Remanufacturing mode choice considering consumer’s dual preference under patent licensing

Abstract

Many original equipment manufacturers (OEMs) do not remanufacture by themselves, but instead license the remanufacturing business to other companies for a royalty. This paper examines to whom OEM should outsource remanufacturing when both distributors and independent third-party remanufacturers (TPR) are capable of remanufacturing. We have established three models: OEM licenses remanufacturing to the distributor (model DR); OEM licenses remanufacturing to TPR (model TR); OEM licenses remanufacturing to both the distributor and TPR (model DTR). We compared the optimal strategies of each member of the supply chain in the three models and analyzed the influence of consumers' dual preference for the remanufactured product on the decision making. The results showed that when consumers are reluctant to purchase a certain type of remanufactured product, OEM prefers to outsource the remanufacturing business to TPR. Otherwise, OEM prefers to license remanufacturing to both the distributor and TPR. In terms of two single remanufacturing modes, whether for OEM or TPR, TPR remanufacturing mode is always better than the distributor remanufacturing mode.

1 Introduction

With severe environmental pollution and resource depletion, remanufacturing as an effective method has become one of the hot spots among academic and industry community in recent years (Guo et al. 2020; Zhou et al. 2021). Some studies have shown that companies in some industries can save about 40–65% in manufacturing costs by remanufacturing compared to the production of original new products (Ding et al. 2020).

Remanufacturing can reduce the environmental burden and the production cost (Atasu et al. 2008; Wu 2012; Zheng et al. 2021). Due to the limitations of market and business strategy, such as the lack of advantages of the OEM in the recycling of waste products (Chen and Chang 2012), the cannibalization of remanufactured products on new products (Ferguson and Toktay 2006; Atasu et al. 2010) or the expensive investment in the remanufacturing fixed costs, many OEMs do not participate in the remanufacturing, but outsource it to other agents (Karakayali et al. 2007; Zou et al. 2016). An American survey of more than 2000 remanufacturing enterprises shows that only 6% of remanufacturing enterprises are OEMs (Yan et al. 2015).

It is generally believed that the entry of TPRs will harm the interests of OEM manufacturers, because remanufacturing will cannibalize the sales of new products. Patent licensing can not only effectively inhibit this encroachment and enable OEM to profit from the entry of TPR, but also effectively protect the patent of its products from infringement. Recently, several cases of patent disputes have been widely studied by the academia and business community. For example, the famous three-year infringement lawsuit between Canon and Recycle Assist concerning recycled ink cartridges, and the patent infringement case between Canon and the south Korean company Alphachem concerning the remanufacturing of light-sensitive drum, Canvas Car Roof Case, etc. (Zhu et al. 2018). In the patent-perfect market, the remanufacturing of patented products is protected by law. Only with the permission of the original manufacturer that has the patent right can other manufacturers recycle and remanufacture the waste products. Many companies, such as IBM, Dow, Kodak, and Procter & Gamble, generate significant licensing revenue by licensing proprietary technologies to others (Arora et al. 2013).

An interesting question then is to whom does the OEM choose to outsource remanufacturing? In fact, in addition to independent TPR, distributors can also conduct remanufacturing. For example, the world machinery giant Caterpillar has authorized his distributor in China, Lei Shing Hong Machinery, to not only distribute Caterpillar’s equipment, but also engage in the remanufacturing and resale of Caterpillar’s waste equipment (Shen et al. 2015). Recently, the sole distributor of Volvo Group in Guangxi province in China, Hua **ng Zhong Nan Machinery Equipment Limited Company, signs Machinery Remanufacturing Project Agreement with Guangxi ASEAN Economic and Technological Development Zone, and it will be responsible for engineering machinery and component remanufacturing under the Volvo brand in Guangxi province (Wang et al. 2016). As a downstream member of OEM, the distributor is also engaged in the sales of new products, whose remanufacturing business can intuitively ease the cannibalization of remanufactured products on new products.

At present, the view that consumers’ willingness to pay (WTP) for new products and remanufactured products are heterogeneous has been widely accepted by researchers, which has also been adopted in this paper. Liu et al. pointed out that consumers are sensitive to the source of the remanufactured products (Liu et al. 2017). We assume that consumers’ WTP regarding the remanufactured products from the distributor and the TPR are heterogeneous, that is, consumers have a dual preference for remanufactured products. We established three models including: (1) OEM authorizes his distributor for remanufacturing; (2) OEM authorizes an independent TPR to carry out remanufacturing; (3) OEM authorizes his distributor and an independent TPR to conduct remanufacturing.

In this paper, we define the mode with only one remanufacturing enterprise as the single remanufacturing mode, such as model DR and model TR; we define the mode with two remanufacturing enterprises as the hybrid remanufacturing mode, such as model DTR. In this paper, we mainly discuss the following three issues:

• (1) What is the optimal equilibrium decision of the three remanufacturing models?

• (2) How does the dual preference affect the optimal variables and profit of each member of the closed-loop supply chain?

• (3) Which of the three models is optimal for each member of the supply chain?

The rest of this paper is structured as follows. The second part presents the related literature review. The third part introduces the basic assumptions, symbols and their definitions. The fourth part introduces the equilibrium solution and analysis of the three models. The fifth part compares the optimal decision scheme and profit in the three models and analyzes the optimal strategy from the perspective of each stakeholder. The last part is conclusions. For the sake of clarity, the appendix presents the proof of all theorems and propositions.

2 Literature review

In this paper, we mainly focus on three streams of research in the literature: remanufacturing outsourcing strategy, patent licensing, and consumers’ WTP. For the first stream, most of the literature focus on remanufacturing by the TPR (Zou et al. 2016; Majumder and Groenevelt 2001; ** et al. 2017; Wu and Zhou 2016; Yan et al. 2018a, b; Hong et al. 2013; Huang et al. 2013; Zheng and ** 2022). For example, Yan et al. (2018a, b) raised the question of whether the OEM should outsource reverse channels when outsourcing remanufacturing to the TPR and concluded that it is better that the OEM recycles by itself in terms of economy, society, and environment. Zou et al. (2016) compared the two approaches in which the OEM allowed the TPR remanufacturing and showed that the TPR is more likely to be authorized when consumers’ preference for remanufactured products is low; otherwise the TPR prefers the outsourcing approach. Considering that suppliers will lower wholesale prices in the closed-loop supply chain, ** et al. (2017) proposed that the TPR may be beneficial to the OEM regardless of the OEM’s remanufacturing capacity.

There are also articles about outsourcing remanufacturing to supplier. For instance, **ong et al. (2016) compared the OEM remanufacturing model with the supplier remanufacturing model and found that supplier remanufacturing model is more favored by manufacturers, suppliers and consumers when remanufacturing costs are low. Yan et al. (2018b) assumed that the OEM could outsource remanufacturing to a supplier or a TPR, and proposed that the TPR remanufacturing model is superior to the supplier remanufacturing model in terms of social welfare and environment. However, only a few articles are concerned with the distributor remanufacturing. For example, Shen et al. (2011) studied a remanufacturing mode of distributor under patent protection. They built centralized decision model and decentralized decision model respectively and obtained the equilibrium solution in both cases, and finally realized the coordination of the closed-loop supply chain by adopting a revenue sharing contract. Huang and Wang (2017) assumed that part of the remanufacturing business would be outsourced to a distributor or TPR while OEM conducts remanufacturing, and discussed the impact of remanufacturing capability on each member of the supply chain.

In recent years, there have been quite a lot of articles on the patent licensing of remanufacturing (Liu et al. 2017; Oraiopoulos et al. 2012; Zhao et al. 2014; Bagchi and Mukherjee 2014; Hong et al. 2017; Zhou et al. 2020). For example, Oraiopoulos et al. (2012) proposed that OEM could control the price of the remanufactured product by adjusting the patent licensing fee, so as to effectively control the remanufactured product market. Zhao et al. (2014) compared the advantages and disadvantages of the three types of licensing under network effect: fixed-fee licensing, royalty licensing and two-part tariff licensing. Bagchi and Mukherjee (2014) proposed that royalty is always better than fixed cost from the perspective of innovation and society whether the market competition is Cournot or Bertrand competition. Liu et al. (2017) studied which manufacturing certification contract is the best in the electric and electronic market by comparing the three types of contracts: one-time payment, profit sharing payment and payment by piece. Hong et al. (2017) studied which authorization mode OEM should choose when there is both positive and negative channel competition between OEM and TPR. However, all of these literatures ignore the competition between remanufactured products from different sources.

Earlier research on remanufacturing assumed that consumers had the same WTP for new products and remanufactured products. With the development of psychological theories, more and more studies have taken into account consumer WTP differences, mostly on the differences between new products and remanufactured products (Oraiopoulos et al. 2012; Guide et al. 2010; Wu 2013; Agrawal et al. 2015). Guide et al. (2010) proved two conclusions by studying product auction cases: one is that the remanufactured product will cannibalize the new product; and the other is that there are differences in consumers' willingness to pay for new products and remanufactured products. Agrawal et al. (2015) proposed that the existence of TPR remanufacturing will lead to the increase of consumers' perceived value of new products by about 7%. Only a few articles mentioned that the consumers’ WTP varies according to the origin of the remanufactured product. For example, Ferrer and Swaminathan (2006) studied the competition between OEM and remanufacturer and pointed out that OEM’s remanufactured products are superior to TPR’s remanufactured products in terms of consumers' perceived value. Liu et al. (2017) took into account the difference in consumers' preference for the uncertified remanufactured product and the certified remanufactured product when studying the best remanufacturing authorization form. In this paper, we pointed that consumers’ WTP has differences between new products, the distributor’s remanufactured products and TPR’s remanufactured products.

According to the aforementioned three streams of literature, in Table 1, we compare our study with related literature in several specific dimensions, such as remanufacturers, methods of relicensing, differences in consumers’ willingness to pay. The main contributions of this paper are listed as follows: First, unlike the above-mentioned literatures focusing on remanufacturing, this paper, in addition to focusing on competition between remanufactured products from different sources and new products independently, also examines direct competition between remanufactured products from distributors and TPR when they are both licensed for remanufacturing by OEM. Second, most of the literature focuses only on the difference in consumers’ willingness to pay between new and remanufactured products, ignoring consumers’ sensitivity to the source of remanufactured products. Therefore, we introduce differences in consumers’ willingness to pay for remanufactured goods from different sources.

Table 1 Comparison between our study with related literature

3 Model description and assumption

Our closed-loop supply chain consists of an OEM, a distributor or a TPR, in which the OEM is responsible for manufacturing new products and wholesaling them to the distributor. The distributor sells the new products to consumers and also sells the remanufactured products if he is engaged in remanufacturing. The TPR only makes remanufactured products and sells them to consumers.

In this paper, we establish three closed-loop supply chain models to study which model the OEM should choose for remanufacturing. As shown in Fig. 1, only the distributor is engaged in remanufacturing in the model DR. Only the TPR carries out the remanufacturing business in the model TR. Both the distributor and TPR participate in the remanufacturing business as remanufacturers in the model DTR.

Fig. 1

The supply chain framework of three models

We assume that the OEMs as patent holders of this product are protected by the Intellectual Property Law, and thus other enterprises must obtain the OEM’s license and technical support if they want to engage in the remanufacturing of the product. We only consider a royalty licensing policy in this paper, where the licensee has to pay the OEM royalties for each unit he remanufactures. In all three models, the supply chain members paly a Stackelberg game with the licensor (the OEM) as the market leader and the other members (the distributor or the TPR) as the followers.

We suppose that the licensees can remanufacture products at the same cost after receiving the OEM’s technical support. The unit manufacturing cost of the new products and the remanufactured products is $c_{n}$ and $c_{r}$ respectively, and $c_{n} > c_{r}$ is satisfied to ensure that remanufacturing is profitable. In reality, remanufactured products are used as substitutes for traditional manufacturing, which requires less materials and energy in the manufacturing process (Zhou et al. 2013). Although the remanufactured products have the same quality and specifications as the new products, consumer’s WTP for two kinds of products is inconsistent.

