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The singly constrained assignment problem: An AP basis algorithm

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Abstract

This manuscript presents a specialized primal simplex algorithm to obtain near optimal integer solutions for the singly constrained assignment problem. An optimal solution for the continuous relaxation of the problem is obtained by generalizing the alternating basis algorithm of Barr, Glover, and Klingman for the pure assignment problem. Near optimal integer solutions are obtained by pivoting into the optimal basis the slack variable associated with the side constraint. Our empirical analysis indicated that for our test problems the soft-ware implementation of this algorithm was six times faster than CPLEX and four times faster than NETSIDE (a specialized code for network problems with side constraints). The integer solutions obtained in our tests were typically within 2% of the optimum of the continuous relaxation.

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References

  1. V. Aggarwal, “A Lagrangean-Relaxation Method for the Constrained Assignment Problem”, Computers and Operations Research, Vol. 12, pp. 97–106, 1985.

    Google Scholar 

  2. M. Balinski, “Signature Methods for the Assignment Problem”. Operations Research, Vol. 33, pp. 527–536, 1985.

    Google Scholar 

  3. M. Balinski, “A Competitive (Dual) Simplex Method for the Assignment Problem”, Mathematical Programming, Vol. 34, pp. 125–141, 1986.

    Google Scholar 

  4. M. Ball, U. Derigs, C. Hilbrand, and A. Metz, “Matching Problems with Generalized Upper Bound Side Constraints”, Networks, Vol. 20, pp. 703–721, 1990.

    Google Scholar 

  5. R. Barr, F. Glover, and D. Klingman, “The Alternating Basis Algorithm for the Assignment Problem”, Mathematical Programming, Vol. 13, pp. 1–13, 1977.

    Google Scholar 

  6. R. Barr, K. Farhangian, and J. Kennington, “Networks with Side Constraints: An LU Factorization Update”, The Annals of the Society of Logistics Engineers, Vol. 1, pp.66–85, 1986.

    Google Scholar 

  7. D. Bertsekas, “A New Algorithm for the Assignment Problem”, Mathematical Programming, Vol. 21, pp. 152–177, 1981.

    Google Scholar 

  8. D. Bertsekas, “The Auction Algorithm: A Distributed Relaxation Method for the Assignment Problem”, Annals of Operations Research, Vol. 4, pp. 105–123, 1988.

    Google Scholar 

  9. R. Bixby, “Implementing the Simplex Method: The Initial Basis”, ORSA Journal of Computing, Vol. 4, pp. 267–284, 1992.

    Google Scholar 

  10. J. Brans, M. Leclercq, and P. Hansen, “An Algorithm for Optimal Reloading of Pressurized Water Reactors”, Operational Research '72, Editor M. Ross, North Holland Publishing Company: Amsterdam, pp. 417–428, 1973.

    Google Scholar 

  11. N. Bryson, “Parametric Programming and Lagrangian Relaxation: The Case of the Network Problem with a Single Side-Constraint”, Computers and Operations Research, Vol. 18, pp. 129–140, 1991.

    Google Scholar 

  12. S. Gass and T. Saaty, “The Computational Algorithm for the Parametric Objective Function”, Naval Research Logistics Quarterly, Vol. 2, pp. 39–45, 1955.

    Google Scholar 

  13. F. Glover, D. Karney, D. Klingman, and R. Russell, “Solving Singly Constrained Transshipment Problems”, Transportation Science, Vol. 12, pp. 277–297, 1978.

    Google Scholar 

  14. A. Gupta and J. Sharma, “Tree Search Method for Optimal Core Management of Pressurized Water Reactors”, Computers and Operations Research Vol. 8, pp. 263–269, 1981.

    Google Scholar 

  15. R. Helgason and J. Kennington, “Network Flows”, Technical Report 92-CSE-26, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275, 1992.

    Google Scholar 

  16. M. Hung and W. Rom, “Solving the Assignment Problem by Relaxation”, Operations Research Vol. 28, pp. 969–982, 1980.

    Google Scholar 

  17. R. Jonker and T. Volgenant, “A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems”, Computing, Vol. 38 pp. 325–340, 1987.

    Google Scholar 

  18. J. Kennington and R. Helgason, Algorithms for Network Programming, John Wiley and Sons: New York, NY, 1980.

    Google Scholar 

  19. J. Kennington and Z. Wang, SEMI Users Guide, Technical Report 90-CSE-20, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275, 1990.

    Google Scholar 

  20. J. Kennington and Z. Wang, “A Shortest Augmenting Path Algorithm for the Semi-Assignment Problem”, Operations Research, Vol. 40 pp. 178–187, 1992.

    Google Scholar 

  21. D. Klingman and R. Russell, “Solving Constrained Transportation Problems”, Operations Research, Vol. 23, pp. 91–106, 1975.

    Google Scholar 

  22. D. Klingman and R. Russell, “A Stream Lined Approach to the Singly Constrained Transportation Problem”, Naval Research Logistics Quarterly, Vol. 25, pp. 681–695, 1978.

    Google Scholar 

  23. H. Kuhn, “The Hungarian Method for the Assignment Problem”, Naval Research Logistics Quarterly, Vol. 2, pp. 83–97, 1955.

    Google Scholar 

  24. J. Mazzola and A. Neebe, “Resource Constrained Assignment Scheduling”, Operations Research, Vol. 34, pp. 560–572, 1986.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, School of Engineering and Applied Science, Southern Methodist University, 75275-0122, Dallas, Texas

    Jeffery L. Kennington & Farin Mohammadi

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  1. Jeffery L. Kennington
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  2. Farin Mohammadi
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Kennington, J.L., Mohammadi, F. The singly constrained assignment problem: An AP basis algorithm. Comput Optim Applic 4, 347–374 (1995). https://doi.org/10.1007/BF01300862

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  • DOI: https://doi.org/10.1007/BF01300862

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