jth service under the ith module in the rth segment as attractive attributes, one-dimensional attributes, must-be attributes, and irrelevant attributes according to customer demand.

Due to the nascent nature of the home care service industry, the market system is imperfect. Additionally, traditional beliefs have led to a relatively low demand for home care services for the older adults. Hence, it is imperative to consider the magnitude of the satisfaction index and dissatisfaction index when assessing the level of customer satisfaction for each service item. This analysis is crucial for enhancing service quality and catering to consumer demands. The satisfaction level is determined by the distance from the origin to the point (csij, |cdij|), where csij and |cdij| are both between (0,1) and serve as the vertical and horizontal coordinates, respectively, (35), distance range between 02. In order to eliminate the effect of physical magnitude, the distance is mapped between (0,1) as the satisfaction value. Let sijr represent the customer satisfaction of the jth service within the ith module in the rth market segment, and let ŝijr represent the value of the customer satisfaction mapping. The sijr and ŝijr calculation formulas are shown in Equation 4 and Equation 5.

sijr=csijr2+cdijr2    (4)

A higher value of sijr indicates that the service has a stronger influence on user satisfaction and is of greater significance. The weight of each candidate module varies across different segments. The weight of a module is determined by standardizing the satisfaction rating of the service item associated with that module. Let wirrepresent the weight of the ith module in the rth segment. The wir calculation formula is shown in Equation 6.

wir=ŝmaxr−ŝminr∑i=1nŝmaxr−ŝminr    (6)

ŝmaxr=maxŝijr|i=1,2,⋯,n,j=1,2,⋯,m, ŝminr=minŝijr|i=1,2,⋯,n,j=1,2,⋯,m, ∑i=1nŝmaxr−ŝminr is the total of the differences between the highest and lowest levels of satisfaction in the related service items of all modules.

Enterprises will carefully evaluate the overall cost of providing services when operating service products. Decreasing service costs is beneficial in reducing the price of service products, thereby enhancing the competitiveness of enterprises. This increases the opportunities for enterprises to operate in emerging markets and mitigates market risks. Cost-based pricing is the simplest method of determining prices, where corporations set product prices based on the predicted profit margins associated with the cost of each service item (39). Let pijrepresent the price of the jth service within the ith module, taking into account the firm’s intended profit margin ε. The pij calculation formula is shown in Equation 7.

ε represents the anticipated profitability of the company, while cijV denotes the variable cost associated with each individual service item.

Adjusting the package price based on the economic capacity of the older adult population in the chosen area, examining the usage of individual services by customers through a questionnaire, and determining that the price of the package in various market segments also influences customer satisfaction. Consequently, customers have distinct target prices for packages, denoted as paim. Additionally, enterprises categorize packages as LE=LE1,⋯,LEh based on the price ceiling, which effectively guarantees the demand for these packages. The price of the package has an impact on the customer’s perceived utility. Each customer segment has a specific price p for the package. If the package price is higher than the target price, it will decrease customer satisfaction. Conversely, if the package price is lower than the target price, it will increase customer satisfaction. Therefore, the package target price can be used as a benchmark to assess the gains and losses (40). Let the benefit be represented by the ek and the damage be represented by the bk. The ek and bk calculation formulas are shown in Equation 8 and Equation 9.

ek=">wirsijrxkij−okwirsijrμ1−ykijM/∑i=1n∑j=1mxij∗1+akr    (17)

wirdenotes the weight of the ith module of the rth market, sijr denotes the customer satisfaction level of the jth service under the ith module of the r th market segment.

3.4 Determination of product requirements

The probabilistic choice rule is a more suitable method for estimating product demand, as it takes into account the size of the target market and the likelihood of customers choosing specific service offerings. This approach is more aligned with choice forecasting, making it probabilistic for customers to select packages. Within the collection of packages PF1,…,PFg, each package is linked to a customer’s likelihood of selection, which is quantified in this study as the customer utility of the service item (34). Let tkr represent the chance of selecting the kth package inside the rth market segment. The tkr calculation formula is shown in Equation 18.

tkr=expukrexpu0r+∑k=1g0expukr    (18)

ukr represents the utility of the kth package for the rth market segment. u0r is the utility of the single service chosen by the customer of the rth market segment. This utility should be a constant as the customer selects the single service based on their specific needs. In this study, the variable u0r is determined by calculating the percentage of respondents who express their willingness to use an individual’s service in a survey. Within a specific market, the proportion of individuals willing to use each service at a specific price is calculated. These proportions are then added together and averaged to determine the utility value of a single service for customers in that market. ∑k=1g0expuk is that the firm ultimately selects the most economically efficient subset of packages PF1′,…,PFg0′ from the complete set of packages.

The questionnaire data provides statistics that allow for determining the size of each market segment. Each segment will have a demand for the package. This paper estimates the demand for the package by multiplying the customer market size with the probability of choosing the package. Let dk be the market demand for the kth package. The dk calculation formula is shown in Equation 19.

dk=∑r=1fqrtkr    (19) 3.5 Profit function calculation

Profits in this study are defined as the difference between revenues and costs. Companies select a specific number of the most optimized subset of packages to offer. These packages combine various individual services and are available for customers to order every month within a particular budget. Each service has a different frequency of use within a month. Implementing monthly subscriptions facilitates cost efficiency, mitigates demand uncertainty, and enhances resource optimization. Companies offering packages to gain high demand will have a discount rate δ to reduce prices and attract more customers. Let the profit of the kth package be denoted as prk. The prk calculation formula is shown in Equation 20.

prk=dk1−δ∑i=1n∑j=1mxkijpijzij−∑i=1n∑j=1mdkcijVxkijzij−ckF    (20)

pijrepresents the price of the service item, dk represents the demand for the kth package, and zij represents the frequency offered by the monthly package of service items.

