The selection of marketplace mode and reselling mode with demand disruptions under cap-and-trade regulation

ABSTRACT This paper studies a cooperating mode selection problem of a manufacturer who sells its products through an offline channel and an online platform under cap-and-trade regulation. The platform can operate with marketplace or reselling mode. We investigate the manufacturer's optimal operational decisions and selection of the platform's modes considering demand disruptions. First, when the carbon cap decreases in the application process, this regulation is firstly easy and then hard to be implemented if the cross-channel effect is small. Even though the platform increases its commission rate, greater cross-channel effect can still bring more production. Second, the increase of demand disruptions brings more profits for the manufacturer and the platform. Specifically, considering demand disruptions, the total profit with reselling mode in decentralised case is larger than that in centralised case under some situations. Third, without demand disruptions, the manufacturer prefers marketplace mode (reselling mode) if the commission rate is low (high). However, demand disruptions make the manufacturer's mode selection a little complicated. Lastly, without demand disruptions, the two firms can be coordinated with reselling mode but cannot be coordinated with marketplace mode. With demand disruptions, the two firms can be coordinated with the two modes in some situations.


Background and motivation
Platform-based e-commerce has been flourishing for the last two decades Dolgui, Ivanov, and Sokolov 2020;). An increasing number of manufacturers cooperate with platforms to add online channels to sell their products (Tian et al. 2018;Choi, Taleizadeh, and Yue 2019;Hosseini, Ivanov, and Dolgui 2019;Zhang, Cao, and He 2019). The cooperation is usually conducted in one of two typical modes: marketplace and reselling (Tan and Carrillo 2017;Liu and Ke 2020;Xu, Zhang, and He 2020). With marketplace mode, a platform offers manufacturers accesses to sell their products to consumers directly, taking a commission for each online sale and a fixed slotting fee for a certain period (Shen, Willems, and Dai 2019;Shen, Yang, and Dai 2019), while the manufacturers retain the right to determine retail prices. With reselling mode, the manufacturers wholesale their products to the platform who further retails them to CONTACT  consumers, wherein the manufacturers determine the wholesale prices and the platform determines the retail prices and the order quantities (Shen, Yang, and Dai 2019;). These two modes exist widely in current e-commerce environment. For example, Tmall.com, which is the largest online platform in China, mainly operates with marketplace mode. Amazon.com and JD.com are famous for their operation with reselling mode. As a new way for the manufacturers to obtain profit, the online channel inevitably has an impact on the offline sales. Following the literature (Abhishek, Jerath, and John Zhang 2016;Nie et al. 2019;Yi, Li, and Ma 2019), we call this impact 'cross-channel' effect. This effect is widespread in practice, with its directions (positive or negative) depending on specific industries. For example, in music industry, the online channel has positive effect on the offline sales (Yan, Zhao, and Liu 2018); while in household appliances industry, the online channel has negative effect on the offline sales. The positive (negative) effect means that the sales by online channel increase (decrease) the sales by offline channel (Abhishek, Jerath, and John Zhang 2016). This 'cross-channel effect' can significantly impact a manufacturer's production and pricing decisions.
Besides the 'cross-channel effect', there are two factors that may also impact the manufacturer's production and pricing decisions. The first is demand disruptions, i.e. the sudden change in sales demand under some specific situations (Cao 2014;Shen and Li 2017;Gupta, Ivanov, and Ming Choi 2020;Choi 2021). Typical examples of such situations include raw material shortage, the outbreak of COVID-19, 'Double Eleven' shopping festival in China, 'Black Friday' in U.S.A., etc. These sudden situations directly lead to demand decrease or increase (Ivanov 2020a(Ivanov , 2020bDolgui 2020a, 2020b;Queiroz et al. 2020;Zhao et al. 2020). For example, in China, during the 'Double Eleven' shopping festival (only one day) of Year 2019, the online sales on Tmall.com reach 268.4 billion CNY 1 , about 17% of the GDP of Hangzhou (where the headquarters of Tmall.com located) in that year 2 . The sudden and significant demand increase definitely makes the manufacturers greatly increase production quantities and thus leads to extra costs for production. The outbreak of COVID-19 in China's Hubei Province at the end of Year 2019 suddenly increased the demands of face mask and medicines, which generated extra costs for related manufacturers.
The second factor is carbon emission regulation. The prosperity of manufacturing and marketing activities is unfriendly to the environment, mainly due to the carbon emissions generated by these activities (Liu et al. 2015). To curb the carbon emissions generated from industrial activities, many governments over the world implement various carbon emission regulations. Cap-and-trade regulation is one of the most effective market-based regulations to control carbon emissions, thus is advocated by more and more practitioners and scholars (Gong and Zhou 2013;Xu et al."Supply chain coordination," 2017;Xu et al."Production and pricing," 2017). For example, European Union Emissions Trading Scheme (EU ETS), which is the largest carbon trading market in the world, covers over 50% of the total carbon emissions in EU (Xu et al."Production and pricing," 2017). The Chinese government also establishes 7 carbon trading pilot areas and implements cap-and-trade regulation there (Xu et al."Supply chain coordination," 2017;Xu et al."Production and pricing," 2017). Under cap-andtrade regulation, the government firstly allocates a certain quantity of free emission credits (i.e. cap) to the manufacturers, who can buy more or sell the allocated emission credits later with a carbon trading price through a carbon trading market (Wang and Wu 2020;Wang, Zhao, and Herty 2018). Under the cap-and-trade regulation, manufacturers have the options of trading carbon emission permits besides investing in green technologies to meet the regulation.
Demand disruption and carbon emission regulations can simultaneously affect the operational decisions of manufacturers. Demand disruption directly affects the total carbon emissions, which further affects the carbon trading decisions of manufacturers. For example, after the outbreak of COVID-19 in China at the end of Year 2019, the demand of steel was reduced dramatically because of the restrictions on steel exports. Many steelmakers had to sell their products at a very low price. Without this demand disruption, the steelmakers might buy emission credits to produce more products. However, after this demand disruption, they produce less products and might sell emission credits to carbon trading market. Demand disruptions can be caused by Natural disasters, such as the tsunami in Sumatra, Indonesia in 2004, the earthquake in Wenchuan, China in 2008, etc.
To better understand the background of the problem to be studied in this paper, we surveyed a food company in Anhui province, China in May 2019 and established a project to explore its operational decisions and cooperations with platforms in dual-channel online supply chain structures. To protect its privacy, we call it Company A in this paper. This company sells its products not only through one offline channel but also through two online channels on Tmall.com and JD.com. We collected its sales data from Tmall.com in 2018, and obtained the following conclusions from this survey. First, when Company A sells its products through Tmall.com, the latter takes a commission at 2% of the revenue from each online order. Second, when this company cooperates with the online platforms, the sales of online channel do affect that of the offline channel. Lastly, since this company is a traditional manufacturing company, the Chinese government takes strict control on its pollution in the production process. In addition, unexpected environmental changes also caused demand disruptions to this company. From the interview to the manager of this company, we interestingly find that the outbreak of COVID-19 has little impact on its offline sales, but has significant impact on its online sales. As we know, the place where COVID-19 firstly broke out in China was in Hubei province. Over 90% of Company A's offline sales are from northeast China, which is far from Hubei province, and thus its offline sales were hardly affected. However, the online sales of Company A cover almost all over China, and thus were significantly affected due to logistics restriction during that special period. With this background, we in this paper assume that the demand disruptions directly affect the online channel sales, and indirectly affect the offline channel sales through cross-channel effect.

