Dynamic Facility Location and Allocation Optimization for Sustainable Product-Service Delivery Using Co-Evolutionary Adaptive Genetic Algorithms

Abstract

Product-service systems contribute to sustainable development through innovative service integration and novel customer value creation. However, the competitive advantage of sustainable product lifecycle service delivery hinges critically on the operational efficiency of service networks. This study addresses dynamic service facility location and allocation challenges in a time-varying demand environment, focusing on the strategic deployment of multiple comprehensive service centers (CSCs) and their dynamic customer allocation across planning horizons. In this study, we develop a 0–1 integer programming model and propose a novel co-evolutionary adaptive multi-objective genetic algorithm (CA-MOGA) with four key enhancements: (1) optimized chromosome representation, (2) adaptive strategy incorporation, (3) genetic operators with gene repair mechanisms, and (4) elite trans-generation migration. Through real-world case validation, CA-MOGA demonstrates significant improvements over conventional genetic algorithms in both convergence speed and solution quality. The performance and adaptability of the proposed algorithm suggest strong potential for customizable applications in solving diverse complex optimization problems.

1. Introduction

One of the dominant trends in manufacturing industries is that the industry manufacturers are trying to extend the value chain and transform themselves from manufacturing and goods-oriented organizations to service-oriented organizations for long-term success [1,2]. The product-service system that uses service strategies to replace product ownership can reduce the waste of materials and resources by product sharing, renting, and leasing to facilitate sustainable production and consumption [3,4]. The product-service system enhances sustainability by adding innovative services and new value creation to customers. Recently, manufacturers’ enterprises have tried to increase sustainability by decreasing the energy consumption of their systems both in production and service offerings [5]. For instance, companies like Rolls-Royce use IoT-enabled “Power-by-the-Hour” services for jet engines, where customers pay per flight hour while Rolls-Royce monitors performance, optimizes maintenance, and reduces downtime. This reduces material waste and extends product life [6]. Ingersoll-Rand, a leading provider of air compressors, is transitioning from a traditional product manufacturer to a comprehensive product-service system provider [7]. Under this model, customers no longer simply purchase equipment but instead subscribe to integrated service packages that include maintenance, monitoring, and optimization. This shift not only reduces the total lifecycle cost for customers but also enhances sustainability by minimizing energy consumption and waste. Through real-time condition monitoring, predictive maintenance, and energy efficiency management, Ingersoll-Rand ensures optimal equipment performance while extending product lifespan and reducing environmental impact.

The competitiveness of sustainable product lifecycle service delivery critically depends on service network efficiency [8,9]. Consequently, traditional after-sales networks lack the flexibility to support evolving and complex product-service delivery. To sustain the product’s ability to continuously create value for customers, the product-service provider must possess robust service delivery capabilities. This capability relies not merely on spare parts, maintenance personnel, or training staff alone, but rather on the integration and optimization of these resources [10]. Such integration necessitates that Comprehensive Service Centers (CSCs) dynamically allocate, combine, and optimize diverse service resources according to evolving customer needs. The quantity and geographic placement of CSCs significantly influence both customer satisfaction and provider competitiveness. A growing number of companies are incorporating sustainability objectives into their service offerings [11,12]. In sustainable product-service delivery, optimizing both the number of service facilities and the allocation of customers to these facilities is crucial for operational efficiency [13]. Facility location problems primarily focus on determining the optimal number and placement of facilities (e.g., warehouses, service centers) and assigning customer demand to these facilities, often with the objective of minimizing transportation or operational costs [14]. In contrast, the supply chain network design for sustainable product-service systems encompasses a holistic integration of product flows, service provisioning, environmental impact, and social responsibility. It addresses multi-layered decisions often including reverse logistics, maintenance operations, life cycle cost, and environmental performance [15].

The studies specifically addressing facility location and allocation problems in the context of sustainable product-service systems remain limited. Moreover, due to challenges in chromosome encoding and genetic operator design, genetic algorithms (GAs) have seen limited application in this domain. This study tackles the dynamic multi-criteria service facility location and allocation (SFLA) problem by proposing a framework for establishing multiple comprehensive service centers (CSCs) in time-varying demand environments. The model dynamically reassigns customers to CSCs across different planning horizons. We formulate a 0–1 integer programming model and develop a co-evolutionary adaptive multi-objective genetic algorithm (CA-MOGA) to solve the problem. A real-world industry case analysis demonstrates that CA-MOGA outperforms conventional GAs and the Particle Swarm Optimization (PSO) algorithm in terms of convergence speed and solution quality.

