Analysis on container port capacity: a Markovian modeling approach

Abstract

Container ports handle outbound, inbound, and transshipment containers plying between the area for vessels on the quay and the storage space in the yard. Port operators typically concentrate their efforts on the container handling process with the aims of increasing the productivity of quay-side operations and reducing the time in port of vessels. Recognizing that operation processes necessitate containers to stay in the storage space for a certain period before moving to other areas, the operational efficiency at the yard (in addition to that at the quayside) plays an influential role in ensuring performance measures of a container port. This study develops analytical models based on the Markov chain to estimate the port capacity under various combinations of resources, namely, quay cranes, yard cranes, and prime movers. Important performance measures representing the capacity in the proposed models are analyzed and sensitivity analyses of the port capacity are conducted through numerical experiments. The results under the suggested operational strategies are also compared.

Appendices

Appendix A: The mean delay time with the different traveling times in the sub-network for a YC

This section explains the formulation for the mean delay time of a PM in the sub-network for $Y_j$ considering the different traveling times between QCs and YCs. Let $t(j,i,j)$, $t(j,i,k)$, $t(k1,i,k2)$, and $t(k,i,j)$ be the corresponding variables for $t_{(j,i,j)}$, $t_{(j,i,k)}$, $t_{(k1,i,k2)}$, and $t_{(k,i,j)}$, respectively, in Sect. 5.1. The difference between $t(j,i,j)$ and $t_{(j,i,j)}$ is that the latter is the average value of the former for all $j$. Each of the delay times is represented as follows:

$$\begin{aligned} t(j,i,j)&= t_P (Y_j ,Q_i )+W^{Q_i }+t_Q +t_P (Q_i ,Y_j )\end{aligned}$$

(43)

$$\begin{aligned} t(j,i,k)&= t_P (Y_j ,Q_i )\!+\!W^{Q_i }\!+\!t_Q \!+\!t_P (Q_i ,Y_k )\!+\!W^{Y_k }\!+\!t_Y \quad \text{ for}\,\text{ any}\,k\ne j\qquad \end{aligned}$$

(44)

$$\begin{aligned} t(k1,i,k2)&= t_P (Y_{k1} ,Q_i )+W^{Q_i }+t_Q +t_P (Q_i ,Y_{k2} )+W^{Y_{k2} }+\,t_Y\nonumber \\&\quad \text{ for}\,\text{ any}\,k1\ne j\,\text{ and}\,k2\ne j\quad \end{aligned}$$

(45)

$$\begin{aligned} t(k,i,j)&= t_P (Y_k ,Q_i )+W^{Q_i }+t_Q +t_P (Q_i ,Y_j )\quad \text{ for}\,\text{ any}\,k\ne j \end{aligned}$$

(46)

The mean delay time of a PM in the sub-network for $Y_j$ can be expressed with reference to Eqs. (43)–(46) and the visit probabilities shown in Fig. 5.

$$\begin{aligned} t_D \left(Y_j \right)&= \frac{1}{N}t(j,i,j) \nonumber \\&+\frac{N-1}{N}\frac{1}{N}\sum _i {\sum _k {\left\{ {t(j,i,k)+t(k,i,j)} \right\} } } \nonumber \\&+\left( {\frac{N-1}{N}} \right)^2\frac{1}{N}\sum _i {\sum _{k1} {\sum _{k2} {\left\{ {t(j,i,k1)+t(k1,i,k2)+t(k2,i,j)} \right\} } } } \nonumber \\&+\left( {\frac{N-1}{N}} \right)^3\frac{1}{N}\sum _i \sum _{k1} \sum _{k2} \sum _{k3} \left\{ t(j,i,k1)+t(k1,i,k2)\right.\nonumber \\&\left.+\,t(k2,i,k3)+t(k3,i,j) \right\} \nonumber \\&+\left( {\frac{N-1}{N}}\right)^4\frac{1}{N}\sum _i {\sum _{k1} {\sum _{k2} {\sum _{k3} {\sum _{k4} {\left\{ {t(j,i,k1)+\cdots +t(k4,i,j)} \right\} } } } } } \nonumber \\&+\left( {\frac{N-1}{N}} \right)^5\frac{1}{N}\sum _i {\sum _{k1} {\sum _{k2} {\sum _{k3} {\sum _{k4} {\sum _{k5} {\left\{ {\cdots } \right\} } } } } } } \nonumber \\&+\cdots \quad \text{ for}\,\text{ all}\,j \end{aligned}$$

(47)

