The inconvenience caused by the harsh environment has been particularly long. How to improve social and economic benefits under the harmonious coexistence of human and nature has become a key topic of concern for the government and firms.
In order to adapt to the environmental protection and sustainable development, lowcarbon products initiated by the government and implemented by firms came into being. In recent years, scholars have tried their best to provide suggestions for lowcarbon supply chain management (see literature review by [1]). In particular, extensive studies are published to discuss the lowcarbon strategies in various polices such as carbon cap and trade (e.g., [2,3]), carbon tax (e.g., [4,5]), low carbon subsidy (e.g., [6,7]), and hybrid low carbon policy (e.g., [8,9]).
Apart form the impact of various policies, the firms’ lowcarbon strategies are also related to the composition of market. For better deal with firms’ decision problems under different power structures, decision makers are often apply Stackelberg game or Nash game models. Zhou et al. [10] construct a social welfare model with Stackelberg game considering carbon emissions, and show that an increasing block carbon tax policy can encourage lowcarbon production. Du et al. [11] propose a carbonrelated pricediscount sharinglike scheme under manufacturer Stackelberg game, and point out that this contract can achieve low carbon supply chain coordination. Xu and Wang [12] consider a twoperiod closedloop supply chain with retail price and emission reduction dependent demand, and obtain the equilibrium decision strategy and profit distribution under the Stackelberg game frame and Nash game frame, respectively. In fact, there are also other literature reviews that explore the lowcarbon strategy of closedloop supply chains under the game framework from multiple perspectives (e.g., [13, 14, 15]). The above studies are conducted under deterministic demand, while those do not take the uncertain risk of demand into account.
In actual transactions, market demand is not static. The variability of the environment, the psychological expectations of consumers, and the diversity of consumer values, all of those will cause market demand become uncertain. In order to better predict the random demand, scholars have contributed their own strength. Cohen et al. [16] study government subsidies for green technology adoption, and show that an increase in demand uncertainty leads to higher production quantities and lower prices. Rong et al. [17] consider product line design under endogenous substitution, and demonstrate that the presence of demand uncertainty can reduce the benefit of market segmentation. Feng and Viswanathan [18] discuss the impact of demand uncertainty on the effectiveness of coordinating such a supply chain, and find that coordination through common replenishment epochs may not always be beneficial when the demand variance is high. Actually, more and more researchers make decisions with random demand in supply chain management (e.g., [19, 20, 21]).
In practice, manufacturers may be a Stackelberg leader, faced a random demand, and taken the maketostock mode. However, the previous literature has not adequately investigated the lowcarbon strategies when these three issues are considered altogether. In summary, different from [22], we focus on lowcarbon strategy. Different from [23] and [24], we discuss a decentralized decision, which includes Stackelberg game and Nash game. Different from [25] concerning the maketoorder mode, we demonstrate the maketostock mode. Different from [26, 27, 28], we investigate the impacts of demand uncertainty. The characteristics and contributions of our study are presented in Table 1.
Summary of relevant literature
Authors  Lowcarbon strategy 
Demand form 
Type of game 
Production mode 


