Abstract
In this paper, an extension of the minimum cost flow problem is considered in which multiple incommensurate weights are associated with each arc. In the minimum cost flow problem, flow is sent over the arcs of a graph from source nodes to sink nodes. The goal is to select a subgraph with minimum associated costs for routing the flow. The problem is tractable when a single weight is given on each arc. However, in many real-world applications, several weights are needed to describe the features of arcs, including transit cost, arrival time, delay, profit, security, reliability, deterioration, and safety. In this case, finding an optimal solution becomes difficult. We propose a heuristic algorithm for this purpose. First, we compute the relative efficiency of the arcs by using data envelopment analysis techniques. We then determine a subgraph with efficient arcs using a linear programming model, where the objective function is based on the relative efficiency of the arcs. The flow obtained satisfies the arc capacity constraints and the integrality property. Our proposed algorithm has polynomial runtime and is evaluated in rigorous experiments.
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Notes
This sentence is a direct quote from Cooper et al. (2006).
See GAMS—A User’s Guide, Frechen, Germany: GAMS Development Corporation, 2021.
References
Ahuja RK, Magnanti TL, Orlin JB, et al (1995) Applications of network optimization. In: Ball MO, Magnanti TL, Monma CL et al (eds) Network models, handbooks in operations research and management science, vol 7. Elsevier B. V., Cambridge, MA, USA, chap 1, pp 1–83. https://doi.org/10.1016/S0927-0507(05)80118-5
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows—theory, algorithms and applications. Prentice Hall, Englewood Cliffs
Amirteimoori A (2011) An extended transportation problem: a DEA-based approach. Central Eur J Oper Res (CEJOR) 19(4):513–521. https://doi.org/10.1007/s10100-010-0140-0
Barabási A, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512. https://doi.org/10.1126/science.286.5439.509
Bazaraa MS, Jarvis JJ, Sherali HD (2011) Linear programming and network flows. Wiley, Hoboken
Belling-Seib K, Mevert P, Müller C (1988) Network flow problems with one side constraint: a comparison of three solution methods. Comput Oper Res 15(4):381–394. https://doi.org/10.1016/0305-0548(88)90022-6
Bollobás B (2011) Random Graphs, 2nd edn. No. 73 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511814068
Bryson N (1991) Parametric programming and Lagrangian relaxation: the case of the network problem with a single side-constraint. Comput Oper Res 18(2):129–140. https://doi.org/10.1016/0305-0548(91)90084-5
Charnes A, Cooper WW (1962) Programming with linear fractional functions. Naval Res Logist 9:181–186. https://doi.org/10.1002/nav.3800090303
Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res (EJOR) 2:429–444. https://doi.org/10.1016/0377-2217(78)90138-8
Chaudhry IA, Khan AA (2016) A research survey: review of flexible job shop scheduling techniques. Int Trans Oper Res (ITOR) 23(3):551–591. https://doi.org/10.1111/itor.12199
Chen P (2021) Effects of the entropy weight on TOPSIS. Expert Syst Appl 168(114):186. https://doi.org/10.1016/j.eswa.2020.114186
Chen L, Lu H (2007) An extended assignment problem considering multiple inputs and outputs. Appl Math Model 31:2239–2248. https://doi.org/10.1016/j.apm.2006.08.018
Chen S, Saigal R (1977) A primal algorithm for solving a capacitated network flow problem with additional linear constraints. Networks 7(1):59–79. https://doi.org/10.1002/net.3230070105
Cook WD, Tone K, Zhu J (2014) Data envelopment analysis: prior to choosing a model. Omega 44:1–4. https://doi.org/10.1016/j.omega.2013.09.004
Cooper WW, Seiford LM, Tone K (2006) Introduction to data envelopment analysis and its uses: with DEA-solver software and references. Springer, New York. https://doi.org/10.1007/0-387-29122-9
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Hoboken
Deo N (1974) Graph theory with applications to engineering and computer science. Prentice Hall Series in Automatic Computation. Prentice-Hall Inc, Upper Saddle River
Doyle J, Green R (1994) Efficiency and cross-efficiency in DEA: derivations, meanings and uses. J Oper Res Soc 45:567–578. https://doi.org/10.1057/jors.1994.84
Du D, Pardalos PM (1993) Network optimization problems: algorithms, applications and complexity, vol 2. Series on applied mathematics. World Scientific, Singapore
Dürr C, Jez L, Vásquez OC (2015) Scheduling under dynamic speed-scaling for minimizing weighted completion time and energy consumption. Discret Appl Math 196:20–27. https://doi.org/10.1016/j.dam.2014.08.001
Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, Berlin. https://doi.org/10.1007/3-540-27659-9
Erdös P, Rényi A (2006) On the evolution of random graphs. In: Newman M, Barabási A, Watts DJ (eds) The structure and dynamics of networks, vol 19. Princeton studies in complexity. Princeton University Press, Princeton, pp 38–82. https://doi.org/10.1515/9781400841356.38
Fang S, Qi L (2003) Manufacturing network flows: a generalized network flow model for manufacturing process modelling. Optim Methods Softw 18(2):143–165. https://doi.org/10.1080/1055678031000152079
Farrell M (1957) The measurement of productive efficiency. J R Stat Soc Ser A (General) 120(3):253–290
Glover F, Klingman D (1981) The simplex SON algorithm for LP/embedded network problems. In: Mulvey JM, Klingman D (eds) Network models and associated applications, mathematical programming studies, vol 15. Springer, New York, pp 148–176. https://doi.org/10.1007/BFb0120942
Glover F, Klingman D, Ross GT (1974) Finding equivalent transportation formulations for constrained transportation problems. Naval Res Log Q 21(2):247–254
Glover F, Klingman D (1985) Basis exchange characterizations for the simplex SON algorithm for LP/embedded networks. In: Cottle RW (ed) Mathematical programming essays in honor of George B. Dantzig, Part I, mathematical programming Studies, vol 24. Springer, New York, pp 141–157. https://doi.org/10.1007/BFb0121048
Guo L (2016) Efficient approximation algorithms for computing k disjoint constrained shortest paths. J Comb Optim 32(1):144–158. https://doi.org/10.1007/s10878-015-9934-2
Hamacher HW, Pedersen CR, Ruzika S (2007) Multiple objective minimum cost flow problems: a review. Eur J Oper Res (EJOR) 176(3):1404–1422. https://doi.org/10.1016/j.ejor.2005.09.033
Hollander M, Wolfe DA, Chicken E (2014) Nonparametric statistical methods, 3rd edn. Wiley series in probability and statistics. Wiley, Hoboken. https://doi.org/10.1002/9781119196037
Holzhauser M, Krumke SO, Thielen C (2016) Budget-constrained minimum cost flows. J Comb Optim 31(4):1720–1745. https://doi.org/10.1007/s10878-015-9865-y
Hu Y, Zhao X, Liu J et al (2020) An efficient algorithm for solving minimum cost flow problem with complementarity slack conditions. Math Probl Eng 2439265:1–5. https://doi.org/10.1155/2020/2439265
Khan RA, Mohammadani KH, Soomro AA et al (2018) An energy efficient routing protocol for wireless body area sensor networks. Wireless Pers Commun 99(4):1443–1454. https://doi.org/10.1007/s11277-018-5285-5
Klingman D (1977) Finding equivalent network formulations for constrained network problems. Manag Sci 23(7):737–744. https://doi.org/10.1287/mnsc.23.7.737
Kovács P (2015) Minimum-cost flow algorithms: an experimental evaluation. Optim Methods Softw 30(1):94–127. https://doi.org/10.1080/10556788.2014.895828
Lehmann EL (2006) Nonparametrics: statistical methods based on ranks. Springer, New York
Li B, Springer J, Bebis G et al (2013) A survey of network flow applications. J Netw Comput Appl 36(2):567–581. https://doi.org/10.1016/j.jnca.2012.12.020
Longaray AA, Ensslin L, Ensslin SR et al (2018) Using MCDA to evaluate the performance of the logistics process in public hospitals: the case of a Brazilian teaching hospital. Int Trans Oper Res (ITOR) 25(1):133–156. https://doi.org/10.1111/itor.12387
Lu H, Yao E, YaoQi L (2006) Some further results on minimum distribution cost flow problems. J Comb Optim 11(4):351–371. https://doi.org/10.1007/s10878-006-8211-9
Mamer JW, McBride RD (2000) A decomposition-based pricing procedure for large-scale linear programs: an application to the linear multicommodity flow problem. Manage Sci 46(5):693–709. https://doi.org/10.1287/mnsc.46.5.693.12042
Mann HB, Whitney DR (1947) On a test of whether one of two random variables is stochastically larger than the other. Ann Math Stat 18:50–60. https://doi.org/10.1214/aoms/1177730491
Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26(6):369–395. https://doi.org/10.1007/s00158-003-0368-6
Mathies S, Mevert P (1998) A hybrid algorithm for solving network flow problems with side constraints. Comput Oper Res 25(9):745–756. https://doi.org/10.1016/S0305-0548(98)00001-X
McBride RD (1985) Solving embedded generalized network problems. Eur J Oper Res (EJOR) 21(1):82–92
McBride RD, Mamer JW (1997) Solving multicommodity flow problems with a primal embedded network simplex algorithm. INFORMS J Comput 9(2):154–163. https://doi.org/10.1287/ijoc.9.2.154
McBride RD, Mamer JW (2001) Solving the undirected multicommodity flow problem using a shortest path-based pricing algorithm. Networks 38(4):181–188. https://doi.org/10.1002/net.1035
