# Using Ant Colony Optimization for Infrastructure Maintenance Scheduling

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2 The monetary objective is to steady the budget flow over the years. Costs can also be decreased by adjusting the maintenance schedule with the maintenance work of third parties also using the building such as public services and public transport organizations. Additional constraints exist which are known to the planner but are too complicated to be modeled in the planning software. So the planning tool should give as result a set of solutions that satisfy the given constraints for the planner to choose from. 3 APPROACH 3.1 Ant Colony Optimization As for the described problem there exists no algorithm to find an exact solution in reasonable time it has to be solved approximately, e.g. by metaheuristics. We are using ant colony optimization (ACO) to construct the schedules, as ACO has performed very well at similar problems (e.g. Lee 2009). ACO is a constructive metaheuristic developed by Dorigo (1992). It is based on the behavior of natural ants that are able to find the shortest way between their nest and a source of food by communicating via a chemical substance called pheromone. Ants are depositing pheromone while walking and prefer routes were more pheromone is present. So this positive feedback guides the majority of ants to use only one path after some time (compare e.g. Dorigo & Stützle 2004; Boysen). The first problem on which ACO was applied was the Traveling Salesman Problem (TSP, Dorigo 1992). Meanwhile ACO has been applied to many other problems, e.g. the bi-criteria TSP (García- Martínez et al. 2004), the Vehicle Routing Problem (Barán & Schaerer 2003; Reimann et al. 2004), the Quadratic Assignment Problem (Stützle & Dorigo 1999), Timetabling Problems (Socha et al. 2002), Construction Site Planning (Ning et al. 2010), and Scheduling Problems (Merkle et al. 2002; Solimanpur et al. 2005; Christodoulou 2010). For an overview see Dorigo & Stützle The basic design of the algorithm is shown in Figure 1. Figure 1. Algorithm 3.2 Objective function Our goal is to minimize the impact on traffic. This can be described by the waiting time produced by establishing a set of working zones that will lead to a closure of some roads. The waiting time is measured by comparing the travel times in an undisturbed net at rush hour to the scenarios set by the maintenance schedule. For each schedule containing n years, n scenarios (each with a number of roads closed at the same time) have to be considered. As evaluation value we choose the highest waiting time produced in one of the n scenarios of an n-year schedule. Another possibility could be to minimize the sum of produced waiting time over all n years. 3.3 Problem model To solve the maintenance scheduling problem by ACO we are structuring it as a graph (see Figure 2). To construct a schedule the ants are moving through the graph from left to right. The vertical layers of the graph represent the years. In each year the ants can chose from a number of buildings that have to be maintained, that is the rows of the graph.

4 same building more than once during the period under consideration. Then once maintained buildings have to be still available for choice but their condition index respectively their maintenance deadline and therefore their η has been changed. This change in condition only exists for the one ant team that has scheduled the maintenance of the building, though. So for each team there has to be maintained its own η-matrix that changes during schedule construction and has to be reset after each iteration. 3.6 Pheromone update After all ant teams have constructed full schedules, i.e. each ant of each team has chosen a building for each year of the period under consideration, the pheromone traces on the graph are updated. First some pheromone evaporates on all nodes to enable the algorithm to forget bad solutions. This is done by decreasing the amount of pheromone by a factor ρ with 0 < ρ 1. Following Dorigo & Stützle (2004) we choose ρ = 0.5. After this all ant teams deposit pheromone on all the nodes belonging to their respective schedules. The amount of pheromone Δτ k deposited by each team k is computed as the inverse square of the highest waiting time produced over the period under consideration in the respective schedule for each node that belongs to this schedule. It is 0 for all nodes that are not part of the schedule k. Also ant teams that constructed infeasible solutions, e.g. solutions that violate the security or the budget constraints, are allowed to deposit pheromone to prevent premature convergence and to exploit the information contained in those solutions. To avoid convergence to such infeasible solutions we are using the Elitist Ant (EA) approach (see also Dorigo & Stützle 2004). In the so called elitist ant the best solution found so far is stored. As long as no better solution is found this elitist ant in each iteration deposits pheromone even if its solution is not part of the set of solutions found in this iteration. The influence of this elitist ant can be weighted by a factor e. So the overall pheromone update is computed as follows: k EA ( 1 ) e (2) k Dorigo & Stützle propose e to be the same as the number of nodes (buildings). As we have teams of ants, e better is chosen same as the number of ant teams. Originally EA has been developed to guide the ants away from the greedy solutions, i.e. the solutions with to high influence of the heuristic information η, of the first few iterations. As in our approach η and τ describe different optimization goals the danger of such greedy solutions is not given. By not allowing infeasible solutions to become the solution of the elitist ant the EA approach is a useful tool to get rid of those solutions while still exploiting possible useful information contained in them. 3.7 Next iteration and termination After the pheromone update the next iteration step starts. The memories of the ant teams respectively the changes to the η-matrix are reset. The construction of the schedules now orients on the new values of the pheromone trails. The algorithm terminates when over a given number of iterations the iteration-best solution does not change. 4 EXPERIMENTAL RESULTS The algorithm described before has been implemented for a simplified model of the problem to test its performance. On the one hand simplifications have been made in the deterioration function: For each building was set the same deterioration function where the condition index is only dependent on the age of the building. Maintenance on a building resets its condition index to 1. Further there is no coupling with a traffic simulator, yet. To model the impact on traffic to each building fictitious values of waiting time produced by a work zone at this building were assigned. The waiting time per building is set to a value between 100,000 and 400,000 units. At a parallel maintenance of some buildings their respective waiting time values are summed up linearly. This of course is far from reality but works well as a simple surrogate model. For the first test only security and budget constraints were considered. Schedules not fulfilling those constraints are set infeasible. The test case is a set of 100 buildings with randomly assigned ages. Per year 5 buildings can be maintained, so the worst possible schedule produces a highest waiting time of 2,000,000 units. The period under consideration is 5 years. Without maintenance after the five years 3 buildings reach condition index 6 (i.e. are likely to collapse). First tests show that the algorithm converges very fast in about 20 iterations to a feasible solution. The solutions found produce a highest waiting time of 900,000 to 1,000,000 units. This is far better than a randomly constructed schedule that produces in average a highest waiting time of 1,500,000 units. 5 CONCLUSIONS Maintenance scheduling is an optimization problem subject to many constraints. With the algorithm

