Decision support and data visualization for liner shipping fleet repositioning

Abstract

Liner carriers move vessels between routes in their networks several times a year in a process called fleet repositioning. There are currently no decision support systems to allow repositioning coordinators to take advantage of recent algorithmic advances in creating repositioning plans. Furthermore, no study has addressed how to visualize repositioning plans and liner shipping services in an accessible manner. Displaying information such as cargo flows and interactions between vessels is a complex task due to the overlap of container demands and long time scales. To this end, we propose a web-based decision support system designed specifically for liner shipping fleet repositioning that integrates an extended version of a state-of-the-art simulated annealing solution approach. Our system supports users in evaluating different strategic settings and scenarios, allowing liner carriers to save money through better fleet utilization and cargo throughput, as well as reduce their environmental impact by using less fuel.

Appendix: LSFRP arc flow model

We now provide the arc flow model from [32] with a brief explanation. The model is based off of a graph with embedded LSFRP constraints, which we do not reproduce here for brevity. It suffices to be given a graph $G=(V^i \cup V^f, A^i \cup A^f)$, where $V^i$ ($A^i$) is the set if inflexible visits (arcs) and $V^f$ ($A^f$) is the set of flexible visits. As a reminder, inflexible visits have fixed enter and exit times for vessels and inflexible arcs have fixed sailing times and costs, whereas the amount of time a vessel spends on a flexible visits or arc is a decision variable. We now present the table of parameters for the mathematical model, followed by the objective function and constraints.

S :

Set of ships.

$V'$ :

Set of visits minus the graph sink.

$V^i, V^f$ :

Set of inflexible and flexible visits, respectively.

$A^i,A^f$ :

Set of inflexible and flexible arcs, respectively.

$A'$ :

Set of arcs minus those arcs connecting to the graph sink, i.e., $(i,j) \in A$, $i,j \in V'$.

Q :

Set of equipment types. $Q = \{ dc , rf \}$.

M :

Set of demand triplets of the form ${(o,d,q)}$, where $o \in V', d \subseteq V'$ and $q \in Q$ are the origin visit, destination visits and the cargo type, respectively.

$V^{q+} \subseteq V'$ :

Set of visits with an equipment surplus of type q.

$V^{q-} \subseteq V'$ :

Set of visits with an equipment deficit of type q.

$V^{q^*} \subseteq V'$ :

Set of visits with an equipment surplus or deficit of type q ($V^{q^*} = V^{q^+} \cup V^{q^-}$).

$u^q_s \in {\mathbb {R}}^+$ :

Capacity of vessel s for cargo type $q \in Q$.

$M^ Orig _i, (M^ Dest _i) \subseteq M$ :

Set of demands with an origin (destination) visit $i \in V$.

$v_s \in V'$ :

Starting visit of ship $s \in S$.

$t^{ Mv }_{si} \in {\mathbb {R}}$ :

Move time per TEU for vessel s at visit $i \in V'$.

$t^E_i \in {\mathbb {R}}$ :

Enter time at inflexible visit $i \in V'$.

$t^X_i \in {\mathbb {R}}$ :

Exit time at inflexible visit $i \in V'$.

$t^P_i \in \mathbb {R}$ :

Pilot time at visit $i \in V'$.

$r^ Var _q \in {\mathbb {R}}^+$ :

Revenue for each TEU of equipment of type $q \in Q$ delivered.

$r^{(o,d,q)}\in {\mathbb {R}}^+$ :

Amount of revenue gained per TEU for the demand triplet.

$c^ Sail _{sij} \in {\mathbb {R}}^+$ :

Fixed cost of vessel s utilizing arc $(i,j) \in A'$.

$c^ VarSail _{sij} \in {\mathbb {R}}^+$ :

Variable hourly cost of vessel $s \in S$ utilizing arc $(i,j) \in A'$.

$c^ Mv _i \in {\mathbb {R}}^+$ :

Cost of a TEU move at visit $i \in V'$.

$c^ Port _{si} \in {\mathbb {R}}$ :

Port fee associated with vessel s at visit $i \in V'$.

$d^{ Min }_{ijs} \in {\mathbb {R}}^+$ :

Minimum duration for vessel s to sail on flexible arc (i, j).

$d^{ Max }_{ijs} \in {\mathbb {R}}^+$ :

Maximum duration for vessel s to sail on flexible arc (i, j).

