Problem description

Due to the particularity of medical products and the high cost of medical logistics, the public mostly pay attention to the safety and cost reduction of medical logistics, so they often ignore the environmental protection of logistics. However, there are a lot of waste of resources and environmental damage in the existing medical logistics, such as repeated packaging, excessive logistics circulation, and exhaust emission of transportation vehicles [24]. On this basis, a secondary green supply chain consisting of a pharmaceutical logistics provider and a pharmaceutical manufacturer is taken as the research object in this paper. The government formulates several different subsidy strategies for green logistics, pharmaceutical logistics provider and pharmaceutical manufacturer make corresponding logistics behavior choices and logistics decisions according to the costs and benefits of green logistics. Pharmaceutical logistics provider can transform green logistics by using green transportation vehicles and green packaging. Retailers have a green logistics preference and are willing to pay extra for pharmaceutical products shipped in a green logistics mode. Pharmaceutical manufacturer mainly contributes to the transformation of green logistics through green packaging and green publicity in the sales process. This paper intends to solve the following three questions as follows:

1. How do different subsidy strategies influence the choice of logistics behavior of pharmaceutical logistics provider and manufacturer?
2. In the case of a certain amount of government subsidies, what are the different effects of different subsidy strategies?
3. Compared with the non subsidy situation, does the government's different subsidy strategies significantly promote the green process of pharmaceutical logistics?

Research hypothesis

Based on the above research questions, the following assumptions were made for the model:

1. This paper only studies the pharmaceutical logistics behavior, considering only the Stackelberg game situation in which the logistics provider is the leader and the manufacturer is the follower. In this supply chain, logistics provider undertakes the task of logistics transportation and collect the corresponding logistics fees. Manufacturers sell products to retailers, and the logistics cost is added to the product price.
2. As green logistics is dependent on products, its costs and benefits are finally reflected in the products, and also affect the greenness and price of medical products. Medical products that use green logistics are slightly more expensive.
3. The promotion of green logistics is related to the use of logistics price, green level and the green investment of pharmaceutical logistics provider and manufacturer. Retailers are willing to pay extra price for the use of green logistics for medical products. The market demand of green pharmaceutical logistics depends on medical products, so we can describe the market demand of "green logistics" as follows: (1)
   Where, d denotes the demand of green logistics; α denotes the overall size of the logistics market, α−βp>0;β denotes the price sensitivity coefficient of logistics demand; p denotes the logistics fee charged by the logistics provider to the retailer; γ denotes the retailer's green logistics preference coefficient; e denotes the green level of logistics contributed by the logistics provider; z represents the green level of logistics contributed by manufacturers [25].
4. When realizing green transformation of logistics, pharmaceutical logistics provider and manufacturer do not change the original logistics unit cost, but need to increase additional green investment (equipment transformation and technology innovation) I. Assuming a quadratic relation between green input cost and greenness, then let I_L = η_Le²/2,I_R = η_Rz²/2, Where, η denotes the coefficient of green input cost [26].
5. Assume that all participants of the green supply chain are completely rational and make corresponding behavioral choices and decisions according to their own interests maximization.

In order to effectively encourage the green logistics transformation of pharmaceutical logistics provider and manufacturers, the government considers various subsidy strategies and studies the influence of different subsidy strategies on the green decision-making of each participant in the supply chain. This paper mainly considers the following situations: anarchic subsidy strategy, the single subsidy strategy of pharmaceutical logistics provider, the single subsidy strategy of pharmaceutical manufacturer and the coordination subsidy strategy. On this basis, a Stackelberg game model with pharmaceutical logistics provider as the leader and manufacturer as the follower is built to explore the behavior choice and optimal decision of each participant under different subsidy strategies. First of all, the s, where s denotes the subsidy coefficient of green logistics is determined by the government. And then the green input and unit price of green logistics are determined by the pharmaceutical logistics provider according to government subsidies, so as to optimize their own profits. According to the unit price and the green level of logistics determined by the logistics provider, the charge of logistics fees and its own logistics green investment (mainly through green packaging and green publicity) are determined by the pharmaceutical manufacturer, so as to optimize its own profit. In order to simplify the description, the anarchic subsidy strategy, the logistics subsidy strategy, the manufacturer subsidy strategy and the coordination subsidy strategy are represented by the subscripts n, l, r and x.