While co-evolutionary and adaptive mechanisms have been explored individually in prior studies, our work integrates four novel enhancements that distinguish our approach: (1) a dynamic chromosome representation scheme suitable for multi-period location-allocation decisions; (2) the adaptive strategies, including the adaptive objective functions and adaptive fitness function, as well as the adaptive crossover and mutation probabilities for performance enhancement of the CA-MOGA; (3) constraint-preserving genetic operators with embedded gene repair mechanisms; and (4) a buffer-based elite trans-generation migration strategy that enhances diversity and convergence.

The remainder of this paper proceeds as follows: The paper provides a brief overview of SFLA problems in Section 2. In Section 3, the assumptions, notations, and the mathematical programming model are proposed. In Section 4, the CA-MOGA is explained in detail. The proposed model is applied to a real case study, and the results are discussed in Section 5. Conclusions are summarized in Section 6.

2. State-of-the-Art Reviews

2.1. Service Facility Location and Allocation

The research on location theory was formally begun by Weber [16] in the decision on how to position a single warehouse to minimize the total distance. Following this initial study, a rich body of articles has addressed the facility location and allocation problems. Turkoglu and Genevois [17] investigated the service facility location problem, considering 19 characteristics of the candidate facilities. Zhang and Mao [18] investigated the rural express logistics network design using an integrated method where the Holt–Winters model was used for demand prediction. The competitive facility location problem was explored by Ahmadi and Ghezavati [19]. In this study, an accelerated Benders’ decomposition was used to solve the proposed model. López et al. [20] proposed an MILP formulation for the multi-period planning problem with minimum purchase commitment contracts faced by the shipper. Zhang et al. [21] investigated the hierarchical facility location problem with drone recharging stations for urban last-mile delivery using a novel mixed-integer programming model. The p-median quadratic facility location problem was explored by Yang et al. [22]. In this study, a Benders decomposition method using a semi-definite programming relaxation within a branch and bound framework was developed to solve the proposed model. A bi-objective generalized mathematical model was developed by Kale et al. [23] for the comprehensive planning of municipal solid waste management facilities while minimizing cost and infrastructure undesirability. A non-linear 0–1 bilevel programming model was established by He et al. [24] to solve the competitive facility location problem involving a leader-follower game under a two-nest nested logit model, aiming to maximize the leader’s revenue while anticipating the follower’s optimal response. The facility location and fortification problem was explored by Hu et al. [25]. In this work, three two-stage robust optimization models with decision-dependent uncertainty were developed and solved using an improved column-and-constraint generation algorithm.

A bi-objective mixed-integer non-linear (MINLP) model was developed by Abbasi et al. [26] to design the hub-and-spoke network while minimizing the cost and time of the supply chain simultaneously. Joneghani et al. [27] addressed the sustainable location-allocation problem in medical waste management and proposed an MILP model under uncertainty. Punyim et al. [28] studied the two-echelon multi-period multi-product location-inventory problem with partial facility closing and reopening. An MINLP was proposed and the tabu search heuristics were adapted to solve this problem. Tang and Wu [29] investigated the reliable capacitated facility location problem with a single source constraint. A multi-objective fuzzy facility location problem with congestion and priority for drone-based emergency deliveries was investigated by Wang et al. [30]. The chance-constrained, second-order conic, fuzzy, and weighted goal programming approaches were used to recast the model as a crisp mixed-integer second-order conic program.

2.2. Supply Chain Network Design for Sustainable Service Delivery

The recent literature is also dedicated to the study of the supply chain network design problem considering sustainable service delivery, like maintenance, repair, and recycling. Mishra and Singh [31] addressed the problem of designing a multi-country production-distribution network that provides services such as repairs and remanufacturing. A MINLP model was developed to design this production-distribution network. The maximal covering location problem was investigated by Medrano-Gómez et al. [32] to design a sustainable recycling network using an MILP model. An MINLP model was established by Shi et al. [33] to design a household e-waste collection network to maximize the amount of collected e-waste and minimize the overall costs of developing the network and its advertising campaign. Gharibi and Abdollahzadeh [34] addressed the green reverse logistics network design problem for after-sales service for mobile phones and digital cameras. In this study, an MILP model was developed to maximize the network’s total profit by calculating the difference between costs and revenue. Altekin et al. [35] presented an MILP model to determine warehouse locations, assign repair vendors to facilities, and choose the mode of transportation while minimizing the total network cost. A robust optimization approach with a penalty limit constraint and robustness index was proposed by Xiang et al. [36] for a robust parcel delivery network design. A decomposition method with valid cuts was developed to solve the problem. Momenitabar et al. [37] investigated a closed-loop blood supply chain network considering blood group compatibility and shelf life constraints, where a fuzzy multi-objective MINLP model was developed to simultaneously minimize total costs and maximize the minimum hospital service level. Wang et al. [38] investigated a community healthcare network design problem with a 2-service framework and hospital congestion structures, aiming to minimize total expected cost. An adaptive distributionally robust optimization approach was proposed to handle uncertainties in resident medical demands. A distributionally robust joint chance constrained programming model was developed by Zang et al. [39] to optimize the service network design under demand uncertainty while controlling the overall service level. Mirzaei et al. [40] examined the integration of service reliability into freight network design under disruption and demand uncertainties, proposing a hybrid two-stage stochastic chance-constrained programming model (RSND) and an accelerated Benders decomposition algorithm with multi-cut enhancements and valid inequalities.

