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.
Preview
Unable to display preview. Download preview PDF.
Refereces
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.
R.M. Arana, “Programming with parametric elements of the matrix coefficient”, R.A.I.R.O. Recherche Opérationnelle/Operations Research 11 (1977) 233–238.
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).
R.L. Armacost and A.V. Fiacco, “Computational experiences in sensitivity analysis for nonlinear programming”, Mathematical Programming 6 (1974) 301–326.
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).
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.
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.
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.
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.
B. Bank, J. Guddat, D. Klatte, B. Kummer und K. Tammer, Nonlinear parametric optimization (Akademic Verlag, Berlin, 1982).
R. Bellman, Adaptive Control Processes: A Guided Tour (Princeton University Press, Princeton, NJ, 1961).
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).
J. Bloech, “Zum Problem der nachträglichen änderung industrieller Produktionsprogramme”, Zeitschrift für Betriebswirtschaft 36 (1966) 186–197.
E. Bodewig, Matrix Calculus, 2nd ed. (North Holland, Amsterdam, 1959).
B. Brosowski, “Zur parametrischen linearen Optimierung. II. Eine hinreichende Bedingung für die Unterhalbstetigkeit”, Operations Research Verfahren 31 (1979) 137–141.
B. Brosowski, “On parametric linear optimization. III. A necessary condition for lower semicontinuity”, Method of Operations Research 36 (1980) 21–30.
B. Brosowski, Parametric semiinfinite optimization (Peter D. Lang Verlag, Frankfurt/Main, 1981).
B. Brosowski and K. Schnatz, “Parametrische Optimierung, differenzierbare parameterfunktionen”, Zeitschrift für angewandte Mathematik und Mechanik 60 (1980) T338–T339.
P.J. Carstensen, “Complexity of some parametric integer and network programing problems”, private communication (March 1981).
T.A. Ciriani, “The free parametrization for linear programming with bounded variables”, Redinconti del Seminario Matematico della Universita di Padova XLVII (1972) 77–107.
T.A. Ciriani, “Studies on postoptimal analysis in linear programming”, Paper presented at the IBM-Conference on MPSX in Essen (West Germany 1981).
R.W. Cottle, “Monotone solutions of the parametric linear complementarity problem”, Mathematical Programming 3 (1972) 210–224.
G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).
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.
L.C. Dennis and J.L. Balintfy, “A linear programming analysis of institutional management policies”, Bulletin of the ORSA Meeting 16 (1968).
W. Dinkelbach, Sensitivitätsanalysen und parametrische Programmierung (Springer, Berlin, New York, 1969).
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.
J.P. Evans and F.J. Gould, “Stability in nonlinear programming”, Operations Research 18 (1) (1970) 107–118.
A.V. Fiacco, “Sensitivity analysis for nonlinear programming using penalty methods”, Mathematical Programming 10 (1976) 287–311.
A.V. Fiacco, “Nonlinear programming sensitivity analysis results using strong second order assumptions”, Numerical Optimization of Dynamic Systems (1980) 327–348.
A.V. Fiacco and G.P. McCormick, Nonlinear programming: Sequential unconstrained minimization techniques (Wiley, New York, 1968).
B.V. Finkelstein, “Generalization of the parametric linear programming problem”, Ekonomicko Matematiceskie Metody 1 (3) (1965) 442–450 (in Russian).
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).
C.O. Fong and M.R. Rao, “Parametric studies in transportation-type problems”, Naval Research Logistics Quarterly 22 (1975) 335–364.
T. Gal, Multiparametric linear programs as an aid for solving farm decision problems, Dissertation for C. Sc., Vysoka Skola Zemedelska. Praha, 1967 (in Czech).
T. Gal, “Changing a row of the A-matrix of a linear parametric problem”, CORE Discussion Paper No. 7018 (Louvain, Belgium, May 1970).
T. Gal, “Homogene mehrparametrische lineare Programmierung”, Unternehmensforschung 16 (1972) 115–136.
T. Gal, “RIM multiparametric linear programming”, Management Science 21 (1975) 167–175.
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.
T. Gal, Postoptimal analysis, parametric programming and related topics (McGraw-Hill, New York, 1979).
T. Gal, “A ‘Historiogramme’ on parametric programming”, Journal of the Operational Research Society 31 (1980) 449–451.
T. Gal, “Letter on [41]”, Journal of the Operational Research Society 34 (2) (1983) 162–163.
T. Gal and H. Leberling, “Postefficient sensitivity analysis in linear vectormaximum problems”, European Journal of Operational Research 8 (1981) 274–282.
T. Gal and J. Nedoma, “Multiparametric linear programming”, Management Science 18 (1972) 406–422.
T. Gal and B. Vogeler, “An interactive procedure for solving a special substitution problem”, Working Paper No 43 (Fern Universität, Hagen/West Germany, 1980).
