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An adaptation of a parametric ant colony optimization (ACO) to multi-objective optimization (MOO) is presented in this paper. In this algorithm (here onwards called MACO) the concept of MOO is achieved using the reference point (or goal vector) optimization strategy by applying scalarization. This method translates the multi-objective optimization problem to a single objective optimization problem. The ranking is done using ϵ-dominance with modified Lp metric strategy. The minimization of the maximum distance from the goal vector drives the solution close to the goal vector. A few validation test cases with multi-objectives have been demonstrated. MACO was found to out perform R-NSGA-II for the test cases considered. This algorithm was then integrated with a meshless computational fluid dynamics (CFD) solver to perform aerodynamic shape optimization of an airfoil. The algorithm was successful in reaching the optimum solutions near to the goal vector on one hand. On the other hand the algorithm converged to an optimum outside the boundary specified by the user for the control variables. These make MACO a good contender for multi-objective shape optimization problems.

There are a few global optimization algorithms that are based on natural processes. Genetic algorithms (GA) are based on Darwin’s theory of natural genetics on one hand. On other hand ant colony algorithms are based on insect behavior usually called swarm algorithms. Simulated annealing is based on physical process. The swarm algorithms are based on collective intelligence, defined as the ability of a group to solve problems more efficiently than its individuals [5]. GA and its variants [7], [14] and [25] have been extensively used for multi-objective optimization (MOO) problems in last few decades. Ant colony optimization (ACO) is a meta-heuristic based global optimization technique introduced by Dorigo [8] and [9] and has proved itself in field of combinatorial optimization problems. Many variants of Ant colony algorithms have been reported in recent few years for combinatorial multi-objective optimization problems [19]. One of the approaches uses multiple ant colonies with exchange of information between them [17]. Cardoso et al. [5] have used a single ant colony with cost vector which was associated to multi-level pheromone trails to solve a multi-objective network ant colony optimization (MONACO) problem. Another variant, the Max-Min ant system [19] was applied in a production process to minimize lead time as well as work required. Garcia-Martinez et al. [11] proposed a taxonomy for ACO algorithms along with an empirical analysis for bi-criteria TSP problem. The automatically configured algorithm by López-Ibáñez and Stützle [12] is reported to outperform the MOACO. Alaya et al. [2] proposed a generic algorithm based on ACO to solve MOO problems. Angus [4] extends the ACO algorithm with a crowding population replacement scheme to increase the search efficiency. Abbaspour et al. [1] has extended ACO to parametric optimization using the route of inverse modeling and named it as ACO-IM. The authors have earlier demonstrated shape optimization capability of ACO-IM when coupled with computational fluid dynamics (CFD) solver using a single objective function [20], [22] and [23]. Authors earlier work on aerodynamic shape optimization using GA [21] using a single objective shows that ACO-IM out performs GA in certain situations [20]. The advantage of ACO-IM method is that, it facilitates the movement of domain space of the variables. Hence the initial domain for the variables defined by the user does not need to contain the final optima [22]. Fainekos and Giannakoglou [10] have demonstrated airfoil optimization using extended ACO (EACO). This algorithm optimizes the path between pairs of control variables using a local criterion over and above the global pheromone based criterion. It is not always possible to define a local criterion for an optimization problem. The movement of domain space of variables is restricted in EACO. Also the best accuracy of final solution is bounded by the user-defined limits and discretization done for each control variables. Discretization is performed once at the beginning and remains same throughout the optimization process. A control variable cannot take values between these discretized values. In contrast, in ACO-IM the discretization is done in all iterations with new bounds for the variables. This process improves the accuracy of the solution as optimization progresses. Definitely for a general optimization problem (like the problems dealt in this paper) ACO-IM is superior to EACO. In this paper the concept of multi-objective optimization using ACO-IM (MACO) is proposed. It is then implemented on a problem of aerodynamic shape optimization with multiple objectives. Next section discusses the concept of MOO in reference to ACO-IM. Section 3 demonstrates the test Cases 1 and 2. Section 4 elaborates the integration of meshless CFD solver with MACO along with results and discussion. Section 5 details conclusion along with future scope.

The parametric ant colony optimization was successfully extended to multi-objective problems. The demonstration of the various test cases shows the robustness of the scheme. MACO was found to outperform MGA for the test cases considered. ϵ-dominance based ranking was able to capture the optimum solution nearest to the goal vector. The SLKNS CFD solver was successfully used along with MACO algorithm to obtain aerodynamic optimal shape of airfoil with multiple objectives. MACO based shape optimizer is a good contender in multi-objective search space as the initial guess for domain of control variables, defined by the user does not need to contain the final optima. The optimum obtained View the MathML sourcex→☆∈X⊂Rn is a strong function of parameterization. Implementing adaptive scalarized ϵ-dominance methodology and development of better parameterization would fall in future scope of work.