# Weighing Efficiency-Robustness in Supply Chain Disruption by Multi-Objective Firefly Algorithm

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Building Models

#### 2.1. Parameters Setting

- Index

- Decision-making variable

- Demand parameter

- Efficiency parameter

- Cost parameter

- Distance parameter

- Probability parameter

- Disrupted product quantity parameter

#### 2.2. Constraints

- Network structure constraint

- Material balance constraint

- Non-negativity constraints

#### 2.3. Objective Functions

## 3. Multi-objective Firefly Algorithm

#### 3.1. Firefly Algorithm

#### 3.2. Multi-Objective Optimization

#### 3.3. Multi-objective Firefly Algorithm

## 4. Performance Test of Multi-objective Firefly Algorithm

## 5. Exemplifications

#### 5.1. Elaboration

#### 5.2. Demand in Client Area

#### 5.3. Distance among Manufacturing Center, Distribution Center and Client Areas

#### 5.4. Probability of Supply Chain Disruptions

#### 5.5. Relevant Costs

## 6. Results and Discussions

#### 6.1. Disruption of Distribution Center

#### 6.2. Linkage Disruption between Manufacturing Center and Distribution Center

## 7. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

- Li, C.; Liu, S. Random Network Models and Sensitivity Algorithms for the Analysis of Ordering Time and Inventory State in Multi-stage Supply Chains. Comp. Ind. Eng.
**2014**, 70, 168–175. [Google Scholar] [CrossRef] - Mizgier, K.J.; Hora, M.; Wagner, S.M.; Jüttner, M.P. Managing Operational Disruptions Through Capital Adequacy and Process Improvement. Eur. J. Oper. Resear.
**2015**, 245, 320–332. [Google Scholar] [CrossRef] - Gorji, M.H.; Setak, M.; Karimi, H. Optimizing Inventory Decisions in a Two-Level Supply Chain with Order Quantity Constraints. Appl. Math. Model.
**2014**, 38, 814–827. [Google Scholar] [CrossRef] - Mizgier, K.J.; Wagner, S.M.; Jüttner, M.P. Disentangling Diversification in Supply Chain Networks. Int. J. Prod. Econ.
**2015**, 162, 115–124. [Google Scholar] [CrossRef] - Wang, W.; Plante, R.D.; Tang, J. Minimum Cost Allocation of Quality Improvement Targets under Supplier Process Disruption. Oper. Resear.
**2013**, 228, 388–396. [Google Scholar] [CrossRef] - Cao, E.; Wan, C.; Lai, M. Coordination of a Supply Chain with One Manufacturer and Multiple Competing Retailers under Simultaneous Demand and Cost Disruptions. Int. J. Prod. Econ.
**2013**, 141, 425–433. [Google Scholar] [CrossRef] - Hishamuddin, H.; Sarker, R.A.; Essama, D. A Recovery Mechanism for a Two Echelon Supply Chain System under Supply Disruption. Econ. Model.
**2014**, 38, 555–563. [Google Scholar] [CrossRef] - Shu, T.; Chen, S.; Wang, S.; Lai, K.K. GBOM-Oriented Management of Production Disruption Risk and Optimization of Supply Chain Construction. Expert Syst. Appl.
**2014**, 41, 59–68. [Google Scholar] [CrossRef] - Rosenberg, R.S. Simulation of Genetic Populations with Biochemical Properties. Diss. Abstr. Int.
**1967**, 28, 27–32. [Google Scholar] - Mizgier, K.J.; Pasia, J.M. Multiobjective Optimization of Credit Capital Allocation in Financial Institutions. Central Eur. J. Oper. Resear.
**2015**. [Google Scholar] [CrossRef] - Yang, X. Multiobjective firefly Algorithm for Continuous Optimization. Eng. Comp.
**2013**, 29, 175–184. [Google Scholar] [CrossRef] - Sayadi, M.K.; Ramezaniana, R.; Ghaffari-Nasaba, N. A Discrete Firefly Meta-heuristic with Local Search for Makespan Minimization in Permutation Flow Shop Scheduling Problems. Ind. Eng. Comp.
**2010**, 1, 1–10. [Google Scholar] [CrossRef] - Yang, X. Nature-inspired Metaheuristic Algorithms; Luniver Press: Frome, UK, 2008. [Google Scholar]
- Sayadi, M.K.; Hafezalkotob, A.; Naini, S.G.J. Firefly-Inspired Algorithm for Discrete Optimization Problems: An Application to Manufacturing Cell Formation. Manuf. Syst.
