Soft Computing

, Volume 23, Issue 9, pp 2979–2993 | Cite as

The risk-averse traveling repairman problem with profits

  • P. Beraldi
  • M. E. BruniEmail author
  • D. Laganà
  • R. Musmanno


In this paper, we study a stochastic variant of the traveling repairman problem with profits in which travel times are random. The introduction of the arrival time in the objective function instead of the travel time, which is common in most vehicle routing problems, poses compelling challenges, emphasized by the incorporation of the stochasticity in travel times and by the presence of profits. A risk-averse perspective is considered in the model, which is then formulated as a nonlinear integer model and heuristically solved by means of a beam search heuristic. Experimental results have been performed on instances adapted from the available deterministic datasets, to show the effectiveness of the solution approach.


Traveling repairman problem Risk Stochastic travel time Chance constraints 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Human and animal rights

This article does not contain any studies with human or animal participants performed by the author.


  1. Adulyasak Y, Jaillet P (2014) Models and algorithms for stochastic and robust vehicle routing with deadlines. Transp Sci 50(2):608–626Google Scholar
  2. Agra A, Christiansen M, Hvattum LM, Figueiredo R, Poss M, Requejo C (2012) Layered formulation for the robust vehicle routing problem with time windows. Lect Notes Comput Sci 7422:249–260MathSciNetzbMATHGoogle Scholar
  3. Agra A, Christiansen M, Hvattum LM, Figueiredo R, Poss M, Requejo C (2013) The robust vehicle routing problem with time windows. Comput Oper Res 40(3):856–866MathSciNetzbMATHGoogle Scholar
  4. Ando N, Taniguchi E (2006) Travel time reliability in vehicle routing and scheduling with time windows. Netw Spat Econ 6(3–4):293–311MathSciNetzbMATHGoogle Scholar
  5. Araya I, Riff MC (2014) A beam search approach to the container loading problem. Comput Oper Res 43:100–107zbMATHGoogle Scholar
  6. Ausiello G, Leonardi S, Marchetti-Spaccamela A (1994) On salesman repairmen spiders and other traveling agents. In: Proceeding of the Italian conference on algorithms and complexity, pp 1–16Google Scholar
  7. Baldi MM, Crainic TG, Perboli G, Tadei R (2014) Branch and price and beam search algorithms for the variable cost and size bin packing problem with optional items. Ann Oper Res 222(1):125–141MathSciNetzbMATHGoogle Scholar
  8. Beraldi P, Bruni ME (2010) An exact approach for solving integer problem under probabilistic constraints with random technology matrix. Ann Oper Res 177(1):127–137MathSciNetzbMATHGoogle Scholar
  9. Beraldi P, Ruszczyński A (2002a) A branch and bound method for stochastic integer problems under probabilistic constraints. Optim Methods Softw 17:359–382MathSciNetzbMATHGoogle Scholar
  10. Beraldi P, Ruszczyński A (2002b) The probabilistic set-covering problem. Oper Res 50(6):956–967MathSciNetzbMATHGoogle Scholar
  11. Beraldi P, Ruszczyński A (2005) Beam search heuristic to solve stochastic integer problems under probabilistic constraints. Eur J Oper Res 167(1):35–47MathSciNetzbMATHGoogle Scholar
  12. Beraldi P, Ghiani G, Laporte G, Musmanno R (2005) Efficient neighborhood search for the probabilistic pickup and delivery travelling salesman problem. Networks 45(4):195–198MathSciNetzbMATHGoogle Scholar
  13. Beraldi P, Bruni ME, Violi A (2012) Capital rationing problems under uncertainty and risk. Comput Optim Appl 51(3):1375–1396MathSciNetzbMATHGoogle Scholar
  14. Beraldi P, Bruni ME, Laganà D, Musmanno R (2015) The mixed capacitated general routing problem under uncertainty. Eur J Oper Res 240(2):382–392MathSciNetzbMATHGoogle Scholar
