The impact of foldable containers on the cost of empty container relocation in the hinterland of seaports

Abstract

This study analyses the impact of using foldable containers in terms of cost savings of truck drayage operations, of both loaded and empty containers, in the hinterland of a seaport. We model a vehicle routing problem to optimise empty container relocation. A simulated annealing algorithm is developed to solve the problem. Numerical experiments are carried out in realistic empty container relocation scenarios. We find that, under certain conditions, foldable containers can offer higher truck productivity compared with standard containers, hence resulting in substantial cost savings. This paper provides managerial insights into how foldable containers can help reduce the costs of hinterland transport.

Appendices

Appendix 1: Formulation

Given the following notation, each of the four scenarios is modelled as a MIP:

1.1 Notation

1.1.1 Sets

$K$ :

set of trucks

$N$ :

set of inland depot and customers, that is $N = \left\{ {0,1, \ldots ,n} \right\}$, where the vertex 0 represents the inland depot (all truck routes start from/end at 0) and $n$ is the number of customers

$R$ :

set of truck routes. Since each track serves multiple routes within a working time (1 day), this set is for identifying each route of each truck

$S$ :

Subset of $N$ where $S \subset N$

$Q_{i}$ :

subset of $N$ where $Q_{i} \in \left\{ {j| \, j \ne 0,i = 0,\forall i,j \in N} \right\} \cup \left\{ {j| \, j = 0,i \ne 0,\forall i,j \in N} \right\}$, necessary for the IX scenario to ensure that empty containers are alternatively transported between the inland depot and customers, not allowing direct transport between customers; For instance, if node $i$ is the inland depot 0, then this subset contains customer sites as node $j$. Conversely, if node $i$ is a customer’s site, then the subset includes the depot 0 as node $j$

1.1.2 Parameters

${\text{DF}}_{i}$ :

number of empty container demand (imported shipment with FLDs) for pickup at customer site $i$

${\text{DS}}_{i}$ :

number of empty container demand (imported shipment with STDs) for pickup at customer site $i$

${\text{PF}}_{i}$ :

number of empty container demand (exporting shipment with FLDs) for delivery at customer site $i$

${\text{PS}}_{i}$ :

number of empty container demand (exporting shipment with STDs) for delivery at customer site $i$

${\text{FF}}$ :

$= \sum\limits_{i \in N} {{\text{DF}}_{i} }$ container fleet size of FLDs

${\text{FS}}$ :

$= \sum\limits_{i \in N} {{\text{DS}}_{i} }$ container fleet size of STDs

$L$ :

working time of trucks

$M$ :

sufficiently large constant

$C^{\text{FF}}$ :

exploitation cost of FLDs

$C^{\text{FS}}$ :

exploitation cost of STDs

$C{}^{\text{H}}$ :

cost of loading and unloading containers to/from trucks at customer sites

$C^{\text{U}}$ :

cost of F/UF FLDs at the inland depot and customer sites

$C^{\text{V}}$ :

fleet-related cost of trucks

$C_{ij}^{\text{T}}$ :

costs of movements of empty containers by trucks from depot/customer site $i$ to $j$

$T_{ij}$ :

travelling time of truck from depot/customer site $i$ to $j$, available as the product of the Euclidean distance between depot/customer sites and an average trucking speed of 40 km/h

Note that $C^{\text{FF}}$ is estimated twice as high as $C^{\text{FS}}$, due to higher purchase, maintenance and repair costs of FLDs. ${\text{FF}}$ and ${\text{FS}}$ specify that the sum of imported shipments corresponds to the total containers in the hinterland.

1.2 Decision variables

$V$ :

Truck fleet size

$W^{k}$ :

= 1 if truck $k$ is used, and 0 otherwise

${\text{EH}}_{i}^{kr}$ :

Number of unfolded empty containers that are loaded/unloaded to/from truck $k$ on route $r$ at customer site $i$

${\text{FH}}_{i}^{kr}$ :

Number of full containers that are loaded/unloaded to/from truck $k$ on route $r$ at customer site $i$

$U_{i}^{kr}$ :

Number of F/UF containers to/from truck $k$ on route $r$ at customer site $i$

${\text{UH}}_{i}^{kr}$ :

Number of folded empty containers that are loaded/unloaded to/from truck $k$ on route $r$ at customer site $i$

${\text{FZ}}_{ij}^{kr}$ :

Number of loaded containers transported by truck $k$ on route $r$ from customer site $i$ to $j$

