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Qualitative Sensitivity Analysis

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Advances in Sensitivity Analysis and Parametic Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 6))

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Abstract

In (post-optimal) sensitivity analysis, one usually draws information from both the objective function and the feasible set. In this chapter, we propose an approach based mainly on the latter, and whose predictions are valid as long as the objective function satisfies some regularity conditions such as additivity, convexity and submodularity. The resulting “pre-optimal” analysis is qualitative and provides sensitivity analysis results prior to solving the problem, which is an attractive feature for large-scale problems where the use of conventional parametric programming methods has a very high computational cost. Among the results discussed, the Ripple Theorem provides an upper bound on the variation in the optimal value of a variable resulting from changes in problem parameters. The theory of substitutes and complements leads to necessary and sufficient conditions for changes in optimal values of two arbitrary decision variables to consistently be of the same (or opposite) sign. The Monotonicity and Smoothing Theorems link changes in problem parameters to changes in optimal variable values, and provide bounds on the absolute change in an optimal solution. Computational issues are discussed, and a qualitative analysis of a production/inventory problem and an asset allocation problem are presented as tutorial examples.

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References

  1. Aho, A.V., J.E. Hopcroft and J.D. Ullman: The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.

    Google Scholar 

  2. Ahuja, R., T. Magnanti and J. Orlin: Network Flows: Theory, Algorithms and Applications, Prentice Hall, 1993.

    Google Scholar 

  3. Arai, T., S. Ueno and Y. Kajitani: Generalization of a theorem on the parametric maximum flow problem, Discrete Applied Mathematics 41 (September 1993), 69–74.

    Article  Google Scholar 

  4. Athey, S.: Monotone comparative statics in stochastic optimization problems. Technical report, Graduate School of Business, Stanford University, January 1995.

    Google Scholar 

  5. Bagchi, U.: Simultaneous minimization of mean and variation of flow time and waiting time in single machine systems, Operations Research 37 (1989), 118–125.

    Article  Google Scholar 

  6. Bagwell, K., G. Ramey: Coordination economies, advertising, and search behavior in retail markets, American Economic Review 84 (1994), 498–517.

    Google Scholar 

  7. Baker, K.: Workforce allocation in cyclical scheduling problems, Operations Research Quarterly 27 (1976), 155–167.

    Article  Google Scholar 

  8. Bartholdi, J.J., J.B. Orlin and H.D. Ratliff: Scheduling via integer programs with circular ones, Operations Research 28 (1980), 1074–1085.

    Article  Google Scholar 

  9. Berge, C.: Graphs and Hypergraphs, North Holland/American Elsevier, 1973.

    Google Scholar 

  10. Brumelle, S., D. Granot: The repair kit problem revisited, Operations Research 41 (1993), 994–1066.

    Article  Google Scholar 

  11. Brumelle, S., D. Granot and L. Liu: Totally ordered optimal solutions and parametric maximum flows. Working paper, Faculty of Commerce, The University of British Columbia, 1996.

    Google Scholar 

  12. Bulow, J.I., J.D. Geanakoplos and P.D. Klemperer: Multimarket oligopoly: strategic substitutes and complements, Journal of Political Economy 93 (1985), 488–511.

    Article  Google Scholar 

  13. Bushnell, T.D., A.D. Shepard: The economics of modern manufacturing: comment, The American Economic Review 85 (1995), 987–990.

    Google Scholar 

  14. Ciurria, I.: Substitutes and Complements in Multicommodity Flows and Inventories. Ph.D. dissertation, Stanford University, Department of Operation Research, 1989.

    Google Scholar 

  15. Ciurria, I., F. Granot and A.F. Veinott, Jr.: Substitutes, complements and ripples in multicommodity production planning. Technical report, Faculty of Commerce and Business Administration, The University of British Columbia, 1989.

    Google Scholar 

  16. Ciurria, I., F. Granot and A.F. Veinott, Jr.: Substitutes, complements and ripples in multicommodity flows on suspension graphs. Technical report, Faculty of Commerce and Business Administration, The University of British Columbia, 1989.

    Google Scholar 

  17. Crawley, P., R.P. Dilworth: Algebraic Theory of Lattices, Prentice-Hall, 1973.

    Google Scholar 

  18. DeCroix, G.A.: Equilibrium Warranties and Prices. Ph.D. dissertation, Stanford University, Department of Operations Research, 1992.

    Google Scholar 

  19. Denardo, E.V., G. Huberman and U.G. Rothblum: Optimal locations on a line are interleaved, Operations Research 30 (1982), 745–754.

    Article  Google Scholar 

  20. Duffin, R.J.: Topology of series-parallel networks, Journal of Mathematical Analysis and Applications 10 (1965), 303–318.

