# A Robust Optimization Approach to the Multiple Allocation p-Center Facility Location Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. A Robust Multiple Allocation p-Center Facility Location Problem

- (i)
- The capacity of all open facilities can satisfy all the demands. The uncapacitated facility location problem (UFLP) is one of the fundamental location problems described in Laporte et al. [6];
- (ii)
- The demand is infinitely divisible and can be fulfilled by one or multiple facilities as addressed in Calik [2].

#### 3.1. Deterministic Problem Formulation

**PCENTER**

#### 3.2. The Robust Problem: Box Uncertainty

#### 3.3. The Robust Problem: Ellipsoid Uncertainty

#### 3.4. The Robust Problem: Cardinality-Constrained Uncertainty

#### 3.5. Robustness of the Multiple Allocation Strategy

**Proposition**

**1.**

**Proof of**

**Proposition 1.**

**case 1**: ${\theta}_{{j}_{1}}\u2a7e0$. Due to Equations (29) and (45), ${x}_{i{j}_{1}}=1$, meaning only one facility ${j}_{1}$ is selected to serve i, only when $({\overline{c}}_{i{j}_{1}}+{\widehat{c}}_{i{j}_{1}}{\alpha}_{{j}_{1}}){d}_{i}+{\theta}_{{j}_{1}}=({\overline{c}}_{i{j}_{2}}+{\widehat{c}}_{i{j}_{2}}{\alpha}_{{j}_{2}}){d}_{i}-{\lambda}_{{j}_{2}}=-\beta $.**case 2**: ${\theta}_{{j}_{1}}=0$. Due to Equations (33), (34), (36) and (45), for all $j\in \overline{J}$, $({\overline{c}}_{ij}+{\widehat{c}}_{ij}{\alpha}_{j}){d}_{i}=\beta $, meaning multiple facilities are choosen to serve demand node i, only when all ${\tilde{c}}_{i{j}_{1}}={\overline{c}}_{i{j}_{1}}+{\widehat{c}}_{i{j}_{1}}{\alpha}_{{j}_{1}}$, ${j}_{1}\in \overline{J}$, are equal.**case 3**: ${\theta}_{{j}_{1}}=0$, ${\delta}_{{j}_{1}}=0$. Due to Equation (37), for all ${j}_{1}\in \overline{J}$, ${\widehat{c}}_{i{j}_{1}}{d}_{i}{x}_{i{j}_{1}}=\gamma $. Therefore, ${x}_{i{j}_{1}}$ is only related to ${\widehat{c}}_{i{j}_{1}}$, ${j}_{1}\in \overline{J}$, which is fixed.**case 4**: ${\theta}_{{j}_{1}}=0$, ${\delta}_{{j}_{1}}\u2a7e0$. Due to Equations (36) and (41), ${\alpha}_{{j}_{1}}=0$, so that all ${\tilde{c}}_{i{j}_{1}}={\overline{c}}_{i{j}_{1}}+{\widehat{c}}_{i{j}_{1}}$, ${j}_{1}\in \overline{J}$, are equal. This requires all ${\overline{c}}_{i{j}_{1}}+{\widehat{c}}_{i{j}_{1}}$ are identical, we assume this is a special case that cannot occur.

