Collaborative emergency berth scheduling based on decentralized decision and price mechanism

Abstract

This paper aims to provide an applicable managerial solution to reduce the losses and impacts from an unexpected shutdown of a terminal by facilitating cooperation between two adjacent terminals that are managed by different operators. In view of the autonomy of different terminals, we focus on not only berth scheduling in the emergency situation, but also interest coordination between different terminals. We propose a novel mechanism based on price adjustment and transfer payment, which can induce the joint optimal scheduling of different terminals in an emergency situation through cooperation between terminals based on decentralized decision making. Extensive experiments show that the berth scheduling achieved under the proposed mechanism can significantly reduce the loss that results from the shutdown of a terminal and provide economic benefits to both terminals.

Appendices

Appendix A

1.1 Proof of Proposition 1

Proof

Let $g = \sum\limits_{{i \in V^{a} }} {w_{i} (y_{i} + b_{i} - d_{i} )^{ + } + } \sum\limits_{{i \in V^{b} }} {[c_{i} u_{i} + c_{i}^{{\prime }} u_{i} + e_{i} (1 - u_{i} )]}$, then

$$ \begin{aligned} f_{1} & = g + \sum\limits_{{i \in V^{b} }} {w_{i} (y_{i} + b_{i} - d_{i} )^{ + } u_{i} } \\ f_{2} & = g + \sum\limits_{{i \in V^{b} }} {w_{i} (y_{i} + b_{i} - d_{i} )^{ + } } = f_{1} + \sum\limits_{{i \in V^{b} \backslash U}} {w_{i} (y_{i} + b_{i} - d_{i} )^{ + } } \\ \end{aligned} $$

We assume that $S$ is not an optimal solution of the model with Objective Function (2). Another solution $S^{{\prime }}$ must be available that satisfies $f_{1} (S^{{\prime }} ) < f_{1} (S)$. We denote the value of $y_{i}$ ($i \in V^{b} \backslash U$) in $S^{{\prime }}$ by $y_{i}^{**}$ ($i \in V^{b} \backslash U$). Then, $f_{2} (S^{{\prime }} ) = f_{1} (S^{{\prime }} ) + \sum\limits_{{i \in V^{b} \backslash U}} {w_{i} (y_{i}^{**} + b_{i} - d_{i} )^{ + } }$.

For $i \in V^{b} \backslash U$, Constraints (3)–(19) do not ensure that the rectangle representing vessel $i$ does not overlap with the rectangles representing other vessels. $y_{i}$ ($i \in V^{b} \backslash U$) can be any value that satisfies $y_{i} \ge 0$ without violating the Constraints (3)–(19). Moreover, from Function (2), the value of $f_{1}$ does not change with $y_{i}$ ($i \in V^{b} \backslash U$). That means we can change the value of $y_{i}^{**}$ ($i \in V^{b} \backslash U$) so that $\sum\limits_{{i \in V^{b} \backslash U}} {w_{i} (y_{i}^{**} + b_{i} - d_{i} )^{ + } }$ = 0 without affecting the feasibility of $S^{{\prime }}$ and the value of $f_{1} (S^{{\prime }} )$. Then, $f_{2} (S^{{\prime }} )$ = $f_{1} (S^{{\prime }} )$.

From $f_{1} (S^{{\prime }} )$ < $f_{1} (S)$, then $f_{2} (S^{{\prime }} )$ < $f_{1} (S)$. From $f_{2} = f_{1} + \sum\limits_{{i \in V^{b} \backslash U}} {w_{i} (y_{i} + b_{i} - d_{i} )^{ + } }$, $f_{1} (S)$$\le f_{2} (S)$, then $f_{2} (S^{{\prime }} )$ < $f_{2} (S)$, which is inconsistent with the definition of $S$.□

Appendix B

2.1 Proof of Proposition 2

Proof

Let polyhedron $\{ \hat{A}s \le \hat{b},s \ge 0\}$ describe the convex hull of feasible solutions of Model PD. Let Model CPD denote Model PD with its constraints replaced with $\{ \hat{A}s \le \hat{b},s \ge 0\}$. Since $s_{a}^{c*}$ is optimal to Model PD if and only if $s_{a}^{c*}$ is optimal to Model CPD which is a linear programming model, it is easy to show $s_{a}^{c*}$ is optimal to Model PD if and only if: (1) the dual problem of Model CPD has a feasible solution and (2) $f_{PD} (s_{a}^{c*} ) \le f_{PD} (s^{0} )$ for all $s^{0} \in \zeta$, according to Lemma 1 in Wang (2009).

