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Ripples, complements, and substitutes in generalized networks

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  • Antoine Gautier
  • Frieda Granot

Abstract

We extend the qualitative theory of sensitivity analysis for minimum‐cost pure network flows of Granot and Veinott [17] to generalized network flow problems, that is, network flow problems where the amount of flow picked up by an arc is multiplied by a (positive) gain while traversing the arc. Three main results are presented. The ripple theorem gives upper bounds on the absolute value of optimal‐flow variations as a function of variations in the problem parameter(s). The theory of substitutes and complements provides necessary and sufficient conditions for optimal‐flow changes to consistently have the same (or the opposite) sign(s) in two given arcs, whereas the monotonicity theorem links changes in the value of the parameters to changes in optimal arc flows. Bounds on the rates of changes are also discussed. Compared with pure networks, the presence of gains makes qualitative sensitivity analysis here a much harder task. We show the profound effect on computational issues caused by the departure from the pure network structure. © 1996 John Wiley & Sons, Inc.

Suggested Citation

  • Antoine Gautier & Frieda Granot, 1996. "Ripples, complements, and substitutes in generalized networks," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(1), pages 1-21, February.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:1:p:1-21
    DOI: 10.1002/(SICI)1520-6750(199602)43:13.0.CO;2-F
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    References listed on IDEAS

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    1. Joyce Elam & Fred Glover & Darwin Klingman, 1979. "A Strongly Convergent Primal Simplex Algorithm for Generalized Networks," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 39-59, February.
    2. Michael Florian & Sang Nguyen, 1976. "An Application and Validation of Equilibrium Trip Assignment Methods," Transportation Science, INFORMS, vol. 10(4), pages 374-390, November.
    3. Richard E. Rosenthal, 1981. "A Nonlinear Network Flow Algorithm for Maximization of Benefits in a Hydroelectric Power System," Operations Research, INFORMS, vol. 29(4), pages 763-786, August.
    4. M. Collins & L. Cooper & R. Helgason & J. Kennington & L. LeBlanc, 1978. "Solving the Pipe Network Analysis Problem Using Optimization Techniques," Management Science, INFORMS, vol. 24(7), pages 747-760, March.
    5. Frieda Granot & Arthur F. Veinott, 1985. "Substitutes, Complements and Ripples in Network Flows," Mathematics of Operations Research, INFORMS, vol. 10(3), pages 471-497, August.
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    1. Gautier, Antoine & Granot, Frieda & Levi, Maurice, 2002. "Alternative foreign exchange management protocols: an application of sensitivity analysis," Journal of Multinational Financial Management, Elsevier, vol. 12(1), pages 1-19, February.

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