**2013**, 32, 78–84. [Google Scholar] [CrossRef] - Chandrasekaran, K.; Simon, S.P.; Padhy, N.P. Binary Real Coded Firefly Algorithm for Solving unit Commitment Problem. Inform. Sci.
**2013**, 249, 67–84. [Google Scholar] [CrossRef] - Talatahari, S.; Gandomi, A.H.; Yun, G.J. Optimum Design of Tower Structures Using Firefly Algorithm. Structural Des. Tall Spec. Build.
**2014**, 23, 350–361. [Google Scholar] [CrossRef] - Marichelvam, M.K.; Prabaharan, T.; Yang, X.S. A Discrete Firefly Algorithm for the Multi-Objective Hybrid Flowshop Scheduling Problems. IEEE Trans. Evol. Comp.
**2014**, 18, 301–305. [Google Scholar] [CrossRef] - Shukla, A.; Lalit, V.A.; Venkatasubramanian, V. Optimizing Efficiency-robustness Trade-offs in Supply Chain Design under Uncertainty due to Disruptions. Phys. Distrib. Logist. Manag.
**2011**, 41, 623–647. [Google Scholar] [CrossRef] - Meepetchdee, Y.; Shah, N. Logistical Network Design with Robustness and Complexity Considerations. Phys. Distrib. Logist. Manag.
**2007**, 37, 201–222. [Google Scholar] [CrossRef] - Peng, P.; Snyder, L.V.; Lim, A.; Liu, Z. Reliable Logistics Networks Design with Facility Disruptions. Transport. Resear. B Methodol.
**2011**, 45, 1190–1211. [Google Scholar] [CrossRef] - Tsiakis, P.; Shah, N.; Pantelides, C.C. Design of Multi-Echelon Supply Chain Networks under Demand Uncertainty. Ind. Eng. Chem. Res.
**2001**, 40, 3585–3604. [Google Scholar] [CrossRef] - Kalaitzidou, M.A.; Longinidis, P.; Tsiakis, P.; Georgiadis, M.C. Optimal Design of Multiechelon Supply Chain Networks with Generalized Production and Warehousing Nodes. Ind. Eng. Chem. Resear.
**2014**, 53, 13125–13138. [Google Scholar] [CrossRef] - Bradfielda, R.; Wright, G.; Burt, G.; Cairns, G.; Heijden, K.V.D. The Origins and Evolution of Scenario Techniques in Long Range Business Planning. Futures
**2005**, 37, 795–812. [Google Scholar] [CrossRef] - Greiner, R.; Puig, J.; Huchery, C.; Collier, N.; Garnett, S.T. Scenario Modelling to Support Industry Strategic Planning and Decision Making. Environ. Model. Softw.
**2014**, 55, 120–131. [Google Scholar] [CrossRef] - Kang, D.; Lansey, K.; ASCE, A.M. Multiperiod Planning of Water Supply Infrastructure Based on Scenario Analysis. Water Resour. Plan. Manag.
**2014**, 140, 40–54. [Google Scholar] [CrossRef] - Menezes, B.C.; Moro, L.F.L.; Lin, W.O.; Medronho, R.A.; Pessoa, F.L.P. Nonlinear Production Planning of Oil-Refinery Units for the Future Fuel Market in Brazil: Process Design Scenario-Based Model. Ind. Eng. Chem. Resear.
**2014**, 53, 4352–4365. [Google Scholar] [CrossRef] - Tsiakis, P.; Shah, N.; Pantelides, C.C. Design of Multi-Echelon Supply Chain Networks under Demand Uncertainty. Ind. Eng. Chem. Res.
**2001**, 40, 3585–3604. [Google Scholar] [CrossRef] - Coello, C.A.C. A Comprehensive Survey of Evolutionary-Based Multiobjective Optimization Techniques. Knowl. Inform.Syst.
**1999**, 1, 269–308. [Google Scholar] [CrossRef] - Coello, C.A.C. An Updated Survey of Evolutionary Multiobjective Optimization Techniques: State of the Art and Future Trends. In Proceedings of 1999 Congress on Evolutionary Computation, Washington, DC, USA, 6–9 July 1999.
- Deb, K. Evolutionary algorithms for multi-criterion optimization in engineering design. In Evolutionary Algorithms in Engineering and Computer Science; Wiley: New York, NY, USA, 1999; pp. 135–161. [Google Scholar]
