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Linear parametric programming—A brief survey

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Sensitivity, Stability and Parametric Analysis

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 21))

Abstract

A brief survey of basic theoretical results and methodological work based on the simplex technique as well as a brief historial development in the field of linear parametric programming is given avoiding pure mathematical considerations. This survey is composed so that it should not interfere with but rather complete introductory parts or general surveys in papers of this Study which deal mainly with the mathematical theory of parametric programming.

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Refereces

  1. J.M. Abadie and A.C. Williams, “Dual and parametric methods in decomposition”, in: R.L. Graves, P. Wolfe, eds., Recent advances in mathematical programming (McGraw-Hill, NY, 1963) pp. 149–158.

    Google Scholar 

  2. R.M. Arana, “Programming with parametric elements of the matrix coefficient”, R.A.I.R.O. Recherche Opérationnelle/Operations Research 11 (1977) 233–238.

    MATH  MathSciNet  Google Scholar 

  3. R.L. Armacost and A.V. Fiacco, “Second order parametric sensitivity analysis in nonlinear programming and estimates by penalty function methods”, Technical Paper No. T-324, George Washington University (Washington, D.C., 1975).

    Google Scholar 

  4. R.L. Armacost and A.V. Fiacco, “Computational experiences in sensitivity analysis for nonlinear programming”, Mathematical Programming 6 (1974) 301–326.

    Article  MATH  MathSciNet  Google Scholar 

  5. R.L. Armacost and A.V. Fiacco, “Nonlinear programming sensitivity for R.R.S. perturbation: A brief survey and recent second-order-extensions”, Technical Paper Series T-334, The George Washington University (Washington D.C., 1976).

    Google Scholar 

  6. V. Balachandran, “An operator theory of parametric programming for the generalized transportation problem. Part 2: RIM, cost and bound operators”, Naval Research Logistics Quarterly 22 (1975) 101–125.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Balachandran, “An operator theory of parametric programming for the generalized transportation problem. Part 3: Weight operators”, Naval Research Logistics Quarterly 22 (1975) 297–315.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Balachandran, “An operator theory of parametric programming for the generalized transportation problem. Part 4: Global operators”, Naval Research Logistics Quarterly 22 (1975) 317–339.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Balachandran and G.L. Thompson, “An operator theory of parametric programming for the generalized transportation problem. Part 1: Basic theory”, Naval Research Logistics Quarterly 22 (1975) 79–100.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Bank, J. Guddat, D. Klatte, B. Kummer und K. Tammer, Nonlinear parametric optimization (Akademic Verlag, Berlin, 1982).

    Google Scholar 

  11. R. Bellman, Adaptive Control Processes: A Guided Tour (Princeton University Press, Princeton, NJ, 1961).

    Google Scholar 

  12. C. Blair and R.G. Jeroslow, “The value function of an integer programme”, Working Paper No. MS-80-12, Georgia Institute of Technology (December 1980).

    Google Scholar 

  13. J. Bloech, “Zum Problem der nachträglichen änderung industrieller Produktionsprogramme”, Zeitschrift für Betriebswirtschaft 36 (1966) 186–197.

    Google Scholar 

  14. E. Bodewig, Matrix Calculus, 2nd ed. (North Holland, Amsterdam, 1959).

    MATH  Google Scholar 

  15. B. Brosowski, “Zur parametrischen linearen Optimierung. II. Eine hinreichende Bedingung für die Unterhalbstetigkeit”, Operations Research Verfahren 31 (1979) 137–141.

    Google Scholar 

  16. B. Brosowski, “On parametric linear optimization. III. A necessary condition for lower semicontinuity”, Method of Operations Research 36 (1980) 21–30.

    MATH  MathSciNet  Google Scholar 

  17. B. Brosowski, Parametric semiinfinite optimization (Peter D. Lang Verlag, Frankfurt/Main, 1981).

    Google Scholar 

  18. B. Brosowski and K. Schnatz, “Parametrische Optimierung, differenzierbare parameterfunktionen”, Zeitschrift für angewandte Mathematik und Mechanik 60 (1980) T338–T339.

    MathSciNet  Google Scholar 

  19. P.J. Carstensen, “Complexity of some parametric integer and network programing problems”, private communication (March 1981).

    Google Scholar 

  20. T.A. Ciriani, “The free parametrization for linear programming with bounded variables”, Redinconti del Seminario Matematico della Universita di Padova XLVII (1972) 77–107.

