Optimizing the Sailing Route for Fixed Groundfish Survey Stations

Transcription

1 International Council for the Exploration of the Sea CM 1996/D:17 Optimizing the Sailing Route for Fixed Groundfish Survey Stations Magnus Thor Jonsson Thomas Philip Runarsson Björn Ævar Steinarsson Presented at the ICES 96 in Reykjavik, Iceland Sept Oct. 4. Abstract Groundfish surveys usually cover relatively large areas. The stations are numerous and often widespread. Determining the best sailing route between stations is not always obvious. An optimization of the sailing routes could reduce cost and research time. The exact sailing path is usually decided during the tour, since it may depend on factors such as weather and other vessels working the stations. Any means of assisting the captain in deciding which route to take should be valuable. The Groundfish Survey Problem (GSP) can be regarded as a variant of the Travelling Salesman Problem (TSP). The TSP is a combinatorial optimization which attempts to minimize the total distance travelled between cities. Unlike the TSP where the cities are entered and exited at the same location, the vessels may enter a station from either end of the towing path and so the number of possible solutions is significantly greater for the GSP. Over the last years Genetic Algorithms (GAs) have been used with some success to solve large TSPs where traditional methods fail or are too slow. Genetic Algorithms work with the coding of the problem rather than the problem itself. Here the GA approach is extended to include a new type of coding. The coding used borrows an idea from nature known as diploidy. Furthermore, the GA performance is enhanced using local search heuristics. A software solution utilizing the global optimization power of GAs, combined with local search has been implemented and tested. Department of Mechanical Engineering, University of Iceland, Hjardarhagi 2-6, 107 Reykjavik, Iceland. University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland. tpr@hi.is The Marine Research Institute, Skulagata 4, 105 Reykjavik, Iceland. 1

2 1 Introduction The condition of groundfish stocks are often explored by fishing with trawlers through a number of predefined stations. Finding the shortest path through a set of stations can be a difficult task and necessary in order to reduce cost and research time. The number of possible solutions is of the order 2 n n! where n is the number of stations to be visited. If a computer system is capable of producing solution every microsecond it takes years, to produce all solutions for 38 stations. The problem is similar to the classical Travelling Salesman Problem (TSP) with one exception: in the TSP the salesman enters and exits a city at the same location, but for the Groundfish Survey Problem (GSP) the vessel enters one location and exits at another, this is again similar to the Hamiltonian Path Problem (HPP) [1]. The GSP like the TSP is a NP-hard optimization problem and so using traditional search techniques such as the Branch-and-Bound Algorithm, it may with modern computer power still take months to find the optimal solution of relatively small problems. Due to limited computer power and time, heuristics have been used for large NP-hard problems, because they produce good solutions in a reasonable time. These methods are however, prone to get trapped in local optima. Evolutionary algorithms unlike heuristics have global convergence properties, but they usually don t utilize problem specific knowledge such as heuristic methods. Recently Frieselben and Merz [2] introduced a Genetic Algorithm (GA) for the TSP, which has shown a remarkable performance on standard benchmark TSPs. Unlike previous GAs their GA works only on local optima found using Lin-Kernighan heuristics [5], that is, the GA is used to find the global optima from a sample of local optima. The concept of working with population based local optima in order to find the true optima is utilized to create an algorithm which solves the GSP. The algorithm is described in chapter 2. In chapter 3 performance results on actual station data, used in the Spring Groundfish Survey conducted by the Marine Research Institute of Iceland are given. The paper concludes with a general discussion and summary. 2 A Local Search Genetic Algorithm Genetic algorithms (GAs) work with a population of individuals or chromosomes. Each individual represents a solution to a problem. The initial population is usually generated randomly. Parents are selected based on their fitness that is, how well they solve the problem. New individuals or offspring are created with a given probability by recombining the selected parents, this is known as crossover. The individuals are then mutated with a given probability, this introduces new genetic material and so guarantees global search. This process is repeated for a number of generations. For a more detailed description of Genetic Algorithms see Goldberg [3]. The limitations of the classical GA is that the algorithm does not use any problem specific knowledge. This has prompted researchers to introduce problem specific heuristics in order to improve their GA, which has been successfully done in [2], and is the basis of the algorithm described as follows. Each gene in the chromosome represents a station in the GSP. The gene order in the chromosome represents a sailing route. Each gene can take a positive of negative value, where the sign represents the towing direction for that station. This is analogous to diploidy found in nature. Using this formulation heuristics designed for the TSP are easily adapted to the GSP. The Local Search GA given in figure 1 (LS-GA) initializes the population using Nearest- Neighbor (NN) heuristics. This is followed by a modified LK improvement heuristic. The LK heuristic breaks iteratively random station links and reconnects them in a greedy manner in order to gain an overall improvement. Individuals are selected randomly but not based on their fitness, since they are all quite fit as they are locally optimal. When working with local optimum solutions premature convergence can be a problem. Diversity can however be maintained by replacing similar individuals by new randomly generated ones. 2

