, Volume 25, Issue 3, pp 413–433 | Cite as

Continuous approximation models in freight distribution management

  • Anna Franceschetti
  • Ola Jabali
  • Gilbert Laporte
Invited Paper


This paper presents an overview of the literature on continuous approximation models in the context of distribution management. It first describes the seminal contributions of Beardwood, Halton and Hammersley, and of Daganzo and Newell. This is followed by a summary of various extensions, and by applications to districting, location, fleet sizing and vehicle routing.


Traveling salesman problem Vehicle routing problem Asymptotic results Partitioning strategies 

Mathematics Subject Classification




The authors gratefully acknowledge funding provided by the Canadian Natural Sciences and Engineering Research Council under Grants 436014-2013 and 2015-06189.


  1. Applegate DL, Bixby RE, Chvátal V, Cook WJ (2011) The traveling salesman problem: a computational study. Princeton University Press, PrincetonGoogle Scholar
  2. Arlotto A, Steele JM (2016) Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: a counterexample. Ann Appl Probab 26(4):2141–2168CrossRefGoogle Scholar
  3. Beardwood J, Halton JH, Hammersley JM (1959) The shortest path through many points. Proc Camb Philos Soc 55(9):299–328CrossRefGoogle Scholar
  4. Blumenfeld DE, Beckmann MJ (1985) Use of continuous space modeling to estimate freight distribution costs. Transp Res Part A Gen 19(2):173–187CrossRefGoogle Scholar
  5. Blumenfeld DE, Burns LD, Daganzo CF, Frick MC, Hall RW (1987) Reducing logistics costs at general motors. Interfaces 17(1):26–47CrossRefGoogle Scholar
  6. Bonomi E, Lutton J-L (1984) The \(N\)-city travelling salesman problem: statistical mechanics and the Metropolis algorithm. SIAM Rev 26(4):551–568CrossRefGoogle Scholar
  7. Bozkaya B, Erkut E, Laporte G (2003) A tabu search heuristic and adaptive memory procedure for political districting. Eur J Oper Res 144(1):12–26CrossRefGoogle Scholar
  8. Burns LD, Hall RW, Blumenfeld DE, Daganzo CF (1985) Distribution strategies that minimize transportation and inventory costs. Oper Res 33(3):469–490CrossRefGoogle Scholar
  9. Campbell JF (1990a) Designing logistics systems by analyzing transportation, inventory and terminal cost tradeoffs. J Bus Logist 11(1):159–179Google Scholar
  10. Campbell JF (1990b) Freight consolidation and routing with transportation economies of scale. Transp Res Part B Methodol 24(5):345–361CrossRefGoogle Scholar
  11. Campbell JF (1992) Location and allocation for distribution systems with transshipments and transportation economies of scale. Ann Oper Res 40(1):77–99CrossRefGoogle Scholar
  12. Campbell JF (1993a) Continuous and discrete demand hub location problems. Transp Res Part B Methodol 27(6):473–482CrossRefGoogle Scholar
  13. Campbell JF (1993b) One-to-many distribution with transshipments: an analytic model. Transp Sci 27(4):330–340CrossRefGoogle Scholar
  14. Campbell JF (1995) Using small trucks to circumvent large truck restrictions: impacts on truck emissions and performance measures. Transp Res Part A Policy Pract 29(6):445–458CrossRefGoogle Scholar
  15. Campbell JF (2013) A continuous approximation model for time definite many-to-many transportation. Transp Res Part B Methodol 54:100–112CrossRefGoogle Scholar
  16. Carlsson JG (2012) Dividing a territory among several vehicles. INFORMS J Comput 24(4):565–577CrossRefGoogle Scholar
  17. Carlsson JG, Behroozi M (2017) Worst-case demand distributions in vehicle routing. Eur J Oper Res 256(2):462–472CrossRefGoogle Scholar
  18. Carlsson JG, Delage E (2013) Robust partitioning for stochastic multivehicle routing. Oper Res 61(3):727–744CrossRefGoogle Scholar
  19. Carlsson JG, Jia F (2014) Continuous facility location with backbone network costs. Transp Sci 49(3):433–451CrossRefGoogle Scholar
  20. Çavdar B, Sokol J (2015) A distribution-free TSP tour length estimation model for random graphs. Eur J Oper Res 243(2):588–598CrossRefGoogle Scholar
  21. Chien TW (1992) Operational estimators for the length of a traveling salesman tour. Comput Oper Res 19(6):469–478CrossRefGoogle Scholar
  22. Chou YH, Chen YH, Chen HM (2014) Pickup and delivery routing with hub transshipment across flexible time periods for improving dual objectives on workload and waiting time. Transp Res Part E Logist Transp Rev 61:98–114CrossRefGoogle Scholar
