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Applications

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 196))

Abstract

In this chapter, we discuss briefly the most important applications of network flow problems.

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Bibliography

  • Adler, I., and Berenguer, S. (1981). Random linear programs (Techincal Report 81-4). Operations Research Center Report, U.C. Berkeley.

    Google Scholar 

  • Adler, I., and Megiddo, N. (1985). A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension. Journal of the ACM, 32, 871–895.

    Article  Google Scholar 

  • Adler, I., Karmarkar, N., Resende, M., and Veiga, G. (1989). An implementation of Karmarkar’s algorithm for linear programming. Mathematical Programming, 44, 297–335.

    Article  Google Scholar 

  • Ahuja, R., Magnanti, T., and Orlin, J. (1993). Network flows: Theory, algorithms, and applications. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Anstreicher, K. (1996). http://www-unix.mcs.anl.gov/otc/InteriorPoint/abstracts/Anstreicher-1.html Potential reduction algorithms (Technical Report). Department of Management Sciences, University of Iowa.

  • Barnes, E. (1986). A variation on Karmarkar’s algorithm for solving linear programming problems. Mathematical Programming, 36, 174–182.

    Article  Google Scholar 

  • Bayer, D., and Lagarias, J. (1989a). The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories. Transactions of the AMS, 314, 499–525.

    Google Scholar 

  • Bayer, D., and Lagarias, J. (1989b). The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories. Transactions of the AMS, 314, 527–581.

    Google Scholar 

  • Bazaraa, M., Jarvis, J., and Sherali, H. (1977). Linear programming and network flows (2nd ed.). New York: Wiley.

    Google Scholar 

  • Bellman, R. (1957). Dynamic programming. Princeton: Princeton University Press.

    Google Scholar 

  • Bendsøe, M., Ben-Tal, A., and Zowe, J. (1994). Optimization methods for truss geometry and topology design. Structural Optimization, 7, 141–159.

    Article  Google Scholar 

  • Bertsekas, D. (1991). Linear network optimization. Cambridge: MIT.

    Google Scholar 

  • Bertsekas, D. (1995). Nonlinear programming. Belmont: Athena Scientific.

    Google Scholar 

  • Bland, R. (1977). New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2, 103–107.

    Article  Google Scholar 

  • Bloomfield, P., and Steiger, W. (1983). Least absolute deviations: Theory, applications, and algorithms. Boston: Birkhäuser.

    Google Scholar 

  • Borgwardt, K. -H. (1982). The average number of pivot steps required by the simplex-method is polynomial. Zeitschrift für Operations Research, 26, 157–177.

    Google Scholar 

  • Borgwardt, K. -H. (1987a). Probabilistic analysis of the simplex method. In Operations research proceedings, 16th DGOR meeting, (pp. 564–576).

    Google Scholar 

  • Borgwardt, K. -H. (1987b). The simplex method—A probabilistic approach. Berlin/Heidelberg/New York: Springer.

    Book  Google Scholar 

  • Bradley, S., Hax, A., and Magnanti, T. (1977). Applied mathematical programming. Reading: Addison Wesley.

    Google Scholar 

  • Carathéodory, C. (1907). Über den variabilitätsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen. Mathematische Annalen, 64, 95–115.

    Article  Google Scholar 

  • Carpenter, T., Lustig, I., Mulvey, J., and Shanno, D. (1993). Higher order predictor-corrector interior point methods with application to quadratic objectives. SIAM Journal on Optimization, 3, 696–725.

    Article  Google Scholar 

  • Charnes, A. (1952). Optimality and degeneracy in linear programming. Econometrica, 20, 160–170.

    Article  Google Scholar 

  • Christofides, N. (1975). Graph theory: An algorithmic approach. New York: Academic.

    Google Scholar 

  • Chvátal, V. (1983). Linear programming. New York: Freeman.

    Google Scholar 

  • Cook, W. (2012). In pursuit of the traveling salesman. Princeton: Princeton University Press.

    Google Scholar 

  • Dantzig, G. (1951a). Application of the simplex method to a transportation problem. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 359–373). New York: Wiley.

    Google Scholar 

  • Dantzig, G. (1951b). A proof of the equivalence of the programming problem and the game problem. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 330–335). New York: Wiley.

