Integrated Optimization of Sustainable Transportation and Inventory with Multiplayer Dynamic Game under Carbon Tax Policy

Growth in environmental sustainability has prompted the logistics industry to seek sustainable development, and carbon tax policies are considered an effective approach to reducing carbon emissions.,is study investigates the optimization of sustainable transportation and inventory under a carbon tax policy and explores effective methods for coordinating the interests of governments and enterprises.,e results can provide insights into sustainable logistics for decision-making by enterprises and policymaking by governments. We first examine a Stackelberg game model and design an iterative solution to optimize sustainable transportation and inventory under the carbon tax policy. We then establish a three-stage dynamic game model to optimize the wholesale price, carbon tax rate, and proportion of sustainable investment shared by the government. Furthermore, we perform a simulation to identify the optimal solution of the three-stage game, and we compare the simulation results with a numerical example. ,e results indicate that a carbon tax policy can improve social welfare and the sustainability of transportation and inventory but could hinder corporate profits. An appropriate sustainable investment-sharing strategy could compensate for the shortcomings of the carbon tax policy and result in positive outcomes for governments and enterprises.


Introduction
With increasingly severe environmental pollution worldwide, people have gradually become aware of the importance of sustainable development. An increasing number of countries are taking measures to control greenhouse gas emissions, and organizations and individuals are undertaking environmental-protection actions to reduce carbon emissions [1]. To address pollution, developed and developing countries have implemented regulations aimed at reducing carbon emissions [2]. A carbon tax is one such emissions-reduction regulation that has been widely implemented to encourage companies to reduce their carbon emissions [3]. Carbon taxes have been adopted in Australia, Japan, Denmark, Austria, Finland, and Ireland [2]. In Australia, the carbon tax was implemented in 2011, with a rate of AU$23 per ton of carbon in 2012 [4]. A carbon tax increases the cost of operations [5], which encourages enterprises to increase their sustainable investments. Dong et al. [6] indicated that sustainable investment can reduce the carbon emissions of manufacturers, and both manufacturers and sellers can benefit from it. Sustainable investment improves not only environmental sustainability but also a firm's long-term profitability and competitiveness [7]. For example, in 2007, Marks & Spencer invested £200 million in carbon-reduction efforts; moreover, Walmart has reduced its carbon emissions by 667,000 m 3 after requiring its 60,000 suppliers to reduce their packaging by 5% [8].
Given this context, enterprises are motivated to optimize their supply chains and invest in clean-energy technologies to reduce carbon emissions [9]. Sustainable supply chain management is a growing concern because production and logistics activities lead to waste and pollution [10]. e logistics industry consumes energy and fossil fuels and is a major source of carbon emissions in the supply chain [11]. A survey of the supply chain operations of 2,500 large enterprises revealed that logistics-induced emissions exceeded 20% of total supply chain emissions [12]. is result demonstrates that logistics has a significant negative impact on both environmental sustainability and economic growth. To protect the environment, governments should implement strict environmental-protection policies and encourage sustainable investment in the logistics industry [13,14].
Transportation and inventory management are the main sources of carbon emissions in the logistics industry [15]. Transportation is the largest source of carbon emissions. Inventory control determines logistics factors, such as inventory level, warehousing activities, and transportation frequency [12], which are crucial for decision-making in supply chain operations. In Europe, transportation is the sector with the second highest carbon emissions (the energy supply sector has the highest), accounting for 23% of total carbon emissions [5]. Transportation and inventory are the two most critical elements of logistics in terms of both economics and ecology. e efficient management of a logistics system requires an integrated approach that combines various logistics functions [16]. Furthermore, few studies have explored the optimization of sustainable investment through the integration of transportation and inventory decision-making on a global scale. erefore, research on integrating decision-making in sustainable transportation and inventory would facilitate the reduction of carbon emissions in the logistics industry and promote sustainable development.
is study investigated the relationships between government, suppliers, and retailers ( Figure 1). Governments adopt carbon tax policies to optimize social welfare. Suppliers and retailers make decisions concerning transportation and inventory management, respectively, with the aim of maximizing profits. Carbon emissions in transportation and inventory management have been analyzed, but carbon emissions due to other activities in the supply chain have not been considered. erefore, we investigated the following research questions: (1) Under a carbon tax, how can suppliers and retailers maintain sustainable levels of transportation and inventory? (2) How do governments set carbon tax rates and incentives? (3) What effect does a carbon tax have on supply chain performance, and how can government and enterprise objectives be harmonized?
is study aimed to achieve three specific goals: (1) to establish an integrated optimization model for sustainable levels of transportation and inventory under a carbon tax; (2) to construct a multiplayer dynamic game model to determine the optimal carbon tax rate and the sustainable investment-sharing strategy; and (3) to solve this multiplayer dynamic game model through a MATLAB simulation and to explore the effect of a carbon tax on sustainable transportation and inventory, enterprise profits, and social welfare. e remainder of this paper is organized as follows. Relevant studies are discussed in Section 2. Section 3 proposes assumptions and details of the notations used. Section 4 presents the model for sustainable transportation and inventory optimization under a carbon tax, along with an analysis of and solution to the model. A three-stage game model, which was designed to optimize the carbon tax rate and sustainable investment-sharing strategies, is presented in Section 5. Section 6 provides the results of simulations performed to solve the three-stage model numerically, along with an analysis of the effect of a carbon tax on decisionmaking. Conclusion is presented in Section 7.

