The Journal of the Institute of Internet, Broadcasting and Communication (한국인터넷방송통신학회논문지)

The Institute of Internet, Broadcasting and Communication (한국인터넷방송통신학회)
권호 표지
DOI QR코드

DOI QR Code

AThe Simplified Solution for Assignment Problem

할당 문제의 단순한 해법

  • Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
  • 이상운 (강릉원주대학교 과학기술대학 멸티미디어공학과)
  • Received : 2012.05.18
  • Accepted : 2012.10.12
  • Published : 2012.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper suggests more simple algorithm than Hungarian algorithm for assignment problem. Hungarian algorithm selects minimum cost of row and column, and subtracts minimum cost from each cost. Then, performs until the number of minimum lines with 0 equals the number of rows. But, the proposed algorithm selects the minimum cost for each rows only. From the start point with over 2 to the target point with null selects in column, fixes the maximum opportunity cost that the difference of the cost of starting point and target point, and moves the cost less than opportunity cost th more than previous cost. For the 25 balance and 7 unbalance assignment problems, This algorithm gets the optimal solution same as Hungarian algorithm. This algorithm improves the time complexity $O(n^3)$ of Hungarian algorithm to $O(n^2)$, and do not performs the transformation process from unbalance to balance assignment in Hungarian algorithm. Therefore, this algorithm can be alter Hungarian algorithm in assignment problem.

본 논문은 할당 문제의 최적해를 찾는 대표적인 Hungarian 보다 간단히 최적해를 구하는 알고리즘을 제안하였다. Hungarian 알고리즘은 행과 열의 최소 비용을 선택하고 각 비용에서 최소 비용을 뺀다. 다음으로 0을 모두 포함하는 최소한의 선이 행의 개수 가 될 때까지 수행한다. 반면에 제안된 알고리즘은 단지 행에 대해 최소 비용을 선택한다. 다음으로 열에 대해 2개 이상 선택된 열을 출발지로, 하나도 선택되지 않은 열을 목적지로하여 출발지의 비용에서 목적지의 최소 비용과의 차이인 기회비용이 가장 큰 비용은 고정시키고 기회비용이 보다 작은 비용들을 다음으로 큰 비용으로 이동시키는 방법을 적용하였다. 제안된 방법을 25개의 균형 할당과 7개의 불균형 할당 문제에 적용하여 Hungarian 알고리즘과 동일한 최적해를 구하는데 성공하였다. 제안된 알고리즘은 Hungarian 알고리즘의 수행 복잡도 $O(n^3)$를 $O(n^2)$로 개선하였으며, 불균형을 균형 할당 문제로 변환시키는 과정도 수행하지 않는 단순한 알고리즘이다. 따라서 할당 문제의 Hungarian 알고리즘을 대체시킬 수 있을 것이다.

Keywords

References

  1. Wikipedia, "Assignment Problem," http://en.wikipedia.org/wiki/Assignment_problem, Wikimedia Foundation Inc., 2008.
  2. Wikipedia, "Hungarian Algorithm," http://en.wikipedia.org/wiki/Hungarian_algorithm, Wikimedia Foundation Inc., 2008.
  3. L. Ntaimo, "Introduction to Mathematical Programming: Operations Research: Transportation and Assignment Problems", Vol. 1, 4th edition, by W. L. Winston and M. Venkataramanan, http://ie.tamu.edu/INEN420/INEN420 _2005Spring/SLIDES/Chapter 7.pdf, 2005.
  4. K. Kinahan and J. Pryor, "Algorithm Animations for Practical Optimization: A Gentle Introduction," http://optlab-server.sce.carleton.ca/POAnimations2007/Default.html, 2007.
  5. R. Burkard, M. D. Amico, and S. Martello, "Assignment Problems, http://www.assignmentproblems.com/HA4 applet.htm, SIAM Monographs on Discrete Mathematics and Applications, 2006.
  6. S. U. Lee, "The Optimal Algorithm for Assignment Problem," Journal of Korea Society of Computer Information, Vol. 17, No. 9, pp. 139-147, Sep, 2012. https://doi.org/10.9708/jksci/2012.17.9.139
  7. S. U. Lee, "A Reverse-delete Algorithm for Assignment Problem," Journal of Korean Institute of Information Technology, Vol. 10, No. 8, pp. 117-126, Aug, 2012.
  8. W. Snyder, "The Linear Assignment Problem," Department of Electrical and Computer Engineering, North Carolina State University, http://www4.ncsu.edu/-wes/Assignment Problem.pdf, 2005.
  9. M. E. Salassi, "AGEC 7123: Operations Research Methods in Agricultural Economics: Standard LP Form of the Generalized Assignment Problem," Department of Agricultural Economics and Agribusiness, Louisiana State University, http://www.agecon.lsu.edu/WebClasses/AGEC_7123/2004-Materials/Ovhd-18.pdf, 2005.
  10. A. Dimitrios, P. Konstantinos, S. Nikolaos, and S. Angelo, "Applications of a New Network-enabled Solver for the Assignment Problem in Computeraided Education," Journal of Computer Science, Vol. 1, No. 1, pp. 19-23, 2005.
  11. S. Noble, "Lectures 15: The Assignment Problem," Department of Mathematical Sciences, Brunel University, http://people.brunel.ac.uk/-mastsdn/combopt/handout8.html, 2000.
  12. S. C. Niu, "Introduction to Operations Research," http://www.utdallas.edu/-scniu/OPRE-6201/documents/TP2-Initialization.pdf, School of Management, The University of Texas at Dallas, 2004.
  13. Rai Foundation Colleges, "Information Research," Bachelor of Business Administration, Business Administration, http://www.rocw.raifoundation.org/management/bba/OperationResearch/lecture-notes/, 2008.
  14. J. E. Beasley, "Operations Research and Management Science: OR-Notes," Department of Mathematical Sciences, Brunel University, West London, http://people.brunel.ac.uk/-mastjjb/jeb/or /contents.html,2004.
  15. J. Havlicek, "Introduction to Management Science and Operation Research," http://orms.czu.cz/text/transproblem. html, 2007.
  16. D. Doty, "Munkres' Assignment Algorithm: Modified for Rectangular Matrices," KCVU, Murray State University, Dept. of Computer Science and Information Systems, http://www.public.iastate.edu/-ddoty/Hungarian Algorithm.html, 2008.
  17. M. S. Radhakrishnan, "AAOC C222: Optimization," Birla Institute of Technology & Science, http://discovery.bits-pilani.ac.in/discipline/math/msr/aaoc222/ppt/assgn1.ppt, 2006.
  18. K. Wayne, "Algorithm Design," http://www.cs.princeton.edu/-wayne/kleinberg-tardos/07assignment.pdf, 2005.
  19. R. M. Berka, "A Tutorial on Network Optimization,"http://home.eunet.cz/berka/o/English/networks/ node8.html, 1997.
  20. R. Sedgewick and K. Wayne, " Computer Science 226: Data Structures and Algorithms, Princeton University, http://www.cs.princeton.edu/courses/archive/spr02/cs226/assignments/assign.html, 2002.
  21. M. A. Trick, "Network Optimizations for Consultants," http://mat.gsia.cmu.edu/mstc/networks/ networks.html, 1996.
  22. Optimalon Software, "Transportation Problem (Minimal Cost)," http://www.optimalon.com/examples/transport. htm, 2008.

Cited by

  1. Linear Bottleneck Assignment Problem Based on Reverse-delete Algorithm vol.13, pp.6, 2013, https://doi.org/10.7236/JIIBC.2013.13.6.211