Research questions and contribution statement
In this paper, we consider a dual-channel supply chain consisting of a manufacturer and a platform under cap-and-trade regulation, considering demand disruptions. The manufacturer sells its products through an offline channel and an online platform. The manufacturer chooses to cooperate with the platform who operates with one of two modes: marketplace or reselling. We intend to investigate the manufacturer's optimal operational decisions and its selection of the platform's mode considering demand disruptions. The problem we target in this paper has three special features. First, the platform can operate with one of the two modes: with marketplace mode, it charges commissions from the manufacturer's online sales; with reselling mode, there is a double marginalisation issue. Second, the manufacturer faces demand disruptions on its online channel, which has indirect impact on its offline sales through cross-channel effect. Lastly, cap-and-trade regulation has certain impact on the production of the manufacturer. To the best of our knowledge, there is no previous research investigating a problem incorporating the above three features simultaneously. We try to fill this gap by exploring the following research questions: (1) With demand disruptions or not, how does the manufacturer make production decision with marketplace or reselling mode under cap-and-trade regulation? (2) With demand disruptions or not, which mode (marketplace or reselling) does the manufacturer prefer to adopt? (3) Can the manufacturer and the platform achieve coordination?
We build Stackelberg games to model the relationship between the two firms, and solve the games to answer the above questions. First, we examine the manufacturer's optimal production decisions with the two modes under cap-and-trade regulation when there are no demand disruptions. In addition, we discuss the manufacturer's selection of marketplace mode and reselling mode in this situation. Then, we study the manufacturer's optimal decisions when there are demand disruptions, and discuss the selection of the two modes in this situation. Lastly, we explore the coordination issue of the manufacturer and the platform with and without demand disruptions.
Some novel results are found in this paper. Without demand disruptions, the optimal production quantity is increasing in the commission rate if the crosschannel effect is great. With demand disruptions, we find that: (i) the optimal production decision is robust when demand disruptions are slight; (ii) with reselling mode, the total profit of the two firms in decentralised case is larger than that in the centralised case under some situations; (iii) with marketplace mode, the optimal production quantity is increasing in the commission rate under some situations. We further compare the manufacturer's profits with the two modes, and find that the relative magnitude of the profits depends on the commission rate, the demand disruptions and the slotting fee.
(1) When the slotting fee is sufficiently high, reselling mode is always preferred by the manufacturer. (2) When the slotting fee is low, we find that: (i) under low commission rate, the manufacturer prefers marketplace mode (reselling mode) if demand disruptions are low (high); (ii) under high commission rate, as demand disruptions increase, the manufacturer firstly prefers reselling mode, then marketplace mode and lastly reselling mode again. Besides, we explore the coordination issue of the manufacturer and the platform, and reveal that with demand disruptions, the two firms with marketplace mode or reselling mode can all be coordinated when the cross-channel effect and demand disruptions meet some conditions.
To the best of our knowledge, this paper is one of the first to discuss the selection of platform's mode with demand disruptions under cap-and-trade regulation. Some novel findings are revealed. These findings not only contribute to the current literature, but also throw light on the impacts of demand disruptions, commission rate and cross-channel effect on the supply chain members' operational decisions, and the coordination issue of the manufacturer and the platform. In addition, the numerical studies using real data can provide direct guide for a designated firm.
The remainder of this paper is organised as follows. In Section 2, we review the related literature, and position our research. In Section 3, we provide the basic assumptions and formulate the model. In Section 4, we solve the model for the optimal operational decisions without demand disruptions. In Section 5, we solve the model for the optimal operational decisions with demand disruptions, and further make the comparison of marketplace mode and reselling mode. In Section 6, we discuss the coordination issue of the manufacturer and the platform. Section 7 presents the numerical studies based on the real data of Company A. Section 8 concludes the paper and provides possible research directions in the future. All the proofs are presented in Appendix.

Literature review
Our work is closely related to three streams of literature. The first stream studies the operational decisions with online platforms. The second stream studies the operational decisions with demand disruptions. The last stream studies the operational decisions under cap-andtrade regulation.