${\text{EZ}}_{ij}^{kr}$ :

Number of empty containers transported by truck $k$ on route $r$ from customer site $i$ to $j$

${\text{US}}_{ij}^{kr}$ :

Number of unfolded empty containers transported by truck $k$ on route $r$ from customer site $i$ to $j$

$X_{ij}^{kr}$ :

= 1 if arc $\left( {i,j} \right)$ is used by truck $k$ on route $r$, and 0 otherwise

1.3 Auxiliary variables

$\delta_{i}^{krp}$ :

= 1 if quantity $p$ ($\le 4$) of containers are loaded/unloaded to/from truck $k$ on route $r$ at customer site $i$, and 0 otherwise

$\varphi_{ij}^{krp}$ :

= 1 if quantity $p$ of containers are transported by truck $k$ on route $r$ from customer site $i$ to $j$, and 0 otherwise

$\lambda_{ij}^{krq}$ :

= 1 if truck $k$’s transport state is $q$ while the truck travels on a series of arcs $\left( {i,j} \right)$ and the incoming/outgoing arcs to/from it on route $r$, and 0 otherwise

$\eta_{ij}^{kr}$ :

= 3 if truck $k$ is with a single FLD as unfolded while passing arc $\left( {i,j} \right)$ on its route $r$ and is without any containers before and after arc $\left( {i,j} \right)$, and < 3 otherwise

1.3.1 Formulation for DX with FLDs

The formulation for DX with FLDs is as follows:

[DX_FLD]

$$\begin{aligned} &{\text{Minimize}}\quad \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\sum\limits_{j \in N} {C_{ij}^{\text{T}} X_{ij}^{kr} } } } }+ C^{\text{H}} \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\left( {2{\text{FH}}_{i}^{kr} + {\text{UH}}_{i}^{kr} } \right)} } } \hfill \\ &\quad+ C^{\text{U}} \left( {\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{{i \in N/\left\{ 0 \right\}}} {U_{i}^{kr} } - 2\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{{i \in N/\left\{ 0 \right\}}} {\sum\limits_{{j \in N/\left\{ 0 \right\}}} {{\text{US}}_{ij}^{kr} } } } } } } } \right) + C^{\text{V}} \cdot V + C^{\text{FF}} \cdot {\text{FF}} \hfill \\ \end{aligned}$$

(3)

subject to

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{j \in N} {X_{ij}^{kr} } \ge 1} } \quad \forall i \in N,$$

(4)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {X_{ij}^{kr} } \ge 1} } \quad \forall j \in N,$$

(5)

$$\sum\limits_{j \in N} {X_{0j}^{kr} \le 1} \quad \forall k \in K,\;r \in R,$$

(6)

$$\sum\limits_{i \in N} {X_{i0}^{kr} \le 1} \quad \forall k \in K,\;r \in R,$$

(7)

$$\sum\limits_{h \in N} {X_{hi}^{kr} } - \sum\limits_{j \in N} {X_{ij}^{kr} } = 0\quad \forall i \in N,\;k \in K,\;r \in R,$$

(8)

$$\sum\limits_{i \in S} {\sum\limits_{j \in S} {X_{ij}^{kr} } } \le \left| S \right| - 1\quad \forall S \subset N/\left\{ 0 \right\},\;\left| S \right| \ge 2,\;k \in K,\;r \in R,$$

(9)

$$\sum\limits_{r \in R} {\sum\limits_{i \in N} {\sum\limits_{j \in N} {T_{ij} } X_{ij}^{kr} \le L} } \quad \forall k \in K,$$

(10)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{FZ}}_{0j}^{kr} } = } {\text{DF}}_{j} \quad \forall j \in N,$$

(11)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{FZ}}_{i0}^{kr} } = } {\text{PF}}_{i} \quad \forall i \in N,$$

(12)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{j \in N} {{\text{EZ}}_{ij}^{kr} } } - } \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{h \in N} {{\text{EZ}}_{hi}^{kr} } = {\text{DF}}_{i} - {\text{PF}}_{i} } } \quad \forall i \in N\backslash \left\{ 0 \right\},$$

(13)

$${\text{FZ}}_{ij}^{kr} + \frac{1}{4}{\text{EZ}}_{ij}^{kr} \le X_{ij}^{kr} \quad \forall i, \, j \in N,\;k \in K,\;r \in R,$$