Literature Review
Numerous studies have explored sustainability in the supply chain. ose highly related to the present topic were divided into three categories according to the topic: carbon tax, integrated environmental optimization of transportation and inventory management, and sustainable investment in the supply chain.

Carbon Tax.
A carbon tax is an effective approach to mitigate climate change and is one of the two main instruments for redesigning the supply chain (the other being cap-and-trade programs) [2]. A succession of studies have focused on the effect of a carbon tax on emission reduction. Shu et al. [17] reported that a carbon tax can place a heavier financial burden on companies to meet certain emission targets than other carbon-reduction policies can. Chen and Nie [18] demonstrated that a specific number of carbon taxes in the production process can improve social welfare, whereas consumption and redistribution reduce social welfare. Moreover, Olsen et al. [19] indicated that if a carbon tax is implemented after properly analyzing its impact, its revenue can be used to support government investment in projects that reduce emissions. Ma et al. [20] argued that economists and policymakers prefer a carbon tax because its implementation requires less management than that of other carbon-reduction policies. However, Xie et al. [21] indicated that a carbon tax harms economic growth because it inevitably entails companies incurring additional costs. Many studies have investigated the significant impact of a carbon tax on supply chain performance. Bao and Zhang [22] developed a mixed linear programming model to explore sustainable procurement relationships in a supply chain under a carbon tax scheme. Fahimnia et al. [4] proposed a tactical supply-planning model that optimizes carbon emissions and economic targets under a carbon tax. Xin et al. [23] discussed the problem of sustainably scheduling a shuttle tanker fleet with variable tanker speed under a carbon tax. Wang et al. [24] examined decisions about a carbon tax in decentralized and centralized supply chains by using the Stackelberg game model. ey proved that a carbon tax in a decentralized supply chain should be higher than that in a centralized supply chain. Wang et al. [25] derived the optimal level of supply chain emission reduction with stochastic demand under a carbon tax. ey reported that optimal emission reductions gradually increased with an increase in the carbon tax rate. Ding et al. [26] developed a diffusion model of energy technology to explore the possible impacts of various carbon tax conditions on the diffusion of energy technologies in China. Alizadeh et al. [27] studied the impact of the uncertainty of the carbon tax rate on emissions in a supply chain. eir results demonstrated that increasing the uncertainty of the carbon tax rate from 0 to 30 reduced total network emissions by 2.8%. However, integrating sustainable investment into logistics operating decisions has not been considered in most of these studies.

Integrated Environmental Optimization of Transportation
and Inventory Management. Employees usually make decisions regarding inventory management and transportation in an organization. Collaboration is the optimal supply chain strategy [28]. e interaction between inventory parameter decisions and transportation choices indicates that, to optimize transportation and inventory management, they must be integrated. Integrated optimization problems can be categorized as strategic or tactical. Gaur and Fisher [29] divided integration optimization problems into those with finite and infinite periods. Four main types of logistics networks exist, namely, one-to-one, one-to-many, many-toone, and many-to-many [30]. is study focused on one-toone networks. One-to-one networks have not a routing problem, whereas the other three network types do.
Recently, increasing numbers of scholars have included carbon emission factors in integrated decision models for transportation and inventory management. Economic order quantity (EOQ) models with various carbon emission regulations are the predominant tool employed in most studies on transportation and inventory management. Bonney and Jaber [31] presented an EOQ model associated with carbon emissions and waste produced by transportation and inventory operations. Tang et al. [32] integrated emissions, transportation costs, and inventory costs into the (R, Q) policy with stochastic demand. Hovelaque and Bironneau [33] considered the carbon emission problem in an EOQ model with demand dependent on price and carbon emissions. Hua et al. [34] constructed an EOQ model for a carbon cap-and-trade policy and analyzed the trends of replenishment decisions, costs, and carbon emissions with changes in the carbon trading price. Chen et al. [35] proposed a condition for achieving a carbon emission-reduction target by adjusting the order quantity based on an EOQ model. Gautam et al. [36] integrated defect management into a sustainable supply chain model to jointly optimize the number of shipments and quantities of orders and backorders. Konur and Schaefer [12] developed integration models for transportation and inventory management for four carbon policies. ey compared and analyzed the impacts of these policies on the optimal order quantity. Wang et al. [39] studied the optimization of refined oil distribution under a carbon tax. eir results revealed that a carbon tax effectively reduced the carbon emissions due to refined oil distribution. Reddy et al. [38] established a multiperiod reverse logistics network design model and analyzed the impact of a carbon tax on the optimal decisions. Konur and Schaefer [12] examined the EOQ model for lessthan-truckload and truckload shipping under cap-and-offset, cap-and-trade, carbon cap, and carbon tax policies. Micheli and Mantella [5] designed a comprehensive model of transportation and inventory under the cap, cap-andoffset, cap-and-trade, and carbon tax. ey involved a comprehensive emission model with vehicles.
In-depth research has been conducted on optimizing the integration of transportation and inventory management. In such research, the main decision-making objectives are to optimize transportation and inventory management under certain economic and social goals. Studies on environmental factors have focused on the impact of carbon emissions and  Mathematical Problems in Engineering carbon-reduction policies on transportation and inventory integration decisions. e present study extends the literature by introducing sustainable investment factors into the decision-making model for transportation and inventory integration, and we focused on the optimal decisions to reach sustainable levels of emissions due to transportation and inventory.

