Stochastic programming approach for unidirectional quay crane scheduling problem with uncertainty

Abstract

Quay crane scheduling is a key aspect of container terminal operation, which can be regarded as a decision-making process with uncertainty. Each task involves stochastic loading and unloading operation times owing to the existence of uncertainty. In this study, we investigate the unidirectional quay crane scheduling problem for a stochastic processing time, which requires that all the quay cranes move in the same direction either from bow to stern, or vice versa, throughout the planning horizon. The problem is formulated as a two-stage stochastic mixed-integer programming model, where the binary first-stage decision variables correspond to the assignment of tasks to quay cranes, and the mixed-integer second-stage decision variables are related to the generation of detailed schedules. To make the model solvable, we develop an alternative equivalent reformulation with a special structure that involves binary variables in the first stage and continuous variables in the second stage. To solve this reformulated model, an integer L-shaped method is presented for small-size instances, and a simulated annealing algorithm is presented for large-size instances to obtain near-optimal solutions. Numerical experiments show that the integer L-shaped method and simulated annealing algorithm could efficiently solve the unidirectional quay crane scheduling problem with uncertainty. The results also indicate that the stochastic model has distinct advantages in terms of shortening the completion time of vessels and improving the service level of container terminals compared with the expected value problem solutions.

Appendices

Appendix A. Notation

See Table 12.

Table 12 Notation

Appendix B. Data support for research motivation

In this section, we showed relevant data to support our research motivation. First, we measured the processing time of 231 unloading and loading movements for cranes operating at Tianjin Port, China, as shown in Fig. 10. We found that the processing times of the cranes during loading and unloading varied drastically (i.e., from 1 to 20 min). According to Moccia et al. (2006), a task is a collection of containers, and an average of 30 containers per task is common in practice. The uncertainty in the processing time of a single move will result in a wider range of fluctuations in the completion time of each task. Second, we fitted the distribution of the processing time and found that it obeys a log-normal distribution (red line in Fig. 10), which is important to describe the container terminal production process. To further verify this finding, we used SPSS to plot the PP diagram of the processing time, as shown in Fig. 11, which shows that most of the points are located around the diagonal; thus, the processing time is consistent with the log-normal distribution. A log-normal process is justified by considering the central limit theorem in the log domain, which is positive. Third, we analyzed the reason why the task processing time obeys a log-normal distribution. For example, consider the time taken by workers to complete a particular task; because each worker is different, we obtain a distribution. However, all the values must be positive (because time cannot be negative). Furthermore, we can predict the distribution of the possible shapes. No one can perform the work in the minimum time, and there are a few “best” workers who are very fast. The most representative values of the common workers’ completion time form a peak. Finally, the tail represents a long list of workers who “lag behind.” Obviously, the normal distribution does not describe this distribution very well because in the normal distribution, a random variable can be defined as positive or negative; it is symmetric, and the tail is extremely short. As the processing times of tasks are the main uncertainty parameters, the optimal solutions obtained by solving the deterministic models using the estimated task processing times might not remain optimal and could sometimes even become infeasible in practice; for further details, readers may refer to Sect. 3.1. In summary, the presence of uncertainty in the real-world processing of operational schedules adds to the complexity of the problem, which motivates us to study the quay crane scheduling problem with uncertainty.

Fig. 10

Processing time statistics and distribution fitting

Fig. 11

PP diagram of the processing time

Appendix C. The scenario data for the example section

See Table 13.

Table 13 The scenario data for the example section

In order to make the scenario easier to understand, the following describes the way the scenario data is generated in the example. For scenario 1, we assume that the processing time of all tasks could increase at most 50% due to bad weather or other unknown reasons. For scenario 2, we assume that the processing time of all tasks could be reduced by at most 50% due to fine weather or other unknown reasons. The other scenarios follow the same setting.

Appendix D. Classical L-shaped algorithm

Appendix E. Sample generation technique by the Metropolis–Hastings algorithm

A sampling technique is used to create scenarios for the stochastic programming model. In our model, it mainly refers to the processing time of the task. Because the constraints of non-crossover and interference between cranes are considered in the model, each task can be treated as independent when it is processed. On the basis of this assumption, we sampled the processing time of each task and then randomly assembled it. In addition, the scenarios with a larger size were assumed to include all scenarios with a smaller size. This assumption was made to ensure that the scenario size was the only factor that influences the results of the experiments, and it is also widely employed in stochastic programming (Higle and Sen 1991; Fu et al. 2017) and questionnaire-based surveys (Wu et al. 2012).

Sampling procedures based on the Markov Chain Monte Carlo (MCMC) method were designed to generate scenarios, where MCMC methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its stationary distribution. In our study, for a given probability density function (pdf) $f\left( \cdot \right)$, we wish to have a convenient way to generate a corresponding sample. Following Metropolis et al. (1953), we constructed a Markov chain such that its limiting pdf is $f$. As Markov chains can converge to a steady distribution, the state of the chain after a number of steps is then used as a sample. For further details on the MCMC method, readers may refer to the work of Kroese et al. (2014). Our sample generation procedure uses the Metropolis–Hastings algorithm, which is widely used in MCMC. The specific steps are as follows:

For the processing time of each task, given a transition density $q({\mathbb{y}}|{\mathbb{x}})$ and starting from an initial state ${\mathbb{X}}_{0}$, repeat the following steps from $t = 1\;{\text{to}}\;N$:

• 1. Generate ${\mathbb{Y}} \sim q\left( {{\mathbb{y}}|{\mathbb{X}}_{t} } \right)$.

