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Research Article

Algorithms for generating Pareto fronts of multi-objective integer and mixed-integer programming problems

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Pages 1413-1425 | Received 05 Oct 2020, Accepted 19 Apr 2021, Published online: 24 Jun 2021

References

  • Achterberg, T. 2009. “Solving Constraint Integer Programs.” Mathematical Programming Computation 1: 1–41.
  • Antunes, C. H., M. J. Alves, and J. Climaco. 2016. Multiobjective Linear and Integer Programming. Cham, Switzerland: Springer.
  • Belotti, P., B. Soylu, and M. M. Wiecek. 2013. “A Branch-and-Bound Algorithm for Biobjective Mixed-Integer Programs.” Optimization Online. http://www.optimization-online.org/DB_FILE/2013/01/3719.pdf.
  • Boland, N., H. Charkhgard, and M. Savelsbergh. 2016. “The L-Shape Search Method for Triobjective Integer Programming.” Mathematical Programming Computation 8: 217–251.
  • Boland, N., H. Charkhgard, and M. Savelsbergh. 2017. “A New Method for Optimizing a Linear Function over the Efficient Set of a Multiobjective Integer Program.” European Journal of Operational Research 260: 904–919.
  • Bonami, P., L. T. Biegler, A. R. Conn, G. Cornuéjols, I. E. Grossmann, C. D. Laird, J. Lee, A. Lodi, F. Margot, and A. Wächter. 2008. “An Algorithmic Framework for Convex Mixed Integer Nonlinear Programs.” Discrete Optimization 5 (2): 25–57.
  • Burachik, R. S., C. Y. Kaya, and M. M. Rizvi. 2014. “A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets.” Journal of Optimization Theory and Applications 162: 428–446.
  • Burachik, R. S., C. Y. Kaya, and M. M. Rizvi. 2017. “A New Scalarization Technique and New Algorithms to Generate Pareto Fronts.” SIAM Journal on Optimization 27: 1010–1034.
  • Cabrera, G., M. Ehrgott, A. Mason, and A. Philpott. 2014. “Multi-Objective Optimisation of Positively Homogeneous Functions and an Application in Radiation Therapy.” Operations Research Letters 42: 268–272.
  • Ceyhan, G., M. Koksalan, and B. Lokman. 2019. “Finding a Representative Nondominated Set for Multi-Objective Mixed Integer Programs.” European Journal of Operational Research 272: 61–77.
  • Chankong, V., and Y. Y. Haimes. 1983. Multiobjective Decision Making: Theory and Methodology. Amsterdam: North-Holland.
  • Current, J. R., C. S. ReVelle, and J. L. Cohon. 1985. “The Maximum Covering/Shortest Path Problem: A Multiobjective Network Design and Routing Formulation.” European Journal of Operational Research 21: 189–199.
  • Dächert, K., and K. Klamroth. 2014. “A Linear Bound on the Number of Scalarizations Needed to Solve Discrete Tricriteria Optimization Problems.” Journal of Global Optimization 61: 643–676.
  • Das, I., and J. E. Dennis. 1998. “Normal–Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems.” SIAM Journal on Optimization 8: 631–657.
  • de Santis, M., G. Eichfelder, J. Niebling, and S. Rocktäschel. 2020. “Solving Multiobjective Mixed Integer Convex Optimization Problems.” SIAM Journal on Optimization 30 (4): 3122–3145.
  • Dutta, J., and C. Y. Kaya. 2011. “A New Scalarization and Numerical Method for Constructing Weak Pareto Front of Multi-Objective Optimization Problems.” Optimization 60: 1091–1104.
  • Eichfelder, G. 2008. Adaptive Scalarization Methods in Multiobjective Optimization. Berlin: Springer.
  • Gass, S., and T. Saaty. 1955. “The Computational Algorithm for the Parametric Objective Function.” Naval Research Logistics Quarterly 2: 39–45.
  • Gleixner, A., L. Eifler, T. Gally, G. Gamrath, P. Gemander, R. L. Gottwald, and G. Hendel, et al. 2017. The SCIP Optimization Suite 5.0, ZIB-Report 17-61, Zuse Institute, Berlin, Germany.
