Skip to main content
SpringerLink
Log in

The Quadratic Assignment Problem

  • Chapter
  • First Online:
Location Science

Abstract

The quadratic assignment problem is reviewed in this chapter. Weights between pairs of facilities and distances between the same number of locations are given. The problem is to find the assignment of facilities to locations that minimizes the weighted sum of distances. This problem is considered to be one of the most difficult combinatorial optimization problems. The construction of efficient solution algorithms (exact or heuristic) is challenging and has been extensively investigated by the communities working in Operations Research/Management Science, Industrial Engineering, or Computer Science. Examples of applications are given, the related layout problem is briefly described, exact and heuristic solution algorithms are reviewed, and a list of test problem instances and results are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Adams W, Guignard M, Hahn P, Hightower W (2007) A level-2 reformulation-linearization technique bound for the quadratic assignment problem. Eur J Oper Res 180:983–996

    Google Scholar 

  • Ahuja R, Orlin J, Tiwari A (2000) A descent genetic algorithm for the quadratic assignment problem. Comput Oper Res 27:917–934

    Google Scholar 

  • Amin S (1999) Simulated jumping. Ann Oper Res 84:23–38

    Google Scholar 

  • Anstreicher K, Brixius N, Gaux JP, Linderoth J (2002) Solving large quadratic assignment problems on computational grids. Math Program 91:563–588

    Google Scholar 

  • Anstreicher KM, Brixius NW (2001) A new bound for the quadratic assignment problem based on convex quadratic programming. Math Program 89:341–357

    Google Scholar 

  • Armour GC, Buffa ES (1963) A heuristic algorithm and simulation approach to relative location of facilities. Manag Sci 9:294–309

    Google Scholar 

  • Battiti R, Tecchiolli G (1994) The reactive tabu search. ORSA J Comput 6:126–140

    Google Scholar 

  • Bos J (1993) Zoning in forest management: a quadratic assignment problem solved by simulated annealing. J Environ Manag 37:127–145

    Google Scholar 

  • Buffa ES, Armour GC, Vollmann TE (1962) Allocating facilities with CRAFT. Harv Bus Rev 42:136–158

    Google Scholar 

  • Burer S, Vandenbussche D (2006) Solving lift-and-project relaxations of binary integer programs. SIAM J Optimiz 16:726–750

    Google Scholar 

  • Burkard R, Rendl F (1984) A thermodynamically motivated simulation procedure for combinatorial optimization problems. Eur J Oper Res 17:169–174

    Google Scholar 

  • Burkard RE (1990) Locations with spatial interactions: the quadratic assignment problem. In: Mirchandani PB, Francis RL (eds) Discrete location theory. Wiley, New York, pp 387–437

    Google Scholar 

  • Burkard RE, Cela E (1999) Linear assignment problems and extensions. In: Pardalos P, Du D-Z (eds) Handbook of combinatorial optimization. Springer, Dordrecht, pp 75–149

    Google Scholar 

  • Burkard RE (2013) Quadratic assignment problems. In: Pardalos P, Du D-Z (eds) Handbook of combinatorial optimization, 2nd edn. Springer, New York, pp 2741–2814

    Google Scholar 

  • Burkard RE, Offermann J (1977) Entwurf von schreibmaschinentastaturen mittels quadratischer zuordnungsprobleme. Math Method Oper Res 21:121–132

    Google Scholar 

  • Cantú-Paz E (2001) Migration policies, selection pressure, and parallel evolutionary algorithms. J Heuristics 7:311–334

    Google Scholar 

  • de Carvalho Jr SA, Rahmann S (2006) Microarray layout as a quadratic assignment problem. In: Huson D, Kohlbacher O, Lupas A, Nieselt K, Zell A (eds) Proceedings of the German conference on bioinformatics, vol 83. Gesellschaft für Informatik, Bonn, pp 11–20

    Google Scholar 

  • Cela E (1998) The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Connoly D (1990) An improved annealing scheme for the QAP. Eur J Oper Res 46:93–100

    Google Scholar 

  • Cordeau JF, Gaudioso M, Laporte G, Moccia L (2006) A memetic heuristic for the generalized quadratic assignment problem. INFORMS J Comput 18:433–443