- Geem, Z.W. Music-inspired Harmony Search Algorithm: Theory and Applications; Springer: Heidelberg, Berlin, 2009. [Google Scholar]
- Talbi, E.G. Metaheuristics: From Design to Implementation; Wiley: New York, NY, USA, 2009. [Google Scholar]
- Yang, X. Firefly Algorithms for Multimodal Optimization; Springer: Heidelberg, Berlin, 2009; pp. 169–178. [Google Scholar]
- Yang, X. Firefly Algorithm, Lévy Flights and Global Optimization; Springer: London, UK, 2010; pp. 209–218. [Google Scholar]
- Yang, X.; Deb, S. Eagle Strategy Using Lévy Walk and Firefly Algorithms for Stochastic Optimization. Stud. Comp. Intell.
**2010**, 284, 101–111. [Google Scholar] - Yang, X. Firefly Algorithm, Stochastic Test Functions and Design Optimisation. Int. J. Bio-Inspir. Comp.
**2012**, 2, 78–84. [Google Scholar] [CrossRef] - Yang, X. Nature-Inspired Mateheuristic Algorithms: Success and New Challenges. Comp. Eng. Inform. Tech.
**2012**, 1, 1–3. [Google Scholar] [CrossRef] - Yang, X.; Hosseini, S.S.S.; Gandomi, A.H. Firefly Algorithm for Solving Non-Convex Economic Dispatch Problems with Valve Loading Effect. Appl. Soft Comp.
**2012**, 12, 1180–1186. [Google Scholar] [CrossRef] - Pareto, V. Cours D'Economie Politique; F. Rouge: Lausanne, Swizerland, 1897. (In Franch) [Google Scholar]
- Burachik, R.S.; Kaya, C.Y.; Rizvi, M.M. A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets. J. Optim. Theor. Appl.
**2014**, 162, 428–446. [Google Scholar] [CrossRef] - Campigotto, P.; Passerini, A.; Battiti, R. Active Learning of Pareto Fronts. IEEE Trans. Neural Netw. Learn. Syst.
**2014**, 25, 506–519. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Zou, X. Runtime Analysis of a Multi-objective Evolutionary Algorithm for Obtaining Finite Approximations of Pareto Fronts. Inform.Sci.
**2014**, 262, 62–77. [Google Scholar] [CrossRef] - Khorram, E.; Khaledian, K.; Khaledyan, M. A Numerical Method for Constructing the Pareto Front of Multi-objective Optimization Problems. J. Comp. Appl. Math.
**2014**, 261, 158–171. [Google Scholar] [CrossRef] - Apostolopoulos, T.; Vlachos, A. Application of the Firefly Algorithm for Solving the Economic Emissions Load Dispatch Problem. Int. J. Comb.
**2011**. [Google Scholar] [CrossRef] - Zhang, L.; Zhou, C.; Ma, M.; Liu, X. Solutions of Multi-Objective Optimization Problems based on Particle Swarm Optimization. J. Comp. Resear. Dev.
**2004**, 41, 1286–1291. [Google Scholar] - Wang, E. Particle Swarm Optimization and Its Application in SAT and Multi-object Planning; Jilin University Press: Changchun, China, 2004. [Google Scholar]
- Deb, K.; Agrawal, S.; Pratap, A.; Meyarivan, T. A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II; Springer: Heidelberg, Berlin, 2000. [Google Scholar]
- Fisher, M.L. What is the right supply chain for your product? Harv. Bus. Rev.
**1997**, 75, 105–116. [Google Scholar] - National Bureau of Statistics of China. China Statistic Yearbook; 2013; China Statistics Press: Beijing, China, 2014. (In Chniese) [Google Scholar]
- Li, N.; Liu, X.; Xie, W.; Wu, J.; Zhang, P. The Return Period Analysis of Natural Disasters with Statistical Modeling of Bivariate Joint Probability Distribution. Risk Anal.
**2013**, 33, 134–145. [Google Scholar] [CrossRef] [PubMed]