Mo J, Qi L, Wei Z (2005) A manufacturing supply chain optimization model for distilling process. Appl Math Comput 171(1):464–485. https://doi.org/10.1016/j.amc.2005.01.041
Niu S, Matsuhisa N, Beker L et al (2019) A wireless body area sensor network based on stretchable passive tags. Nat Electron 2(8):361–368. https://doi.org/10.1038/S41928-019-0286-2
Orlin JB (1993) A faster strongly polynomial minimum cost flow algorithm. Oper Res 41(2):338–350. https://doi.org/10.1287/opre.41.2.338
Orlin JB (1997) A polynomial time primal network simplex algorithm for minimum cost flows. Math Program 78(2):109–129. https://doi.org/10.1007/BF02614365
Raayatpanah MA (2017a) Multicast routing based on data envelopment analysis with multiple quality of service parameters. Int J Commun Syst. https://doi.org/10.1002/dac.3084
Raayatpanah MA (2017b) Subgraph selection over coded packet networks with multiple QOS parameters based on DEA. Trans Emerg Telecommun Technol. https://doi.org/10.1002/ett.3079
Raayatpanah MA, Fathabadi HS, Khalaj BH et al (2014) Bounds on end-to-end statistical delay and jitter in multiple multicast coded packet networks. J Netw Comput Appl 41:217–227. https://doi.org/10.1016/j.jnca.2013.12.004
Sedeño-Noda A, Alonso-Rodríguez S (2015) An enhanced K-SP algorithm with pruning strategies to solve the constrained shortest path problem. Appl Math Comput 265:602–618. https://doi.org/10.1016/j.amc.2015.05.109
Sexton TR, Silkman RH, Hogan AJ (1986) Data envelopment analysis: critique and extensions. New Direct Program Eval 32:73–105. https://doi.org/10.1002/ev.1441
Singh AK, Mishra SK, Dixit S (2019) Energy efficiency in wireless sensor networks: Cooperative MIMO-OFDM. In: Recent trends in communication, computing, and electronics: select proceedings of IC3E 2018, Lecture notes in electrical engineering, vol 524. Springer, Singapore, pp 147–154. https://doi.org/10.1007/978-981-13-2685-1_16
Sokkalingam PT, Ahuja RK, Orlin JB (2000) New polynomial-time cycle-canceling algorithms for minimum-cost flows. Netw: Int J 36(1):53–63
Song L, Liu F (2018) An improvement in DEA cross-efficiency aggregation based on the Shannon entropy. Int Trans Oper Res (ITOR) 25(2):705–714. https://doi.org/10.1111/itor.12361
Spälti SB, Liebling TM (1991) Modeling the satellite placement problem as a network flow problem with one side constraint. Operations-Research-Spektrum 13(1):1–14. https://doi.org/10.1007/BF01719766
Tran CTT, Villano RA (2018) Measuring efficiency of Vietnamese public colleges: an application of the DEA-based dynamic network approach. Int Trans Oper Res (ITOR) 25(2):683–703. https://doi.org/10.1111/itor.12212
Tzeng G, Huang J (2011) Multiple attribute decision making: methods and applications. Chapman and Hall, London
Venkateshan P, Mathur K, Ballou RH (2008) An efficient generalized network-simplex-based algorithm for manufacturing network flows. J Comb Optim 15(4):315–341. https://doi.org/10.1007/s10878-007-9080-6
Vogiatzis C, Pardalos PM (2013) Combinatorial optimization in transportation and logistics networks. In: Pardalos PM, Du D, Graham RL (eds) Handbook of combinatorial optimization, 2nd edn. Springer, New York, pp 673–722. https://doi.org/10.1007/978-1-4419-7997-1_63
Watts DJ (2003) Small worlds: the dynamics of networks between order and randomness. Princeton studies in complexity. Princeton University Press, Princeton
Waxman BM (1988) Routing of multipoint connections. IEEE J Sel Areas Commun 6(9):1617–1622. https://doi.org/10.1109/49.12889
Yang C, Zhao X (2014) A new delay-constrained multicast routing algorithm based on shared edges. Commun Netw. https://doi.org/10.4236/cn.2014.61006
Yang G, Yang J, Liu W et al (2013) Cross-efficiency aggregation in DEA models using the evidential-reasoning approach. Eur J Oper Res (EJOR) 231(2):393–404. https://doi.org/10.1016/j.ejor.2013.05.017
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We acknowledge funding support by the National Natural Science Foundation of China under Grants 61673359 and 71520107002. P.M. Pardalos was conducted within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE).
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Dr. Raayatpanah has conceptualized and conducted the research. Dr. Khodayifar has helped developing the mathematical modules. Dr. Weise and Dr. Pardalaos both have helped writing, rewriting, and revising the manuscript.
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Raayatpanah, M.A., Khodayifar, S., Weise, T. et al. A novel approach to subgraph selection with multiple weights on arcs. J Comb Optim 44, 242–268 (2022). https://doi.org/10.1007/s10878-021-00823-0
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DOI: https://doi.org/10.1007/s10878-021-00823-0