Similar to the research of Liu et al. (2017), consumer’s WTP for a product will change depending on whether the product is remanufactured or not, and will also change according to the source of the product if it is remanufactured. We suppose that consumers prefer new products to remanufactured products, and compared with remanufactured products from the distributor, they prefer the remanufactured products from the TPR because of its higher professionalism in the remanufacturing field. We use $\alpha$ and $\beta$ to represent the consumer’s preference for the remanufactured product compared with a new one and for a remanufactured product from the distributor compared with the remanufactured one from a TPR respectively. $\alpha$ and $\beta$ can be viewed as factors of quality perceived by consumers, including attributes such as warranty, appearance, and technical specifications (Vorasayan and Ryan 2006). We set consumers’ preference for new products as 1 and their preference for remanufactured products as against the new products as $\alpha$. Similarly, we set that consumers’ preference for TPR’s remanufactured products is 1 and their preference for distributor’s remanufactured products is $\beta$. Therefore, we assume that consumers’ preferences for new products, remanufactured products from TPR, and remanufactured products from distributor are 1, $\alpha$ and $\alpha \beta$.

We define the unit retail prices and quantities of new products, remanufactured products from the TPR and the distributor are $p_{n}$, $p_{t}$, $p_{d}$, $q_{n}$, $q_{t}$ and $q_{d}$, respectively. Like the finding of Yan et al. (2018a, b), we obtain the following demand functions:

$$q_{n} = 1 - \frac{{p_{n} - p_{t} }}{1 - \alpha }$$

(1)

$$q_{t} = \frac{{p_{n} - p_{t} }}{1 - \alpha } - \frac{{p_{t} - p_{d} }}{{\alpha \left( {1 - \beta } \right)}}$$

(2)

$$q_{d} = \frac{{p_{t} - p_{d} }}{{\alpha \left( {1 - \beta } \right)}} - \frac{{p_{d} }}{\alpha \beta }$$

(3)

All the notations and definitions used in this paper are shown in Table 2.

Table 2 Notations and definitions

4 Models

4.1 The DR model

In this model, the OEM manufactures new products and sells them to distributor, who sells them to consumers. At the same time, the distributor engages in remanufacturing and sells the remanufactured products to consumers. And the distributor has to pay the OEM royalties $f_{d}$, for each unit he remanufactures.

In this process, the OEM sets $f_{d}$ first, and the distributor decides whether to accept the royalties. We suppose the distributor accept the royalties if remanufacturing can be profitable for him. Then OEM determines $w$ to maximize his own profit $\pi_{m}^{{}}$, and the distributor finally sets $q_{n}^{{}}$ and $q_{d}^{{}}$ to maximize his own profit $\pi_{d}^{{}}$.

The profit functions of the OEM and the distributor are given as follows.

$$\pi_{m}^{DR} = \left( {w - c_{r} } \right)q_{n} + f_{d} q_{d}$$

(4)

$$\pi_{d}^{DR} = \left( {p_{n} - w} \right)q_{n} + \left( {p_{d} - c_{r} - f_{d} } \right)q_{d}$$

(5)

From Eqs. (1) and (2), we can derive the optimal values of DR model.

Theorem 1.

In the DR model, the optimal decisions and constraints are given by $q_{n}^{DR*} = \frac{{c_{r} - c_{n} + 1 - \alpha \beta }}{{4\left( {1 - \alpha \beta } \right)}}$, $q_{d}^{DR*} = \frac{{ - c_{r} + \alpha \beta c_{n} }}{{4\alpha \beta \left( {1 - \alpha \beta } \right)}}$, $f_{d}^{DR*} = \frac{{\alpha \beta - c_{r} }}{2}$, $w^{DR*} = \frac{{1 + c_{n} }}{2}$, $p_{n}^{DR*} = \frac{{c_{n} + 3}}{4}$, $p_{d}^{DR*} = \frac{{c_{r} + 3\alpha \beta }}{4}$.

In order to ensure that the parameters are non-negative and the distributor can accept the royalties, we have $A_{1} \le c_{r} \le B_{1}$, and $c_{n} \le 1$, where $A_{1} = c_{n} + \alpha \beta - 1$, and $B_{1} = \alpha \beta c_{n}$.

As shown in Table 3, in order to clarify the impact of consumers’ dual preferences on this model, we obtained the change of the optimal decision variables with the increase of $\alpha$ or $\beta$ through simple calculation (the proof process is in the appendix). $\uparrow$, $\downarrow$, and $\bot$ in Table 3 respectively represents monotonic increase, monotonic decrease and invariant.

Table 3 The optimal decisions in the DR model change as $\alpha$ or $\beta$ changes

Proposition 1.

We can get $\frac{{\partial q_{n}^{DR*} }}{\partial \alpha } \le 0$, $\frac{{\partial q_{d}^{DR*} }}{\partial \alpha } \ge 0$, $\frac{{\partial p_{n}^{DR*} }}{\partial \alpha } = 0$, $\frac{{\partial p_{d}^{DR*} }}{\partial \alpha } \ge 0$, $\frac{{\partial w_{{}}^{DR*} }}{\partial \alpha } \ge 0$, and $\frac{{\partial f_{d}^{DR*} }}{\partial \alpha } \ge 0$. Besides, $\alpha$ and $\beta$ have similar effects on the model DR.

Obviously, the increase of $\alpha$ and $\beta$ will directly lead to the increase of demand for the distributor’s remanufactured products. Since we mentioned in the hypothesis that consumers only have one product, the demand for new products will decrease.

As shown in Table 3, we can see that with the increase of $\alpha$ and $\beta$, $p_{n}^{DR*}$ remains unchanged and $p_{d}^{DR*}$ increases. This indicates that the distributor's optimal decision is to keep the unit retail price of the new product unchanged and increase that of the remanufactured product when the demand for the new product is reduced and that for the remanufactured product is increased. When OEM authorizes remanufacturing to the distributor, there are only two types of products in the market, the new product and the remanufactured product, and these two products are sold by the distributor. At this time, the threat to the new product is only the remanufactured product. Only in this way can distributor keep his maximum profit. For OEM, his profit comes from two parts, sales profit of new products and patent license fee of remanufactured products. In the case of reduced profit of new products, OEM can only increase his total profit by raising patent license fee.

4.2 The TR model

In the TR model, the OEM manufactures new products and sells them to distributor, who resells them to consumers. As an independent third-party remanufacturer, the TPR collects waste products, remanufactures and sells them to consumers, and pays royalties $f_{t}$ to the OEM. Similar to the DR model above, the decision process in this game model is as follows: the OEM as the Stackelberg leader first announces $f_{t}$ to the TPR, who decides whether to accept the royalties. Then, the OEM determines the wholesale price $w$. Finally, the distributor and the TPR simultaneously and competitively decide their production quantities $q_{n}^{TR}$ and $q_{t}^{TR}$ respectively.

The profit functions of the channel members are identified as:

$$\pi_{m}^{TR} = \left( {w - c_{n} } \right)q_{n} + f_{t} q_{t}$$

(6)

$$\pi_{d}^{TR} = \left( {p_{n} - w} \right)q_{n}$$

(7)

$$\pi_{t}^{TR} = \left( {p_{t} - c_{r} - f_{t} } \right)q_{t}$$

(8)

Theorem 2.

In the TR model, the optimal decisions and constraints are given by $f_{t}^{TR*} = \frac{{\alpha - c_{r} }}{2}$, $w^{TR*} = \frac{{1 + c_{n} }}{2}$, $q_{n}^{TR*} = \frac{{c_{r} - 2c_{n} + 2 - \alpha }}{{2\left( {4 - \alpha } \right)}}$, $q_{t}^{TR*} = \frac{{ - 2c_{r} + \alpha c_{n} + \alpha }}{{2\alpha \left( {4 - \alpha } \right)}}$,$p_{n}^{TR*} = \frac{{c_{r} + \left( {2 - \alpha } \right)c_{n} + 6 - 2\alpha }}{{2\left( {4 - \alpha } \right)}}$, and $p_{t}^{TR*} = \frac{{\left( {2 - \alpha } \right)c_{r} + \alpha c_{n} + \alpha \left( {5 - \alpha } \right)}}{{2\left( {4 - \alpha } \right)}}$.

In order to ensure that the decision variables are non-negative and the TPR can accept the royalty proposed by the OEM, we have $A_{2} \le c_{r} \le B_{2}$, and $c_{n} \le 1$, where $A_{2} = 2c_{n} + \alpha - 2$, $B_{2} = \left( {\alpha + \alpha c_{n} } \right)/2$.

In this process, only the TPR is involved in the remanufacturing business, while the distributor is only responsible for the sales of new products. There is no remanufactured product from the distributor in the market, so $\beta$ will not have an impact on this model. Therefore, we only discuss the influence of $\alpha$ on the optimal decision variables.

Proposition 2.

We can get $\frac{{\partial q_{n}^{TR*} }}{\partial \alpha } \le 0$, $\frac{{\partial q_{t}^{TR*} }}{\partial \alpha } \le 0$, $\frac{{\partial p_{n}^{TR*} }}{\partial \alpha } \le 0$, and $\frac{{\partial p_{t}^{TR*} }}{\partial \alpha } \ge 0$.

Similar to the demand change in model DR, with the increase of consumers’ preference for the remanufactured products $\alpha$, the sales volume of the remanufactured products will inevitably increase, which will correspondingly lead to a decrease in the sales volume of the new products. In model TR, the unit retail price of new products will decrease with the increase of $\alpha$, which means that when TPR participates in the remanufacturing activities, the threat of new products comes from the remanufactured products sold by TPR. At this time, the distributor will choose to lower the unit retail price of new product $p_{n}^{TR*}$ to ensure that its sales volume $q_{n}^{TR*}$ will not decrease significantly. Faced with the same demand change, the reason why the decision-making parties in the two models have different responses is that the competing products in the model DR all come from the same enterprise (the distributor), while in the model TR, the competing products belong to different interest parties. Accordingly, TPR will increase the unit retail price of its products to seek higher profits while consumers' preference for remanufactured products increases.

4.3 The DTR model

In this model, the OEM also produce new produces and sells them to the distributor, which is the same with that in DR model and TR model. Besides, the OEM authorizes the distributor and the TPR for remanufacturing and charge them royalties respectively. In addition to reselling new products from the OEM, the distributor competes with the TPR in the secondary market. The sequence of events is as follow: the OEM first announces the royalties $f_{t}$ and $f_{d}$ to the TPR and the distributor respectively, and then they decide whether to accept it or not. After that, the OEM chooses $w$ to maximize his profit, based on which the distributor and the TPR simultaneously and non-cooperatively decides their production quantities $q_{n}^{DTR}$, $q_{d}^{DTR}$ and $q_{t}^{DTR}$ respectively.

The OEM’s, the distributor’s and the TPR’s problems can be stated as:

$$\pi_{m}^{DTR} = \left( {w - c_{n} } \right)q_{n} + f_{t} q_{t} + f_{d} q_{d}$$

(9)

$$\pi_{d}^{DTR} = \left( {p_{n} - w} \right)q_{n} + \left( {p_{d} - c_{r} - f_{d} } \right)q_{d}$$

(10)

$$\pi_{t}^{DTR} = \left( {p_{t} - c_{r} - f_{t} } \right)q_{t}$$

(11)

Theorem 3.

