IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 4, AUGUST 2014 503 Ant Colony Optimization for Mixed-Variable Optimization Problems Tianjun Liao, Krzysztof Socha, Marco A. Montes de Oca, Thomas St¨ utzle Senior Member, IEEE, and Marco Dorigo, Fellow, IEEE Abstract —In this paper, we introduce ACO MV : an ant colony optimization (ACO) algorithm that extends the ACO R algorithm for continuous optimization to tackle mixed-variable optimization problems. In ACO MV , the decision variables of an optimization problem can be explicitly declared as continuous, ordinal, or categorical, which allows the algorithm to treat them adequately. ACO MV includes three solution generation mechanisms: a con- tinuous optimization mechanism (ACO R ), a continuous relaxation mechanism (ACO MV -o) for ordinal variables, and a categorical optimization mechanism (ACO MV -c) for categorical variables. Together, these mechanisms allow ACO MV to tackle mixed- variable optimization problems. We also deﬁne a novel procedure to generate artiﬁcial, mixed-variable benchmark functions, and we use it to automatically tune ACO MV ’s parameters. The tuned ACO MV is tested on various real-world continuous and mixed- variable engineering optimization problems. Comparisons with results from the literature demonstrate the effectiveness and robustness of ACO MV on mixed-variable optimization problems. Index Terms—Ant colony optimization, artiﬁcial mixed- variable benchmark functions, automatic parameter tuning, en- gineering optimization, mixed-variable optimization problems. I. Introduction M ANY REAL-WORLD optimization problems can be modeled using combinations of continuous and discrete variables. Due to the practical relevance of these mixed- variable problems, a number of optimization algorithms for Manuscript received October 17, 2012; revised February 20, 2013 and May 31, 2013; accepted July 22, 2013. Date of publication September 11, 2013; date of current version July 29, 2014. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) by ERC Grant No. 246939, in part by the Meta-X project from the Scientiﬁc Research Directorate of the French Community of Belgium. Thomas St¨ utzle and Marco Dorigo acknowledge support from the Belgian F.R.S.- FNRS, of which they are a Research Associate and a Research Director, respectively. Tianjun Liao acknowledges a fellowship from the China Scholarship Council. T. Liao is with the Institut de Recherches Interdisciplinaires et de D´ eveloppements en Intelligence Artiﬁcielle (IRIDIA), Université Libre de Bruxelles, Brussels 1050, Belgium, and also with the State Key Laboratory of Complex System Simulation, Beijing Institute of System Engineering, China (e-mail: tliao@ulb.ac.be). K. Socha is with the CERT-EU Pre-Conﬁguration Team, European Com- mission, Brussels 1040, Belgium. M. A. Montes de Oca is with the Department of Mathematical Sciences, University of Delaware, Newark, DE 19716 USA. (e-mail: mmontes@math.udel.edu). T. Stützle, and M. Dorigo are with the Institut de Recherches Interdisci- plinaires et de Développements en Intelligence Artiﬁcielle (IRIDIA), Univer- sité Libre de Bruxelles, Brussels 1050, Belgium (e-mail: stuetzle@ulb.ac.be; mdorigo@ulb.ac.be). Digital Object Identiﬁer 10.1109/TEVC.2013.2281531 tackling them have been proposed. These algorithms are mainly based on genetic algorithms [1], differential evolu- tion [2], particle swarm optimization [3], and pattern search [4]. The discrete variables in these problems can be ordinal or categorical. Ordinal variables exhibit a natural ordering rela- tion (e.g., integers) and are usually handled using a continuous relaxation approach [5]–[12]. Categorical variables take their values from a ﬁnite set of categories [13], which often identify non-numeric elements of an unordered set (e.g., colors, shapes or types of material). Categorical variables do not have a natural ordering relation and therefore require the use of a categorical optimization approach [13]–[19] that assumes no ordering relation. To the best of our knowledge, the approaches to mixed-variable problems available in the literature are targeted to either handle mixtures of continuous and ordinal variables or mixtures of continuous and categorical variables. In other words, they do not consider the possibility that the formulation of a problem may involve, at the same time, the three types of variables. Hence, there is a need for algorithms that allow the explicit declaration of each variable as either continuous, ordinal, or categorical. In this paper, we extend an ant colony optimization al- gorithm for continuous optimization (ACO R ) [20] to tackle mixed-variable optimization problems. Ant colony optimiza- tion (ACO) was originally introduced to solve discrete op- timization problems [21]–[23], and its adaptation to solve continuous or integer optimization problems has reveived in- creasing attention [20], [24]–[29]. Our ACO algorithm, called ACO MV , allows the user to explicitly declare each variable of a mixed-variable optimization problem as continuous, or- dinal, or categorical. Continuous variables are handled with a continuous optimization approach (ACO R ), ordinal variables are handled with a continuous relaxation approach (ACO MV - o), and categorical variables are handled with a categorical optimization approach (ACO MV -c). We also introduce a new set of artiﬁcial, mixed-variable benchmark functions and describe the method to construct them. These benchmark functions provide a ﬂexible envi- ronment for investigating the performance of mixed-variable optimization algorithms and the effect of different parameter settings on their performance. They are also useful as a training set for deriving high-performance parameter settings through the usage of automatic conﬁguration methods. Here, we use Iterated F-Race [30], [31] to automatically tune the parameters 1089-778X c  2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.