Research on the operational decisions with online platforms
There is a growing body of literature discussing the optimal operational decisions with online platforms. These studies can be roughly divided into three groups.
The first group of studies analyze the selection of marketplace mode and reselling mode. Considering network effects, Hagiu and Wright (2016) investigate the selection of marketplace mode and reselling mode of an intermediary by establishing some fundamental trade-offs. They uncover that marketing activities, such as the spillovers across products play an important role in the selection. Chen et al. (2019) discuss a manufacturer's selection of marketplace mode and reselling mode, where the manufacturer sells its products only through offline channel in peak season, and through both offline channel and an online platform in off-season. They find that a Pareto improvement can be achieved when the manufacturer adopts the two modes simultaneously. Zhang and Zhang (2020) consider an e-tailer's demand information sharing policy, where the e-tailer firstly chooses marketplace mode and reselling mode, and then determines whether to join an offline channel. They show that reselling mode (marketplace mode) is preferred when the commission rate is low (moderate). Liu and Ke (2020) analyze the effect of cap-and-trade regulation on a manufacturer's selection of marketplace mode and reselling mode. They find that marketplace mode is preferred when the carbon emission per unit product is moderate.
The second group of studies investigate the coordination problem of supply chain members. Shen, Qian, and Choi (2017) explore the coordination problem of a supplier and an online retailer, considering demand disruptions. They provide three contracts, namely, a quantity discount contract, a linear price contract, and a profitsharing contract. They uncover that all three contracts can coordinate the supplier and the online retailer. Ren, Herty, and Zhao (2020) consider a supply chain consisting of a manufacturer and an online platform. The manufacturer sells products to the platform who further leases them to consumers with a per-use price and a service level. They find that the coordination of the two firms can enhance their profits and the platform's service level. Considering delivery time, Xu, Zhang, and He (2020) discuss the coordination problem of a dualchannel supply chain, where a manufacturer sells its products to a retailer, who further sells them through an offline channel and an online platform. They find that wholesale price contract and delivery cost-sharing contract can coordinate the supply chain under some situations.
The last group of studies deal with the operational decisions considering cross-channel effect. Abhishek, Jerath, and John Zhang (2016) investigate the pricing decisions of a manufacturer who sells its products through two platforms, with cross-channel effect present. They consider three kinds of structures of the two platforms, i.e. wholesale arrangement (both reselling modes), hybrid arrangement (reselling mode and marketplace mode) and agency arrangement (both marketplace modes). They find that the manufacturer should choose marketplace mode when the cross-channel effect is negative. Considering manufacturer's retailing inefficiency, Yan, Zhao, and Liu (2018) explore the production decisions of a manufacturer who sells its products with an online platform operating with marketplace and reselling modes simultaneously. They find that whether the retailing inefficiency can benefit the platform is determined by the platform's commission rate. Chen et al. (2018) explore the pricing decisions of a retailer and analyze the influence of cross-channel effect on the retailer's profit. They find that all the channel's members can achieve a win-win result when the cross-channel effect is slightly positive or negative. Nie et al. (2019) investigate the production and pricing decisions of two retailers, where one retailer sells products only through an offline channel, and the other sells products through both online and offline channels. They find that it is not necessary for the offline retailer to join an online platform when the cross-channel effect is markedly negative.
We, in this paper, investigate the pricing decisions with marketplace mode and reselling mode, and study the selection of the two modes. Compared to the literature on this area, our work has three contributions. First, we extend previous models by considering demand disruptions. This phenomenon is increasingly common in practice such as the outbreak of COVID-19 all over the world. Second, we incorporate cap-and-trade regulation into the model. The presence of this regulation directly affects the manufacturer's operational decisions and the selection of the platform's modes. Lastly, we discuss the impact of demand disruptions on the coordination of the two firms, and find some new results that enrich the related literature.

research on the operational decisions with demand disruptions
As the earliest study to consider demand disruptions, Qi, Bard, and Yu (2004) explore optimal pricing decisions and supply chain coordination with a quantity discount contract. They arrive at the conditions to coordinate the supply chain when demand disruptions are present. Chen and Xiao (2009) investigate the optimal pricing and service investment level decisions with demand disruptions under quantity discount contract and wholesale price contract. They show that the quantity discount contract is better for the manufacturer when the demand is highly increased and the production cost is low. Lei, Li, and Liu (2012) extend the study of Qi, Bard, and Yu (2004), and analyze the optimal production decisions and supply chain contract design with asymmetric information of demand disruptions. They find that the supply chain can achieve more profits under some situations even though the demand disruption information is asymmetric. Considering a dual-channel supply chain with demand disruptions, Cao (2014) investigates the pricing decision and supply chain coordination with a revenue sharing contract. He finds that this contract can coordinate the supply chain when demand disruptions exist. After that, Cao, Zhou, and Kevin (2015) further consider the demand and production cost disruptions and explore the supply chain coordination with a revenue-sharing contract. They also find that this contract can coordinate the supply chain. Liu et al. (2016) discuss the optimal pricing and service effort level decisions of a supply chain consisting of a logistics service integrator and two logistics service providers. They compare their optimal decisions with different modes. Wu, Chen, and Ji (2020) analyze the optimal pricing and production decisions of a supply chain with demand disruptions under different promotion strategies. They show that the supply chain members should adopt aggressive pricing strategies with increasingly intensified demand disruptions. Zhao et al. (2020) investigate the supply chain coordination of a fashion supply chain with quantity discount contract and revenue sharing contracts. They interestingly find that demand disruptions can promote supply chain coordination.
Although the above studies explore the operational decisions with demand disruptions, our work further takes into account the selection of the platform's operational modes. To the best of our knowledge, there is no research exploring this selection with demand disruptions. Our work uncovers the impacts of demand disruptions on the manufacturer's operational decisions, the selection of the platform's operational modes and the coordination of the manufacturer and the platform.
In addition, we consider cap-and-trade regulation and cross-channel effect, which provide some guidelines for the manufacturer's optimal operational decisions and the coordination of the two firms.

Research on the operational decisions under cap-and-trade regulation
Many studies investigate the optimal operational decision under cap-and-trade regulation with a fixed carbon trading price (Xu et al."Supply chain coordination," 2017;Zhang and Xu 2013). Here, we mainly review the studies considering unfixed carbon trading prices, since they are more closely related to our paper. Those studies can be divided into two groups.
The first group of studies assume that the buying price of emission credits is no less than the selling price of emission credits. With this assumption, Gong and Zhou (2013) analyze the optimal production plan and technology choice under cap-and-trade regulation. They find that the carbon trading decisions follow a twothreshold decision. Xu, Xu, and He (2016) investigate the joint production and pricing decisions under cap-andtrade regulation. They further compare this regulation with carbon tax regulation and find that the social welfare under cap-and-trade regulation is not larger than that under carbon tax regulation. Xu et al."Supply chain coordination," (2017) investigate the pricing and production decisions of two products under cap-and-trade regulation. They show that the optimal production quantities may be decreasing in the cap. He, Dou, and Zhang (2017) explore the firm's production decision and the government's optimal cap decision under cap-and-trade regulation. They find that the optimal cap may increase with the carbon emission of unit product.
The other group of studies assume a linear relationship between the carbon trading price and the cap. Hua, Cheng, and Wang (2011) investigate the optimal order quantities under cap-and-trade regulation by assuming a decreasing relationship between the carbon trading price and the cap. They find that the optimal order quantities can increase or decrease when the cap increases. Extending Hua, Cheng, and Wang (2011)'s model into multiple periods, Benjaafar, Li, and Daskin (2013) also explore the optimal order quantities by assuming a linear relationship between the carbon trading price and the cap. They find that the cap can be used to control carbon emissions which is different from the findings of previous works. Ji et al. (2020) investigate the production decisions with wholesale price contract and revenue sharing contract under cap-and-trade regulation. They further compare the social welfare under the two contracts.
The previous studies mainly concentrate on the operational decisions of a manufacturer with only a single channel. Our work contributes to the literature on this area in two aspects. One is that our work involves two channels, and we further consider demand disruptions, which are shown to have significant impacts on the supply chain members' decisions. The other is that we introduce two modes of the online platform and discuss the impacts of cross-channel effect on the manufacturer's operational decisions and mode selection. Some new findings are found regarding the coordination of the manufacturer and the platform.