In the DTR model, the optimal decisions and constraints are given by $f_{t}^{DTR*} = \frac{{\alpha - c_{r} }}{2}$, $f_{d}^{DTR*} = \frac{{\alpha \beta - c_{r} }}{2}$, $w^{DTR*} = \frac{{1 + c_{n} }}{2}$,$q_{n}^{DTR*} = \frac{{\left( {5 - 2\beta } \right)c_{r} - \left( {4 - \beta } \right)c_{n} + 4 - 2\alpha - \beta - \alpha \beta }}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $q_{t}^{DTR*} = \frac{{ - \left( {1 + \alpha - 2\alpha \beta } \right)c_{r} + \alpha \left( {1 - \beta } \right)c_{n} + \alpha \left( {1 - \alpha \beta } \right)}}{{2\alpha \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $q_{d}^{DTR*} = \frac{{ - \left( {4 - \alpha - 2\beta + 2\alpha \beta } \right)c_{r} + 3\alpha \beta c_{n} - \alpha \beta \left( {1 - \alpha } \right)}}{{4\alpha \beta \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $p_{n}^{DTR*} = \frac{{\left( {1 + \alpha - 2\alpha \beta } \right)c_{r} + \left( {4 - 2\alpha - \beta - \alpha \beta } \right)c_{n} + 12 - 4\alpha - 3\beta - 6\alpha \beta + \alpha^{2} \beta }}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $p_{t}^{DTR*} = \frac{{\left( {3 - 2\alpha - \beta } \right)c_{r} + \alpha \left( {1 - \beta } \right)c_{n} + \alpha \left( {5 - \alpha - \beta - 3\alpha \beta } \right)}}{{2\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $p_{d}^{DTR*} = \frac{{\left( {4 - \alpha - \alpha \beta - 2\alpha \beta^{2} } \right)c_{r} - \alpha \beta \left( {1 - \beta } \right)c_{n} + \alpha \beta \left( {11 - 3\alpha - 3\beta - 5\alpha \beta } \right)}}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$.

Similar to the DR model and the TR model, to ensure the decision variables are non-negative and the distribution and the TPR can accept the royalties from the OEM, we have obtained the following conditions:

$A_{3} \le c_{r} \le B_{3}$, and $\frac{4 - 2\alpha - \beta - \alpha \beta }{{4 - \beta }} \le c_{n} \le 1$, where $A_{3} = \frac{{(4 - \beta )c_{n} - 4 + 2\alpha + \beta + \alpha \beta }}{5 - 2\beta }$, and $B_{3} = \frac{{3\alpha \beta c_{n} - \alpha \beta (1 - \alpha )}}{4 - \alpha - 2\beta + 2\alpha \beta }$.

Through simple calculation, we discuss the impact of $\alpha$ and $\beta$ on the decision variables in model DTR. The calculation results are shown in Table 4, and the proof process is shown in the appendix. Since the influence of $\alpha$ and $\beta$ on the decision variables of supply chain members is not completely consistent, we analyze them in turn.

Table 4 The optimal decisions in the DTR model change as $\alpha$ or $\beta$ changes

Proposition 3.

We can get $\frac{{\partial q_{n}^{DTR*} }}{\partial \alpha } \le 0$, $\frac{{\partial q_{t}^{DTR*} }}{\partial \alpha } \le 0$, $\frac{{\partial q_{d}^{DTR*} }}{\partial \alpha } \ge 0$, $\frac{{\partial p_{n}^{DTR*} }}{\partial \alpha } \le 0$, $\frac{{\partial p_{t}^{DTR*} }}{\partial \alpha } \le 0$, and $\frac{{\partial p_{d}^{DTR*} }}{\partial \alpha } \ge 0$.

In model DTR, with the increase of consumers’ preference for the remanufactured products $\alpha$, the demand for both types of remanufactured products will increase. In this process, there are three types of products in the market, namely new products, remanufactured products from the distributor and the TPR. Therefore, new products will be encroached by the two kinds of remanufactured products, resulting in a decrease in sales volume. At this time, the distributor should choose to focus on remanufactured products rather than new products, that is, to reduce the unit retail price of new products to ensure that the sales volume of new products will not be reduced significantly, and to increase that of remanufactured products to make up for the loss of profit of new products. However, for TPR, the profit from his remanufactured products is the only source of profit. When the demand for remanufactured products increases, TPR has to lower the unit retail price and choose the strategy of small profits but quick turnover. This is different from Proposition 2, mainly because TPR’s remanufactured products is affected by the increase in sales volume of the distributor’s remanufactured products. Similar to Propositions 1 and 2, OEM will still choose to increase the royalties to increase its total profit under the condition that TPR makes profit per unit of remanufactured products.

Proposition 4.

We can get $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \le 0$, $\frac{{\partial q_{t}^{DTR*} }}{\partial \beta } \le 0$, $\frac{{\partial q_{d}^{DTR*} }}{\partial \beta } \ge 0$, $\frac{{\partial p_{n}^{DTR*} }}{\partial \beta } \ge 0$, $\frac{{\partial p_{t}^{DTR*} }}{\partial \beta } \le 0$, and $\frac{{\partial p_{d}^{DTR*} }}{\partial \beta } \ge 0$.

When both the distributor and TPR are authorized by OEM for remanufacturing, the increase of consumers' preference for the distributor’s remanufactured products $\beta$ can result in an increase in the demand for such products, while the sales volume of the other two types of products will decrease. It is easy to understand that the unit retail price of the distributor’s remanufactured products also increases accordingly. Similar to Proposition 3, TPR should choose the strategy of small profits but quick turnover. The difference is that an increase of $\alpha$ can significantly improve the competitiveness of the two types of remanufactured products in the market, but an increase of $\beta$ can only improve the competitiveness of the distributor’s remanufactured products, so TPR should rely more on price reduction to ensure the sales volume of his products. Interestingly, the improvement of $\beta$ can bring about a higher price of new products. We can understand that in the face of the only competitor, TPR, whose unit retail price and sales volume of remanufactured product decrease, the distributor chooses to increase the price of the two types of products it operates to ensure the maximum profit.

5 Comparison and managerial implications

In the previous section we have obtained the equilibrium solution of the three models, and discussed the relationship between them with dual preference ($\alpha$ and $\beta$). In this section, we compare the sizes of the optimal decision variables in the three models, and use a numerical example to compare the optimal profit of each supply chain member in the three models. Finally, we draw some conclusions and management implications.

5.1 Comparison of product price and quantity

As both model DR and model TR fall into the single remanufacturing mode, while model DTR falls into the hybrid remanufacturing mode, we first make a comparative analysis between model DR and model TR.

Proposition 5

Through the comparative analysis of model DR and model TR, we have found that $p_{d}^{DR*} \le p_{t}^{TR*}$, $p_{n}^{DR*} \ge p_{n}^{TR*}$; when $c_{n} \le 1 + \alpha \left( {\beta - 1} \right)$, and $c_{r} \in \left[ {A_{1} ,C_{1} } \right]$, we have $q_{n}^{DR*} \le q_{n}^{TR*}$; when $c_{n} \le 1 + \alpha \left( {\beta - 1} \right)$, and $c_{r} \in [C_{1} ,B_{3} ]$, we have $q_{n}^{DR*} \ge q_{n}^{TR*}$; when $c_{n} \ge 1 + \alpha \left( {\beta - 1} \right)$, and $c_{r} \in [A_{2} ,B_{3} ]$, we have $q_{n}^{DR*} \le q_{n}^{TR*}$. When $c_{n} \le C_{3}$, and $c_{r} \in \left[ {A_{1} ,B_{1} } \right]$, we have $q_{d}^{DR*} \le q_{t}^{TR*}$; when $c_{n} \ge C_{3}$, and $c_{r} \in \left[ {\max (A_{1} ,A_{2} ),C_{2} } \right]$, we have $q_{d}^{DR*} \ge q_{t}^{TR*}$; when $c_{n} \ge C_{3}$, and $c_{r} \in \left[ {C_{2} ,B_{1} } \right]$, we have $q_{d}^{DR*} \le q_{t}^{TR*}$.

We find that the unit retail price of the remanufactured products in model DR is always lower than that in model TR, while the unit retail price of the new products is always higher than that in model TR. There are two main reasons for a lower unit retail price of the remanufactured products in model DR. First, the remanufactured products in model DR are manufactured by the distributor, while the remanufactured products in model TR are manufactured by TPR. We have mentioned in the hypothesis that compared with remanufactured products from the distributor, consumers are more willing to purchase the remanufactured products from the TPR. Second, both new and remanufactured products in the DR model are sold by dealers. The distributor can sell the remanufactured products at a lower price when the remanufactured product is not dominant, and raise the price of the new products to make a higher profit. In model TR, the only profit source of TPR is the remanufactured product, so he sets a higher price for the remanufactured product without worrying about competition between the new product and the remanufactured product.

In order to more clearly discuss the quantity differences of new products, remanufactured products between model DR and model TR, we assume that $c_{n} = 0.4$ and $c_{r} = 0.2$. It can be seen from Fig. 2 that only when both the values of $\alpha$ and $\beta$ are high (satisfying $- 0.8 + 0.2\alpha + 0.8\beta - 1.2\alpha \beta - 0.4\alpha^{2} \beta - 0.8\alpha \beta^{2} + 2.8\alpha^{2} \beta^{2} \le 0$), the quantity of new products in model DR is lower than that in model TR. From Fig. 3 we can draw that only when both the values of $\alpha$ and $\beta$ are high (satisfying $\beta \ge \frac{0.4 + 1.2\alpha }{{\alpha (1.2 + \alpha )}}$), the quantity of remanufactured products in model DR is higher than that in model TR.

Fig. 2

Simultaneous effects of $\alpha$ and $\beta$ on $q_{n}$

Fig. 3

Simultaneous effects of $\alpha$ and $\beta$ on $q_{d}$ or $q_{t}$

Proposition 6

Through the comparative analysis of the single remanufacturing mode and the hybrid remanufacturing mode, we find that $q_{d}^{DTR*} \le q_{d}^{DR*} \le q_{d}^{DTR*} + q_{t}^{DTR*}$, $q_{n}^{DR*} \ge q_{n}^{DTR*}$, $p_{d}^{DR*} \ge p_{d}^{DTR*}$, $q_{t}^{DTR*} \le q_{t}^{TR*} \le q_{d}^{DTR*} + q_{t}^{DTR*}$, $q_{n}^{TR*} \ge q_{n}^{DTR*}$, $p_{t}^{TR*} \ge p_{t}^{DTR*}$.

Proposition 6 points out that the quantity of the remanufactured products from the remanufacturer in the single remanufacturing mode is always higher than that in the hybrid remanufacturing mode, and the quantity of the new products in the single remanufacturing mode is higher, but the total quantity of the remanufactured products is lower, no matter whether the OEM subcontracts the remanufacturing to the distributor or TPR.

We analyze the reasons on the basis of the comparison between model DR and model DTR. In model DR, only the distributor operates the remanufactured products and there is no competitor in the secondary market, so consumers who want to buy the remanufactured products can only buy from the distributor. In model DTR, TPR also has the qualification to enter the remanufacturing market, and his appearance attracts some consumers who want to buy the remanufactured products. The competition between the distributor and TPR in the secondary market leads to decrease in the sales volume of their remanufactured products, and both parties will lower the price to ensure the sales volume of the remanufactured products, which also leads to a lower price of the remanufactured products in model DR.

Since we assume that a consumer can own only one type of product, the sales volume of new products in model DR are correspondingly higher. It is the competition between the two parties and the lower price of the remanufactured product that greatly stimulates the vitality of the remanufactured product market, resulting in a higher total sales volume of the remanufactured products in model DTR. Therefore, from the perspective of environment and consumers, the hybrid remanufacturing mode is better than the single remanufacturing mode.