With  Without  Deterministic  Random  Stackelberg  Nash  Make to order  Make to stock  
Shi et al. [22]  ✓  ✓  ✓  ✓  ✓  
Jiang et al. [23]  ✓  ✓  ✓  
Zhang et al. [24]  ✓  ✓  ✓  
Bai et al. [25]  ✓  ✓  ✓  ✓  
Liu et al. [26]  ✓  ✓  ✓  ✓  
Meng et al. [27]  ✓  ✓  ✓  ✓  ✓  
Shi et al. [28]  ✓  ✓  ✓  ✓  
This paper  ✓  ✓  ✓  ✓  ✓ 
Motivated by the research gap, this paper establishes a supply chain with one riskneutral manufacturer and one riskneutral retailer and considers Stackelberg modes to derive the optimal lowcarbon strategies under random demand. Before that, we firstly summarize our main contribution as follows and unfold them in the remaining part of this paper.
First, we establish three Stackelberg models, including Manufacture Stackelberg (MS), Retailer Stackelberg (RS) and Nash. We demonstrate the related equilibrium stocking factor, emission reduction level, wholesale price and retail price of those three models that maximize the firm’s expected profit, respectively. Second, we illustrate effects of the power structures. We show that the MS model has the highest wholesale price, while it has the lowest stocking factor. Third, we reveal the impacts of the demand uncertainty via numerical analysis. We find out that as the market potential or the lowcarbon sensitivity coefficient increases, the emission reduction level, retail price, the firms’ expected profits increase. While those decisions and expected profits are decreasing in the price sensitivity coefficient. The remainder of the paper is organized as follows. In Section 2, we introduce assumptions and notions. Section 3 analysis three Stackelberg models, including MS, RS and Nash. Section 4 discusses the effects of the power structure. Section 5 investigates the impacts of demand uncertainty. Finally, Section 6 concludes and presents future research. All technical proofs are presented in Appendix.
We consider a singleperiod supply chain consisting of a riskneutral manufacturer (she) and a riskneutral retailer (he). In order to reduce carbon emissions and achieve environmental protection, the manufacturer need employ lowcarbon technologies. In this case, the cost of lowcarbon technologies for the manufacturer is
After purchasing ecofriendly products from the manufacturer at the wholesale price
The notions and parameters.
Stocking factor  
Retail price  
Wholesale price  
Emission reduction level  
Retail margin, 

Cost of emergency procurement  
Government subsidies  
Expectation operators  
Represent the firms, 

Represent the models, 

Market potential  
Price sensitivity coefficient  
Lowcarbon sensitivity coefficient  
Emission reduction cost coefficient  
Unit production cost  
Salvaged value  
Probability density function of ·  
Distribution function of ·  
Expected profit  
Order quantity 
For tractability, some assumptions are provided as follows.
The shock demand
4
The retailer’s order quantity
In this section, three cases are discussed according to the different power structure of the firms. Case 1, the manufacturer is a Stackelbergleader and the retailer is the follower. Case 2, the retailer acts as a Stackelbergleader and the manufacturer acts as the follower. Case 3, the manufacture and the retailer have equal status. Throughout this paper, the expected profits of the manufacturer and the retailer are as following, respectively.
In this case, the manufacturer acts as the leader and sets the wholesale price and the emission reduction level. The retailer acts as the follower and sets the retail price and the stocking factor.
Considering the retailer’s decisions, for a given wholesale price
Taking the Eqs. (4) and (5) into Eq. (3), the manufacturer decides on the wholesale price and the emission reduction level that maximizes its profit. We can obtain that
According to Eq. (5), we have that
According to Lemma 1, by substituting
Similarly, by substituting
In this case, the retailer acts as the leader and sets the retail margin and stocking factor firstly. The manufacturer responds by choosing the wholesale price and emission reduction level.
Recall
The manufacturer’s expected profit is
From Eq. (12), we know that
Lemma 2 reflects that the manufacturer’s response, we now examine the retailer’s setting in terms of
Thus the equilibrium stocking factor
Therefore the equilibrium retail price is
In this case, the manufacturer and the retailer choose the wholesale price, retail margin, and emission reduction level simultaneously. For a given wholesale price
The manufacturer’s expected profit is
The retailer’s expected profit is
Lemma 3 shows that the Nash game also exist a unique purestrategy Nash equilibrium. What are the differences of those equilibrium strategies among the MS, RS, and Nash models? Next, we will try to discuss this issue.
In this section, we compare the equilibrium strategies of the three games, and show the effects of power structure.
Theorem 4 reflects that the stocking factor decrease as the power shifts from the retailer to the manufacturer. It lies in the fact that in order to prevent the loss of potential customers under the unknown random demand, the more power of the retailer has, the greater his willingness to order.
Theorem 5 reflects that the wholesale price increase as the power shifts from the retailer to the manufacturer. It is in line with the fact that as the increase in the power of the manufacturer, she pays more attention to her own income, and at this time, appropriately increasing the wholesale price has become one of her means.
Theorem 6 reflects that compared to the Nash game, the emission reduction level in retailerled model is in a weak position, while retail price is higher. The reason may be found from that he retailer’s profit from the market through retail price, and the manufacturer is the main body implementing lowcarbon strategy.
According to Theorem 6, it also find that twice the emission reduction level and retail price in Manufacturerled model are higher than that in Nash game, respectively. So one interesting question is appeared: which one is higher of the emission reduction level in MS game and Nash game? Similarly, how different of the retail price in MS game and Nash game? The next section may give the answers.
In this section, a series of numerical investigations are considered to present new management insights. We suppose that the demand risk is uniformly distributed in the region [−5,5], and the other exogenous variables follow that
On the one hand, we consider the impact of random demand on the equilibrium strategies. Table 3 reflects that as the market potential increases, the stocking factor decreases, while the emission reduction level, wholesale price and retail price increase. Table 4 reflects that the stocking factor is increasing with respect to the price sensitivity coefficient, while the emission reduction level, wholesale price and retail price are decreasing with respect to the price sensitivity coefficient. Table 5 reflects that as the lowcarbon sensitivity coefficient continues to increase, the stocking factor shows a gradual decreasing trend, while the emission reduction level, wholesale price and retail price show a gradually increasing trend.
Impact of
a  

