*3.1. Model Validation*

To validate the finite element model, the simulated ROM and IVD stress in the present intact T10-L3 model were compared with the literature. First, the ROM of the intact thoracolumbar model in flexion-extension, lateral bending, and axial rotation were compared with three in vitro experiments by Chin et al., Rustenburg et al., and Obid et al. [31–33]. The global ROM of the present intact model was as follows: flexion–extension, 6.86 degrees; lateral bending, 3.04 degrees; and axial rotation, 1.54 degrees. Compared with the literature, the results were all within one standard deviation (SD) (Figure 3A). Next, the intact model was also compared with the in vitro intradiscal pressure measurements at L2/L3 IVD conducted by Cunningham et al., Brinckmann et al., and Wilke et al. [34–36]. The maximal IVD stress of this model under sagittal flexion and extension was 0.49 Mpa, which was within one SD compared to the results of Cunningham et al. and Wilke et al. but slightly larger (1.24 SD) than the result reported by Brinckmann et al. (Figure 3B). An extended explanation of the differences is given in the Discussion section.

**Figure 3.** Validation of the present FE model. Comparisons between the (**A**) ROM and (**B**) IVD pressure of the present intact model with the literature [31–36] (presented in mean and standard deviation).

#### *3.2. Global Range of Motion in the TL Spine*

The global flexion, extension, lateral bending, and axial rotation ROMs of the intact and surgical models were shown in Figure 4A. The failure of the L1 vertebra resulted in 7.4, 10.1, 18.0, and 11.5% increases in motion under flexion, extension, lateral flexion, and axial rotation, respectively. All four surgical constructs reduced the global ROM in all directions. The LSPI (U2L2) had the most significant reductions in global ROM in the flexion, extension, lateral flexion, and axial rotation of 88.6, 70.7, 81.1, and 40.7%, respectively. The comparison among U1L1+IS, U1L1+PMMA, and U1L1+Cage showed that TCSR with a titanium cage (U1L1+Cage) results in a slightly larger reduction in ROM than the other two structures (U1L1+IS and U1L1+PMMA), but the differences between each other were all less than 0.5 degrees in all motions.

**Figure 4.** The global ROM, ROM in L1, and ROM distributions. The simulated global ROM (**A**) and the pathological ROM in the L1 vertebra (**B**). ROM distributions (**C**) among the T10–L3 levels. The pathological motion in L1 was indicated in red. \* The asterisks indicate the motions in the supradjacent levels.

#### *3.3. Motion in the Fractured L1 Vertebral Body and Motion Distributions*

Our simulations showed that failure of the anterior and middle spinal column resulted in increased motion in the affected L1 vertebra. The pathological intravertebral motion within the failed L1 was shown in Figure 4B. The largest motion was observed in lateral bending with 0.77 degrees, followed by 0.61, 0.38, and 0.34 degrees in flexion, axial rotation, and extension, respectively. Comparisons between the constructs revealed that TCSR constructs (U1L1+PMMA and U1L1+Cage) had a greater percentage of motion reduction than PI alone. In flexion and lateral bending, U1L1+Cage had the most ROM reduction by 98.6 and 98.1%, respectively. In extension and axial rotation, U1L1+PMMA had the best ROM reduction by 94,9 and 79.1%, respectively.

The ROM distribution was shown in Figure 4C. The pathological motion in L1 was indicated in red and the motions in the supradjacent levels were indicated by the asterisks. In flexion, extension, and lateral bending, all constructs reduced the percentage of motion in L1. Comparisons of the surgical models showed U2L2 had increased ROM distributed

in the supradjacent level. In axial rotation, U2L2 had an increased percentage of motion in L1 while other constructs had decreased percentages of motion in L1. The difference in the percentage of the supradjacent ROM in the axial rotation was not significant.

