IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 28, NO. 5, OCTOBER 1998 629 Varying Fitness Functions in Genetic Algorithm Constrained Optimization: The Cutting Stock and Unit Commitment Problems Vassilios Petridis, Member, IEEE, Spyros Kazarlis, and Anastasios Bakirtzis, Senior Member, IEEE Abstract— In this paper, we present a specific varying fitness function technique in genetic algorithm (GA) constrained opti- mization. This technique incorporates the problem’s constraints into the fitness function in a dynamic way. It consists in forming a fitness function with varying penalty terms. The resulting varying fitness function facilitates the GA search. The performance of the technique is tested on two optimization problems: the cutting stock, and the unit commitment problems. Also, new domain- specific operators are introduced. Solutions obtained by means of the varying and the conventional (nonvarying) fitness function techniques are compared. The results show the superiority of the proposed technique. Index Terms—Constrained optimization, cutting stock, genetic algorithms, genetic operators, unit commitment. I. INTRODUCTION G ENETIC algorithms (GA’s) turned out to be powerful tools in the field of global optimization [7], [10], [13]. They have been applied successfully to real-world prob- lems and exhibited, in many cases, better search efficiency compared with traditional optimization algorithms. GA’s are based on principles inspired from the genetic and evolution mechanisms observed in natural systems and populations of living beings [13]. Their basic principle is the maintenance of a population of encoded solutions to the problem (genotypes) that evolve in time. They are based on the triangle of genetic solution reproduction, solution evaluation and selection of the best genotypes. Genetic reproduction is performed by means of two basic genetic operators: Crossover [10], [13], [30] and Mutation [10]. Many other genetic operators are reported in the literature, including problem specific ones [10], [15], [20], [25]. Evaluation is performed by means of the Fitness Function which depends on the specific problem and is the optimization objective of the GA. Genotype selection is performed according to a selection scheme, that selects parent genotypes with probability proportional to their relative fitness [11]. In a GA application the formulation of the fitness function is of critical importance and determines the final shape of the hypersurface to be searched. In certain real-world problems, there is also a number of constraints to be satisfied. Such Manuscript received October 8, 1994; revised August 27, 1996 and August 5, 1997. The authors are with the Department of Electrical and Computer Engineer- ing, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece. Publisher Item Identifier S 1083-4419(98)07312-9. constraints can be incorporated into the fitness function by means of penalty terms which further complicate the search. The authors of this paper were among the first that proposed the varying fitness function technique [9], [14], [25], [28], [29], [33]. The purpose of this paper is to present a specific varying fitness function technique. This technique incorporates the problem’s constraints into the fitness function as penalty terms that vary with the generation index, resulting thus in a varying fitness function that facilitates the location of the general area of the global optimum. This technique is applied to two small-scale versions of two hard real-world constrained optimization problems, that are used as benchmarks: the cutting stock and unit commitment problems. The cutting stock problem consists in cutting a number of predefined two-dimensional shapes out of a piece of stock material with minimum waste. It is a problem of geometrical nature with a continuous-variable encoding. The unit commitment problem consists in the determination of the optimum operating schedule of a number of electric power production units, in order to meet the forecasted demand over a short term period, with the minimum total operating cost. It is clearly a scheduling problem. It is not our intent to present complete solutions of the above problems but to demonstrate the effectiveness of the varying fitness function technique. We have chosen these particular problems because they are diverse in nature (each problem exhibits unique search space characteristics) and therefore they provide a rigorous test for the efficiency and robustness of our tech- nique. Section II discusses various methods that enable the applica- tion of GA’s to constrained optimization problems. Section III contains a detailed analysis of the varying fitness function technique proposed in this paper. The application of this tech- nique to the cutting stock and the unit commitment problems are presented in Sections IV and V, respectively. Finally conclusions are presented in Section VI. II. GENERAL CONSIDERATIONS In the sequel, without loss of generality, we assume that we deal with minimization problems. As it has been mentioned before, in many optimization problems there are a number of constraints to be satisfied. As far as we know, six basic methods have been reported in the literature that enable GA’s to be applied to constrained optimization problems. 1083–4419/98$10.00  1998 IEEE