Model formulation
In this paper, we consider the pricing and production decisions of a manufacturer with demand disruptions under cap-and-trade regulation, where the manufacturer sells its products through an online platform and an offline channel. The online platform can operate with marketplace mode or reselling mode. With marketplace mode, the manufacturer sells its products directly to consumers but should pay a commission rate φ besides a fixed slotting fee F to the platform for its online sales (Shen, Yang, and Dai 2019;Shen, Willems, and Dai 2019). With reselling mode, the manufacturer sells its products to the platform with a wholesale price ω, who further sells them to consumers with a retail price p. We denote the retail price of the offline channel as p 0 . Following the studies of Abhishek, Jerath, and John Zhang (2016), Yan, Zhao, and Liu (2018) and Nie et al. (2019), we assume that the demand of the offline channel is composed of a base demand Q and the impact from the online channel. That is, the demand of the offline channel is Q + rq where q is the online sales, and r ∈ [−1, 1] captures the cross-channel effect.
Under cap-and-trade regulation, the manufacturer receives an allocated cap C (i.e. the free cap) and he can buy or sell emission credits if necessary. In addition, the carbon trading price has an inverse relationship with the cap, which has been reported by empirical research (Benz and Truck 2009) and modelling research (Benjaafar, Li, and Daskin 2013;Ji et al. 2020). Following these studies, we assume the carbon trading price is a 0 − b 0 C where a 0 is the maximal carbon trading price and b 0 is the price sensitivity of the cap. The carbon emission for unit product is e 0 . Similar to Xu et al."Production and pricing," (2017), Xu et al."Supply chain coordination," (2017) and Ji et al. (2020), we define the demand of the online channel as follows: ( 1 ) where a is the maximal market size and k is the sensitivity of the retail price. Figure 1 presents the channel structures of the supply chain in our research. Stackelberg game is used to model the research problem, where the manufacturer is the leader and the platform is the follower. The sequence of events is as follows. With reselling mode, the manufacturer firstly determines the wholesale price and the platform further determines the retail price. With marketplace mode, the manufacturer determines the production quantity and the retail price. In practice, many online platforms, such as Tmall.com, take a 2%-5% commission rate for most kinds of products and the commission rate is rarely changed along the time. Thus, we in this paper assume that the commission rate is exogenous, following Geng, Tan, and Wei (2018) and Tian et al. (2018). Therefore, the game with marketplace mode is degenerated to an optimal decision problem of the manufacturer. All the key notations are summarised in Table 1.

Without demand disruptions
In this section, we first investigate the optimal pricing and production decisions in centralised case, and then explore the optimal pricing and production decisions in  The sensitivity of the retail price. p 0 The retail price of the offline channel. φ The commission rate with marketplace mode. Q The base demand of offline channel. q The demand of online channel. r The cross-channel effect. a 0 The maximal carbon trading price. b 0 The price sensitivity of the cap. C The allocated cap (i.e. the free cap). e 0 The carbon emission for unit product. The retail price. ω The wholesale price with reselling mode.
decentralised case with marketplace mode and reselling mode, respectively.

Centralised case
In centralised case, we consider the manufacturer and the platform as an integrated entity. Then, the total profit of this 'entity' is obtained by computing π SC−0 * = max π SC−0 (p), where The first term is the profit from offline channel. The second term is the profit from online channel. The last term is the carbon trading cost or revenue. We then get the optimal decisions, as shown in Lemma 4.1.

Lemma 4.1: Without demand disruptions,
(i) the optimal decisions in the centralised case are as follows, , π SC−0 * is firstly increasing then decreasing in C.
From Lemma 4.1(i), it is easy to verify that the optimal production quantity and the optimal total profit are increasing in the cross-channel effect. In addition, the optimal production quantity is increasing in the allocated cap, which seems to be intuitive but is an important result. It indicates that the cap can control carbon emission directly and effectively. Many previous papers uncover that (a) the cap cannot directly control carbon emission (Benjaafar, Li, and Daskin 2013); (b) the cap partly controls carbon emission (Xu, Xu, and He 2016;He, Dou, and Zhang 2017;Xu et al."Supply chain coordination," 2017;Xu et al."Production and pricing," 2017). In this work, we consider the inverse relationship of the cap and emission trading price, and interestingly uncover that the cap can control carbon emission directly and effectively. However, from Lemma 4.1(ii), the optimal total profit is increasing in the allocated

it firstly increases and then decreases with the allocated cap if
It reflects the impact of the crosschannel effect on the monotonicity of the total profit and the allocated cap. To the best of our knowledge, this is the first time to find the impact of the crosschannel effect on the relationship of the cap and the total profit. Previous studies which consider a fixed carbon emission trading price or unequal carbon trading prices show that, the optimal total profit (or the manufacturer's profit) is always increasing in the cap (Gong and Zhou 2013;Xu et al."Supply chain coordination," 2017;Xu et al."Production and pricing," 2017;He, Dou, and Zhang 2017). It is intuitive that the optimal total profit is increasing in the allocated cap. However, the result here presents an opposite conclusion if the cross-channel effect is small. The logic behind this is that with the increase of allocated cap, the carbon trading price decreases. Thus, the revenue decreases when the manufacturer sells emission credits to the carbon trading market. Thus, in this aspect, cap-and-trade regulation is firstly easy and then hard to be implemented when the cross-channel effect is small.