#### *3.4. The Effect of PI and TCSR on the Proximal Junctional Level*

The maximum von Mises stresses exerted on the vertebral body immediately proximal to the constructs (T10 vertebra in U2L2; T11 vertebra in U1L1+IS, U1L1+PMMA, and U1L1+Cage) were shown in Figure 5A. For all constructs, the maximum stress at the proximal junctional vertebra ranged from 0.95 to 5.04 MPa. The highest stress occurred in lateral bending (4.80–5.04 Mpa) in all constructs, followed by flexion (2.77–3.81 Mpa). The greatest differences in stress at the proximal vertebra between the constructs occurred in flexion, in which U2L2 resulted in larger stresses by 1.04, 0.92, and 0.88 Mpa than U1L1+IS, U1L1+Cage, and U1L1+PMMA, respectively. The differences in stress at the proximal vertebra in extension, lateral bending, and axial rotation were all less than 0.5 MPa.

**Figure 5.** The maximum von Mises stress in the proximal junctional level. The maximum von Mises stress in the proximal vertebral body (**A**) and the proximal junctional IVD (**B**) in flexion, extension, lateral bending, and axial rotation.

The maximum von Mises stresses exerted on the IVD immediately proximal to the constructs (T10/11 disc in U2L2; T11/12 disc in U1L1+IS, U1L1+PMMA, and U1L1+Cage) were shown in Figure 5B. Comparison between the constructs showed a similar trend in all motions, with U2L2 having the largest stress at the proximal IVD and U1L1+IS having the smallest stress at the proximal IVD. The differences in the proximal IVD stresses between U1L1+PMMA and U1L1+Cage were all within 0.01 MPa.

#### *3.5. The Effect of PI and TCSR on the Proximal Articular Facets*

The maximum contact force exerted on articular facets immediately proximal to the constructs (T10/11 facets in U2L2; T11/12 facets in U1L1+IS, U1L1+PMMA, and U1L1+Cage) were shown in Figure 6. In flexion, U2L2 had 2.8, 2.7, and 2.7 N less contact forces on the proximal facet joints compared to U1L1+IS, U1L1+PMMA, and U1L1+Cage, respectively. In extension, U2L2 had 5.6, 5.5, and 5.4 N more contact forces on the proximal facet joints compared to U1L1+IS, U1L1+PMMA, and U1L1+Cage, respectively. The differences in the proximal facet contact forces in lateral bending and axial rotation were all less than 1.2 N.

#### *3.6. Von Mises Stress and Strain Energy Density on the Screw and Rod Construct*

The maximum von Mises stress and strain energy density of the pedicle screws in each construct were presented in Table 3. The maximum stress of the pedicle screws occurred in axial rotation in all constructs. U2L2 had the highest pedicle screw stress of 27.98 MPa, followed by 27.31, 24.01, and 16.78 MPa in U1L1+PMMA, U1L1+IS, and U1L1+Cage, respectively. The maximum stress was observed at L2 in constructs involving PI alone (U2L2

and U1L1+IS) but was observed at T12 in TCSR constructs (U1L1+PMMA and U1L1+Cage). The stress distributions were shown in Figure 7. The maximum strain energy density of the pedicle screws occurred in axial rotation in U2L2, U1L1+IS, and U1L1+PMMA, while U1L1+Cage had the highest strain energy density in flexion. U2L2 had the highest strain energy density of 12.41 mJ/mm3 , followed by 8.05, 5.72, and 4.55 mJ/mm3 in U1L1+IS, U1L1+PMMA, and U1L1+Cage, respectively.

**Figure 6.** The maximum contact force at the proximal articular facets. The maximum facet contact force in the proximal level in flexion, extension, lateral bending, and axial rotation.

**Table 3.** Maximum von Mises stress and strain energy in the pedicle screws.

**Figure 7.** The stress distributions of the pedicle screw and rod constructs in the motions of maximum von Mises stress detected.

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#### **4. Discussion**

In the present study, we systemically evaluate the biomechanical performance of different TL reconstruction constructs using FE analysis. Our results showed that TCSR constructs provided better stabilization in the fracture of L1 compared to PI alone. Further, there were decreased intradiscal and intravertebral pressures in the proximal level in U1L1+IS, U1L1+PMMA, and U1L1+Cage compared to U2L2. The stress and strain energy of the pedicle screws were lower in TCSR constructs than in PI alone. We showed that TCSR with anterior reconstruction and SSPI provided sufficient immobilization while offering additional advantages in the preservation of physiological motion, a decreased burden on the proximal junctional level, and lower mechanical stress and strain in the implants.

