A cutting plane method for risk-constrained traveling salesman problem with random arc costs

  • 188 Accesses


In this manuscript, we consider a stochastic traveling salesman problem with random arc costs and assume that the travel cost of each arc follows a normal distribution. All the other parameters in the problem are considered deterministic. In the presence of uncertainty, the optimal route achieved from solving the deterministic model might be exposed to a high risk that the actual cost exceeds the available resource. In this respect, we present the stochastic model incorporating risk management, and the Value at Risk and Conditional Value at Risk techniques are applied as the risk measures to assess and control the risk associated with the uncertainty. A novel cutting plane algorithm is developed to deal with the difficulty of solving such model, and exhibits superior computational performance in our numerical experiments over other solution approaches.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Applegate, D., Bixby, R., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)

  2. 2.

    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)

  3. 3.

    Berman, O.: The traveling salesman location problem on stochastic networks. Transp. Sci. 23(1), 54–57 (1989)

  4. 4.

    Birge, J.R., Dempstert, M.A.H.: Stochastic programming approaches to stochastic scheduling. J. Glob. Optim. 9(3), 417–451 (1996)

  5. 5.

    Campbell, A., Thomas, B.: Probabilistic traveling salesman problem with deadlines. Transp. Sci. 42(1), 1–21 (2008)

  6. 6.

    Carraway, R., Morin, T., Moskowit, H.: Generalized dynamic programming for stochastic combinatorial optimization. Oper. Res. 37(5), 819–829 (1989)

  7. 7.

    Çezik, M., Iyengar, G.: Cuts for mixed 0–1 conic programming. Math. Program. 104(1), 179–202 (2005)

  8. 8.

    Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)

  9. 9.

    Cord, J.: A method for allocating funds to investment projects when returns are subject to uncertainty. Manag. Sci. 10(2), 335–341 (1964).

  10. 10.

    Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)

  11. 11.

    Duffie, D., Pan, J.: An overview of value at risk. J. Deriv. 4(3), 7–49 (1997)

  12. 12.

    Flood, M.: The traveling-salesman problem. Oper. Res. 4(1), 61–75 (1956)

  13. 13.

    Frieze, A.: On random symmetric travelling salesman problems. Math. Oper. Res. 29(4), 878–890 (2004).

  14. 14.

    Frieze, A., Sorkin, G.B.: The probabilistic relationship between the assignment and asymmetric traveling salesman problems. SIAM J. Comput. 36(5), 1435–1452 (2007)

  15. 15.

    Gurvich, I., Luedtke, J., Tezcan, T.: Staffing call centers with uncertain demand forecasts: a chance-constrained optimization approach. Manag. Sci. 56(7), 1093–1115 (2010).

  16. 16.

    Huang, Z., Zheng, Q.P., Pasiliao, E.L., Simmons, D.: Exact algorithms on reliable routing problems under uncertain topology using aggregation techniques for exponentially many scenarios. Ann. Oper. Res. 249(1), 141–162 (2017)

  17. 17.

    Jaillet, P.: A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Oper. Res. 36(6), 929–936 (1988)

  18. 18.

    Kao, E.: A preference order dynamic program for a stochastic traveling salesman problem. Oper. Res. 26(6), 1033–1045 (1978)

  19. 19.

    Karp, R.M.: A patching algorithm for the nonsymmetric traveling-salesman problem. SIAM J. Comput. 8, 561–573 (1979)

  20. 20.

    Kenyon, A., Morton, D.: Stochastic vehicle routing with random travel times. Transp. Sci. 37(1), 69–82 (2003)

  21. 21.

    Krauth, W., Mézard, M.: The cavity method and the traveling-salesman problem. Europhys. Lett. 8(3), 213–218 (1989)

  22. 22.

    Laporte, G.: The traveling salesman problem: an overview of exact and approximate algorithms. Eur. J. Oper. Res. 59, 231–247 (1992)

  23. 23.

    Laporte, G., Louveaux, F., Ercure, H.: A priori optimization of the probabilistic traveling salesman problem. Oper. Res. 42(3), 543–549 (1994)

  24. 24.

    Leipälä, T.: On the solutions of stochastic traveling salesman problems. Eur. J. Oper. Res. 2, 291–297 (1978)

  25. 25.