$a^{(o,d,q)}\in {\mathbb {R}}^+$ :

Amount of demand available for the demand triplet.

$In (i) \subseteq V'$ :

Set of visits with an arc connecting to visit $i \in V$.

$Out (i) \subseteq V'$ :

Set of visits receiving an arc from $i \in V$.

$\tau \in V$ :

Graph sink, which is not an actual visit.

$w^s_{ij} \in {\mathbb {R}}^+_0$ :

The duration that vessel $s \in S$ sails on flexible arc $(i,j) \in A^f$.

$x^{{(o,d,q)}}_{ij} \in {\mathbb {R}}^+_0$ :

Amount of flow of demand triplet ${(o,d,q)}\in M$ on $(i,j) \in A'$.

$x^q_{ij} \in {\mathbb {R}}^+_0$ :

Amount of equipment of type $q \in Q$ flowing on $(i,j) \in A'$.

$y^s_{ij} \in \{0,1\}$ :

Indicates whether vessel s is sailing on arc $(i,j) \in A$.

$z^E_{i} \in {\mathbb {R}}^+_0$ :

Defines the enter time of a vessel at visit i.

$z^X_{i} \in {\mathbb {R}}^+_0$ :

Defines the exit time of a vessel at visit i.

$$\begin{aligned} \max -\sum _{s \in S} \left( \sum _{(i,j) \in A'} c^ Sail _{sij} y^s_{ij} + \sum _{(i,j) \in A^f} c^ VarSail _{sij} w^s_{ij} \right) \end{aligned}$$

(10)

$$\begin{aligned}+ \; \sum _{{(o,d,q)}\in M} \left( \sum _{j \in d} \sum _{i \in In(j)} \left( r^{(o,d,q)}- c^ Mv _o - c^ Mv _j \right) x_{ij}^{(o,d,q)}\right) \end{aligned}$$

(11)

$$\begin{aligned}+ \; \sum _{q \in Q} \left( \sum _{i \in V^{q+}} \sum _{j \in Out (i)} \left( r^{Eqp}_q - c^ Mv _i \right) x^q_{ij} - \sum _{i \in V^{q-}} \sum _{j \in In (i)} c^ Mv _i x^q_{ji} \right) \end{aligned}$$

(12)

$$\begin{aligned}- \sum _{j \in V'} \sum _{i \in In(j)} \sum _{s \in S} c^ Port _{sj} y^s_{ij} \end{aligned}$$

(13)

$$\begin{aligned} \hbox{subject}\;\hbox{to}\;\sum_{s \in S} \sum _{i \in In (j)} y^s_{ij} \le 1 \quad \forall j \in V' \end{aligned}$$

(14)

$$\begin{aligned}\sum _{j \in Out (i)} y^s_{ij} = 1 \quad \forall s \in S, i = v_s \end{aligned}$$

(15)

$$\begin{aligned}\sum _{i \in In (\tau)} \sum _{s \in S} y^s_{i\tau } = |S| \end{aligned}$$

(16)

$$\begin{aligned}\sum _{i \in In (j)} y^s_{ij} - \sum _{i \in Out (j)} y^s_{ji} = 0 \quad \forall j \in \{ V' \setminus \bigcup _{s \in S} v_s \}, s \in S \end{aligned}$$

(17)

$$\begin{aligned}\sum _{(o,d, rf ) \in M} x^{(o,d, rf)}_{ij} \le \sum _{s \in S} u^ rf _s y^s_{ij}\quad \forall (i,j) \in A' \end{aligned}$$

(18)

$$\begin{aligned}\sum _{{(o,d,q)}\in M} x^{{(o,d,q)}}_{ij} + \sum _{q' \in Q} x^{q'}_{ij} \le \sum _{s \in S} u^ dc _s y^s_{ij} \quad \forall (i,j) \in A' \end{aligned}$$

(19)

$$\begin{aligned}\sum _{i \in Out (o)} x^{(o,d,q)}_{oi} \le a^{(o,d,q)} \sum _{i \in Out (o)} \sum _{s \in S} y^s_{oi}\quad \forall {(o,d,q)}\in M \end{aligned}$$

(20)

$$\begin{aligned} \sum _{i \in In (j)} x^{(o,d,q)}_{ij} - \sum _{k \in Out (j)} x^{(o,d,q)}_{jk} = 0\quad \forall {(o,d,q)}\in M, j \in V' \setminus (o \cup d) \end{aligned}$$