  15. Beraldi P, Bruni ME, Manerba D, Mansini R (2017) A stochastic programming approach for the traveling purchaser problem. IMA J Manag Math 28(1):41–63MathSciNetGoogle Scholar
  16. Bigras LP, Gamache M, Savard G (2008) The time-dependent traveling salesman problem and single machine scheduling problems with sequence dependent setup times. Discrete Optim 5(4):685–699MathSciNetzbMATHGoogle Scholar
  17. Birge J, Louveaux F (1997) Introduction to stochastic programming. Springer, BerlinzbMATHGoogle Scholar
  18. Birgin EG, Ferreira JE, Ronconi DP (2015) List scheduling and beam search methods for the flexible job shop scheduling problem with sequencing flexibility. Eur J Oper Res 247(2):421–440MathSciNetzbMATHGoogle Scholar
  19. Blum C, Miralles C (2011) On solving the assembly lineworker assignment and balancing problem via beam search. Comput Oper Res 38:328–339MathSciNetzbMATHGoogle Scholar
  20. Bock S (2015) Solving the traveling repair problem on a line with general processing times and deadlines. Eur J Oper Res 24(3):690–703zbMATHGoogle Scholar
  21. Bruni ME, Beraldi P, Laganà D (2013) The express heuristic for probabilistically constrained integer problems. J Heuristics 19(3):423–441Google Scholar
  22. Bruni ME, Guerriero F, Beraldi P (2014) Designing robust routes for demand-responsive transport systems. Transp Res Part E Logist Transp Rev 70:1–16Google Scholar
  23. Bruni ME, Beraldi P, Khodaparasti S (2018) A heuristic approach for the k-traveling repairman problem with profits under uncertainty. Electron Notes Discrete Math 69:221–228MathSciNetGoogle Scholar
  24. Bruni ME, Beraldi P, Khodaparasti S (2018) A fast heuristic for routing in post-disaster humanitarian relief logistics. Transp Res Procedia 30:304–313Google Scholar
  25. Caceres H, Hwang H, He Q (2016) Estimating freeway route travel time distributions with consideration to time-of-day, inclement weather, and traffic incidents. J Adv Transp 50:967–987Google Scholar
  26. Calafiore G, El Ghaoui L (2006) On distributionally robust chance-constrained linear programs. J Optim Theory Appl 130(1):1–22MathSciNetzbMATHGoogle Scholar
  27. Campbell A, Vandenbussche D, Hermann W (2008) Routing for relief efforts. Transp Sci 42(2):127–145Google Scholar
  28. Dewilde T, Cattrysse D, Coene S, Spieksma FCR, Vansteenwegen P (2013) Heuristics for the traveling repairman problem with profits. Comput Oper Res 40(7):1700–1707MathSciNetzbMATHGoogle Scholar
  29. Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman & Hall, London. ISBN: 9781315897943Google Scholar
  30. Fernandez-Viagas V, Framinan JM (2017) A beam-search-based constructive heuristic for the PFSP to minimise total flowtime. Comput Oper Res 81:167–177MathSciNetzbMATHGoogle Scholar
  31. Fischetti M, Laporte G, Martello S (1993) The delivery man problem and cumulative matroids. Oper Res 41(6):1055–1064MathSciNetzbMATHGoogle Scholar
  32. Gomez A, Marino R, Akhavan-Tabatabaei R, Medaglia AL, Mendoza JE (2016) On modeling stochastic travel and service times in vehicle routing. Transp Sci 50(2):627–641Google Scholar
  33. Henrion R, Strugarek C (2008) Convexity of chance constraints with independent random variables. Comput Optim Appl 412(2):263–276MathSciNetzbMATHGoogle Scholar
  34. Jaillet P, Qi J, Sim M (2016) Routing optimization under uncertainty. Oper Res 64(1):186–200MathSciNetzbMATHGoogle Scholar
  35. Jung S, Haghani A (2001) Genetic algorithm for the time-dependent vehicle routing problem. Transp Res Rec J Transp Res Board 1771(1):164–171Google Scholar
  36. Kenyon A, Morton D (2003) Stochastic vehicle routing with random travel times. Transp Sci 37(1):69–82Google Scholar
