Typically, multi-objective optimization problems give rise to a large number of optimal solutions. However, this information can be overwhelming to a decision maker. This article introduces a technique to find a representative subset of optimal solutions, of a given bounded cardinality for an unconstrained bi-objective combinatorial optimization problem in terms of \(\epsilon \)-indicator. This technique extends the Nemhauser–Ullman algorithm for the knapsack problem and allows to find a representative subset in a single run. We present a discussion on the representation quality achieved by this technique, both from a theoretical and numerical perspective, with respect to an optimal representation.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Bazgan, C., Jamain, F., Vanderpooten, D.: Approximate Pareto sets of minimal size for multi-objective optimization problems. Oper. Res. Lett. 43(1), 1–6 (2015)
Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. In: Larmore, L.L., Goemans, M.X. (eds.) Proceedngs of the 35rd Annual ACM Symposium on Theory of Computing (STOC), pp. 232–241. ACM Press, New york (2003)
Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–288 (1957)
Diakonikolas, I., Yannakakis, M.: Small approximate Pareto sets for bi-objective shortest paths and other problems. SIAM J. Comput. 39(4), 1340–1371 (2009)
Ebem-Chaime, M.: Parametric solution for linear bricriteria knapsack models. Manag. Sci. 42(11), 1565–1575 (1996)
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)
Eusébio, A., Figueira, J.R., Ehrgott, M.: On finding representative non-dominated points for bi-objective integer network flow problems. Comput. Oper. Res. 48, 1–10 (2014)
Guerreiro, A.P., Fonseca, C.M., Paquete, L.: Greedy hypervolume subset selection in low dimensions. Evol. Comput. 24(3), 521–544 (2016)
Hamacher, H.W., Pedersen, C.R., Ruzika, S.: Finding representative systems for discrete bicriterion optimization problems. Oper. Res. Lett. 35(3), 336–344 (2007)
Jesus, A. D.: Implicit enumeration for representation systems in multiobjective optimization. Master’s thesis, University of Coimbra, Portugal (2015)
Kuhn, T., Fonseca, C.M., Paquete, L., Ruzika, S., Duarte, M.M., Figueira, J.R.: Hypervolume subset selection in two dimensions: formulations and algorithms. Evol. Comput. 24(3), 411–425 (2016)
Nemhauser, G., Ullman, Z.: Discrete dynamic programming and capital allocation. Manag. Sci. 15(9), 494–505 (1969)
Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS ’00, pp. 86–92, Washington, DC, USA, 2000. IEEE Computer Society
Sayın, S.: Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math. Program. 87(3), 543–560 (2000)
Sayın, S.: A procedure to find discrete representations of the efficient set with specified coverage errors. Oper. Res. 51(3), 427–436 (2003)
Sayın, S., Kouvelis, P.: The multiobjective discrete optimization problem: a weighted min-max two-stage optimization approach and a bicriteria algorithm. Manag. Sci. 51(10), 1572–1581 (2005)
Serafini, P.: Some considerations about computational complexity for multiobjective combinatorial optimization. In: Recent Advances and Historical Development of Vector Optimization, Lecture Notes in Economics and Mathematics, pp. 221–231, Berlin, Germany. Springer, Berlin (1986)
Vaz, D., Paquete, L., Fonseca, C.M., Klamroth, K., Stiglmayr, M.: Representation of the non-dominated set in biobjective discrete optimization. Comput. Oper. Res. 63, 172–186 (2015)
Verel, S., Liefooghe, A., Jourdan, L., Dhaenens, C.: Analyzing the effect of objective correlation on the efficient set of MNK-landscapes. In: Proceedings of the 5th Conference on Learning and Intelligent OptimizatioN (LION 5), Lecture Notes in Computer Science, pp. 116–130. Springer, Berlin (2011)
Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithmsa comparative case study. In: Proceedings of the International Conference on Parallel Problem Solving from Nature PPSN V, pp. 292–301. Springer, Berlin (1998)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)
This work was partially supported by the European Regional Development Fund (FEDER), through the COMPETE 2020—Operational Program for Competitiveness and Internationalization (POCI).
About this article
Cite this article
Jesus, A.D., Paquete, L. & Figueira, J.R. Finding representations for an unconstrained bi-objective combinatorial optimization problem. Optim Lett 12, 321–334 (2018). https://doi.org/10.1007/s11590-017-1129-6
- Multiobjective combinatorial optimization
- Representation problem
- Dynamic programming