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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.
Optimization Problems01:26

Optimization Problems

Optimization problems often involve identifying maximum or minimum values under specific constraints. A well-known example is determining the longest horizontal pipe that can be moved around a right-angled corner, where a 3-meter-wide hallway meets a 2-meter-wide hallway. This scenario, common in architectural design and industrial transport, can be understood conceptually through geometric and trigonometric reasoning.To visualize the problem, consider the pipe as a straight line that touches...
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 =...
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...
Planes in Space01:31

Planes in Space

A plane in three-dimensional space is fundamentally characterized by a point that lies on the plane and a normal vector that is perpendicular to its surface. This normal vector uniquely determines the orientation of the plane, making it an essential geometric descriptor. In architectural applications, such as the installation of a sloped glass panel on a building façade, this mathematical model provides a precise representation of the panel’s position and orientation in space.Let r₀ be the...
Design Example: Alignment of a Road Line Using GIS01:17

Design Example: Alignment of a Road Line Using GIS

The alignment of a road line using Geographic Information Systems (GIS) is a critical process in civil engineering, combining advanced technology with practical decision-making. This methodology begins with the collection of geospatial data, including information on land cover, geomorphology, drainage patterns, slope, and contour details. Such data is typically acquired through satellite imagery and GIS tools, offering a comprehensive understanding of the terrain.Once the data is gathered, it...

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Related Experiment Video

Updated: Jun 25, 2026

Evaluation of an Exclusive Spur Dike U-Turn Design with Radar-Collected Data and Simulation
11:41

Evaluation of an Exclusive Spur Dike U-Turn Design with Radar-Collected Data and Simulation

Published on: February 1, 2020

Solving the multiple competitive facilities location and design problem on the plane.

Juana López Redondo1, José Fernández, Inmaculada García

  • 1Department of Computer Architecture and Electronics, University of Almería, Almería, Spain. juani@ace.ual.es

Evolutionary Computation
|February 12, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces evolutionary algorithms to solve complex continuous location problems for new facilities. These algorithms outperform other methods, effectively finding optimal solutions for profit maximization in competitive markets.

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Operation of the Collaborative Composite Manufacturing (CCM) System
10:09

Operation of the Collaborative Composite Manufacturing (CCM) System

Published on: October 1, 2019

Related Experiment Videos

Last Updated: Jun 25, 2026

Evaluation of an Exclusive Spur Dike U-Turn Design with Radar-Collected Data and Simulation
11:41

Evaluation of an Exclusive Spur Dike U-Turn Design with Radar-Collected Data and Simulation

Published on: February 1, 2020

Operation of the Collaborative Composite Manufacturing (CCM) System
10:09

Operation of the Collaborative Composite Manufacturing (CCM) System

Published on: October 1, 2019

Area of Science:

  • Operations Research
  • Computational Optimization

Background:

  • Firms face challenges in locating new facilities within competitive markets.
  • Existing facilities create a complex environment for new entrants.
  • Optimizing both location and quality is crucial for maximizing profit.

Purpose of the Study:

  • To analyze and compare various heuristics for solving continuous location problems.
  • To identify the most effective method for profit maximization in competitive facility location.
  • To develop adaptable evolutionary strategies for continuous location problems.

Main Methods:

  • Analysis of three multistart local search heuristics.
  • Implementation of a multistart simulated annealing algorithm.
  • Development and testing of two variants of an evolutionary algorithm.

Main Results:

  • Evolutionary algorithms demonstrated superior performance compared to other heuristics.
  • Only evolutionary algorithms successfully identified optimal solutions for all tested instances.
  • The proposed evolutionary strategies showed high adaptability.

Conclusions:

  • Evolutionary algorithms are highly effective for solving continuous location problems.
  • These algorithms provide a robust approach for profit maximization in competitive environments.
  • The presented strategies offer a flexible framework for related optimization challenges.