Rank-Based Ant Colony Algorithm For A Thermal Generator Maintenance Scheduling Problem

The maintenance scheduling of thermal generators is a large-scale combinatorial optimization problem with constraints. In this paper we introduce the Rank-Based Ant System algorithm based version of the Ant System. This algorithm reinforces local search in neighborhood of the best solution found in each iteration while implementing methods to slow convergence and facilitate exploration.Rank-Based Ant System (RBAS) algorithm has been proved to be very effective in finding optimum solution to hard combinational optimization problems . To show its efficiency and effectiveness, the proposed Rank-Based Ant System algorithm is applied to a real-scale system, and further experimenting leads to results that are commented. Key-Words: thermal generator maintenance scheduling problem; ant colony optimization; ant system,rank-based ant system algorithm 1.Introduction The Thermal Generator Maintenance Scheduling Problem is a complex multivariable problem that is necessary for the reliability and right operation of a generator system, given that the whole production cost is dependent on the maintenance and operation cost. Thus, the maintenance procedure has to be scheduled and complied with the best possible way, minimizing these two costs and at the same time, covering the energy demands, so as every constraint of the problem is satisfied. The problem has been studied in the past with a variety of modeling methods. The initial formulation was made by Gruhl [1], [2]. He presented an umbrella of scheduling problems, one of which was the generator maintenance scheduling problem, with a linear approach. Two years later, Dopazo and Merill [3] developed a model which was claimed to have the ability of finding the best solution, but this approach was lacking in real-scale problems application, something that Zurn and Quintana [4] later achieved to do using computational methods. In 1983, Yamayee and Sidenblad [5] improved the cost function that was used till then, with great improvements in execution time. In 1991, Satoh and Nara [6] applied for the first time a stochastic method, called Simulated Annealing with very good results in large-scale systems as well, that were impossible to be solved with linear methods in the past. They also investigated the problem with genetic algorithms [7] and tabu-list methods [8] with similar results, but with the ability to solve realscale problems, too. In 1993, Charest and Ferland [9] tried to modify the linear method with successful results in execution time, while Dahal and McDonald [10] applied a genetic algorithm in Boolean representation [11] which had WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Aristidis Vlachos E-ISSN: 2224-266X 273 Issue 9, Volume 12, September 2013 also some good results. In 1997, Burke and Smith [12] tried to create a hybrid model of the simulated annealing and the tabu-list method without success, following another attempt to make another hybrid model with memetic and tabu-list methods three years later, which resulted in better results, but with a small increase in execution time. In 2010 Y.Yare.,G.K.Venayagamoorthy [13] using multiple swarms-MDPSO framework with good results for Optimal maintenance scheduling of generators problem. In 2011, Saraiva, Pereiva, Mendes, and Sousa [14] solved the generator maintenance scheduling problem using a simulated annealing algorithm. In this paper, we introduce Rank-Based Ant System algorithm [15], an imported version of basic Ant System [16] of the family algorithms: Ant Colony Optimization (ACO) [17], which was inspired by the observation of ant colonies. 2.Ant Colony Optimization 2.1 Generally Analogy The Ant Colony Optimization (ACO) [18] is a metaheuristic to solve combinatorial optimization problems, is motivated by the behavior of real ant colonies. When ants attempt to find short paths between their nest and food sources, they communicate indirectly by using pheromone (pheromone trail) to mark the decisions they made when building their respective paths. Within ACO algorithms, the optimization problem is represented as a complete weighted graph G = (N,A) with N being the set of nodes and A the set of edges fully connecting the nodes N. In the Travelling Salesman Problem (TSP) application, edges have a cost associated (e.g. their length) and the problem is to find a minimal-length closed tour that visits all the nodes once and only once. In order to solve the problem, random walks of a fixed number of ants through the graph take place. The transition probabilities of each ant are governed by two parameters associated to the edges of the graph: the pheromone values (or pheromone trail) τij, representing the learned desirability of choosing node j when in node i. inverse of the distance between two nodes i and j: 1 ij ij n = d where ij d is the distance between these two nodes. The more distinctive feature of ACO is the management of pheromone trails that are used, in conjunction with the objective function, to construct new solutions. Informally, the pheromone trails are used for exploration and exploitation. Exploration representing the probabilistic choice of the components used to construct a solution. A higher probability is given to elements with a strong pheromone trail. Exploitation is based on the choice of the component that maximizes ablend of pheromone-trail values and partial objective function evaluations. The mathematical formulations of the ACO algorithms presented in this paper named Ant System (AS) and Max-Min Ant System (MMAS), are given in the following sections. 2.2 Ant System Ant System (AS) [19] is the original and most simplistic ACO algorithm. The decision policy used within AS is as follows: The probability with which ant k, currently at node i, chooses to go to node j is given [16] by: (1) [ ] [ ] ( ) 1 α β ij ij k ij α β ιl il k l Ji τ (t) n p (t)= τ (t) n ∈     ⋅     ⋅ ∑ If j є k i J and 0 if k i j J ∉ WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Aristidis Vlachos E-ISSN: 2224-266X 274 Issue 9, Volume 12, September 2013 : k i J is the feasible neighborhood of ant k, that is, the set of nodes which ant k has not yet visited. ij τ (t) : is the concentration of pheromone associated with edge (i,j) in iteration t. : ij n is the inverse of the length of the edge known as visibility α and β: are parameters that control the relative importance of pheromone intensity versus visibility Upon conclusion of an iteration (i.e. each ant has generated a solution) the pheromone on each edge is updated, according to the following formula: (2) ( ) 2 ij ij ij τ (t)= ρτ (t)+ Δτ (t) Where ρ is the coefficient representing pheromone persistence (0 ≤ ρ < 1), and ij Δτ , is a function of the solutions found at iteration t, given by: (3) ( ) 1 3 n k ij ij k= Δτ = Δτ (t) ∑ n: number of ants : k ij Δτ is the quantity per unit of length of pheromone addition laid on edge (i,j) by the k ant at the end of iteration t, is given by: Where k T (t) is the tour done by ant k at iteration t, k L (t) , is its length and Q is a constant parameter, used for defining to be of high quality solutions with low cost. (4) ( ), if ( , ) ( ) ( ) (4) 0, if ( , ) ( ) k k k ij k Q L t i j T t t i j T t τ   ∈ ∆ =   ∉  2.3 Rank-Based Ant System Algorithm (Bullnheimer,Hartl and Strauss) [15],proposed the RBAS algorithm which is a modification of the AS algorithm. The RBAS algorithm uses a rank idea to update the pheromone trail. The RBAS search mechanism can be divided into : (1) Initialization. (2) Transition rule. (3) Pheromone update rule. (1) Initialization. The pheromone initialization strategy is to initialize the pheromone matrix of RBAS algorithm. As elements using a constant value [19,20,21]: (5) ( ) 0 5 ij τ τ , i, j ← ∀ (2) Transition rule. Let denote for each arc (i,j) in the TSPinstance graph a heuristic value and pheromone value. Ant k at node I chooses next node j according the following transition rule .