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Joint chance constrained shortest path problem with Copula theory

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Abstract

In this paper, we investigate the constrained shortest path problem where the arc resources of the problem are dependent normally distributed random variables. A model is presented to maximize the probability of all constraints, while not exceeding a certain amount. We assume that the rows of the constraint matrix are dependent, so we use a marginal distribution of the Copula functions, instead of the distribution functions and the dependency is driven by an appropriate Archimedean Copula. Then, we transform the joint chance-constrained problems into deterministic problems of second-order cone programming. This is a new approach where considers the dependency between resource consumptions and connects Copulas to stochastic resource constrained shortest path problem (SRCSPP). The results indicate that the effect of marginal probability levels is considerable. Moreover, the linear relaxation of SRCSPP is generally not convex; thus we can use lower and upper bounds of the second-order cone programming approximation to solve the relaxation problem. The experimental results show that the SRCSPP with Copula theory can achieve efficient performance.

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References

  • Ahuja RK, Magnanti TL, Orlin JB, Reddy MR (1995) Applications of network optimization. Handb Oper Res Manag Sci 7:1–83

    MathSciNet  MATH  Google Scholar 

  • Bavaud F, Guex G (2012) Interpolating between random walks and shortest paths: a path functional approach. In: International conference on social informatics. Springer, Berlin, Heidelberg

  • Bertsekas DP (1977) Monotone mappings with application in dynamic programming. SIAM J Control Optim 15:438–464

    MathSciNet  MATH  Google Scholar 

  • Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98:49–71

    MathSciNet  MATH  Google Scholar 

  • Chen F, Huang GH, Fan YR, Wang S (2016) A Copula-based chance-constrained waste management planning method: an application to the city of Regina, Saskatchewan, Canada. J Air Waste Manag Assoc 66(3):307–328

    Google Scholar 

  • Chen P, Zeng W, Chen M, Yu G, Wang Y (2019) Modeling arterial travel time distribution by accounting for link correlations: a copula-based approach. J Intell Transp Syst 23(1):28–40

    Google Scholar 

  • Cheng J, Lisser A (2015) Maximum probability shortest path problem. Discrete Appl Math 192:40–48

    MathSciNet  MATH  Google Scholar 

  • Cheng J, Houda M, Lisser A (2015) Chance constrained 0–1 quadratic programs using Copulas. Optim Lett 9(7):1283–1295

    MathSciNet  MATH  Google Scholar 

  • Cheng J, Houda M, Lisser A (2014) Second–order cone programming approach for elliptically distributed joint probabilistic constraints with dependent rows. Technical report, Optimization

  • Cooper C, Frieze A, Mehlhorn K, Priebe V (2000) Average-case complexity of shortest paths problems in the vertex potential model. Random Struct Algorithms 16(1):33–46

    MathSciNet  MATH  Google Scholar 

  • Cornnejols G, Fisher M, Nemhauser G (1977) Location of bank accounts of optimize float: an analytic study of exact and approximate algorithm. Manag Sci 23:789–810

    MathSciNet  MATH  Google Scholar 

  • Dantzig GB (2010) Linear programming under uncertainty, Stochastic programming. Springer, New York, pp 1–11

    MATH  Google Scholar 

  • Darabi R, Baghban M (2018) Application of Clayton Copula in Portfolio optimization and its comparison with Markowitz mean-variance analysis. Adv Math Finance Appl 3(1):33–51

    Google Scholar 

  • De Brún A, McAuliffe E (2018) Social network analysis as a methodological approach to explore health systems: a case study exploring support among senior managers/executives in a hospital network. Int J Environ Res Public Health 15(3):511

    Google Scholar 

  • De La Barriére RP (1980) Optimal control theory: a course in automatic control theory. Dover Pubns, Mineola

    MATH  Google Scholar 

  • Fisher ML (1981) The Lagrangian relaxation method for solving integer programming problems. Manag Sci 27(1):1–18