## 4. Numerical Study

#### 4.1. Effect of Robustness Parameters

#### 4.2. Effect of Overlapping Uncertain Cost

#### 4.3. Comparison of RO Approaches

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Jia, H.; Ordóñez, F.; Dessouky, M.M. A modeling framework for facility location of medical services for large-scale emergencies. IIE Trans.
**2007**, 39, 41–55. [Google Scholar] [CrossRef] [Green Version] - Calik, H. Exact Solution Methodologies for the p-Center Problem Under Single and Multiple Allocation Strategies. Ph.D. Thesis, Bilkent University, Ankara, Turkey, 2013. [Google Scholar]
- Nemhauser, G.L.; Wolsey, L.A. Integer and Combinatorial Optimization; Wiley: New York, NY, USA, 1988. [Google Scholar]
- Sarkar, B.; Majumder, A. A study on three different dimensional facility location problems. Econ. Model.
**2013**, 30, 879–887. [Google Scholar] [CrossRef] - Daskin, M.S. Network and Discrete Location: Models, Algorithms, and Applications; Wiley: New York, NY, USA, 2013. [Google Scholar]
- Laporte, G.; Nickel, S.; da Gama, F.S. Location Science; Springer: Berlin, Germany, 2015. [Google Scholar]
- Snyder, L.V. Facility location under uncertainty: A review. IIE Trans.
**2006**, 38, 547–564. [Google Scholar] [CrossRef] - Louveaux, F.V. Discrete stochastic location models. Ann. Oper. Res.
**1986**, 6, 23–34. [Google Scholar] [CrossRef] - Louveaux, F.V.; Peeters, D. A dual-based procedure for stochastic facility location. Oper. Res.
**1992**, 40, 564–573. [Google Scholar] [CrossRef] - Kouvelis, P.; Yu, G. Robust Discrete Optimization and Its Applications; Kluwer Academic Publishers: Boston, MA, USA, 1997. [Google Scholar]
- Soyster, A.L. Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res.
**1973**, 21, 1154–1157. [Google Scholar] [CrossRef] - Ben-Tal, A.; Nemirovski, A. Robust solutions of uncertain linear programs. Oper. Res. Lett.
**1999**, 25, 1–13. [Google Scholar] [CrossRef] [Green Version] - Ben-Tal, A.; Nemirovski, A. Robust solutions of linear programming problems contaminated with uncertain data. Math. Programm.
**2000**, 88, 411–424. [Google Scholar] [CrossRef] - Bertsimas, D.; Sim, M. The price of robustness. Oper. Res.
**2004**, 52, 35–53. [Google Scholar] [CrossRef] - Büsing, C.; D’Andreagiovanni, F. New results about multi-band uncertainty in robust optimization. In Proceedings of the 11th International Symposium on Experimental Algorithms, Bordeaux, France, 7–9 June 2012; pp. 63–74. [Google Scholar]
- Gabrel, V.; Murat, C.; Thiele, A. Portfolio optimization with pw-robustness. EURO J. Comput. Optim.
**2018**, 6, 267–299. [Google Scholar] [CrossRef] - Goh, J.; Sim, M. Distributionally robust optimization and its tractable approximations. Oper. Res.
**2010**, 58, 902–917. [Google Scholar] [CrossRef] - Zeng, B.; Zhao, L. Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper. Res. Lett.
**2013**, 41, 457–461. [Google Scholar] [CrossRef] - Hanasusanto, G.A.; Kuhn, D.; Wiesemann, W. K-adaptability in two-stage robust binary programming. Oper. Res.
**2015**, 63, 877–891. [Google Scholar] [CrossRef] - Ben-Tal, A.; Ghaoui, L.E.; Nemirovski, A. Robust Optimization; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar]
- Bertsimas, D.; Brown, D.B.; Caramanis, C. Theory and applications of robust optimization. SIAM Rev.
