Heuristics for the Quadratic Assignment Problem (QAP).

R package qap - Heuristics for the Quadratic Assignment Problem (QAP)

This package implements heuristics for the Quadratic Assignment Problem (QAP) first introduced by Koopmans and Beckmann (1957). Although, the QAP was introduced as a combinatorial optimization problem for the facility location problem in operations research, it also has many applications in data analysis (see Hubert and Schultz; 1976).

The problem is NP-hard and the package implements the simulated annealing heuristic described in Burkard and Rendl (1984).

Installation

Stable CRAN version: Install from within R with

install.packages("qap")

Current development version: Install from r-universe.

install.packages("qap", repos = "https://mhahsler.r-universe.dev")

Usage

The package contains a copy of the problem instances and solutions from QAPLIB. We load the had20 QAPLIB problem. The problem contains the A and B matrices and the optimal solution and the optimal objective function value.

library(qap)
set.seed(1000)

p <- read_qaplib(system.file("qaplib", "had20.dat", package = "qap"))
p$solution

##  [1]  8 15 16 14 19  6  7 17  1 12 10 11  5 20  2  3  4  9 18 13

p$opt

## [1] 6922

We run the simulated annealing heuristic 10 times and use the best solution.

a <- qap(p$A, p$B, rep = 10)
a

##  [1]  8 15 16 14 19  6  7 12  1 11 10  5  3 20  2 17  4  9 18 13
## attr(,"obj")
## [1] 6926

Compare the solution with known optimum (% above optimum).

(attr(a, "obj") - p$opt)/p$opt * 100

## [1] 0.058

References

• Hahsler M (2022). qap: Heuristics for the Quadratic Assignment Problem (QAP). R package version 0.1-2, https://github.com/mhahsler/qap.
• R.E. Burkard and F. Rendl (1984). A thermodynamically motivated simulation procedure for combinatorial optimization problems. European Journal of Operations Research, 17(2):169-174. https://doi.org/10.1016/0377-2217(84)90231-5
• Koopmans TC, Beckmann M (1957). Assignment problems and the location of economic activities. Econometrica 25(1):53-76. https://doi.org/10.2307/1907742
• Hubert, L., and Schultz, J. (1976). Quadratic assignment as a general data analysis strategy. British Journal of Mathematical and Statistical Psychology, 29(2), 190–241. https://doi.org/10.1111/j.2044-8317.1976.tb00714.x.