Coordination Mechanism of Revenue Sharing Contracts in Port Supply Chains: A Case Study of China’s Nantong Port

Abstract

Considering the port function’s importance, the supply chain’s overall and individual revenues, and the port supply chain’s uniqueness, a game model for a two-stage port supply chain composed of port service providers and port enterprises was developed. Using revenue sharing contracts (RSCs) as a coordination mechanism, game equilibriums were investigated under two conditions: price-sensitive or price-insensitive port logistics service demand. The results suggested that RSCs can achieve Pareto improvement in both cases, thereby coordinating port supply chain revenues. Led by the port logistics service provider and port enterprise, the coordination mechanisms based on RSCs of port supply chain were discussed using logistics service capacity as the coordination link. Despite differing decision orders, the RSC model can coordinate the port’s supply chain with its service provider, while the port and logistics service provider determine the retention ratio. This paper also proposed coordination measures for the Nantong Port, one of the most important ports in China, to maximize the logistics capacity of iron ore supply chain and enhance the port logistics’ value-added services.

1. Introduction

In the product supply chain, the supply chain contract identifies the physical product as the contract object and accomplishes the contract’s optimization objective through inventory coordination among the various supply chain actors. Then, in the port supply chain, the object of the contract between actors shifts to the order and delivery of service capacity, resulting in a fundamental difference between the coordination mechanisms of the port supply chain contract and the product supply chain [1].

The revenue sharing contract (RSC), which regulates the distribution of revenues and losses among participants in a supply chain, has grown in popularity over the last two decades [2]. It has been demonstrated that this type of contract highly coordinates the supply chain [3], with the intention of encouraging separate actors to develop efficiencies or innovate in ways that are mutually beneficial. The stable and coordinated operation of all parties (or actors) in port supply chain is the major concern of supply chain management. This model can effectively solve the coordination problem between service providers and enterprises in supply chains. Generally speaking, one way is to consider it as a generalization of the conventional wholesale-price contract with an added revenue-sharing mechanism (WCRS). Another way is to omit the wholesale-price component and interact via consignment contract with revenue sharing (CCRS). The second one is growing in favor among retailers in recent years. Mortimer and Julie [4] employed the empirical method to study the RSC in the video rental industry. Gerchak et al. [5] added the license fee to redistribute supply chain revenues and accomplished coordination. Giannoccaro and Pontrandolfo [6] extended the RSC to coordinate a three-stage supply chain, and they believed that the revenues of supply chain actors could be regulated by tuning the contract parameters. Lee [7] applied the social network theory to port supply chain coordination and achieved optimization. Song and Panayides [8] proposed six parameters, including information and communication technologies and value-added services, to reflect the degree of port integration, which better presented logical relationships. Tian [9] established a return subcontract model of logistics service supply chain. Cui and Liu [10] proposed the option contract coordination mechanism of logistics service supply chain. Liu [11] proposed the coordination method of RSC in the three-echelon logistics service supply chain. Zhu et al. [12] established the coordination model of quantitative cooperative alliance. Meng and Huang [13] studied the contract choice strategy of logistics service supply chain. Liu et al. [14] established a quality coordination model of multi-period logistics service supply chain. He et al. [15] established a joint contract coordination mechanism consisting of option contract and cost-sharing contract. Wang and Wang [16] established the revenue coordination model of port supply chain in Cloud Environment. Yang [17] analyzed the logistics service supply chain in the principle-agent relation and believed that the Sharpley Value method is the most representative method to allocate revenues according to the contribution of stakeholders to the supply chain. Zhang et al. [18] assessed the impact of demand disruption on the centralized supply chain and established the coordinated RSC for numerical validation. Wang et al. [19] created the coal-power supply chain RSC coordination mechanism by analyzing coal-power enterprises’ coal production and order quantity in both decentralized and centralized decision-making, as well as their respective carbon emission reduction and profits in the absence of a contract, and then validated their theoretical results with numerical simulations and parameter sensitivity analysis. In practice, both the iTunes App Store and the Google Play Store have a unified contract reserving 30% of app sales revenue for themselves and 70% for developers, and Amazon Marketplace is another example, with sellers paying Amazon a percentage of their sales and a fixed fee [2].

To date, studies on supply chain management primarily focused on manufacturing enterprises, but there were few studies on service providers, particularly port enterprises (as the nodal enterprise in the port supply chain) [20,21,22]. As for the studies on port supply chain, the majority of attention was paid to port supply chain development, function upgrade, and service innovation, while the coordination of port supply chain has not been systematically investigated. Most of the methodologies were qualitative, with only a few involving mathematical methods. Port supply chain differs from general logistics service supply chain due to the specificity of port assets, the uniqueness of port location, and the high correlation with hinterland economy. Revenue and risk sharing are the major demands of port supply chain coordination. However, in practice, port service providers’ resources, particularly their specific assets, are underutilized. This has a negative influence on port function performance, the entire supply chain, and all actors’ revenues.