Since the mean delay time in the sub-network for $Y_j$ could not be derived in closed form, we attempt to approximate it using the average delay times. The approximate mean delay time is as follows:

$$\begin{aligned} t_D \left(Y_j \right)&= \frac{1}{N}t_{(j,i,j)} +\frac{N-1}{N}\frac{1}{N}\left( {t_{(j,i,k)} +t_{(k,i,j)} }\right) \nonumber \\&+ \left( {\frac{N-1}{N}}\right)^2\frac{1}{N}\left( {t_{(j,i,k)} +t_{(k1,i,k2)} +t_{(k,i,j)} }\right)\nonumber \\&+\left( {\frac{N-1}{N}}\right)^3\frac{1}{N}\left( {t_{(j,i,k)} +2t_{(k1,i,k2)} +t_{(k,i,j)} }\right) \nonumber \\&+ \left( {\frac{N-1}{N}}\right)^4\frac{1}{N}\left( {t_{(j,i,k)} +3t_{(k1,i,k2)} +t_{(k,i,j)} }\right)+\cdots \nonumber \\&= \frac{1}{N}t_{(j,i,j)} +\frac{N-1}{N}\frac{1}{N}\left[ {\sum _{r=0} {\left( {\frac{N-1}{N}}\right)^r \left( {t_{(j,i,k)} +rt_{(k1,i,k2)}+t_{(k,i,j)} }\right)} } \right] \nonumber \\&= \frac{1}{N}t_{(j,i,j)} \nonumber \\&+\frac{N-1}{N}\left[ t_{(j,i,k)} \sum _{r=0} {\left( {\frac{N-1}{N}}\right)^r\frac{1}{N}} +t_{(k1,i,k2)} \sum _{r=0} {\left( {\frac{N-1}{N}}\right)^r\frac{1}{N}r}\right.\nonumber \\&\left.+\,t_{(k,i,j)} \sum _{r=0} {\left( {\frac{N-1}{N}}\right)^r\frac{1}{N}} \right] \nonumber \\&= \frac{1}{N}t_{(j,i,j)} +\frac{N-1}{N}\left[ {t_{(j,i,k)} +(N-1)t_{(k1,i,k2)} +t_{(k,i,j)} } \right]\quad \text{ for}\,\text{ all}\,j \end{aligned}$$

(48)

If all the distances are the same, which means $t_{(j,i,j)} =t_{(j,i,k)} =t_{(k1,i,k2)} =t_{(k,i,j)}$, then the mean delay time, $t_D (Y_j )$, will be exactly identical to that of the approximate capacity model described in section 4.

Appendix B: The mean delay time with the different deployment probabilities in the sub-network for a YC

Let $p(j,i,j)$, $p(j,i,k)$, $p(k1,i,k2)$, and $p(k,i,j)$ be the deployment probabilities that can be used for expressing the mean delay time. $p_{(j,i,j)}$, $p_{(j,i,k)}$, $p_{(k1,i,k2)}$, and $p_{(k,i,j)}$ are the average values of $p(j,i,j)$, $p(j,i,k)$, $p(k1,i,k2)$, and $p(k,i,j)$, respectively, as described in Sect. 5.2. By definition,

$$\begin{aligned} p(j,i,j)&= p(Y_j ,Q_i )p(Q_i ,Y_j ) \end{aligned}$$

(49)

$$\begin{aligned} p(j,i,k)&= p(Y_j ,Q_i )p(Q_i ,Y_k )\quad \text{ for}\,\text{ any}\,k\ne j \end{aligned}$$

(50)

$$\begin{aligned} p(k1,i,k2)&= p(Y_{k1} )p(Y_{k1} ,Q_i )p(Q_i ,Y_{k2} )\quad \text{ for}\,\text{ any}\,k1\ne j\quad \text{ and}\quad k2\ne j\end{aligned}$$

(51)

$$\begin{aligned} p(k,i,j)&= p(Y_k )p(Y_k ,Q_i )p(Q_i ,Y_j )\quad \text{ for}\,\text{ any}\,k\ne j \end{aligned}$$

(52)

Thus, the mean delay time of a PM in the sub-network for $Y_j$ is as follows:

$$\begin{aligned} t_D (Y_j )&= \sum _i {p(j,i,j)t(j,i,j)} \nonumber \\&+\sum _i {\sum _k {\left\{ {p(j,i,k)t(j,i,k)+p(k,i,j)t(k,i,j)} \right\} } } \nonumber \\&+\sum _i \sum _{k1} \sum _{k2} \left\{ p(j,i,k1)t(j,i,k1)\right.\nonumber \\&\left.+p(k1,i,k2)t(k1,i,k2)+p(k2,i,j)t(k2,i,j) \right\} \nonumber \\&+\sum _i {\sum _{k1} {\sum _{k2} {\sum _{k3} {\left\{ {p(j,i,k1)t(j,i,k1)+\cdots +p(k3,i,j)t(k3,i,j)} \right\} } } } }\nonumber \\&+\sum _i {\sum _{k1} {\sum _{k2} {\sum _{k3} {\sum _{k4} {\left\{ {\cdots } \right\} } } } } } \nonumber \\&+\cdots \quad \text{ for}\,\text{ all}\,j \end{aligned}$$

(53)

