# Difference between revisions of "Quadratic assignment problem"

NatashaRice (talk | contribs) (Updated formula) |
NatashaRice (talk | contribs) (Finalized backboard problem) |
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In his paper he derives the below formula: | In his paper he derives the below formula: | ||

− | <math>min \sum_{1\leq i \leq j \leq n }^{} C_{ij}(d_{s(i)s(j))}) | + | <math>min \sum_{1\leq i \leq j \leq n }^{} C_{ij}(d_{s(i)s(j))})</math> |

− | |||

<math>C_{ij}</math> represents the number of wires connecting components <math>E_{i}</math> and <math>E_{i}</math> | <math>C_{ij}</math> represents the number of wires connecting components <math>E_{i}</math> and <math>E_{i}</math> | ||

Line 37: | Line 36: | ||

<math>d_{s(i)s(j))} = \surd(\bigl(x_{\alpha} - x_{\beta}\bigr)^{2} + \bigl(y_{\alpha} - y_{\beta}\bigr)^{2} + \bigl(z_{\alpha} - z_{\beta}\bigr)^{2}) </math> | <math>d_{s(i)s(j))} = \surd(\bigl(x_{\alpha} - x_{\beta}\bigr)^{2} + \bigl(y_{\alpha} - y_{\beta}\bigr)^{2} + \bigl(z_{\alpha} - z_{\beta}\bigr)^{2}) </math> | ||

− | + | <math>s(x)</math> represents a particular placement of a component. | |

+ | In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation he selected a random initial placement of s1 and chose a random family of 25 unconnected sets. | ||

− | + | The initial placement of components is shown below: | |

− | + | [[File:Initial wire length.jpg|center|thumb|412x412px|The initial placement of components in the backboard from Steinberg's ]] | |

− | + | After the initial placement of elements it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440. | |

− | + | [[File:Final wire length.jpg|center|thumb|377x377px|Final component position in Steinberg's Backboard optimization problem]] | |

− | |||

=== Hospital Layout === | === Hospital Layout === |

## Revision as of 15:39, 21 November 2020

Author: Thomas Kueny, Eric Miller, Natasha Rice, Joseph Szczerba, David Wittmann (SysEn 5800 Fall 2020)

## Introduction

## Theory, Methodology, and/or Algorithmic Discussions

Joe Szczerba and Tom Kueny to provide.

## Example

Eric Miller to provide.

## Applications

Natasha Rice and David Wittmann to provide.

### The Backboard Wiring Problem

As the quadratic assignment problem is focused on minimizing the cost of traveling from one location to another, it is an ideal approach determining placement of components in many modern electronics. Leon Steinberg proposed a QAP solution optimize the layout of elements on a blackboard by minimizing the total amount of wiring required.

When defining the problem Steinberg states that we have a set of n elements

as well as a set of r points

where

In his paper he derives the below formula:

represents the number of wires connecting components and

is the length of wire needed to connect tow components if one is placed at and the other is placed at . If has coordinates then can be calculated by

represents a particular placement of a component.

In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.

The initial placement of components is shown below:

After the initial placement of elements it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.

### Hospital Layout

### Campus Building Arrangement

### Molecular Confrontation Problem

## Conclusion

## References

- Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem.
*Operational Research Quarterly (1970-1977)*. 1977;28(1):167. doi:10.2307/3008789 - Dickey JW, Hopkins JW. Campus building arrangement using topaz.
*Transportation Research*. 6(1):59-68. doi:10.1016/0041-1647(72)90111-6 - Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm.
*SIAM Review*. 1961;3(1):37. - Phillips AT, Rosen JB. A quadratic assignment formulation of the molecular conformation problem.
*Journal of Global Optimization: An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management, and Engineer*. 1994;4(2):229. doi:10.1007/bf01096724