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Research article

On a data-driven mathematical model for prostate cancer bone metastasis

  • Prostate cancer bone metastasis poses significant health challenges, affecting countless individuals. While treatment with the radioactive isotope radium-223 (223Ra) has shown promising results, there remains room for therapy optimization. In vivo studies are crucial for optimizing radium therapy; however, they face several roadblocks that limit their effectiveness. By integrating in vivo studies with in silico models, these obstacles can be potentially overcome. Existing computational models of tumor response to 223Ra are often computationally intensive. Accordingly, we here present a versatile and computationally efficient alternative solution. We developed a PDE mathematical model to simulate the effects of 223Ra on prostate cancer bone metastasis, analyzing mitosis and apoptosis rates based on experimental data from both control and treated groups. To build a robust and validated model, our research explored three therapeutic scenarios: no treatment, constant 223Ra exposure, and decay-accounting therapy, with tumor growth simulations for each case. Our findings align well with experimental evidence, demonstrating that our model effectively captures the therapeutic potential of 223Ra, yielding promising results that support our model as a powerful infrastructure to optimize bone metastasis treatment.

    Citation: Zholaman Bektemessov, Laurence Cherfils, Cyrille Allery, Julien Berger, Elisa Serafini, Eleonora Dondossola, Stefano Casarin. On a data-driven mathematical model for prostate cancer bone metastasis[J]. AIMS Mathematics, 2024, 9(12): 34785-34805. doi: 10.3934/math.20241656

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  • Prostate cancer bone metastasis poses significant health challenges, affecting countless individuals. While treatment with the radioactive isotope radium-223 (223Ra) has shown promising results, there remains room for therapy optimization. In vivo studies are crucial for optimizing radium therapy; however, they face several roadblocks that limit their effectiveness. By integrating in vivo studies with in silico models, these obstacles can be potentially overcome. Existing computational models of tumor response to 223Ra are often computationally intensive. Accordingly, we here present a versatile and computationally efficient alternative solution. We developed a PDE mathematical model to simulate the effects of 223Ra on prostate cancer bone metastasis, analyzing mitosis and apoptosis rates based on experimental data from both control and treated groups. To build a robust and validated model, our research explored three therapeutic scenarios: no treatment, constant 223Ra exposure, and decay-accounting therapy, with tumor growth simulations for each case. Our findings align well with experimental evidence, demonstrating that our model effectively captures the therapeutic potential of 223Ra, yielding promising results that support our model as a powerful infrastructure to optimize bone metastasis treatment.



    The rapid development of e-commerce platforms in recent years has led to more and more large online capital flows and offline commodity flows, which have resulted in not only an increasingly rapid rate of product updating but also environmental pollution, resource waste, and other problems. With the advancement of the global low-carbon economy, there is an increasing demand for remanufacturing and low-carbon technologies by enterprises. As a result, closed-loop supply chain is gradually becoming an important mode of operation in the context of a low-carbon circular economy. The closed-loop supply chain (CLSC) is developed from the traditional forward supply chain by adding the reverse logistics of recycling and remanufacturing of used products [1]. As an important operation mode of enterprises, CLSC can save resources through the recycling of used products, and is an important way to achieve the development of a green economy with a low carbon cycle [2]. The CLSC operation mode can bring good economic benefits for enterprises [3]. Specifically, an enterprise can establish a good image and improve market competitiveness by recycling used products. In addition, the recycling and remanufacturing activities can reduce carbon emissions and save costs. For example, Bosch can save 40% of costs annually through recycling and remanufacturing activities, and reduce 23,000 tons of carbon dioxide emissions [4,5]. Through recycling and remanufacturing activities, Kodak can save 40%–60% [6] and IBM saves up to 80% [7] of production costs.

    In essence, CLSC involves the incorporation of a reverse feedback process into the conventional forward supply chain, thereby establishing a complete, closed-loop system. At its core, CLSC operates as a feedback system that encompasses the entire lifecycle of products, spanning from resource extraction and production to consumption, and ultimately to the recycling and renewal of resources, through both forward delivery and reverse recycling processes [8]. CLSC members are committed to recycling and reusing used products, obtaining the surplus value of the used products, reducing production cost, and achieving the cyclic flow of resources, which will help indirectly reduce negative impacts on the environment [9]. Since pricing decisions are also pivotal in shaping the overall operational efficiency of CLSC, it has received extensive attention from scholars [10]. Vorasayan et al. analyzed the effect of changes in the returned product quality and refurbishment cost on the pricing decision [11]. Thereafter, the pricing strategies of the CLSC have been extensively explored [12,13,14], focusing on various aspects such as incentive strategies, product quality, power asymmetry, and demand uncertainty. Existing studies have discussed pricing strategies, reverse logistics, and remanufacturing in CLSC, but most of them have ignored a key reality: there is a significant difference in consumer acceptance of new and remanufactured products, especially in the context of the gradual increase in low-carbon awareness. To enhance the acceptance of remanufactured products in the market [15,16], this study introduces low-carbon technologies in the hope of increasing consumer demand for remanufactured products by reducing carbon emissions.