(14)

$${\text{FH}}_{i}^{kr} = \sum\limits_{j \in N} {{\text{FZ}}_{ij}^{kr} } \quad \forall i \in N,\;k \in K,\;r \in R,$$

(15)

$$- {\text{EH}}_{i}^{kr} \le \sum\limits_{j \in N} {{\text{EZ}}_{ij}^{kr} } - \sum\limits_{h \in N} {{\text{EZ}}_{hi}^{kr} } \le {\text{EH}}_{i}^{kr} \quad \forall i \in N,\;k \in K,\;r \in R,$$

(16)

$$\sum\limits_{p = 1}^{4} {\left( {p \cdot \delta_{i}^{krp} } \right)} = {\text{EH}}_{i}^{kr} \quad \forall i \in N,\;k \in K,\;r \in R,$$

(17)

$$\sum\limits_{p = 1}^{4} {\delta_{i}^{krp} } \le 1\quad \forall i \in N,\;k \in K,\;r \in R,$$

(18)

$${\text{UH}}_{i}^{kr} = \sum\limits_{p = 1}^{4} {\delta_{i}^{krp} } \quad \forall i \in N,\;k \in K,\;r \in R,$$

(19)

$$U_{i}^{kr} = {\text{EH}}_{i}^{kr} \quad \forall i \in N,\;k \in K,\;r \in R,$$

(20)

$$\sum\limits_{p = 1}^{4} {\left( {p \cdot \varphi_{ij}^{krp} } \right)} = {\text{EZ}}_{ij}^{kr} \quad \forall i,j \in N,\;k \in K,\;r \in R,$$

(21)

$$\sum\limits_{p = 1}^{4} {\varphi_{ij}^{krp} } \le 1\quad \forall i,j \in N,\;k \in K,\;r \in R,$$

(22)

$$\eta_{ij}^{kr} = \left( {1 - \sum\limits_{h \in N} {\sum\limits_{p = 1}^{4} {\varphi_{hi}^{krp} } } } \right) + \varphi_{ij}^{kr1} + \left( {1 - \sum\limits_{l \in N} {\sum\limits_{p = 1}^{4} {\varphi_{jl}^{krp} } } } \right)\quad \forall i,j \in N\backslash \left\{ 0 \right\},\;k \in K,\;r \in R,$$

(23)

$$\sum\limits_{q = 1}^{3} {\left( {q \cdot \lambda_{ij}^{krq} } \right)} = \eta_{ij}^{kr} \quad \forall i,j \in N\backslash \left\{ 0 \right\},\;k \in K,\;r \in R,$$

(24)

$$\sum\limits_{q = 1}^{3} {\lambda_{ij}^{krq} } \le 1\quad \forall i,j \in N\backslash \left\{ 0 \right\},\;k \in K,\;r \in R,$$

(25)

$${\text{US}}_{ij}^{kr} = \lambda_{ij}^{kr3} \quad \forall i,j \in N\backslash \left\{ 0 \right\},\;k \in K,\;r \in R,$$

(26)

$$\sum\limits_{r \in R} {\sum\limits_{i \in N} {\sum\limits_{j \in N} {X_{ij}^{kr} } } } \le M \cdot W^{k} \quad k \in K,$$

(27)

$$V = \sum\limits_{k \in K} {W^{k} } ,$$

(28)

$$V \ge 0\;{\text{and}}\;{\text{integer}},$$

(29)

$${\text{EH}}_{i}^{kr} ,{\text{FH}}_{i}^{kr} ,{\text{UH}}_{i}^{kr} ,U_{i}^{kr} \ge 0\;{\text{and}}\;{\text{integer}}\quad \forall i \in N,\;k \in K,\;r \in R,$$

(30)

$${\text{EZ}}_{ij}^{kr} ,{\text{FZ}}_{ij}^{kr} ,{\text{US}}_{ij}^{kr} ,\eta_{ij}^{kr} \ge 0\;{\text{and}}\;{\text{integer}}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,$$

(31)

$$W^{k} \in \left\{ {0, \, 1} \right\}\quad k \in K,$$

(32)

$$X_{ij}^{kr} \in \left\{ {0, \, 1} \right\}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,$$

(33)

$$\delta_{ij}^{krp} ,\varphi_{ij}^{krp} \in \left\{ {0, \, 1} \right\}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,\;p \in \left\{ {1, \ldots ,4} \right\},$$

(34)

$$\lambda_{ij}^{krq} \in \left\{ {0, \, 1} \right\}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,\;q \in \left\{ {1, \ldots ,3} \right\}.$$

