Abstract

The research outlined in this paper aims the development of a methodology to arrive at critical path calculations in construction networks using Ant Colony Optimization (ACO) algorithms. Ant Colony Optimization is a population-based, artificial multiagent, general-search technique for the solution of difficult combinatorial problems. The method's theoretical roots are based on the behaviour of real ant colonies and the collective trail-laying and trail-following of its members in searching for optimal solutions in traversing multiple paths. In essence, ACO is inspired by the foraging behaviour of natural ant colonies which optimize their path from an origin (ant nest) to a destination (food source) by taking advantage of knowledge acquired by other ants that previously traversed the possible paths. In computer implementations of the ACO algorithms, artificial ants are both agents and solution-construction procedures that stochastically build solutions by considering (1) artificial pheromone trails which change dynamically at run time to reflect the agents' acquired search experience, and (2) heuristic information on the problem/network being solved. The paper outlines the fundamental mathematical background of the ACO method and a suggested possible implementation strategy for solving for longest (critical) paths in construction schedule networks. Keywords: ant colony optimization, construction scheduling, critical path. INTRODUCTION Currently, the most widely used method for construction scheduling is the critical path method (CPM) which, in essence, is based on forward and backward passes of the activity network to arrive at longest path and total duration calculations. The method provides planners and constructors with a formalized way of calculating the earliest and latest dates each construction activity can start and finish at, subject to the envisioned network topology (activity relationships) and the imposed time, resource and logic constraints. Undoubtedly, over the years CPM has grown in functionality, accessibility and acceptability. Yet, some limitations of the method have recently given rise to the search for complimentary methodologies that could address some of either the perceived limitations or the desired additional features. Among the most notable limitations are: (i) the inability of CPM-based tools to calculate longest (or shortest) paths from a node to any node, (ii) the inability of CPMbased tools to account for resource-driven activity relationships ("AND"/"OR" combinations of resources), and (iii) the computational inefficiency of the critical path method. Item (i) refers to the need to identify the longest (or shortest) path to a desired activity from any other activity in a project network. This feature would enable constructors identify the paths to start (or completion) of any desired activity from any other activity in the network and act accordingly to recover or accelerate the project and to redirect resources in the most efficient manner. Similarly, this added feature would enable planners and managers interested at successful timely completion of specific activities identify the longest, or shortest, path to that activity and mediate associated risks as needed. Item (ii) refers to the need to account for resource-based activity relationships. In this scenario, tasks are performed and schedule advancement is made upon additive or conditional combinations of resources. Even though CPM schedules can nowadays be "resource-loaded", so that resources are accounted for during the network calculation phase, such schedules do not enable accounting for real-life situations where activity sequencing is based on resource (rather than activity) relationships. In other words, how can a planner account for the situation where an activity can start only when a resource from a predecessor activity is freed and then combined with a resource from another predecessor activity to enable the start of the successor activity without completion of the two predecessor activities? Item (iii) refers to the perceived computational inefficiency of the traditional CPM algorithms. If one considers that solution of the activity network can be achieved by solving the underlying equations which represent the activity interrelationships, then possible solution methodologies could include solution of linear equations, linear programming, and forward/backward passes. The first two approaches imply that the applied solver is capable of handling a large number of equations and constraints, as well as optimization. Despite that scheduling problems within certain categories in general have polynomial-type exact solution algorithms and their solution is simple, in its most general form, though, resource-constrained scheduling problems are "NPhard", meaning that there are no known algorithms for finding optimal solutions in polynomial time. This class of problems (NP-class) requires computational power that increases exponentially with the size of the problem and has no exact solution. In such cases a heuristic approach is warranted. The last approach implies exhaustive enumeration of the possible paths in a project network, and calculation through network-traversing. In essence, calculation in this case is achieved by starting from the first activity and exhaustively identifying all possible paths (successive activities) until reaching the last activity, adding the duration of each link to the total duration of the identified path up to that point (EarlyStart, EarlyFinish dates) and then identifying the longest path (forward pass). A reverse pass (same exhaustive approach) is used to calculate the latest dates each activity can start and finish subject to keeping the project end date fixed (as calculated during the forward pass phase). The combination of the two network passes provides the TotalFloat of each activity (LateStartEarlyStart, or LateFinish -EarlyFinish) and therefore the critical path (critical are the activities with TotalFloat = 0). In summary, despite the methods' relative ease of application they possess inefficiencies (i.e. the exhaustive enumeration of network paths) and computational constraints (i.e. the maximum number of equations and inequalities the applied solver can solve). Construction Scheduling with Artificial Agents and the ACO Metaheuristic 775 ANT COLONY OPTIMIZATION Introduction Ant Colony Optimization (ACO) is a population-based general search technique inspired by the foraging behaviour exhibited by real ant colonies. The method was first proposed by The underlying methodology is modelled after the behaviour exhibited by real-life ant colonies as they search for optimal solutions in node-to-node path traversing situations, during which a shortest path in a static or dynamic topology is sought. The behaviour exhibited by such ants is characterized by a reinforcement mechanism that helps steer succeeding ants to the most frequently previously traversed path. In particular, an ant randomly traversing possible paths can help find the shortest path between food sources and a nest and in doing so deposit a chemical substance called "pheromone", forming "pheromone trails" which can then be followed by other ants in the colony. When choosing their way through the possible path routes, ants smell the deposited pheromone and tend to follow those paths marked by stronger pheromone concentrations. Therefore, while an isolated ant moves essentially at random, an ant encountering a previously traversed path and pheromone-laid trail can detect such, decide with high probability to follow it and subsequently reinforce the trail with its own pheromone. The collective behaviour is therefore characterized by a positive (reinforcing) feedback loop where the probability with which each ant chooses the path to follow increases with the number of ants having chosen the same path in the preceding steps. The pheromone trail is reinforced with each successive pass until the ant population and path traversing converge to the shortest path between source and destination, and the final result is the relatively quick convergence of the path-traversing to the shortest path. Theoretical Framework of the ACO Metaheuristic A number of ACO algorithms, starting from the original work by The generic problem topology was outlined by • Ants build solutions by moving on the construction graph G=(C, R), where C is the set of components in the network, and R is the set of relationships (connections) fully connecting the components. Even though both feasible and infeasible solutions can be built, artificial ants, in general, try to build feasible solutions. The problem constraints, Ω, are implemented during the networktraversing and the policy followed by the artificial ants. • The components, ci ⊆ C, and connections, rij ⊆ R, can have a pheromone trail, τ, associated with them which allows for the implementation of a long-term memory policy about the ant search process. Similarly, the components, ci ⊆ C, and connections, rij ⊆ R, can have a heuristic value, η, which allows incorporation of problem-specific information. • The path that each artificial ant, k, follows can be stored in the ant's memory M k .