    MathSciNet  Google Scholar 

  21. T.A. Ciriani, “Studies on postoptimal analysis in linear programming”, Paper presented at the IBM-Conference on MPSX in Essen (West Germany 1981).

    Google Scholar 

  22. R.W. Cottle, “Monotone solutions of the parametric linear complementarity problem”, Mathematical Programming 3 (1972) 210–224.

    Article  MATH  MathSciNet  Google Scholar 

  23. G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).

    MATH  Google Scholar 

  24. G.B. Dantzig, J. Folkman and N. Shapiro, “On the continuity of the minimum set of a continuous function”, Journal of Mathematical Analysis and Application 17 (1967) 519–548.

    Article  MATH  MathSciNet  Google Scholar 

  25. L.C. Dennis and J.L. Balintfy, “A linear programming analysis of institutional management policies”, Bulletin of the ORSA Meeting 16 (1968).

    Google Scholar 

  26. W. Dinkelbach, Sensitivitätsanalysen und parametrische Programmierung (Springer, Berlin, New York, 1969).

    MATH  Google Scholar 

  27. I. Dragan, “Un algorithme pour la resolution de certains problemes parametriques, avec un seul parametre contenue dans la fonction economique”, Revue de Roumain Mathematique pure et Application 11 (4) (1966) 447–451.

    MATH  MathSciNet  Google Scholar 

  28. J.P. Evans and F.J. Gould, “Stability in nonlinear programming”, Operations Research 18 (1) (1970) 107–118.

    Article  MATH  MathSciNet  Google Scholar 

  29. A.V. Fiacco, “Sensitivity analysis for nonlinear programming using penalty methods”, Mathematical Programming 10 (1976) 287–311.

    Article  MATH  MathSciNet  Google Scholar 

  30. A.V. Fiacco, “Nonlinear programming sensitivity analysis results using strong second order assumptions”, Numerical Optimization of Dynamic Systems (1980) 327–348.

    Google Scholar 

  31. A.V. Fiacco and G.P. McCormick, Nonlinear programming: Sequential unconstrained minimization techniques (Wiley, New York, 1968).

    MATH  Google Scholar 

  32. B.V. Finkelstein, “Generalization of the parametric linear programming problem”, Ekonomicko Matematiceskie Metody 1 (3) (1965) 442–450 (in Russian).

    MathSciNet  Google Scholar 

  33. B.V. Finkelstein and L.P. Gumenok, “Algorithm for solving a linear parametric program when the A-matrix depends upon a parameter”, Ekonomicko Matematiceskie Metody 13 (1977) 342–347 (in Russian).

    Google Scholar 

  34. C.O. Fong and M.R. Rao, “Parametric studies in transportation-type problems”, Naval Research Logistics Quarterly 22 (1975) 335–364.

    Article  MathSciNet  Google Scholar 

  35. T. Gal, Multiparametric linear programs as an aid for solving farm decision problems, Dissertation for C. Sc., Vysoka Skola Zemedelska. Praha, 1967 (in Czech).

    Google Scholar 

  36. T. Gal, “Changing a row of the A-matrix of a linear parametric problem”, CORE Discussion Paper No. 7018 (Louvain, Belgium, May 1970).

    Google Scholar 

  37. T. Gal, “Homogene mehrparametrische lineare Programmierung”, Unternehmensforschung 16 (1972) 115–136.

    MathSciNet  Google Scholar 

  38. T. Gal, “RIM multiparametric linear programming”, Management Science 21 (1975) 167–175.

    Article  MathSciNet  Google Scholar 

  39. T. Gal, “A general method for determining the set of all efficient solutions to a linear vectormaximum problem”, European Journal for Operational Research 1 (1977) 307–322.

    Article  MATH  MathSciNet  Google Scholar 

  40. T. Gal, Postoptimal analysis, parametric programming and related topics (McGraw-Hill, New York, 1979).

    Google Scholar 

  41. T. Gal, “A ‘Historiogramme’ on parametric programming”, Journal of the Operational Research Society 31 (1980) 449–451.

    Article  Google Scholar 

  42. T. Gal, “Letter on [41]”, Journal of the Operational Research Society 34 (2) (1983) 162–163.

    Article  Google Scholar 

  43. T. Gal and H. Leberling, “Postefficient sensitivity analysis in linear vectormaximum problems”, European Journal of Operational Research 8 (1981) 274–282.