(35)

The objective function (3) minimises the total costs, which consist of: the costs of the movements of containers by trucks, the costs of loading and unloading containers to/from trucks, F/UF costs at customer sites, the fleet-related costs of trucks and exploitation costs of the FLD fleet. Constraints (4) and (5) ensure that each customer is served at least once by trucks. Constraints (6) and (7) ensure that each truck route consists of a maximum of one truck. Constraints (8) guarantee flow conservation. Constraints (9) relate to sub-tour elimination. Constraints (10) guarantee the feasibility of the working time of trucks. Constraints (11) and (12) state that the import/export loaded shipment with FLDs is carried directly between the inland depot and customer sites. Constraints (13) define empty container demand as the difference between the number of import and export shipments with FLDs of the customer. Constraints (14) specify the transport capacity of trucks for loaded and empty containers. The factor 1/4 means that up to four FLDs can be folded and bundled into one package. Four folded and bundled FLDs correspond to the dimensions of one STD. Furthermore, this constraint guarantees that loaded and empty containers cannot be moved by the same truck at the same time. Equations (15) and (16) define the absolute number of loaded and empty containers that are loaded or unloaded to/from trucks at each site, respectively. Constraints (17)–(19) specify the substantive number of F/UF containers that are loaded or unloaded to/from trucks at each site, incorporating a binary auxiliary variable. These constraints mean that up to four folded and bundled FLDs can be loaded or unloaded to/from trucks as a single STD likewise; For instance, if the absolute number of FLDs handled is $\text{EH}_{i}^{kr} = 3$, then the binary auxiliary variable is $\delta_{i}^{kr3} = 1$. Hence, the practical number of FLDs handled is $\text{UH}_{i}^{kr} = 1$. Equation (20) defines the absolute number of FLDs that are folded/unfolded at each site. Constraints (21)–(26) determine the number of empty containers which are moved alone without being folded between customer sites. Specifically, these constraints count the number of unfolded containers which are distributed as a single STD between customer sites, because the F/UF process is unnecessary when a single FLD is directly exchanged between customers. Figure 5 demonstrates an example of a single FLD movement without F/UF processes between customer sites. In this situation, if $\eta_{12}^{kr}$ in constraint (23) has accordingly the specific value (= 3), then $\lambda_{12}^{kr3} = 1$ and $\text{US}_{12}^{kr} = 1$. In other words, if $\sum\limits_{p = 1}^{4} {\varphi_{01}^{krp} } = 0$, $\varphi_{12}^{kr1} = 1$ and $\sum\limits_{p = 1}^{4} {\varphi_{20}^{krp} } = 0$, then $\eta_{12}^{kr} = 3$. The situation $\eta_{12}^{kr} = 3$ means that truck $k$ carries a single empty FLD on arc $\left( {1,2} \right)$ of route $r$. Then the truck moves without cargo on arcs $\left( {0,1} \right)$ and $\left( {2,0} \right)$. If $\eta_{12}^{kr} = 3$, then $\lambda_{12}^{kr3} = 1$. Moreover, if $\lambda_{12}^{kr3}$ has the specific value ($= 1$), it identifies a single container which is not needed to be folded and unfolded on arc $\left( {1,2} \right)$, $\text{US}_{12}^{kr} = 1$. Equations (27) and (28) define the truck fleet size. Note that the characteristics of foldable containers are mainly reflected in constraints (14) and (17)–(19). These constraints affect the reductions in the number of trucks used, the trip length of truck haulage and the number of handlings of foldable containers.

Fig. 5

Example of single FLD moving without folding/unfolding

1.3.2 Formulation for DX with STDs

The formulation for DX with STDs is as follows:

[DX_STD]

$${\text{Minimize}}\quad \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\sum\limits_{j \in N} {C_{ij}^{T} X_{ij}^{kr} } } } } + C^{H} \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\left( {2FH_{i}^{kr} + \text{EH}_{i}^{kr} } \right)} } } + C^{V} \cdot V + C^{FS} \cdot FS$$

(36)

subject to (6)–(12),

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{FZ}}_{0j}^{kr} } = } {\text{DS}}_{j} \quad \forall j \in N,$$

(37)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{FZ}}_{i0}^{kr} } = } {\text{PS}}_{i} \quad \forall i \in N,$$

(38)

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{j \in N} {{\text{EZ}}_{ij}^{kr} } } - } \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{h \in N} {{\text{EZ}}_{hi}^{kr} } = {\text{DS}}_{i} - {\text{PS}}_{i} } } \quad \forall i \in N\backslash \left\{ 0 \right\},$$