5.2 Comparison of profit

Considering the complexity of the profit function of each member of the supply chain in the three models, we adopt a numerical example to further discuss the profit relationship between the three models and the impacts of $\alpha$ and $\beta$ on them. As with proposition 5, we assume that $c_{n} = 0.4$ and $c_{r} = 0.2$. According to Theorems 1, 2, and 3, we obtain that the parameters in the three models must satisfy $\frac{0.5}{\alpha } \le \beta \le \frac{0.8}{\alpha }$ in model DR, $0.5 \le \alpha \le 1$ and $0.5 \le \beta \le 1$ in model TR, and $\frac{0.8 - 0.2\alpha }{{0.4 - 0.2\alpha + \alpha^{2} }} \le \beta \le \frac{3.4 - 2\alpha }{{1 + \alpha }}$ in model DTR respectively.

We plot on X-axis $\alpha$ and Y-axis $\beta$ to establish the rectangular coordinate system. As can be seen from Fig. 4, the applicable scope of the model TR is the most extensive, followed by the model DR, and finally the model DTR. This is because consumers’ WTP for TPR’s remanufactured products is higher than that for the distributor’s remanufactured products, and the conditions for TPR to accept remanufacturing authorization in model TR are looser. Compared with the single remanufacturing mode, the conditions for ensuring that both the distributor and TPR will accept remanufacturing licenses are stricter in the hybrid remanufacturing mode.

Fig. 4

Constraints for the three models

Proposition 7

We can get by comparing OEM’s profit in the three models:

• (1) When $D_{1} \ge 0$, $\pi_{m}^{DTR*} \ge \pi_{m}^{TR*}$;

• (2) When $D_{1} < 0$, $\pi_{m}^{DTR*} < \pi_{m}^{TR*}$;

(3)$\min (\pi_{m}^{DTR*} ,\pi_{m}^{TR*} ) > \pi_{m}^{DR*}$.

In order to clarify the common impact of $\alpha$ and $\beta$ on OEM’s profits in the three models, we established a three-dimensional coordinate system. Proposition 7 shows that model TR and model DTR are always better than model DR for OEM. When $\alpha$ is large enough and $\beta$ is large enough ($D_{1} \ge 0$), model DTR will outperform model TR. Conversely, model TR has more advantages. We know that in model TR, OEM only earns profits from the wholesale of new products and the remanufacturing license to TPR, while OEM can also get a certain license fee from the remanufacturing of the distributor in model DTR. A larger $\alpha$ indicates that consumers' purchase intention for the remanufactured products is closer to that for the new products, and a larger $\beta$ indicates that consumers' purchase intention for the distributor’s remanufactured products is closer to that for TPR’s remanufactured products, while the higher $\alpha$ and $\beta$ indicate that consumers' purchase intention for the three types of products in the market is very similar. Although OEM's profit from new products and remanufactured products in model DTR is lower than model TR (Proposition 6), there will be a good market environment for the distributor’s remanufactured products when $\alpha$ and $\beta$ are both high, thus bringing higher profits to OEM.

As can be seen from Fig. 5, OEM’s profit increases in all three models as $\alpha$ increases. This is consistent with what we expected, and the reasons are similar in the three models. We focused on the relationship between $\alpha$ and OEM’s profit in model DTR. Although a higher $\alpha$ results in a lower profit from new products for OEM, along with higher sales volume of both types of remanufactured products, OEM also benefits more from licensed remanufacturing activities by increasing patent licensing fee (Proposition 3). Similarly, OEM’s profit in model DR and model TPR also increases as $\beta$ increases.

Fig. 5

Simultaneous effects of $\alpha$ and $\beta$ on $\pi_{m}$

Proposition 8

We can get by comparing TPR’s profit in the three models:

• (1) $D_{2} \ge 0$, and $\pi_{t}^{DTR*} \ge \pi_{t}^{TR*}$;

• (2) $D_{2} < 0$, and $\pi_{t}^{DTR*} < \pi_{t}^{TR*}$.

Proposition 8 shows that from the perspective of TPR, model DTR is superior to model TR only when $\alpha$ is large enough and $\beta$ is small enough ($D_{{2}} \ge 0$). Otherwise, TPR can get higher returns in model TR. Compared with model TR, the entry of the distribution into the remanufactured products market will definitely cannibalize TPR’s remanufactured products in model DTR. When $\alpha$ is larger and $\beta$ is smaller, the more attractive TPR’s remanufactured products are to consumers, the more competitive they will be in the secondary market. Only in this case is it possible for model DTR to outperform model TR (Fig. 6).

Fig. 6

Simultaneous effects of $\alpha$ and $\beta$ on $\pi_{t}$

As $\alpha$ grows, TPR’s total profit in both model TR and model DTR will increase. Whether in the single or hybrid remanufacturing mode, the sales of remanufactured products are always TPR’s only source of profit. In model TR, a higher $\alpha$ will increase both the sales volume and unit retail price of TPR’s remanufactured products (Proposition 2). Although OEM will charge higher patent licensing fees, it will not affect the trend of TPR's overall profit increase. In model DTR, an increase in $\alpha$ results in an increase in TPR’s remanufactured products quantity (Proposition 3). Although TPR has to reduce the price of its remanufactured products due to the fact that the sales volume of the distributor’s remanufactured products is also increasing in the secondary market, this will not affect the increase of his total profit.

As $\beta$ increases, the total profit of each member model TR remains the same, which we will not repeat here. In model DTR, with a higher $\beta$, the distributor’s remanufactured products will pose a more serious threat to TPR’s remanufactured products, and the unit retail price of TPR’s remanufactured products will also be reduced, which directly leads to a decrease in the total profit of TPR.

Proposition 9.

We can get by comparing the distributor’s profit in the three models:

• (1) $D_{3} \ge 0$, and $\pi_{d}^{DTR*} \ge \pi_{d}^{TR*}$;

• (2) $D_{3} < 0$, and $\pi_{d}^{DTR*} < \pi_{d}^{TR*}$;

• (3) $\max (\pi_{d}^{DTR*} ,\pi_{d}^{TR*} ) < \pi_{d}^{DR*}$.

Proposition 9 shows that the distributor prefers model DR regardless of the value of the consumers’ dual preference. This is because in model DR, the only two types of products in the market are directly sold to consumers by the distributor. Compared with model TR, the business entity engaged in remanufacturing activities changed from the distributor to TPR. In model DTR, although the distributor also participates in the remanufacturing, the entry of TPR into the secondary market will inevitably cause an adverse impact on the new products and the distributor’s remanufactured products.

Now we will shift our focus on the comparison between model TR and model DTR. In general, the distributor might prefer model DTR. This is because in model DTR, the distributor can earn profits from the sales of new and remanufactured products, while he can only make profits from new products in model TR. But the reality is that the distributor prefers model TR when $\alpha$ is large enough and $\beta$ is small enough (as shown in the black circle in Fig. 7). In this case, compared with TPR’s remanufactured products, the distributor's remanufactured products are not competitive at all, and they will also cannibalize his own new products. Therefore, it is reasonable that the total profit of the distributor in model DTR is lower than that in model TR.

Fig. 7

Simultaneous effects of $\alpha$ and $\beta$ on $\pi_{d}$

As can be seen from Fig. 7, an increase in $\alpha$ will lead to an increase of the distributor’s total profit in model DR. In this case, the distributor's profit comes from two parts: the profit of new product and remanufactured products. Proposition 1 tells us that as $\alpha$ increases, the sales volume of new product will decrease. Although the distributor can keep the unit retail price of new product unchanged, the decline of retail profits of new products is inevitable. Both the price and sales volume of the remanufactured products are rising, and the profit increase from his remanufactured products is higher than the profit loss from his new product, so the total profit of the distributor will present an upward trend.

With an increase of $\alpha$, the profit of the distributor in model TR will decrease. This is because a higher $\alpha$ makes TPR’s remanufactured products more competitive, resulting in a lower unit retail price and sales volume of new products (Proposition 2).

The difference is that with an increase of $\alpha$, the profit of the distributor decreases firstly and then increases in model DTR. In this model, in addition to selling new products, the distributor also undertakes part of the remanufacturing business. According to Proposition 3, with an increase of $\alpha$, both the sales volume and the unit retail price of the new products decrease, while the price and sales volume of the remanufactured products increase. There is a threshold $\alpha_{{1}}$, and when $\alpha < \alpha_{1}$, the profit growth from remanufactured products cannot make up for the profit loss from new products. When $\alpha > \alpha_{1}$, the profit from remanufactured products is higher. With the rise of $\beta$, the distributor’s profit of both model DR and model DTR will increase, which is consistent with what we expected. Although a rise of $\beta$ is not conducive to the sales of new products, it does not affect the overall trend of rising profit.

6 Conclusions and suggestions for further research

In this paper, we developed three closed-loop supply chain models to study how OEM should choose the best remanufacturing model when remanufacturing activities can be implemented by the distributor and TPR. We proposed the optimal solution for each member in the three models under patent protection, and discussed the impacts of consumer’s dual preference on their decision variables and profits.

Our main conclusions are as follows: (1) Due to the stimulating effect of competition between the two types of remanufactured products on the market, the total quantity of remanufactured products in the hybrid remanufacturing mode is always higher than that in the single remanufacturing modes, and the unit retail price of remanufactured products and the sales volume of new products in the hybrid remanufacturing mode are lower. In terms of two single remanufacturing mode, the unit retail price of new products in model DR is always higher than that in model TR, but the unit retail price of remanufactured products in model DR is lower. (2) The wholesale price of OEM’s new products and the patent licensing fee charged to licensee (the distributor or TPR) will not change with the change of remanufacturing mode. (3) When consumers are very reluctant to purchase a certain type of remanufactured products, whether the remanufactured products come from the distributor or TPR, OEM prefers to outsource the remanufacturing business to TPR. Otherwise, OEM prefers to subcontract remanufacturing to both the distributor and TPR. In terms of two single remanufacturing modes, whether for OEM or TPR, TPR remanufacturing is always better than the distributor remanufacturing mode. (4) No matter which mode OEM chooses for remanufacturing, the higher consumers' preference for the remanufactured products, the greater OEM’s total profit will be. (5) The entry of TPR into the secondary market will always have an adverse impact on the distributor, whether the distributor participates in remanufacturing business or not. TPR will not be adversely affected by the distributor’s remanufacturing activities only when the former’s remanufactured products are very competitive. Otherwise, the distributor's entry into the secondary market will have a negative impact on TPR.

Our research conclusions provide the following managerial insights for companies that may be interested in remanufacturing: (1) When only distributor is licensed to remanufacture, distributor should keep the price of the new product unchanged and increase the price of remanufactured products in order to earn higher profits if consumers’ preference for remanufactured products increases. At this point, OEM should increase the total profits by increasing the royalty licensing fee. (2) When only TPR is licensed to remanufacture, TPR should take the opportunity to raise the price of the remanufactured product if consumers’ willingness to purchase remanufactured product increases, and distributor should lower the price of the new product in order to maximize profits. (3) When there is competition between distributor and TPR in the market for the remanufactured products, distributor should raise the price of remanufactured products and lower the price of new products if consumers’ perceived preference for remanufactured products increases, while TPR should lower the price of remanufactured products. When consumers’ preference for the remanufactured products from distributor increases, distributor should raise the price of the new products while raising the price of remanufactured products. (4) From the perspective of consumer welfare and environmental protection, OEM should license remanufacturing to both distributor and TPR, rather than to a single one. This is because the demand for remanufactured products is higher in a hybrid remanufacturing model than that in a single remanufacturing model, and the price of remanufactured products is lower in a hybrid remanufacturing model, regardless of whether the remanufactured products come from a distributor or a TPR. (5) In general, OEM should encourage both distributor and TPR to enter the remanufactured products market, especially if consumers are not sensitive to the differences between new and remanufactured products, as well as the source of remanufactured products. In particular, OEM should encourage TPR to remanufacture. TPR should welcome distributor into the remanufactured products market when consumers have a low preference for distributor’s remanufactured products. Distributors should seek exclusive patent licenses for remanufacturing, which can significantly increase their profits. At the same time, they should actively resist the entry of TPR whether or not they are involved in the remanufacturing business.

