Optimization of truck appointments in container terminals

Abstract

Truck appointment has proved to be an efficient tool in reducing congestion at container terminals. To make a reasonable appointment quota plan, it is necessary to take terminal operations into consideration. We develop a novel approach (model) for optimizing a truck appointment system with the objective of decreasing external trucks’ waiting times, at the gate and yard, and internal trucks’ waiting times at the yard. The vacation queuing model is used to describe the coordinated service process of yard cranes. Based on non-stationary queuing theory, truck waiting times are estimated more accurately. Numerical experiments are conducted to illustrate the validity of the model and algorithm. Results show that the model reflects the characteristics of the service process of yard cranes and it improves the calculation accuracy of the truck waiting time.

Appendix: Solution algorithm

A heuristic approach is designed to optimize the truck appointment quotas of each time slot. To calculate the waiting time of ETs and ITs, a non-stationary queue model with policies is used. The main operations of the algorithm follow.

Step 1:

Randomly generate 168 numbers to represent the appointment quota of each time period. Each of the random numbers should satisfy constraint (5). Therefore, the boundary of each number is $(\lambda_{m}^{G} - Q,\lambda_{m}^{G} + Q)$. Each group of numbers represents a plan for the truck appointment system. Set the iteration k = 0.

Step 2:

Let $l_{it}^{G} = 0,w_{it}^{G} = 0,d_{it}^{G} = 0$ for all i and t = 0.

Step 3:

Then update $l_{it}^{G} ,w_{it}^{G} ,d_{it}^{G}$ using Eqs. (6)–(9) at the gate, update t by $t = t + 1$.

Step 4:

Repeat Step 3 until all the lengths of each time point are estimated.

Step 5:

Let $l_{t}^{E} = 0,w_{t}^{E} = 0,d_{t}^{E} = 0,l_{t}^{I} = 0,w_{t}^{I} = 0,d_{t}^{I} = 0$ for t = 0.

Step 6:

Update $l_{t}^{I} ,w_{t}^{I} ,d_{t}^{I}$ using Eqs. (11)–(19). The bisection method is used to estimate the queue length of ITs at the yard. At every iteration, the length calculated by Eq. (17) is compared with the length calculated by Eq. (11) to decide the value of $\rho_{t}^{I}$.

Step 7:

Update $l_{t}^{E} ,w_{t}^{E} ,d_{t}^{E}$ using Eqs. (20)–(26). The estimation of queue length of ETs at the yard is the same as Step 6. Update t by $t = t + 1$.

Step 8:

Repeat Step 6 and Step 7 until all the queue lengths of each time point are approximated.

Step 9:

Calculate the objective value by the above approximations. Then let $f_{s} = \frac{1}{{1 + Z_{s} }}$ represent the fitness value of the plan s, where $Z_{s}$ represents the objective value of the plan s.

Step 10:

Use the roulette wheel selection and multi-point cross method to generate new plans of the next iteration, and then update the iteration $k = k + 1$.

Step 11:

Repeat the above procedures until the iteration reaches the pre-designed iteration.