A growing body of work has examined SFLA problems in service delivery supply chains. While existing research has advanced facility location theory, several critical gaps remain. First, most models neglect the dynamic service needs of product-service systems. Second, sustainability objectives (e.g., servitization-driven revenue) are often excluded from optimization criteria. Third, conventional algorithms lack adaptive mechanisms to accommodate dynamic constraints, often resulting in suboptimal convergence or infeasible solutions. Moreover, research specifically addressing sustainable SFLA remains notably scarce. The provision of sustainable product-service offerings, which integrate service-enhanced products, requires enterprises to develop robust service capabilities. Addressing these gaps is particularly crucial for industries undergoing servitization, where operational efficiency depends on the sustainable reallocation of service resources. This entails strategically deploying comprehensive service centers and implementing dynamic customer allocation mechanisms to meet escalating service demands effectively. Our work addresses these challenges by proposing a dynamic SFLA model with CA-MOGA, integrating time-varying demand, multi-objective sustainability metrics, and adaptive genetic operators.

3. Model Formulation

3.1. Problem Description

When manufacturers sell their equipment, they commonly provide comprehensive product-service packages that incorporate installation and commissioning services (like on-site setup and operational fine-tuning), training programs (like operational and maintenance procedures), maintenance solutions (like on-site repairs and spare parts replacement), and other value-added services like periodic inspections, software and hardware updates, and performance optimization.

These services are provided continuously throughout the equipment’s complete operational lifespan, from initial commissioning through to final decommissioning. It is important to recognize that product-service delivery represents an ongoing engagement rather than a single transaction, developing progressively as the equipment remains in active use. Service requirements fluctuate dynamically during the planning period, being shaped by three primary factors: the existing installed equipment base, new equipment sales volumes, and the implementation level of servitization strategies.

An additional consideration is the geographical dispersion of customers, as shown in Figure 1. To maintain consistent service quality across all locations, manufacturers must strategically establish CSCs during the planning phase and implement dynamic customer allocation systems. The setup and ongoing operational expenses for these CSCs differ significantly by location, while their service capabilities are fundamentally limited by available physical space, specialized tools, qualified personnel, and other critical resources. Before presenting the formulation of the problem, the assumptions are listed below:

• A customer is only served by one CSC in a time period [41].
• Once a CSC is opened, it shouldn’t be closed.
• The total number of the CSCs is not predetermined.

3.2. Parameters and Decision Variables

The assumptions constitute the preconditions of the dynamic SFLA problem formulation. The following notations are presented to describe the 0–1 formulation.

Parameters:

• $I:$ set of customers.
• $J:$ set of CSC.
• $T:$ set of periods.
• $d_{ij}:$ distance between customer j and i.
• $v_{∆ti}$: added product sales in customer i in the time period $t+1$ compared with period $t$.
• $v_{0i}:$ service demand of customer i in the time period $t_0$.
• ${\alpha }_t:$ index indicating the extent of the product servitization in the time period t,${\mathsf{\alpha }}_{\mathrm{t}}\in$ (0,1].

It exhibits a positive correlation with service intensity, where higher values indicate greater service commitments to customers. The scale ranges from α = 0, representing pure product sales with minimal service obligations, to α = 1, denoting complete servitization where the provider’s revenue stream is predominantly service-driven.

• $\beta :$ average profit of a completed servitized product.
• ${cs}_i:$ fixed cost for opening a CSC in customer i.
• ${co}_i:$ average annual operation cost of a completed servitized product in customer i.
• $V_{max}$: maximum service capacity of CSC.
• $d_{max}:$ maximum allowable distance between customer i and CSC.

Variables:

• $m_t$: total number of CSCs in the time period t.
• $x_{ti}$: 1 if an opened CSC is in customer I at the beginning of period $t$, 0 otherwise.
• $S_{tij}:$ 1 if the customer i is assigned to CSC j at the beginning of period $t$, 0 otherwise.

