Special Issue on Latin-American Research: A Time Based Discretization Approach for Ship Routing and Scheduling with Variable Speed

Abstract

In this paper we develop a network based model for the routing and scheduling of a heterogeneous tramp fleet. The objective of the problem is to serve a known set of single trip cargo contracts, observing time window constraints at both origin and destination of cargoes, while minimizing total operating cost. A distinctive aspect of the methodology is that time windows for picking and delivering cargoes are discretized. This approach allows for a broad variety of features and practical constraints to be implicitly included in the model. In particular, we consider problems where navigation speed can be used to control fuel consumption, which is a main operating cost in ocean shipping. We performed a computational study on three set of fifteen problem instances each, involving 30, 40 and 50 cargoes per instance, respectively. Each problem instance was solved with two fleet sizes, three levels of discretization, and with constant and variable speed. The numerical results show that our model presents a much better trade-off between solution quality and computing time than a similar constant speed continuous model. For example, discretizing the time windows in as few as 3 points, we obtained solutions that in average were no more than 0,8% worse than the best solution found by the continuous model. Computing time, on the other side, decreased in at least two orders of magnitude. The results also confirm that significant benefits might be obtained by incorporating the navigation speed as a controllable variable in the model.

Appendix A

1.1 Constant Speed Continuous Optimization Model

In this appendix, we develop a constant speed continuous model for Ship Routing and scheduling, in which time windows are modeled continuously. The modeling approach is similar (although using a different notation) to the one presented in Desrosiers et al. (1995) for general TWVRP. This model, as stated in Section 5, is used as a benchmark model to evaluate the performance of the proposed approach.

This continuous model can also be described in terms of a network with multiple arcs between nodes. In this case, the graph contains exactly one node per cargo contract. There is an arc (I,J,k) from node I to node J, if cargoes I and J are compatible with each other, and with ship k. With arc (I,J,k) we associate two parameters, c _IJk and v _IJk , which represent, respectively, total cost and total time of trip I plus the ballast trip from the destination of cargo I to the origin of cargo J. With each node we associate a time window [L _i ,U _i ] for picking the cargo, and with each ship we associate a per time-unit waiting cost w _k . Finally, we complete the graph with a fictitious node where all ships start from and return to.

The mathematical formulation is the following:

$$ Min\quad \sum\limits_{\left( {I,J,k} \right) \in A} {{c_{IJk}}{x_{IJk}}} + \sum\limits_{k = 1}^B {\sum\limits_{I = 1}^N {{w_k}{z_{jk}}} } $$

(A-1)

$$ s.t.:\quad \sum\limits_{I = 1}^N {{x_{0Ik}}} \leqslant 1\quad k = 1,...,B $$

(A-2)

$$ \sum\limits_{\left( {I,J,k} \right) \in A} {{x_{IJk}}} = 1\quad J = 1,...,N $$

(A-3)

$$ \sum\limits_{\left( {I,J,k} \right) \in A} {{x_{IJk}}} = \sum\limits_{\left( {L,J,k} \right) \in A} {{x_{LIk}}} \quad J = 0,...,N,\quad k = 1,...,B $$

(A-4)

$$ {T_0} = 0 $$

(A-5)

$$ {T_J} \geqslant {T_I} + {v_{IJk}} + M\left( {{x_{IJk}} - 1} \right)\quad \left( {I,J,k} \right) \in A,\quad J \ne 0 $$

(A-6)

$$ {T_I} \geqslant {L_I}\quad I = 1,...,N $$

(A-7)

$$ {T_I} \leqslant {U_I}\quad I = 1,...,N $$

(A-8)

$$ {z_{Jk}} \geqslant {T_J} - \left( {{T_I} + {v_{IJk}}} \right) + M\left( {{x_{IJk}} - 1} \right)\quad \left( {I,J,k} \right) \in A,\quad J \ne 0 $$

(A-9)

$$ {x_{IJk}} \in \left\{ {0,1} \right\}\left( {I,J,k} \right) \in A $$

(A-10)

$$ {T_I} \geqslant 0\quad I = 1,...,N $$

(A-11)

$$ {z_{Ik}} \geqslant 0\quad I = 1,...,N,\quad k = 1,...B $$

(A-12)

Where:

N :

Number of cargoes or contracts to be served.

B :

Number of available ships.

A :

Set of arcs in the network.

$$ {x_{IJk}} = \left\{ {\begin{array}{*{20}{c}} {1\quad {\hbox{if ship }}k{\hbox{ serves cargoes }}I{\hbox{ and }}J{\hbox{ consecutively, i}}{\hbox{.e}}{\hbox{., arc (}}I,J,k{\hbox{) is selected as part of the solution}}} \hfill \\{{0}\quad {\hbox{otherwise}}} \hfill \\\end{array} } \right. $$

T _I :

Starting time of service for cargo I.

z _Ik :

Waiting time of ship k at node I.

The objective function (A-1) represents total traveling plus waiting cost. Constraint (A-2) ensures that each ship is employed at most in one route. Equation (A-3) establishes that exactly one arc must enter to node J, i.e., each cargo must be served exactly once, by exactly one ship. For nodes different than the fictitious, Eq. (A-4) states that exactly one arc must leave node J, and that this arc has to be associated with the same ship that the entering arc. For the fictitious node this constraint states that if ship k leaves the node, then it must return to it . Constraints (A-5) and (A-6) allow for the computation of service starting time at each node (cargo). As usual, M denotes a value large enough such that the constraint becomes non-active when x _IJk  = 0. Constraints (A-7) and (A-8) assure that time windows are satisfied. Constraint (A-9) allows computing the waiting time at each node. Finally, constraints (A-10), (A-11) and (A-12) state decision variable to be binary and nonnegative respectively.

Notice that model (A-1)–(A-12) is not equivalent to model (1)–(5). On one hand, it allows for continuous time windows to be considered. On the other hand, it is restricted to a constant speed and does not admit most of the special features discussed in previous sections, such as soft time windows and port service costs that depend on arrival time. In fact, incorporating these features would, most likely, require the inclusion of nonlinear constraints in the model. Also notice that this model does not consider time windows for delivering the cargoes. This is of no relevance here, since for the purpose of our numerical evaluation, we made the same assumption.