Sustainable Investment in a Supply
Chain. Supply chain emission reduction has become a critical, theoretical, and practical topic, with numerous studies discussing it and proposing investment in clean technologies to reduce carbon emissions (i.e., sustainable investment). Several studies have focused on the effectiveness of sustainable investment. Drake and Spinler [37] noted that the effectiveness of clean technologies in sustainable economic development should not be underestimated. Shi et al. [40] analyzed the comprehensive effects of power structures and sustainable investment in the supply chain on the economy and environment. Su et al. [41] developed a pricing decision model to explore the effects of government subsidies on optimal decisions for sustainable supply chain management under various government subsidies and power structures. Scholars have often studied sustainable investment decisionmaking under multiple carbon-reduction policies. Benjaafar et al. [42] studied the impact of clean technologies on carbon reduction. e results indicated that a carbon cap-and-trade regulation can effectively encourage companies to adopt clean technologies if the benefits of a clean technology are substantial. Krass et al. [43] believed that increasing taxes may prompt companies to switch to clean technology when opting for emission control technology. If the input cost of clean technology is compensated for, the negative environmental impact will be eliminated and the tax will have proved effective. Chen [44] explored the impact of a carbon tax on the choice of energy-saving technologies by enterprises under market competition. Chen's results demonstrated that clean and conventional technologies can coexist and that companies choose technologies according to the tax rate. Shifting to clean technology is not always the optimal choice when dealing with a rising tax rate. Drake et al. [45] compared the impacts of a carbon tax and a carbon cap-andtrade policy on the corporate sustainable technology choice and capacity decisions, and their results revealed that the expected profit under a cap-and-trade policy was higher than that under a carbon tax.
With the deepening of research into sustainable investment, integrating sustainable investment into supply chain activities such as procurement, production, and inventory has received increasing attention. Toptal et al. [46] introduced retailers' sustainable investments into an EOQ model and constructed joint decisions for sustainable investment and inventory management under carbon cap and cap-and-trade policies. e model revealed that sustainable investment can simultaneously reduce carbon emissions and costs. Dong et al. [6] studied the sustainable investment problem under a carbon cap-and-trade policy and constructed a decision-making model for decentralized coordination and centralized control of a supply chain. e conclusions indicated that sustainable investment substantially influenced the optimal supply chain strategy. Based on the results of Dong et al. [6], Cheng et al. [8] introduced big data to the sustainable investment decision-making model. eir results revealed that adopting big data was not always the optimal choice for retailers. Moreover, they showed that whether manufacturers increase sustainable investment depends on the service level set by the retailer.
With continual advancements in sustainability, consumers are increasingly concerned about the environmental impact of goods, and this is referred to as consumer environmental awareness [5]. Consumer environmental awareness has often been integrated into the supply chain optimization model along with sustainable investment. For example, Zhang and Liu [47] examined pricing strategies in a three-level supply chain where demand was relative to the environmental impact of products. Du et al. [48] proposed a carbon-sensitive demand function to investigate the impact of consumers' preferences for reducing carbon emissions on a supply chain, and they designed various supply chain coordination contracts. Consumer environmental awareness has two main impacts on the sustainable supply chain model. e first is the impact on market demand; when carbon emissions decrease, market demand increases [6,8].
e second is the impact on prices; when consumer environmental awareness increases, low-carbon products receive more recognition, even among companies with higher prices [48]. e higher consumer environmental awareness is, the greater is the willingness of consumers to pay for sustainable products [49]. Consumer environmental awareness motivates manufacturers to produce more sustainable products, but this does not necessarily mean higher profits for manufacturers [50].
Toptal et al. [46], Dong et al. [6], and Cheng et al. [8] have conducted similar investigations into retailers' perspectives regarding inventory replenishment and sustainable investment. Dong et al. [6] and Cheng et al. [8] have investigated sustainable investments in production. By contrast, the present study investigated sustainable investments in logistics. Toptal et al. [46] analyzed sustainable investment in inventory management, whereas we considered sustainable investments in both transportation and inventory management. Moreover, Toptal et al. [46] explored deterministic demand. By contrast, the present study included the sales effort factor and sustainable investment in a random demand function to harmonize the optimization model with the real enterprise situation. Table 1 presents the studies related to the present paper.