    Article  Google Scholar 

  21. Fulkerson, D.R.: An out-of-kilter method for minimal cost network flow problems, Journal of the Society for Industrial and Applied Mathematics 9 (1961), 18–27.

    Article  Google Scholar 

  22. Gale, D., T. Politof: Substitutes and complements in network flow problems, Discrete Applied Mathematics 3 (1981), 175–186.

    Article  Google Scholar 

  23. Gallo, G., H. Grigoriadis and R.E. Tarjan: A fast parametric network flow algorithm, SIAM Journal on Computing 18 (1989), 30–35.

    Article  Google Scholar 

  24. Garey, M.R., D.S. Johnson: Computers and intractability: A guide to NP-Completeness, W. H. Freeman and Company, New York, 1979.

    Google Scholar 

  25. Gautier, A., F. Granot: A parametric analysis of a nonlinear asset allocation management model, Operations Research Letters 11 (June 1992), 273–279.

    Article  Google Scholar 

  26. Gautier, A., F. Granot: On the equivalence of constrained and unconstrained flows, Discrete Applied Mathematics 55 (1994), 113–132.

    Article  Google Scholar 

  27. Gautier, A., F. Granot: A parametric analysis of a constrained nonlinear inventory-production model, Management Science 40 (November 1994), 1500–1513.

    Article  Google Scholar 

  28. Gautier, A., F. Granot: Ripples, complements and substitutes in singly-constrained monotropic parametric network flow problems, Networks 24 (1994), 284–296.

    Article  Google Scholar 

  29. Gautier, A., F. Granot: Ripples, complements and substitutes in generalized networks, Naval Research Logistics 43 (february 1996), 1–21.

    Google Scholar 

  30. Gautier, A., F. Granot and H. Zheng: Substitutes, complements and ripples in monotropic programming. Technical report, Faculty of Commerce and Business Administration, The University of British Columbia, January 1995.

    Google Scholar 

  31. Gorman, T.: More scope for qualitative economics, Review of Economic Studies 31 (1964), 65–68.

    Article  Google Scholar 

  32. Granot, F., A.F. Veinott, Jr.: Substitutes, complements and ripples in network flows, Mathematics of Operations Research 10 (1985), 471–497.

    Article  Google Scholar 

  33. Greenberg, H.J.: Measuring complementarity and qualitative determinacy in matricial forms. In Greenberg, H.J., J.S. Maybee (eds.), Proceedings of the Symposium on Computer Assisted Analysis and Model Simplification. Academic Press, New York, 1981.

    Google Scholar 

  34. Greenberg, H.J., J.R. Lundgren and J.S. Maybee: Rectangular matrices and signed graphs, SIAM Journal of Algebraic and Discrete Methods 4 (1983), 50–61.

    Article  Google Scholar 

  35. Greenberg, H.J., J.R. Lundgren and J.S. Maybee: Inverting graphs of rectangular matrices, Discrete Applied Mathematics 8 (1984), 255–265.

    Article  Google Scholar 

  36. Greenberg, H.J., J.R. Lundgren and J.S. Maybee: Inverting signed graphs, SIAM Journal of Algebraic and Discrete Methods 5 (1984), 216–222.

    Article  Google Scholar 

  37. Gries, D., F.B. Schneider: A logical approach to discrete math, Texts and Monographs in Computer Science, Springer-Verlag, 1993.

    Google Scholar 

  38. Hicks, J.R.: Value and Capital, 2nd edition, Oxford Press, Clarendon, 1939.

    Google Scholar 

  39. Hicks, J.R., R.G.D. Allen: A reconsideration of the theory of value, parts I and II, Economica 1 (1934), 52–76

    Article  Google Scholar 

  40. Johnson, C.R.: Sign patterns of inverse nonnegative matrices, Linear Algebra and its Applications 55 (1983), 69–80.

    Article  Google Scholar 

  41. Johnson, C.R., F.T. Leighton and H.A. Robinson: Sign patterns of inverse-positive matrices, Linear Algebra and its Applications 24 (1979), 75–83.

    Article  Google Scholar 

  42. Klee, V., R. Ladner: Qualitative matrices: Strong sign-solvability and weak satisfiability. In Greenberg, H.J., J.S. Maybee (eds.), Computer-assisted analysis and model simplification, 293–320. Associated Press, New York, 1981.

    Google Scholar 

  43. Klee, V., R. Ladner and R. Manber: Sign solvability revisited, Linear Algebra and its Applications 59 (1984), 131–157.

    Article  Google Scholar 

  44. Klingman, D.: Finding equivalent network formulations for constrained network problems, Management Science 23 (1977), 737–744.

