Abstract
This paper proposes a new problem entitled as “the quickest flow over time network interdiction problem”. This problem stands for removing some of network links using a limited interdiction resource with the aim of maximizing the minimum time required to transfer a predefined flow value through a given network. We formulate the quickest flow problem as a linear fractional programming problem and then, we transform it to a linear formulation. Using the linear formulation of the quickest flow problem we formulate the quickest flow network interdiction problem as a mixed integer linear programming problem. We also provide an improved formulation for the quickest flow network interdiction problem which is computationally more efficient than basic linear formulation. Finally, we apply the basic and improved formulations of the quickest flow network interdiction problem on a real world network and several grid networks.
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References
Afshari-Rad M, Taghizadeh-Kakhki H (2013) Maximum dynamic network flow interdiction problem: new formulation and solution procedures. Comput Ind Eng 65(4):531–536
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall Inc, Upper Saddle River
Akgün I, Tansel Bc, Wood R (2011) The multi-terminal maximum-flow network-interdiction problem. Eur J Oper Res 211(2):241–251. https://doi.org/10.1016/j.ejor.2010.12.011
Anderson EJ, Philpott AB (1994) Optimisation of flows in networks over time. In: Kelly FP (ed) Probability, statistics and optimisation, chap. 27. Wiley, New York, pp 369–382
Benders J (1962) Partioning procedures for solving mixed integer variables programming problems. Numer Math 4:238–252
Burkard RE, Dlaska K, Klinz B (1993) The quickest flow problem. ZOR Methods Models Oper Res 37(1):31–58
Charnes A, Cooper W (1962) Programming with linear fractional functionals. Nav Res Logist Q 9:181–186
Corley H, Sha D (1982) Most vital links and nodes in weighted networks. Oper Res Lett 1(4):157–160
Cormican K, Morton D, Wood R (1998) Stochastic network interdiction. Oper Res 46:184–197
Fleischer LK (2001) Faster algorithms for the quickest transshipment problem. SIAM J Optim 12(1):18–35
Fleischer L, Skutella M (2002) The quickest multicommodity flow problem. Springer, Berlin, pp 36–53
Fleischer L, Skutella M (2007) Quickest flows over time. SIAM J Comput 36(6):1600–1630
Fleischer L, Tardos É (1998) Efficient continuous-time dynamic network flow algorithms. Oper Res Lett 23(3–5):71–80
Ford L, Fulkerson D (1958) Constructing maximal dynamic flows from static flows. Oper Res 6:419–433
Fulkerson D, Harding G (1977) Maximizing the minimum source-sink path subject to a budget constraint. Math Program 13:116–118
Ghare P, Montgomery D, Turner T (1971) Optimal interdiction policy for a flow network. Nav Res Logist Q 18:37–45
Golden B (1978) A problem in network interdiction. Nav Res Logist Q 25:711–713
Hoppe B, Tardos É (2000) The quickest transshipment problem. Math Oper Res 25(1):36–62
Ionac D (2001) Some duality theorems for linear-fractional programming having the coefficients in a subfield \(k\) of real numbers. Studia Universitatis Babeş-Bolyai Mathematica XLV I(4):69–73
Israeli E (1999) System interdiction and defense. Ph.D. thesis, Operations Research Department, Naval Postgraduate School, Monterey, California
Israeli E, Wood R (2002) Shortest path network interdiction. Networks 40:97–111
Lim C, Smith C (2007) Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Trans 39:15–26
Lin KC, Chern MS (1993) The fuzzy shortest path problem and its most vital arcs. Fuzzy Sets Syst 58:343–353
Lin M, Jaillet P (2015) On the quickest flow problem in dynamic networks: a parametric min-cost flow approach. In: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms
Lunday BJ, Sherali HD (2010) A dynamic network interdiction problem. Informatica 21(4):553–574
Malik K, Mittal A, Gupta S (1989) The k most vital arcs in the shortest path problem. Oper Res Lett 8:223–227
McMasters A, Mustin T (1970) Optimal interdiction of a supply network. Nav Res Logist 17:261–268
Morowati-Shalilvand S, Mehri-Tekmeh J (2012) Finding most vital links over time in a flow network. Int J Optim Control Theor Appl 2(2):173–186
Murty KG (1983) Linear programming. Wiley, New York
Royset JO, Wood RK (2007) Solving the bi-objective maximum-flow network-interdiction problem. INFORMS J Comput 19(2):175–184
Schrijver A (1986) Theory of linear and integer programming. Wiley, New York
Skutella M (2009) An introduction to network flows over time. In: Cook W, Lovasz L, Vygen J (eds) Research trends in combinatorial optimization. Springer, Berlin
Stancu-Minasian IM, Tigan S (1993) On some methods for solving fractional programming problems with inexact data. Studii si Cercetari Matematice 45(6):517–532
Whiteman P (1999) Improving single strike effectiveness for network interdiction. Mil Oper Res 4:15–30
Wollmer RD (1963) Some methods for determining the most vital link in a railway network. RAND Memorandum RM-3321-ISA
Wollmer RD (1970) Algorithm for targeting strikes in a lines-of-communication network. Oper Res 18:497–515
Wood RK (1993) Deterministic network interdiction. Math Comput Model 17:1–18
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Morowati-Shalilvand, S., Shahmorad, S., Mirnia, K. et al. Quickest flow over time network interdiction: mathematical formulation and a solution method. Oper Res Int J 21, 1179–1209 (2021). https://doi.org/10.1007/s12351-019-00472-6
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DOI: https://doi.org/10.1007/s12351-019-00472-6