# Two Uncertain Programming Models for Inverse Minimum Spanning Tree Problem

• Zhang, Xiang ;
• Wang, Qina ;
• Zhou, Jian
• Accepted : 2013.03.05
• Published : 2013.03.31
• 70 15

#### Abstract

An inverse minimum spanning tree problem makes the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. In this paper, the concept of uncertain ${\alpha}$-minimum spanning tree is initiated for minimum spanning tree problem with uncertain edge weights. Using different decision criteria, two uncertain programming models are presented to formulate a specific inverse minimum spanning tree problem with uncertain edge weights involving a sum-type model and a minimax-type model. By means of the operational law of independent uncertain variables, the two uncertain programming models are transformed to their equivalent deterministic models which can be solved by classic optimization methods. Finally, some numerical examples on a traffic network reconstruction problem are put forward to illustrate the effectiveness of the proposed models.

#### Keywords

Minimum Spanning Tree;Uncertain Minimum Spanning Tree;Inverse Optimization;Uncertain Programming

#### References

1. Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993), Network Flows: Theory, Algorithms, and Applications, Prentice Hall, Englewood Cliffs, NJ.
2. Ahuja, R. K. and Orlin, J. B. (2000), A faster algorithm for the inverse spanning tree problem, Journal of Algorithms, 34(1), 177-193. https://doi.org/10.1006/jagm.1999.1052
3. Chen, X. (2011), American option pricing formula for uncertain financial market, International Journal of Operations Research, 8(2), 27-32.
4. Farago, A., Szentesi, A., and Szviatovszki, B. (2003), Inverse optimization in high-speed networks, Discrete Applied Mathematics, 129(1), 83-98. https://doi.org/10.1016/S0166-218X(02)00235-4
5. Guan, X. and Zhang, J. (2007), Inverse constrained bottleneck problems under weighted $l_{{\infty}}$ norm, Computers and Operations Research, 34(11), 3243-3254. https://doi.org/10.1016/j.cor.2005.12.003
6. He, Y., Zhang, B., and Yao, E. (2005), Weighted inverse minimum spanning tree problems under Hamming distance, Journal of Combinatorial Optimization, 9(1), 91-100. https://doi.org/10.1007/s10878-005-5486-1
7. Kershenbaum, A. (1993), Telecommunication Network Design Algorithms, McGraw-Hill, New York, NY.
8. Li, S. and Peng, J. (2012), A new approach to risk comparison via uncertain measure, Industrial Engineering & Management Systems, 11(2), 176-182. https://doi.org/10.7232/iems.2012.11.2.176
9. Liu, B. (2007), Uncertainty Theory (2nd ed.), Springer-Verlag, Berlin.
10. Liu, B. (2009), Some research problems in uncertainty theory, Journal of Uncertain Systems, 3(1), 3-10.
11. Liu, B. (2010), Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer- Verlag, Berlin.
12. Peng, J. and Li, S. (2011), Spanning tree problem of uncertain network, Proceedings of the 3rd International Conference on Computer Design and Applications, Xi'an, Shaanxi, China.
13. Peng, Z. and Iwamura, K. (2010), A sufficient and necessary condition of uncertainty distribution, Journal of Interdisciplinary Mathematics, 13(3), 277-285. https://doi.org/10.1080/09720502.2010.10700701
14. Sheng, Y. and Yao K. (2012), Fixed charge transportation problem and its uncertain programming model, Industrial Engineering and Management Systems, 11(2), 183-187. https://doi.org/10.7232/iems.2012.11.2.183
15. Sokkalingam, P. T., Ahuja, R. K., and Orlin, J. B. (1999), Solving inverse spanning tree problems through network flow techniques, Operations Research, 47(2), 291-298. https://doi.org/10.1287/opre.47.2.291
16. Wang, Q., Yang, X., and Zhang, J. (2006), A class of inverse dominant problems under weighted $l_{{\infty}}$ norm and an improved complexity bound for Radzik's algorithm, Journal of Global Optimization, 34(4), 551-567. https://doi.org/10.1007/s10898-005-1649-y
17. Xu, X. and Zhu, Y. (2012), Uncertain bang-bang control for continuous time model, Cybernetics and Systems, 43(6), 515-527. https://doi.org/10.1080/01969722.2012.707574
18. Yang, X. and Zhang, J. (2007), Some inverse min-max network problems under weighted l1 and $l_{{\infty}}$ norms with bound constraints on changes, Journal of Combinatorial Optimization, 13(2), 123-135.
19. Zhang, B., Zhang, J., and He, Y. (2006), Constrained inverse minimum spanning tree problems under the bottleneck-type Hamming distance, Journal of Global Optimization, 34(3), 467-474. https://doi.org/10.1007/s10898-005-6470-0
20. Zhang, J., Liu. Z., and Ma, Z. (1996), On the inverse problem of minimum spanning tree with partition constraints, Mathematical Methods of Operations Research, 44(2), 171-187. https://doi.org/10.1007/BF01194328
21. Zhang, J. and Zhou, J. (2006), Models and hybrid algorithms for inverse minimum spanning tree problem with stochastic edge weights, World Journal of Modelling and Simulation, 2(5), 297-311.
22. Zhou, C. and Peng, J. (2011), Models and algorithm ofmaximum flow problem in uncertain network, Proceedingsof the 3rd International Conference on-Artificial Intelligence and Computational Intelligence,Taiyuan, Shanxi, China, 101-109.

#### Cited by

1. Multi-objective optimization in uncertain random environments vol.13, pp.4, 2014, https://doi.org/10.1007/s10700-014-9183-3
2. Uncertain Quadratic Minimum Spanning Tree Problem pp.17962021, 2014, https://doi.org/10.12720/jcm.9.5.385-390
3. Entropy of Uncertain Random Variables wi h Application to Minimum Spanning Tree Problem vol.25, pp.04, 2017, https://doi.org/10.1142/S0218488517500210
4. An interactive satisficing approach for multi-objective optimization with uncertain parameters vol.28, pp.3, 2017, https://doi.org/10.1007/s10845-014-0998-0
5. Minimum spanning tree problem of uncertain random network vol.28, pp.3, 2017, https://doi.org/10.1007/s10845-014-1015-3
6. Uncertain risk aversion vol.28, pp.3, 2017, https://doi.org/10.1007/s10845-014-1013-5
7. The covariance of uncertain variables: definition and calculation formulae pp.1573-2908, 2017, https://doi.org/10.1007/s10700-017-9270-3