  23. Cui T, Ouyang Y, Shen Z-JM (2010) Reliable facility location design under the risk of disruptions. Oper Res 58(4-part-1):998–1011Google Scholar
  24. Daganzo CF (1984a) The distance traveled to visit \({N}\) points with a maximum of \({C}\) stops per vehicle: an analytic model and an application. Transp Sci 18(4):331–350CrossRefGoogle Scholar
  25. Daganzo CF (1984b) The length of tours in zones of different shapes. Transp Res Part B Methodol 18(2):135–145CrossRefGoogle Scholar
  26. Daganzo CF (1987a) Modeling distribution problems with time windows: part I. Transp Sci 21(3):171–179CrossRefGoogle Scholar
  27. Daganzo CF (1987b) Modeling distribution problems with time windows: part II: two customer types. Transp Sci 21(3):180–187CrossRefGoogle Scholar
  28. Daganzo CF (1987c) The break-bulk role of terminals in many-to-many logistic networks. Oper Res 35(4):543–555CrossRefGoogle Scholar
  29. Daganzo CF (1988) A comparison of in-vehicle and out-of-vehicle freight consolidation strategies. Transp Res Part B Methodol 22(3):173–180CrossRefGoogle Scholar
  30. Daganzo CF (1991) Logistics systems analysis: lecture notes in economics and mathematical systems. Springer, BerlinGoogle Scholar
  31. Daganzo CF (2005) Logistics systems analysis, 4th edn. Springer, New YorkGoogle Scholar
  32. Daganzo CF, Hall RW (1993) A routing model for pickups and deliveries: no capacity restrictions on the secondary items. Transp Sci 27(4):315–329CrossRefGoogle Scholar
  33. Daganzo CF, Newell GF (1985) Physical distribution from a warehouse: vehicle coverage and inventory levels. Transp Res Part B Methodol 19(5):397–407CrossRefGoogle Scholar
  34. Daganzo CF, Smilowitz KR (2004) Bounds and approximations for the transportation problem of linear programming and other scalable network problems. Transp Sci 38(3):343–356CrossRefGoogle Scholar
  35. Davis BA, Figliozzi MA (2013) A methodology to evaluate the competitiveness of electric delivery trucks. Transp Res Part E Logist Transp Rev 49(1):8–23CrossRefGoogle Scholar
  36. Drexl A, Haase K (1999) Fast approximation methods for sales force deployment. Manag Sci 45(10):1307–1323CrossRefGoogle Scholar
  37. Eilon S, Watson-Gandy CDT, Christofides N (1971) Distribution management. Griffin, LondonGoogle Scholar
  38. Fairthorne D (1965) The distances between pairs of points in towns of simple geometrical shapes. In: Proceedings of the second international symposium on the theory of road traffic flow. OECD, london, pp 391–406Google Scholar
  39. Few L (1955) The shortest path and the shortest road through \(n\) points. Mathematika 2(2):141–144CrossRefGoogle Scholar
  40. Figliozzi MA (2007) Analysis of the efficiency of urban commercial vehicle tours: data collection, methodology, and policy implications. Transp Res Part B Methodol 41(9):1014–1032CrossRefGoogle Scholar
  41. Figliozzi MA (2008) Planning approximations to the average length of vehicle routing problems with varying customer demands and routing constraints. Transp Res Rec J Transp Res Board 2089:1–8CrossRefGoogle Scholar
  42. Figliozzi MA (2010) The impacts of congestion on commercial vehicle tour characteristics and costs. Transp Res Part E Logist Transp Rev 46(4):496–506CrossRefGoogle Scholar
  43. Finch SR (2003) Mathematical constants. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  44. Fisher ML, Jaikumar R (1981) A generalized assignment heuristic for vehicle routing. Networks 11(2):109–124CrossRefGoogle Scholar
  45. Franceschetti A, Honhon D, Laporte G, Van Woensel T, Fransoo JC (2017) Strategic fleet planning for city logistics. Transp Res Part B Methodol 95:19–40CrossRefGoogle Scholar
  46. Francis P, Smilowitz KR (2006) Modeling techniques for periodic vehicle routing problems. Transp Res Part B Methodol 40(10):872–884CrossRefGoogle Scholar
  47. Gaboune B, Laporte G, Soumis F (1994) Optimal strip sequencing strategies for flexible manufacturing operations in two and three dimensions. Int J Flex Manuf Syst 6(2):123–135CrossRefGoogle Scholar
  48. Galvão LC, Novaes AGN, De Cursi JES, Souza JC (2006) A multiplicatively-weighted Voronoi diagram approach to logistics districting. Comput Oper Res 33(1):93–114CrossRefGoogle Scholar