Nevertheless, there are some flaws in this paper. For example, we assume that OEM charges to the distributor or TPR royalties for each unit he remanufactures, but patent holders usually use the fixed-fee licensing or the two-part tariff licensing in the real world. Moreover, the distributor usually has an advantage in recycling waste products because he is usually closer to consumers, but the competition of reverse channels is not considered in this paper. Future research can be conducted based on our study in the following two aspects. First, what impact will three different licensing strategies have on the decision-making of supply chain members? Second, some interesting findings may be obtained by adding reverse channel competition into our model to study the decisions in supply chain.

Appendix

See Fig. 8

Proof of Eqs. (1–3)

Defining the consumer’s WTP for a new product as $u$, which is uniformly distributed in $\left[ {0,1} \right]$, i.e. $u \sim U[0,1]$. As shown in Fig. a1, the net utility of consumers who are willing to pay for a new product, a manufactured product from the TPR and that from the distribution is $u_{n} = u - p_{n}$, $u_{t} = \alpha u - p_{t}$, $u_{d} = \alpha \beta u - p_{d}$, respectively. By simple calculation, the indifferent point between buying a new product or a remanufactured product from the TPR is at $\frac{{p_{n} - p_{t} }}{1 - \alpha }$. Likewise, the indifferent point between buying a remanufactured product from the TPR or from the distributor is at $\frac{{p_{t} - p_{d} }}{\alpha (1 - \beta )}$. The indifferent point between buying a remanufactured product from the distributor or buying nothing is $\frac{{p_{d} }}{\alpha \beta }$. So we can derive that the quantities of new products, the TPR’s remanufactured products and the distributor’s remanufactured products are $1 - \frac{{p_{n} - p_{t} }}{1 - \alpha }$, $\frac{{p_{n} - p_{t} }}{1 - \alpha } - \frac{{p_{t} - p_{d} }}{{\alpha \left( {1 - \beta } \right)}}$, and $\frac{{p_{t} - p_{d} }}{{\alpha \left( {1 - \beta } \right)}} - \frac{{p_{d} }}{\alpha \beta }$, respectively.

Fig. 8

Net present value functions for consumer

Proof of Theorem 1.

The Hessian matrix of $\pi_{d}^{DR}$with respect to $q_{n}$ and $q_{d}$ is as follows:

$$H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{DR} }}{{\partial q_{n}^{2} }}} & {\frac{{\partial^{2} \pi_{d}^{DR} }}{{\partial q_{n} \partial q_{d} }}} \\ {\frac{{\partial^{2} \pi_{d}^{DR} }}{{\partial q_{d} \partial q_{n} }}} & {\frac{{\partial^{2} \pi_{d}^{DR} }}{{\partial q_{d}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & { - 2\alpha \beta } \\ { - 2\alpha \beta } & { - 2\alpha \beta } \\ \end{array} } \right]$$

Since $1 - \alpha \beta \ge 0$.Hence, $\pi_{d}^{DR}$ is strictly jointly concave in $q_{n}$ and $q_{d}$.Using the first-order conditions of Eq. (5), we can derive that $q_{n}^{DR} = \frac{{ - w + c_{r} + f_{d} + 1 - \alpha \beta }}{{2\left( {1 - \alpha \beta } \right)}}$, and $q_{d}^{DR} = \frac{{\alpha \beta w - c_{r} - f_{d} }}{{2\alpha \beta \left( {1 - \alpha \beta } \right)}}$. (a1).

Substituting $q_{n}^{DR}$ and $q_{d}^{DR}$ into Eq. (4), we can derive that $\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial w^{2} }} = - \frac{1}{1 - \alpha \beta } < 0$, hence $\pi_{m}^{DR}$ is concave in $w$. From the first-order condition of Eq. (4), we have $w^{DR} = \frac{{c_{r} + c_{n} + 2f_{d} + 1 - \alpha \beta }}{2}$. (a2).

Substituting $q_{n}^{DR}$, $q_{d}^{DR}$ and $w^{DR}$ into Eq. (4), we can get that $\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial f_{d}^{2} }} = - \frac{1}{2\alpha \beta } < 0$, hence $\pi_{m}^{DR}$ is concave in $f_{d}$. From the first-order condition of Eq. (4), we get $f_{d}^{DR*} = \frac{{\alpha \beta - c_{r} }}{2}$. (a3).

Substituting $f_{d}^{DR*}$ into Eq. (a2), we have $w^{DR*} = \frac{{1 + c_{n} }}{2}$. (a4).

Substituting $f_{d}^{DR*}$ and $w^{DR*}$, we have $q_{n}^{DR*} = \frac{{c_{r} - c_{n} + 1 - \alpha \beta }}{{4\left( {1 - \alpha \beta } \right)}}$, and $q_{d}^{DR*} = \frac{{ - c_{r} + \alpha \beta c_{n} }}{{4\alpha \beta \left( {1 - \alpha \beta } \right)}}$. (a5).

Plugging $q_{n}^{DR*}$ and $q_{d}^{DR*}$ into Eqs. (1–3), we can derive $p_{n}^{DR*} = \frac{{c_{n} + 3}}{4}$, and $p_{d}^{DR*} = \frac{{c_{r} + 3\alpha \beta }}{4}$. (a6).

We notice that all parameters and variables must satisfy the following constrains: $q_{n}^{DR*} \ge 0$, $q_{d}^{DR*} \ge 0$, $f_{d}^{DR*} \ge 0$, $w^{DR*} \ge 0$, $w^{DR*} - c_{n} \ge 0$, $p_{n}^{DR*} - w^{DR*} \ge 0$, and $p_{d}^{DR*} - c_{r} - f_{d}^{DR*} \ge 0$. (a7).

Substituting Eq. (a4) and Eq. (a5) into inequation (a7), we can have $A_{1} \le c_{r} \le B_{1}$, and $c_{n} \le 1$. Where $A_{1} = c_{n} + \alpha \beta - 1$, and $B_{1} = \alpha \beta c_{n}$. (a8).

Proof of Theorem 2.

Since $\frac{{\partial^{2} \pi_{d}^{TR} }}{{\partial q_{n}^{2} }} = - 2 < 0$, and $\frac{{\partial^{2} \pi_{t}^{TR} }}{{\partial q_{t}^{2} }} = - 2\alpha < 0$, hence, $\pi_{d}^{TR}$ is concave in $q_{n}^{TR}$ and $\pi_{t}^{TR}$ is concave in $q_{t}^{TR}$.

Hence, from the first order conditions of Eq. (7) and Eq. (8), we get $q_{n}^{TR} = \frac{{ - 2w + c_{r} + f_{t} + 2 - \alpha }}{4 - \alpha }$, and $q_{t}^{TR} = \frac{{\alpha w - 2c_{r} - 2f_{t} + \alpha }}{{\alpha \left( {4 - \alpha } \right)}}$. (a9).

Substituting $q_{n}^{TR}$ and $q_{t}^{TR}$ into Eq. (6), we have $\frac{{\partial^{2} \pi_{m}^{TR} }}{{\partial w^{2} }} = - \frac{4}{4 - \alpha } < 0$, hence, $\pi_{m}^{TR}$ is concave in $w^{TR}$. From the first order condition of Eq. (6) with respect to $w^{TR}$, we can derive $w^{TR} = \frac{{c_{r} + 2c_{n} + 2f_{t} + 2 - \alpha }}{4}$. (a10).

Subsitituting $q_{n}^{TR}$, $q_{t}^{TR}$ and $w^{TR}$ into Eq. (6), we get $\frac{{\partial^{2} \pi_{m}^{TR} }}{{\partial f_{t}^{2} }} = - \frac{8 - 2\alpha }{{2\alpha \left( {4 - \alpha } \right)}} < 0$, hence, $\pi_{m}^{TR}$ is concave in $f_{t}^{TR}$. From the first order condition of Eq. (6) with respect to $f_{t}^{TR}$, we can have $f_{t}^{TR*} = \frac{{\alpha - c_{r} }}{2}$. (a11).

Substituting $f_{t}^{TR*}$ into Eq. (a9), we can derive $w^{TR*} = \frac{{1 + c_{n} }}{2}$. (a12).

Substituting $f_{t}^{TR*}$ and $w^{TR*}$ into Eq. (a8), we can derive $q_{n}^{TR*} = \frac{{c_{r} - 2c_{n} + 2 - \alpha }}{{2\left( {4 - \alpha } \right)}}$, and $q_{t}^{TR*} = \frac{{ - 2c_{r} + \alpha c_{n} + \alpha }}{{2\alpha \left( {4 - \alpha } \right)}}$. (a13).

Substituting Eqs. (a11)-(a13) into Eqs. (1–3), we get $p_{n}^{TR*} = \frac{{c_{r} + \left( {2 - \alpha } \right)c_{n} + 6 - 2\alpha }}{{2\left( {4 - \alpha } \right)}}$, and $p_{t}^{TR*} = \frac{{\left( {2 - \alpha } \right)c_{r} + \alpha c_{n} + \alpha \left( {5 - \alpha } \right)}}{{2\left( {4 - \alpha } \right)}}$. (a14).

Similar to the DR model above, the decision variables in the TR model must meet the following constrains:$q_{n}^{TR*} \ge 0$, $q_{t}^{TR*} \ge 0$, $f_{t}^{TR*} \ge 0$, $w^{TR*} \ge 0$, $w^{TR*} - c_{n} \ge 0$, $p_{n}^{TR*} - w^{TR*} \ge 0$, $p_{t}^{TR*} - c_{r} - f_{t}^{TR*} \ge 0$. (a15).

Substituting Eqs. (a10)-(a13) into inequation (a14), we have $A_{2} \le c_{r} \le B_{2}$, and $c_{n} \le 1$. Where $A_{2} = 2c_{n} + \alpha - 2$, and $B_{2} = \left( {\alpha + \alpha c_{n} } \right)/2$. (a16).

Proof of Theorem 3.

The Hessian matrix of $\pi_{d}^{DTR}$ with respect to $q_{n}$ and $q_{d}$ is as follows $H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{DTR} }}{{\partial q_{n}^{2} }}} & {\frac{{\partial^{2} \pi_{d}^{DTR} }}{{\partial q_{n} \partial q_{d} }}} \\ {\frac{{\partial^{2} \pi_{d}^{DTR} }}{{\partial q_{n} \partial q_{d} }}} & {\frac{{\partial^{2} \pi_{d}^{DTR} }}{{\partial q_{d}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & { - 2\alpha \beta } \\ { - 2\alpha \beta } & { - 2\alpha \beta } \\ \end{array} } \right]$.

Since $4\alpha \beta \left( {1 - \alpha \beta } \right) > 0$. Hence, $\pi_{d}^{DTR}$ is jointly concave in $q_{n}$ and $q_{d}$.

Since $\frac{{\partial^{2} \pi_{t}^{DTR} }}{{\partial q_{t}^{2} }} = - 2\alpha \le 0$. Hence, $\pi_{t}^{DTR}$ is concave in $q_{t}$.

From the first order condition of Eq. (10) respect to $q_{n}$ and $q_{d}$, and the first order condition of Eq. (11) respect to $q_{t}$, we have $q_{n}^{DTR} = \frac{{\left( {\beta - 4} \right)w + \left( {5 - 2\beta } \right)c_{r} + 2\left( {1 - \beta } \right)f_{t} + 3f_{d} + 4 - 2\alpha - \beta - \alpha \beta }}{{2\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $q_{d}^{DTR} = \frac{{3\alpha \beta w - \left( {4 - \alpha - 2\beta + 2\alpha \beta } \right)c_{r} + 2\beta \left( {1 - \alpha } \right)f_{t} + \left( {\alpha - 4} \right)f_{d} - \alpha \beta \left( {1 - \alpha } \right)}}{{2\alpha \beta \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, and $q_{t}^{DTR} = \frac{{\alpha \left( {1 - \beta } \right)w - \left( {1 + \alpha - 2\alpha \beta } \right)c_{r} - 2\left( {1 - \alpha \beta } \right)f_{t} + \left( {1 - \alpha } \right)f_{d} + \alpha \left( {1 - \alpha \beta } \right)}}{{\alpha \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$. (a17).