(39)

$${\text{EZ}}_{ij}^{kr} + {\text{FZ}}_{ij}^{kr} \le X_{ij}^{kr} \quad \forall i, \, j \in N,\;k \in K,\;r \in R,$$

(40)

(15), (16), (27), (28),

$$V \ge 0\;{\text{and}}\;{\text{integer}},$$

(41)

$${\text{EZ}}_{ij}^{kr} ,{\text{FZ}}_{ij}^{kr} \ge 0\;{\text{and}}\;{\text{integer}}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,$$

(42)

$${\text{EH}}_{i}^{kr} ,{\text{FH}}_{i}^{kr} \ge 0\;{\text{and}}\;{\text{integer}}\quad \forall i \in N,\;k \in K,\;r \in R,$$

(43)

(32), (33).

The objective function (36) minimises the total costs, which consist of: the costs of the movements of containers by truck, the costs of loading and unloading containers to/from trucks, the fleet-related costs of trucks and exploitation costs of the STD fleet. Constraints (37) and (38) specify that the import/export loaded shipments with STDs are carried directly between the inland depot and customer sites. Constraint (39) specifies the number of empty containers demanded as a fulfilment for the gap between the import and export traffic using STDs. Constraint (40) specifies the transport capacity of trucks for loaded and empty containers. Moreover, this constraint guarantees that loaded and empty containers cannot be moved by the same truck at the same time. Needless to mention, each truck can move only a single STD.

1.3.3 Formulation for IX with FLDs

The formulation for IX with FLDs is as follows:

[IX_FLD]

$$\begin{aligned} &{\text{Minimize}}\;\sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\sum\limits_{j \in N} {C_{ij}^{\text{T}} X_{ij}^{kr} } } } } + C^{\text{H}} \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{i \in N} {\left( {2{\text{FH}}_{i}^{kr} + {\text{UH}}_{i}^{kr} } \right)} } } + C^{\text{U}} \sum\limits_{k \in K} {\sum\limits_{r \in R} {\sum\limits_{{i \in N/\left\{ 0 \right\}}} {U_{i}^{kr} } } } \hfill \\ &\quad +\; C^{\text{V}} \cdot V + C^{\text{FF}} \cdot {\text{FF}} \hfill \\ \end{aligned}$$

(44)

subject to

(4)–(12),

$$\sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{EZ}}_{i0}^{kr} } - } \sum\limits_{k \in K} {\sum\limits_{r \in R} {{\text{EZ}}_{0i}^{kr} = {\text{DF}}_{i} - {\text{PF}}_{i} } } \quad \forall i \in N\backslash \left\{ 0 \right\},$$

(45)

(14), (15),

$$- {\text{EH}}_{i}^{kr} \le \sum\limits_{{j \in Q_{i} }} {{\text{EZ}}_{ij}^{kr} } - \sum\limits_{{h \in Q_{i} }} {{\text{EZ}}_{hi}^{kr} } \le {\text{EH}}_{i}^{kr} \quad \forall i \in N,\;k \in K,\;r \in R,$$

(46)

(17)–(20), (27)–(30),

$${\text{EZ}}_{ij}^{kr} ,{\text{FZ}}_{ij}^{kr} ,{\text{US}}_{ij}^{kr} \ge 0\;{\text{and}}\;{\text{integer}}\quad \forall i, \, j \in N,\;k \in K,\;r \in R.$$

(47)

(32), (33),

$$\delta_{ij}^{krp} \in \left\{ {0, \, 1} \right\}\quad \forall i, \, j \in N,\;k \in K,\;r \in R,\;p \in \left\{ {1, \ldots ,4} \right\}.$$

(48)

The objective function (44) is partially changed in terms of the costs of F/UF containers from the objective function (3), since a single empty FLD is not exchanged between customer sites. Constraint (45) defines empty container demand as the difference between the number of import and export containers, and this constraint only allows empty containers to be exchanged between the inland depot and customers. Inequality (46) determines the absolute number of empty containers that are loaded or unloaded to/from trucks at each site, because empty containers cannot be directly exchanged between customer sites.

1.3.4 Formulation for IX with STDs

The formulation for IX with STDs is as follows:

[IX_STD]

Minimize (36)

subject to

(4)–(10), (37)–(39), (45), (40), (15), (46), (27), (28), (41)–(43), (32), (33).