  49. Hall RW (1986) Discrete models/continuous models. Omega 3:213–220CrossRefGoogle Scholar
  50. Hall RW (1987) Direct versus terminal freight routing on a network with concave costs. Transp Res Part B Methodol 21(4):287–298CrossRefGoogle Scholar
  51. Hall RW (1991) Characteristics of multi-shop/multi-terminal delivery routes, with backhauls and unique items. Transp Res Part B Methodol 25(6):391–403CrossRefGoogle Scholar
  52. Hall RW (1993) Design for local area freight networks. Transp Res Part B Methodol 27(2):79–95CrossRefGoogle Scholar
  53. Hall RW, Du Y, Lin J (1994) Use of continuous approximations within discrete algorithms for routing vehicles: experimental results and interpretation. Networks 24(1):43–56CrossRefGoogle Scholar
  54. Halton JH, Terada R (1982) A fast algorithm for the Euclidean traveling salesman problem, optimal with probability one. SIAM J Comput 11(1):28–46CrossRefGoogle Scholar
  55. Han AFW, Daganzo CF (1986) Distributing nonstorable items without transshipments. Transp Res Rec 1061:32–41Google Scholar
  56. Hindle A, Worthington D (2004) Models to estimate average route lengths in different geographical environments. J Oper Res Soc 55(6):662–666CrossRefGoogle Scholar
  57. Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. Stud Appl Math 20(1–4):224–230Google Scholar
  58. Huang M, Smilowitz KR, Balcik B (2013) A continuous approximation approach for assessment routing in disaster relief. Transp Res Part B Methodol 50:20–41CrossRefGoogle Scholar
  59. Jabali O, Erdoğan G (2015) Continuous approximation models for the fleet replacement and composition problem. Technical report, CIRRELT-2015-64Google Scholar
  60. Jabali O, Gendreau M, Laporte G (2012) A continuous approximation model for the fleet composition problem. Transp Res Part B Methodol 46(10):1591–1606CrossRefGoogle Scholar
  61. Jessen RJ (1943) Statistical investigation of a sample survey for obtaining farm facts. PhD thesis, Iowa State CollegeGoogle Scholar
  62. Jordan WC, Burns LD (1984) Truck backhauling on two terminal networks. Transp Res Part B Methodol 18(6):487–503CrossRefGoogle Scholar
  63. Karp RM (1977) Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math Oper Res 2(3):209–224CrossRefGoogle Scholar
  64. Karp RM, Steele JM (1985) Probabilistic analysis of heuristics. In: Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (eds) The traveling salesman problem, chap 6. Wiley, Chichester, pp 181–205Google Scholar
  65. Kirac E, Milburn AB, Wardell C III (2015) The traveling salesman problem with imperfect information with application in disaster relief tour planning. IIE Trans 47(8):783–799CrossRefGoogle Scholar
  66. Klincewicz JG, Luss H, Pilcher MG (1990) Fleet size planning when outside carrier services are available. Transp Sci 24(3):169–182CrossRefGoogle Scholar
  67. Kwon O, Golden BL, Wasil EA (1995) Estimating the length of the optimal TSP tour: an empirical study using regression and neural networks. Comput Oper Res 22(10):1039–1046CrossRefGoogle Scholar
  68. Langevin A, Soumis F (1989) Design of multiple-vehicle delivery tours satisfying time constraints. Transp Res Part B: Methodol 23(2):123–138CrossRefGoogle Scholar
  69. Langevin A, Mbaraga P, Campbell JF (1996) Continuous approximation models in freight distribution: an overview. Transp Res Part B Methodol 30(3):163–188CrossRefGoogle Scholar
  70. Laporte G, Dejax PJ (1989) Dynamic location-routing problems. J Oper Res Soc 40(5):471–482CrossRefGoogle Scholar
  71. Laporte G, Semet F, Dadeshidze VV, Olsson LJ (1998) A tiling and routing heuristic for the screening of cytological samples. J Oper Res Soc 49(12):1233–1238CrossRefGoogle Scholar
  72. Larsen C, Turkensteen M (2014) A vendor managed inventory model using continuous approximations for route length estimates and Markov chain modeling for cost estimates. Int J Prod Econ 157:120–132CrossRefGoogle Scholar
  73. Lei H, Laporte G, Guo B (2012) Districting for routing with stochastic customers. EURO J Transp Logist 1(1–2):67–85CrossRefGoogle Scholar
  74. Lei H, Laporte G, Liu Y, Zhang T (2015) Dynamic design of sales territories. Comput Oper Res 56:84–92CrossRefGoogle Scholar
  75. Lei H, Wang R, Laporte G (2016) Solving a multi-objective dynamic stochastic districting and routing problem with a co-evolutionary algorithm. Comput Oper Res 67:12–24CrossRefGoogle Scholar
