Pricing and coordination of marine service chain with empty equipment repositioning

Abstract

The empty equipment repositioning (EER) problem is deemed an inefficiency of the marine industry. However, owing to trade imbalance as the fundamental cause of EER, such problem is inevitable and requires consideration in pricing decisions. Accordingly, this study explores a marine service chain with one carrier and two dominant forwarders providing transportation service between two ports. We built a mathematical model to examine how carrier and forwarders determine pricing decisions with and without EER cost sharing. We find that the smaller one between empty equipment balancing and EER costs determines pricing decisions. We also investigate whether the EER cost-sharing mechanism can coordinate the service chain and subsequently found that the EER cost-sharing mechanism aggravates double marginalization. Then, we design a two-part tariff coordination mechanism to coordinate the service chain. Lastly, we study the effect of EER cost sharing and coordination mechanism on the environment. Interestingly, we find that EER cost-sharing and coordination should be avoided from the perspective of environmental protection.

Appendix

Proof of Proposition 1

Due to forwarders lead the Stackelberg game, we first solve decisions problem of the carrier. For Karush–Kuhn–Tucker condition of Eq. (2), we can obtain the carrier’s optimal pricing strategy as follows:

• 1. when ${T^C} \leq - {e_A} + \frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}$, we have $w_A^{FD} = \frac{{{D_A} + {\alpha_A}\left({{c_A} - {e_A}} \right) - {\alpha_A}m_A^{FD}}}{{2{\alpha_A}}}$, $w_B^{FD} = \frac{{{D_B} + {\alpha_B}\left({{c_B} + {e_A}} \right) - {\alpha_B}m_B^{FD}}}{{2{\alpha_B}}}$, the corresponding realized demands are $d_A^{FD} = \frac{{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right) - {\alpha_A}m_A^{FD}}}{2}$, $d_B^{FD} = \frac{{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right) - {\alpha_B}m_B^{FD}}}{2}$;

• 2. when $- {e_A} + \frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}} < {T^C} < {e_B} + \frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}$, we have $w_A^{FD} = \frac{{{D_A} + {\alpha_A}\left({{c_A} + {T^C}} \right) - {\alpha_A}m_A^{FD} - {\alpha_A}\frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}}}{{2{\alpha_A}}}$, $w_B^{FD} = \frac{{{D_B} + {\alpha_B}\left({{c_B} - {T^C}} \right) - {\alpha_B}m_B^{FD} + {\alpha_B}\frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}}}{{2{\alpha_B}}}$, the corresponding realized demands are $d_A^{FD} = \frac{{{D_A} - {\alpha_A}\left({{c_A} + {T^C}} \right) - {\alpha_A}{\alpha_B}\frac{{m_A^{FD} + m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}}}{2}$, $d_B^{FD} = \frac{{{D_B} - {\alpha_B}\left({{c_B} - {T^C}} \right) - {\alpha_A}{\alpha_B}\frac{{m_A^{FD} + m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}}}{2}$;

• 3. when ${T^C} \geq {e_B} + \frac{{{\alpha_A}m_A^{FD} - {\alpha_B}m_B^{FD}}}{{{\alpha_A} + {\alpha_B}}}$, we have $w_A^{FD} = \frac{{{D_A} + {\alpha_A}\left({{c_A} + {e_B}} \right) - {\alpha_A}m_A^{FD}}}{{2{\alpha_A}}}$, $w_B^{FD} = \frac{{{D_B} + {\alpha_B}\left({{c_B} - {e_B}} \right) - {\alpha_B}m_B^{FD}}}{{2{\alpha_B}}}$, the corresponding realized demands are $d_A^{FD} = \frac{{{D_A} - {\alpha_A}\left({{c_A} + {e_B}} \right) - {\alpha_A}m_A^{FD}}}{2}$, $d_B^{FD} = \frac{{{D_B} - {\alpha_B}\left({{c_B} - {e_B}} \right) - {\alpha_B}m_B^{FD}}}{2}$.

Then, substituting the above $d_A^{FD}$ and $d_B^{FD}$ into decision problems of the two forwarders respectively. By the first-order condition of Eq. (3), we can obtain $m_A^{FD*}$ and $m_B^{FD*}$ in Proposition 1. Substituting $m_A^{FD *}$ and $m_B^{FD *}$ of the above carrier’s pricing decision $w_A^{FD}$ and $w_B^{FD}$, we have $w_A^{FD*}$ and $w_B^{FD*}$ in Proposition 1. Lastly, we substitute $m_i^{FD*}$ and $w_i^{FD*}$ ($i = A,B$)into the realized demand and profit functions which is shown in Table 1 and Table 2.