3.3. Objective Functions and Constraints

Thus, the formulation of the dynamic SFLA problem is given as follows:

$$\mathrm{M}\mathrm{i}\mathrm{n} Z_1=\sum _{t=1}^s\sum _{j=1}^{m_t}\sum _{i=1}^n{\left(v_{0i}+v_{∆ti}\right)d}_{ij}{S}_{tij}$$

(1)

where

$$m_t=\sum _{i=1}^n{x}_{ti}$$

(2)

$$\mathrm{M}\mathrm{a}\mathrm{x} Z_2=\sum _{t=1}^s\sum _{i=0}^n(v_{0i}{\alpha }_0+v_{∆ti}{\alpha }_t)\beta -\sum _{i=1}^n{cs}_i{x}_{ti}-\sum _{t=1}^s\sum _{i=1}^n{co}_i\left(v_{0i}+v_{∆ti}\right)S_{tij}$$

(3)

Subject to

$$\sum _{i=1}^n(v_{0i}+v_{∆ti}) S_{tij}\le V_{max}, \forall j\in J, t\in T$$

(4)

$$\sum _{j=1}^{m_t}{S}_{tij}=1,\forall i\in I, t\in T$$

(5)

$$S_{tij}\le x_{ti}, \forall i\in I,j\in J, t\in T$$

(6)

$$d_{ij}{S}_{tij}\le d_{max},\forall i\in I,j\in J, t\in T$$

(7)

$$x_{ti}\left(1-x_{ti}\right)=0,\forall i\in I, t\in T$$

(8)

$$S_{tij}\left(1-S_{tij}\right)=0,\forall i\in I,j\in J, t\in T$$

(9)

$$S_{tij}=1,\forall i\in I,j\in J,i=j$$

(10)

$$\sum _{t=k}^s{x}_{ti}=s+1-k,\forall x_{ki}=1$$

(11)

Objective function (1) minimizes the total weighted service distance between CSCs and their assigned customers (Equation (1)). Equation (2) calculates the total number of CSCs established per period. Objective function (2) maximizes total service profit (Equation (3)), comprising three components: service revenue from delivered services (first term), fixed costs of opening CSCs (second term), and operational costs that scale with servitized product volume (third term). Constraint (4) enforces CSC capacity limits. Constraint (5) ensures single-period single-assignment for each customer. Constraint (6) mandates customer allocation exclusively to operational CSCs. Constraint (7) requires CSC presence at any customer location serving others. Constraint (8) maintains service-level agreements through maximum distance guarantees. Constraint (9) restricts CSC colocation to at most one per customer site. Constraint (10) enforces single-source service provision. Constraint (11) binds customers to receive service from collocated CSCs. Constraint (12) prevents CSC decommissioning after initial opening.

4. Proposed Algorithm: A Co-Evolutionary-Based Adaptive Multi-Objective Genetic Algorithms

GAs represent a prominent class of evolutionary algorithms extensively employed for multi-objective optimization problems. This study develops a CA-MOGA to address the dynamic SFLA problem. Drawing from biological co-evolution principles [42], namely the reciprocal evolutionary changes among interdependent species, we implement a multi-group co-evolution strategy at the population level. The algorithm incorporates adaptive mechanisms to optimize the genetic search process, with specific enhancements implemented to improve computational efficiency.

4.1. Chromosome Representation

The solution encoding employs a chromosome structure consisting of s real-valued sub-chromosomes, each containing n genes (Figure 2). Each sub-chromosome t encodes three key elements: (1) the quantity of CSCs, (2) their spatial locations, and (3) customer-CSC assignments for time period t. The gene value specifies the CSC location, while the gene position indicates the corresponding served customer. For instance, in Figure 2, the second gene value of 26 in sub-chromosome t′ denotes that customer 2 receives service from the CSC located at customer 26. According to constraints (7) and (8), the value of a gene in sub-chromosome t + 1 is equal to that in sub-chromosome t′ if there is a CSC opened in the customer.

4.2. Adaptive Strategy

4.2.1. Adaptive Objective Functions

The proposed chromosome representation satisfies all constraints except (4) and (7). Following the constraint-handling approach proposed by Joines and Houck [43], we first convert constraints to standard form:

$$f_p\left(x\right)\le 0, p=\mathrm{1,2},\cdots q.$$

(12)

q is the number of constraints. ${\mathrm{p}}_{\left(\mathrm{x}\right)}$ is the dynamic penalty defined by Joines and Houck [43],

$$p_{\left(x\right)}={\left(c\times g\right)}^{\phi }\sum _{p=1}^q{f_{p\left(x\right)}}^{\omega }$$

(13)

$c,\phi$ and $\omega$ are constant. g is the generation number.

