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A priority based unbalanced time minimization assignment problem

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Abstract

This paper discusses a priority based unbalanced time minimization assignment problem which deals with the allocation of n jobs to \(m~(<n)\) persons such that the project is executed in two stages, viz. Stage-I and Stage-II. Stage-I is composed of \(n_1(<m)\) primary jobs and Stage-II is composed of the remaining \((n-n_1)\) secondary jobs which are commenced only after Stage-I jobs are completed. Each person has to do at least one job whereas each job is to be assigned to exactly one person. It is assumed that the nature of primary jobs is such that one person can perform exactly one job whereas a person is free to perform more than one job in Stage-II. Also, persons assigned to primary jobs cannot perform secondary jobs. In a particular stage, all persons start performing the jobs simultaneously. However, if a person is performing more than one job, he does them one after the other. The objective of the proposed study is to find the feasible assignment that minimizes the overall completion time (i.e., the sum of Stage-I and Stage-II time) for the two stage implementation of the project. In this paper, an iterative algorithm is proposed that solves a constrained unbalanced time minimization assignment problem at each iteration and generates a pair of Stage-I and Stage-II times. In order to solve this constrained problem, a solution strategy is developed in the current paper. An alternative combinations based method to solve the priority based unbalanced problem is also analysed and a comparative study is carried out. Numerical demonstrations are provided in support of the theory.

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References

  1. 1.

    Aggarwal, V.: The assignment problem under categorized jobs. Eur. J. Oper. Res. 14, 193–195 (1983)

  2. 2.

    Aggarwal, V., Tikekar, V.G., Hsu, L.F.: Bottleneck assignment problems under categorization. Comput. Oper. Res. 13(1), 11–26 (1986)

  3. 3.

    Arora, S., Puri, M.C.: A variant of time minimizing assignment problem. Eur. J. Oper. Res. 110, 314–325 (1998)

  4. 4.

    Bhatia, H.L.: Time minimizing assignment problem. SCIMA 6(3), 75–83 (1977)

  5. 5.

    Brandt, A., Intrator, Y.: The assignment problem with three job categories. Cas. Pro Pest. Mat. 96, 8–11 (1971)

  6. 6.

    Burkard, R.E., Rendl, F.: Lexicographic bottleneck problems. Oper. Res. Lett. 10, 303–308 (1991)

  7. 7.

    Carpaneto, G., Toth, P.: Algorithm for the solution of the bottleneck assignment problem. Computing 27, 179–187 (1981)

  8. 8.

    Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Inf. Process. Lett. 100(4), 162–166 (2006)

  9. 9.

    Derigs, U.: Alternate strategies for solving bottleneck assignment problems: analysis and computational results. Computing 33(2), 95–106 (1984)

  10. 10.

    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp 611–620 (2006)

  11. 11.

    Frieze, A.M., Yadagar, J.: On the quadratic assignment problem. Discrete Appl. Math. 5, 89–98 (1983)

  12. 12.

    Garfinkel, R.S.: An improved algorithm for the bottleneck assignment problem. Oper. Res. 19, 1747–1751 (1971)

  13. 13.

    Gevezes, T.P., Pitsoulis, L.S.: A new greedy algorithm for the quadratic assignment problem. Optim. Lett. 7, 207–220 (2013)

  14. 14.

    Ghose, S.: The maximum capacity routes: a lexisearch approach. Opsearch 8(3), 209–225 (1971)

  15. 15.

    Gross, O.: The Bottleneck Assignment Problem. The Rand Corporation, Santa Monica, CA, Technical report P-1630 (1959)

  16. 16.

    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling non identical processors. J. Assoc. Comput. Mach. 23, 317–327 (1976)

  17. 17.

    Iampang, A., Boonjing, V., Chanvarasuth, P.: A cost and space efficient method for unbalanced assignment problems. In: 2010 IEEE International Conference on Industrial Engineering and Engineering Management, Macao, pp 985–988 (2010)

  18. 18.

    Kaur, P., Sharma, A., Verma, V., Dahiya, K.: A priority based assignment problem. Appl. Math. Model. 40(7), 7784–7795 (2016)

  19. 19.

    Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83–97 (1955)

  20. 20.

    Kuhn, H.W.: Variants of the Hungarian method for assignment problems. Nav. Res. Logist. Q. 3, 253–258 (1956)

  21. 21.

    Kumar, A.: A modified method for solving the unbalanced assignment problems. Appl. Math. Comput. 176, 76–82 (2006)

  22. 22.

    Lawler, E.L.: The quadratic assignment problem. Manag. Sci. 9(4), 586–599 (1963)

  23. 23.

    Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R., Schmoys, D.B.: Sequencing and scheduling: algorithms and complexity. Handb. OR MS 4, 445–522 (1993)

  24. 24.

    Malhotra, R., Bhatia, H.L.: Variants of the time minimization assignment problem. Trabajos De Estadistica Y De Investigacian Operativa 35(3), 331–338 (1984)

  25. 25.

    Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)

  26. 26.

    Pandit, S.N.N., Murthy, M.S.: Allocation of sources and destinations. In: \(8^{th}\) Annual Convention of Operational Research Society of India, pp 22–24 (1975)

  27. 27.

    Pandit, S.N.N., Subrahmanyam, Y.V.: Enumeration of all optimal job sequence. Opsearch 12(1–2), 35–39 (1975)

  28. 28.

    Pferchy, U.: Solution methods and computational investigations for the linear bottleneck assignment problem. Computing 59, 237–258 (1997)

  29. 29.

    Ravindran, A., Ramaswamy, V.: On the bottleneck assignment problem. J. Optim. Theory Appl. 21(4), 451–458 (1977)

  30. 30.

    Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976)

  31. 31.

    Sahu, A., Tapadar, R.: Solving the assignment problem using genetic algorithm and simulated annealing. In: IMECS, pp 762–765 (2006)

  32. 32.

    Seshan, C.R.: Some generalizations of time minimizing assignment problem. J. Oper. Res. Soc. 32, 489–494 (1981)

  33. 33.

    Sonia, Puri, M.C.: Two stage time minimizing assignment problem. Omega 36, 730–740 (2008)

  34. 34.

    Subrahmanyam, Y.V.: Some special cases of assignment problems. Opsearch 16(1), 45–47 (1979)

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Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions and comments which helped in improving the manuscript considerably. The first author is thankful to Council of Scientific and Industrial Research, India (Sanction No. 09/135/(0724)/2015-EMR-I) for providing financial support for carrying out this research.

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Correspondence to Kalpana Dahiya.

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Jain, E., Dahiya, K. & Verma, V. A priority based unbalanced time minimization assignment problem. OPSEARCH 57, 13–45 (2020). https://doi.org/10.1007/s12597-019-00399-8

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Keywords

  • Assignment
  • Time minimization
  • Unbalanced
  • Priority
  • Constrained