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Scalar notation is a useful method for simplifying calculations involving vectors. When vectors are added or subtracted, their components can be added or subtracted separately using scalar notation. For instance, force, a vector quantity, can be broken down into its x and y components, called rectangular components, and then the magnitude and direction of these components can be determined using trigonometric functions.
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The Artificial Neural Networks Based on Scalarization Method for a Class of Bilevel Biobjective Programming Problem.

Tao Zhang1, Zhong Chen1, June Liu2

  • 1School of Information and Mathematics, Yangtze University, Jingzhou 434023, China.

Computational Intelligence and Neuroscience
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A novel two-stage artificial neural network (ANN) effectively solves bilevel biobjective programming problems (BLBOP). This method identifies the complete efficient set by first finding minimal solutions using a scalarization approach.

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Area of Science:

  • Operations Research
  • Artificial Intelligence
  • Optimization Theory

Background:

  • Bilevel biobjective programming problems (BLBOP) present significant computational challenges.
  • Existing methods often struggle to derive the complete efficient set for BLBOP.

Purpose of the Study:

  • To propose a novel two-stage artificial neural network (ANN) for solving BLBOP.
  • To effectively derive the entire efficient set of BLBOP using the proposed ANN.

Main Methods:

  • The proposed method utilizes a scalarization approach to express the induced set of BLBOP as minimal solutions of a biobjective problem.
  • A two-stage ANN is employed to explore the induced set and derive the complete efficient set of BLBOP.

Main Results:

  • The proposed two-stage ANN method successfully derived the efficient set for tested BLBOP instances.
  • Numerical examples demonstrated the efficacy of the ANN approach, with results comparable to classical literature.
  • A practical problem was effectively solved using the developed algorithm.

Conclusions:

  • The two-stage ANN based on scalarization is a viable and effective method for solving bilevel biobjective programming problems.
  • The approach provides a robust framework for exploring and deriving the efficient set in complex optimization scenarios.