However, traditional penalty methods face two key limitations [44]: (1) difficulty in parameter tuning and (2) significant value disparity between penalty and objective functions. Moreover, chromosomes with identical penalties may contain different quality genetic patterns. To address these issues, we propose an innovative adaptive penalty scheme. The revised objective function ${Z_{1\left(x\right)}}^{\prime }$ and ${Z_{2\left(x\right)}}^{\prime }$ are as follows:

$${Z_{1\left(x\right)}}^{\prime }=Z_{1\left(x\right)}+{\displaystyle \frac{p_{\left(x\right)}}{\sqrt{{p_{\left(x\right)}}^2+1}}}{Z}_{1\left(x\right)}{(u1+u2)}^{\beta } ,\beta \in (\mathrm{0,1})$$

(14)

$${Z_{2\left(x\right)}}^{\prime }=Z_{2\left(x\right)}+{\displaystyle \frac{p_{\left(x\right)}}{\sqrt{{p_{\left(x\right)}}^2+1}}}{Z}_{2\left(x\right)}{(u1+u2)}^{\beta } ,\beta \in (\mathrm{0,1})$$

(15)

The variables $u1\mathrm{a}\mathrm{n}\mathrm{d}u2$ represent the violation frequencies for constraints (4) and (7), respectively. The penalty term $p_{\left(x\right)}$ quantifies the degree of constraint violation in the current solution. To mitigate the penalty’s impact, we introduce a damping coefficient ${\displaystyle \frac{{\mathrm{p}}_{\left(\mathrm{x}\right)}}{\sqrt{{{\mathrm{p}}_{\left(\mathrm{x}\right)}}^2+1}}}$, which asymptotically limits the maximum penalty influence. The term ($u1 \mathrm{ a}\mathrm{n}\mathrm{d } u2$) serves as a violation counter, where higher values indicate more severe constraint violations. This formulation reflects that solutions with greater violations typically contain inferior genetic patterns. The attenuation factor $\beta$ is incorporated to proportionally scale the penalty’s effect on the objective function values.

4.2.2. Adaptive Fitness Function

Following the principle of Pareto-optimal solutions [45], we evaluate chromosomes by comparing their objective function values and assigning population ranks. To account for varying objective importance in real-world decision-making, we propose an adaptive fitness function to enhance algorithm convergence speed. For each objective, individuals are ranked according to their modified objective function values (Equations (14) and (15)). The fitness function is defined as:

$$F_z\left(X_j\right)=\left\{\begin{array}{c}{(N+1-Y_z\left(X_j\right))}^2 Y_z\left(X_j\right)>1\\ ({\left({\displaystyle \frac{g}{G}}\right)}^k{+1)N}^2 Y_z\left(X_j\right)=1.\end{array}\right. z=\mathit{1},\mathit{2}\cdots ,W$$

(16)

$$F\left(X_j\right)=\sum _{z=1}^W{\gamma }_z{F}_z\left(X_j\right), j=1, 2, \cdots , n$$

(17)

z is the number of objectives. N is the size of the population. $Y_z\left(X_j\right)$ is the rank of the individual $X_j$ referring to the objective z. g is the current generation number, and G is the maximal generation number. $F_z\left(X_j\right)$ represents the fitness of the individual. $X_j$ referring to the objective z. $F\left(X_j\right)$ is the total fitness of the individual ${ X}_j$. ${\gamma }_z$ is the weight of the objective z. The coefficient $\left({\left({\displaystyle \frac{g}{G}}\right)}^k+1\right)$ is borrowed to improve the fitness of the best solution, where $k>1$. In the initial phase, the difference in the fitness between the optimal individual and others is supposed to be small to prevent an earlier convergence. In the last phase, the difference is suggested to be big to accelerate the convergence of the algorithm.