Anarchy subsidy strategy

Under the anarchic subsidy strategy, the government does not conduct subsidy in different forms. Under this strategy, pharmaceutical logistics provider and manufacturer invest green costs independently to improve the green level of logistics.

The profit function of pharmaceutical logistics provider (L) is as follows: (2)

The profit function of pharmaceutical manufacturer (R) is as follows: (3)

Total profit function of green supply chain (B) function is as follows: (4)

The decision order is determined after the model is established. This paper constructs a single supply chain consisting of pharmaceutical logistics provider and manufacturer, and only considers their behavior selection and decision-making. First of all, pharmaceutical logistics provider, as leader, should first set logistics carrier price v and contribution e_n of logistics green level to provide corresponding logistics services to manufacturer. Then, as a follower, the manufacturer decides its logistics cost plus price and green investment effort level according to the carrier price and green degree of the logistics provider, so that the pharmaceutical products can be delivered to the retailer through green logistics. Based on the above decision order, this paper use backward induction to solve: (5)

The above simultaneous equations can be solved as follows: (6)

When the manufacturer's green logistics plus price p and the green logistics level z_n contributed by the manufacturer are both greater than 0, the extreme value is taken here. Then, p and z_n in (6) are substituted into Eq (2) to find the partial derivatives of v and e_n respectively. The simultaneous equations can be solved as follows: (7)

In this paper, the hessian matrix is used to verify the solving conditions of the equilibrium solution. According to hessian matrix, in order to obtain the above equilibrium solution, β²η_R[γ²η_R+2η_L(γ²−2βη_R)]≻0 is needed. Then, the above equilibrium solution v* and e_n* are substituted into Eq (7) to obtain the optimal solution: (8)

According to the actual problem, Eq (8) and Eq (7) can be substituted into Eq (3) and Eq (2) respectively to obtain the optimal profit of the medical logistics provider and manufacturer: (9)

At this point, the optimal profit of the supply chain is as follows: (10)

Single subsidy strategy for pharmaceutical logistics provider

Due to the high requirements of the pharmaceutical industry for the safety, timeliness and professionalism of logistics, and the strict cold chain transport to vaccines, blood products and other products, the difficulty and cost of green transformation of pharmaceutical logistics is greater than that of general commodity logistics. Pharmaceutical logistics provider plays an important role in the green transformation of medical logistics, but the high cost investment reduces the enthusiasm. Therefore, in order to encourage the green transformation of pharmaceutical logistics companies, the government carries out a single subsidy strategy. The subsidy amount is affected by the green level e_L contributed by the logistics provider. Assume that the subsidy coefficient of the government is s(1≥ s≥0). On this basis, under the single subsidy mode of pharmaceutical logistics provider, the profit functions of logistics provider and manufacturer are as follows: (11) (12)

The total profit of green supply chain is as follows: (13)

Similarly, the inverse induction method is used to verify the existence conditions of the equilibrium solution according to hessian matrix. When 2βη_R−γ²≻0 and β²η_R[γ²η_R−2(s−1)η_L(γ²−2βη_R)]≺0, the following equation is obtained: (14) (15)

By substituting Eqs (14) and (15) into Eqs (11) and (12) respectively, the optimal profits of pharmaceutical logistics provider and manufacturer can be obtained as follows: (16)

At this time, the optimal profit of the supply chain is as follows: (17)

Proposition 1: By comparing and analyzing the changes of green level contribution, optimal profit and overall benefit of supply chain of pharmaceutical logistics provider under single subsidy strategy and anarchy subsidy strategy, the results are as follows:

Proof: Because so , In the same way but card .

Because so . In the same way but card , .

