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**A composite algorithm for a concave-cost network flow problem.**
*(English)*
Zbl 0673.90034

Summary: We consider the problem of routing multiple commodities between various origin-destination pairs in a network, at minimum total cost. Economies of scale in arc flow costs are modeled using piecewise linear, concave total-cost functions for each arc. This model applies to a variety of medium- and long-term planning contexts including transportation planning, design of communication networks, plant location and capacity expansion planning, production planning, and water resource management. We formulate the general problem as a mixed-integer program and develop a composite algorithm to generate both good lower bounds and heuristic solutions. We also report on computational results for several randomly generated general networks (with up to 40 nodes, 359 arcs, and 60 commodities) and layered networks (with up to 60 nodes, 372 arcs, and 60 commodities). These tests demonstrate that even for relatively large problems, the composite algorithm is very effective in generating solutions with small gaps between the upper and lower bounds (1.7% on average for general networks, and 0.4% on average for layered networks); for 19 out of the 25 layered network problems, the method generated and verified the optimal solution.

### MSC:

90B10 | Deterministic network models in operations research |

90C11 | Mixed integer programming |

65K05 | Numerical mathematical programming methods |

90B05 | Inventory, storage, reservoirs |

90C90 | Applications of mathematical programming |

90B30 | Production models |

90C35 | Programming involving graphs or networks |

### Keywords:

routing multiple commodities; origin-destination pairs; minimum total cost; Economies of scale in arc flow costs; transportation planning; design of communication networks; plant location; capacity expansion; lower bounds; heuristic; computational results
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\textit{A. Balakrishnan} and \textit{S. C. Graves}, Networks 19, No. 2, 175--202 (1989; Zbl 0673.90034)

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