5 shown in this paper near optimal solutions can be found. The algorithm is based on Ant Colony Optimization. Modifications have been made to model parallel maintenance work and to guide the search in the first iteration step without knowing the benefit of certain choices to the objective function. A first implementation could show that the algorithm is able to find good solutions. A next step will be the coupling with a traffic simulator to get realistic values for the produced waiting time. Future studies could further improve the performance of the algorithm by testing different settings of the ACO-parameters α, β and ρ. Further adaptations, should make the algorithm fit even better to the needs of building managers. Solimanpur, M., Vrat, P. & Shankar, R An ant algorithm fort he single row layout problem in flexible manufacturing systems, Computers and Operations Research 32: Stützle, T. & Dorigo, M ACO algorithms for the Quadratic Assignment Problem, New Ideas in Optimization, REFERENCES Barán, B. & Schaerer, M A multiobjective ant colony system for vehicle routing problem with time windows, Proceedings of the 21 st IASTED International Conference Applied Informatics: Boysen, N. no year. Ameisenalgorithmen, Hamburg: rithmen.pdf. Christodoulou, S Scheduling Resource-Constrained projects with Ant Colony Optimization Artificial Agents, Journal of Computing in Civil Engineering January/February 2010: Dorigo, M Optimization, learning and natural algorithms. Milan: Politecnico di Milano. Dorigo, M. & Stützle, T Ant Colony Optimization. Cambridge: The MIT Press. García-Martínez, C., Cordón, O. & Herrera, F. 2004, An empirical analysis of Multiple Objective Ant Colony Optimization Algorithms for the Bi-criteria TSP, Proceedings of the 4 th International Workshop on Ant Colony Optimization and Swarm Intelligence: Lee, H.-Y Optimizing schedule for improving the traffic impact of work zone on roads, Automation in Construction 18: Merkle, D., Middendorf, M. & Schmeck, H Ant Colony Optimization for Resource-Constrained Project Scheduling, IEEE Trans. Evolut. Comput. 6, 4: Ning, X., Lam, K.-C. & Lam, M. C.-K. 2010, Dynamic construction site layout planning using max-min ant system, Automation in Construction 19: Reimann, M., Doerner, K. & Hartl, R. F. 2004, D-Ants: Savings Based Ants divide and conquer the vehicle routing problem, Computers & Operations Research 31: Schießl, P. & Mayer, T Lebensdauermanagement von Stahlbetonbauwerken, Statusseminar zum Verbundforschungsvorhaben Nachhaltig Bauen mit Beton : Schießl, P. & Mayer, T Forschungsbericht 31006/05, DAfStB-Verbundforschungsvorhaben Nachhaltig Bauen mit Beton, Teilprojekt A3 Wesentliche Bausteine eines Bauwerksmanagementsystems. Berlin: Deutscher Ausschuss für Stahlbetonbau (DAfStb), Heft 572. Socha, K., Knowles, J., Samples, M A MAX-MIN Ant System for the University Course Timetabling Problem, Proceedings of the 3 rd International Workshop on Ant Colony Optimization and Swarm Intelligence: 1-13.

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