Model Assumptions and Notation
In this study, we considered a supply chain consisting of one supplier and one retailer selling a single product. Over a single cycle, the supplier sells the product to the retailer at the wholesale price, and the supplier is responsible for shipping the product from its factory to the retailer's warehouse. e retailer is responsible for product ordering and inventory management, and the retailer sells the product to consumers at the market retail price. In a carbon tax scenario, decision-making involves two consecutive steps. In the first step, the supplier determines the appropriate extent of sustainable investment for transportation given the goal of maximizing profit. In the second step, given a sustainable level of transportation, the retailer determines the sustainable inventory level and order quantity for the product. e notations used in the models are summarized in Table 2. e models are limited by the following assumptions: (1) A supplier trades with a retailer during a single period. e retailer can only order once during the period. If the retailer's order quantity exceeds market demand, the leftover stock is disposed at the unit salvage value v. Without loss of generality, we assume that v < C s , where C s is the unit production cost. Moreover, we assume that the explicit shortage cost is zero if the market demand exceeds the retailer's order quantity [6]. When market demand exceeds the retailer's order quantity, the retailer sells out its inventory; urgent orders are not allowed. (2) e sustainability level is a dimensionless index that measures carbon emissions. e higher the sustainability level is, the lower are the carbon emissions. Assume that the sustainable transportation level s p ∈ [0, (a p /b p )] and the sustainable inventory level s l ∈ [0, (a l /b l )]. When s p � 0, carbon emitted per unit of the product for transportation is a p . When s l � 0, carbon emitted per unit of the product for transportation is a l . Carbon emitted per unit of the product for inventory is a p − b p s p . Carbon emitted per unit of the product for inventory is a l − b l s l .
(3) Sales effort x is a comprehensive index that reflects the retailer's efforts to promote market demand through activities such as advertising, services, and sales promotions. Let x ≥ 0; we assume that represents the demand generated by sales effort and g(x) represents the costs incurred by sales effort. e assumption indicates that f(x) and g(x) are increasing functions. However, the rate of demand increase decreases with x, and the rate of cost increase is zero or increasing with x [52].
(4) Product demand D is uncertain; it depends not only on the retailer's sales effort x, the sustainable inventory level s l , and the sustainable transportation level of the supplier s p but also on the market's random demand factor ξ, as expressed in the fol- Moreover, β is the coefficient of the effect of sustainability on increasing demand, ξ is the random demand with mean μ, and the probability density function and probability distribution function are ϕ(y) and F(y), respectively. Furthermore, β(s p + s l ) represents the demand generated by the sustainable transportation level s p and sustainable inventory level s l . If β > 0, then the sustainability level has a positive effect on the demand. When s p � 0 and s l � 0, the demand generated by the sustainability level is zero. e maximum values of demand generated by s p and s l are βa p /b p and βa l /b l , respectively. (5) e higher the sustainability level is, the higher is the cost of sustainable investment and the faster is the cost increase. Similar to [6,8,53], we assume that the sustainable investment costs for transportation and inventory are quadratic functions, specifically δ l s 2 l /2 and δ p s 2 p /2, respectively, where δ l and δ p are the sustainable investment coefficients for transportation and inventory, respectively. In practice, the costs of sustainable investment are usually high. erefore, we assume that δ l and δ p are sufficiently high that δ l ≥ 2rb l β and δ p ≥ 2rb p β, where r is the carbon tax rate. Otherwise, the lower bound and upper bound of the sustainability level interval are the optimal values of s p and s l ; that is, s p � 0 or (a p /b p ) and s l � 0 or (a l /b l ). e assumption is that δ l ≥ 2rb l β and δ p ≥ 2rb p β can avoid these simple solutions [6].

Optimization Model for Sustainable Transportation and Inventory under a Carbon Tax
Based on the Stackelberg game between the supplier and the retailer, this paper uses a backward sequential decisionmaking approach. First, the retailer's sales effort, sustainable  [21]. When considering the carbon tax policy, the expected profit of retailers is calculated as follows: (1) In equation (1), the first term is the income of the sales product, the second term is the purchasing cost of the product from the supplier and the logistics cost, the third term is the sales cost of the retailer, the fourth term is the sustainable investment cost in the inventory, the fifth term is the remaining cost, the sixth term is the carbon tax cost, and (a l − b l s l )Q is the carbon emissions due to inventory. Based on equation (1), when considering the carbon tax, the retailer's sales effort x is affected by the sustainable inventory level s l . e retailer must integrate decisions related to the sales effort x and the sustainable inventory level s l , which add complexity to the decision process.
Given x, s p , and s l , the first and second derivatives of equation (1) with respect to Q yield the following proposition.

Proposition 1.
Under the carbon tax, the retailer's optimal order quantity, denoted by Q * , for given x, s p , and s l , is as follows: e optimal order quantity Q * increases with an increase in the remaining cost v and decreases with increases in the market price p, the wholesale price w, and the inventory cost c l . e first derivatives of Q * with respect to x and s p are shown in equation (3).
ese equations indicate that the optimal order quantity Q * increases with increases in x and s p . dQ By substituting Q * into equation (1) and incorporating the first-order and second-order conditions of equation (1) Proposition 3. Under the carbon tax, the retailer's optimal sustainable inventory level, denoted by s * l , for given x, is as follows:

Supplier's Decision.
e supplier supplies products according to the retailer's order quantity and determines the sustainable transportation level. When considering the carbon tax cost, the supplier's profit function is shown in equation (6), and (a p − b p s p )Q represents the carbon emissions due to transportation: By substituting the retailer's optimal order quantity Q * into equation (6), we can obtain Proposition 4 based on the first-order and second-order derivatives of equation (6) with respect to s p .