TCSR with PI and anterior vertebral augmentation or intermediate screw fixation has been shown to provide the immediate stabilization and restoration of spinal integrity [2]. Although previous studies have reported the advantages of TCSR in terms of better neurological improvement, stability, restoration of sagittal balance, and fewer implant failures [2–4], the rigid constructs in the TL region pose a significant risk for PJF [5], and the ideal strategy for TL reconstruction remains controversial. This study evaluated and compared the biomechanics of different reconstruction strategies using FE analysis.

The T10-L3 FE model in this study was validated against previously published in vitro measurements of the ROM and intradiscal pressure. The majority of our simulation results remained compatible and within one SD compared to the literature [31–34,36]. Some differences were noted in the intradiscal pressure between our results and previous experiments by Brinckmann et al. (1.24 SD) [35]. Factors such as the anatomical variation between the present model and the cadavers in the literature could result in the differences. Moreover, the location where the intradiscal pressure was measured in the cadaveric experiments could also contribute to the difference since the pressure measured at the periphery of a degenerated disc tends to be greater than the pressure in the center [37,38]. Further, the assumption of isotropic material properties in the present FE model and the difference in the loading application technique might also contribute to the differences since the mechanical responses of the spine to moments in different planes may not be the same. Despite these variations, the difference between our results and that of Brinckmann et al. remained small and within 1.24 SD [35].

To achieve adequate immobilization at the failure level and prevent PJF, the present analysis was aimed to optimize TL reconstruction constructs to minimize motion in the failed L1 level as well as lessen the impact or burden of the constructs on the proximal junctional level. The relation between excessive motion and pseudarthrosis has been established, especially in the TL area, where T12–L2 is susceptible to premature micromotion due to its transitional biomechanics [39,40]. Our current analysis showed that although all constructs successfully reduced the pathological motion at L1, TCSR constructs were shown to provide better ROM reduction compared to PI alone. This is consistent with the clinical results showing better clinical satisfaction, improved fusion rates, and reduced segmental kyphosis in patients receiving TCSR [41]. In addition, the construct of TCSR with SSPI can also provide sufficient stability to the fractured vertebral body, thereby reducing the number of fixed vertebral segments compared to conventional LSPI. As demonstrated in our study and in the literature, this configuration provides the additional advantage of preserving more vertebral motion segments with better physiologic motion and less overall ROM reduction [24].

The present study highlighted the effect of TL constructs on the proximal junctional level by investigating the intravertebral pressure, intradiscal pressure, and facet contact force of the proximal level adjacent to the fixation. PJF remained a significant complication after TL fusion, with associated neurological injury reported in 11-19% of patients [5,42,43]. A major risk factor for PJF was an excessively long fixed spinal motion segment, which is consistent with our results that U2L2 had a higher risk than (U1L1+IS, U1L1+PMMA, and U1L1+cage) [6,7]. We found a reduced intradiscal pressure at the supradjacent disc in all motions and a decreased intravertebral pressure in the supradjacent vertebra in flexion in

the constructs with SSPI (U1L1+IS, U1L1+PMMA, and U1L1+Cage) compared to U2L2. It is important to note that although the stimulation of the highest intravertebral pressure occurs in lateral bending, the orientation of the thoracic facet joints and the presence of the ribs and thoracic cage limit the lateral motion of the thoracic segment. Therefore, intravertebral pressure exerted during flexion may be more clinically relevant than lateral bending, so most PJFs are associated with compression and kyphosis in the sagittal plane [5,6].