Result analysis of proposition 1: When the government gives green subsidies to pharmaceutical logistics provider with subsidy coefficient s, the contribution of green level of pharmaceutical logistics provider and manufacturer is greater than that without subsidy, and the optimal profit of pharmaceutical logistics provider and manufacturer is greater than that under the anarchy subsidy strategy. In the meantime, profit of whole supply chain also increases accordingly. The results show that the implementation of single subsidy strategy by the government for pharmaceutical logistics provider can improve the self-interest and social welfare of each participant in the green supply chain, and encourage each participant to increase their contribution to the green level of logistics. Compared with the anarchy subsidy strategy, it has a more obvious promoting effect on the motivation of each participant to implement green behaviors and the improvement of the green level of logistics, and the overall effect of the strategy is better. On this basis, it is easier for pharmaceutical logistics company and manufacturer to accept this subsidy strategy.

Single subsidy strategy for pharmaceutical manufacturer

In reality, most of the attention on the green transformation effect of pharmaceutical logistics is focused on logistics provider, and manufacturers' efforts are often ignored. However, as logistics runs through the whole process of pharmaceutical products circulation, manufacturer can also contribute to the development of green logistics by optimizing packaging and promoting green products. Especially in the pharmaceutical industry, consumers almost focus on the efficacy and safety of the product, ignoring the greenness of the product. Therefore, there are a lot of excessive packaging and waste of resources in the pharmaceutical market, which can be fundamentally changed through the green consciousness and green behavior of manufacturers. Therefore, the government should start from the reality, focus on the overall situation, pay attention to the contribution of pharmaceutical manufacturer in the process of logistics greening. In order to further stimulate the efforts of pharmaceutical manufacturer in green logistics, the government implemented a single subsidy strategy for them. It is assumed that the government subsidy amount is related to the logistics green level z contributed by the manufacturer, and the subsidy coefficient is s (1≥ s ≥0). On this basis, the profit functions of pharmaceutical logistics provider and manufacturer are established, which are as follows: (18) (19)

The total profit of green supply chain is as follows: (20)

Similarly, using the inverse induction, under the condition β²(s−1)η_R{−γ²(s−1)η_R+2η_L[γ²+2β(s−1)η_R]}≻0,−γ²−2β(s−1)η_R≻0: (21) (22)

By substituting Eq (21) and Eq (22) into Eq (18) and Eq (19), the optimal profits of pharmaceutical logistics provider and manufacturer can be obtained as follows: (23)

At this time, the optimal profit of the supply chain is as follows: (24)

Proposition 2: By comparing and analyzing the changes of green level contribution, optimal profit and overall effect of supply chain of pharmaceutical manufacturers under single subsidy strategy and anarchy subsidy strategy, we can get the results as follows: (When the 4βη_L−γ²≻0)、 (When the ).

Proof: Because so . In the same way but card .

Because so [γ²η_R+2η_L(γ²−2βη_R)]≺0, So whether is greater than 0 depends on 4βη_L−γ²,when 4βη_L−γ²≻0, ; And vice, . In the same way but card, when , , and vice, .

Result analysis of Proposition 2: When the government only subsidizes pharmaceutical manufacturer, the contribution of pharmaceutical logistics provider to the green level of logistics is greater than that without subsidies. At the same time, the optimal profit of medical logistics companies is also greater than that under the condition of no subsidies. This indicates that the single subsidy strategy of pharmaceutical manufacturer also has a certain incentive effect on the logistics companies and has achieved good results. However, for manufacturer, the contribution of logistics green level under this strategy is greater than that under the anarchy subsidy strategy only when 4βη_L−γ²≻0 is met. Similarly, the optimal profit is greater than that under the anarchy subsidy strategy only when is satisfied. To sum up, when the government only provides green subsidies to manufacturer, the promotion effect of green logistics is greater than that of anarchy subsidy strategy, but the effect of subsidies is limited. Among them, the strategy has an obvious positive effect on the green logistics behavior of pharmaceutical logistics provider, but has a conditional limit on the incentive effect of the subsidy object itself. On this basis, under these circumstances, pharmaceutical logistics companies tend to accept this kind of subsidy arrangement. In order to maximize their own interests, manufacturers tend not to accept this situation. Because, in the process of logistics green logistics business always assume the main task, play a more critical role. Under this subsidy strategy, because only manufacturers get the subsidy, at this time, the logistics companies are more dependent on transferring the task of logistics greening to the manufacturers, which makes the manufacturers get the government subsidy, but need to pay more green input, leading to their profits are not necessarily increased. At the same time, manufacturers will have the conditions to improve their contribution to the green logistics. In such cases, pharmaceutical manufacturer will be cautious about accepting the policy.