Proposition 4.
Under the carbon tax, the supplier's optimal sustainable transportation level for given x and s l is as follows: Proposition 4 indicates that s * p monotonically increases with increases in w and β and monotonously decreases with increases in c s , c p , and δ p .

Analysis of Decision Results.
e decision model results support the following conclusions.

Corollary 1.
e carbon tax reduces the retailer's sales effort s l .

Corollary 2.
Under the carbon tax, increasing the retailer's sustainable inventory level s l can increase its sales effort x * .
Equation (4) illustrates that x * increases monotonically with increase in s l . is means that the higher the sustainable inventory level is, the lower is the carbon tax cost and the higher is the profit per unit product. erefore, the retailer increases its sales effort to increase profit. Equation (5) indicates that s * l increases monotonically with increase in x. By increasing its sales effort x, a retailer can enhance the sustainable inventory level s * l . Equation (5) shows that if the supplier increases the sustainable transportation level, then the retailer will increase the sustainable inventory level. Moreover, equation (7) shows that if the retailer increases the sustainable inventory level, then the supplier will increase the sustainable transportation level. erefore, when considering the carbon tax, the retailer's sustainable inventory investment decisions and the supplier's sustainable transportation investment decisions interact with and promote each other.

Solution for Optimal Sustainable Levels under a Carbon
Tax. In the preceding analysis, equations (4), (5), and (7) provide analytical expressions of x * , s * l , and s * p , respectively. However, these three decision variables are nested in each other in these three equations, and they cannot be directly obtained from the corresponding formula. An iterative method is required to solve for these three decision variables.
A careful analysis revealed three implicit mathematical results from the previous conclusions:

) e supplier's profit increases monotonically with the sustainable inventory level s l
According to the aforementioned information, we can conclude that, in the interval [(p − w 1 − c l − ra l ) β/(δ l − 2rb l β), (a l /b l )], an optimal sustainable inventory level s * l must maximize the expected profits of the retailer and the supplier. erefore, the following iterative method is introduced to obtain the optimal solutions: Step 1: let i � 1 and s l (i) � (p − w 1 − c l − ra l ) β/(δ l − 2rb l β); initialize π R (0) as zero.
Step 3: calculate π R (i) by substituting s l (i), x(i), s p (i), and Q(i) into equation (1). If π R (i) ≤ π R (i − 1), the procedure terminates, and s l (i), x(i), and s p (i) are the optimal values; thus, s * l � s l (i), x * � x(i), and s * p � s p (i). Otherwise, let i � i + 1 and go to Step 4.
Step 4: calculate s l (i) by substituting x(i − 1) and s p (i − 1) into equation (13) and return to Step 2.

Multiplayer Dynamic Game
Optimization Model e optimal sustainable levels for transportation and inventory can be determined using the optimization method described in Section 4 given the wholesale price w and carbon tax rate r. However, sustainable transportation and inventory levels are related to carbon tax rates and wholesale prices, which affect retailers' and suppliers' profits.
erefore, appropriate carbon tax rates and wholesale prices are crucial for the coordinated operation of the supply chain. e dynamic game method with complete information was used to construct a three-stage game optimization model to determine the optimal wholesale prices and carbon tax rates. Table 3 displays the basic information of the game, and the optimization processes of each stage of the game simulation are illustrated in Figure 2.

Wholesale Price Game (Stage 1).
In stage 1, the effect of the carbon tax is neglected (i.e., r � 0). By substituting r � 0 into equations (5) and (7), we can determine the optimal sustainable levels for inventory and transportation (denoted by s 0 * l and s 0 * p , respectively) as follows: We then calculate the retailer's profit and supplier's profit by substituting r � 0, s 0 * l , and s 0 * p into equations (1) and (6). In the supplier-retailer relationship, the supplier is the dominant party because wholesale prices are set by the supplier. If the retailer accepts this price, both parties trade at this price, and the game is finished. If the retailer does not accept this price, then the supplier gradually reduces the price until the retailer accepts it. When considering the opportunity cost, the equilibrium condition for a retailer is that its profit rate is higher than the social average. Otherwise, the retailer would choose another investment opportunity. When a retailer's profit rate is higher than the social average, the supplier will not reduce the wholesale price because it can easily find other retailers in the market. erefore, retailers no longer require suppliers to reduce wholesale prices, and the Nash equilibrium is achieved.