In addition to investigating the disc and facet joint pressures at the proximal junction near the spinal fixation device for TL burst fractures, this study also investigated the maximum von Mises stress and strain energy of pedicle screws since one of the main problems of SSPI is increased pedicle screw stress, which may contribute to the risk of early implant failure [44]. Since material failure occurs when the von Mises stress surpasses the tensile yield [45], the maximum von Mises stress in the pedicle screws is associated with the risk of acute screw breakage. However, since the tensile yield stress of titanium is approximately 880 MPa and the maximum stress in the present analysis was 29.78 MPa in U2L2, acute screw breakage is unlikely unless there is major trauma. On the other hand, cyclic strain energy during repetitive motion is related to material fatigue, so the strain energy density in pedicle screws may be an indicator of the constructs' susceptibility to implant failure due to long-term wear [46]. Our result showed that the strain energy of U1L1+PMMA and U1L1+Cage is lower than that of U2L2 and U1L1+IS, while U1L1+Cage has the least strain energy of 4.55 mJ/mm3 . A plausible explanation for this finding is the effect of stress shielding [47], where part of the axial load is transferred to the anteriorly reconstructed constructs of the PMMA or titanium cage. These results suggested that the TCSR constructs might have a lower risk of implant failure than PI alone and that the titanium cage may provide better stress shielding than PMMA. Further, among the PI constructs, our result also showed that the addition of IS to SSPI also lowers the strain energy density in pedicle screws, but the effect was less compared to TCSR constructs. Taken together, our biomechanical assessments demonstrate that TCSR with SSPI provides adequate stability for an A3 burst fracture at L1 with additional advantages in the preservation of more physiologic motion and reducing the burden on the proximal junctional level to the spinal fixation. Anterior reconstruction with PMMA or a titanium cage also provides stress shielding for pedicle screws, which may lower the risk of screw loosening or wear.

There are some limitations in the present study. First, since the transitional anatomy of the thoracolumbar junction between the rigid thoracic spine and mobile lumbar spine featured unique biomechanics, changing the level of the construct was likely to alter the biomechanical response of the TL segments. With this in mind, considering burst fracture was one of the most common indications requiring thoracolumbar reconstruction, we selected the level with the highest incidence of burst fracture, L1, for simulation [48]. A different location of burst fracture would yield different outcomes in our model. Second, the simplification of the material properties including the assumption of linear isotropic materials might not reflect the real-world behavior of the tissues and the surgical constructs. Third, the position and configuration of the implants including the pedicle screws, PMMA cement, and titanium cage are likely to have variations. Changes in the position and orientation of the implants may vary the motion and stress; however, this is very challenging to simulate since multiple real-world factors including anatomical variation, surgical approach, and surgeon's preference could all influence the positioning of the hardware. In addition, the bone quality of the spine as well as the decision on whether spinal canal decompression would be performed may also be important issues that affect the overall success of internal fixation surgery. The assumption of the thickness of the cortical bone and endplate might also influence the simulation results. Previous studies have shown that aging and degeneration resulted in decreased endplate thickness [20], and their effect that spinal biomechanics requires future studies to investigate. It should be noted that in the present FE model, convergence tests were performed separately on the spinal model and implant models, and the instrumented model was built based on modifications of the intact model after the convergence tests were performed and the mesh

size was reduced. This approach of performing convergence tests prior to the addition of implants was also utilized in previous FE publications [49–52] and had an advantage in the consistency among the FE models since only part of the model was modified in each surgical construct and the other parts remained unaltered. Finally, perfect contact with tie constraints was achieved between implants and bone. However, the main conclusions of this study were based on comparisons among the surgical construct models. The abovementioned model simplifications were equally applied to all models, yet their impacts may artificially influence the comparative analyses.

#### **5. Conclusions**

In this study, we utilized a validated FE model to investigate the biomechanics of different thoracolumbar reconstruction strategies for TL burst fracture and compared their effect on the proximal junctional level. Our results showed that TCSR constructs provided better stabilization in the fracture L1. Further, there were decreased intradiscal and intravertebral pressures in the proximal level in U1L1+IS, U1L1+PMMA, and U1L1+Cage compared to U2L2. The stress and strain energy of the pedicle screws were lower in TCSR constructs than in PI alone. We showed that TCSR with anterior reconstruction and SSPI provided sufficient immobilization while offering additional advantages in the preservation of physiological motion, the decreased burden on the proximal junctional level, and lower mechanical stress and strain in the implants. The knowledge gained from this study can provide help spine surgeons select an optimal TL reconstruction construct to minimize proximal junctional complications.