Proof to Corollary 1

This Corollary can be derived straight forwardly from Proposition 1.

Proof of Proposition 2

Due to forwarders lead the Stackelberg game, we first solve decisions problem of the carrier. By the first-order condition of Eq. (6), we have $w_A^{FE} = \frac{{{D_A} + {\alpha_A}\left[{{c_A} - \left({1 - {\eta^{FE}}} \right){e_A}} \right] - {\alpha_A}m_A^{FE}}}{{2{\alpha_A}}}$, $w_B^{FE} = \frac{{{D_B} + {\alpha_B}\left[{{c_B} + \left({1 - {\eta^{FE}}} \right){e_A}} \right] - {\alpha_B}m_B^{FE}}}{{2{\alpha_B}}}$, $d_A^{FE} = \frac{{{D_A} - {\alpha_A}\left[{{c_A} - \left({1 - {\eta^{FE}}} \right){e_A}} \right] - {\alpha_A}m_A^{FE}}}{2}$, $d_B^{FE} = \frac{{{D_B} - {\alpha_B}\left[{{c_B} + \left({1 - {\eta^{FE}}} \right){e_A}} \right] - {\alpha_B}m_B^{FE}}}{2}$.

Then, substituting the above $d_A^{FE}$ and $d_B^{FE}$ into Eq. (5), by Karush–Kuhn–Tucker condition, we can have

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial L^{FE} }}{{\partial m_{B}^{FE} }} = \frac{{D_{B} - 2\alpha_{B} m_{B}^{FE} - \alpha_{B} \left[ {c_{B} + \left( {1 - 2\eta^{FE} } \right)e_{A} } \right]}}{2} = 0} \hfill \\ \begin{aligned} \frac{{\partial L^{FE} }}{{\partial \eta^{FE} }} = - e_{A} \left[ {\frac{{D_{B} - \alpha_{B} \left[ {c_{B} + \left( {1 - 2\eta^{FE} } \right)e_{A} } \right] - 2\alpha_{B} m_{B}^{FE} }}{2} - \frac{{D_{A} - \alpha_{A} \left[ {c_{A} - \left( {1 - 2\eta^{FE} } \right)e_{A} } \right] - \alpha_{A} m_{A}^{FE} }}{2}} \right] + \lambda_{1}^{FE} - \lambda_{2}^{FE} = 0 \hfill \\ \frac{{\partial L^{FE} }}{{\partial \lambda_{1}^{FE} }} = \eta^{FE} \ge 0,\lambda_{1}^{FE} \ge 0\& \lambda_{1}^{FE} \frac{{\partial L^{FE} }}{{\partial \lambda_{1}^{FE} }} = 0 \hfill \\ \end{aligned} \hfill \\ {\frac{{\partial L^{FE} }}{{\partial \lambda_{2}^{FE} }} = 1 - \eta^{FE} \ge 0,\lambda_{2}^{FE} \ge 0\& \lambda_{2}^{FE} \frac{{\partial L^{FE} }}{{\partial \lambda_{2}^{FE} }} = 0} \hfill \\ \end{array} } \right. $$

According to the Karush–Kuhn–Tucker condition, four cases are considered as following:

• 1. when $\lambda_1^{FE*} > 0$ and $\lambda_2^{FE*} > 0$, no solution;

• 2. when $\lambda_1^{FE*} > 0$ and $\lambda_2^{FE*} = 0$, ${\eta^{FE*}} = 0$, $m_A^{FE*} = \frac{{\frac{D_A}{\alpha_A} - \left({{c_A} - {e_A}} \right)}}{2}$, $m_B^{FE*} = \frac{{\frac{D_B}{\alpha_B} - \left({{c_B} + {e_A}} \right)}}{2}$. Under this case the carrier solely undertakes EER cost, the result is the same to the result of Proposition 1;

• 3. when $\lambda_1^{FE*} = 0$ and $\lambda_2^{FE*} > 0$, ${\eta^{FE*}} = 1$, $m_A^{FE*} = \frac{{\frac{D_A}{\alpha_A} - {c_A}}}{2}$, $m_B^{FE*} = \frac{{\frac{D_B}{\alpha_B} - \left({{c_B} - {e_A}} \right)}}{2}$;

• 4. when $\lambda_1^{FE*} = 0$ and $\lambda_2^{FE*} = 0$, ${\eta^{FE*}} = \frac{{\frac{D_A}{\alpha_A} - \left({{c_A} - {e_A}} \right)}}{{3{e_A}}}$, $m_A^{FE*} = \frac{{\frac{D_A}{\alpha_A} - \left({{c_A} - {e_A}} \right)}}{3}$, $m_B^{FE*} = \frac{{\frac{D_B}{\alpha_B} - {c_B}}}{2} + \frac{{\frac{D_A}{\alpha_A} - {c_A}}}{6} + \frac{{\frac{D_A}{\alpha_A} - \left({{c_A} + {e_A}} \right)}}{6}$. Due to ${\eta^{FE*}}$ is incompatible with $0 \leq {\eta^{FE*}} \leq 1$, this is not the K-T point.