    Google Scholar 

  • Dantzig, G. (1955). Upper bounds, secondary constraints, and block triangularity in linear programming. Econometrica, 23, 174–183.

    Article  Google Scholar 

  • Dantzig, G. (1963). Linear programming and extensions. Princeton: Princeton University Press.

    Book  Google Scholar 

  • Dantzig, G., and Orchard-Hayes, W. (1954). The product form for the inverse in the simplex method. Mathematical Tables and Other Aids to Computation, 8, 64–67.

    Article  Google Scholar 

  • Dantzig, G., Orden, A., and Wolfe, P. (1955). The generalized simplex method for minimizing a linear form under linear inequality constraints. Pacific Journal of Mathematics, 5, 183–195.

    Article  Google Scholar 

  • den Hertog, D. (1994). Interior point approach to linear, quadratic, and convex programming. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Dijkstra, E. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1, 269–271.

    Article  Google Scholar 

  • Dikin, I. (1967). Iterative solution of problems of linear and quadratic programming. Soviet Mathematics Doklady, 8, 674–675.

    Google Scholar 

  • Dikin, I. (1974). On the speed of an iterative process. Upravlyaemye Sistemi, 12, 54–60.

    Google Scholar 

  • Dodge, Y. (Ed.). (1987). Statistical data analysis based on the L 1 -norm and related methods. Amsterdam: North-Holland.

    Google Scholar 

  • Dorn, W., Gomory, R., and Greenberg, H. (1964). Automatic design of optimal structures. Journal de Mécanique, 3, 25–52.

    Google Scholar 

  • Dresher, M. (1961). Games of strategy: Theory and application. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  • Duff, I., Erisman, A., and Reid, J. (1986). Direct methods for sparse matrices. Oxford: Oxford University Press.

    Google Scholar 

  • Elias, P., Feinstein, A., and Shannon, C. (1956). Note on maximum flow through a network. IRE Transactions on Information Theory, 2, 117–119.

    Article  Google Scholar 

  • Farkas, J. (1902). Theorie der einfachen Ungleichungen. Journal für die reine und angewandte Mathematik, 124, 1–27.

    Google Scholar 

  • Fiacco, A., and McCormick, G. (1968). Nonlinear programming: Sequential unconstrained minimization techniques. McLean: Research Analysis Corporation. Republished in 1990 by SIAM, Philadelphia.

    Google Scholar 

  • Ford, L., and Fulkerson, D. (1956). Maximal flow through a network. Canadian Journal of Mathematics, 8, 399–404.

    Article  Google Scholar 

  • Ford, L., and Fulkerson, D. (1958). Constructing maximal dynamic flows from static flows. Operations Research, 6, 419–433.

    Article  Google Scholar 

  • Ford, L., and Fulkerson, D. (1962). Flows in networks. Princeton: Princeton University Press.

    Google Scholar 

  • Forrest, J., and Tomlin, J. (1972). Updating triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical Programming, 2, 263–278.

    Article  Google Scholar 

  • Fourer, R., and Mehrotra, S. (1991). Solving symmetric indefinite systems in an interior point method for linear programming. Mathematical Programming, 62, 15–40.

    Article  Google Scholar 

  • Fourer, R., Gay, D., and Kernighan, B. (1993). AMPL: A modeling language for mathematical programming. Belmont: Duxbury Press.

    Google Scholar 

  • Fulkerson, D., and Dantzig, G. (1955). Computation of maximum flow in networks. Naval Research Logistics Quarterly, 2, 277–283.

    Article  Google Scholar 

  • Gal, T. (Ed.). (1993). Degeneracy in optimization problems (Vol. 46/47 of annals of operations research). Basel: J.C. Baltzer.

    Google Scholar 

  • Gale, D., Kuhn, H., and Tucker, A. (1951). Linear programming and the theory of games. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 317–329). New York: Wiley.

    Google Scholar 

  • Garey, M., and Johnson, D. (1977). Computers and intractability. San Francisco: W.H. Freeman.

    Google Scholar 

  • Garfinkel, R., and Nemhauser, G. (1972). Integer programming. New York: Wiley.

    Google Scholar 

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

    Article  Google Scholar 

  • Gay, D. (1985). Electronic mail distribution of linear programming test problems. Mathematical Programming Society COAL Newslettter, 13, 10–12.