  76. Mahalanobis PC (1940) A sample survey of the acreage under jute in Bengal. Sankhyā Indian J Stat Proc Indian Stat Conf 1939 4:511–530Google Scholar
  77. Marks ES (1948) A lower bound for the expected travel among \(m\) random points. Ann Math Stat 19(3):419–422CrossRefGoogle Scholar
  78. Mehrotra A, Johnson EL, Nemhauser GL (1998) An optimization based heuristic for political districting. Manag Sci 44(8):1100–1114CrossRefGoogle Scholar
  79. Monge G (1781) Mémoire sur la théorie des déblais et des remblais. Imprimerie Royale, ParisGoogle Scholar
  80. Newell GF (1986) Design of multiple-vehicle delivery tours—III valuable goods. Transp Res Part B Methodol 20(5):377–390CrossRefGoogle Scholar
  81. Newell GF, Daganzo CF (1986a) Design of multiple-vehicle delivery tours—I a ring-radial network. Transp Res Part B Methodol 20(5):345–363CrossRefGoogle Scholar
  82. Newell GF, Daganzo CF (1986b) Design of multiple vehicle delivery tours—II other metrics. Transp Res Part B Methodol 20(5):365–376CrossRefGoogle Scholar
  83. Nourinejad M, Roorda MJ (2016) A continuous approximation model for the fleet composition problem on the rectangular grid. OR Spectr 39(2):1–29Google Scholar
  84. Novaes AGN, De Cursi JES, Graciolli OD (2000) A continuous approach to the design of physical distribution systems. Comput Oper Res 27(9):877–893CrossRefGoogle Scholar
  85. Novaes AGN, De Cursi JES, da Silva ACL, Souza JC (2009) Solving continuous location-districting problems with Voronoi diagrams. Comput Oper Res 36(1):40–59CrossRefGoogle Scholar
  86. Novaes AGN, Graciolli OD (1999) Designing multi-vehicle delivery tours in a grid-cell format. Eur J Oper Res 119(3):613–634CrossRefGoogle Scholar
  87. Ouyang Y (2007) Design of vehicle routing zones for large-scale distribution systems. Transp Res Part B Methodol 41(10):1079–1093CrossRefGoogle Scholar
  88. Ouyang Y, Daganzo CF (2006) Discretization and validation of the continuum approximation scheme for terminal system design. Transp Sci 40(1):89–98CrossRefGoogle Scholar
  89. Pang G, Muyldermans L (2013) Vehicle routing and the value of postponement. J Oper Res Soc 64(9):1429–1440CrossRefGoogle Scholar
  90. Rosenfield DB, Engelstein I, Feigenbaum D (1992) An application of sizing service territories. Eur J Oper Res 63(2):164–172CrossRefGoogle Scholar
  91. Royden HL (1968) Real analysis, 2nd edn. Macmillan, New YorkGoogle Scholar
  92. Saberi M, Verbas Ö (2012) Continuous approximation model for the vehicle routing problem for emissions minimization at the strategic level. J Transp Eng 138(11):1368–1376CrossRefGoogle Scholar
  93. Sankaran JK, Wood L (2007) The relative impact of consignee behaviour and road traffic congestion on distribution costs. Transp Res Part B Methodol 41(9):1033–1049CrossRefGoogle Scholar
  94. Skiera B, Albers S (1998) COSTA: contribution optimizing sales territory alignment. Mark Sci 17(3):196–213CrossRefGoogle Scholar
  95. Steele JM (1981a) Complete convergence of short paths and Karp’s algorithm for the TSP. Math Oper Res 6(3):374–378CrossRefGoogle Scholar
  96. Steele JM (1981b) Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann Probab 9(3):365–376CrossRefGoogle Scholar
  97. Stein DM (1978) An asymptotic, probabilistic analysis of a routing problem. Math Oper Res 3(2):89–101CrossRefGoogle Scholar
  98. Teitz MB, Bart P (1968) Heuristic methods for estimating the generalized vertex median of a weighted graph. Oper Res 16(5):955–961CrossRefGoogle Scholar
  99. Turkensteen M, Klose A (2012) Demand dispersion and logistics costs in one-to-many distribution systems. Eur J Oper Res 223(2):499–507CrossRefGoogle Scholar
  100. Verblunsky S (1951) On the shortest path through a number of points. Proc Am Math Soc 2(6):904–913CrossRefGoogle Scholar
  101. Xie W, Ouyang Y (2015) Optimal layout of transshipment facility locations on an infinite homogeneous plane. Transp Res Part B Methodol 75:74–88CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  • Anna Franceschetti
    • 1
  • Ola Jabali
    • 2
  • Gilbert Laporte
    • 1
  1. 1.Canada Research Chair in Distribution ManagementHEC MontréalMontréalCanada
  2. 2.Dipartimento di Elettronica, Informazione e BioingegneriaPolitecnico di MilanoMilanItaly

Personalised recommendations