Decentralised case without demand disruptions
In the decentralised case, we consider two modes of the platform. One is reselling mode and the other is marketplace mode.
Reselling mode. With reselling mode, the manufacturer sells its products to the platform with a wholesale price ω, and the platform further sells them to consumers with a retail price p. Then, the profits of the manufacturer and the platform are as follows: We have the following lemma about the optimal decisions and profits.
Lemma 4.2: Without demand disruptions, we obtain optimal decisions in the decentralised case with reselling mode as follows: Since Lemma 4.2 presents similar results in Lemma 4.1, we do not give further analyses for conciseness.
Marketplace mode. With marketplace mode, the platform offers online marketplace service to the manufacturer and charges a commission rate φ. Then, the profits of the manufacturer and the platform are as follows: By maximising π DM−0 m (p), we have the following lemma.

Lemma 4.3:
Without demand disruptions, the optimal decisions of the manufacturer in the decentralised case with marketplace mode are as follows: From Lemma 4.3, it is easy to verify that, if −1 ≤ r < e 0 (a 0 − b 0 C)/(p 0 − e 0 (a 0 − b 0 C)), the optimal retail price (production quantity) is increasing (decreasing) in the commission rate, and if e 0 (a 0 − b 0 C)/(p 0 − e 0 (a 0 − b 0 C)) < r ≤ 1, the optimal retail price (production quantity) is decreasing (increasing) in the commission rate. Intuitively, the optimal production quantity should be decreasing in the commission rate. However, we interestingly find that the optimal production quantity is increasing in the commission rate if the cross-channel effect is great. Larger value of cross-channel effect indicates that the manufacturer can obtain more profits from offline channel. Thus, the manufacturer has the motivation to increase production quantity although the platform increases the commission rate. It is worth noting that the optimal production quantity is decreasing in the commission rate when there is no cross-channel effect. Some previous studies do not consider the cross-channel effect, and find that the optimal production quantity is independent of the commission rate (Tian et al. 2018;Shen, Willems, and Dai 2019;Shen, Yang, and Dai 2019). Please note that the reason is that they assume the production cost to be zero. Actually, if they consider nonzero production cost, the optimal production quantity will be decreasing in the commission rate. Some previous studies consider the cross-channel effect, and find that the optimal production quantity is increasing in the commission rate (Abhishek, Jerath, and John Zhang 2016; Yan, Zhao, and Liu 2018). In this work, we consider capand-trade regulation and uncover the new results on the relationship between the optimal production quantity and the commission rate.
In addition, we obtain that (1) the manufacturer's profit is decreasing in the commission rate, and (2) the platform's profit firstly increases and then decreases in the commission rate. Comparing Lemma 4.2 with Lemma 4.3, we find that, (1) if 0 < φ ≤ (a − M)/(a + M), then p DR−0 * ≥ p DM−0 * ; (2) if (a − M)/(a + M) < φ < 1 then p DR−0 * < p DM−0 * . Hence, the double marginalisation with reselling (marketplace) mode is more serious than that with marketplace (reselling) mode when the commission rate is low (high).
Corollary 4.1: When 0 < φ < φ 0 , the profit of the manufacturer with marketplace mode is larger than that with reselling mode; otherwise, the profit of the manufacturer with reselling mode is larger than that with marketplace Corollary 4.1 shows that, the manufacturer prefers marketplace mode (reselling mode) if the commission rate is low (high). With marketplace mode, the manufacturer needs to pay a commission to the platform for its online sales, and with reselling mode, there is double marginalisation which reduces the manufacturer's profit. Consequently, if the commission rate is low, the manufacturer prefers marketplace mode; otherwise, the manufacturer prefers reselling mode. This indicates that the impact of the commission rate is smaller than that of double marginalisation.

With demand disruptions
Similar to Qi, Bard, and Yu (2004), Wu, Chen, and Ji (2020) and Zhao et al. (2020), we divide the decision process into two stages. The first stage is before the selling season, and the second stage is during the selling season. Before the selling season, the manufacturer and the platform make decisions without demand disruptions. In this stage, the manufacturer's production plan or the platform's order quantity is determined. During the selling season in which demand disruptions occur, the platform with reselling mode may revise the order, or the manufacturer with marketplace mode may change its production quantity, both of which are deviated from those in the first stage.
In this section, we first explore the optimal decisions with demand disruptions in centralised case, and then investigate these decisions in decentralised case with the two modes, respectively.

Centralised case with demand disruptions
Demand disruptions mean the sudden changes of the demand that result in a certain deviation costs, which do not occur under normal environment. In this paper, the models with demand disruptions are to be built in two stages. In the first stage, a production plan is made by assuming online demand to be d = q = a − kp. In the second stage, the realised online demand is found to be d = q = a + a − kp, where demand disruptions are captured by the term a (we assume that a ≥ −a to ensure demand d ≥ 0). a > 0 means positive disruptions where the maximal demand will increase, and a < 0 means negative disruptions where the maximal demand will decrease. Note that we do not consider the demand disruptions on offline channel. The reason is that the places where emergencies occur may not be the main market of the firm. In other words, the firm mainly sell their products in place A while the emergencies may occur in place B. For example, COVID-19 has little effect on offline channels of many firms such as Qiaqia Food Company 3 , Babyonlinedress Company. However, the places that the products on online channels can be sold to are much wider. When emergencies occurred, the online channels will be directly affected since there are no geographical restrictions for the online channels. Just as mentioned in the Introduction section, the outbreak of COVID-19 has little impact on Company A's offline sales but has significant impact on its online sales.
We first assume that there is a central decision-maker who seeks to maximise the total profit of the manufacturer and the platform. When demand disruptions occur, the manufacturer has to undertake more cost. On one side, if the actual production quantity is larger than the planned production quantity (i.e. the production quantity given in Lemma 4.1), the manufacturer has to undertake the cost related to some casual and expensive resources, which means higher production cost for the increased demand. On the other side, if the actual production quantity is less than the planned production quantity, the manufacturer has to undertake the inventory cost for the unsold products or the loss caused by selling them in a secondary market. Then the total profit with demand disruptions can be written as, (7) where (x) + = max{x, 0}. λ 1 > 0 and λ 2 > 0 are the marginal extra costs of the increased demand and decreased demand, respectively. Proposition 5.1: With demand disruptions, the optimal decisions in the centralised case are as follows, Case 1: When −a ≤ a ≤ −λ 2 k(1 + r), then, Case 2: When −λ 2 k(1 + r) < a ≤ λ 1 k(1 + r), then, Case 3: When a > λ 1 k(1 + r), then, From Proposition 5.1, we also find that the optimal production quantity is increasing in the cap. As a consequence, the demand disruptions have no impact on the monotonicity of the optimal production quantity and the cap. From Case 2 of Proposition 5.1, we find that q SC * = a−M 2 = q SC−0 * . That is, the original production plan q SC−0 * is robust when demand disruptions vary slightly. In this case, an adjustment of the retail price can compensate the costs related to demand disruptions. When demand disruptions are greater than a threshold (i.e. λ 1 k(1 + r)) or smaller than a threshold (i.e. −λ 2 k(1 + r)), the manufacturer should modify its production decisions and the retail price. In addition, the total profit in the centralised case is increasing in a. This finding is in accordance with Qi, Bard, and Yu (2004), Cao (2014) and Zhao et al. (2020). By comparing Lemma 4.1 and Proposition 5.1, we discover that, if demand disruptions are positive (negative), the total profit with demand disruptions in centralised case is larger (less) than that without demand disruptions in centralised case.