• 2. Generate $U \sim U\left( {0,1} \right)$.

• 3. Set ${\mathbb{X}}_{t + 1} = \left\{ {\begin{array}{*{20}l} {\mathbb{Y}} \hfill & {{\text{if}}\; U \le \alpha \left( {{\mathbb{X}}_{t} ,{\mathbb{Y}}} \right)} \hfill \\ {{\mathbb{X}}_{t} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$, where $\alpha \left( {{\mathbb{x}},{\mathbb{y}}} \right) = { \hbox{min} }\left\{ {\frac{f\left( {\mathbb{y}} \right)q({\mathbb{x}}|{\mathbb{y}})}{f\left( {\mathbb{x}} \right)q({\mathbb{y}}|{\mathbb{x}})},1} \right\}$.

In particular, we adopt the random walk sampler to deal with the transition density $q({\mathbb{y}}|{\mathbb{x}})$. In the random walk sampler, the proposal state ${\mathbb{Y}}$ is given by ${\mathbb{Y}} = {\mathbb{X}} + {\mathbb{Z}}$, where ${\mathbb{Z}}$ is generated from a symmetric distribution. Therefore, the proposed transition density is symmetric, i.e., $q({\mathbb{y}}|{\mathbb{x}}) = q({\mathbb{x}}|{\mathbb{y}})$. It follows that the acceptance probability is $\alpha \left( {{\mathbb{x}},{\mathbb{y}}} \right) = { \hbox{min} }\left\{ {\frac{f\left( {\mathbb{y}} \right)}{f\left( {\mathbb{x}} \right)},1} \right\}$. The above algorithm produces a sequence ${\mathbb{X}}_{1}$, ${\mathbb{X}}_{2}$,… of dependent random vectors, with ${\mathbb{X}}_{\text{t}}$ approximately distributed according to $f\left( {\mathbb{x}} \right)$ for large $t$. In addition, the probability of each scenario is generated both independently and equally with their sum equal to one.

Appendix F. Implementation of the log-normal distribution

In this section, we explain how to use the log-normal distribution to generate the required stochastic processing time. Specifically, the mean value of the stochastic processing time of each task $i \in \varOmega$ over all scenarios $E\left[ {p_{i} } \right]$ is equal to the processing time $\hat{p}_{i}$ of Kim and Park (2004) and Meisel and Bierwirth (2011). The standard deviation $\sqrt {\text{var} \left[ {p_{i} } \right]}$ is obtained as the product of the mean value $E\left[ {p_{i} } \right]$ and the coefficient of variation $C_{\text{v}}$, $\sqrt {\text{var} \left[ {p_{i} } \right]}$ = $C_{\text{v}} \cdot E\left[ {p_{i} } \right]$. Accordingly, the log-normal distribution can be determined by the parameters $\mu = \ln \left( {E\left[ {p_{i} } \right]} \right) - 1/2\sigma^{2}$ and $\sigma^{2} = \ln \left( {1 + \text{var} \left[ {p_{i} } \right]/\left( {E\left[ {p_{i} } \right]} \right)^{2} } \right)$.

Appendix G. Parameter sensitivity analysis of the simulated annealing algorithm

In the following, we will attempt to select a set of parameters that are expected to produce good solutions for the simulated annealing algorithm. Five different problem instances are tested. As for each problem instance, we solve it with six different initial temperatures $\tau^{0}$ (1000, 2000, 4000, 6000, 8000, and 10000), seven different stopping temperatures $\tau^{\text{s}}$ (0.1, 0.5, 1, 2, 3, 4, and 5), six different cooling rates cr (0.55, 0.65, 0.75, 0.85, 0.95, and 0.99), and five different local search iterations $I_{\hbox{max} }$(1, 2, 5, 10, and 20). In addition, these five instances are solved five times, and we normalize each objective value by calculating 100 $\times$ (objective value/lowest objective value). The average, best, and worst normalized values were plotted for different parameters, as shown in Fig. 12. The best performance was found when $\tau^{0} = 4000$, $\tau^{\text{s}} = 1$, ${\text{cr}} = 0.95$, and $I_{\hbox{max} } = 10$.

Fig. 12

Sensitivity analysis of the simulated annealing algorithm parameters

Appendix H. Results obtained by the simulated annealing algorithm for medium- and large-sized instances

See Tables 14 and 15.

Table 14 Results obtained by the simulated annealing algorithm for medium- and large-sized instances of Kim and Park (2004)

Table 15 Results obtained by the simulated annealing algorithm for medium- and large-sized instances of Meisel and Bierwirth (2011)