The port supply chain coordination mechanism remains a gap in logistics service supply chain research. This paper analyzed the optimization and coordination of port supply chain capacity and obtained the proportional value range of port enterprises’ revenue retention using an RSC as a coordination mechanism. Numerical simulation was also integrated with theoretical derivation to provide theoretical reference and method guidance for port supply chain revenue coordination. By establishing the revenue-sharing coordination mechanism model, this paper can effectively solve the coordination problem in the port supply chain. This may change the port logistics management process, reduce the expense of port logistics, improve the quality of logistics service and the satisfaction of each node enterprise in the supply chain, enhance the core competitiveness of port enterprises, and thus improve the overall performance of the port supply chain.

2. Modelling Notations and Hypothesis

In order to illustrate the logical relationships and highlight the research focus, a simple two-stage single-cycle port supply chain model is considered in Figure 1. The main game side, port service providers (mainly providing logistics services such as loading and unloading, processing, and transportation for port enterprises), present RSCs to port enterprises (mainly providing various comprehensive port logistics services). Port enterprises order logistics capacity from service providers in proportion to market service demand, contract service price, and retention, so as to maximize their own revenues while satisfying logistics service demand.

Notations:

• q: logistics capacity of port enterprises;
• S(q): expected logistics capacity of the port:

$$S\left(q\right)=E\left[\mathit{min}\left(q,x\right)\right]=q -{{\displaystyle \int }}_0^{q}F\left(x\right)\mathit{dx}$$

(1)

• L(q): expected loss of logistics capacity of port:

$$L\left(q\right)=E\left[\mathit{max}\left(x-q,0\right)\right]=\mu -S\left(q\right)$$

(2)

• p: unit logistics service price of port enterprises;
• w: purchase price of port logistics capacity per unit;
• c: operation cost of unit logistics capacity of service providers, where p > w > c > 0;
• $g_p$: loss caused by lack of capacity of port units;
• $g_s$: loss caused by lack of unit capacity of port service providers;
• ${\pi }_p$: expected revenue of the port when no RSC is implemented;
• ${\pi }_s$: expected revenue of the service provider when no RSC is implemented;
• π: expected revenue of integrated supply chain system;
• ${\pi }_{p\lambda }$: expected revenue of the port when the RSC is implemented;
• ${\pi }_{s\lambda }$: expected revenue of the service provider when the RSC is implemented;
• ${\pi }_{\lambda }$: expected revenue of supply chain system when implementing RSC.

Notations of symbols cited in the coordination mechanism based on RSC led by port enterprises:

• $Q$: the maximum logistics capacity of the port.
• $\eta$: sensitivity coefficient of port logistics capacity order quantity relative to logistics service provider capacity transfer price, $\eta > 0$.
• $Z_p$: expected revenue of port based on RSC led by ports.
• $Z_s$: expected revenue of port service providers based on RSC led by ports.
• $Z$: expected revenue of port supply chain system based on RSC led by ports. The rest of the model symbols are the same as above.

Model hypothesis [23]:

• Both service providers and ports are bounded rationality and risk neutral (that refers to a manner in which an individual neglects risk when making investment decisions), taking the expected revenue maximization as the decision-making objective;
• The logistics demand faced by the port is a non-negative and continuous random variable. The mean value $\mu =E\left(x\right)={\int }_0^{\infty }\mathit{xf}\left(x\right)\mathit{dx}$, and F(x) and f(x) are the distribution function and density function, respectively;
• F(x) is a strictly increasing function that is continuously differentiable and invertible, F(0) = 0;
• Insufficient logistics capacity leads to opportunities loss;
• There is no information asymmetry, i.e., price, cost, and demand are public information;
• For a coordination mechanism based on RSC led by port enterprises, there is a negative linear correlation between the actual quantity of port logistics capacity ordering and the transfer price of logistics service providers: $q=Q-\eta w$.

3. Modelling Formulation and Calculation

3.1. RSC Design

3.1.1. Establishing the Demand Function and Model

There are two modes to establish a demand function: addition and multiplication. According to Petruzzi and Dada [24], considering that the unit logistics service price p of port enterprises is an endogenous variable, the multiplication mode is used to establish the demand function.