Substituting $q_{n}^{DTR}$, $q_{d}^{DTR}$ and $q_{t}^{DTR}$ into Eq. (9), we have $\frac{{\partial^{2} \pi_{m}^{DTR} }}{{\partial w^{2} }} = - \frac{4 - \beta }{{2\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \le 0$.

Hence, $\pi_{m}^{DTR}$ is concave in $w$. From the first condition of Eq. (9) respect to $w$, we have $w^{DTR} = \frac{{\left( {5 - 2\beta } \right)c_{r} + \left( {4 - \beta } \right)c_{n} + 4\left( {1 - \beta } \right)f_{t} + 6f_{d} + 4 - 2\alpha - \beta - \alpha \beta }}{{2\left( {4 - \beta } \right)}}$. (a18).

Substituting $q_{n}^{DTR}$, $q_{d}^{DTR}$, $q_{t}^{DTR}$ and $w^{DTR}$ into Eq. (9), we get the Hessian matrix of $\pi_{m}^{DR}$ respect to $f_{t}$ and $f_{d}$, $H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial f_{t}^{2} }}} & {\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial f_{t} \partial f_{d} }}} \\ {\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial f_{d}^{2} }}} & {\frac{{\partial^{2} \pi_{m}^{DR} }}{{\partial f_{d} \partial f_{t} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \frac{4}{{\alpha \left( {4 - \beta } \right)}}} & {\frac{2}{{\alpha \left( {4 - \beta } \right)}}} \\ {\frac{2}{{\alpha \left( {4 - \beta } \right)}}} & { - \frac{4}{{\alpha \beta \left( {4 - \beta } \right)}}} \\ \end{array} } \right]$.

Since $- \frac{4}{{\alpha \left( {4 - \beta } \right)}} \le 0$ and $\frac{4}{{\alpha^{2} \left( {4 - \beta } \right)^{2} }}\left( {\frac{4}{\beta } - 1} \right) > 0$. Hence, $\pi_{m}^{DR}$ is jointly concave in $f_{t}$ and $f_{d}$. From the first order conditions of Eq. (9) respect of $f_{t}$ and $f_{d}$, we have $f_{t}^{DTR*} = \frac{{\alpha - c_{r} }}{2}$, and $f_{d}^{DTR*} = \frac{{\alpha \beta - c_{r} }}{2}$. (a19).

Substituting $f_{t}^{DTR*}$ and $f_{d}^{DTR*}$ into the Eq. (a18), we get $w^{DTR*} = \frac{{1 + c_{n} }}{2}$. (a20).

Substituting $f_{t}^{DTR*}$, $f_{d}^{DTR*}$ and $w^{DTR*}$ into (a17), we have $q_{n}^{DTR*} = \frac{{\left( {5 - 2\beta } \right)c_{r} - \left( {4 - \beta } \right)c_{n} + 4 - 2\alpha - \beta - \alpha \beta }}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $q_{d}^{DTR*} = \frac{{ - \left( {4 - \alpha - 2\beta + 2\alpha \beta } \right)c_{r} + 3\alpha \beta c_{n} - \alpha \beta \left( {1 - \alpha } \right)}}{{4\alpha \beta \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, and $q_{t}^{DTR*} = \frac{{ - \left( {1 + \alpha - 2\alpha \beta } \right)c_{r} + \alpha \left( {1 - \beta } \right)c_{n} + \alpha \left( {1 - \alpha \beta } \right)}}{{2\alpha \left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$. (a21).

Substituting Eqs. (a19–a21), we have $p_{n}^{DTR*} = \frac{{\left( {1 + \alpha - 2\alpha \beta } \right)c_{r} + \left( {4 - 2\alpha - \beta - \alpha \beta } \right)c_{n} + 12 - 4\alpha - 3\beta - 6\alpha \beta + \alpha^{2} \beta }}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, $p_{d}^{DTR*} = \frac{{\left( {4 - \alpha - \alpha \beta - 2\alpha \beta^{2} } \right)c_{r} - \alpha \beta \left( {1 - \beta } \right)c_{n} + \alpha \beta \left( {11 - 3\alpha - 3\beta - 5\alpha \beta } \right)}}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$, and $p_{t}^{DTR*} = \frac{{\left( {3 - 2\alpha - \beta } \right)c_{r} + \alpha \left( {1 - \beta } \right)c_{n} + \alpha \left( {5 - \alpha - \beta - 3\alpha \beta } \right)}}{{2\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}}$. (a22).

Similar to the DR model and the TR model, the parameters must satisfy the following constrains:$q_{n}^{DTR*} \ge 0$, $q_{d}^{DTR*} \ge 0$, $q_{t}^{DTR*} \ge 0$, $w^{DTR*} \ge 0$, $f_{d}^{DTR*} \ge 0$, $f_{t}^{DTR*} \ge 0$, $w^{DTR*} - c_{n} \ge 0$, $p_{n}^{DTR*} - w^{DTR*} \ge 0$, $p_{t}^{DTR*} - c_{r} - f_{t}^{DTR*} \ge 0$, $p_{d}^{DTR*} - c_{r} - f_{d}^{DTR*} \ge 0$. (a23).

From $q_{n}^{DTR*} \ge 0$, we have $c_{r} \ge A_{3}$, where $A_{3} = \frac{{(4 - \beta )c_{n} - 4 + 2\alpha + \beta + \alpha \beta }}{5 - 2\beta }$. From $q_{d}^{DTR*} \ge 0$, we have $c_{r} \le B_{3}$, where $B_{3} = \frac{{3\alpha \beta c_{n} - \alpha \beta (1 - \alpha )}}{4 - \alpha - 2\beta + 2\alpha \beta }$. From $q_{t}^{DTR*} \ge 0$, and $p_{t}^{DTR*} - c_{r} - f_{t}^{DTR*} \ge 0$, we have $c_{r} \le B_{4}$, where $B_{4} = \frac{{\alpha (1 - \beta )c_{n} + \alpha \left( {1 - \alpha \beta } \right)}}{1 + \alpha - 2\alpha \beta }$. From $p_{n}^{DTR*} - w^{DTR*} \ge 0$, we have $c_{r} \ge A_{4}$, where $A_{4} = \frac{{(4 - \beta - 3\alpha \beta )c_{n} - 4 + 2\alpha + \beta + 2\alpha \beta - \alpha^{2} \beta }}{1 + \alpha - 2\alpha \beta }$. From $p_{d}^{DTR*} - c_{r} - f_{d}^{DTR*} \ge 0$, we have $c_{r} \le B_{5}$, where $B_{5} = \frac{{\alpha \beta ( - 1 + \beta )c_{n} + \alpha \beta \left( {3 - \alpha - \beta - \alpha \beta } \right)}}{{4 - \alpha - 2\beta - 3\alpha \beta + 2\alpha \beta^{2} }}$. From $w^{DTR*} - c_{n} \ge 0$, we have $c_{n} \le 1$. From $f_{d}^{DTR*} \ge 0$, we have $c_{r} \le \alpha \beta$. From $f_{t}^{DTR*} \ge 0$, we have $c_{r} \le \alpha$.

Because $B_{3} - B_{4} = \frac{{ - \alpha \left( {4 - \alpha - \beta - 2\alpha \beta } \right)\left[ {1 + \left( {1 - 2\beta } \right)c_{n} } \right]}}{{\left( {1 + \alpha - 2\alpha \beta } \right)\left( {4 - \alpha - 2\beta + 2\alpha \beta } \right)}} \le 0$,$B_{3} - B_{5} \le \frac{{\alpha \beta (8 - 10\beta - 2\alpha - 2\alpha \beta + 2\beta^{2} + 4\alpha \beta^{2} )}}{{(4 - \alpha - 2\beta - 3\alpha \beta + 2\alpha \beta^{2} )(4 - \alpha - 2\beta + 2\alpha \beta )}} \le 0$, $B_{3} - 1 \le 0$, $B_{3} - \alpha \le 0$, and $B_{3} - \alpha \beta \le 0$, and $A_{3} - A_{4} \ge \frac{{\alpha ( - 8 + 10\beta + 2\alpha + 2\alpha \beta - 2\beta^{2} - 4\alpha \beta^{2} )}}{(5 - 2\beta )(1 + \alpha - 2\alpha \beta )} \ge 0$, we have $A_{3} \le c_{r} \le B_{3}$.

Since $A_{3} \ge {0}$, we have $c_{n} \ge \frac{4 - 2\alpha - \beta - \alpha \beta }{{4 - \beta }}$.

From the above all, the parameters must satisfy the following conditions:

$A_{3} \le c_{r} \le B_{3}$, and $\frac{4 - 2\alpha - \beta - \alpha \beta }{{4 - \beta }} \le c_{n} \le 1$, where $A_{3} = \frac{{(4 - \beta )c_{n} - 4 + 2\alpha + \beta + \alpha \beta }}{5 - 2\beta }$, and $B_{3} = \frac{{3\alpha \beta c_{n} - \alpha \beta (1 - \alpha )}}{4 - \alpha - 2\beta + 2\alpha \beta }$.

Proof of proposition 1.

It can be easily prove that $\frac{{\partial q_{n}^{DR*} }}{\partial \alpha } = \frac{{4\beta (c_{r} - c_{n} )}}{{16(1 - \alpha \beta )^{2} }} \le 0$, $\frac{{\partial q_{d}^{DR*} }}{\partial \alpha } = \frac{{(1 - 2\alpha \beta )c_{r} + \alpha^{2} \beta^{2} c_{n} }}{{4\alpha^{2} \beta (1 - \alpha \beta )^{2} }} \ge 0$, $\frac{{\partial p_{n}^{DR*} }}{\partial \alpha } = 0$, $\frac{{\partial p_{d}^{DR*} }}{\partial \alpha } = \frac{3\beta }{4} \ge 0$, $\frac{{\partial q_{n}^{DR*} }}{\partial \beta } = \frac{{\alpha (c_{r} - c_{n} )}}{{4(1 - \alpha \beta )^{2} }} \le 0$, $\frac{{\partial q_{d}^{DR*} }}{\partial \beta } = \frac{{(1 - 2\alpha \beta )c_{r} + \alpha^{2} \beta^{2} c_{n} }}{{4\alpha \beta^{2} (1 - \alpha \beta )^{2} }} \ge 0$, $\frac{{\partial p_{n}^{DR*} }}{\partial \beta } = 0$, and $\frac{{\partial p_{d}^{DR*} }}{\partial \beta } = \frac{3\alpha }{4} \ge 0$.

Proof of proposition 2.

We can prove that $\frac{{\partial q_{n}^{TR*} }}{\partial \alpha } = \frac{{c_{r} - 2c_{n} - 2}}{{2(4 - \alpha )^{2} }} \le 0$, $\frac{{\partial q_{t}^{TR*} }}{\partial \alpha } = \frac{{4(2 - \alpha )c_{r} + \alpha^{2} (1 + c_{n} )}}{{2\alpha^{2} (4 - \alpha )^{2} }} \le 0$,$\frac{{\partial p_{n}^{TR*} }}{\partial \alpha } = \frac{{c_{r} - c_{n} - 2}}{{2(4 - \alpha )^{2} }} \le 0$, and $\frac{{\partial p_{t}^{TR*} }}{\partial \alpha } = \frac{{ - 2c_{r} + 4c_{n} + \alpha^{2} - 2\alpha + 20}}{{2(4 - \alpha )^{2} }} \ge 0$.

