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Primal transportation and transshipment algorithms

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Summary

Capacitated transshipment problems constitute the most general class of single commodity network flow problems. During the last ten years the emphasis has shifted away from the primal-dual solution methods back to specializations of the primal simplex algorithm as the more efficient approach. In this survey numerous variations in data structures, selection rules and implementations of individual steps are extracted from the relevant literature. The exposition is centered around one basic algorithm which is explained in complete detail with particular attention to the intricacies of the basis exchange part. Specialized solution methods for transportation and assignment problems are taken into account.

Zusammenfassung

Die kapazitierten Transshipmentprobleme bilden die allgemeinste Klasse von Flußproblemen in Netzwerken, die sich mit der Verteilung einer einzigen Ware beschäftigen. Während der letzten zehn Jahre hat sich die Betonung von den primal-dualen Lösungsmethoden zurückverlagert zu den Spezialisierungen des primalen Simplexalgorithmus als der effizienteren Prozedur. Für diesen Übersichtsartikel werden die zahlreichen Variationen in den Datenstrukturen, in den Auswahlregeln und in den Verwirklichungen der einzelnen Schritte aus der entsprechenden Literatur ausgewählt. Die Ausführungen haben als Mittelpunkt einen Algorithmus, der in allen Einzelheiten erklärt wird, wobei den Schwierigkeiten des Basiswechsels besondere Aufmerksamkeit gewidmet wird. Spezielle Lösungsmethoden für Transport- und Zuordnungsprobleme werden mitberücksichtigt.

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References

  • Ahrens, J.H.: Counting Basic Feasible Solutions of Transportation Problems. Methods of Operations Research. Proc. III Symposium on Ops. Res., 1978.

  • -: A Conjecture of E.D. Bolker. To appear.

  • Ali, A.I., R. V. Helgason, J.L. Kennington andH.S. Lall: Primal Simplex Network Codes: State-of-the-Art Implementation Technology. Networks8, 1978, 315–339.

    Google Scholar 

  • Balas, E., andP.L. Hammer: On the Transportation Problem — Part 1. Cahiers du Centre d'Etudes de Recherche Opérationelle4 (2), 1962, 98–116.

    Google Scholar 

  • Balinski, M.L.: On Two Special Classes of Transportation Polytopes. Math. Prog. Study1, 1974, 43–58.

    Google Scholar 

  • Barr, R.S., F. Glover andD. Klingman: An Improved Version of the Out-of-Kilter Method and a Comparative Study of Computer Codes. Math. Prog.7 (1), 1974, 60–86.

    Article  Google Scholar 

  • —: The Alternating Basis Algorithm for Assignment Problems. Math. Prog.13 (1), 1977, 1–13.

    Article  Google Scholar 

  • —: The Generalized Alternating Path Algorithm for Transportation Problems. European Journal of Ops. Res.2, 1978, 137–144.

    Article  Google Scholar 

  • —: Enhancements of Spanning Tree Labelling Procedures for Network Optimization. INFOR17 (1), 1979, 16–34.

    Google Scholar 

  • Bazaraa, M.S., andJ.J. Jarvis: Linear Programming and Network Flows. New York 1977.

  • Bennington, G.E.: An Efficient Minimal Cost Flow Algorithm. Man. Sci.19 (9), 1973, 1042–1051.

    Google Scholar 

  • Bland, R.G.: New Finite Pivoting Rules for the Simplex Method. Math. of Ops. Res.2 (2), 1977, 103–107.

    Google Scholar 

  • Bolker, E.D.: Transportation Polytopes. J. Comb. Theory (B)13, 1972, 251–262.

    Article  Google Scholar 

  • Bradley, G.H., G.G. Brown andG.W. Graves: Design and Implementation of Large Scale Primal Transshipment Algorithms. Man. Sci.24 (1), 1977, 1–34.

    Google Scholar 

  • Brandt, A., andJ. Intrator: Fast Algorithms for Long Transportation Problems. Comp. & Ops. Res.5, 1978, 263–271.

    Google Scholar 

  • Burkard, R.E.: A General Hungarian Method for the Algebraic Transportation Problem. Discrete Math.22, 1978, 219–232.

    Article  Google Scholar 

  • Busacker, R.G., andP.J. Gowen: A Procedure for Determining a Family of Minimum-Cost Flow Patterns. ORO Technical Report15, Ops. Res. Office, John Hopkins University, 1961.

  • Charnes, A., andW.W. Cooper: Management Models and Industrial Applications of Linear Programming. Vol. I and II. New York 1961.

  • Charnes, A., F. Glover andD. Klingman: The Lower Bounded and Partial Upper Bounded Distribution Model. NRLQ18, 1971, 277–281.