**Author Contributions:** Conceptualization, C.-E.W. and K.-Y.H.; methodology, H.-T.H. and K.-Y.H.; software, H.-T.H. and Y.-H.H.; validation, C.-E.W. and K.-Y.H.; formal analysis, C.-E.W.; investigation, C.-E.W.; resources, H.-T.H.; data curation, C.-E.W. and Y.-H.H.; writing—original draft preparation, C.-E.W.; writing—review and editing, C.-E.W. and K.-Y.H.; visualization, C.-E.W.; supervision, H.-T.H. and K.-Y.H.; project administration, H.-T.H. and K.-Y.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Science and Technology (Taiwan) under grant numbers MOST 110-2314-B-006-020 and MOST 108-2314-B-006-048-MY2.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**1. Introduction**

In last two decades, a closed-loop supply chain (CLSC) is gaining increasing attention from both industry practitioners and academics due to alleged benefits for sustainable development and growing environmental awareness among consumers and environmental regulations [1–7]. Government organizations play an impressive role in the development of sustainable product manufacturing and re-manufacturing decisions; they can enforce strict legislation, as well as offer support through various subsidy policies. In Japan, the Ministry of Environment approved 5 billion yen in 2019 as a subsidy for the manufacturer to cover 33% to 50% of their equipment price to produce products with biodegradable bio-plastics (https://bioplasticsnews.com/2018/08/27/japangovernment-bioplastics/). The government of China provides a subsidy in different ways. For example,

a manufacturer in Hunan province receives a one-time subsidy to improve re-manufacturing activities, whereas in Hubei province, Sevalo Construction Machinery Re-manufacturing Co. Ltd. receives 1 million RMB as a research and development (R&D) subsidy to improve re-manufacturing activities [8]. In 2016, the Chinese government introduced the "Guidance on Promoting Green Consumption" program to achieve the long-term goal of stimulating green product consumption. Through the Technology & Quality Up-gradation Support scheme for MSMEs (TEQUP), the government of India provides a subsidy up to 25% of the project cost for implementation of energy efficient technology. The support became one of the key factors influencing growth for the companies like Banyan Nation, Karma Recycling; in fact, the former company received the Dell People's choice award for circular economy entrepreneur at the world economic forum in Davos (www.standupmitra.in/ Home/SubsidySchemesForAll). To encourage the consumer to procure an energy-efficient green vehicle, the United States government provides subsidies up to \$7500 for the purchase of a plug-in electric vehicle [9]. Government organizations in the USA, such as The Ohio Environmental Protection Agency (EPA), has awarded a total of \$1.24 million in recycling market development grant money to upgrade and install new equipment to increase the amount of recyclables (www.recyclingtoday.com/ article/ohio-epa-recycling-grants/). The European Union also put significant efforts to promote green product manufacturing and expedite product reuse, as well as introduce various financial packages to encourage the circular economy (European Commission, 2015). Recently, an innovative and flexible pan-European network of research funding organizations, supported by EU Horizon 2020, proposed a funding of 14.530 million euros (www.era-min.eu/sites/default/files/docs/call\_text\_2018\_0.pdf). The above evidences explain that government subsidy policies are made in different ways.

Therefore, it is imperative to conduct comparative analysis for highlighting the pros and cons among those policies. Despite the necessity to explore the effect of different government subsidy policies in CLSCs under government social welfare (SW) maximization objective, comparative study is relatively sparse. This study considers omnipotence of three subsidy policies under the manufacturer-Stackelberg (MS) and retailer-Stackelberg (RS) games to pinpoint their effects and explores the answers to the following research issues:

In an attempt to answer the above questions and provide insights, we examined outcomes of eight scenarios and compared corresponding optimal decisions. For tractability, and in line with the CLSC configuration considered by previous studies [10], we mainly focused on single period optimal decision. However, product collection and network design [11] is an important aspect, we limited this study on the manufacturer collection mode only to keep our focus on the assessment of three subsidy policies under a three-stage game framework. In Policy C, the government provides a subsidy directly to the consumer [12] to stimulate a green product purchase. In Policy RE, the government shares a fraction of manufacturer investment effort to encourage used product return. In Policy T, the manufacturer receives a fraction of R&D investment from the government as a subsidy to improve the greening level (GL) [13]. To compare outcomes of subsidy policies, we derived an optimal decision under no subsidy as the benchmark, called Policy N. To explore the influence of a powerful retailer, models were formulated under both the MS and RS game frameworks; and results are compared. This study contributes to the present literature as follows: First, comparative analysis will help practitioners to understand the behavior of pricing, investment decision for the manufacturer in used product return, and R&D to improve GL in a CLSC. Second, the government sets the SW optimization goal and decides