Proof of proposition 3.

We can prove that $\frac{{\partial q_{n}^{DTR*} }}{\partial \alpha } = - \frac{{(1 + 2\beta )[(5 - 2\beta )c_{r} - (4 - \beta )c_{n} ] + 3(4 - \alpha - \beta - 2\alpha \beta )(1 + \beta )}}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }} \le 0$, $\frac{{\partial q_{t}^{DTR*} }}{\partial \alpha } = \frac{{(4 - 2\alpha - \beta - 4\alpha \beta - \alpha^{2} + 4\alpha^{2} \beta^{2} )c_{r} + \alpha^{2} (1 + \beta )(1 - \beta )c_{n} + \alpha^{2} (1 - \beta )^{2} }}{{2\alpha^{2} (4 - \alpha - \beta - 2\alpha \beta )^{2} }} \le 0$,

$\frac{{\partial q_{d}^{DTR*} }}{\partial \alpha } = \frac{{[ - 4\alpha^{2} \beta^{2} + \alpha^{2} + 8\alpha \beta^{2} - 12\alpha \beta - 8\alpha + 2\beta^{2} - 12\beta + 16]c_{r} + 3\alpha^{2} \beta [(1 + 2\beta )c_{n} + 1 - \beta ]}}{{4\alpha^{2} \beta (4 - \alpha - \beta - 2\alpha \beta )^{2} }} \ge 0$, and $\frac{{\partial p_{n}^{DTR*} }}{\partial \alpha } = \frac{{(5 - 2\beta )(1 - \beta )c_{r} - (4 - \beta )(1 - \beta )c_{n} - 4 + \beta + 8\alpha \beta - \alpha^{2} \beta - 2\alpha \beta^{2} - 2\alpha^{2} \beta^{2} }}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }}$. When $c_{r} \ge C_{4} = \frac{{(4 - \beta )(1 - \beta )c_{n} + 4 - \beta - 8\alpha \beta + 2\alpha \beta + \alpha^{2} \beta + 2\alpha^{2} \beta^{2} }}{(5 - 2\beta )(1 - \beta )}$, we have $\frac{{\partial p_{n}^{DTR*} }}{\partial \alpha } \ge 0$.When $c_{r} \le C_{4}$, we have $\frac{{\partial p_{n}^{DTR*} }}{\partial \alpha } \le 0$.

$\begin{gathered} C_{4} - B_{3} = [(1 - \beta )(16 - 4\alpha - 12\beta - 6\alpha \beta + 2\beta^{2} + 4\alpha \beta^{2} )c_{n} + (4 - \beta - 8\alpha \beta + \alpha^{2} \beta + 2\alpha \beta^{2} + 2\alpha^{2} \beta^{2} ) \hfill \\ (4 - \alpha - 2\beta + 2\alpha \beta ) + a\beta (1 - \alpha )(1 - \beta )(5 - 2\beta )]/[(5 - 2\beta )(1 - \beta )(4 - \alpha - 2\beta + 2\alpha \beta ) \ge 0 \hfill \\ \end{gathered}$, we get $C_{4} \ge B_{3}$. Hence we obtain $\frac{{\partial p_{n}^{DTR*} }}{\partial \alpha } \le 0$.

$\frac{{\partial p_{t}^{DTR*} }}{\partial \alpha } = \frac{{ - (5 - 2\beta )(1 - \beta )c_{r} + (1 - \beta )(4 - \beta )c_{n} + {6}\alpha^{2} \beta^{2} { + 5}\alpha^{2} \beta + \alpha^{2} + {6}\alpha \beta^{2} - {22}\alpha \beta - 8\alpha + \beta^{2} - 9\beta + 20}}{{2(4 - \alpha )^{2} }}$. When $c_{r} \le C_{5} = \frac{{(1 - \beta )(4 - \beta )c_{n} + {6}\alpha^{2} \beta^{2} { + 5}\alpha^{2} \beta + \alpha^{2} + {6}\alpha \beta^{2} - {22}\alpha \beta - 8\alpha + \beta^{2} - 9\beta + 20}}{(5 - 2\beta )(1 - \beta )}$, we have $\frac{{\partial p_{t}^{DTR*} }}{\partial \alpha } \ge 0$.

$C_{5} - B_{3} = \left( \begin{gathered} \{ [(1 - \beta )(4 - \beta )(4 - \alpha - 2\beta + 20\alpha \beta ) - 3\alpha \beta (5 - 2\beta )(1 - \beta )]c_{n} + [(5 - \beta )(4 - \beta ) - \alpha (1 \hfill \\ + 3\beta )(8 - \alpha - 2\beta - 2\alpha \beta )](4 - \alpha - 2\beta + 2\alpha \beta ) + \alpha \beta (1 - \alpha )(1 - \beta )(5 - 2\beta )\} /[(4 - \alpha - 2\beta + 2\alpha \beta ) \hfill \\ (1 - \beta )(5 - 2\beta )] \hfill \\ \end{gathered} \right) \ge 0$ When $c_{r} \in [A_{3} ,B_{3} ]$, we have $\frac{{\partial p_{t}^{DTR*} }}{\partial \alpha } \le 0$.

$\frac{{\partial p_{d}^{DTR*} }}{\partial \alpha } = \frac{{\beta [(5 - 2\beta )(1 - \beta )c_{r} - (1 - \beta )(4 - \beta )c_{n} + {10}\alpha^{{2}} \beta^{{2}} { + 11}\alpha^{{2}} \beta { + 3}\alpha^{{2}} { - 34}\alpha \beta { - 24}\alpha { + 3}\beta { + 44}]}}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }}$.

From $c_{r} \ge A_{3}$, $\frac{{\partial p_{d}^{DTR*} }}{\partial \alpha } \ge \frac{{10\alpha^{2} \beta^{2} + 11\alpha^{2} \beta + 3\alpha^{2} + 9\alpha \beta^{2} - 35\alpha \beta - 22\alpha + 2\beta^{2} - 18\beta + 40}}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }} \ge 0$.

When $c_{r} \in [A_{3} ,B_{3} ]$, we have $\frac{{\partial p_{d}^{DTR*} }}{\partial \alpha } \ge 0$.

Proof of proposition 4.

$\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } = \frac{{3[(4\alpha - 1)c_{r} - 3\alpha c_{n} + \alpha (1 - \alpha )]}}{4(4 - \alpha - \beta - 2\alpha \beta )}$. When $0 \le \alpha \le 1/4$, and $c_{r} \le C_{6} = \frac{{ - 3\alpha c_{n} + \alpha (1 - \alpha )}}{1 - 4\alpha }$, we have $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \ge 0$.

$C_{6} - A_{3} = \frac{{(1 - 3\alpha - c_{n} )(4 - \alpha - \beta - 2\alpha \beta )}}{(5 - 2\beta )(1 - 4\alpha )}$.

From $c_{n} \ge \frac{4 - 2\alpha - \beta - \alpha \beta }{{4 - \beta }}$, we have $C_{6} - A_{3} \le 0$. When $c_{r} \in [A_{3} ,B_{3} ]$, we have $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \le 0$.

When $1/4 \le \alpha \le 1$, and $c_{r} \ge C_{7} = \frac{{3\alpha c_{n} - \alpha (1 - \alpha )}}{4\alpha - 1}$, we have $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \ge 0$.

$C_{7} - B_{3} = \frac{{\alpha (3c_{n} + \alpha - 1)(4 - \alpha - \beta - 2\alpha \beta )}}{(4 - \alpha - 2\beta + 2\alpha \beta )(4\alpha - 1)}$.

From $c_{n} \ge \frac{4 - 2\alpha - \beta - \alpha \beta }{{4 - \beta }}$, we have $C_{6} - A_{3} \ge \frac{{2\alpha (4 - \alpha - \beta - 2\alpha \beta )^{2} }}{(4 - \alpha - 2\beta + 2\alpha \beta )(4\alpha - 1)(4 - \beta )} \ge 0$.

When $c_{r} \in [A_{3} ,B_{3} ]$, we have $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \le 0$.

From the above, we can obtain $\frac{{\partial q_{n}^{DTR*} }}{\partial \beta } \le 0$.

Similarly, $\frac{{\partial q_{t}^{DTR*} }}{\partial \beta } = \frac{{(1 - \alpha )[(4\alpha - 1)c_{r} - 3\alpha c_{n} + \alpha (1 - \alpha )]}}{{2\alpha (4 - \alpha - \beta - 2\alpha \beta )^{2} }} \le 0$.

$$\begin{gathered} \frac{{\partial q_{d}^{DTR*} }}{\partial \beta } = \{ 3\alpha \beta^{2} (1 + 2\alpha )c_{n} + (2\alpha^{2} \beta^{2} - 13\alpha^{2} \beta^{2} + 4\alpha^{2} \beta + \alpha^{2} + 11\alpha \beta^{2} - 14\alpha \beta - 8\alpha - 8\beta + 16)c_{r} \hfill \\ - \alpha \beta^{2} (1 - \alpha )(1 + 2\alpha )]\} /[2\alpha (4 - \alpha - \beta - 2\alpha \beta )^{2} ] \hfill \\ \end{gathered}$$

From $c_{r} \le B_{3}$, we have $\frac{{\partial q_{d}^{DTR*} }}{\partial \beta } \ge \frac{{(4 - \alpha )c_{r} }}{{4\alpha \beta^{2} (4 - \alpha - \beta - 2\alpha \beta )}} \ge 0$.

$\frac{{\partial p_{n}^{DTR*} }}{\partial \beta } = \frac{{(1 - \alpha )[(1 - 4\alpha )c_{r} + 3\alpha c_{n} - \alpha (1 - \alpha )]}}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }} \ge 0$.

$\frac{{\partial p_{t}^{DTR*} }}{\partial \beta } = \frac{{ - (1 - \alpha )[(1 - 4\alpha )c_{r} + 3\alpha c_{n} - \alpha (1 - \alpha )]}}{{2(4 - \alpha - \beta - 2\alpha \beta )^{2} }} \ge 0$.

$\frac{{\partial p_{d}^{DTR*} }}{\partial \beta } = \frac{{C_{8} c_{r} + \alpha C_{9} c_{n} + C_{10} }}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }}$.

When $\alpha \in [0,1]$, and $\beta \in [0,1]$, we have $C_{8} \in [ - 0.34,6]$, $C_{9} \in [ - 4,3]$, and $C_{10} \in [0,24]$.

Form $c_{r} \le c_{n} \le 1$, we have $\frac{{\partial p_{d}^{DTR*} }}{\partial \beta } \ge \frac{{C_{8} + \alpha C_{9} + C_{10} }}{{4(4 - \alpha - \beta - 2\alpha \beta )^{2} }} \ge 0$.

Proof of proposition 5.

$p_{d}^{DR*} - p_{t}^{TR*} = \frac{{\alpha \left( {c_{r} - 2c_{n} - 10 + 2\alpha + 2\beta - 3\alpha \beta } \right)}}{{4\left( {4 - \alpha } \right)}} \le 0$,

$p_{n}^{DR*} - p_{n}^{TR*} = \frac{{ - 2c_{r} + \alpha c_{n} + \alpha }}{4(4 - \alpha )}$, because $c_{r} \le B_{1}$, we have $p_{n}^{DR*} - p_{n}^{TR*} \ge 0$.

$q_{n}^{DR*} - q_{n}^{TR*} = \frac{{(2 - \alpha + \alpha \beta )c_{r} + \alpha (1 - 4\beta )c_{n} + \alpha (1 - \alpha \beta )}}{4(1 - \alpha \beta )(4 - \alpha )}$, the condition is $\max (A_{1} ,A_{2} ) \le c_{r} \le B_{3}$.