Likewise in the previous section, this mean delay time could not be derived in closed form. To approximate it further, the approximated deployment probabilities (i.e., $p_{(j,i,j)}$, $p_{(j,i,k)}$, $p_{(k1,i,k2)}$, and $p_{(k,i,j)} )$ and the approximated delay times (i.e., $t_{(j,i,j)}$, $t_{(j,i,k)}$, $t_{(k1,i,k2)}$, and $t_{(k,i,j)} )$ are used for obtaining the closed form expression. Hence, the approximate mean delay time in the sub-network for $Y_j$ is as follows:

$$\begin{aligned} t_D (Y_j )&= p_{(j,i,j)} t_{(j,i,j)} +p_{(j,i,k)} p_{(k,i,j)} \left( {t_{(j,i,k)} +t_{(k,i,j)} }\right) \nonumber \\&+p_{(j,i,k)} p_{(k1,i,k2)} p_{(k,i,j)} \left( {t_{(j,i,k)} +t_{(k1,i,k2)} +t_{(k,i,j)} }\right) \nonumber \\&+p_{(j,i,k)} p_{(k1,i,k2)} ^2p_{(k,i,j)} \left( {t_{(j,i,k)} +2t_{(k1,i,k2)} +t_{(k,i,j)} }\right) \nonumber \\&+p_{(j,i,k)} p_{(k1,i,k2)} ^3p_{(k,i,j)} \left( {t_{(j,i,k)} +3t_{(k1,i,k2)} +t_{(k,i,j)} }\right)+\cdots \nonumber \\&+p_{(j,i,k)} p_{(k1,i,k2)} ^rp_{(k,i,j)} \left( {t_{(j,i,k)} +rt_{(k1,i,k2)} +t_{(k,i,j)} }\right)+\cdots \nonumber \\&= p_{(j,i,j)} t_{(j,i,j)} +\sum _{r=0} {p_{(j,i,k)} p_{(k1,i,k2)} ^rp_{(k,i,j)} \left( {t_{(j,i,k)} +rt_{(k1,i,k2)} +t_{(k,i,j)} }\right)} \nonumber \\&= p_{(j,i,j)} t_{(j,i,j)} +p_{(j,i,k)} p_{(k,i,j)}\nonumber \\&\left[ {t_{(j,i,k)} \sum _{r=0} {p_{(k1,i,k2)} ^r} +t_{(k1,i,k2)} \sum _{r=0} {p_{(k1,i,k2)} ^rr} +t_{(k,i,j)} \sum _{r=0} {p_{(k1,i,k2)} ^r} } \right] \nonumber \\&= p_{(j,i,j)} t_{(j,i,j)} +p_{(j,i,k)} p_{(k,i,j)} \frac{1}{1-p_{(k1,i,k2)} }\nonumber \\&\left[ {t_{(j,i,k)} +t_{(k1,i,k2)} \frac{p_{(k1,i,k2)} }{1-p_{(k1,i,k2)} }+t_{(k,i,j)} } \right]\quad \text{ for}\,\text{ all}\,j \end{aligned}$$

(54)

If we assume that the number of QCs connected to a YC is same for all YCs and the probabilities to select a QC are same for the YC, then the mean delay time proposed in the Eq. (54) would be equivalent to the Eq. (48) in Appendix B.

Appendix C: Requirement for developing dispatching strategies

As shown in Tables 1–5, the port capacity is influenced by the dispatching strategy for PMs. This section conducts a simulation experiment that provides a guide to develop efficient dispatching strategies which can possibly improve the port capacity. Thus, another dispatching strategy is introduced for comparing the results with those of the balance and random strategies. This strategy is referred to as the distribution strategy. Under the distribution strategy, a PM selects a QC with the shortest queue length whenever a PM is available. The purpose of this strategy is to minimize the queue length differences among QCs.

Table 6 shows the simulation results for different number of resources and dispatching strategies. The balance and distribution strategies outperform the random strategy in terms of the mean throughput. As explained in section 7, this is because the balance and distribution strategies make a conscientious effort to minimize the difference of the queue length among QCs. By comparing the balance strategy with the distribution strategy, the balance strategy still produces slightly better results. Since this experiment constrains the total number of PMs to be evenly allocated to QCs under the balance strategy, the difference of queue length among QCs is less than that under the distribution strategy in the proposed capacity model. Although a PM selects a QC with the shortest queue length under the distribution strategy, other PMs can select the same QC before the PM arrives at the QC. This is because the distribution strategy is a kind of PM-initiated task assignment rules. However, the capacity model does not provide such kind of QC initiated task assignment rules as referred to Egbelu and Tanchoco (1984). Therefore, although different numbers of PMs are assigned to a QC under the balance strategy, the port capacity would be improved when an efficient dispatching strategy that balances workloads among QCs on the quay is employed.

Table 6 Comparison of the three dispatching strategies in a simulation study