Then, we substitute ${\eta^{FE*}} = 1$, $m_A^{FE*} = \frac{{\frac{D_A}{\alpha_A} - {c_A}}}{2}$ and $m_B^{FE*} = \frac{{\frac{D_B}{\alpha_B} - \left({{c_B} - {e_A}} \right)}}{2}$ into $w_A^{FE}$ and $w_B^{FE}$, we get results shown in Proposition 2. Lastly, we substitute $m_i^{FD*}$ and $w_i^{FD*}$ ($i = A,B$)into the realized demand and profit functions which is shown in Table 1 and Table 2.

Proof to Corollary 2 and 3

These Corollary can be derived straight forwardly from Proposition 2.

Proof of proposition 3

According to Table 1, if the carrier solely undertakes EER cost, when ${T^C} \leq - {e_A}$ and $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, the optimal profit of the forwarder B is $\pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} < d_B^{FE*}}} = \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{8{\alpha_B}}}$ and $\pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}} = \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} - {T^C}} \right)} \right]}^2}}}{{8{\alpha_B}}}$ respectively. If the forwarder B undertakes EER cost completely, when ${T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, the optimal profit of the forwarder B is $\pi_B^{FE*}{|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} = \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{8{\alpha_B}}} + \frac{{{D_A} - {\alpha_A}{c_A}}}{4}{e_A}$. Then to compare two feasible EER sharing mechanism, ${\eta^{FE*}} = 0$ and ${\eta^{FE*}} = 1$, we have to compare optimal profit of forwarder B under both mechanisms in the interval ${T^C} \leq - {e_A}$ and $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$.

When ${T^C} \leq - {e_A}$, apparently we have $\pi_B^{FE*}{|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} > \pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} < d_B^{FE*}}}$. When $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, we have

$$ \begin{aligned} & \pi_{B}^{FE*} |_{{\eta^{FE*} = 1,d_{A}^{FE*} < d_{B}^{FE*}}} - \pi_{B}^{FE*} |_{{\eta^{FE*} = 0,d_{A}^{FE*} = d_{B}^{FE*}}} \\ & \quad = - \frac{{\left({e_{A} + T^{C}} \right)\left({\left[{D_{B} - \alpha_{B} \left({c_{B} + e_{A}} \right)} \right] + \left[{D_{B} - \alpha_{B} \left({c_{B} - T^{C}} \right)} \right]} \right)}}{8} + \frac{{D_{A} - \alpha_{A} c_{A}}}{4}e_{A} \\ \end{aligned} $$

Due to $\pi_B^{FE*}{|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} - \pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}}$ is strictly decreasing function of ${T^C}$ and $\left({\pi_B^{FE*}{|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} - \pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}}} \right){|_{{T^C} = - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} > 0$, we can obtain $\pi_B^{FE*}{|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} > \pi_B^{FE*}{|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}}$. In summary, we get $\pi_B^{FE*}{|_{{\eta^{FE*}} = 1}} > \pi_B^{FE*}{|_{{\eta^{FE*}} = 0}}$.

Proof of Proposition 4

Similarly to proof of Proposition 3, to investigate the effect of EER cost-sharing mechanism on double marginalization, we compare total optimal profit of the service chain in the interval ${T^C} \leq - {e_A}$ and $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$.

When ${T^C} \leq - {e_A}$, we have

$$ \begin{aligned} & \left({\pi_{A}^{FE*} + \pi_{B}^{FE*} + \pi_{C}^{FE*}} \right)|_{{\eta^{FE*} = 1,d_{A}^{FE*} < d_{B}^{FE*}}} - \left({\pi_{A}^{FE*} + \pi_{B}^{FE*} + \pi_{C}^{FE*}} \right)|_{{\eta^{FE*} = 0,d_{A}^{FE*} < d_{B}^{FE*}}} \\ & \quad = - \frac{{2\left({D_{A} - \alpha_{A} c_{A}} \right) + 3\alpha_{A} e_{A}}}{16}e_{A} < 0 \\ \end{aligned} $$