    Article  Google Scholar 

  45. Lady, G.M.: The structure of qualitative determinate linear systems. Technical report, Rice University Systems Report, Houston, Texas, 1967.

    Google Scholar 

  46. Lady, G.M.: The structure of qualitative relationships, Econometrica 51 (1983), 197–218.

    Article  Google Scholar 

  47. Lancaster, K.: The scope of qualitative economics, Review of Economic Studies 29 (1962), 99–132.

    Article  Google Scholar 

  48. Lancaster, K.: The theory of qualitative linear systems, Econometrica 33 (1965), 395–408.

    Article  Google Scholar 

  49. Lancaster, K.: The solution of qualitative comparative static problems, Quarterly Journal of Economics 80 (1966), 278–295.

    Article  Google Scholar 

  50. Lippman, S.A., J.W. Mamer and K. McCardle: Comparative statics in non-cooperative games via transfinitely iterated play, Journal of Economic Theory 41 (1987), 288–303.

    Article  Google Scholar 

  51. Manber, R.: Graph-theoretical approach to qualitative solvability of linear systems. Technical report, University of Washington, Department of Mathematics, 1981.

    Google Scholar 

  52. Markowitz, H.: Portfolio Selection, John Wiley & Sons, 1959.

    Google Scholar 

  53. May, R.M.: Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, 1973.

    Google Scholar 

  54. Maybee, J., J.P. Quirk: Qualitative problems in matrix theory, SIAM Review 11 (1969), 30–51.

    Article  Google Scholar 

  55. Maybee, J.S.: Sign-solvability. In Greenberg, H.J., J.S. Maybee (eds.), Computer-assisted analysis and model simplification, 201–258. Associated Press, New York, 1981.

    Google Scholar 

  56. Milgrom, P., J. Roberts: The economics of modern manufacturing technology, strategy, and organization, American Economic Review 80 (1990), 511–528.

    Google Scholar 

  57. Milgrom, P., J. Roberts: Rationalization, learning and equilibrium in games with strategic complementarities, Econometrica 58 (1990), 1255–1277.

    Article  Google Scholar 

  58. Milgrom, P., J. Roberts: Comparing equilibria, American Economic Review 84 (1994), 441–459.

    Google Scholar 

  59. Milgrom, P., J. Roberts: Complementaries and fit: strategy, structure and organizational change in manufacturing, Journal of Accounting and Economics 19 (1995), 179–208.

    Article  Google Scholar 

  60. Milgrom, P., J. Roberts: The economics of modern manufacturing technology: reply, American Economic Review 85 (1995), 997–999.

    Google Scholar 

  61. Milgrom, P., C. Shannon: Monotone comparative statics, Econometrica 62 (1994), 157–180.

    Article  Google Scholar 

  62. Mulvey, J.M.: Nonlinear network models in finance. In Advances in Mathematical Programming and Financial Planning, volume 1, 253–271, Greenwich, Conn., 1987. J.A.I. Press.

    Google Scholar 

  63. Orlin, J.B.: Minimum convex cost dynamic network flows. Technical report, Sloan School of Management, M.I.T., 1982.

    Google Scholar 

  64. Orlin, J.B., U.G. Rothblum: Computing optimal scalings by parametric network algorithms, Mathematical Programming 10 (1985), 1–10.

    Article  Google Scholar 

  65. Pinheiro Mello, M.: Generalized Leontieff Systems and Lattice Programming. Ph.D. dissertation, Department of Operations Research, Stanford University, 1993.

    Google Scholar 

  66. Provan, J.S.: Determinacy in linear systems and networks, SIAM Journal of Algebraic and Discrete Methods 4 (1983), 262–278.

    Article  Google Scholar 

  67. Provan, J.S.: Substitutes and complements in constrained linear models. Technical Report UNC/ORSA/TR-86/23, University of North Carolina at Chapel Hill, Curriculum in Operations Research and Systems Analysis, december 1986.

    Google Scholar 

  68. Provan, J.S., A.S. Kydes: Determinacy in network models: A study of structure and correlation. BNL Report 51243, Brookhaven National Laboratory, Upton, NY, 1980.

    Google Scholar 

  69. Quirk, J.P., R. Ruppert: Qualitative economics and the stability of equilibrium, Review of Economic Studies 32 (1965), 311–326.

    Article  Google Scholar 

  70. Ritschard, G.: Computable qualitative comparative static techniques, Econometrica 51 (1983), 1145–1168.

    Article  Google Scholar 

  71. Rockafellar, R.T.: The elementary vectors of a subspace of ℜn .In R.C. Bose & T.A. Dowling, editor, Combinatorial Mathematics and its Applications, Proceedings of the Chapel Hill Conference, 104–127. University of North Carolina Press, 1967.