**2011**, 53, 464–501. [Google Scholar] [CrossRef] - Gabrel, V.; Murat, C.; Thiele, A. Recent advances in robust optimization: An overview. Eur. J. Oper. Res.
**2014**, 235, 471–483. [Google Scholar] [CrossRef] - Gorissen, B.L.; Yanıkoğlu, İ.; den Hertog, D. A practical guide to robust optimization. Omega
**2015**, 53, 124–137. [Google Scholar] [CrossRef] [Green Version] - Baron, O.; Milner, J.; Naseraldin, H. Facility location: A robust optimization approach. Prod. Oper. Manag.
**2011**, 20, 772–785. [Google Scholar] [CrossRef] - Nikoofal, M.E.; Sadjadi, S.J. A robust optimization model for p-median problem with uncertain edge lengths. Int. J. Adv. Manuf. Technol.
**2010**, 50, 391–397. [Google Scholar] [CrossRef] - Lu, C.C.; Sheu, J.B. Robust vertex p-center model for locating urgent relief distribution centers. Comput. Oper. Res.
**2013**, 40, 2128–2137. [Google Scholar] [CrossRef] - Lu, C.C. Robust weighted vertex p-center model considering uncertain data: An application to emergency management. Eur. J. Oper. Res.
**2013**, 230, 113–121. [Google Scholar] [CrossRef] - Gülpinar, N.; Pachamanova, D.; Çanakoglu, E. Robust strategies for facility location under uncertainty. Eur. J. Oper. Res.
**2013**, 225, 21–35. [Google Scholar] [CrossRef] - D’Andreagiovanni, F.; Mett, F.; Nardin, A.; Pulaj, J. Integrating LP-guided variable fixing with MIP heuristics in the robust design of hybrid wired-wireless FTTx access networks. Appl. Soft Comput.
**2017**, 61, 1074–1087. [Google Scholar] [CrossRef] - Atamturk, A.; Zhang, M. Two-stage robust network flow and design under demand uncertainty. Oper. Res.
**2007**, 55, 662–673. [Google Scholar] [CrossRef] - Hervet, C.; Faye, A.; Costa, M.C.; Chardy, M.; Francfort, S. Solving the two-stage robust FTTH network design problem under demand uncertainty. Electron. Notes Discrete Math.
**2013**, 41, 335–342. [Google Scholar] [CrossRef] - Gabrel, V.; Lacroix, M.; Murat, C.; Remli, N. Robust location transportation problems under uncertain demands. Discrete Appl. Math.
**2014**, 164, 100–111. [Google Scholar] [CrossRef] - An, Y.; Zeng, B.; Zhang, Y.; Zhao, L. Reliable p-median facility location problem: Two-stage robust models and algorithms. Transp. Res. Part B Methodol.
**2014**, 64, 54–72. [Google Scholar] [CrossRef] - Álvarez-Miranda, E.; Fernández, E.; Ljubić, I. The recoverable robust facility location problem. Transp. Res. Part B Methodol.
**2015**, 79, 93–120. [Google Scholar] [CrossRef] [Green Version] - Peng, C.; Li, J.; Wang, S. Two-stage robust facility location problem with multiplicative uncertainties and disruptions. In Proceedings of the 14th International Conference on Services Systems and Services Management (ICSSSM 2017), Dalian, China, 16–18 June 2017; pp. 1–6. [Google Scholar]
- Lu, M.; Ran, L.; Shen, Z.j.M. Reliable facility location design under uncertain correlated disruption. Manuf. Serv. Oper. Manag.
**2015**, 17, 445–455. [Google Scholar] [CrossRef] - Kariv, O.; Hakimi, S.L. An algorithmic approach to network location problems. I: The p-centers. SIAM J. Appl. Math.
**1979**, 37, 513–538. [Google Scholar] [CrossRef]

