An integrated multi-ship crane allocation in Beirut Port container terminal

Abstract

This paper investigates the integration between the quay and yard sides for multiple berthing ships with transshipment containers. This paper is motivated by the experience of an operator at Beirut Port. An integer linear programming model is formulated to minimize the total number of cranes used in both quay and yard sides for all berthing ships with transshipment containers unloading during a finite and discretized time horizon. The number of containers to be unloaded is determined in each time period, by each quay crane, at each ship bay location, along with the designated storage location at the yard side. The number of yard cranes needed at each storage yard block is also determined over the time horizon. Major capacity, time, and spatial constraints related to transshipment operations are taken into consideration. One insight from our numerical results is that restricting resources at the yard side will lead to an increase in required cranes at the quay side, and vice versa, which confirm results in earlier literature on single ship. However, we argue, via several counter examples, that single-ship solutions are not easily adaptable to multi-ship situations, which justifies the purpose of integrated formulations such as ours.

Appendix 1. Proof that the problem is NP hard

We show that the problem is NP hard by proving that it reduces to the multi-dimensional knapsack problem in a special case. Then, we discuss the number of integer decision variables needed in the solution. Consider a special case of the problem in Sect. 3 with one ship having one bay, a yard having one sub-block, and no restriction on the number of available cranes and storage capacity. That is, V = 1, B₁ = 1, F = 1, S = 1, QC_t, YC_t, K_st, and C_fs → ∞. In this special case, denote the total number of containers by N, the number of quay and yard cranes utilized in period t by y_t and z_t, and the number of unloaded containers in period t by x_t. The model in Sect. 3 simplifies to,

$$ \begin{aligned} & {\text{Minimize}}\quad w\sum\limits_{t = 1}^{T} {y_{t} } + (1 - w)\sum\limits_{t = 1}^{T} {z_{t} } \\ & {\text{Subject}}\;{\text{to}} \\ & \frac{{x_{t} }}{{C_{Q} }} \le y_{t} ,\quad \forall t \\ & \sum\limits_{t = 1}^{T} {x_{t} } = N \\ & \frac{{x_{t} }}{{C_{Y} }} \le z_{t} ,\quad \forall t \\ & x_{t} ,\;z_{t} \;{\text{integer}},\;y_{t} ,\;{\text{binary}} .\\ \end{aligned} $$

Replacing the value of $x_{t}$ from the second constraint, $x_{t} = N - \sum\nolimits_{{i \in \varUpsilon |\{ t\} }} {x_{i} } ,$ where $\varUpsilon = \{ 1,2, \ldots ,T\}$ in the first and third constraints, the problem becomes equivalent to

$$ \begin{aligned} & {\text{Minimize}}\quad w\sum\limits_{i \in \varUpsilon } {y_{i} } + (1 - w)\sum\limits_{i \in \varUpsilon } {z_{i} } \\ & {\text{Subject}}\;{\text{to}} \\ & C_{Q} y_{t} + \sum\limits_{{i \in \varUpsilon |\{ t\} }} {x_{i} } \ge N,\quad \forall t \\ & \sum\limits_{i \in \varUpsilon } {x_{i} } = N \\ & C_{Y} z_{t} + \sum\limits_{{i \in \varUpsilon |\{ t\} }} {x_{i} } \ge N,\quad \forall t \\ & x_{t} ,\;z_{t} \;{\text{integer}},\;y_{t} ,\;{\text{binary}}. \\ \end{aligned} $$

This model is equivalent to a multi-dimensional knapsack problem with decision variables, x_t, y_i, and z_i. Since the knapsack problem is NP hard as discussed by Wolsey (1998), we conclude that our transshipment ILP model is NP hard problem.