Coordinate subsidy strategy

In practice, the government's single subsidy strategy often fails to achieve the desired effect. Because pharmaceutical logistics provider plays a major role in green logistics transformation, they receive more direct subsidies, and manufacturer's contributions are often ignored. However, as manufacturer in the green packaging and green logistics promotion of the positive role gradually prominent, the government has also paid more and more attention to it. On this basis, in order to promote the greening of pharmaceutical logistics, the government formulates a coordinated subsidy strategy according to the actual situation, and subsidizes both pharmaceutical logistics provider and manufacturer. Under the internal coordination subsidy strategy, the profit functions of medical logistics provider and manufacturer are as follows: (25) (26)

The total profit function of green supply chain is as follows: (27)

And similarly, we're going to do it by backward induction, when −γ²−2β(s−1)η_R≻0, {γ²η_R+2η_L[γ²+2β(s−1)η_R]}≺0, the result is as follows: (28) (29)

Eq (28) and Eq (29) are substituted into Eq (25) and Eq (26) respectively to obtain the optimal profit: (30)

At this time, the optimal profit of the supply chain is as follows: (31)

Proposition 3: By comparing and analyzing the contribution of each participant to the green level of logistics under the subsidy strategy and the anarchy subsidy strategy, the change of the optimal profit is as follows:

((γ²−2βη_R){γ²η_R+2η_L[γ²+2β(s−1)η_R]}²≻(1−s)[γ²+2β(s−1)η_R][γ²η_R+2η_L(γ²−2βη_R)]²)Proof: From the foregoing we know that: −γ²−2β(s−1)η_R≻0,[γ²η_R+2η_L(γ²+2β(s−1)η_R)]≺0,2βη_R−γ²≻0, so when (γ²−2βη_R){γ²η_R+2η_L[γ²+2β(s−1)η_R]}²≻(1−s)[γ²+2β(s−1)η_R][γ²η_R+2η_L(γ²−2βη_R)]², . Same as proposition 2, proposition 3 can be proved .

Result analysis of Proposition 3: Compared with anarchy subsidy strategy, under the coordinated subsidy strategy of the government, the contribution of pharmaceutical logistics provider to the green level of logistics and their corresponding optimal profits are greater than that of manufacturer. And the optimal profit of the manufacturer is greater than the optimal profit without subsidies only at (γ²−2βη_R){γ²η_R+2η_L[γ²+2β(s−1)η_R]}²≻(1−s)[γ²+2β(s−1)η_R][γ²η_R+2η_L(γ²−2βη_R)]². This indicates that the government coordination subsidy strategy has a relatively obvious positive incentive effect on the green logistics transformation behavior of pharmaceutical logistics provider and a restrictive incentive positive effect on manufacturer. Since the contribution of the number of participants in this strategy is greater than that of the anarchy subsidy strategy, this subsidy strategy can effectively strengthen the motivation of pharmaceutical logistics provider and manufacturer to implement green logistics behavior, but it has certain restrictions on the improvement of the overall green level of logistics. On this basis, pharmaceutical logistics companies are more willing to accept this subsidy, while pharmaceutical manufacturer tends to reject this subsidy strategy with the goal of profit maximization. This is not consistent with the conventional wisdom. Because under the subsidy policy, both parties are subsidized, each party is willing to increase their contribution to the green level of pharmaceutical logistics. However, due to the strong position of logistics providers in the supply chain, manufacturers will undertake more green tasks, such as increasing the green requirements for the packaging of the goods themselves, and reducing the transportation packaging requirements of logistics companies The cost of flow business will be reduced while the cost of manufacturer will be further increased, which will hit the enthusiasm of pharmaceutical manufacturers.