    While it is true that low-carbon technologies can increase the demand for remanufactured products, many companies are reluctant to actively participate in carbon reduction activities due to concerns about high investment costs [17]. To address this issue, this paper proposes a program where retailers and manufacturers share the investment costs of low-carbon technologies. Based on the literature discussing recycling activities [18,19], a differential response is constructed in which the manufacturer is responsible for product recycling, and the retailer and the manufacturer share the costs of recycling and low-carbon technology development. Unlike traditional studies, this paper focuses on analyzing cost-sharing mechanisms and pricing strategies under different power structures to find out the optimal decision-making of supply chain members with product differentiation and the way to maximize the overall profit.

    Based on the above discussion, this paper addresses the following questions for a dynamic supply chain with product differentiation:

    (1) What are the equilibrium pricing strategies and low-carbon technology levels of CLSC members under different power structures?

    (2) How does the retailer's optimal cost-sharing ratio affect the pricing decision under different power structures?

    (3) How do substitutability coefficients and low-carbon preferences affect pricing strategies? Under which power structure is the system profit maximized?

    To solve the above questions, we construct a CLSC consisting of a manufacturer and a retailer, since most of the literature assumes no difference between new and used products and does not consider the dynamic characteristics in the selling and recycling activities of used and end-of-life products. Therefore, this study presents a differential game model that captures the dynamic demand patterns specific to both new and remanufactured products. The focus of this study is a CLSC comprising a manufacturer and a retailer, which jointly bear the costs associated with remanufacturing and the low-carbon technology investment. With the differential game model, this study investigates the pricing strategies within three distinct game scenarios: the manufacturer-dominated Stackelberg game, the Nash game, and the retailer-dominated Stackelberg game. In addition, the effects of substitutability coefficient, low-carbon preference, and power structures on pricing decisions are analyzed. Furthermore, the optimal profits attainable within the CLSC system are examined under varying power structures. The main contributions of this study are as follows:

    (1) For the first time, we incorporate a dynamic perspective into a CLSC study of low-carbon technology R & D and cost-sharing mechanisms under different power structures.

    (2) We innovatively construct a differential response model that covers the dynamic demand for both new and remanufactured products. This model accurately determines the optimal pricing strategy, the level of low-carbon technology upgrading, and predicts market demand for new and remanufactured products under the three different power structures.

    (3) We reveal how substitutability coefficients and consumers' low-carbon preferences profoundly affect the formulation of pricing strategies in the context of different power structures. In addition, we clarify how the retailer optimizes its cost-sharing ratio to maximize its own benefits under different power structures.

    (4) In this paper, we explore in depth the optimal power structure configuration corresponding to the maximization of the overall profit of the CLSC system.

    The above contributions not only fill the research gap in resource recycling and reuse in supply chain management but also provide a new perspective on the research of the low-carbon economy. Moreover, they show how enterprises can achieve a balance between economic and environmental benefits through a CLSC. This is a key challenge under the current dual pressures of market competition and environmental protection.

    This study is organized as follows: Section 2 is the literature review; Section 3 describes the problems and basic assumptions; Section 4 shows the model construction and solution under different power structures; Section 5 is about numerical simulation; Section 6 presents the conclusions and prospects.

    The literature related to this study focuses on three areas: the pricing decision problem, the cost sharing and power structure problem, and the application of differential games in CLSC. Table 1 sets out a comparison between previous literature and our work.

    Table 1.  Most relevant literature.
    Literature Cost-sharing Power structure Low-carbon technology Distinguish between new and remanufactured products Differential game
    [26] × × ×
    [35] × × ×
    [38] × × ×
    [32] × × ×
    [28] × ×
    [24] × ×
    This study

     | Show Table
    DownLoad: CSV

    Currently, many scholars have profoundly studied pricing decisions in CLSC. Liu et al. explored the pricing strategies that took into account different reverse channels, quality uncertainty, and socially responsible investment behaviors [20,21,22]. But, the above studies do not consider the recovery cost, and all assume that the remanufacturing cost is zero. In fact, these costs affect the pricing decisions of CLSC members. Herein, we add the investment cost of low-carbon technology to the above costs, and study the pricing strategies of CLSC under the sharing of the recycling cost of the remanufacturing process and the investment cost of low-carbon technology between the manufacturer and the retailer.

    Meanwhile, the current society presents a diversified CLSC dominance situation that involves a model where large-scale manufacturers like Volkswagen and first automobile works dominate the market, and a model where large-scale retailers such as Wal-Mart, Carrefour, and Gome dominate the market. There is also a model in which electronic equipment manufacturers and Jingdong, Suning, and other retailers dominate the market simultaneously, which is also known as an evenly matched model [23]. Different power structures determine different decision-making orders of CLSC members, which will impact the pricing decision-making results of the whole CLSC and the benefit distribution structure of CLSC members in different ways [24]. Wei et al. studied the optimal decision-making problem of CLSC with symmetric and asymmetric information structures under manufacturer-led or retailer-led scenarios [25]. Results show that firms in the leadership of the supply chain have the advantage of obtaining higher profits under both symmetric and asymmetric information. Under varying power structures, some researchers studied the optimal pricing strategies, remanufacturing strategies, and coordination mechanisms within CLSC [17,26,27]. Nevertheless, the dynamic factors in product sales and waste product recycling activities were not considered in their research. In reality, the favorable reputation of an enterprise throughout its operations can be translated into consumers' positive perception of the product as a potential value, thereby influencing the sales and recycling of the product in subsequent periods [28]. If it is seen as a static state, the resulting strategy is only the company's local short-term optimal strategy, rather than the global optimal strategy. Therefore, given the dynamic influence of members' decisions, it is more suitable to investigate the effects of low-carbon technology and cost burdens under varying power structures from a dynamic perspective.