**Figure 5.**True value of Test Function 2 (E. Wang,2004) [46].

**Figure 15.**Pareto front of multiple target in weighing efficiency and robustness in the distribution center disrupted.

**Figure 16.**The most effective supply chain network in linkages disrupted between manufacturing center and distribution center.

**Figure 17.**Supply chain network on the optimal objective functions in manufacturing and distribution centers disrupted.

**Figure 18.**Pareto front of multiple target in weighing efficiency and robustness in linkages disrupted.

Problems | Dimensions | Range | Objective functions |
---|---|---|---|

Zhang et al. (2004) [45] | 1 | [−5,7] | ${f}_{1}\left(x\right)={x}^{2}$, ${f}_{2}\left(x\right)={\left(x-2\right)}^{2}$ |

E. Wang (2004) [46] | 2 | [−5,10] | ${f}_{1}\left(x,y\right)={\left({x}^{2}+{y}^{2}\right)}^{\frac{1}{8}}$ |

${f}_{2}\left(x,y\right)={({\left(x-0.5\right)}^{2}+{\left(y-0.5\right)}^{2})}^{\frac{1}{4}}$ | |||

FON (Deb et al., 2000) [47] | 3 | [−4,4] | ${f}_{1}\left(x\right)=1-\text{exp}(-\sum _{i=1}^{3}{({x}_{i}-\frac{1}{\sqrt{3}})}^{2})$ |

${f}_{2}\left(x\right)=1-\text{exp}(-\sum _{i=1}^{3}{({x}_{i}+\frac{1}{\sqrt{3}})}^{2})$ | |||

KUR (Deb et al., 2000) [47] | 3 | [−5,5] | ${f}_{1}\left(x\right)=\sum _{i=1}^{n-1}(-10\text{exp}(-0.2\sqrt{{x}_{i}^{2}+{x}_{i+1}^{2}}))$ |

${f}_{2}\left(x\right)=\sum _{i=1}^{n}({\left|{x}_{i}\right|}^{0.8}+5\mathrm{sin}{x}_{i}^{3})$ | |||

ZDT3 (Deb et al., 2000) [47] | 30 | [0,1] | ${f}_{1}\left(x\right)={x}_{1}$ |

${f}_{2}\left(x\right)=g\left(x\right)\left[1-\sqrt{{x}_{1}/g\left(x\right)}-\frac{{x}_{1}}{g\left(x\right)}\mathrm{sin}\left(10\pi {x}_{1}\right)\right]$ | |||

$g\left(x\right)=1+9(\sum _{1=2}^{n}{x}_{i})/\left(n-1\right)$ |

Client Area | Provincial Administrative Region | Demand for Ordinary Computers ${N}_{1}$ (Sets) | Demand for New Computers ${N}_{1}$ (Sets) |
---|---|---|---|