    Article  MATH  MathSciNet  Google Scholar 

  44. T. Gal and J. Nedoma, “Multiparametric linear programming”, Management Science 18 (1972) 406–422.

    Article  MATH  MathSciNet  Google Scholar 

  45. T. Gal and B. Vogeler, “An interactive procedure for solving a special substitution problem”, Working Paper No 43 (Fern Universität, Hagen/West Germany, 1980).

    Google Scholar 

  46. S.I. Gass and T.L. Saaty, “The parametric objective function 2”, Operations Research 3 (1955) 395–401

    Article  MathSciNet  Google Scholar 

  47. S.I. Gass and T.L. Saaty, “The computational algorithm for the parametric objective function”, Naval Research Logistics Quarterly 2 (1955) 39–45.

    Article  MathSciNet  Google Scholar 

  48. J. Gauvin and J.W. Tolle, “Differential stability in nonlinear programming”, SIAM Journal of Control and Optimization 15 (1977) 294–311.

    Article  MATH  MathSciNet  Google Scholar 

  49. A.M. Geoffrion, “Strictly concave parametric programming. I. Basic Theory”, Management Science 13 (1966) 244–253.

    Article  MathSciNet  Google Scholar 

  50. A.M. Geoffrion, “Strictly concave parametric programming. II. Additional theory and computational considerations”, Management Science 13 (1967) 359–370.

    Article  MathSciNet  Google Scholar 

  51. A.M. Geoffrion, “Solving bicriterion mathematical programs”, Operations Research 15 (1967) 39–54.

    Article  MATH  MathSciNet  Google Scholar 

  52. A.M. Geoffrion and R. Nauss, “Parametric and postoptimality analysis in integer linear programming”, Management Science 13 (1977) 453–466.

    Article  Google Scholar 

  53. J.J. Glen, “A parametric programming method for beef cottle ration formulation”, Journal of the Operational Research Society 31 (1980) 689–698.

    Article  Google Scholar 

  54. R.L. Graves, “Parametric linear programming”, in: R.L. Graves and P. Wolfe, eds., Recent advances in mathematical programming (McGraw-Hill, New York, 1963) pp. 201–210.

    Google Scholar 

  55. H.J. Greenberg and W.P. Pierskalla, “Extension of the Evans-Gould stability theorems for mathematical programming”, Operations Research 20 (1972) 143–153.

    Article  MATH  MathSciNet  Google Scholar 

  56. P.E. Hildebrand, “Farm organization and resource fixity-modification of the LP-model”, Ph.D. Thesis, Michigan State University (1959).

    Google Scholar 

  57. R.R. Hocking and R.L. Sheppard, “Parametric solution of a class of nonconvex programs”, Operations Research 19 (1971) 1742–1747.

    Article  MATH  MathSciNet  Google Scholar 

  58. W. House, “Sensitivity analysis—a case study of the pipeline industry”, The Engineering Economist 12 (1966) 155–166.

    Article  Google Scholar 

  59. W. House, “Use of sensitivity analysis in capital budgeting”, Management Services 4 (1967) 37–40.

    Google Scholar 

  60. P. Huard, ed., Point-to-Set maps and mathematical programming, Mathematical Programming Study 10 (North-Holland, Amsterdam, 1979).

    Google Scholar 

  61. E.L. Hullander, “Parametric linear programming analysis of state aid plants to non-public schools”, Bulletin of the ORSA Meeting 19 Suppl. 1 (1971) WP 7.18.

    Google Scholar 

    Google Scholar 

  62. R. Jeroslow, “Bracketing discrete problems by two problems of linear optimization”, Working Paper, Carnegie-Mellon University (July, 1976).

    Google Scholar 

  63. C.H. Jones, “Parametric production planning”, Management Science 19 (1967) 843–866.

    Article  Google Scholar 

  64. I.K. Kaneko, “A maximization problem related to parametric linear complementarity”, SIAM Journal of Control and Optimization 16 (1978) 41–55.

    Article  MATH  Google Scholar 

  65. U. Kausmann, K. Lommatzsch and F. Nozicka, Lineare parametrische Optimierung (Akademie Verlag, Berlin, 1976).

    MATH  Google Scholar 

  66. K.H. Khalil and P.U. Kokotovic, “D-Stability and multi-parameter singular perturbation”, SIAM Journal of Control and Optimization 17 (1979) 56–65.