It then suffices to show that the dual problem of Model CPD has a feasible solution if and only if the dual problem of Model LPD has a feasible solution. Since $s_{a}^{c*}$ is a feasible solution of Model LPD and Model CPD, the dual problem of Model CPD has a feasible solution if and only if Model CPD is bounded, namely Model PD is bounded; similarly, the dual problem of Model LPD has a feasible solution if and only if Model LPD is bounded. Then, from Corollary 6.8 of Nemhauser and Wolsey (1999), since Model PD and Model LPD both posses a feasible solution $s_{a}^{c*}$, then Model PD is bounded if and only if Model LPD is bounded.□

Appendix C

3.1 Constraints of the dual problem of model LPD

$$a_{i} \gamma_{i} + M\sum\limits_{j = i + 1}^{{n^{a} + n^{s} }} {\psi_{ji}^{{\prime }} } + M\sum\limits_{{j \in V^{a} }} {\psi_{ij}^{{\prime \prime }} } + \varepsilon_{i} + M\sum\limits_{{j \in V^{0} }} {\varphi_{ij} } \ge p_{i} - c_{i} ,\quad i = n^{a} + 1$$

$$\begin{aligned}& a_{i} \gamma_{i} + M\sum\limits_{{j = n^{a} + 1}}^{i - 1} {\psi^{\prime}_{ji} } + M\sum\limits_{j = i + 1}^{{n^{a} + n^{s} }} {\psi_{ij}^{{\prime }} } + M\sum\limits_{{j \in V^{a} }} {\psi_{ij}^{{\prime \prime }} } + \varepsilon_{i} + M\sum\limits_{{j \in V^{0} }} {\varphi_{ij} } \ge p_{i} - c_{i}\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad i = n^{a} + 2, \ldots ,n^{a} + n^{s} - 1 \end{aligned}$$

$$a_{i} \gamma_{i} + M\sum\limits_{{j = n^{a} + 1}}^{i - 1} {\psi_{ji}^{{\prime}} } + M\sum\limits_{{j \in V^{a} }} {\psi_{ij}^{{\prime \prime }} } + \varepsilon_{i} + M\sum\limits_{{j \in V^{0} }} {\varphi_{ij} } \ge p_{i} - c_{i} ,\quad i = n^{a} + n^{s}$$

$${M\tau_{ij} - \psi_{ij} + \eta_{ij} \ge 0,\quad i,j \in V^{a} ,\quad i < j}$$

$${M\tau_{ij} - \psi_{ji} + \eta_{ij} \ge 0,\quad i,j \in V^{a} ,\quad i > j}$$

$${M\tau_{ij} - \psi_{ij}^{{\prime }} + \eta_{ij} \ge 0,\quad i,j \in V^{s} ,\quad i < j}$$

$${M\tau_{ij} - \psi_{ji}^{{\prime }} + \eta_{ij} \ge 0,\quad i,j \in V^{s} ,\quad i > j}$$

$${M\tau_{ij} - \psi_{ij}^{{\prime \prime }} + \eta_{ij} \ge 0,i \in V^{s} ,\quad j \in V^{a} }$$

$${M\tau_{ij} - \psi_{ji}^{{\prime \prime }} + \eta_{ij} \ge 0,\quad i \in V^{a} ,\quad j \in V^{s} }$$

$${M\pi_{ij} - \psi_{ij} + \eta_{ij}^{{\prime }} \ge 0,\quad i,j \in V^{a} ,\quad i < j}$$

$${M\pi_{ij} - \psi_{ji} + \eta_{ij}^{{\prime }} \ge 0,\quad i,j \in V^{a} ,\quad i > j}$$

$${M\pi_{ij} - \psi_{ij}^{{\prime }} + \eta_{ij}^{{\prime }} \ge 0,\quad i,j \in V^{s} ,\quad i < j}$$

$${M\pi_{ij} - \psi_{ji}^{{\prime }} + \eta_{ij}^{{\prime }} \ge 0,\quad i,j \in V^{s} ,\quad i > j}$$

$${M\pi_{ij} - \psi_{ij}^{{\prime \prime }} + \eta_{ij}^{{\prime }} \ge 0,\quad i \in V^{s} ,\quad j \in V^{a} }$$

