Coordination of port service chain with an integrated contract

Abstract

Port, carriers and many other departments are involved in the whole water transportation service system, and these departments tend to have conflicts of interest in the process of service, which results in a difficult coordination phenomenon. We propose an integrated contract that combines the revenue sharing and service cost allocation to coordinate the port service chain. We explore the effects of the contract decision variables in the different scenarios. The results show that two sharing factors exist “blind zone,” but the improved contract reveals that revenue sharing and cost allocation contract combining with the fixed payment mechanism can be more effective to coordinate the port service chain.

Appendix

1.1 A: Proof proposition 1

Proof

We can calculate the first derivative of p₁, p₂, s₁, s₂ and s₃ in Eqs. (A1–A5), and let it be equal to zero; then, we get the optimal value of w and s₃:

$$ \frac{{\partial \pi^{\text{CC}} }}{{\partial p_{1} }} = a + s_{1} - \beta s_{2} + \gamma s_{3} - 2p_{1} + 2\rho p_{2} + c(1 - \rho ) = 0 $$

(A1)

$$ \frac{{\partial \pi^{\text{CC}} }}{{\partial p_{2} }} = a - \beta s_{1} + s_{2} + \gamma s_{3} - 2p_{2} + 2\rho p_{1} + c(1 - \rho ) = 0 $$

(A2)

$$ \frac{{\partial \pi^{\text{CC}} }}{{\partial s_{1} }} = - \beta p_{2} - c(1 - \beta ) + p_{1} - ms_{1} = 0 $$

(A3)

$$ \frac{{\partial \pi^{\text{CC}} }}{{\partial s_{2} }} = - \beta p_{1} - c(1 - \beta ) + p_{2} - ms_{2} = 0 $$

(A4)

$$ \frac{{\partial \pi^{\text{CC}} }}{{\partial s_{3} }} = \gamma (p_{1} + p_{2} - 2c) - ns_{3} = 0 $$

(A5)

Firstly, we use the formula (A1) and formula (A2) to calculate the second derivative of p₁, p₂, which forms Hessian matrix:

$$ \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial p_{2} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2} \partial p_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & {2\rho } \\ {2\rho } & { - 2} \\ \end{array} } \right] $$

(A6)

The first-order leading principal minor is $\Delta_{1} = - 2 < 0$; the second-order leading principal minor $\Delta_{2} = 4(1 - \rho^{2} )$, because of $0 < \rho < 1$ ,so $\Delta_{2} > 0$. The total profit of the port under centralized decision ($\pi^{\text{CC}}$) is strict concave function of $p_{1}$ and $p_{2}$, respectively.

Then, we use the formula (A3), (A4) and (A5) to calculate the second derivative of $s_{1}$, $s_{2}$ and $s_{3}$, which forms Hessian matrix:

$$ \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{1}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{1} \partial e_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{1} \partial e_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{2} \partial e_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{2}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{2} \partial e_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{3} \partial e_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{3} \partial e_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial e_{3}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - m} & 0 & 0 \\ 0 & { - m} & 0 \\ 0 & 0 & { - n} \\ \end{array} } \right] $$

(A7)

The first-order leading principal minor is $\Delta_{1} = - m < 0$; the second-order leading principal minor is $\Delta_{2} = m^{2} > 0$; the third-order leading principal minor is $\Delta_{3} = - m^{2} n < 0$. So, the total profit of the port under centralized decision ($\pi^{\text{CC}}$) is strict concave function of $s_{1}$, $s_{2}$ and $s_{3}$, respectively.

Combining the formula (A6) and formula (A7):

$$ \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial p_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial s_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial s_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial s_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2} \partial p_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2} \partial s_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{1} \partial s_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial p_{2} \partial s_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{1} \partial p_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{1} \partial p_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{1}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{1} \partial s_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{1} \partial s_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{2} \partial p_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{2} \partial p_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{2} \partial s_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{2}^{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{2} \partial s_{3} }}} \\ {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{3} \partial p_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{3} \partial p_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{3} \partial s_{1} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{3} \partial s_{2} }}} & {\frac{{\partial^{2} \pi^{\text{CC}} }}{{\partial s_{3}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & {2\rho } & 1 & { - \beta } & \gamma \\ {2\rho } & { - 2} & { - \beta } & 1 & \gamma \\ 1 & { - \beta } & { - m} & 0 & 0 \\ { - \beta } & 1 & 0 & { - m} & 0 \\ \gamma & \gamma & 0 & 0 & { - n} \\ \end{array} } \right] $$

(A8)

The first-order leading principal minor is $\Delta_{1} = - 2 < 0$; the second-order leading principal minor is $\Delta_{2} = 4(1 + \rho )(1 - \rho )$, because $0 < \rho < 1$, $\Delta_{2} > 0$; the third-order leading principal minor is $\Delta_{3} = 2 - 4\beta \rho + 2\beta^{2} - 4m(1 - \rho^{2} )$; the positive and negative of $\Delta_{3}$ are not sure. So, the total profit of the port under centralized decision ($\pi^{\text{CC}}$) may not be the strict concave function of $p_{1}$, $p_{2}$, $s_{1}$, $s_{2}$ and $s_{3}$.□