    Google Scholar 

  72. Rockafellar, R.T.: Network Flows and Monotropic Optimization, John Wiley & Sons, 1984.

    Google Scholar 

  73. Samuelson, P.A.: Foundations of Economic Analysis, Anthenum, 1971. (Originally published by Harvard University Press, 1947).

    Google Scholar 

  74. Samuelson, P.A.: Complementarity: An essay on the fortieth anniversary of the Hicks-Allen revolution in demand theory, J. Econ. Lit. (1974), 1255–1289.

    Google Scholar 

  75. Schrijver, A.: Theory of Linear and Integer Programming, John Wiley & Sons, 1986.

    Google Scholar 

  76. Shapley, L.S.: On network flow functions, Naval Research Logistics Quarterly 8 (1961), 151–158.

    Article  Google Scholar 

  77. Shapley, L.S.: Complements and substitutes in the optimal assignment problem, Naval Research Logistics Quarterly 9 (1962), 45–48.

    Article  Google Scholar 

  78. Shiloach, Y.: A polynomial solution to the undirected two paths problem, Journal of the ACM 27 (1980), 445–456.

    Article  Google Scholar 

  79. Sobel, M.J.: Isotone comparative statics in supermodular games. Technical Report HAR-88–003, W. Averall Harriman School, State University of New York, Stonybrook, 1988.

    Google Scholar 

  80. Spãlti, S., T. Liebling: A special case of a network problem with one side constraint. Technical Report R.O. 890429, Département de Mathématiques, EPF Lausanne, Switzerland, 1989.

    Google Scholar 

  81. Tarski, A.: A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 (1955), 285–309.

    Article  Google Scholar 

  82. Topkis, D.M.: Equilibrium points in nonzero-sum n-person subadditive games. Technical Report Operations Research Center Technical Report ORC 70–38, University of California, 1970.

    Google Scholar 

  83. Topkis, D.M.: Minimizing a submodular function on a lattice, Operations Research 26 (1978), 305–321.

    Article  Google Scholar 

  84. Topkis, D.M.: Equilibrium points in nonzero-sum n-person subadditive games, SIAM Journal on Control and Optimization 17 (1979), 773–787.

    Article  Google Scholar 

  85. Topkis, D.M.: Comparative statics of the firm, Journal of Economic Theory 67 (1995), 370–401.

    Article  Google Scholar 

  86. Topkis, D.M.: The economics of modern manufacturing, comment, American Economic Review 85 (1995), 991–996.

    Google Scholar 

  87. Topkis, D.M., A.F. Veinott, Jr.: Monotone solutions of extremal problems on lattices. In Seventh International Mathematical Programming Abstracts. Stanford University, 1973. (abstract).

    Google Scholar 

  88. Truemper, K.: Unimodular matrices of flow problems with additional constraints, Networks 7 (1977), 343–358.

    Article  Google Scholar 

  89. Truemper, K.: Algebraic characterizations of unimodular matrices, SIAM Journal on Applied Mathematics 35 (1978), 328–332.

    Article  Google Scholar 

  90. Tutte, W.T.: Connectivity in Graphs, University of Toronto Press, Toronto, 1966.

    Google Scholar 

  91. Veinott, Jr., A.F.: Production planning with convex costs: a parametric study, Management Science 10 (1964), 441–460.

    Article  Google Scholar 

  92. Veinott, Jr., A.F.: Monotone solutions of extremal problems. Unpublished manuscript, 1965.

    Google Scholar 

  93. Veinott, Jr., A.F.: Substitutes and complements in network flow problems. Presentation to the ORSA meeting, San Francisco, CA, 1968.

    Google Scholar 

  94. Veinott, Jr., A.F.: Lattice Programming, Forthcoming.

    Google Scholar 

  95. Vives, X.: Nash equilibrium with strategic complementarities, Journal of Mathematical Economics 19 (1990), 305–321.

    Article  Google Scholar 

  96. Whitney, H.: Non-separable and planar graphs, Transactions of the American Mathematical Society 34 (1932), 339–362.

    Article  Google Scholar 

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Authors
  1. Antoine Gautier
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  2. Daniel Granot
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  3. Frieda Granot
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Editors and Affiliations

  1. FernUniversität, Hagen, Germany

    Tomas Gal

  2. University of Colorado at Denver, USA

    Harvey J. Greenberg

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Gautier, A., Granot, D., Granot, F. (1997). Qualitative Sensitivity Analysis. In: Gal, T., Greenberg, H.J. (eds) Advances in Sensitivity Analysis and Parametic Programming. International Series in Operations Research & Management Science, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6103-3_8

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  • DOI: https://doi.org/10.1007/978-1-4615-6103-3_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7796-2

  • Online ISBN: 978-1-4615-6103-3

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