    Article  MATH  MathSciNet  Google Scholar 

  67. Ch. Kim, “Parametrizing an activity vector in linear programming”, Operations Research 19 (1971) 1632–1646.

    Article  MATH  MathSciNet  Google Scholar 

  68. D. Klatte, “Lineare Optimierungsprobleme mit Parametern in allen Koeffizienten der Zielfunktion und der Restriktionen”, Wissenschaftliche Zeitschrift der Humboldt-Universität Berlin, mathematisch-naturwissenschaftliche Reihe 26 (1977) Heft 5.

    Google Scholar 

    Google Scholar 

  69. D. Klatte, “On the lower semicontinuity of optimal sets in convex parametric optimization”, Mathematical Programming Study 10 (1979) 104–109.

    MATH  MathSciNet  Google Scholar 

  70. D. Klein and S. Holm, “Integer programming post-optimal analysis with cutting planes”, Management Science 25 (1979) 64–72

    Article  MATH  MathSciNet  Google Scholar 

  71. W.T. Knowles, I. Gupta and M. Zia Hasan, “Decomposition of water distribution networks”, AIIE Transactions 8 (1976) 443–448.

    Google Scholar 

  72. K.O. Kortanek and A.L. Soyster, “On equating the difference between the optimal and marginal values of general convex programs”, Journal of Optimization Theory and Applications, to appear, loc. cit. [117].

    Google Scholar 

  73. W. Krabs, “Zur stetigen Abhängigkeit des Extremalwertes eines konvexen Optimierungsproblems von einer stetigen änderung des Problems”, Zeitschrift für angewandte Mathematik und Mechanik 52 (1972) 359–368.

    Article  MATH  MathSciNet  Google Scholar 

  74. B. Kummer, “Globale Stabilität in der quadratischen Optimierung”, Wissenschaftliche Zeitschrift der Humboldt-Universität Berlin, mathematisch-naturwissenschaftliche Reihe 26 (1977) Heft 5.

    Google Scholar 

    Google Scholar 

  75. S. Kurcyusz and S. Zowe, “Regularity and stability for the mathematical programming problem in Banach-spaces”, Journal for applied Mathematics and Optimization 5 (1979) 49–62.

    Article  MATH  MathSciNet  Google Scholar 

  76. Ch. Laperche, “Multiparametric linear programming algorithmic aspects”, Memoire presente de l’obtenue du grade d’ingenieur civil ou mathematiques appliquees, Université Catholique de Louvain (1970).

    Google Scholar 

  77. F. Lempio and H. Maurer, “Differential stability in infinite-dimensional nonlinear programming”, Journal of Applied Mathematics and Optimization 6 (1980) 139–152.

    Article  MATH  MathSciNet  Google Scholar 

  78. E.S. Levitin, “On differential properties of the optimal value of parametric problems of mathematical programming”, Sowiet Mathematical Doklady Akademii Nauk 15 (1974) 603–608.

    MATH  Google Scholar 

  79. E.S. Levitin, “On the local perturbation theory of a problem of mathematical programming in Banach-space”, Sowiet Mathematical Doklady Akademii Nauk 16 (1975) 1954.

    Google Scholar 

  80. E.S. Levitin, “Differentiability with respect to a parameter of the optimal value in parametric problems of mathematical programming”, Kibernetika (1976) 44–59.

    Google Scholar 

  81. K. Lommatzsch, ed., Anwendungen der linearen parametrischen Optimierung (Birkhäuser Verlag, Basel und Stuttgart, 1979).

    MATH  Google Scholar 

  82. G. Lorenzen, Parametrische Optimierung und einige Anwendungen (R. Oldenburg Verlag, München-Wien, 1974).

    MATH  Google Scholar 

  83. D.F. Lyons and V.A. Dodd, “The mix-feed problem”, in: K.B. Haley, ed., Proceedings of the 7th IFORS International Conference of Operations Research (North-Holland, Amsterdam, 1975) pp. 1–15.

    Google Scholar 

  84. A. S. Manne, “Notes on parametric linear programming”, RAND Corporation Report No p-468 (1953).

    Google Scholar 

  85. A.S. Manne, Scheduling of petroleum refinery operations (Harvard University Press, Cambridge, Massachusetts, 1956).

    Google Scholar 

  86. R.E. Marsten and T.L. Morin, “Parametric integer programming: The right-hand-side-case”, Annales of Discrete Mathematics 1 (1977) 375–390.