Appendix 2: The virtual truck route with dummy nodes

The SA requires the design of the solution structure for solving the problems. Furthermore, to explicitly distinguish the shipment states of trucks, i.e. loaded container, empty container or no carriage, we assume a virtual truck route associated with some types of dummy nodes. The sets of dummy nodes are, namely: for consignee dummy nodes ($N^{\text{D}}$: delivery point $D$ of loaded (imported) container, $N^{\text{Es}}$: supply point ${\text{Es}}$ of empty container), for shipper dummy nodes ($N^{\text{P}}$: pickup point $P$ of loaded (exporting) container, $N^{\text{Ed}}$: demand point ${\text{Ed}}$ of empty container), and for the inland depot dummy nodes ($N^{{{\text{Es}}'}}$: supply point ${\text{Es}}^{\prime}$ of empty container, $N^{{{\text{Ed}}'}}$: demand point ${\text{Ed}}^{\prime}$ of empty container). We assume $\left| {N^{\text{D}} } \right| = \left| {N^{\text{Es}} } \right| = \left| {N^{{{\text{Ed}}'}} } \right|$ and $\left| {N^{\text{P}} } \right| = \left| {N^{\text{Ed}} } \right| = \left| {N^{{{\text{Es}}'}} } \right|$, where $\left| {N^{\text{D}} } \right|$ is the cardinality of $N^{\text{D}}$.

Figure 6 shows examples of DX and IX relocation scenarios with dummy nodes in case of using STDs. At the delivery point $D$, once the imported container from the inland depot 0 becomes empty, it is automatically transferred to the supply point ${\text{Es}}$, and the empty container is ready to be repositioned from ${\text{Es}}$ to ${\text{Ed}}$ (DX) or ${\text{Ed}}^{\prime}$ (IX). Moreover, after the empty is repositioned from ${\text{Es}}$ (DX) or ${\text{Es}}^{\prime}$ (IX) to ${\text{Ed}}$, then the empty container is automatically transferred to $P$. The empty is loaded with an exporting cargo at $P$, and that loaded container is transported from $P$ to the inland depot 0. In the IX scenario, empty containers are moved through ${\text{Es}} \to {\text{Ed}}^{\prime}$ or ${\text{Es}}' \to {\text{Ed}}$, because empty containers enable only indirect exchange via the inland depot. Note that, in case of using FLDs, since each truck has a transport capacity of four empty FLDs, trucks serve through the DX and IX scenarios within the capacity between ${\text{Es}}$, ${\text{Es}}^{\prime}$, ${\text{Ed}}$ and ${\text{Ed}}^{\prime}$.

Fig. 6

Example virtual truck routes with dummy nodes in case of using STDs

Appendix 3: Solution representation and generating neighbourhood solutions

Figure 7a and b illustrate an example of truck movements and loaded/empty container distributions in a solution for DX_STD and DX_FLD, respectively. In DX_FLD, loaded (import/export) containers are transported through the node pair 0 → D or P → 0, then empty containers can be carried between ${\text{Es}}$, ${\text{Es}}^{\prime}$, ${\text{Ed}}$ and ${\text{Ed}}^{\prime}$, because trucks can freely serve between customers as long as their carrying capacity allows. Surely, trucks run with no cost in the arc (node pair), if the arc consists of the originally identical node.

Fig. 7

Example service routes for SA

Figure 7c and d show solution representations for DX_STD and DX_FLD. In DX_STD, truck A serves 0–2–6–12–9–10–0–1–0 and truck B serves 0–5–8–4–0–3–7–11–0. On the other hand, in DX_FLD, truck A serves 0–3–9–12–0–1–4–0 and truck B serves 0–2–6–5–7–8–10–11–0. If a truck working time reaches the limit, then the artificial node pair 0–0, which means the end of the service route by the specific truck, is inserted into the previous node pair to obtain a feasible solution. To distinguish the end of the service route by either the specific truck or an identical truck, we employ this representation, namely the artificial node pair 0–0.

To generate a neighbourhood solution, randomly choose two node pairs from the current solution and then exchange them, or insert another node pair chosen randomly to lie between the two node pairs. Additionally, it is effective to swap two nodes which are contained within a node pair.

Appendix 4: Total costs in various cases (NC = 30 and 60 FEUs)

See Tables 5 and 6.

Table 5 Total costs in various cases (NC = 30 FEUs)

Table 6 Total costs in various cases (NC = 60 FEUs)