    MathSciNet  MATH  Google Scholar 

  • Garg D (2018) Dynamizing Dijkstra: a solution to dynamic shortest path problem through retroactive priority queue. J King Saudi Univ Comput Inf Sci. https://doi.org/10.1016/j.jksuci.2018.03.003

    Article  Google Scholar 

  • Guan L, Jianping L, Weidong L, Junran L (2019) Improved approximation algorithms for the combination problem of parallel machine scheduling and path. J Comb Optim 38:689–697

    MathSciNet  MATH  Google Scholar 

  • Hall CA, Meyer WW (1976) Optimal error bounds for cubic spline interpolation. J Approx Theory 16(2):105–122

    MathSciNet  MATH  Google Scholar 

  • Handler GY, Zang I (1980) A dual algorithm for the constrained shortest path problem. Networks 10:293–310

    MathSciNet  MATH  Google Scholar 

  • Held M, Karp RM (1970) The travelling-salesman problem and minimum spanning trees. Oper Res 18:11–38

    MATH  Google Scholar 

  • Held M, Karp RM (1971) The traveling-salesman problem and minimum spanning trees: Part II. Math Program 1:6–26

    MATH  Google Scholar 

  • Henrion R, Strugarek C (2008) Convexity of chance constraints with independent random variables. Comput Optim Appl 41:263–276

    MathSciNet  MATH  Google Scholar 

  • Hernández-Lerma O, Lasserre JB (2012) Further topics on discrete-time Markov control processes, vol 42. Springer, Berlin

    MATH  Google Scholar 

  • Houda M (2014) A note on the use of Copulas in chance-constrained programming. In: Proceedings of 32nd international conference on mathematical methods in economics MME, pp 327–332

  • Houda M, Lisser A (2014) On the use of Copulas in joint chance–constrained programming, In: Proceedings of the 3rd international conference on operations research and enterprise systems. SCITEPRESS-Science and Technology Publications, Lda, pp 72–79

  • Jaworski P, Durante F, Hardle WK, Rychlik T (2010) Copula theory and its applications. Springer, New York, p 198

    MATH  Google Scholar 

  • Kolliopoulos SG, Stein C (1998) Finding real-valued single-source shortest paths in \(o(n^3)\) expected time. J Algorithms 28(1):125–141

    MathSciNet  MATH  Google Scholar 

  • Mincer M, Niewiadomska-Szynkiewicz E (2012) Application of social network analysis to the investigation of interpersonal connections. J Telecommun Inf Technol 2:83–91

    Google Scholar 

  • Mirino AE (2017) Best paths selection using Dijkstra and Floyd-Warshall algorithm. In: 11th international conference on information and communication technology and system (ICTS), IEEE

  • Mohemmed AW, Sahoo NC, Geok TK (2008) Solving shortest path problem using particle swarm optimization. Appl Soft Comput 8(4):1643–1653

    Google Scholar 

  • Nelsen RB (2007) An introduction to Copulas. Springer, Berlin

    MATH  Google Scholar 

  • Nie YM, Wu X (2009) Shortest path problem considering on-time arrival probability. Transp Res Part B: Methodolog 43(6):597–613

    Google Scholar 

  • Nikolova E, Kelner JA, Brand M, Mitzenmacher M (2006) Stochastic shortest paths via quasi-convex maximization. In: European symposium on algorithms. Springer, Berlin, pp 552–563

  • Papaefthymiou G, Kurowicka D (2008) Using Copulas for modeling stochastic dependence in power system uncertainty analysis. IEEE Trans Power Syst 24(1):40–49

    Google Scholar 

  • Peyer S, Rautenbach D, Vygen J (2009) A generalization of Dijkstra’s shortest path algorithm with applications to VLSI routing. J Discrete Algorithms 7(4):377–390

