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**A parametric algorithm for convex cost network flow and related problems.**
*(English)*
Zbl 0532.90034

Summary: The context cost network flow problem is to determine the minimum cost flow in a network when cost of flow over each arc is given by a piecewise linear convex function. In this paper, we develop a parametric algorithm for the convex cost network flow problem. We define the concept of optimum basis structure for the convex cost network flow problem The optimum basis structure is then used to parametrize v, the flow to be transshipped from source to sink. The resulting algorithm successively augments the flow on the shortest paths from source to sink which are implicitly enumerated by the algorithm. The algorithm is shown to be polynomially bounded. Computational results are presented to demonstrate the efficiency of the algorithm in solving large size problems. We also show how this algorithm can be used to (i) obtain the project cost curve of a CPM network with convex time-cost tradeoff functions; (ii) determine maximum flow in a network with concave gain functions; (iii) determine optimum capacity expansion of a network having convex arc capacity expansion costs.

### MSC:

90B10 | Deterministic network models in operations research |

65K05 | Numerical mathematical programming methods |

68Q25 | Analysis of algorithms and problem complexity |

90C35 | Programming involving graphs or networks |

### Keywords:

computational complexity; polynomial boundedness; computational results; context cost network flow problem; minimum cost flow; parametric algorithm; optimum basis structure; shortest paths; large size problems; project cost curve; CPM network; optimum capacity expansion
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\textit{R. K. Ahuja} et al., Eur. J. Oper. Res. 16, 222--235 (1984; Zbl 0532.90034)

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