  37. Khokhlov V (2016) Conditional value at risk for elliptical distributions. Evropsky Casopis Ekonomiky a Managementu 2(6):70–79Google Scholar
  38. Krokhmal P, Zabarankin M, Uryasev S (2011) Modeling and optimization of risk. Surv Oper Res Manag Sci 16(2):49–66Google Scholar
  39. Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transp Sci 26(3):161–70zbMATHGoogle Scholar
  40. Lecluyse C, Van Woensel T, Peremans H (2009) Vehicle routing with stochastic time-dependent travel times. 4OR-Q J Oper Res 7:363–377MathSciNetzbMATHGoogle Scholar
  41. Lee C, Lee K, Park S (2004) Robust vehicle routing problem with deadlines and travel time/demand uncertainty. J Oper Res Soc 63(9):1294–306Google Scholar
  42. Li X, Stephen PT, Leung CH (2010) Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm. Int J Prod Econ 125(1):137–145Google Scholar
  43. Lucena A (1990) Time-dependent traveling salesman problem—the deliveryman case. Networks 20(6):753–763MathSciNetzbMATHGoogle Scholar
  44. Maggioni F, Perboli G, Tadei R (2014) The multi-path traveling salesman problem with stochastic travel costs: building realistic instances for city logistics applications. Transp Res Procedia 3:528–536Google Scholar
  45. Oyola J, Arntzen H, Woodruff DL (2017) The stochastic vehicle routing problem, a literature review, part II: solution methods. EURO J Transp Logist 6(4):349–388Google Scholar
  46. Oyola J, Arntzen H, Woodruff DL (2018) The stochastic vehicle routing problem, a literature review, part I: models. EURO J Transp Logist 7(3):193–221Google Scholar
  47. Perboli G, Gobbato L, Maggioni F (2017) A progressive hedging method for the multi-path traveling salesman problem with stochastic travel times. IMA J Manag Math 28(1):65–86MathSciNetGoogle Scholar
  48. Solano-Charris EL, Prins C, Santos AC (2016) Solving the bi-objective robust vehicle routing problem with uncertain costs and demands. RAIRO Oper Res 50(4–5):689–714MathSciNetzbMATHGoogle Scholar
  49. Sungur I, Ordóñez F, Dessouky M (2008) A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty. IIE Trans 40(5):509–523Google Scholar
  50. Susilawati S, Taylor M, Somenahalli S (2013) Distributions of travel time variability on urban roads. J Adv Transp 47(8):720–736Google Scholar
  51. Taş D, Dellaert N, Van Woensel T, de Kok AG (2013) Vehicle routing problem with stochastic travel times including soft time windows and service costs. Comput Oper Res 40(1):214–224MathSciNetzbMATHGoogle Scholar
  52. Taş D, Gendreau M, Dellaert N, Van Woensel T, de Kok AG (2014) Vehicle routing with soft time windows and stochastic travel times: a column generation and branch-and-price solution approach. Eur J Oper Res 236(3):789–799MathSciNetzbMATHGoogle Scholar
  53. Tulabandhula T, Rudin C, Jaillet P (2011) The machine learning and traveling repairman problem. Algorithm Decis Theory Lect Notes Comput Sci 6992:262–276MathSciNetzbMATHGoogle Scholar
  54. Yavuz M (2017) An iterated beam search algorithm for the green vehicle routing problem. Networks 69(3):317–328MathSciNetGoogle Scholar
  55. Zhang T, Chaovalitwongse WA, Zhang Y (2012) Scatter search for the stochastic travel-time vehicle routing problem with simultaneous pick-ups and deliveries. Comput Oper Res 39(10):2277–2290MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • P. Beraldi
    • 2
  • M. E. Bruni
    • 1
    Email author
  • D. Laganà
    • 2
  • R. Musmanno
    • 2
  1. 1.Department of Mechanical, Energy and Management EngineeringUniversity of CalabriaRendeItaly
  2. 2.Department of Mechanical, Energy and Management EngineeringUniversity of CalabriaRendeItaly

Personalised recommendations