When $c_{r} \ge C_{1} = \frac{{\alpha (4\beta - 1)c_{n} + \alpha (\alpha \beta - 1)}}{2 - \alpha + \alpha \beta }$, we have $q_{n}^{DR*} \ge q_{n}^{TR*}$. When $c_{r} \le C_{1}$, we have $q_{n}^{DR*} \le q_{n}^{TR*}$.

We obtain $C_{1} - B_{1} = \frac{{\alpha (\alpha \beta - 1)\left[ {1 + \left( {1 - 2\beta } \right)c_{n} } \right]}}{2 - \alpha + \alpha \beta } \le 0$, and $C_{1} \le B_{1}$.

When $c_{n} \le 1 + \alpha \left( {\beta - 1} \right)$, we have $C_{1} - A_{1} = \frac{{2(1 - \alpha \beta )\left[ {1 - c_{n} + \alpha \left( {\beta - 1} \right)} \right]}}{2 - \alpha + \alpha \beta } \ge 0$, $C_{1} - A_{2} = \frac{{(4 - \alpha )\left[ {1 - c_{n} + \alpha \left( {\beta - 1} \right)} \right]}}{2 - \alpha + \alpha \beta } \ge 0$, $A_{1} - A_{2} = 1 - c_{n} + \alpha \left( {\beta - 1} \right) \ge 0$, and $A_{1} \le c_{r} \le B_{3}$.

Hence, when $A_{1} \le c_{r} \le C_{1}$, we have $q_{n}^{DR*} \le q_{n}^{TR*}$. When $C_{1} \le c_{r} \le B_{3}$, we have $q_{n}^{DR*} \ge q_{n}^{TR*}$.

When $c_{n} \ge 1 + \alpha \left( {\beta - 1} \right)$, we have $C_{1} - A_{1} \le 0$, $C_{1} - A_{2} \le 0$, and $A_{1} - A_{2} = 1 - c_{n} + \alpha \left( {\beta - 1} \right) \le 0$, $A_{2} \le c_{r} \le B_{3}$, and $q_{n}^{DR*} \le q_{n}^{TR*}$.

$q_{d}^{DR*} - q_{t}^{TR*} = \frac{{( - 4 + \alpha + 4\beta - 4\alpha \beta^{2} )c_{r} + \alpha \beta (2 - \alpha + 2\alpha \beta )c_{n} - 2\alpha \beta (1 - \alpha \beta )}}{4\alpha \beta (1 - \alpha \beta )(4 - \alpha )}$.

When $c_{r} \le C_{2} = \frac{{\alpha \beta (2 - \alpha + 2\alpha \beta )c_{n} - 2\alpha \beta (1 - \alpha \beta )}}{{4 - \alpha - 4\beta + 4\alpha \beta^{2} }}$, we have $q_{d}^{DR*} \ge q_{t}^{TR*}$. when $c_{r} \ge C_{2}$, we have $q_{d}^{DR*} \le q_{t}^{TR*}$.

We obtain $C_{2} - A_{1} = \frac{{( - 4 + \alpha + 4\beta + 2\alpha \beta - \alpha^{2} \beta - 4\alpha \beta^{2} + 2\alpha^{2} \beta^{2} )c_{n} - \left( {1 - \alpha \beta } \right)(4 - \alpha - 4\beta + 2\alpha \beta + 4\alpha \beta^{2} )}}{{4 - \alpha - 4\beta + 4\alpha \beta^{2} }} \le 0$, and $C_{2} - A_{2} = \frac{{( - 8 + 2\alpha + 8\beta + 2\alpha \beta - \alpha^{2} \beta - 8\alpha \beta^{2} + 2\alpha^{2} \beta^{2} )c_{n} - 2\alpha^{2} \beta^{2} + \alpha^{2} + 8\alpha \beta^{2} - 6\alpha - 8\beta + 8}}{{4 - \alpha - 4\beta + 4\alpha \beta^{2} }}$.

When $c_{n} \le C_{3} = \frac{{ - 2\alpha^{2} \beta^{2} + \alpha^{2} + 8\alpha \beta^{2} - 6\alpha - 8\beta + 8}}{{8 - 2\alpha - 8\beta - 2\alpha \beta + \alpha^{2} \beta + 8\alpha \beta^{2} - 2\alpha^{2} \beta^{2} }}$, we have $C_{2} \ge A_{2}$. When $c_{n} \ge C_{3}$, we have $C_{2} \le A_{2}$.

$C_{2} - B_{1} = \frac{{2\alpha \beta \left( {1 - \alpha \beta } \right)[(2\beta - 1)c_{n} - 1]}}{{4 - \alpha - 4\beta + 4\alpha \beta^{2} }} \le 0$.

From above all, when $c_{n} \le C_{3}$, we have $q_{d}^{DR*} \le q_{t}^{TR*}$. When $c_{n} \ge C_{3}$, and $c_{r} \in \left[ {\max (A_{1} ,A_{2} ),C_{2} } \right]$, we have $q_{d}^{DR*} \ge q_{t}^{TR*}$. When $c_{n} \ge C_{3}$, and $c_{r} \in \left[ {C_{2} ,B_{1} } \right]$, we have $q_{d}^{DR*} \le q_{t}^{TR*}$.

Proof of proposition 6.

$q_{d}^{DR*} - q_{d}^{DTR*} = (1 - \alpha )\beta \frac{{\alpha (1 - \beta )c_{n} + (2\alpha \beta - 1 - \alpha )c_{r} + \alpha (1 - \alpha \beta )}}{4\alpha \beta (1 - \alpha \beta )(4 - \alpha - \beta - 2\alpha \beta )} \ge \frac{{\beta \left( {1 - \alpha } \right)\left( {\alpha - c_{r} } \right)}}{4\alpha \beta (4 - \alpha - \beta - 2\alpha \beta )} \ge 0$,

$\begin{gathered} q_{d}^{DTR*} + q_{t}^{DTR*} - q_{d}^{DR*} = \frac{{\beta \left( {1 + \alpha - 2\alpha \beta } \right)[\left( { - 1 - \alpha + 2\alpha \beta } \right)c_{r} + \alpha \left( {1 - \beta } \right)c_{n} + \alpha (1 - \alpha \beta )]}}{4\alpha \beta (1 - \alpha \beta )(4 - \alpha - \beta - 2\alpha \beta )} \ge \frac{{\left( {1 + \alpha - 2\alpha \beta } \right)\left( {\alpha - c_{r} } \right)}}{4\alpha (4 - \alpha - \beta - 2\alpha \beta )} \ge 0 \hfill \\ \hfill \\ \end{gathered}$, $q_{n}^{DR*} - q_{n}^{DTR*} = \left( {1 - \beta } \right)\frac{{ - \left[ {1 + \alpha \left( {1 - 2\beta } \right)} \right]c_{r} + \alpha (1 - \beta )c_{n} + \alpha \left( {1 - \alpha \beta } \right)}}{{4(1 - \alpha \beta )\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \ge \frac{{(1 - \beta )(\alpha - c_{r} )}}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \ge 0$,

$p_{d}^{DR*} - p_{d}^{DTR*} = \frac{{\beta (1 - \alpha \beta )(\alpha - c_{r} )}}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \ge 0$,

$q_{t}^{TR*} - q_{t}^{DTR*} = (1 - \alpha )\frac{{3\alpha \beta c_{r} - (4 - \alpha - 2\beta + 2\alpha \beta )c_{r} - \alpha \beta (1 - \alpha )}}{2\alpha (4 - \alpha )(4 - \alpha - \beta - 2\alpha \beta )} \ge 0$,

$q_{d}^{DTR*} + q_{t}^{DTR*} - q_{t}^{TR*} = \frac{{3\alpha \beta c_{n} - \left( {4 - \alpha - 2\beta + 2\alpha \beta } \right)c_{r} - \alpha \beta (1 - \alpha )]}}{4\alpha \beta (4 - \alpha )(4 - \alpha - \beta - 2\alpha \beta )} \ge 0$,

$q_{n}^{TR*} - q_{n}^{DTR*} = \frac{{3[( - 4 + \alpha + 2\beta - 2\alpha \beta )c_{r} + 3\alpha \beta c_{n} - \alpha \beta \left( {1 - \alpha } \right)]}}{{4(4 - \alpha )\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \ge \frac{{\alpha - c_{r} }}{{4\left( {4 - \alpha - \beta - 2\alpha \beta } \right)}} \ge 0$,

$p_{t}^{TR*} - p_{t}^{DTR*} = \frac{{\left( {1 - \alpha } \right)\left( { - 4 + \alpha + 2\beta - 2\alpha \beta } \right)c_{r} + 3\alpha \beta \left( {1 - \alpha } \right)c_{n} - \alpha \beta \left( {1 - \alpha } \right)^{2} }}{2(4 - \alpha )(4 - \alpha - \beta - 2\alpha \beta )}$, because $c_{r} \le B_{3}$, we can obtain $p_{t}^{TR*} - p_{t}^{DTR*} \ge 0$.

Proof of proposition 7.

We obtain $\pi_{m}^{DTR*} - \pi_{m}^{TR*} = D_{1} /[200\alpha \beta (4 - \alpha - \beta - 2\alpha \beta )]$, where $D_{1} = 25\alpha^{4} \beta^{2} - 50\alpha^{3} \beta^{2} - 110\alpha^{3} \beta - 7\alpha^{2} \beta + 38\alpha \beta + \alpha^{2} - 12\alpha \beta^{2} + 26\alpha \beta - 8\alpha + 2\beta - 8\beta^{2} + 16$. When $D_{1} \ge 0$, we have $\pi_{m}^{DTR*} \ge \pi_{m}^{TR*}$.

Proof of proposition 8.

We obtain $\pi_{t}^{DTR*} - \pi_{t}^{TR*} = D_{2} /[100\alpha (4 - \alpha - \beta - 2\alpha \beta )^{2} (4 - \alpha )^{2} ]$, where $\begin{gathered} D_{2} = 25\alpha^{6} \beta^{2} - 8\alpha^{5} \beta^{2} - 60\alpha^{5} \beta + 204\alpha^{4} \beta^{2} + 294\alpha^{4} \beta - 13\alpha^{4} - 84\alpha^{3} \beta^{2} - 242\alpha^{3} \beta + 100\alpha^{3} \hfill \\ + 47\alpha^{2} \beta^{2} + 144\alpha^{2} \beta - 399\alpha^{2} + 12\alpha \beta^{2} - 16\alpha \beta + 280\alpha - 4\beta^{2} + 32\beta - 48 \hfill \\ \end{gathered}$. When $D_{2} \ge 0$, we have $\pi_{t}^{DTR*} \ge \pi_{t}^{TR*}$.

Proof of proposition 9.

We obtain $\pi_{d}^{DTR*} - \pi_{d}^{TR*} = D_{3} /[400\alpha \beta (4 - \alpha - \beta - 2\alpha \beta )^{2} (4 - \alpha )^{2} ]$, where $\begin{gathered} D_{3} = - 50\alpha^{6} \beta^{3} - 75\alpha^{6} \beta^{2} + 285\alpha^{5} \beta^{3} + 770\alpha^{5} \beta^{2} + 65\alpha^{5} \beta - 470\alpha^{4} \beta^{3} - 2823\alpha^{4} \beta^{2} - 718\alpha^{4} \beta + \alpha^{4} \hfill \\ - 496\alpha^{3} \beta^{3} + 4022\alpha^{3} \beta + 2740\alpha^{3} \beta - 80\alpha^{3} + 474\alpha^{3} \beta^{3} - 830\alpha^{2} \beta^{2} - 4112\alpha^{2} \beta + 96\alpha^{2} + 31\alpha \beta^{3} \hfill \\ - 696\alpha \beta^{2} + 1984\alpha \beta - 256\alpha + 64\beta^{2} - 256\beta + 256 \hfill \\ \end{gathered}$.when $D_{3} \ge 0$, we have $\pi_{d}^{DTR*} \ge \pi_{d}^{TR*}$.