4.2.3. Adaptive Crossover and Mutation Probabilities

The crossover and mutation probabilities must adapt to fitness variations to prevent algorithm stagnation or premature convergence. The cosine-based functions in Equations (18) and (19) were adopted due to their smooth and non-linear transition properties, which ensure a balanced trade-off between exploration and exploitation. This design mitigates abrupt changes in probabilities, thereby preventing premature convergence while maintaining population diversity [46]. Compared to linear or exponential adaptive schemes, the cosine-based approach offers a more gradual adjustment, which is particularly beneficial for complex, multi-objective optimization problems like the dynamic SFLA. Here, $f^{\prime }$ represents the lower fitness value between two parents during crossover, while it also denotes the selected individual’s fitness during mutation. The parameter $f_{avg}$ indicates the population’s average adaptive fitness. The terms $p_c \mathrm{a}\mathrm{n}\mathrm{d} p_s$ correspond to crossover and mutation probabilities, respectively, where $p_{c1} \mathrm{a}\mathrm{n}\mathrm{d} p_{c2}$ represent the maximum and minimum crossover probabilities, and $p_{s1} \mathrm{a}\mathrm{n}\mathrm{d} p_{s2}$ indicate the maximum and minimum mutation probabilities. For chromosomes with low fitness values, elevated crossover and mutation probabilities ensure robust global search capability. Conversely, for high-fitness chromosomes, reduced probabilities help preserve optimal solutions.

$$p_c=\left\{\begin{array}{c}{\displaystyle \frac{p_{c1}-p_{c2}}{2}}\mathit{cos}\left({\displaystyle \frac{f^{\prime }-f_{avg}}{f_{max}-f_{avg}}}\pi \right)+{\displaystyle \frac{p_{c1}+p_{c2}}{2}}, f^{\prime } \ge f_{avg}\\ p_{c1}, f^{\prime } < f_{avg}\end{array}\right.$$

(18)

$$p_s=\left\{\begin{array}{c}{\displaystyle \frac{p_{s1}-p_{s2}}{2}}\mathit{cos}\left({\displaystyle \frac{f^{\prime }-f_{avg}}{f_{max}-f_{avg}}}\pi \right)+{\displaystyle \frac{p_{s1}+p_{s2}}{2}}, f^{\prime } \ge f_{avg}\\ p_{s1}, f^{\prime } < f_{avg}\end{array}\right.$$

(19)

4.3. Genetic Operators with Gene Repair

4.3.1. Crossover Operator

The complexity of the constraints requires checking and repairing the genes in crossover and mutation to avoid illegitimate individuals. The multi-point crossover operator is performed in three steps:

Step 1: Select the same sub-chromosome from the parents and determine the crossing sections randomly (see Figure 3).

Step 2: Exchange the genes with the same position in the sub-chromosomes.

Step 3: Check the serial number and the changed value of each gene in children. If they are equal, perform gene repair I: given that the kth gene in sub-chromosome t′ is changed, the value of the kth gene in the sub-chromosome t′ + 1 to s should be revised to be k.

The fact that the serial number is equal to the value of the changed gene in the child implies that there is a CSC in the customer, such as the third and the fifth CSC in child 1 in the t = t′ period in Figure 3. Following constraints (7) and (8), the values of the third and fifth genes in sub-chromosome t′ + 1 to s are revised to be three and five, respectively.

Step 4. If they are not equal and the value of the gene is equal to the serial number in the parent, perform gene repair II: given that the crossover happens to the kth gene in sub-chromosome t′, the gene value, which is equal to k from sub-chromosome 1 to t′, should be changed to be the serial number, which promises that a CSC is available with the shortest distance between the customer and existing CSCs in the time period.

For example, there is a CSC in customer four in parent 1 (t = t′). But the CSC is missing in child 1 (t = t′) after crossover. By carrying out the gene repair II, the second and the fourth gene values in sub-chromosome 1 are revised to be eight and thirty-two, respectively, because the CSCs in customers eight and thirty-two are the nearest CSCs for customers two and four in the time period t = 1. Using this method, the legality of chromosomes can be maintained.

4.3.2. Mutation Operator

Mutation generates unexpected features for the children to increase the variability of the population. The mutation is performed with the following Steps 1–3. The result of the mutation is to open new CSCs or change the service relationships between the customers and the CSCs, as illustrated in Figure 4.

Step 1: Select a sub-chromosome from the individual and determine the genes for mutation randomly.

Step 2: Change the values of the chosen genes to be random numbers ($\in [1,\mathrm{n}]$).

Step 3: Check the serial number and the changed value of each gene in children. If they are equal, gene repair I should be performed. If they are not equal and the value of the gene is equal to the serial number before mutation, gene repair II should be performed.

4.4. Buffer-Based Elite Trans-Generation Migration

The elite-preserving strategy is commonly employed to safeguard high-quality individuals from elimination [44,47]. A fundamental assumption in GAs is that optimal individuals possess robust survival capabilities. However, chromosomes containing superior genetic material may not always persist through evolutionary processes [48], particularly during initial generations. To simultaneously maintain promising genetic solutions and enhance population diversity, this study implements a buffer-based elite trans-generation migration approach, which operates through the following procedure:

Step 1: According to the fitness, select the same number of individuals from the current groups separately and construct the elite group (group n + 1).