    Luedtke, J.: An integer programming and decomposition approach to general chance-constrained mathematical programs. In: Eisenbrand, F., Shepherd, F.B. (eds.) Integer Programming and Combinatorial Optimization, pp. 271–284. Springer, Berlin (2010)

  26. 26.

    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

  27. 27.

    Matai, R., Singh, S., Mittal, M.: Traveling salesman problem: an overview of applications, formulations, and solution approaches, chap. 1. In: Traveling Salesman Problem, Theory and Applications. InTech (2010)

  28. 28.

    Miyashiro, R., Takano, Y.: Mixed integer second-order cone programming formulations for variable selection in linear regression. Eur. J. Oper. Res. 247(3), 721–731 (2015)

  29. 29.

    Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia (1994)

  30. 30.

    Orman, A., Williams, H.: A survey of different integer programming formulations of the traveling salesman problem. Optim. Econ. Financ. Anal. 9, 91–104 (2007)

  31. 31.

    Pagnoncelli, B., Ahmed, S., Shapiro, A.: Computational study of a chance constrained portfolio selection problem. J. Optim. Theory Appl. 142, 399–416 (2009)

  32. 32.

    Prekopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, vol. 113, p. 138 (1970)

  33. 33.

    Qiu, F., Ahmed, S., Dey, S., Wolsey, L.: Covering linear programming with violations. INFORMS J. Comput. 26, 531–546 (2014)

  34. 34.

    Roberti, R., Toth, P.: Models and algorithms for the asymmetric traveling salesman problem: an experimental comparison. EURO J. Transp. Logist. 1(1), 113–133 (2012)

  35. 35.

    Rockafellar, R., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

  36. 36.

    Sarykalin, S., Serraino, G., Uryasev, S.: Value-at-risk vs. conditional value-at-risk in risk management and optimization, chap. 13. In: State-of-the-Art Decision-Making Tools in the Information-Intensive Age, pp. 270–294. INFORMS Institute for Operations Research (2008)

  37. 37.

    Schrijver, A.: On the History of Combinatorial Optimization Till 1960. Elsevier, New York City (2005)

  38. 38.

    Schultz, R., Tiedemann, S.: Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math. Program. 105(2), 365–386 (2006)

  39. 39.

    Sniedovich, M.: Analysis of a preference order traveling salesman problem. Oper. Res. 29(6), 1234–1237 (1981)

  40. 40.

    Stubbs, R.A., Mehrotra, S.: Generating convex polynomial inequalities for mixed 0–1 programs. J. Glob. Optim. 24(3), 311–332 (2002)

  41. 41.

    Takyi, A.K., Lence, B.J.: Surface water quality management using a multiple-realization chance constraint method. Water Resour. Res. 35(5), 1657–1670 (1999)

  42. 42.

    Tong, X., Sun, H., Luo, X., Zheng, Q.: Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems. J. Glob. Optim. 70(1), 131–158 (2018)

  43. 43.

    Toriello, A., Haskell, W.B., Poremba, M.: A dynamic traveling salesman problem with stochastic arc costs. Oper. Res. 62(5), 1107–1125 (2014)

  44. 44.

    Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: a computational study. Comput. Optim. Appl. 24(2), 289–333 (2003)

  45. 45.

    Yuan, Y., Li, Z., Huang, B.: Robust optimization approximation for joint chance constrained optimization problem. J. Glob. Optim. 67(4), 805–827 (2017)

  46. 46.

    Zeng, B., An, Y., Kuznia, L.: Chance constrained mixed integer program: bilinear and linear formulations, and benders decomposition (2014). arXiv:1403.7875

  47. 47.

    Zheng, Q.P., Shen, S., Shi, Y.: Loss-constrained minimum cost flow under arc failure uncertainty with applications in risk-aware kidney exchange. IIE Trans. 47(9), 961–977 (2015)

  48. 48.

    Zheng, Q.P., Wang, J., Liu, A.L.: Stochastic optimization for unit commitment—a review. IEEE Trans. Power Syst. 30(4), 1913–1924 (2015)

Download references

Author information

Correspondence to Zhouchun Huang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huang, Z., Zheng, Q.P., Pasiliao, E. et al. A cutting plane method for risk-constrained traveling salesman problem with random arc costs. J Glob Optim 74, 839–859 (2019).

Download citation