S.I. Gass and T.L. Saaty, “The parametric objective function 2”, Operations Research 3 (1955) 395–401
S.I. Gass and T.L. Saaty, “The computational algorithm for the parametric objective function”, Naval Research Logistics Quarterly 2 (1955) 39–45.
J. Gauvin and J.W. Tolle, “Differential stability in nonlinear programming”, SIAM Journal of Control and Optimization 15 (1977) 294–311.
A.M. Geoffrion, “Strictly concave parametric programming. I. Basic Theory”, Management Science 13 (1966) 244–253.
A.M. Geoffrion, “Strictly concave parametric programming. II. Additional theory and computational considerations”, Management Science 13 (1967) 359–370.
A.M. Geoffrion, “Solving bicriterion mathematical programs”, Operations Research 15 (1967) 39–54.
A.M. Geoffrion and R. Nauss, “Parametric and postoptimality analysis in integer linear programming”, Management Science 13 (1977) 453–466.
J.J. Glen, “A parametric programming method for beef cottle ration formulation”, Journal of the Operational Research Society 31 (1980) 689–698.
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.
H.J. Greenberg and W.P. Pierskalla, “Extension of the Evans-Gould stability theorems for mathematical programming”, Operations Research 20 (1972) 143–153.
P.E. Hildebrand, “Farm organization and resource fixity-modification of the LP-model”, Ph.D. Thesis, Michigan State University (1959).
R.R. Hocking and R.L. Sheppard, “Parametric solution of a class of nonconvex programs”, Operations Research 19 (1971) 1742–1747.
W. House, “Sensitivity analysis—a case study of the pipeline industry”, The Engineering Economist 12 (1966) 155–166.
W. House, “Use of sensitivity analysis in capital budgeting”, Management Services 4 (1967) 37–40.
P. Huard, ed., Point-to-Set maps and mathematical programming, Mathematical Programming Study 10 (North-Holland, Amsterdam, 1979).
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.
R. Jeroslow, “Bracketing discrete problems by two problems of linear optimization”, Working Paper, Carnegie-Mellon University (July, 1976).
C.H. Jones, “Parametric production planning”, Management Science 19 (1967) 843–866.
I.K. Kaneko, “A maximization problem related to parametric linear complementarity”, SIAM Journal of Control and Optimization 16 (1978) 41–55.
U. Kausmann, K. Lommatzsch and F. Nozicka, Lineare parametrische Optimierung (Akademie Verlag, Berlin, 1976).
K.H. Khalil and P.U. Kokotovic, “D-Stability and multi-parameter singular perturbation”, SIAM Journal of Control and Optimization 17 (1979) 56–65.
Ch. Kim, “Parametrizing an activity vector in linear programming”, Operations Research 19 (1971) 1632–1646.
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.
D. Klatte, “On the lower semicontinuity of optimal sets in convex parametric optimization”, Mathematical Programming Study 10 (1979) 104–109.
D. Klein and S. Holm, “Integer programming post-optimal analysis with cutting planes”, Management Science 25 (1979) 64–72
W.T. Knowles, I. Gupta and M. Zia Hasan, “Decomposition of water distribution networks”, AIIE Transactions 8 (1976) 443–448.
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].
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.
B. Kummer, “Globale Stabilität in der quadratischen Optimierung”, Wissenschaftliche Zeitschrift der Humboldt-Universität Berlin, mathematisch-naturwissenschaftliche Reihe 26 (1977) Heft 5.
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.
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).
F. Lempio and H. Maurer, “Differential stability in infinite-dimensional nonlinear programming”, Journal of Applied Mathematics and Optimization 6 (1980) 139–152.
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.
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.
E.S. Levitin, “Differentiability with respect to a parameter of the optimal value in parametric problems of mathematical programming”, Kibernetika (1976) 44–59.
K. Lommatzsch, ed., Anwendungen der linearen parametrischen Optimierung (Birkhäuser Verlag, Basel und Stuttgart, 1979).
G. Lorenzen, Parametrische Optimierung und einige Anwendungen (R. Oldenburg Verlag, München-Wien, 1974).
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.
A. S. Manne, “Notes on parametric linear programming”, RAND Corporation Report No p-468 (1953).
A.S. Manne, Scheduling of petroleum refinery operations (Harvard University Press, Cambridge, Massachusetts, 1956).
R.E. Marsten and T.L. Morin, “Parametric integer programming: The right-hand-side-case”, Annales of Discrete Mathematics 1 (1977) 375–390.
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.
P. van Moeseke and G. Tintner, “Base duality theorem for stochastic and parametric programming”, Unternehmensforschung 8 (1964) 75–79.
H. Müller-Merbach, “Sensibilitätsanalyse der Losgrößenbestimmung”, Unternehmensforschung 6 (1962) 79–88.
K.G. Murty, “Computational complexity of parametric linear programming”, Mathematical Programming 19 (1980) 213–219.
