## Discrete and Fractional Programming Techniques for Location ModelsAt first sight discrete and fractional programming techniques appear to be two com pletely unrelated fields in operations research. We will show how techniques in both fields can be applied separately and in a combined form to particular models in location analysis. Location analysis deals with the problem of deciding where to locate facilities, con sidering the clients to be served, in such a way that a certain criterion is optimized. The term "facilities" immediately suggests factories, warehouses, schools, etc. , while the term "clients" refers to depots, retail units, students, etc. Three basic classes can be identified in location analysis: continuous location, network location and dis crete location. The differences between these fields arise from the structure of the set of possible locations for the facilities. Hence, locating facilities in the plane or in another continuous space corresponds to a continuous location model while finding optimal facility locations on the edges or vertices of a network corresponds to a net work location model. Finally, if the possible set of locations is a finite set of points we have a discrete location model. Each of these fields has been actively studied, arousing intense discussion on the advantages and disadvantages of each of them. The usual requirement that every point in the plane or on the network must be a candidate location point, is one of the mostly used arguments "against" continuous and network location models. |

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### Contents

Introduction | xvii |

Discrete Location Models | 3 |

22 Multilevel Uncapacitated Facility Location Problems | 9 |

222 The 2echelon Uncapacitated Facility Location Problem | 11 |

23 Submodularity | 14 |

24 A General Uncapacitated Facility Depot Location Model | 18 |

242 Formulations | 20 |

243 Linear Relaxation | 22 |

321 1level Fractional Location Problems | 79 |

322 2level Fractional Location Problems | 89 |

33 Conclusions | 93 |

Generalized Fractional Programming | 95 |

41 A Primal Approach | 97 |

411 The Parametric Approach | 98 |

412 An Allocation Model | 106 |

413 A Nonstandard Class of Generalized Fractional Programs | 110 |

244 Lagrangian Relaxation | 25 |

245 Heuristics | 41 |

246 Branch and Bound | 43 |

247 Computational Results | 47 |

25 Conclusions | 56 |

Location Models and Fractional Programming | 59 |

31 Fractional Programming | 62 |

311 Continuous Fractional Programming | 63 |

312 Integer Fractional Programming | 76 |

32 Fractional Location Models | 78 |

### Other editions - View all

Discrete and Fractional Programming Techniques for Location Models A. I. Barros No preview available - 2014 |

Discrete and Fractional Programming Techniques for Location Models A. I. Barros No preview available - 2013 |

### Common terms and phrases

2-echelon uncapacitated facility 2-level uncapacitated facility Algorithm 4.3 applied associated parametric problem assumption bound algorithm branch and bound branching rule clients concave fractional programming consider constraints convergence rate convex generalized fractional convex set Crouzeix derived Dinkelbach-type algorithm Dinkelbach's algorithm discrete location dual It Sec dual problem duality gap facilities and depots facility location problem feasible set fixed costs follows formulation fractional location problems fractional programming problem function F Gao and Robinson given greedy heuristic Hence integer fractional programming iteration point Lagrangian dual Lagrangian relaxation Lemma linear fractional programming linear relaxation location models lower bound modified dual Moreover net present value node nonempty nonnegative number of iterations objective function Observe obtain open facilities optimal solution optimal value optimization problem parametric function primal profitability index Proposition Random examples Section solve standard dual subgradient method submodular superlinear test problems Theorem type algorithms uncapacitated facility location upper bounds variables