Assume that port logistics demand function is x(p) = d(p)·ε, where d(p) = θp^−η, $\theta$ is constant, $\theta > 0$, $\eta$ is market demand elasticity, $\epsilon$ is random variable, and $g\left(\epsilon \right)$ and $G\left(\epsilon \right)$ are its density function and distribution function, respectively. Other hypotheses are consistent with preceding paragraphs.

The port revenue function can be expressed as $R\left(p,q\right)=\mathit{pE}\left[\mathit{min}\left(q,x\left(p\right)\right)\right]$, let $z=q/d\left(p\right)$, and then,$R\left(p,z\right)=\mathit{pd}\left(p\right)E\left[\mathit{min}\left(z,\epsilon \right)\right]$. Under the RSC mechanism, the expected revenue function of port enterprises can be expressed as

$${\pi }_{p\lambda }\left(p,z,\lambda \right)=\lambda R\left(p,z\right)-w_{\lambda }q$$

(3)

The expected revenue function of port service provider is

$${\pi }_{s\lambda }=\left(p,z,\lambda \right)=\left(1-\lambda \right)R\left(p,z\right)+\left(w_{\lambda }-c\right)q$$

(4)

The expected revenue function of port supply chain is

$${\pi }_{\lambda }\left(p,z\right)=R\left(p,z\right)-\mathit{cq}$$

(5)

3.1.2. Port Enterprise Decision Model and Analysis

The expected revenue function of port enterprises is

$$\begin{array}{c}{\pi }_{p\lambda }\left(p,z,\lambda \right)=\lambda R\left(p,z\right)-w_{\lambda }q\\ =\left(\lambda p-w_{\lambda }\right)d\left(p\right)z-\lambda \mathit{pd}\left(p\right){{\displaystyle \int }}_0^{z}G\left(\epsilon \right)d\epsilon \end{array}$$

(6)

If taking the first partial derivative of ${\pi }_{p\lambda }\left(p,z,\lambda \right)$ with respect to p and z, we obtain

$$\frac{\partial {\pi }_{p\lambda }\left(p,z,\lambda \right)}{\partial p}=\left(1-\eta \right){\lambda \theta p}^{-\eta }\left(z-{{\displaystyle \int }}_0^{z}G\left(\epsilon \right)d\epsilon \right){+\eta \theta \mathrm{zw}}_{\lambda }{p}^{-\eta -1}$$

(7)

$$\frac{\partial {\pi }_{p\lambda }\left(p,z,\lambda \right)}{\partial z}=\left(\lambda p-w_{\lambda }\right)d\left(p\right)-\lambda \mathit{pd}\left(p\right)G\left(z\right)$$

(8)

Let the two first-order partial derivatives be zero, and obtain the optimal decision of the port enterprise when implementing the RSC:

$$p_{\lambda }=\frac{{\eta w}_{\lambda }}{\lambda \left(\eta -1\right)}·\frac{z_{\lambda }}{z_{\lambda }-{{\displaystyle \int }}_0^{z_{\lambda }}G\left(\epsilon \right)d\epsilon }$$

(9)

$$G\left(z_{\lambda }\right)=\frac{{\lambda p}_{\lambda }-w_{\lambda }}{{\lambda p}_{\lambda }}$$

(10)

At this time, the expected revenue of port enterprises is

$${\pi }_{p\lambda }\left(p_{\lambda }{,z}_{\lambda },\lambda \right)=\left({\lambda p}_{\lambda }-w_{\lambda }\right)d\left(p_{\lambda }\right)z_{\lambda }-{\lambda p}_{\lambda }d\left(p_{\lambda }\right){{\displaystyle \int }}_0^{z_{\lambda }}G\left(\epsilon \right)d\epsilon$$

(11)

The expected revenue of port supply chain is

$$\pi \left(p_{\lambda }{,z}_{\lambda }\right)=\left(p_{\lambda }-c\right)d\left(p_{\lambda }\right)z_{\lambda }-p_{\lambda }d\left(p_{\lambda }\right){{\displaystyle \int }}_0^{z_{\lambda }}G\left(\epsilon \right)d\epsilon$$

(12)

And the port supply chain is investigated when the centralized decision-making mechanism is implemented. At this time, the expected revenue of port supply chain is

$$\pi \left(p,z\right)=\left(p-c\right)d\left(p\right)z-\mathit{pd}\left(p\right){{\displaystyle \int }}_0^{z}G\left(\epsilon \right)d\epsilon$$

(13)

With the same method, the optimal decision of port supply chain can be obtained when the centralized decision mechanism is implemented:

$$p_{\mathrm{I}}=\frac{\eta c}{\eta -1}·\frac{z_{\mathrm{I}}}{z_{\mathrm{I}}-{{\displaystyle \int }}_0^{z_{\mathrm{I}}}G\left(\epsilon \right)d\epsilon }$$

