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Exact algorithm for solving a special fixed-charge linear programming problem

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Abstract

A branch-and-bound algorithm (A) for solving a fixed-charge linear programming problem (P) involving identical fixed charges, one equality constraint, and explicit bounds on the variables is presented. Problem (P) can serve as a mathematical model for profit optimization in sawn timber production. Some theoretical considerations upon a fixed-charge problem (P′), arising from (P) by permitting the fixed charges to be different for each variable, are carried out. A basic algorithm (A0) is stated, and it is proved that Algorithm (A0) finds an optimal solution of Problem (P) [resp., (P′)] within a finite number of steps. Algorithm (A0), combined with bounds developed with regard to Problem (P), yields Algorithm (A), which operates on a subset of all vertices of the feasible region. Finally, computational results concerning the numerical solution of Problem (P) by Algorithm (A) are stated.

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Authors and Affiliations

  1. Institute of Mathematics, University of Klagenfurt, Klagenfurt, Austria

    J. Haberl (Lecturer)

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  1. J. Haberl
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Communicated by J. Abadie

A part of this work was carried out in connection with the project “Optimierung der Schnittholzproducktion auf Zerspaneranlagen,” which was done at the Institute of Mathematics of the University of Klagenfurt in cooperation with the firm J. Offner, Holzindustrie GmbH, Wolfsberg. This project was partially supported by “Forschungsförderungsfonds für die gewerbliche Wirtschaft.” The author would like to thank Professor H. Stettner, C. Nowak, and H. Woschitz for their support and G. Stoiser for his help in achieving the numerical results.

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Haberl, J. Exact algorithm for solving a special fixed-charge linear programming problem. J Optim Theory Appl 69, 489–529 (1991). https://doi.org/10.1007/BF00940686

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