  • Goel, T., R. Vaidyanathan, R. T. Haftka, W. Shyy, N. V. Queipo, and K. Tucker. 2007. “Response Surface Approximation of Pareto Optimal Front in Multi-Objective Optimization.” Computer Methods in Applied Mechanics and Engineering 196: 879–893.
  • Göpfert, A., H. Riahi, C. Tammer, and C. Zalinescu. 2003. Variational Methods in Partially Ordered Spaces. New York: Springer.
  • Gourion, D., and D. T. Luc. 2008. “Generating the Weakly Efficient Set of Nonconvex Multiobjective Problems.” Journal of Global Optimization 41 (4): 517–538.
  • Gutiérrez, C., and R. López. 2020. “On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems.” Journal of Optimization Theory and Applications 185: 880–902.
  • Jahn, J. 2004. Vector Optimization: Theory, Applications, and Extensions. Berlin: Springer.
  • Kaur, D., S. Mukherjee, and K. Basu. 2015. “Solution of a Multi-Objective and Multi-Index Real-Life Transportation Problem Using Different Fuzzy Membership Functions.” Journal of Optimization Theory and Applications 164: 666–678.
  • Kaya, C. Y., and H. Maurer. 2014. “A Numerical Method for Nonconvex Multi-Objective Optimal Control Problem.” Computational Optimization and Applications 57: 685–702.
  • Kirlik, G., and S. Sayın. 2014. “A New Algorithm for Generating all Nondominated Solutions of Multiobjective Discrete Optimization Problems.” European Journal of Operational Research 232: 479–488.
  • Luc, D. 1989. Theory of Vector Optimization. Berlin: Springer.
  • Marler, R. T., and J. S. Arora. 2004. “Survey of Multi-Objective Optimization Methods for Engineering.” Structural and Multidisciplinary Optimization 26: 369–395.
  • Mavrotas, G., and K. Florios. 2013. “An Improved Version of the Augmented ϵ-Constraint Method (AUGMECON2) for Finding the Exact Pareto Set in Multiobjective Integer Programming Problems.” Applied Mathematics and Computation 219: 9652–9669.
  • Mela, K., J. Koski, and R. Silvennoinen. 2007. “Algorithm for Generating the Pareto Optimal Set of Multiobjective Nonlinear Mixed-Integer Optimization Problems.” In Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Conference on Structures, Structural Dynamics, and Materials. Reston, VA: American Institute of Aeronautics and Astronautics.
  • Miettinen, K. M. 1999. Nonlinear Multiobjective Optimization. Boston, MA: Kluwer Academic.
  • Mueller-Gritschneder, D., H. Graeb, and U. Schlichtmann. 2009. “A Successive Approach to Compute the Bounded Pareto Front of Practical Multiobjective Optimization Problems.” SIAM Journal on Optimization 20: 915–934.
  • Niebling, J., and G. Eichfelder. 2019. “A Branch-and-Bound-Based Algorithm for Nonconvex Multiobjective Optimization.” SIAM Journal on Optimization 29: 794–821.
  • Ozlen, M., and M. Azizoglu. 2009. “Multi-Objective Integer Programming: A General Approach for Generating all Non-Dominated Solutions.” European Journal of Operational Research 199: 25–35.
  • Ozlen, M., B. A. Burton, and C. A. G. MacRae. 2014. “Multi-Objective Integer Programming: An Improved Recursive Algorithm.” Journal of Optimization Theory and Applications 160: 470–482.
  • Pardalos, P., A. Žilinskas, and J. Žilinskas. 2017. Non-Convex Multi-Objective Optimization. Cham, Switzerland: Springer.
  • Pascoletti, A., and P. Serafini. 1984. “Scalarizing Vector Optimization Problems.” Journal of Optimization Theory and Applications 42: 499–524.
  • Pettersson, W., and M. Ozlen. 2017. “A Parallel Approach to Bi-Objective Integer Programming.” ANZIAM Journal 58: 69–81.
  • Przybylski, A., X. Gandibleux, and M. Ehrgott. 2010a. “A Recursive Algorithm for Finding all Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme.” INFORMS Journal on Computing 22: 371–386.