(21)

$$\begin{aligned}\sum _{i \in In (j)} x^q_{ij} - \sum _{k \in Out (j)} x^q_{jk} = 0\quad \forall q \in Q, j \in V' \setminus V^{q^*} \end{aligned}$$

(22)

$$\begin{aligned}d^{ Min }_{ijs} y^s_{ij} \le w^s_{ij} \le d^{ Max }_{ijs} y^s_{ij}\quad \forall (i,j) \in A^f, s \in S \end{aligned}$$

(23)

$$\begin{aligned}z^E_{i} = t^E_i \sum _{s \in S} \sum _{j \in In (i)} y^s_{ij}\quad \forall i \in V^i \end{aligned}$$

(24)

$$\begin{aligned}z^X_{i} = t^X_i \sum _{s \in S} \sum _{j \in Out (i)} y^s_{ij}\quad \forall i \in V^i \end{aligned}$$

(25)

$$\begin{aligned}z^X_{i} + \sum _{s \in S} w^s_{ij} \le z^E_{j}\quad \forall (i,j) \in A^f \end{aligned}$$

(26)

$$\begin{aligned}&{\sum _{{(o,d,q)}\in M^ Orig _{i}} \sum _{j \in Out(o)} t^{ Mv }_{so} x^{(o,d,q)}_{oj} + \sum _{{(o,d,q)}\in M^ Dest _{i}} \sum _{d' \in d} \sum _{j \in In (d')} t^ Mv _{sd} x^{(o,d,q)}_{jd'}} \\ &\quad +{\sum _{q \in Q} \left( \sum _{i' \in \{V^{q^+} \cap \{i\}\}} \sum _{j \in Out(i')} t^ Mv _{si'} x^q_{i'j} + \sum _{i' \in \{V^{q^-} \cap \{i\} \}} \sum _{j \in In (i')} t^ Mv _{sj} x^q_{ji'} \right) } \\ &\quad - z^X_i + z^E_i + t^P_i \sum _{j \in In (i)} y^s_{ij} \le 0 \quad \forall i \in V^f, s \in S \end{aligned}$$

(27)

The objective first consists of the sailing cost (10) that takes into account the precomputed sailing costs on arcs between inflexible visitations, as well as the variable cost for sailings to and from flexible visitations. Note that the fixed sailing cost on an arc includes fuel costs, canal fees or even the time-charter bonus for entering an SoS. The profit from delivering cargo (11) is computed based on the revenue from delivering cargo minus the cost to load and unload the cargo from the vessel. Equipment profit is taken into account in (12), and, finally, port fees are deducted in (13).

Multiple vessels are prevented from visiting the same visitation in (14). The flow of each vessel from its source node to the graph sink is handled by (15), (16) and (17), with (16) ensuring that all vessels arrive at the sink.

Arcs are assigned capacities if a vessel utilizes the arc in (18), which assigns the reefer container capacity, and in (19), which assigns the total container capacity, respectively. Note that constraints (18) do not take into account empty reefer equipment, since empty containers do not need to be turned on, and can therefore be placed anywhere on the vessel. Cargo is only allowed to flow on arcs with a vessel in (20). The flow of cargo from its source to its destination, through intermediate nodes, is handled by (21). Constraints (22) balance the flow of equipment in to and out of nodes. In contrast to the way cargo is handled, equipment can flow from any port where it is in supply to any port where it is in demand. Since the amount of equipment carried is limited only by the capacity of the vessel, no flow source/sink constraints are required.

Flexible arcs have a duration constrained by the minimum and maximum sailing time of the vessel on the arc in (23). The enter and exit time of a vessel at inflexible ports is handled by (24) and (25), and we note that in practice these constraints are only necessary if one of the outgoing arcs from an inflexible visitation ends at a flexible visitation. Constraints (26) sets the enter time of a visitation to be the duration of a vessel on a flexible arc plus the exit time of the vessel at the start of the arc. Constraints (27) controls the amount of time a vessel spends at a flexible visitation. The first part of the constraint computes the time required to load and unload cargo and equipment, with the final term of the constraint adding the piloting time to the duration only if one of the incoming arcs is enabled (i.e., the flexible visitation is being used).