3 Initialize population P using Nearest-Neighbor heuristic foreach individual i do modified iterative LK heuristic repeat for number of crossovers do select two parents i a, i b P randomly i c = DP X(i a, i b ) perform modified LK heuristic on i c mutate i c using double bridge kick move for a given probability perform modified LK heuristic on i c replace an individual of P by i c end for until converged Figure 1: Pseudocode for the LS-GA. Offsprings are formed by crossing two selected parents using the Distance Preserving Crossover (DPX) operator [2]. The DPX finds fragments that are common to both parents and uses them to construct an offspring. The distance then between the offspring and the parents is equal. The offspring will not be a local optimum and so LK-heuristics are applied to it. The offspring is then mutated using the double bridge kick move [5] and by randomly rotating stations. The mutated individual is again improved using the LK-heuristic. The new offspring then replaces an inferior individual in the population. This process is repeated for a number of generations until convergence or a certain time has lapsed. 3 Implementation and Results The algorithm described in the previous section was implemented in C on a Pentium PC under Windows 95. The graphical user interface allows the user to easily select stations and constraints for a problem definition. Stations may have a fixed towing direction or be free to rotate. Fixing towing directions or possibly deciding which station must follow another, depends on factors such as weather and fishing conditions. Figure 2: The 38-station problem Two problems were formulated using fixed station data from the Spring Fishing Expedition conducted by the Marine Research Institute of Iceland. The first test problem contains 38 stations which normally would take less than a week to complete. The second problem is larger containing 100 stations which involves approximately two weeks of fishing. In all cases the towing directions are not fixed. Figure 2 shows the 38-station test problem. Included in the figure are the path s starting 3

4 (left mark) and finish location (right mark). From the figure it is easy to imagine that choosing the shortest path can be a difficult task for the captain. Figure 3: The solution found to the 38-station problem Ten independent runs for this problem were performed resulting each time in the same solution which is depicted in figure 2. The computation time was ten minutes on a Pentium 75. Figure 4: The 100-station problem The next problem is slightly larger as can be seen in figure 4. This problem includes the stations used in the previous example, which are in the north-west corner. This test problem was also run ten times and resulted in solutions which are within 2% of the best solution obtained. This percentage corresponds to approximately one hour of sailing between stations. The best found solution can be seen in figure 5. 4

5 4 Conclusions Figure 5: The best found solution to the 100-station problem We have presented a novel approach to the Groundfish Survey Problem. The approach is based on a combination of local search methods and a Genetic Algorithm. The new coding scheme simplifies the GSP significantly. The algorithm has produced high quality solutions in a reasonable time. GAs are a blackbox optimization method and so it is easy to include special requirements, such as fixing the towing directions for a station. A software implementation of the algorithm was used by one trawler in the last Spring Groundfish Survey with good results. Acknowledgment This work has been supported by the University of Iceland and the Marine Research Institute of Iceland, hereby gratefully acknowledged. References [1] L. M. Adleman. Molecular computation of solutions to combinatorial problems. Science, 266: , Nov [2] B. Frieselben and P. Merz. A genetic local search algorithm for solving symmetric and asymmetric travelling salesman problems. Proceedings of the 3rd IEEE International Conference of Evolutionary Computing, pages , May [3] D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison- Wesley, New York, [4] S. Lin and B. Kernighan. An effective heuristic algorithm for the travelling salesman problem. Operations Research, 21: ,