  • Przybylski, A., X. Gandibleux, and M. Ehrgott. 2010b. “A Two Phase Method for Multiobjective Integer Programming and its Application to the Assignment Problem with Three Objectives.” Discrete Optimization 7: 149–165.
  • Rais, A., F. Alvelos, and M. S. Carvalho. 2014. “New Mixed Integer-Programming Model for the Pickup-and-Delivery Problem with Transshipment.” European Journal of Operational Research 235: 530–539.
  • Rizvi, M. M. 2013. “New Optimality Conditions for Nonlinear Multiobjective Optimization Problems and New Scalarization Techniques for Constructing Pathological Pareto fronts.” PhD thesis, University of South Australia, Australia. https://ap01-a.alma.exlibrisgroup.com/view/delivery/61USOUTHAUS_INST/12146673610001831.
  • Romanko, O. 2010. “Parametric and Multiobjective Optimization with Applications in Finance.” PhD thesis, McMaster University, Canada.
  • Ruzika, S., and M. M. Wiecek. 2005. “Approximation Methods in Multiobjective Programming.” Journal of Optimization Theory and Applications 126: 473–501.
  • Sahinidis, N. V. 2017. BARON 17.8.9 (Global Optimization of Mixed-Integer Nonlinear Programs); user's manual and software. http://archimedes.cheme.cmu.edu/?q=baron.
  • Schenker, S. 2019. “Multicriteria Linear Optimisation with Applications in Sustainable Manufacturing.” PhD thesis, Berlin Institute of Technology, Berlin.
  • Smimou, K. 2014. “International Portfolio Choice and Political Instability Risk: A Multi-Objective Approach.” European Journal of Operational Research 234: 546–560.
  • Statnikov, R., J. Matusov, and A. Statnikov. 2012. “Multicriteria Engineering Optimization Problems: Statement, Solution and Applications.” Journal of Optimization Theory and Applications 155: 355–375.
  • Stewart, T. J., and R. C. van den Honert, eds. 1997. Trends in Multicriteria Decision Making, Vol. 465 of Lecture Notes in Economics and Mathematical Systems. Berlin: Springer.
  • Sun, L. G., W. De Puy, and G. W. Evans. 2014. “Multi-Objective Optimization Models for Patient Allocation During a Pandemic Influenza Outbreak.” Computers and Operations Research 51: 350–359.
  • Tawarmalani, M., and N. V. Sahinidis. 2005. “A Polyhedral Branch-and-Cut Approach to Global Optimization.” Mathematical Programming 103: 225–249.
  • Ulungu, E. L., and J. Teghem. 1994. “Multi-Objective Combinatorial Optimization Problems: A Survey.” Journal of Multi-Criteria Decision Analysis 3: 83–104.
  • Vaidyanathan, R., P. K. Tucker, N. Papila, and W. Shyy. 2004. “CFD-Based Design Optimization for a Single Element Rocket Injector.” Journal of Propulsion and Power 20: 705–717.
  • Wächter, A., and L. T. Biegler. 2006. “On the Implementation of a Primal–Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming.” Mathematical Programming 106 (1): 25–57.
  • Yang, X. Q., and C. J. Goh. 1997. “A Method for Convex Curve Approximation.” European Journal of Operational Research 97: 205–212.
  • Yu, P. L. 1985. Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions. New York: Plenum Press/Springer US.
  • Yu, S., S. Zheng, S. Gao, and J. Yang. 2017. “A Multi-Objective Decision Model for Investment in Energy Savings and Emission Reductions in Coal Mining.” European Journal of Operational Research 260: 335–347.
  • Zadeh, L. 1963. “Optimality and Non-Scalar-Valued Performance Criteria.” IEEE Transactions on Automatic Control 8: 59–60.
  • Zhang, H., and G. Zhao. 1999. “CMEOC—An Expert System in the Coal Mining Industry.” Expert Systems with Applications 16: 73–77.

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