Step 2: Copy the chromosomes in the elite group to the buffer, which is used to store the elite temporally (group n + 2).

Step 3: Conduct the elite mitigation after x generations: (1) select part of the best chromosomes in the buffer and mix the selections with the n groups separately; (2) remove the bad chromosomes until the size of each undated group is equal to the original group.

4.5. The Procedure of the Proposed Algorithm

The procedures of the proposed CA-MOGA are presented as follows (see Figure 5):

Step 1: Encode the solutions according to the rules mentioned above.

Step 2: Initialize the populations and construct the original elite group (group n + 1), as well as the buffer (group n + 2). The populations are initialized by opening a set of CSCs randomly. If there is a CSC for a customer, it should be assigned to the CSC. Otherwise, the customer is served by the nearest CSC.

Step 3: Perform the co-evolution using genetic operators. As to the selection operator, the tournament selection is employed.

Step 4: Update the elite group and rank the individuals.

Step 5: If g $\le$G, check the irritation times g of the evolution:

• If g $\ne$ x × y, go to Step 3. x is the interval between the migrations. y is a positive integer.
• If g $=$ x × y, carry out the elite migration and update the individuals in the buffer. And then, go to Step 3.

Step 6: If g $=$ G, present the optimal solutions in the elite group.

5. Case Study

5.1. Problem Statement

Company SY ranks among the world’s largest manufacturers of concrete machinery. As a global industry leader, the company is currently transitioning from a traditional product-focused business model to becoming a sustainable product-service provider. The CSCs operated by Company SY deliver various product-related services, including spare parts distribution, maintenance services, and energy consumption management solutions. Service demand is primarily determined by two factors: (1) the total number of operational equipment units in the market and (2) the equipment’s years in service. Industry projections indicate a significant increase in equipment servitization levels in the coming years. The company’s customer base is geographically concentrated across thirty-nine cities, each exhibiting distinct characteristics in terms of service demand patterns, facility opening costs, and operational expenses. The case study analysis incorporates parameter values sourced directly from Company SY’s marketing division, with detailed specifications provided in

Table 1. Part of the initial parameter values of customer locations.

Number	Customers	Location		Product Demand in the Planning Horizon				Cost (Thousand CNY)
		Longitude	Latitude	${\mathbf{v}}_{0\mathbf{i}}$	${\mathbf{v}}_{∆1\mathbf{i}}$	${\mathbf{v}}_{∆2\mathbf{i}}$	${\mathbf{v}}_{∆3\mathbf{i}}$	${\mathbf{c}\mathbf{s}}_{\mathbf{i}}$	${\mathbf{c}\mathbf{o}}_{\mathbf{i}}$
1	Shenzhen	114.09	22.55	1407	59	44	30	6000	100
2	Nanning	108.31	22.83	1415	75	56	88	4000	67
3	Guangzhou	113.24	23.13	2340	81	61	41	6000	100
4	Xiamen	118.08	24.45	512	21	16	11	5000	83
…
36	Beijing	116.5	39.9	1261	70	53	39	6000	100
37	Huhehaote	111.65	40.81	613	60	45	59	3000	50
38	Luoyang	112.27	34.41	650	57	43	69	3000	50
39	Dongguan	113.45	23.02	790	37	28	40	4000	67

and

Table 2. Initial values of the other parameters.

Parameter	t = 0	t = 1	t = 2	t = 3
$\mathsf{\alpha }$	0.2	0.6	0.8	0.9
$\mathsf{\beta }$	CNY 150 thousand
${\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	$1500$   $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{s}$
${\mathrm{d}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	$650$ km

.

5.2. Computational Results

The computational implementation of CA-MOGA was developed in MATLAB R2018b, with the algorithm executed across three experimental groups, each maintaining a population size of 200 chromosomes. All simulations were performed on a PC equipped with an Intel Core i5 processor and 4GB DDR2 RAM operating at 2.4 GHz (Intel, Santa Clara, CA, USA). The parameter configuration was set as follows: Equation (13) parameters $c=0.5,\phi =0.5,and \omega$ = 2; Equations (14) and (15) used $\beta =$ 0.1; Equation (16) employed $k=$ 2. Reflecting the prioritization of service quality in decision-making, objective weights were assigned as 0.6 for total weighted distance and 0.4 for service profit. The algorithm’s adaptive parameters were bounded with crossover rates constrained to [0.6, 0.9] and mutation rates to [0.01, 0.1]. The elite preservation strategy implemented trans-generation migration every 15 generations, with tournament selection conducted among 20 candidates.