    There is an extensive body of research on dynamic characteristics-based analyses of CLSC [29,30]. For instance, Ma et al. constructed a dynamic differential equation for product goodwill and explored its influence on the decision-making processes of each participant within a CLSC [31,32,33]. Yang et al. built a dynamic differential equation for carbon emission reduction and thereby investigated the emission reduction decision-making of the manufacturer [34,35,36]. Ma et al. established a dynamic differential equation for carbon emission reduction by considering the product quality and the dynamic differential equation of product greenness and compared the dynamic pricing model with the static pricing model [37]. Nonetheless, the existing literature fails to differentiate between new and remanufactured products. In contrast, this study proposes a differential game model with product differentiation that captures dynamic demand patterns specific to new and remanufactured products. Moreover, the supply chain includes a manufacturer and a retailer who share the costs of remanufacturing and investing in low-carbon technologies. Finally, we give the optimal cost-sharing ratio for the retailer under different power structures to maximize its own benefit.

    This study focuses on the continuous time t[0,+), and is aimed to analyze the decisions related to pricing and low-carbon technology within CLSC under distinct power structures (Figure 1). The production of new products and the recycling and remanufacturing activities of used products are handled by the manufacturer. Additionally, the manufacturer conducts research and development to enhance the low-carbon technology used in remanufacturing processes. Simultaneously, the retailer assumes the role of selling both new and remanufactured products to consumers and collaborates with the manufacturer in sharing the recycling price and the investment cost associated with low-carbon technology within the recycling and remanufacturing activities.

    Figure 1.  CLSC model.

    Table 2 shows the specific symbols and corresponding meanings of relevant parameters. The notation πji denotes the profit by member i within the CLSC. i{m,r,T} represents the manufacturer, the retailer, and the CLSC system, respectively. j{MLM,NM,RLM} represents the manufacturer-dominated Stackelberg game, the Nash game, and the retailer-dominated Stackelberg game, respectively.

    Table 2.  Control variables and parameters.
    Category Symbol Definitions
    Control variables pN(t) retail price of new products
    pR(t) retail price of remanufactured products
    ωN(t) wholesale price of new products
    ωR(t) wholesale price of remanufactured products
    E(t) Manufacturer's low-carbon technology level
    Parameters γ1 natural growth factor of new products
    γ2 natural growth factor of remanufactured products
    a substitutability coefficient
    b low-carbon preference coefficient of consumers
    η investment cost coefficient of low-carbon technology
    I recycling price
    cN production cost of new products
    cR production cost of remanufactured products
    ρ discount rate, ρ>γ1,γ2
    φ cost-sharing ratio

     | Show Table
    DownLoad: CSV

    The basic assumptions of the CLSC model are specified below:

    Assumption 1. Both parties are rational economic people, and all decisions are made from rational principles.

    Assumption 2. Under the assumption of zero inventory cost and out-of-stock cost for both the manufacturer and the retailer, and considering an infinite time horizon, it is assumed that they share the same discount rate at any given point in time.

    We develop three models for decentralized decision-making: the Nash game (a market game without a leader), the Stackelberg game where the manufacturer dominates, and the Stackelberg game where the retailer dominates. Within these three decentralized decision-making models, both the retailer and the manufacturer exhibit risk-neutral and rational behavior, and their objectives are to maximize their profits throughout the decision-making procedure.

    The demands for new and remanufactured products are denoted as QN and QR, respectively, and satisfy the following differential equations

    QN(t)=γ1QN(t)pN(t)ωN(t)+apR(t),QR(t)=γ2QR(t)pR(t)ωR(t)+apN(t)+bE(t). (1)

    Where the demand naturally decreases over time. Meanwhile, the prices of both new and remanufactured products influence their respective demands. Specifically, as the price of a particular product increases, its demand tends to be suppressed. Correspondingly, when the price of new products rises, consumers are more likely to purchase remanufactured goods, and vice versa. Additionally, higher wholesale prices negatively affect retailers' procurement of similar products. An increase in the level of low-carbon technology can increase the recognition of remanufactured products. Therefore, it positively affects the change in demand for remanufactured products. At the same time, the research and development cost of low-carbon technology is highly correlated with its level, which is set as a quadratic function in this paper, i.e.,

    cm=12ηE2.

    For convenience, the time variable t is omitted below. The necessary theorem on the existence of equilibrium solutions for differential games (Theorem 2.4.2) please refer to [39].

    Manufacturer, as the core company in a CLSC, enjoys leadership over the supply chain. In this scenario, the manufacturer first decides on the wholesale price of products ωN and ωR, and low-carbon technology level E. The retailer then gives the retail price of products pN and pR, based on the manufacturer's decisions. The profit functions for each side are as follows:

    maxωN,ωR,EπMLMm=max0eρt[(ωNcN)QN+(ωRcR)QR(1φ)(IQR+12ηE2)]dt,
    maxpN,pR πMLMr=max0eρt[(pNωN)QN+(pRωR)QRφ(IQR+12ηE2)]dt.

    Based on the above decision sequence, each equilibrium solution is found according to the principle of backward derivation, as in Proposition 1.