(14)

$$G\left(z_{\mathrm{I}}\right)=\frac{p_{\mathrm{I}}-c}{p_{\mathrm{I}}}$$

(15)

At this time, the maximum expected revenue of port supply chain is

$$\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)=\left(p_{\mathrm{I}}-c\right)d\left(p_{\mathrm{I}}\right)z_{\mathrm{I}}-p_{\mathrm{I}}d\left(p_{\mathrm{I}}\right){{\displaystyle \int }}_0^{z_{\mathrm{I}}}G\left(\epsilon \right)d\epsilon$$

(16)

Under the RSC mechanism, in order to make the optimal decision of the port enterprise $\left(p_{\lambda }{,z}_{\lambda }\right)$ conform to the optimal decision of the port supply chain, i.e., $\pi \left(p_{\lambda }{,z}_{\lambda }\right)=\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)$, let $p_{\lambda }{=p}_{\mathrm{I}}$,${ z}_{\lambda }{=z}_{\mathrm{I}}$, $w_{\lambda }=\lambda c$ can be obtained.

This result is consistent with the previous section, indicating that for the port supply chain, the RSC mechanism is applicable regardless of whether the port logistics service demand is price sensitive. When $w_{\lambda }=\lambda c$, the expected revenue of port enterprises is

$$\begin{array}{c}{\pi }_{p\lambda }\left(p_{\lambda }{,z}_{\lambda },\lambda \right)=\left({\lambda p}_{\lambda }-\lambda c\right)d\left(p_{\lambda }\right)z_{\lambda }-{\lambda p}_{\lambda }d\left(p_{\lambda }\right){{\displaystyle \int }}_0^{z_{\lambda }}G\left(\epsilon \right)d\epsilon \\ =\lambda \pi \left(p_{\lambda }{,z}_{\lambda }\right)=\lambda \pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)\end{array}$$

(17)

The expected revenue of port service provider is

$${\pi }_{s\lambda }\left(p_{\lambda }{,z}_{\lambda },\lambda \right)=\left(1-\lambda \right)\pi \left(p_{\lambda }{,z}_{\lambda }\right)=\left(1-\lambda \right)\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)$$

(18)

3.1.3. Port Service Provider Decision Model and Contract Design

When port logistics service demand is price sensitive, the expected revenue of port enterprises under decentralized decision-making is

$${\pi }_{\mathit{pd}}\left(p,z\right)=\left(p-w\right)d\left(p\right)z-\mathit{pd}\left(p\right){{\displaystyle \int }}_0^{z}G\left(\epsilon \right)d\epsilon$$

(19)

Similarly, the optimal decision of port enterprises in the implementation of decentralized decision mechanism can be obtained:

$$p_d=\frac{\eta w}{\eta -1}·\frac{z_d}{z_d-{\int }_0^{z_d}G\left(\epsilon \right)d\epsilon }$$

(20)

$$G\left(z_d\right)=\frac{p_d-w}{p_d}$$

(21)

Thus, the expected revenue of port enterprises in the implementation of decentralized decision-making mechanism can be obtained:

$${\pi }_{\mathit{pd}}\left(p_d{,z}_d\right)=\left(p_d-w\right)d\left(p_d\right)z_d{-p}_{d}d\left(p_d\right){{\displaystyle \int }}_0^{z_d}G\left(\epsilon \right)d\epsilon$$

(22)

The expected revenue of port supply chain is

$${\pi }_d\left(p_d{,z}_d\right)=\left(p_d-c\right)d\left(p_d\right)z_d{-p}_{d}d\left(p_d\right){{\displaystyle \int }}_0^{z_d}G\left(\epsilon \right)d\epsilon$$

(23)

The expected revenue of port service providers is

$${\pi }_{\mathit{sd}}\left(p_d{,z}_d\right){=\pi }_d\left(p_d{,z}_d\right){-\pi }_{\mathit{pd}}\left(p_d{,z}_d\right)=\left(w-c\right)d\left(p_d\right)z_d$$

(24)

Game theory holds that rational economic clients always pursue the maximization of their own revenues, and only by satisfying individual rationality can they realize collective rationality through institutional arrangement. Therefore, the self-execution condition of RSC is that when the RSC mechanism is implemented, the revenue of port enterprises and port business service providers outnumber the revenue when the decentralized decision-making mechanism is implemented, as follows:

$${\pi }_{p\lambda }\left(p_{\lambda }{,z}_{\lambda },\lambda \right) \ge { \pi }_{\mathit{pd}}\left(p_d{,z}_d\right)$$