Decentralised case of reselling mode with demand disruptions
In the decentralised case, the manufacturer sells its products to the platform with a wholesale price ω and the platform further sells them with a retail price p. The profits of the manufacturer and the platform are as follows: Proposition 5.2: With demand disruptions, we obtain the optimal decisions in the decentralised case with reselling mode as follows: Case 1: When −a ≤ a ≤ −λ 2 k(1 + r), then, (1 + r), Case 2: When −λ 2 k(1 + r) < a ≤ λ 1 k(1 + r), then, Case 3: When a > λ 1 k(1 + r), then, From Case 2 in Proposition 5.2, we find that the profit of the platform remains constant when demand disruptions vary slightly. Thus, with reselling mode, the platform's profit is robust when demand disruptions vary slightly. When demand disruptions are greater than a threshold (i.e. λ 1 k(1 + r)) or smaller than a threshold (i.e.−λ 2 k(1 + r)), the profits of the manufacturer and the platform are increasing in demand disruptions a. From Proposition 5.2, it is easy to verify that the optimal wholesale price and the optimal retail price are increasing in demand disruptions a. We also uncover that the optimal production quantity is increasing in the cap, which is similar to Lemma 4.1 and Proposition 5.1. < λ 2 < a k(1+r) , the profit with reselling mode in the decentralised case is larger than that in the centralised case.
Corollary 5.1 presents an interesting result. It shows that the decentralised case can bring more profit than the centralised case under some situations. Corollary 5.1 shows that the case with demand disruptions differs from the case without demand disruptions where the total profit of the manufacturer and the platform in the centralised case is always larger than that in the decentralised case. In addition, in traditional supply chain, the centralised case is considered to have more advantages than the decentralised case in terms of the total profit. From Equation (8), we know that the manufacturer will generate extra costs of handling the leftover inventory when a < M−a 4 < 0. Moreover, when λ 2 is high, this extra costs is relatively high. By comparing Propositions 5.1 and 5.2, we can easily verify that the optimal production quantity in the centralised case is larger than that in the decentralised case. It indicates that the centralised case can generate more costs. Thus, the profit with reselling mode in the decentralised case is larger than that in the centralised case. To the best of our knowledge, this paper is among the first to find this interesting result. Previous studies considering demand disruptions, such as Qi, Bard, and Yu (2004); Cao, Zhou, and Kevin (2015) and Zhao et al. (2020), mainly discuss the coordination of a manufacturer and a retailer. Note that if we do not consider cap-and-trade regulation and cross-channel effect, we will find that the centralised case is always larger than the decentralised case with reselling mode, which is in line with the findings in traditional supply chain research.

Decentralised case of marketplace mode with demand disruptions
The profits of the manufacturer and the platform are as follows: In marketplace mode, commission rate is a critical variable. In practice, many platforms take a low commission rate. For example, as a typical example of platform operating with marketplace mode, Tmall.com adopts a commission rate of 2%-5% for most of the products sold on its platform. In our work, in order to focus on valuable results, we make the following assumption 0 < φ < min{ a−M 2a , a−λ 2 k(1+r) a }.