$${M\pi_{ij} - \psi_{ji}^{{\prime \prime }} + \eta_{ij}^{{\prime }} \ge 0,\quad i \in V^{a} ,\quad j \in V^{s} }$$

$$ \theta_{i} + \sum\limits_{{j \in V^{a} \cup V^{s} ,\;j \ne i}} {\tau_{ij} } - \sum\limits_{{j \in V^{a} \cup V^{s} ,\;j \ne i}} {\tau_{ji} } - \sum\limits_{{j \in V^{0} }} {\upsilon_{ij} } + \sum\limits_{{j \in V^{0} }} {\upsilon_{ij}^{{\prime }} } \ge 0,\quad i \in V^{a} \cup V^{s} , $$

$${\lambda_{i} - \lambda_{i}^{{\prime }} - \gamma_{i} + \sum\limits_{{j \in V^{a} \cup V^{s} ,\;j \ne i}} {\pi_{ij} } - \sum\limits_{{j \in V^{a} \cup V^{s} ,\;j \ne i}} {\pi_{ji} } - \sum\limits_{{j \in V^{0} }} {\mu_{ij} } \ge 0,\quad i \in V^{a} \cup V^{s} ,}$$

$${ - \,\lambda_{i} + \lambda_{i}^{{\prime }} \ge - w_{i} ,\quad i \in V^{a} }$$

$${ - \,\lambda_{i} + \lambda_{i}^{{\prime }} \ge - w_{i}^{{\prime }} ,\quad i \in V^{s} }$$

$${\lambda_{i} - \lambda_{i}^{{\prime }} \ge 0,\quad i \in V^{a} \cup V^{s} }$$

$${M\mu_{ij} - \varphi_{ij} + \zeta_{ij} \ge 0,\quad i \in V^{a} \cup V^{s} ,\quad j \in V^{0} }$$

$${M\upsilon_{ij} - \varphi_{ij} + \zeta^{\prime}_{ij} \ge 0,\quad i \in V^{a} \cup V^{s} ,\quad j \in V^{0} }$$

$${M\upsilon_{ij}^{{\prime }} - \varphi_{ij} + \zeta_{ij}^{{\prime \prime }} \ge 0,\quad i \in V^{a} \cup V^{s} ,\quad j \in V^{0} }$$

$${\lambda_{i} ,\lambda^{\prime}_{i} ,\gamma_{i} ,\theta_{i} \ge 0,\quad i \in V^{a} \cup V^{s} }$$

$${\pi_{ij} ,\tau_{ij} ,\eta_{ij} ,\eta^{\prime}_{ij} \ge 0,\quad i,j \in V^{a} \cup V^{s} ,\quad i \ne j}$$

$${\mu_{ij} ,\upsilon_{ij} ,\upsilon^{\prime}_{ij} ,\varphi_{ij} ,\zeta_{ij} ,\zeta^{\prime}_{ij} ,\zeta^{\prime\prime}_{ij} \ge 0,\quad i \in V^{a} \cup V^{s} ,\quad j \in V^{0} }$$

$${\psi_{ij} \ge 0,\quad i,j \in V^{a} ,\quad i < j}$$

$${\psi^{\prime}_{ij} \ge 0,\quad i,j \in V^{s} ,\quad i < j}$$

$${\psi^{\prime\prime}_{ij} \ge 0,\quad i \in V^{s} ,\quad j \in V^{a} }$$

$${\varepsilon_{i} \ge 0,\quad i \in V^{s} }$$

where $\lambda_{i} ,\lambda_{i}^{{\prime }} ,\gamma_{i} ,\theta_{i} ,\pi_{ij} ,\tau_{ij} ,\eta_{ij} ,\eta_{ij}^{{\prime }} ,\mu_{ij} ,\upsilon_{ij} ,\upsilon_{ij}^{{\prime }} ,\varphi_{ij} ,\zeta_{ij} ,\zeta^{\prime}_{ij} ,\zeta_{ij}^{{\prime \prime }} ,\psi_{ij} ,\psi_{ij}^{{\prime }} ,\psi_{ij}^{{\prime \prime }} ,\varepsilon_{i} \ge 0$ are dual variables of Model LPD.

Appendix D

4.1 Proof of Proposition 3

Proof

The proof is similar to Theorem 2 in Wang (2009).