(25)

$${\pi }_{s\lambda }\left(p_{\lambda }{,z}_{\lambda },\lambda \right) \ge { \pi }_{\mathit{sd}}\left(p_d{,z}_d\right)$$

(26)

By substituting Equations (17) and (18) into the above two inequalities, we can obtain

$$\frac{{\pi }_{\mathit{pd}}\left(p_d{,z}_d\right)}{\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)} \le \lambda \le 1-\frac{{\pi }_{\mathrm{sd}}\left(p_d{,z}_d\right)}{\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)}$$

(27)

This is the value range of $\lambda$ when the port service provider designs the RSC ${(w}_{\lambda }, \lambda$). The final value of $\lambda$ is mainly determined by the resource endowment and position of port enterprises and port service providers in the port supply chain.

3.2. Coordination Mechanism Based on RSC Led by Port Enterprises

3.2.1. Problem Description and Contract Design

The coordination mechanisms of the port supply chain addressed in the preceding paragraphs assumed that cooperation should be led by port logistics service providers. However, ports have distinct advantages in terms of capital, scale, and location in the port supply chain, and they all have roles in leading enterprises in coordination. This section discusses how to coordinate the entire supply chain led by port enterprises through RSC.

Assuming that the port influences the capacity transfer price of the service provider by highlighting the logistics capacity order quantity of the potential market [25], the initiative can be gained in decision-making. In practice, the port declares its logistics capacity order quantity before the logistics service provider proclaims the transfer price per unit of capacity. The actual capacity order quantity is sensitive to the capacity transfer price, and a port’s actual logistics capacity cannot surpass its potential logistics capacity. Therefore, the maximum logistics capacity of a port is its potential logistics capacity. The decision-making process of both is as follows:

• The port determines the maximum logistics capacity order quantity and retains the λ(0 ≤ λ ≤ 1) part of its revenue; the rest is returned to the service provider. And the port claims that its actual capacity order quantity is sensitive to the capacity transfer price of the logistics service provider.
• The logistics service provider decides its capacity transfer price according to the decision and information of the port.
• In the case that the capacity transfer price of the logistics service provider is specified, the port determines the final logistics capacity order quantity based on the demand information prediction and the maximization of its own profit.
• When the logistics demand arises, the logistics service provider will have to provide the ordered quantity of logistics capacity to the port at the agreed-upon capacity transfer price and perform the required logistics services.

As evidenced by the decision-making processes of both parties, the port achieves a dominant position in the supply chain by making a declaration before the service provider determining the capacity transfer price.

3.2.2. Analysis of Decision Model of Port Service Provider

The port service provider determines the capacity transfer price it provides to the port with the goal of maximizing its own revenue in accordance with the port’s declaration. The decision model is

$$\begin{array}{c}{\mathrm{maxZ}}_s=\left(w-c \right)q+\left(1-\lambda \right)\mathit{pS}\left(q\right)\\ =\left(w-c\right)\left(Q-\eta w\right)+\left(1-\lambda \right)p\left(q-{{\displaystyle \int }}_0^{q}F\left(x\right)\mathit{dx}\right)\end{array}$$

(28)

S.t. $w > 0$, where $q=Q-\eta w$.

This allows us to calculate the optimal capacity transfer price $w_{\lambda }$ of the service provider, which satisfies the following conditions:

$$w_{\lambda }=\frac{Q+\eta c-\left(1-\lambda \right)p\eta \left(1-F\left(q\right)\right)}{2\eta }$$

(29)

$$\frac{\partial w^{\lambda }}{\partial Q}=\frac{1+\left(1-\lambda \right)p\eta F’\left(q\right)}{2\eta +\left(1-\lambda \right){p\eta }^{2}F’\left(q\right)}$$

(30)

If $\varpi =\left(1-\lambda \right)p\eta F’\left(q\right)$, $\frac{\partial w_{\lambda }}{\partial Q}=\frac{1+\varpi }{\eta \left(2+\varpi \right)}$.

Clearly, $\frac{\partial w_{\lambda }}{\partial Q} > 0$, indicating that the optimal capacity transfer price $w_{\lambda }$ of service providers and the maximal logistics capacity order quantity of ports $Q$ are positively correlated.

$\frac{\partial w_{\lambda }}{\partial \lambda }=\frac{p}{2}\left(1-F\left(q\right)\right) > 0$ shows that the service capacity of the optimal transfer price $w_{\lambda }$ and the proportion of revenue retained $\lambda$ are positively correlated.

