Modelling and solving the multi-quays berth allocation and crane assignment problem with availability constraints

Abstract

In this work, we are interested in the multi-quays berth allocation and crane assignment problem under availability restrictions. Availability restrictions may arise due weather conditions, or when, for example, cranes must undergo planned maintenance in order to stay in good performance. This problem was inspired by a real-case of a bulk port in Morocco. To solve the problem we propose at first a mixed-integer programming model. Then, in view of the limitations of the proposed model, we investigate a set of heuristics based on general variable neighborhood search with three variants of variable neighborhood descent as a local search. To validate the proposed model and the proposed heuristic approach, real-world instances are used. The computational results reveal that CPLEX MIP solver consumes a lot of CPU time to solve this model, even sometimes failing to guarantee the optimality of the provided solution. On the other hand, the proposed GVNS heuristic turns out to be very efficient in solving the considered problem.

Appendices

Appendix A1

Here we present the procedure to construct a feasible planning for each vessel (see Fig. 1).

Fig. 1

Constructive heuristic for generating a feasible planning

Appendix A2

Here, we provide the detailed comparison of three different VNDs (basic sequential VND, pipe VND and cyclic VND) using different orders of neighborhood classes. We distinguish 6 different orders of neighborhood classes because we have three neighborhood classes. The distinguished orders are as follows:

• order 1: inter-quay neighborhoods, intra-quay neighborhoods, and crane assignment neighborhood;

• order 2: inter-quay neighborhoods, crane assignment neighborhood, and intra-quay neighborhoods;

• order 3: intra-quay neighborhoods, inter-quay neighborhoods, and crane assignment neighborhood;

• order 4: intra-quay neighborhoods, crane assignment neighborhood, and inter-quay neighborhoods;

• order 5: crane assignment neighborhood, intra-quay neighborhoods, and inter-quay neighborhoods;

• order 6: crane assignment neighborhood, inter-quay neighborhoods, and intra-quay neighborhoods;

As stated in Sect. 4, the order of neighborhoods within each class is set according to the increasing cardinality. In the intra-quay neighborhoods class the order is: Intra-quay insert, Intra-quay swap , Intra-quay reverse neighborhood; while in the inter-quay the order is Inter-quay insert and Inter-quay swap neighborhood.

Each variant is employed within the GVNS scheme (Algorithm 7) using 6 different orders. For testing purposes we used the set of largest instances with the number of vessels equal to 14. The summary results are provided in Table 4. For each variant and each order of neighborhood classes we report the average solution value and average CPU time consumption in minutes over the set of largest instances.

Table 4 Comparison of VNDs with different orders of neighborhoods

From the presented results, we infer that the best order for the basic sequential VND is “order 6”, while for the others the best order is “order 2”. Among all tested variants, the best overall performance has been exhibited by pipe VND employing “order 2”.

Appendix A3

Detailed results for entire data set are presented in Tables 5, 6, 7, 8, 9, 10. The emphasized values in the tables represent solution values found by CPLEX, but which optimality were not proven within 2 h of computation time.

Table 5 Results on instances with $n=6$

Table 6 Results on instances with $n=7$

Table 7 Results on instances with $n=8$

Table 8 Results on instances with $n=10$

Table 9 Results on instances with $n=12$

Table 10 Results on instances with $n=14$