When $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, we have

$$ \begin{aligned} & \left({\pi_{A}^{FE*} + \pi_{B}^{FE*} + \pi_{C}^{FE*}} \right)|_{{\eta^{FE*} = 1,d_{A}^{FE*} < d_{B}^{FE*}}} - \left({\pi_{A}^{FE*} + \pi_{B}^{FE*} + \pi_{C}^{FE*}} \right)|_{{\eta^{FE*} = 0,d_{A}^{FE*} = d_{B}^{FE*}}} \\ & \quad = \frac{{3T^{C} \left({\left[{D_{A} - \alpha_{A} c_{A}} \right] + \left[{D_{A} - \alpha_{A} \left({c_{A} + T^{C}} \right)} \right]} \right)}}{16} \\ & \quad \quad - \frac{{3\left({e_{A} + T^{C}} \right)\left({\left[{D_{B} - \alpha_{B} \left({c_{B} + e_{A}} \right)} \right] + \left[{D_{B} - \alpha_{B} \left({c_{B} - T^{C}} \right)} \right]} \right)}}{16} + \frac{{D_{A} - \alpha_{A} c_{A}}}{4}e_{A}. \\ \end{aligned} $$

Due to $\frac{{\partial \left({\left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} - \left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}}} \right)}}{{\partial {T^C}}} = 0$ as well as $\left({\left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} - \left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}}} \right){|_{{T^C} = - {e_A}}} < 0$, we obtain $\left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 1,d_A^{FE*} < d_B^{FE*}}} - \left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 0,d_A^{FE*} = d_B^{FE*}}} < 0$. In summary, we get $\left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 1}} < \left({\pi_A^{FE*} + \pi_B^{FE*} + \pi_C^{FE*}} \right){|_{{\eta^{FE*}} = 0}}$.

Proof of Proposition 5

Through solving centralized marine service chain, when ${T^C} \leq - {e_A}$, the pricing decision is $p_A^{C*} = \frac{{\frac{D_A}{\alpha_A} + {c_A} - {e_A}}}{2}$ and $p_B^{C*} = \frac{{\frac{D_B}{\alpha_B} + {c_B} + {e_A}}}{2}$.

For Karush–Kuhn–Tucker condition of Eq. (7), when ${T^C} \leq - {e_A}$, we have $\frac{D_A}{\alpha_A} + {c_A} - {e_A} + m_A^{FC} = 2\left({w_A^{FC} + m_A^{FC}} \right)$ and $\frac{D_B}{\alpha_B} + {c_B} + {e_A} + m_B^{FC} = 2\left({w_B^{FC} + m_B^{FC}} \right)$. The coordination of the service chain requires $w_A^{FC} + m_A^{FC} = p_A^{C*}$ and $w_B^{FC} + m_B^{FC} = p_B^{C*}$, which means $\frac{D_A}{\alpha_A} + {c_A} - {e_A} + m_A^{FC} = \frac{D_A}{\alpha_A} + {c_A} - {e_A}$ and $\frac{D_B}{\alpha_B} + {c_B} + {e_A} + m_B^{FC} = \frac{D_B}{\alpha_B} + {c_B} + {e_A}$. So we obtain $m_A^{FC} = 0$ and $m_B^{FC} = 0$.

Similarly, we prove that when $- {e_A} < {T^C} < {e_B}$ and ${T^C} \geq {e_B}$, the coordination of the service chain also requires $m_A^{FC} = 0$ and $m_B^{FC} = 0$.

Proof of Proposition 6

When ${T^C} \leq - {e_A}$, we substitute $m_i^{FD*}$ and $w_i^{FD*}$ ($i = A,B$)into Eqs. (7) and (8) to solve optimal profit of the carrier and forwarders. We obtain that $\pi_A^{FC*} = F_A^{FC}$, $\pi_B^{FC*} = F_B^{FC}$, $\pi_C^{FC*} = \frac{{{{\left[{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right)} \right]}^2}}}{{4{\alpha_A}}} + \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{4{\alpha_B}}} - F_A^{FC} - F_B^{FC}$. Then by comparing service chain members’ optimal profit with and without two-part tariff coordination mechanism, we have $\pi_A^{FC*} - \pi_A^{FD*} = F_A^{FC} - \frac{{{{\left[{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right)} \right]}^2}}}{{8{\alpha_A}}}$, $\pi_B^{FC*} - \pi_B^{FD*} = F_B^{FC} - \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{8{\alpha_B}}}$, $\pi_C^{FC*} - \pi_C^{FD*} = \frac{{3{{\left[{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right)} \right]}^2}}}{{16{\alpha_A}}} + \frac{{3{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{16{\alpha_B}}} - F_A^{FC} - F_B^{FC}$.