    Proposition 1. The optimal equilibrium strategies under the MLM model are given by

    pMLMN=(z1+z5)QMLMN+(z2+z6)QMLMR+cN,pMLMR=(z3z7)QMLMN+(z4+z8)QMLMR+cR+I,ωMLMN=z5QMLMN+z6QMLMR+cN,ωMLMR=(z7)QMLMN+z8QMLMR+cR+(1φ)I,EMLM=z9QMLMR,

    where

    QMLMN=(z2az4+2z6az8)(2cR+2IφIacN)+(γ2+az2z4+az62z8+bz9)(2cNaaRaI)(z2az4+az6az8)(az1z3+az5+2z7)+(γ1z1+az32z5az7)(γ2+az2z4+az62z8+bz9),QMLMR=(z3az1az52z7)(2cNacRaI)+(γ1z1+az32z5az7)(2cR+2IφIacN)(z2az4+az6az8)(az1z3+az5+2z7)+(γ1z1+az32z5az7)(γ2+az2z4+az62z8+bz9),

    and

    z1=ργ11a2,z2=a(ργ1)1a2,z3=a(ργ2)1a2,z4=ργ21a2,z7=z6,z9=b2η(1φ),z5={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}(1a2)[(ργ1)(2a2)a2(ργ2)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2},z6={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}[a(1a2)(γ1γ2)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2},z8={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}(1a2)[(ργ2)(2a2)a2(ργ1)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2}.

    Proof. The Hamiltonian function for the follower is obtained as

    HMLMr(t,pN,pR)=(pNωN)QN+(pRωR)QRφ(IQR+12ηE2)+β1[γ1QNpNωN+apR]+β2[γ2QRpRωR+apN+bE], (2)

    where β1,β2 are the adjoint variables of the corresponding state variables. The HJB equations are obtained by taking a first-order partial derivative for pN,pR, respectively,

    HMLMrpN=QNβ1+aβ2=0,HMLMrpR=QR+aβ1β2=0. (3)

    The adjoint variables satisfy the following adjoint equations

    β1=ρβ1HMLMrQN=(ργ1)β1+ωNpN,β2=ρβ2HMLMrQR=(ργ2)β2+ωRpR+φI.

    And the limiting transversality conditions

    limteρtQNβ1=0,limteρtQRβ2=0.

    Then, in order not to violate the transversality condition, we choose a set of stable solutions

    β1=β1(0)=pNωNργ1,β2=β2(0)=pRωRφIργ2. (4)

    The definition of a stable solution is one solution that satisfies

    βi(t)=0

    and is itself a solution of the adjoint equation. Substituting Eq (4) into Eq (3), we obtain

    QNpNωNργ1+apRωRφIργ2=0,QR+apNωNργ1pRωRφIργ2=0.

    We can obtain

    pN=z1QN+z2QR+ωN,pR=z3QN+z4QR+ωR+φI, (5)

    where

    z1=ργ11a2,z2=a(ργ1)1a2,z3=a(ργ2)1a2,z4=ργ21a2.

    Substituting Eq (5) into the leader's objective function yields the leader's Hamiltonian function

    HMLMm(t,ωN,ωR,E)=(ωNcN)QN+(ωRcR)QR(1φ)(IQR+12ηE2)+λ1[(γ1z1+az3)QN(t)+(az4z2)QR(t)+aωR(t)2ωN(t)+aφI]+λ2[(az1z3)QN(t)+(γ2z4+az2)QR(t)2ωR(t)+aωN(t)+bE(t)φI]. (6)

    The HJB equations are obtained by taking a first-order partial derivative for ωN,ωR,E, respectively

    HMLMmωN=QN2λ1=0,HMLMmωR=QR2λ2=0,HMLMmE=(1φ)ηE+bλ2=0. (7)

    Similarly, in order not to violate the transversality conditions, we choose a set of stable solutions. Then

    λ1=ρλ1HMLMmQN=0,λ2=ρλ2HMLMmQR=0.

    According to Eq (6), the constant adjoint variables

    λ1=λ1(0)

    and

    λ2=λ2(0)

    are obtained,

    λ1={[(ργ2)(2a2)a2(ργ1)](ωNcN)a(γ1γ2)(ωRcRI+φI)}(1a2)[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2,λ2={[(ργ1)(2a2)a2(ργ2)](ωRcRI+φI)+a(γ1γ2)(ωNcN)}(1a2)[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2. (8)

    Substituting Eq (8) into Eq (7), we can obtain

    ωMLMN=z5QMLMN+z6QMLMR+cN,ωMLMR=(z7)QMLMN+z8QMLMR+cR+(1φ)I,EMLM=z9QMLMR, (9)

    where

    z5={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}(1a2)[(ργ1)(2a2)a2(ργ2)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2},z6={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}[a(1a2)(γ1γ2)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2},z7=z6,z8={[(ργ1)(2a2)a2(ργ2)][(ργ2)(2a2)a2(ργ1)]+a2(γ1γ2)2}(1a2)[(ργ2)(2a2)a2(ργ1)]2{(1a2)2[(ργ2)(2a2)a2(ργ1)]+[a(1a2)(γ1γ2)]2},z9=b2η(1φ).