Proposition 5.3:
With demand disruptions, the optimal decisions of the manufacturer in the decentralised case with marketplace mode are as follows: Case 1: When −a ≤ a ≤ − λ 2 k(1+r) 1−φ , then, Case 3: When a > λ 1 k(1+r) 1−φ , then, Proposition 5.3 presents an important and interesting result about the relationship between the optimal production quantities and the commission rate. If −1 ≤ r < e 0 (a 0 −b 0 C)−λ 2 p 0 −e 0 (a 0 −b 0 C)+λ 2 , the optimal production quantities in all three cases are decreasing in the commission rate.
, the optimal production quantity in Case 1 (Case 2 and Case 3) is increasing (decreasing) in the commission rate.
, the optimal production quantities in Case 1 and Case 2 (Case 3) are increasing (decreasing) in the commission rate. If e 0 (a 0 −b 0 C)+λ 1 p 0 −e 0 (a 0 −b 0 C)−λ 1 ≤ r < 1, then the optimal production quantities in all three cases are increasing in the commission rate. It is intuitive that the optimal production quantities are decreasing in the commission rate. However, they are actually increasing in the commission rate under some situations. With the increase of demand disruptions, the manufacturer can obtain more profits. In addition, with the increase of the cross-channel effect, it can also generate more profits for the manufacturer from offline channel. These lead to the increase of production quantities with the increase of Commission rate. After comparing Proposition 5.3 and Lemma 5.3, we find that positive demand disruptions induce the manufacturer to reduce the optimal production quantity when the platform increases the commission rate. We also find that the optimal production quantity is increasing in the cap, which is similar to Lemma 4.1 and Propositions 5.1 and 5.2.
Proposition 5.4: When F ≥ F, reselling mode can bring more profit for the manufacturer; otherwise, when F < F, , φ}, there is an unique a * 2 that, if a ≥ a * 2 , reselling mode can bring more profit for the manufacturer; otherwise, marketplace mode can bring more profit for the manufacturer; , there are a * 1 and a * 2 that ( a * 2 > a * 1 ), if a ≤ a * 1 or a ≥ a * 2 , reselling mode can bring more profit for the manufacturer; otherwise, marketplace mode can bring more profit for the manufacturer, where F = .
Proposition 5.4 presents an interesting result. Without demand disruptions, we know that the manufacturer prefers marketplace mode (reselling mode) if the commission rate is low (high). However, with demand disruptions, the manufacturer's preference depends on the commission rate, demand disruptions and the slotting fee. (1) When the slotting fee is sufficiently high, reselling mode is always preferred. (2) When the slotting fee is low, we uncover that (i) under low commission rate, the manufacturer prefers marketplace mode (reselling mode) if demand disruptions are low (high); (ii) under high commission rate, with the increase of demand disruptions, the manufacturer firstly prefers reselling mode, then marketplace mode and lastly reselling mode. In previous studies, there are two main conclusions for the selection of marketplace mode and reselling mode. Some studies, such as Shen, Willems, and Dai (2019) and Shen, Yang, and Dai (2019), show that marketplace mode (reselling mode) is preferred when the slotting fee is low (high). Some other studies, such as Liu and Ke (2020), find that marketplace mode (reselling mode) is preferred when the commission rate is low (high). Our work contributes to the literature by analyzing the impact of demand disruptions on the selection of the two modes. As we know, the outbreak of COVID-19 generates substantial demand for face mask and some special medicines. Proposition 5.4 suggests that the manufacturer should adopt reselling mode. From the threshold of the commission rate, we can easily find that the presence of cap-and-trade regulation decreases the value of this threshold. It indicates that the presence of cap-and-trade regulation directly affects the selection of marketplace mode and reselling mode if demand disruptions are relatively small. Without capand-trade regulation, we know the manufacturer may select marketplace mode if the demand disruptions are relatively small. However, with cap-and-trade regulation, the threshold of the commission rate is decreased. From Proposition 5.4 (ii), we know the manufacturer selects reselling mode if the demand disruptions are relatively small. Therefore, cap-and-trade regulation has impact on the manufacturer's selection if the demand disruptions are relatively small. Figure 2 describes the results of Proposition 5.4 if the slotting fee is relatively small.

Coordination of the manufacturer and the platform
In this section, we explore the coordination problem of the manufacturer and the platform with and without demand disruptions. Similar to Chen, Zhang, andSun (2012), Xu et al."Supply chain coordination," (2017) and , we define the coordination of the manufacturer and the platform as follows.
Definition 6.1 (Coordination): With reselling mode (marketplace mode), the manufacturer and the platform can be coordinated if there exists a wholesale price (commission rate) which can realise that the optimal production quantity in the decentralised case is equal to that in the centralised case. We analyze the coordination problem without and with demand disruptions, respectively. The analysis is divided into two steps. In the first step, we explore the optimal decision in the centralised case, and in the second step, we analyze the coordination problem of the two firms by looking for a wholesale price or a commission rate that can make the optimal production quantity in the decentralised case equal to that in the centralised case. The optimal production decisions in the centralised case without and with demand disruptions are presented in Lemma 4.1 and Proposition 5.1, respectively. The corresponding results on the coordination of the two firms without and with demand disruptions are given in Subsections 6.1 and 6.2, respectively.

Without demand disruptions
Proposition 6.1: Without demand disruptions, if −1 ≤ r < min{r 0 , 1}, the two firms can be coordinated by setting ω = M k with reselling mode; otherwise, the two firms can not be coordinated, where r 0 = e 0 (a 0 −b 0 C) p 0 −e 0 (a 0 −b 0 C) . The two firms can not be coordinated with marketplace mode.
Proposition 6.1 presents some important results of coordination without demand disruptions. First, reselling mode can coordinate the manufacturer and the platform if the cross-channel effect is small. As we know, reselling mode is actually operated with wholesale price contract. Thus, this finding is an interesting result and is different from the studies on traditional wholesale price contract. Many previous studies show that wholesale price contract cannot coordinate supply chain members due to double marginalisation (Xu, Zhang, and He 2020). The reason behind our finding is that the presence of cross-channel effect and cap-and-trade regulation helps to increase the wholesale price which is larger than zero. Note that the production cost is assumed to be zero for simplification. Second, the coordination of the two firms with reselling mode sacrifices the manufacturer's profit, and improves the platform's profit. Lastly, marketplace mode cannot coordinate the manufacturer and the platform. In other words, if the manufacturer affiliates with an online platform operating with reselling mode (such as JD.com), the two firms can try to be coordinated to obtain more profits. However, if the manufacturer cooperates with an online platform (such as Tmall.com), it is impossible for them to be coordinated to obtain more profits.
where, r 1 = e 0 (a 0 −b 0 C)−λ 2 p 0 −e 0 (a 0 −b 0 C)+λ 2 , r 2 = e 0 k(a 0 −b 0 C)+ a k(p 0 −e 0 (a 0 −b 0 C)) , Proposition 6.2 shows the impact of cross-channel effect and demand disruptions on the coordination of the manufacturer and the platform with reselling mode. Proposition 6.2 uncovers some important results. First, when the cross-channel effect is sufficiently low, the manufacturer and the platform can be coordinated. Second, when the cross-channel effect is moderate, the manufacturer and the platform can be coordinated with the presence of demand disruptions under some conditions. In other words, with the increase of cross-channel effect, the two firms can only be coordinated when the demand disruptions are greater. Third, if the cross-channel effect is sufficiently large, the two firms cannot be coordinated, whether demand disruptions exist or not. Lastly, the coordination sacrifices the manufacturer's profit. However, it does not mean that the manufacturer's profit cannot be improved. As shown by many traditional supply chain studies, a two-part tariff agreement in which the downstream firm (such as a retailer) pays a lump fee to the manufacturer can coordinate the supply chain (Chen, Zhang, and Sun 2012;Supply chain coordination," 2017; Choi and Guo 2020). When exploring the coordination of the manufacturer and the platform, a two-part tariff agreement can also be used to improve the profits of the manufacturer and the platform.