40  6.44  6.01  5.08  2.02  2.59  1.71  4.21  4.59  5.43  7.03  6.55  6.88 
50  5.96  5.41  4.29  2.64  3.13  2.07  4.64  5.13  6.14  8.49  7.88  8.28 
60  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
70  5.00  4.20  2.69  3.50  4.22  2.78  5.50  6.22  7.58  11.42  10.53  11.07 
Impact of
b  

3  3.44  2.48  0.87  4.90  5.77  3.61  6.90  7.77  9.22  16.06  14.85  15.21 
4  4.67  3.87  2.40  3.79  4.52  2.92  5.79  6.52  7.84  12.27  11.32  11.79 
5  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
6  6.05  5.48  4.31  2.56  3.07  2.06  4.56  5.07  6.13  8.41  7.79  8.23 
Impact of
t  

0.5  5.64  4.97  3.63  1.46  1.77  1.18  4.92  5.53  6.73  9.71  8.85  9.43 
1  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
1.5  5.15  4.51  3.23  5.05  5.91  3.82  5.36  5.94  7.09  10.43  9.86  10.12 
2  4.53  4.01  2.84  7.85  8.78  5.45  5.92  6.39  7.45  11.25  10.95  10.81 
All tables from table 3 to table 5 reveal that
Figure 1 and Figure 2 reflect that the firms’ expected profits are increasing in the market potential. Figure 3 and Figure 4 reflect that the firms’ expected profits are decreasing regarding the price sensitivity coefficient.
Figure 5 and Figure 6 reflect that the firms’ expected profits are increasing with respect to the lowcarbon sensitivity coefficient. According to the figures from Figure 1 to Figure 6, it indicates that when the power is shifted from the retailer to the manufacturer, the manufacturer’s expected profit increase while the retailer’s expected profit decrease. This further reveals the fact that firms are selfserving.
In order to further cherish life, environmental protection becomes more important, and a lowcarbon economy is once again mentioned as one of the main means of environmental protection. This article considers the manufacturerretailer supply chain and discusses the manufacturer’s lowcarbon strategy under the setting that market demand is uncertain. The results are as follows:
Firstly, three Stackelberg models, including MS, RS and Nash, are established. We obtain the equilibrium stocking factor, emission reduction level, wholesale price and retail price, respectively.
Secondly, the effects of power structure are discussed. We show that as the power shifts from the retailer to the manufacturer, the stocking factor and the retailer’s expected profit decrease while the wholesale price and the manufacturer’s expected profit increase. Meanwhile, the retail price appears as a Ushaped curve, while the emission reduction level appears as an inverted Ushaped curve.
Thirdly, the impacts of demand uncertainty are demonstrated. We find that a higher market potential or a higher lowcarbon sensitivity, leads to a higher emission reduction level, wholesale price, retail price and firms’ expected profits, while leads to a lower stocking factor. Meanwhile, a higher price sensitivity leads to a lower emission reduction level, wholesale price, retail price and firms’ expected profits, while leads to a higher stocking factor.
For further research, an interesting direction is to consider the issue of competition when multiple manufactures are involved. Moreover, given that risk aversion can effectively influence the decisions of firms, the other possible direction may be consider this scenario.
Impact of b on the equilibrium strategies (where a = 60 and t = 1)
b  