From Proposition 2 and constraints in “Appendix C” which are used in Step 2 of Algorithm AP, it is easy to show when (48) is satisfied in Step 4 of Algorithm AP, $s_{a}^{c*}$ is optimal to Model PD. And then from (44) in Step 2 of Algorithm AP, $p_{i}$ and $w^{\prime}_{i}$ ($i \in V^{s}$) obtained currently are optimal to InvPD. Showing that Algorithm AP terminates finitely is easy because the number of extreme points is finite for the convex hull of feasible solutions of Model PD.□

Appendix E

5.1 Lagrange relaxation algorithm for calculating the lower bound of objective value of model PC

5.1.1 E.1. Alternative formulation for Model PC

We suppose that the wharf line of the assisting terminal is partitioned into small sections with lengths that are one length unit (i.e., Sections $1,2, \ldots ,L$ from left to right), and the planning horizon is partitioned into small segments with lengths that are one time unit (i.e., segments $1,2, \ldots ,T$ in chronological order). The following additional notations are introduced.

5.2 Parameters

$$ r_{t} \quad r_{t} = t,\quad t = 1,2, \ldots ,T $$

$\varpi_{ijt}$ :

1 if berth section $j$ at time segment $t$ is unavailable; 0 otherwise

5.3 Decision variables

$v_{ijt}$ :

1 if section $j$ is the left-most berth section allocated to vessel $i$ and segment $t$ is the earliest time segment allocated to vessel $i$; 0 otherwise

$h_{ijt}$ :

1 if section $j$ is one of the berth sections allocated to vessel $i$ and segment $t$ is one of the time segments allocated to vessel $i$; 0 otherwise

Model PC can be reformulated as follows:

$$ \hbox{min} \sum\limits_{{i \in V^{a} \cup V^{b} }} {w_{i} \beta_{i}^{ + } } - \sum\limits_{{i \in V^{b} }} {(e_{i} - c_{i} - c^{\prime}_{i} )u_{i} } $$

(E-1)

which is subject to

$$ \sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {r_{t} v_{ijt} } } + b_{i} - 1 - d_{i} = \beta_{i}^{ + } - \beta_{i}^{ - } ,\quad i \in V^{a} \cup V^{b} $$

(E-2)

$$ \sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {v_{ijt} } } = 1 ,\quad i \in V^{a} $$

(E-3)

$$ \sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {v_{ijt} } } = 1 ,\quad i \in V^{a} $$

(E-4)

$$ \sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {v_{ijt} } } = u_{i} ,\quad i \in V^{b} $$

(E-5)

$$ \sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {v_{ijt} } } = u_{i} ,\quad i \in V^{b} $$

(E-6)

$$ \sum\limits_{{j^{\prime} = j}}^{{j + l_{i} - 1}} {\sum\limits_{{t^{\prime} = t}}^{{t + b_{i} - 1}} {h_{{ij^{{\prime }} t^{{\prime }} }} } } + M(1 - v_{ijt} ) \ge l_{i} b_{i} ,\quad \forall i \in V^{a} \cup V^{b} ,\quad j \le L - l_{i} + 1,\quad t \ge a_{i} + 1 $$

(E-7)

$$ h_{ijt} + \varpi_{ijt} \le 1,\quad \forall i,j,t $$

(E-8)

$$ \sum\limits_{i = 1}^{{n^{a} + n^{b} }} {h_{ijt} } \le 1,\quad \forall j,t $$

(E-9)

$$ u_{i} \in \{ 0,1\} ,\quad i \in V^{b} $$

(E-10)

$$ v_{ijt} ,h_{ijt} \in \{ 0,1\} ,\quad \forall i \in V^{a} \cup V^{b} ,\,j,\,t $$

(E-11)

$$ \beta_{i}^{ + } ,\beta_{i}^{ - } \ge 0 ,\quad i \in V^{a} \cup V^{b} $$

(E-12)

Constraint (E-2) is related to the definition of $\beta_{i}^{ + }$ and $\beta_{i}^{ - }$. Constraints (E-3) and (E-4) ensure that any vessel of the assisting terminal’s own customers can be handled and cannot berth before its arrival time, and its berthing position is restricted by the length of the wharf. Constraints (E-5) and (E-6) ensure that any vessel diverted to the assisting terminal can be handled and cannot berth before its arrival time, and its berthing position is restricted by the length of the wharf. Constraint (E-7) ensures that $h_{ijt} = 1$ if section $j$ is one of the berth sections occupied by vessel $i$, and segment $t$ is one of the time segments occupied by vessel $i$. Constraint (E-8) ensures that if berth section $j$ at time segment $t$ is unavailable, it cannot be allocated to any vessel. Constraint (E-9) ensures that a berth section can be allocated to at most one vessel in a time segment.

