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Bi-Objective Optimization over the Efficient Set of Multi-Objective Integer Quadratic Problem

Ali Bencheikh, Mustapha Moulai, Ilies Badaoui

Abstract

In this paper, we present an exact algorithm for optimizing two linear fractional over the efficient set of a multi-objective integer quadratic problem. This type of problems arises when two decision-makers, such as firms, each have a preference function to optimize over the efficient set of a multi-objective problem. The algorithm employs a branch-and-cut approach, which involves: (1) exploring the solution space using a branch-and-bound strategy in the decision space, and (2) eliminating inefficient solutions using a cutting plane technique with efficient cuts constructed from the non-increasing directions of objective functions. Additionally, integral tests are incorporated to further ensure the efficiency of the obtained solutions.We present a comprehensive example, accompanied by a step-by-step resolution, to demonstrate the functioning of the algorithm.

Bi-Objective Optimization over the Efficient Set of Multi-Objective Integer Quadratic Problem

Abstract

In this paper, we present an exact algorithm for optimizing two linear fractional over the efficient set of a multi-objective integer quadratic problem. This type of problems arises when two decision-makers, such as firms, each have a preference function to optimize over the efficient set of a multi-objective problem. The algorithm employs a branch-and-cut approach, which involves: (1) exploring the solution space using a branch-and-bound strategy in the decision space, and (2) eliminating inefficient solutions using a cutting plane technique with efficient cuts constructed from the non-increasing directions of objective functions. Additionally, integral tests are incorporated to further ensure the efficiency of the obtained solutions.We present a comprehensive example, accompanied by a step-by-step resolution, to demonstrate the functioning of the algorithm.
Paper Structure (10 sections, 4 theorems, 12 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 4 theorems, 12 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation & Notations
  3. Problem formulation
  4. Notations
  5. Theoretical background & Main results
  6. Efficiency Tests
  7. Description of Method
  8. BLF/MOIQP Algorithm
  9. Illustrative example
  10. Conclusion

Key Result

Theorem 2

The feasible solution $x^{*(l)}$ is an optimal solution for the problem $(LFP)_l$ if and only if the vector $\gamma$ is such that $\bar{\gamma_j} \ge 0$ for all $j \in \mathcal{N}_l.$ see martos1975nonlinear

Figures (1)

  • Figure 1: Search tree of the example

Theorems & Definitions (8)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof