Nonlinear goal programming using multi-objective genetic algorithms
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Nonlinear goal programming using multi-objective genetic algorithms K Deb* Indian Institute of Technology Kanpur, Kanpur, India Goal programming is a technique often used in engineering design activities primarily to ®nd a compromised solution which will simultaneously satisfy a number of design goals. In solving goal programming problems, classical methods reduce the multiple goal-attainment problem into a single objective of minimizing a weighted sum of deviations from goals. This procedure has a number of known dif®culties. First, the obtained solution to the goal programming problem is sensitive to the chosen weight vector. Second, the conversion to a single-objective optimization problem involves additional constraints. Third, since most real-world goal programming problems involve nonlinear criterion functions, the resulting single-objective optimization problem becomes a nonlinear programming problem, which is dif®cult to solve using classical optimization methods. In tackling nonlinear goal programming problems, although successive linearization techniques have been suggested, they are found to be sensitive to the chosen starting solution. In this paper, we pose the goal programming problem as a multi-objective optimization problem of minimizing deviations from individual goals and then suggest an evolutionary optimization algorithm to ®nd multiple Pareto-optimal solutions of the resulting multiobjective optimization problem. The proposed approach alleviates all the above dif®culties. It does not need any weight vector. It eliminates the need of having extra constraints needed with the classical formulations. The proposed approach is also suitable for solving goal programming problems having nonlinear criterion functions and having a non-convex tradeoff region. The ef®cacy of the proposed approach is demonstrated by solving a number of nonlinear goal programming test problems and an engineering design problem. In all problems, multiple solutions (each corresponding to a different weight vector) to the goal programming problem are found in one single simulation run. The results suggest that the proposed approach is an effective and practical tool for solving real-world goal programming problems. Keywords: goal programming; genetic algorithms; engineering design; multi-objective optimization
Introduction Developed in the year 1955, goal programming has enjoyed innumerable applications in engineering design activities.1 ± 5 Goal programming is different in concept from nonlinear programming or optimization techniques in that the goal programming attempts to ®nd one or more solutions which satisfy a number of goals to the extent possible. Instead of ®nding solutions which absolutely minimize or maximize objective functions, the task is to ®nd solutions that, if possible, satisfy a set of goals, and otherwise violate the goals minimally. This makes the approach more appealing to designers compared to optimizat
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