    Google Scholar 

  • Charnes, A., D. Karney, D. Klingman, J. Stutz andF. Glover: Past, Present and Future of Large Scale Transshipment Computer Codes and Applications. Comp. and Ops. Res.2, 1975, 71–81.

    Article  Google Scholar 

  • Cunningham, W.H.: A Network Simplex Method. Math. Prog.11, 1976, 105–116.

    Article  Google Scholar 

  • Dantzig, G.B.: Application of the Simplex Method to a Transportation Problem. Activity Analysis of Production and Allocation. Ed. by T.C. Koopmans. New York 1951.

  • -: Linear Programming and Extensions. Princeton, New Jersey, 1963.

  • Dennis, J.B.: A High-Speed Computer Technique for the Transportation Problem. J. ACM5, 1958, 132–153.

    Article  Google Scholar 

  • Derigs, U., andU. Zimmermann: An Augmenting Path Method for Solving Linear Bottleneck Assignment Problems. Computing19 (4), 1978, 285–295.

    Google Scholar 

  • —: An Augmenting Path Method For Solving Linear Bottleneck Transportation Problems. Computing22, 1979, 1–15.

    Google Scholar 

  • Dömölky, B., andS. Frivaldszky: Solving the Transportation Problem by Means of Graph Representation (in Hungarian). Információ-Elektronika, 1966, 47–49.

  • Dwyer, P.S.: Transportation Problems with Some xij Negative and Transshipment Problems. NRLQ22, 1975, 751–776.

    Google Scholar 

  • Finke, G.: A Unified Approach to Reshipment, Overshipment and Post-Optimization Problems. Lecture Notes in Control and Information Sciences. Vol. 7, Optimization Techniques, Part 2. Berlin 1978, 201–208.

  • Finke, G., andJ.H. Ahrens: A Variant of the Primal Transportation Algorithm. INFOR16 (1), 1978, 35–46.

    Google Scholar 

  • Finke, G., andP.A. Smith: Primal Equivalents to the Threshold Algorithm. Methods of Operations Research31, 1978, 185–198.

    Google Scholar 

  • Flood, M.M.: A Transportation Algorithm and Code. NRLQ8, 1961, 257–276.

    Google Scholar 

  • Ford, L.R., andD.R. Fulkerson: A Primal-Dual Algorithm for the Capacitated Hitchcock Problem. NRLQ4 (1), 1957, 47–54.

    Google Scholar 

  • -: Flows in Networks. Princeton, New Jersey, 1962.

  • Fulkerson, D.R.: An Out-of-Kilter Method for Minimal-Cost Flow Problems. SIAM J. Appl. Math.9 (1), 1961, 18–27.

    Article  Google Scholar 

  • Gassner, B.J.: Cycling in the Transportation Problem. NRLQ11, 1964, 43–58.

    Google Scholar 

  • Gavish, B., P. Schweitzer andE. Shlifer: The Zero Pivot Phenomenon in Transportation and Assignment Problems and its Computational Implications. Math. Prog.12, 1977, 226–240.

    Article  Google Scholar 

  • Glicksman, S., L. Johnson andL. Eselson: Coding the Transportation Problem. NRLQ7 (2), 1960, 169–183.

    Google Scholar 

  • Glover, F., D. Karney andD. Klingman: The Augmented Predecessor Index Method for Locating Stepping-Stone Paths and Assigning Dual Prices in Distribution Problems. Transp. Sci.6, 1972, 171–180.

    Google Scholar 

  • —: Implementation and Computational Comparisons of Primal, Dual and Primal-Dual Computer Codes for Minimum Cost Network Flow Problems. Networks4 (3), 1974, 191–212.

    Google Scholar 

  • Glover, F., D. Karney, D. Klingman andA. Napier: A Computational Study on Start Procedures, Basis Change Criteria, and Solution Algorithms for Transportation Problems. Man. Sci.20 (5), 1974, 793–813.

    Google Scholar 

  • Glover, F., andD. Klingman: Locating Stepping-Stone Paths in Distribution Problems Via the Predecessor Index Method. Transp. Sci.4, 1970, 220–226.

    Google Scholar 

  • Glover, F., D. Klingman andJ. Stutz: Augmented Threaded Index Method for Network Optimization. INFOR12 (3), 1974, 293–298.

    Google Scholar 

  • Hadley, G.F.: Linear Programming. Reading 1962.

  • Harris, B.: A Code for the Transportation Problem of Linear Programming. J. ACM23, 1976, 155–157.

    Article  Google Scholar 

  • —: A New Algorithm for Tree Modification in the Primal Transportation Problem. Transp. Sci.12 (4), 1978, 271–276.