Carbon Tax Rate Game (Stage 2).
e dominant party in stage 2 is the government, the goal of which is to maximize social welfare. Social welfare is expressed in equation (10). e first term in equation (10) is economic utility, and the second term is environmental utility. Economic utility is a positive utility that is determined using equation (11). In equation (11), the first term is the supplier's profit, the second term is the retailer's profit, and the third term is the carbon tax revenue. Environmental utility is a disutility and reflects the environmental damage caused by carbon emissions. A quadratic environmental damage function [54] is used to represent the environmental utility, as illustrated in equation (12).
Stage 2 involves three players. Based on the Stackelberg game, a government sets a carbon tax rate, and suppliers and retailers select sustainable levels of transportation and inventory by using the optimization model presented in Section 4. If profits are lower than expected, suppliers and retailers will abandon the investments, which will result in no social welfare. In the simulation game, the government gradually reduces the carbon tax rate from a high level until it is accepted by suppliers and retailers. e game equilibrium maximizes social welfare while suppliers and retailers achieve their expected profit rates.

Sustainable Investment-Sharing Strategy Game (Stage 3).
Carbon tax policy imposes a negative financial burden on companies [17]. To encourage enterprises to strengthen their efforts to reduce carbon emissions, a government can implement a sustainable investment-sharing strategy. e sharing proportions of the government and enterprises are k and 1 − k, respectively. In this strategy, government sustainable investment is represented as k(δ l s 2 l + δ p s 2 p )/2, and sustainable investments of suppliers and retailers are (1 − k)δ l s 2 l /2 and (1 − k)δ p s 2 p /2, respectively. e expected profit functions of retailers and suppliers are as follows:

Mathematical Problems in Engineering
By using the same derivation as in Section 4, we can obtain the optimal sustainable levels of inventory and transportation as follows: Equations (14) and (15) indicate that the optimal sustainable levels of inventory and transportation increase monotonically with the sharing proportion of government k. e sustainable investment-sharing strategy is optimized based on the Stackelberg game between the government and the enterprises (i.e., suppliers and retailers). Suppliers and retailers will accept any government strategy. However, higher proportions of government investment induce higher sustainable levels of transportation and inventory, which improve social welfare but reduce carbon tax revenue. erefore, a balance between social welfare, investment, and carbon tax revenue is required. However, the primary aim of the carbon tax policy is to improve the environment rather than raise revenue. is model assumes that governments aim to maximize social welfare when carbon tax revenue is higher than the investment. In this game, the government first sets the investment-sharing proportion to a relatively high value and then gradually reduces it. When the government achieves the decision objectives, the game reaches its Nash equilibrium.

Simulation Method and Accuracy Analysis.
To illustrate the simulation method for game optimization and analyze the effect of a carbon tax on supply chain performance, supply chains were studied in specific situations. In the following numerical examples, let ξ ∼ N(200, 20), we set f(x) � 10x 1/2 and g(x) � x [55]. e expression f(x) � 10x 1/2 shows that the regularity of demand generated by sales effort varies with sales effort, and g(x) � x indicates that when the retailer increases sales effort by one unit, the cost incurred for sales effort is 1 CNY. e constant operating costs of the supplier and the retailer were set at 20, 000 CNY and 15, 000 CNY, respectively. e simulation calculations for each stage of the threestage game optimization process were programmed separately in MATLAB (Figure 2). A total of 500 simulation cycles were implemented. e relative error ε of the simulation result was calculated using equation (16), where X and σ are the mean and standard deviation, respectively, of 500 simulation results. e simulation accuracy was set to 99%, and the allowable relative error was set to 5%. Social welfare was selected as the test index, and statistical analyses of the simulation results are illustrated in Table 4. According to Table 4, the relative errors of the results for the 500 simulations were all <5%. e accuracy of the simulation results satisfied research requirements, and these results were used for decision analysis.

Simulation Results of the Wholesale Price Game (Stage 1).
Let r � 0, and suppose that the average social profit rate for the product is 15%. Set w � 30, 31, . . . , 65 CNY, and simulate the supply chain operations for each of the 36 wholesale price situations. Table 5 summarizes the results of stage 1. When w � 49 CNY, retailer profit was 15.34%. When w > 49 CNY, retailer profit was <15%. erefore, the maximum acceptable wholesale price for retailers was 49 CNY. e relationship between the supplier and the retailer indicated that the optimal wholesale price determined by the supplier was 49 CNY.

Simulation Results of the Carbon Tax Rate Game (Stage 2).
Let w � 49 CNY, and suppose that the average social profit rate for the product is 10% after the implementation of the carbon tax. Simulation experiments revealed that the sustainable transportation and inventory levels were maximized when r � 623 CNY/ton. For r > 623 CNY/ton, increases in the carbon tax rate had no effect on the operation of the supply chain. erefore, we investigated tax rates in the range of 1-623 CNY/ton and simulated the supply chain operations for each of the 623 carbon tax rates. Table 6 summarizes the results of stage 2.
According to the simulation results, social welfare exhibits an inverted U-shaped trend with increases in the carbon tax rate. When r � 529 CNY/ton, social welfare was maximized at a value of 51,276 CNY. However, the profits of the supplier and the retailer were −6.86% and 9.02%, respectively. If the government sets the tax rate at 529CNY/ton, both suppliers and retailers abandon their businesses because profits do not satisfy expectations. erefore, after considering the effect on the supplier and the retailer, social welfare was 0. erefore, tax rates must be reduced to ensure that the supplier and the retailer achieve their expected profit rates. When the carbon tax rate was reduced to 163 CNY/ton, both supplier and retailer achieved their expected profits.
us, the carbon tax rate game reached equilibrium with an optimal carbon tax rate of 163 CNY/ton. Under a carbon tax, the supplier bears more costs for sustainable investment and the carbon tax than the retailer does, resulting in a considerable decrease in the supplier's profit. Hence, the supplier attempts to increase the wholesale price w to shift some costs to retailers. We set r � 163 CNY/ton; consider w � 30, 31, . . . , 65 CNY; and simulate the supply chain operations for each of the 36 wholesale prices. e results are summarized in Table 7. When w � 51 CNY, the retailer's profit was 10.45%. For w ≥ 52 CNY, the retailer's profit was <10%. erefore, the supplier could adjust the wholesale price w to 51 CNY.