    Google Scholar 

  • Gill, P., Murray, W., and Wright, M. (1991). Numerical linear algebra and optimization (Vol. 1). Redwood City: Addison-Wesley.

    Google Scholar 

  • Gill, P., Murray, W., Ponceleón, D., and Saunders, M. (1992). Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13(1), 292–311.

    Article  Google Scholar 

  • Goldfarb, D., and Reid, J. (1977). A practicable steepest-edge simplex algorithm. Mathematical Programming, 12, 361–371.

    Article  Google Scholar 

  • Golub, G., and VanLoan, C. (1989). Matrix computations (2nd ed.). Baltimore: The Johns Hopkins University Press.

    Google Scholar 

  • Gonin, R., and Money, A. (1989). Nonlinear L p -norm estimation. New York/Basel: Marcel Dekker.

    Google Scholar 

  • Gordan, P. (1873). Über die Auflösung linearer Gleichungen mit reelen coefficienten. Mathematische Annalen, 6, 23–28.

    Article  Google Scholar 

  • Hall, L., and Vanderbei, R. (1993). Two-thirds is sharp for affine scaling. OR Letters, 13, 197–201.

    Google Scholar 

  • Harris, P. (1973). Pivot selection methods of the Devex LP code. Mathematical Programming, 5, 1–28.

    Article  Google Scholar 

  • Hemp, W. (1973). Optimum structures. Oxford: Clarendon Press.

    Google Scholar 

  • Hillier, F., and Lieberman, G. (1977). Introduction to mathematical programming (2nd ed.). New York: McGraw-Hill.

    Google Scholar 

  • Hitchcock, F. (1941). The distribution of a produce from several sources to numerous localities. Journal of Mathematical Physics, 20, 224–230.

    Article  Google Scholar 

  • Hoffman, A. (1953). Cycling in the simplex algorithm (Techincal Report 2974). National Bureau of Standards.

    Google Scholar 

  • Howard, R. (1960). Dynamic programming and Markov processes. New York: Wiley.

    Google Scholar 

  • Huard, P. (1967). Resolution of mathematical programming with nonlinear constraints by the method of centers. In J. Abadie (Ed.), Nonlinear programming (pp. 209–219). Amsterdam: North-Holland.

    Google Scholar 

  • Jensen, P., and Barnes, J. (1980). Network flow programming. New York: Wiley.

    Google Scholar 

  • John, F. (1948). Extremum problems with inequalities as subsidiary conditions. In K. Fredrichs, O. Neugebauer, and J. Stoker (Eds.), Studies and essays: Courant anniversary volume (pp. 187–204). New York: Wiley.

    Google Scholar 

  • Kantorovich, L. (1960). Mathematical methods in the organization and planning of production. Management Science, 6, 550–559. Original Russian version appeared in 1939.

    Google Scholar 

  • Karlin, S. (1959). Mathematical methods and theory in games, programming, and economics (Vols. 1 and 2). Reading: Addison-Wesley.

    Google Scholar 

  • Karmarkar, N. (1984). A new polynomial time algorithm for linear programming. Combinatorica, 4, 373–395.

    Article  Google Scholar 

  • Karush, W. (1939). Minima of functions of several variables with inequalities as side conditions (Technical report, M.S. thesis) Department of Mathematics, University of Chicago.

    Google Scholar 

  • Kennington, J., and Helgason, R. (1980). Algorithms for network programming. New York: Wiley.

    Google Scholar 

  • Khachian, L. (1979). A polynomial algorithm in linear programming. Doklady Academiia Nauk SSSR, 244, 191–194 (in Russian) (English translation: Soviet Mathematics Doklady, 20, 191–194)

    Google Scholar 

  • Klee, V., and Minty, G. (1972). How good is the simplex algorithm? In O. Shisha (Ed.), Inequalities–III (pp. 159–175). New York: Academic.

    Google Scholar 

  • Kojima, M., Mizuno, S., and Yoshise, A. (1989). A primal-dual interior point algorithm for linear programming. In N. Megiddo (Ed.), Progress in mathematical programming (pp. 29–47). New York: Springer.

    Chapter  Google Scholar 

  • Kotzig, A. (1956). Súvislost’ a Pravideliná Súvislots’ Konečných Grafov (Techincal Report). Bratislava: Vysoká Škola Ekonomická.

