A Note on Alternative Objectives for the Blocks Relocation Problem

Abstract

The Blocks Relocation Problem (BRP) is an important problem arising at container terminals when containers have to be transshipped. Recently, we have seen the upcoming discussion whether some objectives are more appropriate than others. While most authors minimize the number of relocations as objective, some minimize the crane’s working time. After adapting a given BRP formulation by including the second objective in two variants, a computational study is carried out on benchmark instances from literature applying some sensitivity analysis indicating to which extent optimal solutions change if the objective is replaced. If the number of relocations is minimized, often larger crane working times may be obtained. Conversely, if the crane working time is minimized, often solutions achieving also a minimum number of relocations are found. Based on our analysis, the consideration of alternative objectives like crane working time is recommended.

A Mathematical Model

To make this paper self-contained we repeat the mathematical model of [18] with modifications as described above. Whenever applicable, the numbering of constraints and equations follows the original appended by a small z for distinction.

Parameters

Variables

$$\begin{aligned} b_{ijnt}&= {\left\{ \begin{array}{ll} 1&{} \text {if block { n} is at stack { i} and row { j} at the beginning of period { t},}\\ 0&{} \text {otherwise;} \end{array}\right. } \\&\forall i=1,\ldots ,W, j=1,\ldots ,H, t=1,\ldots ,T, n=t,\ldots ,N&\\ x_{ijklnt}&= {\left\{ \begin{array}{ll} 1&{} \text {if block { n} is relocated from stack { i}, row { j} to stack { k}, row { l} in period { t},}\\ 0&{} \text {otherwise;} \end{array}\right. } \\&\forall i=1,\ldots ,W, j=1,\ldots ,H, k=1,\ldots ,W,l=1,\ldots ,H, t=1,\ldots , T - 1, n=t+1,\ldots ,N&\\ y_{ijtt}&= {\left\{ \begin{array}{ll} 1&{} \text {if block n=t is retrieved from stack { i} and row { j} in period { t},}\\ 0&{} \text {otherwise;} \end{array}\right. } \\&\forall i=1,\ldots ,W, j=1,\ldots ,H, t=1,\ldots ,T&\end{aligned}$$

Objectives

Objective 1: Minimize number of relocations

$$\min f_1= \sum _{i,k=1}^W \sum _{j,l=1}^H \sum _{n=1}^N \sum _{t=1}^N x_{ijklnt} $$

Objective 2: Minimize crane time

$$ \min f_2=\sum _{i=1}^{W}\sum _{j=1}^{H}\sum _{t=1}^{T}(2 t_s i+t_{pp})y_{ijtt} + \sum _{i,k=1}^{W}\sum _{j,l=1}^{H}\sum _{n=1}^{N}\sum _{t=1}^{T}(2 t_s |i-k|+t_{pp})x_{ijklnt} $$

Objective $2^{vert}$: Minimize crane time with tier-dependent pick-up/place-down effort

$$\begin{aligned} \min f_{2^{vert}}= & {} \sum _{i=1}^{W}\sum _{j=1}^{H}\sum _{t=1}^{T}(2 t_s i+\left( 2 h_{max}-j-h_{out}\right) t_{r})y_{ijtt} \\+ & {} \sum _{i,k=1}^{W}\sum _{j,l=1}^{H}\sum _{n=1}^{N}\sum _{t=1}^{T}(2 t_s |i-k|+ \left( 2 h_{max}-j-l\right) t_{r})x_{ijklnt} \end{aligned}$$

Objective 3: Minimize weighted sum of objectives 1 and 2

$$ \min f_1 + \bar{W} f_2 $$

Constraints and Preprocessing

Constraints as given in [18] (they need to be appended by Constraints (5) and (6) from Sect. 2):

$$\begin{aligned}&\sum _{n=t}^N b_{ijnt} \le 1 \quad \forall i=1,\dots ,W, j=1,\dots H, t=1,\dots ,T-1&(2_z)\\&\sum _{n=t}^N b_{ijnt} \ge \sum _{n=t}^N b_{ij+1nt} \quad \forall i=1,\dots ,W, j=1,\dots H-1, t=1,\dots ,T&(3_z)\\&b_{ijnt+1} = b_{ijnt} + \sum _{k=1}^W \sum _{l=2}^H x_{klijnt} - \sum _{k=1}^W \sum _{l=1}^H x_{ijklnt}\\&\quad \forall i=1,\dots ,W, j=1,\dots H, t=1,\dots ,T-1, n=t+1,\dots ,N&(6a_z)\\&b_{ijnt} - y_{ijtt} = 0\\&\quad \forall i=1,\dots ,W, j=1,\dots H, t=1,\dots ,T-1, n=t&(6b_z)\\&\sum _{i=1}^W \sum _{j=1}^H y_{ijtt} = 1 \quad \forall t=1,\dots ,T-1&(7''_z)\\&M \cdot \left( 1 - \sum _{n=t+1}^N x_{ijklnt}\right) \ge \sum _{n=t+1}^N \sum _{j'=j+1}^H \sum _{l'=l+1}^H x_{ij'kl'nt} \\&\quad \forall i=1,\dots ,W, j=2,\dots H-1, k=1,\dots ,W,l=1,\dots H-1, t=1,\dots ,T-1&(8'_z)\\&\sum _{j'=1}^{j-1} y_{ij'tt} \ge \sum _{k=1}^W \sum _{l=1}^H \sum _{n=t+1}^N x_{ijklnt} \quad \forall i=1,\dots ,W, j=2,\dots H, t=1,\dots ,T-1&(A'_z) \end{aligned}$$

$$\begin{aligned} \begin{array}{ll} \displaystyle (i_n,j_n) &{}\qquad \text {Position of container { n} in the initial bay}\\ \displaystyle \pi _n &{}\qquad \text {Time period when { n} has to be relocated the first time, i.e.,}\\ \displaystyle &{}\qquad \pi _n = \min n' \text {such that n' is placed at any position} (i_n,1) \,\, to \,\, (i_n, j_n)\\ \end{array} \end{aligned}$$