    Article  MathSciNet  Google Scholar 

  87. R.D. McBride and J.S. Yormark, “Finding all solutions of a Class of parametric quadratic integer programming problems”, Management Science 26 (1980) 784–795.

    Article  MATH  MathSciNet  Google Scholar 

  88. P. van Moeseke and G. Tintner, “Base duality theorem for stochastic and parametric programming”, Unternehmensforschung 8 (1964) 75–79.

    Article  MATH  MathSciNet  Google Scholar 

  89. H. Müller-Merbach, “Sensibilitätsanalyse der Losgrößenbestimmung”, Unternehmensforschung 6 (1962) 79–88.

    Article  Google Scholar 

  90. K.G. Murty, “Computational complexity of parametric linear programming”, Mathematical Programming 19 (1980) 213–219.

    Article  MATH  MathSciNet  Google Scholar 

  91. G. Nemhauser, Introduction to dynamic programming (Wiley, New York, 1966).

    Google Scholar 

  92. N. Noltemeier, Sensitivitätsanalyse bei diskreten linearen Optimierungsproblemen, Lecture Notes in Operations Research and Mathematical Systems (Springer, Berlin, 1970).

    MATH  Google Scholar 

  93. F. Nozicka, “über eine Klasse von linearen einparametrischen Optimierungsproblemen”, Mathematik, Operationsforschung und Statistik 3 (1972) 159–194.

    MathSciNet  Google Scholar 

  94. F. Nozicka, J. Guddat, H. Hollatz and B. Bank, Theorie der linearen parametrischen Optimierung (Akademie-Verlag, Berlin, 1974).

    MATH  Google Scholar 

  95. W.-D. Oberhoff, Integrierte Produktionsplanung. Deterministische Entscheidungsmodelle zur Planung optimaler Losgrößen und des optimalen Produktionsprogramms (Studienverlag Dr. N. Brockmeyer, Bochum, 1975).

    Google Scholar 

  96. D.T. O’Laoghaire and D.M. Himmelblau, “Modelling and sensitivity analysis for planning decision in water resource expanses”, Water Research Bulletin 8 (1972) 653–668.

    Google Scholar 

  97. W. Orchard-Hays, Advanced linear programming computing technique, (McGraw-Hill, New York, 1968).

    Google Scholar 

  98. J.-S. Pang, “A parametric linear complementarity techniques for optimal portfolio selection with a risk-free asset”, Operations Research 28 (1980) 927–941.

    Article  MATH  MathSciNet  Google Scholar 

  99. C. van de Panne, “Parametrizing an activity vector in linear programming”, Technical Notes (1972) 389–391.

    Google Scholar 

  100. C. van de Panne, Methods for linear and quadratic programming (North-Holland, Amsterdam, 1975).

    MATH  Google Scholar 

  101. C. van de Panne and J. Popp, “Minimum cost cattle feed under probabilistic protein constraints”, Management Science 9 (1968) 405–430.

    Article  Google Scholar 

  102. E.L. Peterson, “The duality between suboptimization and parameter deletion with application to parametric programming and decomposition theory in geometric programming”, Mathematics of Operations Research 2(1977) 311–319.

    Article  MATH  MathSciNet  Google Scholar 

  103. H. Raiffa, Decision analysis—introductory lectures on choices under uncertainty (Addison-Wesley, Reading, MA 1968).

    MATH  Google Scholar 

  104. A. Rappoport, “Sensitivity analysis—validating link between information systems and decision systems”, Management Science 12 (1966) C–271.

    Google Scholar 

  105. A. Rappoport, “Sensitivity analysis in decision making”, Accounting Review 42 (1967) 441–456.

    Google Scholar 

  106. K. Ritter, “über Probleme parameterabhängiger Planungsrechnung”, DVL Bericht Nr. 238 (Porz-Wahn, 1963).

    Google Scholar 

  107. S.M. Robinson, “Stability theory for systems of inequalities. Part I: Linear systems”, SIAM Journal of Numerical Analysis 12 (1975) 754–769.

    Article  MATH  Google Scholar 

  108. S.M. Robinson, “Stability theory for systems of inequalities. Part II. Differentiable nonlinear systems”, SIAM Journal of Numerical Analysis 13 (1976) 497–513.