    Plugging Eq (9) into Eq (5)

    pMLMN=(z1+z5)QMLMN+(z2+z6)QMLMR+cN,pMLMR=(z3z7)QMLMN+(z4+z8)QMLMR+cR+I.

    In order to find the Stackelberg solutions, it is necessary to solve QN,QR. Therefore, by substituting pMLMN,pMLMR,ωMLMN,ωMLMR into (1), we obtain

    QN=(γ1z1+az32z5az7)QN+(az4z2+az82z6)QR(2cNaaRaI),QR=(az1z3+az5+2z7)QN+(γ2+az2z4+az62z8+bz9)QR(2cR+2IφIacN).

    When the autonomy equations are stable,

    QN=(γ1z1+az32z5az7)QN+(az4z2+az82z6)QR(2cNaaRaI)=0,QR=(az1z3+az5+2z7)QN+(γ2+az2z4+az62z8+bz9)QR(2cR+2IφIacN)=0.

    Then the equilibrium solutions for the demand for the new and remanufactured products are

    QMLMN=(z2az4+2z6az8)(2cR+2IφIacN)+(γ2+az2z4+az62z8+bz9)(2cNaaRaI)(z2az4+az6az8)(az1z3+az5+2z7)+(γ1z1+az32z5az7)(γ2+az2z4+az62z8+bz9),QMLMR=(z3az1az52z7)(2cNacRaI)+(γ1z1+az32z5az7)(2cR+2IφIacN)(z2az4+az6az8)(az1z3+az5+2z7)+(γ1z1+az32z5az7)(γ2+az2z4+az62z8+bz9).

    The Stackelberg solutions for the manufacturer and retailer control variables are obtained

    pMLMN=(z1+z5)QMLMN+(z2+z6)QMLMR+cN,pMLMR=(z3z7)QMLMN+(z4+z8)QMLMR+cR+I,ωMLMN=z5QMLMN+z6QMLMR+cN,ωMLMR=(z7)QMLMN+z8QMLMR+cR+(1φ)I,EMLM=z9QMLMR.

    The Nash game, a leaderless market game, assumes that both parties make decisions at the same time, but the manufacturer decides on the wholesale price of the products ωN and ωR, and the level of low-carbon technology E, and the retailer decides on the retail price of the products pN and pR. The profit functions for each side are as follows:

    maxωN,ωR,EπNMm=max0eρt[(ωNcN)QN+(ωRcR)QR(1φ)(IQR+12ηE2)]dt,maxpN,pR πNMr=max0eρt[(pNωN)QN+(pRωR)QRφ(IQR+12ηE2)]dt.

    Proposition 2. The optimal equilibrium strategies under the NM model are given by

    pNMN=y1QNMN+y2QNMR+cN,pNMR=y3QNMN+y4QNMR+cR+I,ωNMN=y5QNMN+cN,ωNMR=y6QNMN+cR+(1φ)I,ENM=y7QNMR,

    where

    QNMN=(γ2y4y6+ay2+by7)(2cNacRaI)+(y2ay4)(2cR+2IφIacN)(γ1y1y5+ay3)(γ2y4y6+ay2+by7)+(ay1y3)(y2ay4),QNMR=(γ1y1y5+ay3)(2cR+2IφIacN)+(y3ay1)(2cNacRaI)(γ1y1y5+ay3)(γ2y4y6+ay2+by7)+(ay1y3)(y2ay4),

    and

    y1=(ργ1)(2a2)(1a2),y2=a(ργ1)(1a2),y3=a(ργ2)(1a2),y4=(ργ2)(2a2)(1a2),y5=(ργ1),y6=bη(1φ).

    Proof. The Hamiltonian functions for manufacturer and retailer, respectively, are as follows:

    HNMm(t,ωN,ωR,E)=(ωNcN)QN+(ωRcR)QR(1φ)(IQR+12ηE2)+λ1[γ1QNpNωN+apR]+λ2[γ2QRpRωR+apN+bE],HNMr(t,pN,pR)=(pNωN)QN+(pRωR)QRφ(IQR+12ηE2)+β1[γ1QNpNωN+apR]+β2[γ2QRpRωR+apN+bE].

    The HJB equations are obtained by taking a first order partial derivative for ωN,ωR,E and pN,pR,

    HNMrpN=QNβ1+aβ2=0,HNMrpR=QR+aβ1β2=0,HNMmωN=QN2λ1=0,HNMmωR=QR2λ2=0,HNMmE=(1φ)ηE+bλ2=0. (10)

    The adjoint variables satisfy the following adjoint equations:

    λ1=ρλ1HNMrQN=(ργ1)λ1+cNωN,λ2=ρλ2HNMrQR=(ργ2)λ2+cRωR+(1φ)I,β1=ρβ1HNMrQN=(ργ1)β1+ωNpN,β2=ρβ2HNMrQR=(ργ2)β2+ωRpR+φI.

    And the limiting transversality conditions

    limteρtQNλ1=0,limteρtQRλ2=0,limteρtQNβ1=0,limteρtQRβ2=0.

    In order not to violate the transversality condition, we choose a set of stable solutions

    λ1=ωNcNργ1,λ2=ωRcR(1φ)Iργ2,β1=pNωNργ1,β2=pRωRφIργ2. (11)

    Substituting Eq (11) into Eq (10), we obtain

    1ργ1pN+aργ2pR+1ργ1ωNaργ2ωR=aφIργ2QN,aργ1pN1ργ2pRaργ1ωN+1ργ2ωR=φIργ2QR,1ργ1ωN=cNργ1+QN,1ργ2ωR=(1φI)ργ2+cRργ2+QR,bργ2ωR(1φ)ηE=b(1φI)+bcRργ2.

