Abstract
In this paper, a bi-objective integer programming problem is analysed using the characteristic equation that was developed to solve a single-objective pure integer program. This equation can also provides other ranked solutions i.e. 2nd, 3rd,... best solutions. These solutions are potential non-dominated points for a bi-objective integer program, which is being investigated in this paper. A “C” code is developed to solve the characteristic equation, a tool which is not available in the IBM ILOG CPLEX library. Two versions of this algorithm are developed to identify the non-dominated points for the bi-objective integer programming problem. The second version improves on the first by reducing the number of search steps. Computational experiments are carried out with respect to the two algorithms developed in this paper and comparisons have also been carried out with one of the recently developed method, the balanced box method. These computational experiments indicate that the second version of the algorithm developed in this paper performed significantly better than the first version and out performed the balanced box method with respect to both CPU time and the number of iterations.
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References
Abounacer R, Rekik M, Renaud J (2014) An exact solution approach for multi-objective location-transportation problem for disaster response. Comput Oper Res 41:83–93
Al-Hasani A, Al-Rabeeah M, Kumar S, Eberhard E (2018) An improved cpu time in triangle splitting method for solving a biobjective mixed integer program. Int J Math Eng Manag Sci 3(4):351–364
Aneja YP, Nair KP (1979) Bicriteria transportation problem. Manag Sci 25(1):73–78
Antunes CH, Alves MJ, Clímaco J (2016) Multiobjective linear and integer programming. Springer, Berlin
Bérubé J-F, Gendreau M, Potvin J-Y (2009) An exact \(\epsilon \)-constraint method for bi-objective combinatorial optimization problems: application to the traveling salesman problem with profits. Eur J Oper Res 194(1):39–50
Boland N, Charkhgard H, Savelsbergh M (2015) A criterion space search algorithm for biobjective integer programming: the balanced box method. INFORMS J Comput 27(4):735–754
Chalmet L, Lemonidis L, Elzinga D (1986) An algorithm for the bi-criterion integer programming problem. Eur J Oper Res 25(2):292–300
Chankong V, Haimes YY (2008) Multiobjective decision making: theory and methodology. Courier Dover Publications, New York
Ehrgott M (2006) A discussion of scalarization techniques for multiple objective integer programming. Ann Oper Res 147(1):343–360
Ehrgott M, Gandibleux X (2000) A survey and annotated bibliography of multiobjective combinatorial optimization. OR-Spektrum 22(4):425–460
Ehrgott M, Ruzika S (2008) Improved \(\varepsilon \)-constraint method for multiobjective programming. J Optim Theory Appl 138(3):375
Hamacher HW, Pedersen CR, Ruzika S (2007) Finding representative systems for discrete bicriterion optimization problems. Oper Res Lett 35(3):336–344
Kumar S, Munapo E (2012) Some lateral ideas and their applications for developing new solution procedures for a pure integer programming model, keynote address. In: Marriappan, Srinivasan, Amritraj (eds) Proceedings of herbal international conference on applications of mathematics and statistics for intelligent solutions through mathematics and statistics, Excell India Publisher, pp 13–21
Kumar S, Munapo E, Jones B (2007) An integer equation controlled descending path to a protean pure integer program. Indian J Math 49(2):211–237
Parragh SN, Tricoire F (2018) Branch-and-bound for bi-objective integer programming. arXiv preprint arXiv:1809.06823
Sourd F, Spanjaard O, Perny P (2006) Multi-objective branch and bound. Application to the bi-objective spanning tree problem. In: 7th international conference in multi-objective programming and goal programming
Soylu B, Yıldız GB (2016) An exact algorithm for biobjective mixed integer linear programming problems. Comput Oper Res 72:204–213
Stanimirovic IP (2012) Compendious lexicographic method for multi-objective optimization. Ser Math Inform. 27(1):55–66
Stidsen T, Andersen KA, Dammann B (2014) A branch and bound algorithm for a class of biobjective mixed integer programs. Manag Sci 60(4):1009–1032
Vincent T, Seipp F, Ruzika S, Przybylski A, Gandibleux X (2013) Multiple objective branch and bound for mixed 0–1 linear programming: corrections and improvements for the biobjective case. Comput Oper Res 40(1):498–509
Vira C, Haimes YY (1983) Multiobjective decision making: theory and methodology. In: North Holland series in system science and engineering, number 8. North-Holland
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Al-Rabeeah, M., Kumar, S., Al-Hasani, A. et al. Bi-objective integer programming analysis based on the characteristic equation. Int J Syst Assur Eng Manag 10, 937–944 (2019). https://doi.org/10.1007/s13198-019-00824-7
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DOI: https://doi.org/10.1007/s13198-019-00824-7