    Google Scholar 

  • Coxeter HSM (1973) Regular polytopes. Dover Publications, New York

    Google Scholar 

  • Cung VD, Mautor T, Michelon P, Tavares AI (1997) A scatter search based approach for the quadratic assignment problem. In: Proceedings of the IEEE international conference on evolutionary computation and evolutionary programming (ICEC’97), Indianapolis, pp 165–170

    Google Scholar 

  • Dickey JW, Hopkins JW (1972) Campus building arrangement using topaz. Transp Res 6:59–68

    Google Scholar 

  • Drezner T, Drezner Z (2005) Genetic algorithms: mimicking evolution and natural selection in optimization models. In: Bar-Cohen Y (ed) Biomimetics—biologically inspired technologies. CRC Press, Boca Raton, pp 157–175

    Google Scholar 

  • Drezner T, Drezner Z (2006) Gender specific genetic algorithms. INFOR Inform Syst Oper Res 44:117–127

    Google Scholar 

  • Drezner Z (1975) Problems in non-linear programming (the allocation problem). Ph.D. thesis, The Technion, Haifa

    Google Scholar 

  • Drezner Z (1980) DISCON—a new method for the layout problem. Oper Res 28:1375–1384

    Google Scholar 

  • Drezner Z (1987) A heuristic procedure for the layout of a large number of facilities. Manag Sci 33:907–915

    Google Scholar 

  • Drezner Z (2002) A new heuristic for the quadratic assignment problem. J Appl Math Decis Sci 6:163–173

    Google Scholar 

  • Drezner Z (2003) A new genetic algorithm for the quadratic assignment problem. INFORMS J Comput 15:320–330

    Google Scholar 

  • Drezner Z (2005a) A distance based rule for removing population members in genetic algorithms. 4OR-Q J Oper Res 3:109–116

    Google Scholar 

  • Drezner Z (2005b) The extended concentric tabu for the quadratic assignment problem. Eur J Oper Res 160:416–422

    Google Scholar 

  • Drezner Z (2006) Finding a cluster of points and the grey pattern quadratic assignment problem. OR Spectr 28:417–436

    Google Scholar 

  • Drezner Z (2008a) Extensive experiments with hybrid genetic algorithms for the solution of the quadratic assignment problem. Comput Oper Res 35:717–736

    Google Scholar 

  • Drezner Z (2008b) Tabu search and hybrid genetic algorithms for quadratic assignment problems. In: Jaziri W (ed) Tabu search, in-tech, pp 89–108. Available free on: http://books.i-techonline.com

  • Drezner Z (2010) On the unboundedness of facility layout problems. Math Method Oper Res 72:205–216

    Google Scholar 

  • Drezner Z, Marcoulides GA (2003) A distance-based selection of parents in genetic algorithms. In: Resende MGC, de Sousa JP (eds) Metaheuristics: computer decision-making. Kluwer Academic Publishers, Boston, pp 257–278

    Google Scholar 

  • Drezner Z, Misevičius A (2013) Enhancing the performance of hybrid genetic algorithms by differential improvement. Comput Oper Res 40:1038–1046

    Google Scholar 

  • Drezner Z, Suzuki A (2010) Covering continuous demand in the plane. J Oper Res Soc 61:878–881

    Google Scholar 

  • Drezner Z, Zemel E (1992) Competitive location in the plane. Ann Oper Res 40:173–193

    Google Scholar 

  • Drezner Z, Hahn PM, Taillard ÉD (2005) Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann Oper Res 139:65–94

    Google Scholar 

  • Drezner Z, Misevičius A, Palubeckis G (2014) Exact algorithms for the solution of the grey pattern quadratic assignment problem. In review

    Google Scholar 

  • Duman E, Uysal M, Alkaya AF (2012) Migrating birds optimization: a new metaheuristic approach and its performance on quadratic assignment problem. Inform Sci 217:65–77

    Google Scholar 

  • Elshafei AN (1977) Hospital layout as a quadratic assignment problem. Oper Res Q 28:167–179

    Google Scholar 

  • Fischetti M, Monaci M, Salvagnin D (2012) Three ideas for the quadratic assignment problem. Oper Res 60:954–964

    Google Scholar 

  • Fleurent C, Ferland J (1994) Genetic hybrids for the quadratic assignment problem. In: Pardalos P, Wolkowicz H (eds) Quadratic assignment and related problems, DIMACS series in discrete mathematics and theoretical computer science, vol 16. American Methematical Society, Providence, pp 173–187