    We can obtain

    pNMN=y1QNMN+y2QNMR+cN,pNMR=y3QNMN+y4QNMR+cR+I,ωNMN=y5QNMN+cN,ωNMR=y6QNMN+cR+(1φ)I,ENM=y7QNMR,

    and

    y1=(ργ1)(2a2)(1a2),y2=a(ργ1)(1a2),y3=a(ργ2)(1a2),y4=(ργ2)(2a2)(1a2),y5=(ργ1),y6=bη(1φ).

    In order to find the Nash solutions, it is necessary to solve QN,QR. Therefore, by substituting pMLMN,pMLMR,ωMLMN,ωMLMR,E into Eq (1), we get

    QN=(γ1y1+ay3y5)QN+(ay4y2)QR(2cNaaRaI),QR=(ay1y3)QN+(γ2+ay2y4y6+by7)QR(2cR+2IφIacN).

    When the autonomy equations are stable,

    (γ1y1+ay3y5)QN+(ay4y2)QR(2cNaaRaI)=0,(ay1y3)QN+(γ2+ay2y4y6+by7)QR(2cR+2IφIacN)=0.

    Then the optimal demands for the new and remanufactured products are

    QNMN=(γ2y4y6+ay2+by7)(2cNacRaI)+(y2ay4)(2cR+2IφIacN)(γ1y1y5+ay3)(γ2y4y6+ay2+by7)+(ay1y3)(y2ay4),QNMR=(γ1y1y5+ay3)(2cR+2IφIacN)+(y3ay1)(2cNacRaI)(γ1y1y5+ay3)(γ2y4y6+ay2+by7)+(ay1y3)(y2ay4).

    And the Nash equilibrium for the manufacturer and retailer control variables are obtained

    pNMN=y1QNMN+y2QNMR+cN,pNMR=y3QNMN+y4QNMR+cR+I,ωNMN=y5QNMN+cN,ωNMR=y6QNMN+cR+(1φ)I,ENM=y7QNMR.

    The retailer, as the core company in a CLSC, enjoys leadership over the supply chain. In this scenario, the retailer first decides the retail price of the products pN and pR. The manufacturer then gives the wholesale price of the product ωN and ωR, and the level of low-carbon technology E, based on the retailer's decision. The profit functions for each side are as follows:

    maxωN,ωR,EπRLMm=max0eρt[(ωNcN)QN+(ωRcR)QR(1φ)(IQR+12ηE2)]dt,maxpN,pR πRLMr=max0eρt[(pNωN)QN+(pRωR)QRφ(IQR+12ηE2)]dt.

    Proposition 3. The optimal equilibrium strategies under the MLM model are

    pRLMN=x4QRLMN+x5QRLMR+cN,pRLMR=x6QRLMN+x7QRLMR+cR+I,ωRLMN=x1QRLMN+cN,ωRLMR=x2QRLMR+cR+(1φ)I,ERLM=x3QRLMR,

    where

    QNMN=(γ2x2+bx3+ax5x7)(2cNacRaI)+(x5ax7)(2cR+2IφIacN)(γ1x1x4+ax6)(γ2x2+bx3+ax5x7)+(x5ax7)(ax4x6),QNMR=(γ1x1x4+ax6)(2cR+2IφIacN)+(x6ax4)(2cNacRaI)(γ1x1x4+ax6)(γ2x2+bx3+ax5x7)+(x5ax7)(ax4x6),

    and

    x1=ργ1,x2=ργ2,x3=bη(1φ),x4=2(ργ1)(2a2)(1a2),x5=2a(ργ1)(1a2),x6=2a(ργ2)(1φ)ηab2(1a2)(1φ)η,x7=(2a2)[2(ργ2)(1φ)ηb2](1a2)(1φ)η+b2(1+φ)[2(ργ2)(1φ)ηb2][2(ργ2)(1φ)2ηb2](1φ)η.

    Proof. Similar to the proof of Proposition 1.

    In order to be more intuitive and thorough for more in-depth research and analysis, we conduct simulations to explore the impacts of cost-sharing coefficient, product substitutability, and consumers' low-carbon preference on pricing decisions. The parameters in this section include

    γ1=0.9,γ2=0.8,ρ=1,cN=0.8,cR=0.3,I=0.2,η=1.

    In addition to the manufacturer benefiting from recycling, recycling is also a profitable endeavor for the retailer. Therefore, the retailer is actively involved in cost-sharing, both in terms of recycling costs for the manufacturer and investment costs for low-carbon technology, thus incentivizing recycling. Therefore, it is necessary to investigate the optimal cost-sharing ratio for retailers under different power structures to maximize the retailer's profits. The parameters in this section are

    a=0.5,b=0.5.