    MathSciNet  MATH  Google Scholar 

  • Polychronopoulos GH, Tsitsiklis JN (1996) Stochastic shortest path problems with recourse. Netw Int J 27(2):133–143

    MathSciNet  MATH  Google Scholar 

  • Provan JS (2003) A polynomial time algorithm to find shortest paths with recourse. Netw Int J 41(2):115–125

    MathSciNet  MATH  Google Scholar 

  • Sambivasan M, Yahya S (2005) A Lagrangian-based heuristic for multi-plant, multi-item, multi-period capacitated lot-sizing. Comput Oper Res 32:537–552

    MATH  Google Scholar 

  • Sklar M (1959) Fonctions de repártition á n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8:229–231

    MathSciNet  MATH  Google Scholar 

  • Stoer J, Bulirsch R (2013) Introduction to numerical analysis. Springer, Berlin, p 12

    Google Scholar 

  • Stuart A, Voss J (2009) Matrix analysis and algorithms. Springer, Berlin

    Google Scholar 

  • Van Ackooij W, de Oliveira W (2016) Convexity and optimization with Copula structured probabilistic constraints. Optimization 65(7):1349–1376

    MathSciNet  MATH  Google Scholar 

  • Xin C, Qingge L, Wang J, Zhu B (2015) Robust optimization for the hazardous materials transportation network design problem. J Combin Optim 30(2):320–334

    MathSciNet  MATH  Google Scholar 

  • Yang Y, Wang R, Zhang Q (2016) Optimization of shortest path of multiple transportation model based on cost analyses. Int J Simul Syst Sci Technol 17(29):1–6

    Google Scholar 

  • Zeng W, Miwa T, Wakita Y, Morikawa T (2015) Application of Lagrangian relaxation approach to \(\alpha \)-reliable path finding in stochastic networks with correlated link travel times. Transp Res Part C: Emerg Technol 56:309–334

    Google Scholar 

  • Zeng W, Miwa T, Morikawa T (2016) Prediction of vehicle \(\text{ CO }_2\) emission and its application to eco-routing navigation. Transp Res Part C: Emerg Technol 68:194–214

    Google Scholar 

  • Zeng W, Miwa T, Morikawa T (2017) Application of the support vector machine and heuristic k-shortest path algorithm to determine the most eco-friendly path with a travel time constraint. Transpo Res Part D: Transp Environ 57:458–73

    Google Scholar 

  • Zeng W, Miwa T, Morikawa T (2020) Eco-routing problem considering fuel consumption and probabilistic travel time budget. Transpo Res Part D: Transp Environ 78:102219

    Google Scholar 

  • Zhang Y, Shen ZJ, Song S (2017) Lagrangian relaxation for the reliable shortest path problem with correlated link travel times. Transp Res Part B: Methodol 104:501–521

    Google Scholar 

Download references

Acknowledgements

This research is jointly supported by grants from the Institute for Advanced Studies in Basic Sciences (IASBS), University of Tabriz, the Ministry of Science, Research and Technology of the Islamic Republic of Iran, the Paul and Heidi Brown Preeminent Professorship at ISE (University of Florida, USA), and a Humboldt Research Award (Germany).

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Authors and Affiliations

  1. School of Mathematics, University of Tabriz, Tabriz, Iran

    Zohreh Hosseini Nodeh, Ali Babapour Azar & Rashed Khanjani Shiraz

  2. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66731, Iran

    Salman Khodayifar

  3. Center for Applied Optimization, Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

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  1. Zohreh Hosseini Nodeh
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  2. Ali Babapour Azar
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  3. Rashed Khanjani Shiraz
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  4. Salman Khodayifar
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  5. Panos M. Pardalos
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Correspondence to Salman Khodayifar.

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Hosseini Nodeh, Z., Babapour Azar, A., Khanjani Shiraz, R. et al. Joint chance constrained shortest path problem with Copula theory. J Comb Optim 40, 110–140 (2020). https://doi.org/10.1007/s10878-020-00562-8

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