3.2.3. Port Decision Model and Its Coordination Mechanism

After the port logistics service provider has determined the capacity transfer price $w_{\lambda }$, the port follows the maximization of revenue principle to determine the optimal maximum logistics capacity order $Q_{\lambda }$. The port’s expected revenue function is

$$\begin{array}{c}{Z}_p=\lambda \mathit{pS}\left(q\right){-w}_{\lambda }q\\ =\lambda p\left({Q-\eta w}_{\lambda }-{{\displaystyle \int }}_0^{{Q-\eta w}_{\lambda }}F\left(x\right)\mathit{dx}\right){-w}_{\lambda }\left({Q-\eta w}_{\lambda }\right)\end{array}$$

(31)

Similarly, the port’s optimal maximal logistics capacity order quantity $Q_{\lambda }$ can be calculated as follows:

$$Q_{\lambda }=\frac{\varpi \mathit{pc}+p\eta \left(2\lambda -\left(1-\lambda \right)\varpi \right)\left(1-F\left(q\right)\right)}{2+\varpi }$$

(32)

To obtain the analytical formula for $Q_{\lambda }$, it is necessary to know the demand distribution function’s form.

$\frac{\partial Q_{\lambda }}{\partial \lambda }=p\eta \left(1-F\left(q\right)\right) > 0$ shows that the maximum logistics capacity of the port $Q_{\lambda }$ is positive with its revenue retention λ.

For the coordination mechanism under the dominance of port enterprises based on RSC, ports declare in advance that their maximum capacity order quantity is proportional to their revenue retention according to Equation (32) and make it clear that their actual capacity order quantity is sensitive to the capacity transfer price of logistics service providers. The port logistics service provider gives its optimal capacity transfer price according to Equation (29). When the logistics demand arises, the logistics service provider is obligated to provide the ordered quantity of logistics capacity to the port at the agreed-upon capacity transfer price and perform the required logistics services. When designing the RSC, the port enterprise must make the value of λ satisfy Equation (27). Only in this way can we guarantee the interests of all parties in the port and logistics service providers. The specific value of λ depends on the position of the port and service providers in the supply chain and the bargaining power between each other.

4. Numerical Analysis

The purpose of the numerical simulation is to validate the operability of the theoretical model and analyze the influence of different values of $\lambda$ in the implementation of the RSC mechanism on the port supply chain, port enterprises, and port business service providers’ revenues.

It is assumed that the random variable $\epsilon$ obeys uniform distribution $U\left(a,b\right)$, then $G\left(\epsilon \right)=\frac{\epsilon -b}{a-b}$. By substituting Equations (9) and (10), the analytical formula of $p_{\lambda }$ and $z_{\lambda }$ can be obtained:

$$p_{\lambda }=\frac{\eta +1}{\eta -1}\frac{b-a}{b}{c, z}_{\lambda }=\frac{2}{\eta +1}b$$

(33)

The analytic expression of $p_d$ and $z_d$ can also be obtained:

$$p_d=\frac{\eta +1}{\eta -1}\frac{b-a}{b}{w, z}_d=\frac{2}{\eta +1}b$$

(34)

$${\pi }_{\mathit{pd}}\left(p_d{,z}_d\right)=\left(p_d-w\right)d\left(p_d\right)z_d{-p}_{d}d\left(p_d\right){{\displaystyle \int }}_0^{z_d}G\left(\epsilon \right)d\epsilon$$

(35)

$${\pi }_{\mathit{sd}}\left(p_d{,z}_d\right){=\pi }_d\left(p_d{,z}_d\right){-\pi }_{pd}\left(p_d{,z}_d\right)=\left(w-c\right)d\left(p_d\right)z_d$$

(36)

$$\pi \left(p_{\mathrm{I}}{,z}_{\mathrm{I}}\right)=\left(p_{\mathrm{I}}-c\right)d\left(p_{\mathrm{I}}\right)z_{\mathrm{I}}{-p}_{\mathrm{I}}d\left(p_{\mathrm{I}}\right){{\displaystyle \int }}_0^{z_{\mathrm{I}}}G\left(\epsilon \right)d\epsilon$$

(37)

$$p_{\lambda }{=p}_{\mathrm{I}}, z_{\lambda }{=z}_{\mathrm{I}}, d\left(p\right){=\theta p}^{-\eta }$$

(38)

The parameter settings in the model are shown in

Table 1. Model parameter settings.

ε	θ	η	w	c
U(0.4, 2)	100	2	4	2

.

From the above analytic expressions, it can be calculated as $p_d=9.6$, $z_d=1.33$, $p_{\mathrm{I}}=4.8, z_{\mathrm{I}}=1.33$, ${\pi }_{\mathit{pd}}\left(9.6,1.33\right)=5.79, {\pi }_{\mathit{sd}}\left(9.6,1.33\right)=2.89, \pi \left(4.8,1.33\right)=11.58$. The value range of $\lambda$ can be calculated from Equation (27): $0.5 \le \lambda \le 0.75$.