    Google Scholar 

  • Kuhn, H. (1950). A simplified two-person poker. Annals of Mathematics Studies, 24, 97–103.

    Google Scholar 

  • Kuhn, H. (1976). Nonlinear prgramming: A historical view. In R. Cottle and C. Lemke (Eds.), Nonlinear programming, SIAM-AMS proceedings (Vol. 9, pp. 1–26). Providence: American Mathetical Society.

    Google Scholar 

  • Kuhn, H., and Tucker, A. (1951). Nonlinear prgramming. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 481–492). Berkeley: University of California Press.

    Google Scholar 

  • Lawler, E. (1976). Combinatorial optimization: Networks and matroids. New York: Holt, Rinehart and Winston.

    Google Scholar 

  • Lemke, C. (1954). The dual method of solving the linear programming problem. Naval Research Logistics Quarterly, 1, 36–47.

    Article  Google Scholar 

  • Lemke, C. (1965). Bimatrix equilibrium points and mathematical programming. Management Science, 11, 681–689.

    Article  Google Scholar 

  • Luenberger, D. (1984). Introduction to linear and nonlinear programming. Reading: Addison-Wesley.

    Google Scholar 

  • Lustig, I. (1990). Feasibility issues in a primal-dual interior-point method for linear programming. Mathematical Programming, 49(2), 145–162.

    Article  Google Scholar 

  • Lustig, I., Marsten, R., and Shanno, D. (1994). Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6, 1–14.

    Article  Google Scholar 

  • Markowitz, H. (1957). The elimination form of the inverse and its application to linear programming. Management Science, 3, 255–269.

    Article  Google Scholar 

  • Markowitz, H. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.

    Google Scholar 

  • Marshall, K., and Suurballe, J. (1969). A note on cycling in the simplex method. Naval Research Logistics Quarterly, 16, 121–137.

    Article  Google Scholar 

  • Mascarenhas, W. (1997). The affine scaling algorithm fails for λ = 0. 999. SIAM Journal on Optimization, 7, 34–46.

    Google Scholar 

  • Megiddo, N. (1989). Pathways to the optimal set in linear programming. In N. Megiddo (Ed.), Progress in mathematical programming (pp. 131–158). New York: Springer.

    Chapter  Google Scholar 

  • Mehrotra, S. (1989). Higher order methods and their performance (Techincal Report TR 90-16R1) Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston. Revised July 1991.

    Google Scholar 

  • Mehrotra, S. (1992). On the implementation of a (primal-dual) interior point method. SIAM Journal on Optimization, 2, 575–601.

    Article  Google Scholar 

  • Michell, A. (1904). The limits of economy of material in frame structures. Philosophical Magazine Series, 8, 589–597.

    Article  Google Scholar 

  • Mizuno, S., Todd, M., and Ye, Y. (1993). On adaptive-step primal-dual interior-point algorithms for linear programming. Mathematics of Operations Research, 18, 964–981.

    Article  Google Scholar 

  • Monteiro, R., and Adler, I. (1989). Interior path following primal-dual algorithms: Part i: Linear programming. Mathematical Programming, 44, 27–41.

    Article  Google Scholar 

  • Nash, S., and Sofer, A. (1996). Linear and nonlinear programming. New York: McGraw-Hill.

    Google Scholar 

  • Nazareth, J. (1986). Homotopy techniques in linear programming. Algorithmica, 1, 529–535.

    Article  Google Scholar 

  • Nazareth, J. (1987). Computer solutions of linear programs. Oxford: Oxford University Press.

    Google Scholar 

  • Nazareth, J. (1996). The implementation of linear programming algorithms based on homotopies. Algorithmica, 15, 332–350.

    Article  Google Scholar 

  • Nemhauser, G., and Wolsey, L. (1988). Integer and combinatorial optimization. New York: Wiley.

    Book  Google Scholar 

  • Nesterov, Y., and Nemirovsky, A. (1993). Interior point polynomial methods in convex programming: Theory and algorithms. Philadelphia: SIAM.

    Google Scholar 

  • Recski, A. (1989). Matroid theory and its applications in electric network theory and in statics. Berlin/Heidelberg/New York: Springer.

    Book  Google Scholar 

  • Reid, J. (1982). A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. Mathematical Programming, 24, 55–69.

