The berth allocation problem with mobile quay walls: problem definition, solution procedures, and extensions

Abstract

The berth allocation problem (BAP), which defines a processing interval and a berth at the quay wall for each ship to be (un-)loaded, is an essential decision problem for efficiently operating a container port. In this paper, we integrate mobile quay walls into the BAP. Mobile quay walls are huge propelled floating platforms, which encase ships moored at the immobile quay and provide additional quay cranes for accelerating container processing. Furthermore, additional ships can be processed at the seaside of the platform, so that scarce berthing space at a terminal is enlarged. We formalize the BAP with mobile quay walls and provide suitable solution procedures.

Appendices

Appendix 1: Mixed integer programming model for BAP–MQW

See Table 4.

Table 4 Notation

Applying the notation summarized in Table 4 the mixed-integer programming model for BAP–MQW consists of objective function (2) and constraints (3)–(11).

$$\begin{aligned} \mathrm{Minimize}\, Z = \sum _{j=1,\ldots ,n} \left( P_{j} + 2\cdot D\cdot y_j \right) , \end{aligned}$$

(2)

subject to

$$\begin{aligned}&\sum _{j=1,\ldots ,n} \left( x^\mathrm{in}_{i,j}+x^\mathrm{out}_{i,j}\right) = 1 \quad \forall \, i=1,\ldots ,n, \end{aligned}$$

(3)

$$\begin{aligned}&P_{j} \ge \sum _{i=1,\ldots ,n}c_i\cdot f^\mathrm{in}\cdot x^\mathrm{in}_{i,j}\quad \forall \, j=1,\ldots ,n, \end{aligned}$$

(4)

$$\begin{aligned}&P_{j} \ge \sum _{i=1,\ldots ,n}c_i\cdot f^\mathrm{out}\cdot x^\mathrm{out}_{i,j}\quad \forall \, j=1,\ldots ,n, \end{aligned}$$

(5)

$$\begin{aligned}&n\cdot y_{j} \ge \sum _{i=1,\ldots ,n} \left( x^\mathrm{in}_{i,j}+x^\mathrm{out}_{i,j}\right) \quad \forall \, j=1,\ldots ,n, \end{aligned}$$

(6)

$$\begin{aligned}&\sum _{i = 1}^n{x^\mathrm{in}_{i,j}} \le 1 \quad \forall \, j = 1, \ldots , n, \end{aligned}$$

(7)

$$\begin{aligned}&x^\mathrm{in}_{i,j} \in \{0,1\} \quad \forall \, i=1,\ldots ,n, j=1,\ldots ,n, \end{aligned}$$

(8)

$$\begin{aligned}&x^\mathrm{out}_{i,j} \in \{0,1\} \quad \forall \, i=1,\ldots ,n, j=1,\ldots ,n, \end{aligned}$$

(9)

$$\begin{aligned}&y_{j} \in \{0,1\} \quad \forall \, j=1,\ldots ,n, \end{aligned}$$

(10)

$$\begin{aligned}&P_{j} \ge 0 \quad \forall \, j=1,\ldots ,n. \end{aligned}$$

(11)

Objective (2) minimizes the makespan. Equation (3) ensures that each ship is assigned to exactly one slot. Note that we do not need more than $n$ slots. Hence, we consider a fixed number of $n$ slots and allow slots to remain empty. Constraints (4) and (5) force $P_{j}$ to be at least slot $j$’s duration. Inequalities (6) force $y_j$ to equal 1 if at least one ship is assigned to $j$, while constraints (7) restrict the number of inner ships per slot to one. Constraints (8)–(11) define the domain of the variables.

Appendix 2: Instances solved in the computational study

See Table 5.

Table 5 Instances generated for the computational study

Table 5 lists the instances generated for the computational study of this paper. Columns $n$ and $\rho$ describe the parameters used in the instance generation (see Sect. 5 for details), ship container loads lists the number of containers to be (un-)loaded for each of the $n$ ships, and opt. shows the optimal makespan in minutes for $f^\mathrm{in} = 0.25, \, f^\mathrm{out} = 0.35$ and $D = 30$.