The optimal strategy of port supply chain and corresponding expected revenue when different mechanisms are implemented are compared, as shown in

Table 2. The optimal strategy and expected revenue of port supply chain under different mechanisms.

	Centralized Decision-Making Mechanism	Decentralized Decision-Making Mechanism	Revenue Sharing Contract Mechanism
			$\mathit{\lambda }=0.5$	$\mathit{\lambda }=0.6$	$\mathit{\lambda }=0.75$
Optimal service price (Yuan/t)	4.8	9.6	4.8	4.8	4.8
Logistics capability optimal order quantity (104 t)	5.77	1.44	5.77	5.77	5.77
Port expected revenue (104 Yuan)	——	5.79	5.79	6.95	8.69
Expected revenue of service provider (104 Yuan)	——	2.89	5.79	4.63	2.89
The supply chain system expects revenue (104 Yuan)	11.58	8.68	11.58	11.58	11.58

. Obviously, in the implementation of the RSC, as long as $0.5 \le \lambda \le 0.75$, both the expected revenue of the whole supply chain system and the expected revenue of port enterprises and port service providers exceed the expected revenue when the decentralized decision-making mechanism is implemented, and Pareto improvement has been achieved.

5. Case Study: The Nantong Port

5.1. Iron Ore Supply Chain and Its Revenue Coordination

Nantong Port is situated in Nantong, Jiangsu Province, China, adjacent to the Yellow Sea and Yangtze River (Figure 2). It is the Yangtze River port closest to the sea, located 195.5 km away from the Yangtze River estuary. Nantong Port is now open to 312 ports in 96 countries and regions, with an average throughput of 106.54 million tons in the past decade, which makes a significant transmission and distribution center for bulk commodities.

Iron and steel enterprises along the Yangtze River are thriving, and the demand for imported iron ore is increasing with China’s continued economic growth. However, since 2012, the operating efficiency of iron and steel enterprises has continued to decline, and the capital flow of businesses has become increasingly difficult. Figure 3 shows the iron ore supply chain of Nantong Port. Iron ore transported by Nantong Port is typically sourced from Australia, Brazil, South Africa, and other countries, which are among the biggest suppliers of iron ore. Usually, large cargo ships serve to transport supplies from the source to Nantong Port. At the Nantong Port, iron ore is first unloaded to the port’s storage yard and then distributed according to actual logistics needs and cost considerations. If iron ore needs to be transported to a coastal port such as Zhanjiang or Ningbo, the following two approaches are used:

• direct unloading to land transport to other ports;
• discharge to other ships and then proceed by sea to other ports.

For the distribution to the port where the enterprise is located, upon arrival at the port of destination, the iron ore can be unloaded and distributed according to the enterprise’s needs. The specific distribution method may depend on the port’s management regulations, the enterprise’s contractual arrangement, and other factors. According to the RSC coordination mechanism proposed in this paper, the Nantong Port can assist iron and steel enterprises in reducing operating costs and stabilizing production and management by implementing three measures.

First, collaborate with iron and steel enterprises and financial institutions to promote the “business of port inventory pledge financing”. It takes time for the customs clearance and inspection of imported commodities, and this time could be used to assist steel enterprises with their financing. Nantong Port is required to provide a yard with a certain capacity to store imported iron ore (300,000 tons) and to report the inventory information to financial institutions and iron and steel enterprises in real time, in strict accordance with their joint instructions. On the basis of assuring 300,000 tons of inventory, the financial institution shall provide a corresponding credit line; iron and steel enterprises use the loan to replenish raw materials to meet the production needs.

Second, assist enterprises with inventory circulation management. Since 2012, the purchasing and transportation of iron and steel enterprises have been rapid, frequent, and time-efficient due to the increase in total throughput and the decline in port inventory. They will lose the market if they do not adjust to the tempo shift. By increasing loading and unloading equipment, optimizing the operation process, and maximizing port berths, the port of Nantong is able to adapt to the requirements of its customers.

Third, transfer the vat reform revenues to consumers. In China’s tax reform, the tax on ports as logistics auxiliary industries has been changed from a 3% business tax to a 6% value-added tax, which can be mitigated as an input item by production companies. Thus, it is reasonable to increase the customer’s port fee by 6%, as the customer has not increased the direct cost. Nantong Port, however, proactively proposed to give customers a 3% revenue—that is, customers would pay 3% to make up the port fee tax increase, receive 6% of the input deduction, and receive 3% of the net revenue.