Simulation Results of the Sustainable Investment-Sharing Strategy Game (Stage 3).
Let w � 51 CNY and r � 163 CNY/ton. rough simulation experiments, we determined that the sustainable transportation and inventory levels reached their maximum when the government's sharing proportion was k � 57%. At this value, further increases in the shared proportion did not improve social welfare; therefore, the shared proportion set by the government should not exceed 57%. With k � 1%, 2%, . . . , 57%, we simulated supply chain operations for each of the 57 investment-sharing strategies. Table 8 summarizes the results of stage 3. e simulation results indicated that carbon tax revenue decreased with an increase in the shared proportion, whereas government investment increased with an increase in the shared proportion. With k � 44%, social welfare was maximized. For this value, the carbon tax revenue was 12,850 CNY, and the government investment was 31,636 CNY. erefore, the government was required to invest additional 18,786 CNY. For k < 44%, social welfare increased with the shared proportion. For k > 44%, social welfare decreased with the shared proportion. For k < 57%, tax revenues decreased, and the government investment increased with an increasing shared proportion. For k � 35%, tax revenue was higher than government investment. erefore, the optimal shared proportion was 35%.

Summary and Comparison of Optimal
Results. e optimal results of each stage are summarized in Table 9.
Comparing the results of stages 1 and 2 revealed that the effects of the carbon tax on the supply chain were principally manifested in the following four aspects: ( ese results demonstrated that the carbon tax improved the sustainable transportation and inventory levels and reduced carbon emissions in logistics, thus reducing environmental utility. When the supplier and the retailer improve their sustainability levels, their sustainable investments are increased correspondingly. However, the increase in sustainable investment does not result in an increase in profits. By contrast, supplier and retailer profits decreased after the introduction of the carbon tax. e conflict between investment and profit reflected the negative effect of the carbon tax on the supplier and the retailer. e increase in the sustainable transportation level after the introduction of the carbon tax was higher than that of the sustainable inventory level, encouraging sustainable investment by the supplier. Moreover, the carbon tax revenue from the supplier was higher than that from the retailer because emissions from transportation are higher than those from inventory. erefore, the reduction in the supplier's profits was higher than that of the retailer's profits, which adjusted the profit structure in the supply chain. Although the carbon tax had a negative effect on profit in the supply chain, the economic utility of the supply chain did not  decrease because the government received carbon tax revenue. However, with the rapid decline in the environmental utility, the social welfare of the entire supply chain was significantly improved. erefore, although the carbon tax had negative effects on enterprises [17], it induced positive effects for the society. e optimization results of stages 2 and 3 indicated that the effects of sustainable investment-sharing strategies on the supply chain were as follows: (1) e sustainable transportation and inventory levels increased by 123.24% and 46.15%, respectively. e sustainable investment-sharing strategy improved supply chain performance for the government and enterprises. It reduced the marginal investment cost for the supplier and the retailer, thus providing an incentive to improve sustainability. A higher government share proportion induced more sustainable practices. By sharing the investment cost with the government, the supplier and the retailer obtained more revenue at a reduced cost. With the improvement in sustainability, the supplier and the retailer reduced their carbon emissions in logistics, thereby reducing the environmental utility and their carbon tax costs. e reduction in carbon tax costs increased the profits of enterprises, thereby increasing their economic utility. e dual function of economic utility and environmental utility resulted in a rapid increase in social welfare. Notably, the sustainable investment-sharing strategy had a more substantial effect on the supplier because suppliers generate higher carbon emissions than retailers do.
Compared with the results of stage 1, the optimization strategy of stage 3 increased the government revenues (1,335 CNY), social welfare (39,588 CNY), and the profit of the entire supply chain (54,974 CNY), thus coordinating the interests of the government and enterprises. ese results indicated that the sustainable investment-sharing strategy not only promoted the positive role of the carbon tax for the sustainable logistics level, social welfare, and environmental utility but also effectively compensated for the negative effect of the carbon tax on enterprise profits. erefore, the    proposed combined optimization of the carbon tax and the sustainable investment-sharing strategy can increase social welfare, stimulate enthusiasm for carbon emission reduction in enterprise logistics, and facilitate supply chain coordination. Figure 3 shows the mechanisms and effects of the carbon tax and the sustainable investment-sharing strategy as well as the internal relationships between the supply chain elements. e decisions of suppliers and retailers regarding sustainable investment are interlinked. To ensure that enterprises can make optimal investments in transportation and inventory to reduce carbon emissions, both suppliers and retailers must establish communication mechanisms to reach consensus and reduce carbon emissions [46]. Suppliers' sustainable investments in inventory benefit retailers, and retailers' sustainable investment and sales investment benefit suppliers. Suppliers and retailers should use the benefits of supply chain spillovers to reasonably share investment costs and encourage the other parties to enhance their sustainable investment [6,45]. erefore, incentive mechanisms should be implemented to reduce carbon emissions by suppliers and retailers. When formulating policies, governments should compromise between social welfare and corporate goals. High carbon taxes can harm government revenues and enterprises. When implementing a sustainable investment-sharing strategy, the government should prioritize enterprises with high carbon emissions to maximize the effect of emission reduction. When financial expenditure is sufficient or the task of reducing carbon emissions is challenging, governments can increase their share of sustainable investment to reduce environmental utility and improve social welfare.