3  3.44  2.48  0.87  4.90  5.77  3.61  6.90  7.77  9.22  16.06  14.85  15.21 
4  4.67  3.87  2.40  3.79  4.52  2.92  5.79  6.52  7.84  12.27  11.32  11.79 
5  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
6  6.05  5.48  4.31  2.56  3.07  2.06  4.56  5.07  6.13  8.41  7.79  8.23 
The notions and parameters.
Stocking factor  
Retail price  
Wholesale price  
Emission reduction level  
Retail margin, 

Cost of emergency procurement  
Government subsidies  
Expectation operators  
Represent the firms, 

Represent the models, 

Market potential  
Price sensitivity coefficient  
Lowcarbon sensitivity coefficient  
Emission reduction cost coefficient  
Unit production cost  
Salvaged value  
Probability density function of ·  
Distribution function of ·  
Expected profit  
Order quantity 
Impact of a on the equilibrium strategies (where b = 5 and t = 1)
a  

40  6.44  6.01  5.08  2.02  2.59  1.71  4.21  4.59  5.43  7.03  6.55  6.88 
50  5.96  5.41  4.29  2.64  3.13  2.07  4.64  5.13  6.14  8.49  7.88  8.28 
60  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
70  5.00  4.20  2.69  3.50  4.22  2.78  5.50  6.22  7.58  11.42  10.53  11.07 
Summary of relevant literature
Authors  Lowcarbon strategy 
Demand form 
Type of game 
Production mode 


With  Without  Deterministic  Random  Stackelberg  Nash  Make to order  Make to stock  
Shi et al. [ 
✓  ✓  ✓  ✓  ✓  
Jiang et al. [ 
✓  ✓  ✓  
Zhang et al. [ 
✓  ✓  ✓  
Bai et al. [ 
✓  ✓  ✓  ✓  
Liu et al. [ 
✓  ✓  ✓  ✓  
Meng et al. [ 
✓  ✓  ✓  ✓  ✓  
Shi et al. [ 
✓  ✓  ✓  ✓  
This paper  ✓  ✓  ✓  ✓  ✓ 
Impact of t on the equilibrium strategies (where a = 60 and b = 5)
t  

0.5  5.64  4.97  3.63  1.46  1.77  1.18  4.92  5.53  6.73  9.71  8.85  9.43 
1  5.48  4.81  3.49  3.07  3.67  2.43  5.07  5.67  6.86  9.96  9.20  9.67 
1.5  5.15  4.51  3.23  5.05  5.91  3.82  5.36  5.94  7.09  10.43  9.86  10.12 
2  4.53  4.01  2.84  7.85  8.78  5.45  5.92  6.39  7.45  11.25  10.95  10.81 
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Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for EnglishChinese CrossLanguage Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontaltorsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Research on evaluation system of crossborder Ecommerce platform based on the combined model A mathematical model of PCNN for image fusion with nonsampled contourlet transform Nonlinear Differential Equations in ComputerAided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with DiamondSquare Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Crossborder Ecommerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Timedelay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of watersaving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of AirConditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the winlose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises Research on classification of ecommerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Crossborder Ecommerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edgebased finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of ecommerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on ecommerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks A study of local smoothnessinformed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRABiLSTMCRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithmbased congestion control optimisation for mobile data network