3.6 Older adult service product family design optimization mode

Older adult service product family design is mathematically expressed as a mixed-integer programming model. The goal is to maximize profits by choosing the best combination of packages PF1′,…,PFg0′ to offer to customers. Profits are influenced by market demand, variable costs linked to care workers and materials, fixed expenses tied to equipment and staff involved in the provided packages, and the price of each service product. From the customer’s perspective, combining service products into a package is cheaper than purchasing each service product separately. The more variety of packages available, the higher customer satisfaction tends to be. However, as the number of package options increases, the demand for each package naturally decreases. When the demand cannot cover the cost of the package, the company’s profits will decline. From a firm standpoint, higher prices or lower costs associated with designing the senior care service product family can lead to increased profit margins. However, if the product is priced too high or specific features are compromised to cut costs, customer preference for the product and market demand will decrease. Comprehensive analysis: in order to maximize corporate profits, the enterprise must select a specific number of subsets of packages while ensuring market demand. The objective is to design an older adults service product family that takes into account the interaction between fixed costs, variable costs, product prices, and market acceptance, and to find an equilibrium point, establish a service product package structure model.

maxpr=∑k=1gokdk1−δ∑i=1n∑j=1mxkijpijzij−∑k=1g∑i=1n∑j=1mokdkcijVxkijzij−∑k=1gokckF    (21) s.t.∑k=1gok=g0,ok∈01,k=1,2,⋯g    (22) ∑i=1n∑j=1m|xkij−xk′ij|>0,∀k≠k′    (23)

Equation 21 refers packages are ordered on a monthly basis, the frequency of provision of individual service items in the monthly packages is determined to maximize the total profits of all packages. The variable cost is the direct cost assigned by the firm for offering individual services, and the total variable cost is determined based on the demand for the selected packages. The total fixed cost of a package offered by the enterprise remains constant regardless of market demand. It represents the constant expenses incurred by the enterprise in its regular operations, such as replacing old factory equipment with new ones and providing regular salary payments to management staff. By older adult service product family design, the enterprise ensures long-term stability and meets a certain level of demand to obtain profits. According to the changes in market size, the actual demand for packages will change. When the market size of the case increases, the actual demand for packages will increase, and the enterprise needs a larger site to provide services but also needs more caregivers; when the market size of the case is smaller, the situation is reversed. Based on the current status, enterprises can reduce wastage of resources such as personnel wages and rent by making efficient use of resources. This will help in reducing fixed costs and enable the enterprise to offer package deals at a lower price than individual services. Equation 22 indicates that the constraints applied to the optimization include limiting the number of package types in the product portfolio to g0. Equation 23 restricts each package to be unique.

3.7 Model solving

The optimization model presented in this paper constitutes a single-objective optimization problem as the individual modules of the service items are selected randomly. Unlike other manufactured products, the composition of the package does not necessitate specific modules, and the service items associated with these modules are also chosen at random. There are no constraints on the number of modules included in the package or the corresponding service items. Consequently, when the quantity of service product modules and items is substantial, the problem’s size is demonstrated to be extensive through enumeration, categorizing it as an NP-hard optimization problem. Genetic Algorithm (GA) is very suitable for problems with large solution spaces and low efficiency of numerical algorithms.

GA is widely used in many product family design problems (7, 34, 35, 37). This study presents a single-objective optimization genetic algorithm designed to solve the problem of older adult service product family design. The genetic algorithm flow is shown in Figure 3. The specific design steps of the genetic algorithm are as follows:

Figure 3. Genetic algorithm flow.

Step 1: Configuring the parameters. Specify the parameters for the product family design of the older adult service product, such as the number of modules and the matching list of service items for each module. Additionally, establish the population size, maximum number of iterations, crossover chance, and mutation probability.

Step 2: Chromosome encoding. Every chromosome is a binary sequence consisting of 0 and 1 that represents the composition of the package. The length of the chromosome is determined by the cumulative amount of service items across all modules. Each place on the chromosome can have a value of either 0 or 1, indicating whether the related service item is chosen or not.

Step 3: Initialize the population. Generate an initial population randomly, taking into account the decision variables of the optimization problem.

Step 4: Involves performing crossover and mutation operations. Tournament selection is employed to choose a fresh generation of people from the population that has undergone two-point crossover and bit-flip mutation processes. Specifically, in each round, the most successful individuals are chosen from a randomly chosen subset of the population.

Step 5: End the iteration. Firstly, ascertain whether the constraints are met. If they are, proceed to assess the fitness value of the individuals in the population. Secondly, verify if the satisfaction and gain have reached the maximum number of iterations. If they have, terminate the iterations and document the optimal solution and its corresponding value at that point.

4 Example analysis 4.1 Background and data

The case pertains to a home care service firm planning to establish a new home care service station on Huangcheng Street, Shenhe District, Shenyang City, encompassing all neighborhoods inside Huangcheng Street as its service area. China released ‘the Basic Norms for Home-Based Older adults Ho