5.3.1 E.2. Lagrange relaxation and lower bound

Constraint (E-9) is relaxed, and $\phi_{jt}$ is the Lagrange multiplier for given $j$ and $t$. The relaxed problem can be written as

$$ z_{LR} (\phi ) = \hbox{min} \sum\limits_{{i \in V^{a} \cup V^{b} }} {w_{i} \beta_{i}^{ + } } - \sum\limits_{{i \in V^{b} }} {\left( {e_{i} - c_{i} - c_{i}^{{\prime }} } \right)u_{i} + \sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {\phi_{jt} \left( {\sum\limits_{i = 1}^{{n^{a} + n^{b} }} {h_{ijt} } - 1} \right)} } } , $$

(E-13)

which is subject to (E-2)–(E-8) and (E-10)–(E-12).

The preceding relaxed problem can be decomposed into two groups of subproblems as follows:

Group 1: For each $i \in V^{a}$,

$$ \hbox{min} w_{i} \beta_{i}^{ + } + \sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {\phi_{jt} h_{ijt} } } $$

which is subject to (E-11) and (E-12), and

$$ \begin{array}{*{20}l} {\sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {r_{t} v_{ijt} } } + b_{i} - 1 - d_{i} = \beta_{i}^{ + } - \beta_{i}^{ - } ,} \hfill \\ {\sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {v_{ijt} } } = 1 ,} \hfill \\ {\sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {v_{ijt} } } = 1,} \hfill \\ {\sum\limits_{{j^{{\prime }} = j}}^{{j + l_{i} - 1}} {\sum\limits_{{t^{{\prime }} = t}}^{{t + b_{i} - 1}} {h_{{ij^{{\prime }} t^{{\prime }} }} } } + M(1 - v_{ijt} ) \ge l_{i} b_{i} ,\quad \forall j \le L - l_{i} + 1,\quad t \ge a_{i} + 1,} \hfill \\ {h_{ijt} + \varpi_{ijt} \le 1,\quad \forall j,t} \hfill \\ \end{array} $$

Group 2: For each $i \in V^{b}$,

$$ \hbox{min} w_{i} \beta_{i}^{ + } - \left( {e_{i} - c_{i} - c_{i}^{{\prime }} } \right)u_{i} + \sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {\phi_{jt} h_{ijt} } } $$

Subject to (E-10)–(E-12) and

$$ \begin{array}{*{20}l} {\sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {r_{t} v_{ijt} } } + b_{i} - 1 - d_{i} = \beta_{i}^{ + } - \beta_{i}^{ - } } \hfill \\ {\sum\limits_{j = 1}^{{L - l_{i} + 1}} {\sum\limits_{{t = a_{i} + 1}}^{{T - b_{i} + 1}} {v_{ijt} } } = u_{i} } \hfill \\ {\sum\limits_{j = 1}^{L} {\sum\limits_{t = 1}^{T} {v_{ijt} } } = u_{i} } \hfill \\ {\sum\limits_{{j^{{\prime }} = j}}^{{j + l_{i} - 1}} {\sum\limits_{{t^{{\prime }} = t}}^{{t + b_{i} - 1}} {h_{{ij^{{\prime }} t^{{\prime }} }} } } + M(1 - v_{ijt} ) \ge l_{i} b_{i} ,\quad \forall j \le L - l_{i} + 1,\quad t \ge a_{i} + 1,} \hfill \\ {h_{ijt} + \varpi_{ijt} \le 1,\quad \forall j,t} \hfill \\ \end{array} $$

The optimal solution for the relaxed problem can be obtained by solving the preceding subproblems separately. For fixed values of $\phi_{jt}$, the corresponding value of the objective function of the relaxed problem provides a lower bound of the objective value of Model PC.

To obtain an improved lower bound, the classical subgradient optimization method is employed to solve the Lagrangian dual problem $z_{LD} = \mathop {\hbox{max} }\limits_{\phi \ge 0} z_{LR} (\phi )$. Details of the subgradient optimization method is available in the work of Park and Kim (2002).