Ant Colony Optimization for the Ship Berthing Problem

Ant Colony Optimization (ACO) is a paradigm that employs a set of cooperating agents to solve functions or obtain good solutions for combinatorial optimization problems. It has previously been applied to the TSP and QAP with encouraging results that demonstrate its potential. In this paper, we present FF-AS-SBP, an algorithm that applies ACO to the ship berthing problem (SBP), a generalization of the dynamic storage allocation problem (DSA), which is NP-complete. FF-AS-SBP is compared against a randomized first-fit algorithm. Experimental results suggest that ACO can be applied effectively to find good solutions for SBPs, with mean costs of solutions obtained in the experiment on difficult (compact) cases ranging from 0% to 17% of optimum. By distributing the agents over multiple processors, applying local search methods, optimizing numerical parameters and varying the basic algorithm, performance could be further improved. 1 Ant Colony Optimization The Ant Colony Optimization (ACO) paradigm was introduced in [1], [2] and [3] by Dorigo, Maniezzo and Colorni. ACO has been applied effectively to the traveling salesman problem (TSP) [4] and the quadratic assignment problem (QAP) [5], among several other problems. The basic idea of ACO is inspired by the way ants explore their environment in search of a food source, wherein the basic action of each ant is: to deposit a trail of pheromone (a kind of chemical) on the ground as it moves, and to probabilistically prefer moving in directions with high concentrations of pheromone deposit. As an ant moves, the pheromone it leaves on the ground marks the path that it takes. Another ant that passes by later can detect the pheromone and decide to follow the trail with high probability. If it does follow the trail, it leaves its own pheromone on it, thus reinforcing the existing pheromone deposit. By this mechanism, the movement of ants along a path between the nest and the food reinforces the pheromone deposit on it, and this in turn encourages further traffic along the path. This behavior characterized by positive feedback is described as autocatalytic. P.S. Thiagarajan, R. Yap (Eds.): ASIAN’99, LNCS 1742, pp. 359–370, 1999. c © Springer-Verlag Berlin Heidelberg 1999 360 Chia Jim Tong, Hoong Chuin Lau, and Andrew Lim On the other hand, ants may take a direction other than the one with the highest pheromone concentration. In this way, an ant does not always have to travel on the path most traveled. If an ant takes a path less traveled that deviates slightly from a popular path, and also happens to be better (shorter) than other popular paths, the pheromone it deposits encourages other ants to also take this new path. Since this path is shorter, the rate of pheromone deposit per ant that travels on it is higher, as an ant traverses a shorter distance in one trip. In this way, positive feedback can occur on this path and it can start to attract ants from other paths. By the interplay of these two mechanisms, better and better paths emerge as the exploration proceeds. For the purpose of designing an algorithm based on this idea drawn from nature, an analogy can be made of: 1) real ants vs. artificial agents, 2) ants’ spatial environment vs. space of feasible solutions, 3) goodness of a given path vs. objective function of a given solution, 4) desirability of taking a particular direction vs. desirability of making a certain decision in constructing the solution, 5) real pheromone at different parts of the environment vs. artificial pheromone for different solution choices. One of the main ideas behind ACO algorithms is how relatively simple agents can, without explicit communication, cooperate to solve a problem by indirect communication through distributed memory implemented as pheromone. In this paper, we study how ACO can be applied effectively to the ship berthing problem (SBP), through the FF-AS-SBP algorithm, an application of ACO to the SBP. The focus of this study is not on the SBP itself or on finetuning our algorithm for maximum performance. Rather, it is on demonstrating that ACO can be applied effectively to the SBP. In Section 2, we formally describe the SBP. In Section 3, we describe a candidate solution representation, from which we adapt an indirect, first-fit (FF), solution approach in Section 4 so that it becomes more suitable for the complex nature of the SBP. In this section, we also describe a randomized FF algorithm and the basis of FF-AS-SBP. FF-AS-SBP is described in Section 5. By naming the algorithm FF-AS-SBP, we acknowledge that there could be many other ACO algorithms for the SBP. In Section 6, we describe the experiment and report and interpret the results, comparing FF-AS-SBP against the randomized FF algorithm. In this section, we also discuss how results could be further improved, how the algorithm lends itself to parallelization, and possible future work. Finally, Section 7 is the conclusion. 2 The Ship Berthing Problem This problem, which has been studied in [6] and [7], can be defined as follows: ships (S = {Si: i = 1, 2, . . . , n}) are specified to have lengths li, arrive at a port at specified times ti and stay at the port for specified durations di. Each ship that arrives is to be berthed along a wharf line of length L, i. e., it is placed at the interval (bi, bi + li) along the wharf line. Once berthed, its location is fixed for the entire duration of its stay. Also, each ship has a minimum interAnt Colony Optimization for the Ship Berthing Problem 361 ship clearance distance ci and a minimum end-berth clearance distance c b i . Four types of constraints apply: – Ships can only be berthed within the extend of the wharf line. No part of any ship can extend beyond the beginning or the end of the wharf line. More strongly, the distance from either end of a ship to either end of the wharf line cannot be less than the minimum end-berth clearance distance. ∀i ∈ {1, 2, . . . , n} ci ≤ bi ≤ L− li − ci – No two ships can share the same space along the wharf line if the time intervals in which they are berthed intersect. More strongly, the end-to-end distance between them cannot be less than the minimum inter-ship clearance of either one of them. ∀i, j ∈ {1, 2, . . . , n} (ti, ti + di) ∩ (tj , tj + dj) 6= ∅ → (bi −max{csi , csj} , bi + li + max {csi , csj}) ∩ (bj , bj + lj) = ∅ – A ship may be given a fixed berthing location (bi is fixed for some values of i). – A ship may be prohibited from berthing in certain intervals of the wharf line. More precisely, the interval bounded by the location of the two ends of a ship after it has been berthed cannot intersect with any of the prohibited interval. (bi, bi + li) ∩ (p, q) = ∅ if constraint applies to Si, where (p, q) is some forbidden interval The minimization version of the problem is to determine Lo the minimum length of the wharf line needed to berth all the ships subject to the given constraints. The decision version of the problem is to determine whether berthing is possible, given a fixed value of L. The density D is defined as the maximum total length of berthed ships at any one time : D = max t∈(−∞,+∞)   ∑ i∈{i:ti≤t<ti+di} li   It is easy to see that D is a tight lower bound on L. In this paper, we also define a measure F , which we call the fragmentation, defined as: F = 1− ∑ dili (max(ti + di)−min(ti))D The berthing scenario can be visualized as a 2-D plane where the x-axis represents time, the y-axis represents space (along the wharf line), and each ship 1 For convienience, this definition ignores minimum end-berth and inter-ship clearance 362 Chia Jim Tong, Hoong Chuin Lau, and Andrew Lim