**Figure 2.**The choice of facilities to serve demand node i: Ellipsoidal and Cardinality-constrained uncertainty.

**Figure 4.**Solutions to the example with respect to $\mathsf{\Omega}$ and $\mathsf{\Gamma}$: (${x}_{11}=0$, ${x}_{21}=0$, ${x}_{22}=1$, ${x}_{23}=0$, ${x}_{31}=0$, ${x}_{32}=0$, and ${x}_{33}=1$ constantly).

**Figure 6.**Solutions to the example with respect to $\Delta $: (${x}_{11}=0$, ${x}_{21}=0$, ${x}_{22}=1$, ${x}_{23}=0$, ${x}_{31}=0$, ${x}_{32}=0$, and ${x}_{33}=1$ constantly).

Articles | Location Model | Uncertainty Model | |||||
---|---|---|---|---|---|---|---|

Box/Interval | Ellipsoid | CC | 2SRO | Stochastic | Scenario | ||

This paper | p-center | ✓ | ✓ | ✓ | |||

[24] | fixed-charge | ✓ | ✓ | ||||

[25] | p-median | ✓ | |||||

[26] | p-center | ✓ | |||||

[27] | p-center | ✓ | |||||

[28] | fixed-charge | ✓ | ✓ | ||||

[29] | connected | ✓ | |||||

[30] | network design | ✓ | ✓ | ✓ | |||

[31] | network design | ✓ | ✓ | ||||

[32] | location-transportation | ✓ | ✓ | ||||

[33] | p-median | ✓ | ✓ | ||||

[34] | fixed-charge | ✓ | ✓ | ||||

[35] | p-median | ✓ |

$\mathit{\epsilon}$ | Data Range | ${\overline{\mathit{L}}}_{\mathit{b}}$ | ${\overline{\mathit{L}}}_{\mathit{e}}$ | ${\overline{\mathit{L}}}_{\mathit{c}}$ | SA |
---|---|---|---|---|---|

1.5 | [41, 60] | 3577.23 | 3469.80 | 3356.86 | 27 |

[31, 70] | 2999.12 | 2938.17 | 2867.64 | 46 | |

[21, 80] | 2510.58 | 2477.91 | 2428.16 | 58 | |

[11, 90] | 1919.07 | 1902.59 | 1884.71 | 78 | |

[1, 100] | 1477.63 | 1472.94 | 1462.34 | 80 | |

2 | [41, 60] | 4430.49 | 4084.62 | 3758.87 | 13 |

[31, 70] | 3552.13 | 3394.61 | 3206.22 | 22 | |

[21, 80] | 2884.79 | 2802.47 | 2703.26 | 38 | |

[11, 90] | 2154.57 | 2128.01 | 2077.06 | 66 | |

[1, 100] | 1639.75 | 1619.31 | 1584.65 | 68 | |

4 | [41, 60] | 7067.90 | 5826.54 | 4792.86 | 4 |

[31, 70] | 5790.10 | 5075.30 | 4299.58 | 11 | |

[21, 80] | 4683.79 | 4242.60 | 3694.35 | 17 | |

[11, 90] | 3407.91 | 3201.37 | 2876.97 | 28 | |

[1, 100] | 2776.84 | 2611.62 | 2370.89 | 34 | |

8 | [41, 60] | 13,347.64 | 9768.67 | 7051.35 | 4 |

[31, 70] | 11,091.67 | 8770.87 | 6474.97 | 2 | |

[21, 80] | 9728.67 | 8227.88 | 6229.68 | 6 | |

[11, 90] | 7270.50 | 6275.85 | 5029.25 | 7 | |

[1, 100] | 5121.69 | 4573.74 | 3728.96 | 15 | |

16 | [41, 60] | 26,609.43 | 18,203.65 | 11,904.55 | 1 |

[31, 70] | 22,405.20 | 16,604.11 | 11,055.42 | 1 | |

[21, 80] | 17,063.13 | 13,441.57 | 9345.56 | 4 | |

[11, 90] | 13,022.03 | 11,133.35 | 8217.29 | 3 | |

[1, 100] | 9161.57 | 8023.31 | 6112.43 | 6 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Du, B.; Zhou, H.
A Robust Optimization Approach to the Multiple Allocation *p*-Center Facility Location Problem. *Symmetry* **2018**, *10*, 588.
https://doi.org/10.3390/sym10110588

**AMA Style**

Du B, Zhou H.
A Robust Optimization Approach to the Multiple Allocation *p*-Center Facility Location Problem. *Symmetry*. 2018; 10(11):588.
https://doi.org/10.3390/sym10110588

**Chicago/Turabian Style**

Du, Bo, and Hong Zhou.
2018. "A Robust Optimization Approach to the Multiple Allocation *p*-Center Facility Location Problem" *Symmetry* 10, no. 11: 588.
https://doi.org/10.3390/sym10110588