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Related Concept Videos

Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Lagrange Multipliers: Problem Solving01:30

Lagrange Multipliers: Problem Solving

A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.The total volume of the silo is obtained by...
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Lagrange Multipliers: One Constraint01:29

Lagrange Multipliers: One Constraint

In constrained optimization, the objective is to maximize or minimize a quantity while satisfying a fixed condition. A standard example is a rectangular pen built against a barn wall using 100 meters of fencing. Because the wall provides one side of the enclosure, only the other three sides require fencing. The problem is to find the dimensions that produce the greatest possible area.Let L represent the length parallel to the wall and W the width perpendicular to it. The area of the pen is A =...
Quadratic Equations01:29

Quadratic Equations

A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...

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Related Experiment Videos

Hybrid metaheuristic approaches for grey generalized quadratic assignment problem.

Malihe Niksirat1, Mostafa Sabzekar2, Javad Tayyebi3

  • 1Department of Computer Sciences, Birjand University of Technology, Birjand, Iran. niksirat@birjandut.ac.ir.

Scientific Reports
|July 8, 2026
PubMed
Summary

This study introduces a new grey variant of the Generalized Quadratic Assignment Problem (GQAP) to handle cost uncertainty. The proposed OHNN-PSO method effectively solves these complex assignment problems, outperforming other algorithms.

Keywords:
Generalized quadratic assignment problemGrey optimizationHopfield neural networkHybrid metaheuristicParticle swarm optimizationUncertainty modeling

Related Experiment Videos

Area of Science:

  • Operations Research
  • Artificial Intelligence
  • Optimization

Background:

  • The Generalized Quadratic Assignment Problem (GQAP) is complex and often involves uncertainty in cost data.
  • Limited information necessitates the use of grey numbers to represent uncertain installation and transportation costs.

Purpose of the Study:

  • To formulate and solve a novel grey variant of the GQAP (GGQAP) under uncertainty.
  • To evaluate the performance of hybrid metaheuristic approaches for solving the GGQAP.

Main Methods:

  • A grey variant of GQAP (GGQAP) was formulated.
  • Two hybrid metaheuristic algorithms, OHNN-PSO and OHNN-SA, were developed, integrating Optimized Hopfield Neural Network (OHNN) with Particle Swarm Optimization (PSO) and Simulated Annealing (SA).

Main Results:

  • OHNN-PSO demonstrated superior performance, achieving the best average rank across benchmark, GQAP, and GGQAP instances.
  • OHNN-PSO significantly outperformed PSO, standard OHNN, and OHNN-SA on the GGQAP instances.

Conclusions:

  • The proposed OHNN-PSO framework provides a practical and statistically validated solution for GQAP under grey uncertainty.
  • Hybrid metaheuristics are effective for addressing complex optimization problems with uncertain parameters.