Conclusions
7.1. Discussion. Sustainable supply chain management is a widely discussed topic in environmental sustainability, and a carbon tax is an effective method for reducing carbon emissions. Transportation and inventory management are the two core factors that affect economic and environmental utilities in logistics. However, research on joint decisionmaking for sustainable transportation and inventory management when a carbon tax is implemented on a global scale is lacking. We investigated a two-part supply chain in which the supplier first determines the sustainable transportation level and the retailer then places an order and assesses the sustainable inventory level relative to sales effort. We integrated the optimization of sustainable transportation and inventory levels and identified an approach to improve  sustainable supply chain performance when a carbon tax is implemented and the government shares sustainable investment with enterprises.
For suppliers and retailers, they must maintain appropriate sustainable transportation and inventory levels under a carbon tax. We adopted a Stackelberg game to optimize the sustainable transportation and inventory levels. Because the three factors, namely, sales effort, inventory sustainability, and transportation sustainability, influenced each other in the decision-making process, optimal results could not be directly deduced, and thus, an iterative process was designed to solve the optimization problem. e results revealed that the carbon tax policy encouraged suppliers and retailers to increase their sustainable transportation and inventory levels [25], thus reducing carbon emissions in the logistics industry. Increasing the sustainable inventory level improved retailers' sales efforts. However, an increase in carbon tax costs reduced sales motivation.
For the government, the carbon tax rate must be set to improve environmental utility and social welfare. To solve this problem, we developed a three-stage dynamic game optimization model. e wholesale price was determined in the first stage, which included suppliers and retailers. In the second stage, which also included the government, the carbon tax rate was optimized, and the wholesale price was adjusted accordingly. e sustainable investment-sharing strategy was considered in the third stage. We implemented a MATLAB simulation to solve this three-stage game model.
A single carbon tax has a dual effect on the performance of the supply chain, and an appropriate sustainable investment-sharing strategy can harmonize government and enterprise objectives. e results of numerical studies indicated that the carbon tax played a strong role in improving environmental utility and social welfare. However, such a policy hinders enterprises [43]. Enterprises increased their sustainable investments, but their profits decreased. e sustainable investment-sharing strategy can stimulate enterprises to promote sustainability, reduce environmental utility, and increase social welfare. is may enable an increase in profits for enterprises with less investment, which would offset the negative effect of the carbon tax on enterprises. An appropriate sustainable investment-sharing strategy can promote the interests of governments and enterprises and improve sustainability in the supply chain. e novel contributions of this paper are summarized as follows. An integrated optimization model for sustainable transportation and inventory management was developed. To the best of our knowledge, this study is the first to combine two sustainable logistics factors in the supply chain. Furthermore, a three-stage dynamic game model was proposed to optimize wholesale prices, carbon tax rates, and investment-sharing strategies. e proposed model could be used as a framework for governments that intend to implement a carbon tax and enterprises that are committed to developing sustainable logistics.

Limitations and Future
Research. Although this paper makes several novel contributions, certain limitations should be noted, and future studies should address the following: (1) the carbon tax rate was set as a fixed value in the model. Uncertain carbon tax rates should be discussed in future studies. (2) Our study only considered the carbon tax. However, sustainable investment under other carbon emission-reduction policies, such as carbon cap, carbon cap-andtrade, and carbon offset, also warrants further study. (3) e supply chain network structure has certain limitations. Future research should investigate more complex supply chain networks such as one-to-many, many-to-one, and many-tomany. Routing problems in these networks complicate the optimization process. In such networks, the measurement of sustainable transportation levels in different routing schedules and the differences in consumer environmental awareness in different markets would increase the value and challenge of the problem. (4) e spillover effect of sustainable investment has a major influence on decision-making in supply chain enterprises. Future studies should focus on the spillover effect in one-to-many, many-to-one, and many-to-many supply chains. We should further analyze the horizontal and vertical benefit spillovers and the coordination mechanisms within these supply chains.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.