    As demonstrated in Figure 2, the optimal cost-sharing ratio is exclusively present within the RLM model. Conversely, in both the NM and MLM models, the retailer achieves maximum profit without engaging in cost-sharing measures, and the profits decrease as the cost-sharing ratio increases. Under the RLM model, the retailer's profit and its cost-sharing ratio are in an inverted U-shaped curve, as the retailer has the optimal cost-sharing ratio of 0.8, at which the retailer's profit reaches 14.75, which is much higher than the profit under the NM and MLM models. Therefore, as the retailer's power increases, its profit also rises. To maximize their profits, the retailer must establish the most suitable cost-sharing ratio based on their levels of power within the CLSC.

    Figure 2.  Impact of φ on πr under different power structures.

    Within this subsection, the impact of cost-sharing ratios on the equilibrium solution of members within a CLSC is analyzed under varying power structures. This analysis is visually represented in Figures 35. As shown in Figure 3, there is

    ENM>EMLM>ERLM,

    and the low-carbon technology level under the NM and MLM models slowly increase as b rises, indicating the cost-sharing mechanism can improve the low-carbon technology level of remanufacturing. However, the low-carbon technology level under the RLM model is decreasing on the whole, and falls to 0 at φ = 0.21. Therefore, when the retailer possesses greater power within the CLSC, the corresponding consequence is a lower level of low-carbon technology for the manufacturer.

    Figure 3.  Impact of φ on E under different power structures.
    Figure 4.  Impact of φ on pN under different power structures.
    Figure 5.  Impact of φ on pR under different power structures.

    The retail price of new products under the RLM model is consistently higher than under the NM and MLM models regardless of the values of φ (Figure 4). As φ increases, under both the RLM and NM models, the retail price of new products rises, albeit at different rates. Specifically, the retail price rises gradually under the NM model, while it escalates more rapidly under the RLM model. In the MLM model, the retailer's retail prices are slowly decreasing, and there is a threshold point between the RLM and NM models at

    φ=0.12.

    When at

    φ<0.21,

    the retail price of a new product in the MLM model is slightly higher than in the NM model, while the opposite is true at

    φ>0.21.

    This result indicates that as the retailer's power in the CLSC increases, the retail price of the new products rises, because the retailer transfers its inputs for cost-sharing in the recycling and remanufacturing process to the new product. As shown in Figure 5, the retail price of remanufactured products decreases under all three models regardless of the values of φ, and changes in the order of

    pNMR>pMLMR>pRLMR.

    When the manufacturer and retailer simultaneously make their pricing strategies, the retail price of the remanufactured product is higher, but the price decreases as power shifts to the retailer (Figure 5).

    The wholesale price of new products is minimally affected by the sharing ratio in both the NM and MLM models (Figure 6). The wholesale price gradually decreases in the MLM model, while it slowly increases in the NM model. At higher sharing ratios, there is

    ωRLMN>ωNMN>ωMLMN.

    When dominance shifts from the manufacturer to the retailer, the retailer's bargaining power is intensified, while the manufacturer's wholesale price increases. Under all three models, the wholesale price of remanufactured products decreases as the sharing ratio increases:

    ωNMR>ωMLMR>ωRLMR,

    and an increased sharing of investment in recycling activities by retailers, manufacturers demonstrates their willingness to lower the wholesale price of remanufactured products as a means to support retailer (Figure 7).

    Figure 6.  Impact of φ on ωN under different power structures.
    Figure 7.  Impact of φ on ωR under different power structures.

    The substitutability coefficient of the product and the consumers' low-carbon preference under the Nash game little affect the retail price of a new product, which fluctuates between 1.297 and 1.400 with a relatively small range as the two parameters change (Figure 8). In the manufacturer-dominated Stackelberg game, the substitutability coefficient positively impacts the retail price of new products, while low-carbon preference has a negative impact, resulting in a decrease. Compared to the previously mentioned games, the retailer-dominated Stackelberg game results in a higher retail price for new products. The highest retail price is attained when both the substitutability coefficient and low-carbon preference are at larger values.

    Figure 8.  Impact of a,b on pN under different power structures.

    Figure 9 illustrates that the wholesale price of a new product follows a downward trend as the substitutability factor increases under the Nash game. Conversely, when low-carbon preference increases, the wholesale price shows an upward trend. In the manufacturer-dominated Stackelberg game, the impact of both parameters on the wholesale price of a new product is in line with their impact on the retail price. In the context of the retailer-dominated Stackelberg game, the wholesale price of the new products fluctuates between 1.050 and 1.500 with the change of the two parameters, which is a wider range of fluctuation than the two games mentioned above.

    Figure 9.  Impact of a,b on pR under different power structures.

    To further refine the presentation of our model results and minimize the impact of parameter selection, we conducted additional simulations using randomly generated parameters. Ensuring that all parameter values remained meaningful in practice, we generated 500 different parameter sets to analyze the variations in total supply chain profit under different cost-sharing ratios across the three power structures. The total supply chain profit, defined as the sum of the manufacturer's and retailer's profits, is denoted by the subscript T. The results are presented in Table 3 and Figure 10. The findings reveal that when the cost-sharing ratio is low, the MLM structure has a higher probability of achieving greater total profit compared to NM and RLM, while the likelihood of NM outperforming RLM remains relatively low. As the cost-sharing ratio increases, the advantage of MLM becomes more evident. When the ratio reaches around 0.8, the probability of MLM's total profit exceeding that of RLM reaches its peak. At the same time, the chance of NM achieving higher profits than RLM also increases significantly.