    Google Scholar 

  • Fox BR, McMahon MB (1991) Genetic operators for sequencing problems. In: Rawlins G (ed) Foundations of genetic algorithms. Morgan-Kaufmann, San Mateo, pp 284–300

    Google Scholar 

  • Francis RL, McGinnis LF Jr, White JA (1992) Facility layout and location: an analytical approach, 2nd edn. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Gambardella L, Taillard ÉD, Dorigo M (1999) Ant colonies for the quadratic assignment problem. J Oper Res Soc 50:167–176

    Google Scholar 

  • Geoffrion AM, Graves GW (1976) Scheduling parallel production lines with changeover costs: practical application of a quadratic assignment/lp approa ch. Oper Res 24:595–610

    Google Scholar 

  • Gilmore P (1962) Optimal and suboptimal algorithms for the quadratic assignment problem. J SIAM 10:305–313

    Google Scholar 

  • Glover F (1977) Heuristics for integer programming using surrogate constraints. Decis Sci 8:156–166

    Google Scholar 

  • Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13:533–549

    Google Scholar 

  • Glover F, Laguna M (1997) Tabu search. Kluwer Academic Publishers, Boston

    Google Scholar 

  • Hahn P, Grant T (1998) Lower bounds for the quadratic assignment problem based upon a dual formulation. Oper Res 46:912–922

    Google Scholar 

  • Hahn P, Krarup J (2001) A hospital facility problem finally solved. J Intell Manuf 12:487–496

    Google Scholar 

  • Hahn PM, Kim BJ, Guignard M, Smith JM, Zhu YR (2008) An algorithm for the generalized quadratic assignment problem. Comput Optim Appl 40:351–372

    Google Scholar 

  • Hahn PM, Zhu YR, Guignard M, Hightower WL, Saltzman MJ (2012) A level-3 reformulation-linearization technique-based bound for the quadratic assignment problem. INFORMS J Comput 24:202–209

    Google Scholar 

  • Heffley DR (1977) Assigning runners to a relay team. In: Ladany SP, Machol RE (eds) Optimal strategies in sports. North-Holland, Amsterdam, pp 169–171

    Google Scholar 

  • Hilbert D, Cohn-Vossen S (1956) Geometry and the imagination (english translation of Anschauliche Geometrie, 1932). Chelsea Publishing Company, New York

    Google Scholar 

  • Hillier FS, Connors MM (1966) Quadratic assignment problem algorithms and the location of indivisible facilities. Manag Sci 13:42–57

    Google Scholar 

  • Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor

    Google Scholar 

  • Koopmans TC, Beckmann MJ (1957) Assignment problems and the location of economic activities. Econometrica 25:53–76

    Google Scholar 

  • Laporte G, Mercure H (1988) Balancing hydraulic turbine runners: a quadratic assignment problem. Eur J Oper Res 35:378–381

    Google Scholar 

  • Lawler EL (1973) Optimal sequencing of a single machine subject to precedence constraints. Manag Sci 19:544–546

    Google Scholar 

  • Lee CG, Ma Z (2005) The generalized quadratic assignment problem. Department of Mechanical and Industrial Engineering, University of Toronto, MIE OR Technical Reports (TR2005-01)

    Google Scholar 

  • Li Y, Pardalos PM (1992) Generating quadratic assignment test problems with known optimal permutations. Comput Optim Appl 1:163–184

    Google Scholar 

  • Li Y, Pardalos PM, Resende MGC (1994) A greedy randomized adaptive search procedure for the quadratic assignment problem. In: Pardalos PM, Wolkowicz H (eds) Quadratic assignment and related problems, DIMACS series in discrete mathematics and theoretical computer science, vol 16, American Mathematical Society, Providence, pp 237–261

    Google Scholar 

  • Loiola EM, de Abreu NMM, Boaventura-Netto PO, Hahn P, Querido T (2007) A survey for the quadratic assignment problem. Eur J Oper Res 176:657–690

    Google Scholar 

  • Love RF, Wong JY (1976) Solving quadratic assignment problems with rectangular distances and integer programming. Nav Res Logist Q 23:623–627

    Google Scholar 

  • Misevičius A (2003) A modified simulated annealing algorithm for the quadratic assignment problem. Informatica 14:497–514

