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January  2015, 11(1): 13-26. doi: 10.3934/jimo.2015.11.13

A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle

1. 

Materiel Management and Safety Engineering College, Air Force Engineering University, Xi'an, 710051, China, China

2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

3. 

Department of Mathematics, Xijing College, Xi'an, 710236, China

Received  May 2013 Revised  November 2013 Published  May 2014

On the basis of the uncertainty theory, this paper is devoted to the uncertain multiobjective programming problem. Firstly, several principles are provided to define the relationship between uncertain variables. Then a new approach is proposed for obtaining Pareto efficient solutions in uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle, which involves transforming the uncertain multiobjective problem into a problem with only one uncertain objective function, and its validity has been proved. Due to the complexity of this problem, it is very suitable for the use of genetic algorithm. Finally, a numerical example is presented to illustrate the novel approach proposed, and the genetic algorithm is adopted to solve it.
Citation: Zutong Wang, Jiansheng Guo, Mingfa Zheng, Youshe Yang. A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle. Journal of Industrial & Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13
References:
[1]

A. Arturo, A. Graham and G. Stuart, Multi-objective planning of distributed energy resources: A review of the state-of-the-art,, Renewable and Sustainable Energy Reviews, 14 (2010), 1353.

[2]

C. Anthony, K. Juyoung, L. Seungjae and K. Youngchan, Stochastic multi-objective models for network design problem,, Expert Systems with Applications, 37 (2010), 1608.

[3]

Y. Bai and C. Guo, Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems,, Journal of Industrial and Management Optimization, 10 (2014), 543. doi: 10.3934/jimo.2014.10.543.

[4]

C. Chen, T. C. Edwin Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization,, Journal of Industrial and Management Optimization, 7 (2011), 157. doi: 10.3934/jimo.2011.7.157.

[5]

C. M. Fonseca and P. J. Fleming, Genetic algorithms for multiobjective optimization: Formulation,discussion,and generalization,, The Fifth International Conference on Genetic Algorithms, (1993), 416.

[6]

M. Kaisa, Nonlinear Multi-objective Optimization,, Kluwer Academic Publishers, (1999).

[7]

B. Liu, Uncertainty Theory,, $2^{nd}$ edition, (2007).

[8]

B. Liu, Some research problems in uncertainty theory,, Journal of Uncertain System, 3 (2009), 3.

[9]

B. Liu, Theory and Practice of Uncertain Programming,, Springer-Verlag, (2009). doi: 10.1007/978-3-540-89484-1.

[10]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,, An introduction to its axiomatic foundations. Studies in Fuzziness and Soft Computing, (2004). doi: 10.1007/978-3-540-39987-2.

[11]

B. Liu, Uncertainty Theory,, $4^{th}$ edition, (2012).

[12]

B. Liu, Why is there a need for uncertainty theory?,, Journal of Uncertain System, 5 (2012), 3.

[13]

B. Liu and X. W. Chen, Uncertain Multiobjective Programming and Uncertain Goal Programming,, Technical report, (2012).

[14]

C. Rafael, C. Emilio, M. M. Maria del and R. Lourdes, Stochastic approach versus multi-objective approach for obtaining efficient solutions in stochastic multi-objective programming problems,, European Journal of Operational Research, 158 (2004), 633. doi: 10.1016/S0377-2217(03)00371-0.

[15]

Jr. J. Teghem and P. L. Kunsch, Application of multi-objective stochastic linear programming to power systems planning,, Engineering Costs and Production Economics, 9 (1985), 83. doi: 10.1016/0167-188X(85)90013-8.

show all references

References:
[1]

A. Arturo, A. Graham and G. Stuart, Multi-objective planning of distributed energy resources: A review of the state-of-the-art,, Renewable and Sustainable Energy Reviews, 14 (2010), 1353.

[2]

C. Anthony, K. Juyoung, L. Seungjae and K. Youngchan, Stochastic multi-objective models for network design problem,, Expert Systems with Applications, 37 (2010), 1608.

[3]

Y. Bai and C. Guo, Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems,, Journal of Industrial and Management Optimization, 10 (2014), 543. doi: 10.3934/jimo.2014.10.543.

[4]

C. Chen, T. C. Edwin Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization,, Journal of Industrial and Management Optimization, 7 (2011), 157. doi: 10.3934/jimo.2011.7.157.

[5]

C. M. Fonseca and P. J. Fleming, Genetic algorithms for multiobjective optimization: Formulation,discussion,and generalization,, The Fifth International Conference on Genetic Algorithms, (1993), 416.

[6]

M. Kaisa, Nonlinear Multi-objective Optimization,, Kluwer Academic Publishers, (1999).

[7]

B. Liu, Uncertainty Theory,, $2^{nd}$ edition, (2007).

[8]

B. Liu, Some research problems in uncertainty theory,, Journal of Uncertain System, 3 (2009), 3.

[9]

B. Liu, Theory and Practice of Uncertain Programming,, Springer-Verlag, (2009). doi: 10.1007/978-3-540-89484-1.

[10]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,, An introduction to its axiomatic foundations. Studies in Fuzziness and Soft Computing, (2004). doi: 10.1007/978-3-540-39987-2.

[11]

B. Liu, Uncertainty Theory,, $4^{th}$ edition, (2012).

[12]

B. Liu, Why is there a need for uncertainty theory?,, Journal of Uncertain System, 5 (2012), 3.

[13]

B. Liu and X. W. Chen, Uncertain Multiobjective Programming and Uncertain Goal Programming,, Technical report, (2012).

[14]

C. Rafael, C. Emilio, M. M. Maria del and R. Lourdes, Stochastic approach versus multi-objective approach for obtaining efficient solutions in stochastic multi-objective programming problems,, European Journal of Operational Research, 158 (2004), 633. doi: 10.1016/S0377-2217(03)00371-0.

[15]

Jr. J. Teghem and P. L. Kunsch, Application of multi-objective stochastic linear programming to power systems planning,, Engineering Costs and Production Economics, 9 (1985), 83. doi: 10.1016/0167-188X(85)90013-8.

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