Proposition 6.3:
With demand disruptions, we obtain the following results for the coordination of the manufacturer and the platform with marketplace mode: (i) When −1 ≤ r < min{r 2 , 1}, if 0 < a ≤ λ 1 k(1 + r), the two firms can be coordinated by setting φ = a a+M ; otherwise, the two firms cannot be coordinated; (ii) when min{r 2 , 1} ≤ r < min{r 3 , 1}, if −λ 2 k(1 + r) < a ≤ 0, the two firms can be coordinated by setting φ = a a+M ; otherwise, the two firms cannot be coordinated; 1−φ , the two firms can be coordinated by setting φ = λ 1 k(1+r) λ 1 k(1+r)+M ; otherwise, the two firms cannot be coordinated.
Proposition 6.3 also presents some new and important results. First, the manufacturer and the platform cannot always be coordinated with marketplace mode. Comparing Proposition 6.3 with Proposition 6.2, we find that reselling mode has more possibility to coordinate the manufacturer and the platform. Second, the presence of demand disruptions is beneficial to the coordination of the two firms, since the two firms cannot be coordinated with marketplace mode when there is no demand disruption. Lastly, when the cross-channel effect is small or moderate, the two firms can be coordinated with a common commission rate depending on the magnitude of demand disruptions. When the cross-channel effect is great, the commission rate to achieve the coordination is independent of the magnitude of demand disruptions.

Numerical studies
In this section, we conduct a set of numerical studies to illustrate our results, based on the real data from the aforementioned company, i.e. Company A, which is a snack-food company and sells its products through offline channel and two online platforms simultaneously. The two platforms mainly operate with marketplace mode (i.e. Tmall.com) and reselling mode (i.e. JD.com), respectively. When Company A sells its products through Tmall.com, the platform takes a 2% commission rate for each order (i.e. φ = 0.02). We obtain its online and offline sales data from May to November in 2018. Since melon seeds are Company A's core product, we use the average sales and retail prices of melon seeds to illustrate our results. By the regression analysis of the sales to the retail prices, we obtain the results about the relationship in Table 2.
We also derive the value of cross-channel effect through regression analysis, which is presented in Table 3. Based on this result, we set r = 0.54 in the remaining analysis.
The other parameters used in our numerical studies are set as follows: p 0 = 120, Q = 2000, C = 300, e 0 = 2, a 0 = 50, b 0 = 0.01, λ 1 = 12 and λ 2 = 10. Figure 3(a) shows that marketplace mode can bring more profit for Company A if 0 < φ < 0.56, and reselling   mode can bring more profit if φ ≥ 0.56. Since φ = 0.02 in the real data. Therefore, Company A should adopt marketplace mode if there are no demand disruptions. Figure  3(b) shows that Company A should adopt reselling mode if −2000 < a < −1782 and should adopt marketplace mode if a > −1782.

The comparison of the total profits in the decentralised and centralised cases with reselling mode
The results illustrated in Figure  ∈ [−0.133, 7.655]. Since λ 2 = 10, the conditions in Corollary 5.1 are satisfied. Then we can find that the profit with reselling mode in the decentralised case is larger than that in the centralised case. From Figure 4, we observe that with the increase of the demand disruptions, the difference between the two profits tends to be smaller. It indicates that with the increase of demand disruptions, the advantages of decentralised case tend to diminish.

The coordination of the manufacturer and the platform with reselling mode
To better show the benefits of the coordination with reselling mode, we define an efficiency variable, which is calculated by one minus the ratio of the two firms' total  profits before and after the coordination. Figure 5 shows that the efficiency varies in the interval of [0.4%, 10.1%]. Thus, if Company A and the platform operating with reselling mode are coordinated, their total profit can be increased by 0.4%-10.1%, depending on the magnitude of the demand disruptions. In addition, the higher the demand disruptions, the higher the efficiency.

Summary
In this paper, we consider a manufacturer who sells its products through an offline channel and an online platform under carbon emission cap-and-trade regulation. The platform can operate with marketplace mode or reselling mode. Cross-channel effect is considered to reflect the impact of online channel sales on offline channel sales. First, we explore the manufacturer's operational decisions and the selection of the platform's modes without demand disruptions. We analyze three cases, namely, centralised case, decentralised case with reselling mode and decentralised case with marketplace mode. Sensitivity analyses are conducted to uncover the impacts of main parameters (i.e. the cross-channel effect, the emission cap and the commission rate) on the optimal decisions and the manufacturer's profit. The selection of the platform's mode is also analyzed. Second, we study the impact of demand disruptions, and establish the model based on the results without and with demand disruptions, respectively. We analogously analyze the above three cases. We find that some results are robust. For example, (a) the optimal production quantity is increasing in the cap; (b) the optimal production quantity remains unchanged when demand disruptions are slight. However, we also find that the demand disruptions do have impacts on (a) the monotonicity of the optimal decisions and the commission rate; (b) the selection of the platform's mode; (c) the comparison of centralised case and decentralised case. Lastly, we discuss the coordination of the manufacturer and the platform without and with demand disruptions. A set of numerical studies with real data are conducted to illustrate our results.

The impacts of cap-and-trade regulation and cross-channel effect
We characterise the impacts of carbon emission cap-andtrade regulation and cross-channel effect on the manufacturer's operational decisions. We find that the impacts are irrespective of the platform's operating mode and demand disruptions. (1) The optimal production quantities in both centralised and decentralised cases are always increasing in the cap. It indicates that cap-and-trade regulation can always control carbon emission, which is contrary to some common conclusions in previous literature on the implementation of cap-and-trade regulation.
(2) If the cross-channel effect is small, the manufacturer's profit and the total profit of the two firms firstly increase and then decrease in the allocated cap; otherwise, they increase in the allocated cap. In practice, many carbon trading markets, such as EU ETS, gradually reduce the cap to control carbon emission, and the cross-channel effect is usually small in practice. As a result, we can find that cap-and-trade regulation is firstly easy and then hard to be implemented. (3) The optimal production quantity is increasing (decreasing) in the commission rate if the cross-channel effect is great (small). Therefore, for the