    Google Scholar 

  • Misevičius A (2011) Generation of grey patterns using an improved genetic-evolutionary algorithm: some new results. Inf Technol Control 40:330–343

    Google Scholar 

  • Misevičius A (2012) An implementation of the iterated tabu search algorithm for the quadratic assignment problem. OR Spectr 34:665–690

    Google Scholar 

  • Misevičius A, Blonskis J (2005) Experiments with tabu search for random quadratic assignment problems. Inf Technol Control 34:237–244

    Google Scholar 

  • Misevičius A, Guogis E (2012) Computational study of four genetic algorithm variants for solving the quadratic assignment problem. In: Skersys T, Butkienè R, Butleris R (eds) Communications in computer and information science (CCIS). Proceedings of 18th international conference on information and software technologies ICIST 2012, vol 319. Springer, Berlin, pp 24–37

    Google Scholar 

  • Misevičius A, Tomkevičius A, Karbauskas J (2006) Stagnation-protected tabu search variants for unstructured quadratic assignment problems. Inf Technol Control 35:363–370

    Google Scholar 

  • Misevičius A, Guogis E, Stanevičienè E (2013) Computational algorithmic generation of high-quality colour patterns. In: Skersys T, Butkienè R, Butleris R (eds) Communications in computer and information science (CCIS). Proceedings of 19th international conference in information and software technologies ICIST 2013. Springer, Berlin, pp 285–296

    Google Scholar 

  • Misevičius A (2004) An improved hybrid genetic algorithm: new results for the quadratic assignment problem. Knowl-Based Syst 17:65–73

    Google Scholar 

  • Misevičius A (2005) A tabu search algorithm for the quadratic assignment problem. Comput Optim Appl 30:95–111

    Google Scholar 

  • Misevičius A (2008) Restart-based genetic algorithm for the quadratic assignment problem. In: Bramer M, Coenen F, Petridis M (eds) Research and development in intelligent systems. Proceedings of AI-2008, the 28th SGAI international conference on innovative techniques and applications of artificial intelligence. Springer, London, pp 91–104

    Google Scholar 

  • Misevičius A, Rubliauskas D (2009) Testing of hybrid genetic algorithms for structured quadratic assignment problems. Informatica 20:255–272

    Google Scholar 

  • Misevičius A, Rubliauskas D, Barkauskas V (2009) Some further experiments with the genetic algorithm for the quadratic assignment problem. Inf Technol Control 38:325–332

    Google Scholar 

  • Nugent C, Vollman T, Ruml T (1968) An experimental comparison of techniques for the assignment of facilities to locations. Oper Res 16:150–173

    Google Scholar 

  • Nyberg A, Westerlund T (2012) A new exact discrete linear reformulation of the quadratic assignment problem. Eur J Oper Res 220:314–319

    Google Scholar 

  • Nyström M (1999) Solving certain large instances of the quadratic assignment problem: Steinberg’s examples. Technical report, California Institute of Technology. http://resolver.caltech.edu/CaltechCSTR:2001.010

  • Okabe A, Suzuki A (1987) Stability of spatial competition for a large number of firms on a bounded two-dimensional space. Environ Plann A 16:107–114

    Google Scholar 

  • Oliveira CAS, Pardalos PM, Resende MGC (2004) GRASP with path-relinking for the quadratic assignment problem. In: Ribeiro CC, Martins SL (eds) Efficient and experimental algorithms. Springer, Berlin/Heidelberg, pp 237–261

    Google Scholar 

  • Pierce JF, Crowston WB (1971) Tree-search algorithms for quadratic assignment problems. Nav Res Logist Q 18:1–36

    Google Scholar 

  • Pierskalla WP (1967) The tri-substitution method for the three-dimensional assignment problem. CORS J 5:71–81

    Google Scholar 

  • Pierskalla WP (1968) The multidimensional assignment problem. Oper Res 16:422–431

    Google Scholar 

  • Ramakrishnan KG, Resende M, Ramachandran B, Pekny J (2002) Tight QAP bounds via linear programming. In: Pardalos PM, Migdalas A, Burkard R (eds) Combinatorial and global optimization. World Scientific Publishing, Singapore, pp 297–303

    Google Scholar 

  • Rendl F (2002) The quadratic assignment problem. In: Drezner Z, Hamacher H (eds) Facility location: applications and theory. Springer, Berlin