    Google Scholar 

  • Hausmann, D. (ed.): Lecture Notes in Economies and Mathematical System. Vol. 160, Integer Programming and Related Areas. A Classified Bibliography 1976–1978. Berlin-Heidelberg 1978.

  • Hitchcock, F.L.: The Distribution of a Product from Several Sources to Numerous Localities. J. Math. Phys.20, 1941, 224–230.

    Google Scholar 

  • Jacobsen, S.K.: On the Use of Tree-Indexing Methods in Transportation Algorithms. European J. Ops. Res.2, 1978, 54–65.

    Article  Google Scholar 

  • Johnson, E.: Networks and Basic Solutions. Ops. Res.14 (4), 1966, 619–623.

    Google Scholar 

  • Kantorovich, L. V.: On the Translocation of Masses. Compt. rendu. Acad. Sci., U.S.S.R.37, 1942. 199–201.

    Google Scholar 

  • Karney, D., andD. Klingman: Implementation and Computational Study on an In-Core, Out-of-Core Primal Network Code. Ops. Res.24 (6), 1976, 1056–1077.

    Google Scholar 

  • Kastning, C. (ed.): Lecture Notes in Economics and Mathematical Systems. Vol. 128, Integer Programming and Related Areas. A Classified Bibliography 1976. Berlin-Heidelberg 1976.

  • Klingman, D., A. Napier andJ. Stutz: Netgen: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems. Man. Sci.20 (5), 1974, 814–821.

    Google Scholar 

  • Knuth, D.E.: The Art of Computer Programming. Vol. 1. Reading 1968a.

  • —: Another Enumeration of Trees. Can. J. Math.20, 1968b, 1077–1086.

    Google Scholar 

  • Koopmans, T.C.: Optimum Utilization of the Transportation System. Proc. Int. Stat. Conf., Vol. 5. Washington, D.C., 1949.

  • Langley, R.W., J.L. Kennington andC.M. Shetty: Efficient Computational Devices for the Capacitated Transportation Problem. NRLQ21 (4), 1974, 637–647.

    Google Scholar 

  • Müller-Merbach, H.: Die Lösung des Transportproblems auf Rechenautomaten — Ein Algol-Program. Elektronische Datenverarbeitung8 (2), 1966, 49–56.

    Google Scholar 

  • Mulvey, J.M.: Pivot Strategies for Primal-Simplex Network Codes. J. ACM25 (2), 1978, 266–270.

    Article  Google Scholar 

  • Murty, K.G.: Linear and Combinatorial Programming. New York 1976.

  • Orden, A.: The Transshipment Problem. Man. Sci.2 (3), 1956, 276–285.

    Google Scholar 

  • Rao, M.R., andR.S. Garfinkel: The Bottleneck Transportation Problem. NRLQ18, 1971, 465–472.

    Google Scholar 

  • Ross, G. T., D. Klingman andA. Napier: A Computational Study of the Effects of Problem Dimensions on Solution Times of Transportation Problems. J. ACM22, 1975, 413–424.

    Article  Google Scholar 

  • Simonnard, M.: Linear Programming. Englewood Cliffs, N.J., 1966.

  • Srinivasan, V., andG.L. Thompson: Accelerated Algorithms for Labeling and Relabeling of Trees with Applications to Distribution Problems. J. ACM19 (4), 1972, 712–726.

    Article  Google Scholar 

  • —: Benefit-Cost Analysis of Coding Techniques for the Primal Transportation Algorithm. J. ACM20 (2), 1973, 194–213.

    Article  Google Scholar 

  • Szwarc, W.: The Transportation Paradox. NRLQ18, 1971, 185–202.

    Google Scholar 

  • Wagner, H.: The Lower Bounded and Partial Upper Bounded Distribution Problem. NRLQ20, 1973, 265–268.

    Google Scholar 

  • Zimmermann, U.: Duality principles and the algebraic transportation problem. Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen. Bd. 2. Ed. by L. Collatz, G. Meinardus and W. Wetterling. Basel 1979.

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Authors and Affiliations

  1. Mathematisches Seminar, Universität Kiel, Olshausenstr. 40/60, D-2300, Kiel

    J. H. Ahrens

  2. Dept. of Applied Mathematics, Nova Scotia Technical College, P.O.Box 1000, B3J 2X4, Halifax, N.S., Canada

    G. Finke

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  1. J. H. Ahrens
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  2. G. Finke
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An invited survey.

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Ahrens, J.H., Finke, G. Primal transportation and transshipment algorithms. Zeitschrift für Operations Research 24, 1–32 (1980). https://doi.org/10.1007/BF01920269

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