    Table 3.  Optimal profit under different power structures.
    Cost-sharing ratio Power structures Condition Count Percentage
    φ=0.2 πMLMTπNMT > 0 317 63.40%
    < 0 183 36.60%
    πMLMTπRLMT > 0 311 62.20%
    < 0 189 37.80%
    πNMTπRLMT > 0 229 45.80%
    < 0 271 54.20%
    φ=0.4 πMLMTπNMT > 0 357 71.40%
    < 0 143 28.60%
    πMLMTπRLMT > 0 361 72.20%
    < 0 139 27.80%
    πNMTπRLMT > 0 236 47.20%
    < 0 264 52.80%
    φ=0.6 πMLMTπNMT > 0 343 68.60%
    < 0 157 31.40%
    πMLMTπRLMT > 0 362 72.40%
    < 0 138 27.60%
    πNMTπRLMT > 0 245 49.00%
    < 0 255 51.00%
    φ=0.8 πMLMTπNMT > 0 319 63.80%
    < 0 181 36.20%
    πMLMTπRLMT > 0 433 86.60%
    < 0 67 13.40%
    πNMTπRLMT > 0 327 65.40%
    < 0 173 34.60%

     | Show Table
    DownLoad: CSV
    Figure 10.  Optimal profit under different power structures, A: πMLMTπNMT, B: πMLMTπRLMT, C: πNMTπRLMT.

    This indicates that, with a reduced cost burden, the manufacturer can allocate more resources to low-carbon technology development and product innovation. Their technological advantage in both new and remanufactured products enables the supply chain to maintain high operational efficiency. This insight also highlights that, although the manufacturer bears sole responsibility for the recovery process, their ability to optimize the remanufacturing workflow and control costs is stronger, ensuring a high level of profitability across the entire supply chain.

    This study offers valuable insights into the management of CLSC with product differentiation, particularly within the context of a low-carbon economy. Unlike previous research that primarily focused on static models, we combine product differentiation and dynamic game theory to provide a more detailed understanding of long-term strategic interactions among supply chain participants. Our findings demonstrate that integrating low-carbon technologies and cost-sharing mechanisms can significantly enhance both environmental sustainability and economic efficiency. By offering new theoretical perspectives, this study contributes to the field of sustainable supply chain management.

    Theoretically, this study advances the development of CLSC management with product differentiation. Our model highlights how different power structures, such as manufacturer-led and retailer-led scenarios, impact the optimal level of low-carbon technology and cost-sharing strategies. This deepens our understanding of strategic decision-making in the supply chain.

    From a practical perspective, our findings offer actionable insights for policymakers and industry leaders. By implementing the cost-sharing mechanisms and low-carbon technologies proposed in this paper, companies can strengthen their sustainability practices while simultaneously improving their economic performance. This has the potential to drive significant progress toward achieving environmental goals.

    Specifically, the impact of this research extends beyond the specific context of CLSC. By proposing a model that integrates low-carbon technologies and cost-sharing mechanisms, we provide a framework that can be adapted to various industries facing similar sustainability challenges. This approach not only aids in policy development but also offers guidance for practitioners in balancing economic and environmental objectives.

    This paper constructs a differential game model for new and remanufactured products concerning dynamic pricing and low-carbon technology decision-making in CLSC. It fills the gap left by previous studies, which failed to distinguish between new and remanufactured products and primarily relied on static models. By introducing dynamic game analysis, we investigate the pricing strategies and low-carbon technology levels under three power structures: manufacturer-dominated, retailer-dominated, and Nash games.

    The results of the study show that there are significant differences in the decision-making behaviors of manufacturers and retailers under different power structures, particularly in terms of cost-sharing mechanisms. Compared with the static model, this study demonstrates, for the first time, the long-term effect of the manufacturer reducing the wholesale price of remanufactured products for the retailer after the retailer bears more costs, using a dynamic model. This plays a positive role in the overall coordination of the supply chain. The problem of "free-riding", which occurs in some studies, is resolved.

    In addition, we find that an increase in the level of low-carbon technology contributes to the growth of the market for remanufactured products. However, if the level of low-carbon technology is too high, it may undermine the competitiveness of new products, especially if it leads to manufacturer-led price reductions for new products. Therefore, the control of low-carbon technology needs to be moderate to balance the market share between old and new products. This finding addresses gaps in existing studies.

    Through the quantitative analysis of numerical simulation, we find that the retailer-dominated Stackelberg game maximizes profit for the retailer and that there exists an optimal cost-sharing ratio. Compared with previous studies, we not only extend the applicability of the game model but also provide a new quantitative benchmark for power structures and profit distribution in the supply chain, particularly in the context of low-carbon technologies and cost-sharing mechanisms.

    In conclusion, the research in this paper provides a theoretical basis and practical guidance for the strategic management of CLSC with product differentiation in the context of a low-carbon economy. It also proposes an effective cost-sharing mechanism and power structure selection strategy by comparing different game structures.

    Jun Wang: designed project, provided figures and revised the manuscript; Dan Wang: designed the project, wrote the first draft and revised the manuscript; Yuan Yuan: conceptualized and supervised the project, revised the manuscript, and acquired funding. All authors have read and approved the final version of the manuscript for publication.

    The authors would like to thank the reviewer for their valuable comments. This work was supported by the Natural Science Foundation of Jilin Province, under grant number. 20230101288JC.

    The authors declare no conflicts of interest.



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