    Google Scholar 

  • Rendl F, Sotirov R (2007) Bounds for the quadratic assignment problem using the bundle method. Math Program 109:505–524

    Google Scholar 

  • Resende M, Ramakrishnan K, Drezner Z (1995) Computational experiments with the lower bound for the quadratic assignment problem based on linear programming. Oper Res 43:781–791

    Google Scholar 

  • Roupin F (2004) From linear to semidefinite programming: an algorithm to obtain semidefinite relaxations for bivalent quadratic problems. J Comb Optim 8:469–493

    Google Scholar 

  • Sahni S, Gonzalez T (1976) P-complete approximation problems. J ACM 23:555–565

    Google Scholar 

  • Schaffer JD, Caruana RA, Eshelman LJ (1989) A study of control parameters affecting online performance of genetic algorithms. In: Schaffer JD (ed) Proceedings of the 3rd international conference on genetic algorithms. Morgan Kaufmann, San Mateo, pp 51–60

    Google Scholar 

  • Sherali HD, Adams WP (1990) A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J Discret Math 3:411–430

    Google Scholar 

  • Sherali HD, Adams WP (1998) A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Springer, Berlin

    Google Scholar 

  • Skorin-Kapov J (1990) Tabu search applied to the quadratic assignment problem. ORSA J Comput 2:33–45

    Google Scholar 

  • Steinberg L (1961) The backboard wiring problem: a placement algorithm. SIAM Rev 3:37–50

    Google Scholar 

  • Suzuki A, Drezner Z (1996) The p-center location problem in an area. Locat Sci 4:69–82

    Google Scholar 

  • Suzuki A, Okabe A (1995) Using Voronoi diagrams. In: Drezner Z (ed) Facility location: a survey of applications and methods. Springer, New York, pp 103–118

    Google Scholar 

  • Szabo PG, Markot M, Csendes T, Specht E (2007) New approaches to circle packing in a square: with program codes. Springer, New York

    Google Scholar 

  • Taillard ÉD (1991) Robust tabu search for the quadratic assignment problem. Parallel Comput 17:443–455

    Google Scholar 

  • Taillard ÉD (1995) Comparison of iterative searches for the quadratic assignment problem. Locat Sci 3:87–105

    Google Scholar 

  • Taillard ÉD (1998) Fant: fast ant system. Technical report, Istituto Dalle Molle Di Studi Sull Intelligenza Artificiale, iDSIA Technical Report IDSIA-46-98

    Google Scholar 

  • Taillard ÉD (2000) An introduction to ant systems. In: Laguna M, González-Velarde J (eds) Computing tools for modeling, optimization and simulation. Wiley, New York, pp 131–144

    Google Scholar 

  • Talbi EG, Roux O, Fonlupt C, Robillard D (2001) Parallel ant colonies for the quadratic assignment problem. Futur Gener Comput Syst 17:441–449

    Google Scholar 

  • Tate D, Smith A (1995) A genetic approach to the quadratic assignment problem. Comput Oper Res 22:73–83

    Google Scholar 

  • Thonemann U, Bölte A (1994) An improved simulated annealing algorithm for the quadratic assignment problem. Technical report, working paper, School of Business, Department of Production and Operations Research, University of Paderborn, Germany

    Google Scholar 

  • Wilhelm M, Ward T (1987) Solving quadratic assignment problems by simulated annealing. IIE Trans 19:107–119

    Google Scholar 

  • Wu Y, Ji P (2008) Solving the quadratic assignment problems by a genetic algorithm with a new replacement strategy. Int J Comput Intell 4:225–229

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. California State University, Fullerton, CA, 92834, USA

    Zvi Drezner

Authors
  1. Zvi Drezner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zvi Drezner .

Editor information

Editors and Affiliations

  1. HEC Montréal, Montréal, Canada

    Gilbert Laporte

  2. Institute of Operations Research (IOR), Karlsruhe Institute of Technology, Karlsruhe, Germany

    Stefan Nickel

  3. DEIO-FCUL, University of Lisbon, Lisbon, Portugal

    Francisco Saldanha da Gama

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Drezner, Z. (2015). The Quadratic Assignment Problem. In: Laporte, G., Nickel, S., Saldanha da Gama, F. (eds) Location Science. Springer, Cham. https://doi.org/10.1007/978-3-319-13111-5_13

Download citation

Publish with us

Policies and ethics

Search

Navigation