The feasibility of coordination mechanism requires $\pi_A^{FC*} - \pi_A^{FD*} \geq 0$, $\pi_B^{FC*} - \pi_B^{FD*} \geq 0$ and $\pi_C^{FC*} - \pi_C^{FD*} \geq 0$. According to the above conditions, we have $F_A^{FC} \geq \frac{{{{\left[{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right)} \right]}^2}}}{{8{\alpha_A}}} = \pi_A^{FD*}$, $F_B^{FC} \geq \frac{{{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{8{\alpha_B}}} = \pi_B^{FD*}$ and $F_A^{FC} + F_B^{FC} \leq \frac{{3{{\left[{{D_A} - {\alpha_A}\left({{c_A} - {e_A}} \right)} \right]}^2}}}{{16{\alpha_A}}} + \frac{{3{{\left[{{D_B} - {\alpha_B}\left({{c_B} + {e_A}} \right)} \right]}^2}}}{{16{\alpha_B}}} = \frac{3}{2}\left({\pi_A^{FD*} + \pi_B^{FD*}} \right)$.

Similarly, we prove that when $- {e_A} < {T^C} < {e_B}$ and ${T^C} \geq {e_B}$, the feasibility of the two-part tariff coordination mechanism requires the same conditions.

Proof of Proposition 7

According to Table 2, if the carrier solely undertakes EER cost, when ${T^C} \leq - {e_A}$, the EER volume is ${E^{FD}}{ |_{{T^C} \leq - {e_A}}} = - \frac{{\left({{\alpha_A} + {\alpha_B}} \right)\left({{T^C} + {e_A}} \right)}}{4}$. Meanwhile, if the forwarder B undertakes all EER cost, when ${T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, the EER volume is ${E^{FE}}{ |_{{T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} = - \frac{{\left({{\alpha_A} + {\alpha_B}} \right)\left({{T^C} + \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}} \right)}}{4}$. According to Proposition 1, when $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, the EER volume is ${E^{FD}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} = 0$. Through solving centralized marine service chain, when ${T^C} \leq - {e_A}$ and ${T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, the EER volume is ${E^{FC}}{ |_{{T^C} \leq - {e_A}}} = - \frac{{\left({{\alpha_A} + {\alpha_B}} \right)\left({{T^C} + {e_A}} \right)}}{2}$ and ${E^{FC}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} = 0$ respectively.

We first compare the EER volume under scenarios that carrier undertaking EER cost and coordination, apparently we always have ${E^{FC}} > {E^{FD}}$ when ${T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$. Then we compare the EER volume under scenarios that carrier undertaking EER cost and EER cost sharing, when ${T^C} \leq - {e_A}$, we obtain ${E^{FE}}{ |_{{T^C} \leq - {e_A}}} - {E^{FD}}{ |_{{T^C} \leq - {e_A}}} = \frac{{{\alpha_A}{e_A}}}{4} > 0$. When $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, straight forwardly we get ${E^{FE}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} > {E^{FD}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}}$. In summary, we get ${E^{FE}} > {E^{FD}}$.

Lastly, we compare the EER volume under EER sharing and coordination. When ${T^C} \leq - {e_A}$, we have ${E^{FC}}{ |_{{T^C} \leq - {e_A}}} - {E^{FE}}{ |_{{T^C} \leq - {e_A}}} = - \frac{{\left({{\alpha_A} + {\alpha_B}} \right)\left({{T^C} + \frac{{2{\alpha_A} + {\alpha_B}}}{{{\alpha_A} + {\alpha_B}}}{e_A}} \right)}}{4}$. Due to ${E^{FC}}{ |_{{T^C} \leq - {e_A}}} - {E^{FE}}{ |_{{T^C} \leq - {e_A}}}$ is strictly decreasing function of ${T^C}$, we get ${E^{FC}}{ |_{{T^C} \leq - {e_A}}} > {E^{FE}}{ |_{{T^C} \leq - {e_A}}}$ if ${T^C} < - {e_A} - \frac{\alpha_A}{{{\alpha_A} + {\alpha_B}}}{e_A}$. When $- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}$, straight forwardly we get ${E^{FC}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}} < {E^{FE}}{ |_{- {e_A} < {T^C} \leq - \frac{\alpha_B}{{{\alpha_A} + {\alpha_B}}}{e_A}}}$. In summary, when ${T^C} < - {e_A} - \frac{\alpha_A}{{{\alpha_A} + {\alpha_B}}}{e_A}$, we get ${E^{FC}} > {E^{FE}}$.