the amount of the subsidy. Therefore, results can provide a guideline for them before implementing subsidy policies. Third, according to the investment decision, green products can be categorized as development intensive green products (DIGPs); marginal cost-intensive green products (MIGPs); and marginal-development cost-intensive green products (MDIGPs) [14,15]. Examples belonging to the first categories are developing LED bulbs; integrating an adaptive product business model, energy star home appliances, technology for product-life extension, and high-speed electric cars. All of these require a substantial amount of R&D investments. On the other hand, there is installing lithium-ion car batteries or emission control devices, using biodegradable plastics for FMCG packaging, and the manufacturing cost increasing with per unit product, all of which belong to second category. To the best of the authors' knowledge, the effect of MDIGPs on CLSCs has not been explored yet. Therefore, this study provides a complete overview for the manufacturer on the investment decision to produce MDIGPs. Finally, comparative study in the perspective of participating members can assist to formulate a framework to design a subsidy policy for green product manufacturing and re-manufacturing.

This study is organized as follows. The following subsection a brief literature review is reported to highlight the position of our research in the literature. Assumptions and background of the models are presented in Section 2. CLSC decisions under subsidy policies are discussed in Section 3. In Section 4, managerial insights are drawn with numerical illustration. Finally, conclusions, limitations, and future research are presented in Section 5.

#### *1.1. Literature Review*

It is imperative for CLSC members trading with green product to consider the evolving behavior of consumers when making important strategic and operational decisions. Possibly, Ottman [16] first reported the opportunities and the pitfalls in green marketing. The author noted double benefit for the consumer, i.e., 'personal benefit' and 'environmental benefit', in green product procurement. Since then, numerous studies related to green supply chain (GSC) management explored the properties of optimal decision under price-GL sensitive demand in different perspectives [17–19]. We will discuss some recent works focusing on variation of optimal decisions in different games under price-GL sensitive demand. Ghosh and Shah [20] compared optimal decisions obtained for different games and stated that the GL increased in the MS game, but the consumers needs to pay more. Liu and Yi [21] established that pricing and GL changes considerably under various power structures when the manufacturer also invest in knowing consumer preference information in the big data environment, and the manufacturer needs to set the lowest wholesale price under the RS game. Yang and Xiao [22] explored optimal decisions for a GSC under governmental interventions and used triangular fuzzy numbers to describe the imprecise information. The authors found that RS game scenario is the best for all players if governmental interventions are strong enough. Nielsen et al. [23] explored characteristics of the three-level GSC in a two-period setting. The authors found that the manufacturer needs to trade with the product at lower GL if the distributor dominates the GSC. Chen et al. [24] examined pricing, along with the investment effort of both the manufacturer and retailer in a GSC. The authors found that the total GSC profits increased if members share the R&D expenditure but not individual of the manufacturer or retailer simultaneously. Dey et al. [14] found that the manufacturer's decision to produce MIGPs and DIGPs is highly sensitive to the power structure. The authors found that a powerful retailer might want to trade with MIGPs, which leads to less amount of profit for the manufacturer. In this direction, the recent works of Huang et al. [25] and Ranjan and Jha [26] are worthy of mention. The findings of the above cited articles support that the game structures always made an impact on the optimal decisions. However, the above studies explored the characteristics of a forward SC. We extended this stream of research and studied the properties of CLSC under price-GL sensitive demand. CLSC is one of the great interests in both business and academic researchers due to growing consumer awareness on environmental issues and regulations. In existing literature, CLSC models are studied to explore various perspectives. For example, Hong and Yeh [27], Ma et al. [28], and Saha