5.2. Implication

In the logistics and transportation activities of Nantong Port, contract coordination and contractual relationships (Figure 4) are of great importance to ensure transportation efficiency and reduce logistics expenses [26]. A contract between a shipping company and a freight forwarder may clarify both parties’ obligations and rights in the transportation process, as well as how to resolve problems in the event of a dispute [27]. Furthermore, through contract coordination, both parties can collaborate on transportation arrangements to ensure that the supplies reach on time.

Game theory, as a theoretical tool, can be used to evaluate the decision-making behavior of port parties in logistics and transportation activities during the coordination process [28]. For example, while choosing a freight forwarder, a shipping company may consider factors such as the freight forwarder’s reputation, service quality, and transportation cost, whereas the freight forwarder may assess the shipping company’s strength, shipment schedule, etc. [27]. Game theory can benefit understanding the decision-making process, helping planners to strike a balance and produce win-win outcomes for all parties.

Information exchange is critical in port logistics and transportation activities to increase the coordination efficiency of Nantong Port [29]. Port parties can acquire real-time information on ship dynamics, cargo status, storage facilities, and other elements by developing a consolidated information platform, effectively coordinating resources and lowering logistics expenses. Additionally, information exchange helps reduce information asymmetry and promote port management transparency.

Cooperation is a significant means to enhance the coordination effect of Nantong Port’s logistics and transportation [30]. By forming alliances and signing cooperation agreements, port parties can jointly build new transportation lines, optimize shipment schedules, and increase storage facility usage rates. This is beneficial to minimizing competition, realizing effective resource use, and further boosting the port’s transportation efficiency.

In practice, policy support and legal framework are also of great significance to Nantong Port coordination and game. The government can encourage port parties to increase overall transportation efficiency by adopting corresponding policies such as preferred policies and tax policies. At the same time, a solid legal framework may provide a stable operating environment for port logistics and transportation activities while protecting the legitimate rights and interests of all parties.

However, in the process of coordination and game, there are some challenges. For example, the port parties may encounter an issue of interest distribution that makes the cooperation hard to sustain. In addition, issues related to corporate secrecy and personal privacy must be properly addressed when sharing information. To cope with these challenges, port parties constantly need innovation and improvement in the coordination and game processes [31,32]. Potential measures are presented as follows:

• Establish a fair revenue distribution mechanism to ensure that rights and interests of all parties are protected during the cooperation process.
• Strengthen security measures to prevent the disclosure of corporate secrets and personal information.
• Improve port parties’ awareness of policy support and the legal framework by increasing policy publicity and popularizing laws and regulations.
• Conduct regular exchanges and training activities to improve the professional quality of port parties involved in the coordination and game process.
• Establish an effective supervision mechanism to ensure the long-term stability of port logistics and transportation activities based on coordination and game.

In summary, coordination and gaming play a major part in improving port efficiency and lowering logistics costs in Nantong Port’s logistics transportation activities. The port parties can realize effective resource utilization and improve overall transportation efficiency through contract coordination, information sharing, cooperation mode, and other manners. Meanwhile, in order to ensure sustainable development of port logistics and transportation activities, port parties must constantly seek innovation and improvement [32,33].

6. Conclusions

This paper studied the revenue coordination between port and port logistics service providers. Using logistics service capacity as the coordination link, the coordination mechanisms of port supply chain revenue sharing contracts (RSCs) were discussed under the leadership of the port logistics service provider and port enterprise respectively.

Led by the port logistics service provider, the port logistics service provider provides an RSC mechanism (w,λ), and the service provider provides the port with a lower unit capacity service price w. The port retains λ(0 ≤ λ ≤ 1) of its revenue, and the rest is returned to the service provider, so as to achieve perfect coordination of the port supply chain. Then, we investigate the RSC coordination mechanism of port logistics service demand with price sensitivity, determined the value range of λ, and validate the viability of the RSC coordination mechanism through mathematical simulation.

Under the dominance of port enterprises, ports acquire the primary right to make decisions by declaring in advance the maximum capacity of their orders and the proportion of revenues retained. In this case, port enterprises utilize the highest logistics capacity as the decision variable and achieve their optimal logistics capacity ordering objective by influencing the optimal capacity transfer price of logistics service providers. Despite the fact that the decision order differs between the two modes, the RSC mechanism can coordinate the port’s supply chain with its service provider. The specific proportion of retention depends on the status of the port and logistics service provider in the whole supply chain.

Based on a case study of Nantong Port’s operation status and the iron ore supply chain, relevant suggestions have been made to address the issue that Nantong Port lacks sufficient logistics capacity for supply chain management and coordination, and the port logistics’ value-added services require improvement.