The termination criterion was triggered when no improvement occurred in any objective of the best solution for 50 consecutive generations. Empirical observations indicated optimal performance typically emerged between 450 and 500 generations. To ensure statistical reliability, the algorithm executed 550 independent runs. The optimal solution achieved a total weighted distance of 5,006,300 km while generating a service profit of CNY 1,924,627 thousand. As illustrated in Figure 6, CA-MOGA demonstrates effective convergence behavior, successfully identifying optimal CSC locations and customer assignments that balance both spatial efficiency and service profitability objectives. Notably, the evolutionary optimization process resulted in a strategic reduction in CSCs from 19 to 12 facilities, while simultaneously optimizing service allocation patterns to satisfy both capacity constraints and service response time requirements.

The customer’s assignment of four optimal results is selected and presented in

Table 3. The locations of the CSCs and the assigned customers of the four optimal solutions generated by the CA-MOGA.

Customers	Solution 1			Solution 2			Solution 3			Solution 4
	t = 1	t = 2	t = 3	t = 1	t = 2	t = 3	t = 1	t = 2	t = 3	t = 1	t = 2	t = 3
1	39	39	39	39	39	39	39	39	39	39	39	39
2	39	39	39→6	39	39	39→6	39	39	39→6	39	39	39→6
3	39	39	39	39	39	39	39	39	39	39	39	39
…
10	9	9	9→13	8	8	8→13	8	8	8→13	9	9	5
…
23	22	22→18	22→18	22	22→18	22→18	22	22→18	22→18	22	22→18	22→18
24	26	26→28	26→28	26	26→28	26→28	26	26→28	26→28	26	26→28	26→28
…
32	36	36	36→30	36	36	36→30	34	34	34→30	34	34	34→30
…

, where solution 1 is the best solution. Due to the changes in service demand and the capacity constraint of the CSCs, customers 2, 10, 23, 24, and 32 have to obtain service from different CSCs in different periods in solution 1. For instance, customer 23 is served by the CSC located in customer 22 and 18 individually during time periods 1 and 2.

In order to analyze the results of the algorithms, both the GAs and the PSO are also applied to the case. The values of the parameters in GAs are as follows: population size is 400, and crossover and mutation probabilities are 0.8 and 0.5. The values of the parameters in PSO are as follows: the population size is 300, the inertia weight is 0.5, and both cognitive and social acceleration coefficients are set to 1.5. The GA and PSO algorithms terminate after 550 generations. A comparison of their overall performance is presented in Figure 7 and Figure 8. The results demonstrate that the improved CA-MOGA exhibits faster convergence of chromosomes. Specifically, the best-performing individuals obtained by the GAs achieve a total weighted distance of 5,677,144 km and a total service revenue of CNY 1,922,735 thousand, outperforming the PSO algorithm, which yields 5,867,255 km and CNY 1,921,778 thousand, respectively. Compared to the conventional GAs, the CA-MOGA enhances total service revenue by 0.1% while reducing the total weighted distance by 8.3%. Notably, the solutions generated by the CA-MOGA hold substantial practical value for industry practitioners.

6. Conclusions

Life-cycle product-service delivery represents a crucial strategic approach for equipment manufacturers pursuing sustainable development. This study formulates dynamic multi-objective SFLA problems as a 0–1 integer programming model that incorporates practical operational variables and constraints. To solve this NP-hard problem, we developed the CA-MOGA with four key improvements: (1) an optimized chromosome representation scheme, (2) an adaptive strategy integrating penalty functions, fitness evaluation, and dynamic operator probabilities to enhance search efficiency, (3) specialized genetic operators incorporating constraint handling and gene repair mechanisms to ensure solution feasibility, and (4) an elite trans-generation migration strategy to maintain population diversity. Comparative case study results demonstrate CA-MOGA’s superior performance over conventional GAs in both convergence speed and practical outcomes, enabling managers to simultaneously increase total service profit by 0.1% while reducing total weighted distance by 8.3%. The algorithm’s robust performance in this application suggests strong potential for customization and adaptation to other complex dynamic facility location and optimization problems.

Several limitations warrant discussion. First, the current model utilizes a linear weighted sum approach for objective aggregation based on solution rankings within the CA-MOGA framework. Future research will further refine the algorithm to enable direct identification of Pareto-optimal solutions through advanced multi-objective optimization techniques like Non-dominated Sorting Genetic Algorithm II. Second, the product-service demand depends on the extent of the product servitization, which is assumed to be fixed during a time period. It is interesting to explore the product-service facility location and allocation problem with uncertainty about the extent of the product servitization. Finally, the experiment in this study was carried out on a single company (SY) in the engineering machinery industry. Future research can be extended to investigate the diverse product-service delivery cases to verify the proposed approach.