et al. [29] compared optimal decisions in a CLSC under different collection mode. The authors formulated their models mainly under the MS game framework and explored the consequences where the manufacturer, retailer, or a third party, individually or jointly collects used products. On the other hand, CLSC coordination issues were comprehensively studied by Zhang and Ren [30], He et al. [31], and others. For example, Hong et al. [32], Johari and Motlagh [33], and He et al. [31] discussed effect of two-part tariff contract; Zhao et al. [34] used a commission fee contract, while a revenue-and-expense sharing contract is used by Xie et al. [35], and spanning revenue-cost sharing is used by Choi et al. [36]. On the other hand, the optimal decision under different game structures is discussed by Wang et al. [37], Gao et al. [38], Zheng et al. [39], and others. In those studies, the authors made an effort to highlight how the optimal decision changes according to game structure. We refer to recent review articles [2,40–42] on CLSC for detailed discussion. The environmental and operational measures of a CLSC network is another important aspect, where how to reduce number of vehicles to be used and the resulting carbon emissions, as well as the impact of re-manufacturing on environment, are studied extensively under an integer programming framework. We refer to the work by [43–45] for the detailed discussion in this direction. However, as mentioned earlier, it is difficult to ignore the influence of government organizations on the optimal decision in a CLSC, but literature is scanty in that direction. Zhang et al. [46] measured the supply-chain green efficiency (SCGE) of thirty-seven different industrial sectors in China and found that environmental policy management and innovation capacity of the manufacturer are important factors affecting SCGE. Researchers mostly explored the influence of a government subsidy in a forward SC. For example, Hafezalkotob [47], in addition to Sinayi and Rasti-Barzoki [12], explored the optimal decision where the consumer receives a subsidy directly from the government; Chen et al. [48] explored characteristics of optimal decisions when both the manufacturer and retailer receive a subsidy on per unit product; Safarzadeh and Rasti-Barzoki [49] discussed the optimal decision for a two-echelon SC when the manufacturer receives a subsidy on the R&D investment. To establish the position of the present study, we outline existing work on CLSCs under influence of a subsidy in Table 1.

**Table 1.** Comparison of existing studies with the present study. SW = social welfare; GL = greening level; MS = manufacturer-Stackelberg; RS = retailer-Stackelberg.

Table 1 demonstrates that most of the articles focused on the behavior of participants under a single subsidy policy, mostly under the MS game setting. The effect of joint influence of price-GL is also ignored. With growing awareness about green products, the influence of GL needs to be considered to obtain a pragmatic CLSC decision. Comparative study to explore preferences of the CLSC members, consumers, and government organizations are not examined in the previous literature. In this study, the investment and pricing decisions of a CLSC members are explored by correlating the optimal decision of the government organization. This study can help practitioners to understand the pricing and investment patterns for the manufacturer under the MS and RS games in a CLSC setting. Comparative analysis conducted in this study on the efficiency of investment and consumer preference can help government organizations to cultivate a pragmatic subsidy policy.

#### **2. Prerequisites and Assumptions**

We considered eight different scenarios, namely Scenarios *ij*, *i* ∈ {*m*,*r*}, which signifies MS and RS games; *j* ∈ {*C*, *RE*, *T*, *N*}, which refers to the models where consumers receive a subsidy (C), the manufacturer receives a subsidy on the investment effort on recycling (RE), and the manufacturer receives a subsidy on the total R&D investment (T); and the benchmark decision model where the government organizations do not provide a subsidy (N), respectively. Therefore, the first index represents the game structure, and the second one represents the subsidy policy. The following notations presented in Table 2 are used to differentiate the decision and auxiliary variables in different scenarios:

**Table 2.** Decision and auxiliary variables.

The following assumptions are made to establish proposed models:

a R&D subsidy on the total investment. As noted by Dey et al. [14], for MIGPs, the variable cost is directly proportional with the product quality, and it might not possible to recover the cost for the manufacturers. For example, installing emission reduction devices or packaging material are directly proportional to the unit product, but it is difficult for the manufacturer to recover the cost of those in the re-manufacturing process.

	- Step 1: The government decides the subsidy rate (*ρC i* or *ηRE i* or *μT i* ) by maximizing social welfare;
	- Step 2: In the MS game, the manufacturer decides *wj m*, *θ j m*, and *τj m*. In the RS game, the retailer decides the profit margin *mj r* = *p j r* − *wj r*;
	- Step 3: In the MS game, the retailer decides the retail price *p j m*. In the RS game, the manufacturer decides *wj r*, *θ j r*, and *τj r*.

Therefore, the government takes the responses of the CLSC members into consideration, while deciding the subsidy rate [48,59].


**Conflict of interest**

The authors declare that there are no existing or possible conflicts of interest, including financial, personal, or any other relationship which could influence their scientific work, at the time of manuscript submission.