    Article  MATH  Google Scholar 

  109. R.T. Rockafellar, “Duality and stability in extremum problems involving convex functions”, Pacific Journal of Mathematics 21 (1967) 167–187.

    MATH  MathSciNet  Google Scholar 

  110. T.L. Saaty, “Coefficient perturbation of a constrained extremum,” Operations Research 7 (1959) 284–303.

    MathSciNet  Google Scholar 

  111. T.L. Saaty and S.I. Gass, “The parametric objective function. 1”, Operations Research 2 (1954) 316–319.

    Article  MathSciNet  Google Scholar 

  112. T.L. Saaty and K.W. Webb, “Sensitivity and renewals in scheduling aircraft overhaul”, in: J. Banbury and J. Maitland, eds., Proceedings of the second international conference of operations research (English University Press, London, 1961) pp. 708–716.

    Google Scholar 

  113. P.C. Saxena and S.P. Aggarwal, “Parametric linear fractional functional programming”, Economical Computings and Economical Research (1980) 87–97.

    Google Scholar 

  114. P. Schimitzek, “Rezepturoptimierung für Fleischerzeugnisse”, Dissertation, RWTH Aachen, Institut für Wirtschaftswissenschaften (Aachen 1981).

    Google Scholar 

  115. H. Seelbach, “Rentabilitätsmaximierung bei variablem Eigenkapital”, Zeitschrift für Betriebswirtschaft 8 (1968) 237–256.

    Google Scholar 

  116. J.K. Sengupta and B.C. Sanyal, “Sensitivity analysis methods for a crop-mix-problem in LP”, Unternehmensforschung 14 (1970) 2–26.

    Article  MATH  Google Scholar 

  117. I. Sherman and W.J. Morrison, “Adjustment of an inverse matrix corresponding to a change of one element of a given matrix”, Annals of Mathematics and Statistics 21 (1950) 124–127.

    Article  MATH  MathSciNet  Google Scholar 

  118. A.L. Soyster, “An objective function perturbation with economic interpretations”, Management Science 27 (1981) 231–237.

    Article  MATH  MathSciNet  Google Scholar 

  119. W. Stahl, Sensitivitätsanalyse mehrstufiger Standortprobleme (Verlag Anton Hain, Meisenheim am Glan, 1976).

    MATH  Google Scholar 

  120. R.E. Steuer, “Algorithms for linear programming problems with interval objective function coefficients”, Working Paper No. BA 52, University of Kentucky at Lexington (December 1979).

    Google Scholar 

  121. P. Streufert, “Optimale Jahresplanung nach Gewinn und Arbeitszeit in der fischverarbeitenden Industrie”, Wirtschaftswissenschaften 5 (1965) 802–808.

    Google Scholar 

  122. H. Suzuki, “A generalized knapsack problem with variable coefficients”, Mathematical Programming 15 (1978) 162–176.

    Article  MATH  MathSciNet  Google Scholar 

  123. K.B. Tlegenov, K.K. Kaltschaief and P.P. Zapletin, Mathematical programming methods (Nauka, Alma-Ata, 1975) (in Russian).

    Google Scholar 

  124. K.W. Webb, “The mathematical theory of sensitivity”, Paper presented at the 18th National Meeting of ORSA (Detroit, Michigan, 1960).

    Google Scholar 

  125. E. Weickenmeier, “Ein Algorithmus zur systematischen und vollständigen Lösung eines parametrischen linearen Programms mit einem Parameter in allen Koeffizienten”, Zeitschrift für Operations Research 22 (1978) 131–149.

    Article  MATH  MathSciNet  Google Scholar 

  126. H. Weinert, “Doppelt einparametrische lineare Optimierung. I. Unabhängige Parameter”, Mathematik Operations-Forschung und Statistik 2 (1971) 19–39.

    MathSciNet  Google Scholar 

  127. G.R. Wels, “A sensitivity analysis of simulated river basin planning for capital budgeting decisions”, Computings and Operations Research 2 (1975) 49–54.

    Article  Google Scholar 

  128. P.L. Yu and M. Zeleny, “The set of all nondominated solutions in linear cases and a multieriteria simplex method”, Journal of Mathematical Analysis and Applications 49 (1975) 430–468.

    Article  MATH  MathSciNet  Google Scholar 

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Gal, T. (1984). Linear parametric programming—A brief survey. In: Fiacco, A.V. (eds) Sensitivity, Stability and Parametric Analysis. Mathematical Programming Studies, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121210

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