G. Nemhauser, Introduction to dynamic programming (Wiley, New York, 1966).
N. Noltemeier, Sensitivitätsanalyse bei diskreten linearen Optimierungsproblemen, Lecture Notes in Operations Research and Mathematical Systems (Springer, Berlin, 1970).
F. Nozicka, “über eine Klasse von linearen einparametrischen Optimierungsproblemen”, Mathematik, Operationsforschung und Statistik 3 (1972) 159–194.
F. Nozicka, J. Guddat, H. Hollatz and B. Bank, Theorie der linearen parametrischen Optimierung (Akademie-Verlag, Berlin, 1974).
W.-D. Oberhoff, Integrierte Produktionsplanung. Deterministische Entscheidungsmodelle zur Planung optimaler Losgrößen und des optimalen Produktionsprogramms (Studienverlag Dr. N. Brockmeyer, Bochum, 1975).
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.
W. Orchard-Hays, Advanced linear programming computing technique, (McGraw-Hill, New York, 1968).
J.-S. Pang, “A parametric linear complementarity techniques for optimal portfolio selection with a risk-free asset”, Operations Research 28 (1980) 927–941.
C. van de Panne, “Parametrizing an activity vector in linear programming”, Technical Notes (1972) 389–391.
C. van de Panne, Methods for linear and quadratic programming (North-Holland, Amsterdam, 1975).
C. van de Panne and J. Popp, “Minimum cost cattle feed under probabilistic protein constraints”, Management Science 9 (1968) 405–430.
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.
H. Raiffa, Decision analysis—introductory lectures on choices under uncertainty (Addison-Wesley, Reading, MA 1968).
A. Rappoport, “Sensitivity analysis—validating link between information systems and decision systems”, Management Science 12 (1966) C–271.
A. Rappoport, “Sensitivity analysis in decision making”, Accounting Review 42 (1967) 441–456.
K. Ritter, “über Probleme parameterabhängiger Planungsrechnung”, DVL Bericht Nr. 238 (Porz-Wahn, 1963).
S.M. Robinson, “Stability theory for systems of inequalities. Part I: Linear systems”, SIAM Journal of Numerical Analysis 12 (1975) 754–769.
S.M. Robinson, “Stability theory for systems of inequalities. Part II. Differentiable nonlinear systems”, SIAM Journal of Numerical Analysis 13 (1976) 497–513.
R.T. Rockafellar, “Duality and stability in extremum problems involving convex functions”, Pacific Journal of Mathematics 21 (1967) 167–187.
T.L. Saaty, “Coefficient perturbation of a constrained extremum,” Operations Research 7 (1959) 284–303.
T.L. Saaty and S.I. Gass, “The parametric objective function. 1”, Operations Research 2 (1954) 316–319.
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.
P.C. Saxena and S.P. Aggarwal, “Parametric linear fractional functional programming”, Economical Computings and Economical Research (1980) 87–97.
P. Schimitzek, “Rezepturoptimierung für Fleischerzeugnisse”, Dissertation, RWTH Aachen, Institut für Wirtschaftswissenschaften (Aachen 1981).
H. Seelbach, “Rentabilitätsmaximierung bei variablem Eigenkapital”, Zeitschrift für Betriebswirtschaft 8 (1968) 237–256.
J.K. Sengupta and B.C. Sanyal, “Sensitivity analysis methods for a crop-mix-problem in LP”, Unternehmensforschung 14 (1970) 2–26.
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.
A.L. Soyster, “An objective function perturbation with economic interpretations”, Management Science 27 (1981) 231–237.
W. Stahl, Sensitivitätsanalyse mehrstufiger Standortprobleme (Verlag Anton Hain, Meisenheim am Glan, 1976).
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).
P. Streufert, “Optimale Jahresplanung nach Gewinn und Arbeitszeit in der fischverarbeitenden Industrie”, Wirtschaftswissenschaften 5 (1965) 802–808.
H. Suzuki, “A generalized knapsack problem with variable coefficients”, Mathematical Programming 15 (1978) 162–176.
K.B. Tlegenov, K.K. Kaltschaief and P.P. Zapletin, Mathematical programming methods (Nauka, Alma-Ata, 1975) (in Russian).
K.W. Webb, “The mathematical theory of sensitivity”, Paper presented at the 18th National Meeting of ORSA (Detroit, Michigan, 1960).
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.
H. Weinert, “Doppelt einparametrische lineare Optimierung. I. Unabhängige Parameter”, Mathematik Operations-Forschung und Statistik 2 (1971) 19–39.
G.R. Wels, “A sensitivity analysis of simulated river basin planning for capital budgeting decisions”, Computings and Operations Research 2 (1975) 49–54.
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.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/BFb0121210
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00912-9
Online ISBN: 978-3-642-00913-6
eBook Packages: Springer Book Archive