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Chang, C.-T. and Lin, T.-C. (2009) Interval Goal Programming for S-Shaped Penalty Function. European Journal of Operational Research, 199, 9-20.
http://dx.doi.org/10.1016/j.ejor.2008.10.009

has been cited by the following article:

• TITLE: A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming

  AUTHORS: Zhiqing Meng, Rui Shen, Min Jiang

  KEYWORDS: Multi-Objective Programming, Penalty Function, Objective Parameters, Constraint Penalty Parameter, Pareto Weakly-Efficient Solution

  JOURNAL NAME: American Journal of Operations Research, Vol.4 No.6, October 9, 2014

  ABSTRACT: In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.