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Bounds and Approximations for the Transportation Problem of Linear Programming and Other Scalable Network Problems

Author

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  • Carlos F. Daganzo

    (Institute of Transportation Studies, and Department of Civil and Environmental Engineering, University of California, Berkeley, California 94720)

  • Karen R. Smilowitz

    (Institute of Transportation Studies, and Department of Civil and Environmental Engineering, University of California, Berkeley, California 94720)

Abstract

Bounds and approximate formulae are developed for the average optimum distance of the transportation linear programming (TLP) problem with homogeneously, but randomly distributed points and demands in a region of arbitrary shape. It is shown that if the region size grows with a fixed density of points, then the cost per item is bounded from above in 3 + dimensions (3 + -D), but not in 1-D and 2-D. Lower bounds are also developed, based on a mild monotonicity conjecture. Computer simulations confirm the conjecture and yield approximate formulae. These formulae turn out to have the same functional form as the upper bounds. Curiously, the monotonicity conjecture implies that the cost per item does not depend on zone shape asymptotically, as problem size increases, for 2 + -D problems but it does in 1-D. Therefore, the 2-D case can be viewed as a transition case that shares some of the properties of 1-D (unbounded cost) and some of the properties of 3-D (shape independence).The results are then extended to more general network models with subadditive link costs. It is found that if the cost functions have economies of scale, then the cost per item is bounded in 2-D. This explains the prevalence of the “last mile” effect in many logistics applications. The paper also discusses how the results were used to estimate costs under uncertainty for a vehicle repositioning problem.

Suggested Citation

  • Carlos F. Daganzo & Karen R. Smilowitz, 2004. "Bounds and Approximations for the Transportation Problem of Linear Programming and Other Scalable Network Problems," Transportation Science, INFORMS, vol. 38(3), pages 343-356, August.
    • Handle: RePEc:inm:ortrsc:v:38:y:2004:i:3:p:343-356
    DOI: 10.1287/trsc.1030.0037
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    References listed on IDEAS

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    1. Daganzo, Carlos F & Smilowitz, Karen R, 2000. "Asymptotic Approximations for the Transportation LP and Other Scalable Network Problems," University of California Transportation Center, Working Papers qt7wb1g4z7, University of California Transportation Center.
    2. Newell, Gordon F. & Daganzo, Carlos F., 1986. "Design of multiple-vehicle delivery tours--I a ring-radial network," Transportation Research Part B: Methodological, Elsevier, vol. 20(5), pages 345-363, October.
    3. Daganzo, Carlos F. & Smilowitz, Karen R., 2000. "Asymptotic Approximations for the Transportation LP and Other Scalable Network Problems," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt3dn2j66w, Institute of Transportation Studies, UC Berkeley.
    4. Daganzo, Carlos F., 1984. "The length of tours in zones of different shapes," Transportation Research Part B: Methodological, Elsevier, vol. 18(2), pages 135-145, April.
    5. Newell, Gordon F. & Daganzo, Carlos F., 1986. "Design of multiple vehicle delivery tours--II other metrics," Transportation Research Part B: Methodological, Elsevier, vol. 20(5), pages 365-376, October.
    6. Smilowitz, Karen Renee, 2001. "Design and Operation of Multimode, Multiservice Logistics Systems," University of California Transportation Center, Working Papers qt2bb1g26m, University of California Transportation Center.
    7. M. Haimovich & A. H. G. Rinnooy Kan, 1985. "Bounds and Heuristics for Capacitated Routing Problems," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 527-542, November.
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    9. Newell, G. F., 1986. "Design of multiple-vehicle delivery tours--III valuable goods," Transportation Research Part B: Methodological, Elsevier, vol. 20(5), pages 377-390, October.
    10. Carlos F. Daganzo, 1984. "The Distance Traveled to Visit N Points with a Maximum of C Stops per Vehicle: An Analytic Model and an Application," Transportation Science, INFORMS, vol. 18(4), pages 331-350, November.
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