    Article  Google Scholar 

  • Rockafellar, R. (1970). Convex analysis. Princeton: Princeton University Press.

    Book  Google Scholar 

  • Rozvany, G. (1989). Structural design via optimality criteria. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Ruszczyński, A., and Vanderbei, R. (2003). http://www.princeton.edu/~rvdb/tex/lpport/lpport8.pdf Frontiers of stochastically nondominated portfolios. Econometrica, 71(4), 1287–1297.

  • Saigal, R. (1995). Linear programming. Boston: Kluwer.

    Book  Google Scholar 

  • Saunders, M. (1973). The complexity of LU updating in the simplex method. In R. Andersen and R. Brent (Eds.), The complexity of computational problem solving (pp. 214–230). St. Lucia: University of Queensland Press.

    Google Scholar 

  • Smale, S. (1983). On the average number of steps of the simplex method of linear programming. Mathematical Programming, 27, 241–262.

    Article  Google Scholar 

  • Stiemke, E. (1915). Über positive Lösungen homogener linearer Gleichungen. Mathematische Annalen, 76, 340–342.

    Article  Google Scholar 

  • Todd, M. (1986). Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming. Mathematical Programming, 35, 173–192.

    Article  Google Scholar 

  • Todd, M. (1995). http://www-unix.mcs.anl.gov/otc/InteriorPoint/abstracts/Todd.html Potential-reduction methods in mathematical programming (Techincal Report 1112). SORIE, Cornell University, Ithaca.

  • Tsuchiya, T., and Muramatsu, M. (1992). Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3), 525–551.

    Article  Google Scholar 

  • Tucker, A. (1956). Dual systems of homogeneous linear equations. Annals of Mathematics Studies, 38, 3–18.

    Google Scholar 

  • Turner, K. (1991). Computing projections for the Karmarkar algorithm. Linear Algebra and Its Applications, 152, 141–154.

    Article  Google Scholar 

  • Vanderbei, R. (1989). Affine scaling for linear programs with free variables. Mathematical Programming, 43, 31–44.

    Article  Google Scholar 

  • Vanderbei, R. (1994). Interior-point methods: Algorithms and formulations. ORSA Journal on Computing, 6, 32–34.

    Article  Google Scholar 

  • Vanderbei, R. (1995). Symmetric quasi-definite matrices. SIAM Journal on Optimization, 5(1), 100–113.

    Article  Google Scholar 

  • Vanderbei, R. (1999). LOQO: An interior point code for quadratic programming. Optimization Methods and Software, 12, 451–484.

    Article  Google Scholar 

  • Vanderbei, R., and Carpenter, T. (1993). Symmetric indefinite systems for interior-point methods. Mathematical Programming, 58, 1–32.

    Article  Google Scholar 

  • Vanderbei, R., and Shanno, D. (1999). http://www.sor.princeton.edu/~rvdb/ps/nonlin.pdfAn interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13, 231–252.

  • Vanderbei, R., Meketon, M., and Freedman, B. (1986). A modification of Karmarkar’s linear programming algorithm. Algorithmica, 1, 395–407.

    Article  Google Scholar 

  • Ville, J. (1938). Sur la théorie général des jeux ou intervient l’habileté des jouers. In E. Borel (Ed.), Traité du Calcul des probabilités et des ses applications. Paris: Gauthiers-Villars.

    Google Scholar 

  • von Neumann, J. (1928). Zur Theorie der Gesselschaftschpiele. Mathematische Annalen, 100, 295–320.

    Article  Google Scholar 

  • von Neumann, J., and Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.). Princeton: Princeton University Press.

    Google Scholar 

  • Wright, S. (1996). Primal-dual interior-point methods. Philadelphia: SIAM.

    Google Scholar 

  • Xu, X., Hung, P., and Ye, Y. (1993). A simplified homogeneous and self-dual linear programming algorithm and its implementation (Techincal Report). College of Business Administration, University of Iowa. To appear in Annals of Operations Research.

    Google Scholar 

  • Ye, Y., Todd, M., and Mizuno, S. (1994). An \(o(\sqrt{n}l)\)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19, 53–67.

    Article  Google Scholar 

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Vanderbei, R.J. (2014). Applications. In: Linear Programming. International Series in Operations Research & Management Science, vol 196. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-7630-6_15

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