1 | Beijing (BJ) | 2069 | 207 |

2 | Tianjin (TJ) | 1413 | 141 |

3 | Hebei (HEB) | 7288 | 729 |

4 | Shanxi (SAX) | 3611 | 361 |

5 | Neimenggu (NMG) | 2490 | 249 |

6 | Liaoning (LN) | 4389 | 439 |

7 | Jiling (JL) | 2750 | 275 |

8 | Heilongjiang (HLJ) | 3834 | 383 |

9 | Shanghai (SH) | 2380 | 238 |

10 | Jiangsu (JS) | 7920 | 792 |

11 | Zhejiang(ZJ) | 5477 | 548 |

12 | Anhui(AH) | 5988 | 599 |

13 | Fujian (FJ) | 3748 | 375 |

14 | Jiangxi (JX) | 4504 | 450 |

15 | Shandong (SD) | 9685 | 969 |

16 | Henan (HEN) | 9406 | 941 |

17 | Hubei (HUB) | 5779 | 578 |

18 | Hunan (HUN) | 6639 | 664 |

19 | Guangdong (GD) | 10,594 | 1059 |

20 | Guangxi (GX) | 4682 | 468 |

21 | Hainan (HAN) | 887 | 89 |

22 | Chongqing (CQ) | 2945 | 295 |

23 | Sichuan (SC) | 8076 | 808 |

24 | Guizhou (GZ) | 3484 | 348 |

25 | Yunnan (YN) | 4659 | 466 |

26 | Tibet (TB) | 308 | 31 |

27 | Shaanxi (SHX) | 3753 | 375 |

28 | Gansu (GS) | 2578 | 258 |

29 | Qinghai (QH) | 573 | 57 |

30 | Ningxia (NX) | 647 | 65 |

31 | Xinjiang (XJ) | 2233 | 223 |

32 | Hongkong (HK) | 716 | 72 |

33 | Macao (MAC) | 57 | 6 |

$b$ | BJ | LN | SH | HUB | GD | SC | SHX | |
---|---|---|---|---|---|---|---|---|

$a$ | ||||||||

BJ | 0 | 695 | 1261 | 1160 | 2117 | 1800 | 1088 | |

SH | 1257 | 1728 | 0 | 837 | 1475 | 1962 | 1375 |

$b$ | BJ | LN | SH | HUB | GD | SC | SHX | |
---|---|---|---|---|---|---|---|---|

$a$ | ||||||||

BJ | 0 | 695 | 1257 | 1162 | 2114 | 1809 | 1103 | |

TJ | 137 | 669 | 1086 | 1141 | 2094 | 1823 | 1145 | |

HEB | 315 | 1016 | 1138 | 969 | 1922 | 1501 | 794 | |

SAX | 521 | 1210 | 1356 | 949 | 1866 | 1317 | 610 | |

NMG | 485 | 1180 | 1730 | 1384 | 2301 | 1687 | 977 | |

LN | 695 | 0 | 1728 | 1819 | 2771 | 2473 | 1767 | |

JL | 1011 | 333 | 2034 | 2132 | 3085 | 2787 | 2080 | |

HLJ | 1251 | 573 | 2274 | 2372 | 3325 | 3027 | 2320 | |

SH | 1261 | 1720 | 0 | 843 | 1465 | 1965 | 1380 | |

JS | 1051 | 1526 | 303 | 539 | 1359 | 1662 | 1077 | |

ZJ | 1320 | 1785 | 178 | 721 | 1276 | 1856 | 1318 | |

AH | 1059 | 1622 | 465 | 390 | 1214 | 1513 | 928 | |

FJ | 1937 | 2402 | 773 | 910 | 935 | 2045 | 1656 | |

JX | 1431 | 2039 | 729 | 358 | 787 | 1493 | 1104 | |

SD | 446 | 1009 | 859 | 849 | 1817 | 1603 | 897 | |

HEN | 713 | 1369 | 943 | 516 | 1453 | 1193 | 485 | |

HUB | 1160 | 1816 | 837 | 0 | 981 | 1144 | 745 | |

HUN | 1487 | 2136 | 1086 | 354 | 651 | 1211 | 1007 | |

GD | 2117 | 2790 | 1475 | 984 | 0 | 1729 | 1636 | |

GX | 2340 | 2995 | 1902 | 1207 | 560 | 1215 | 1628 | |

HAN | 2759 | 3354 | 2043 | 1626 | 592 | 1700 | 2248 | |

CQ | 1764 | 2497 | 1685 | 871 | 1424 | 318 | 685 | |

SC | 1800 | 2530 | 1962 | 1146 | 1727 | 0 | 712 | |

GZ | 2148 | 2803 | 1843 | 1149 | 1096 | 663 | 1068 | |

YN | 2662 | 3315 | 2355 | 1661 | 1350 | 899 | 1569 | |

TB | 3636 | 4323 | 4196 | 3578 | 3606 | 2095 | 2833 | |

SHX | 1088 | 1821 | 1375 | 746 | 1636 | 712 | 0 | |

GS | 1485 | 2174 | 2010 | 1392 | 2282 | 857 | 646 | |

QH | 1686 | 2375 | 2247 | 1629 | 2519 | 1072 | 883 | |

NX | 1174 | 1843 | 1950 | 1461 | 2350 | 1429 | 736 | |

XJ | 3161 | 3850 | 3899 | 3270 | 4160 | 2792 | 2534 | |

HK | 2204 | 2936 | 1526 | 1110 | 180 | 1894 | 1799 | |

MAC | 2272 | 2916 | 1614 | 1139 | 139 | 1861 | 1792 |

Provincial Administrative Regions | BJ | LN | SH | HUB | GD | SC | SHX |
---|---|---|---|---|---|---|---|

Disruption probability | 0.041 | 0.049 | 0.001 | 0.031 | 0.018 | 0.096 | 0.021 |

**Table 6.**Boundary value of $c\left(O\right)$ and $c\left(E\right)$ in disruption of distribution center.

$c{\left(O\right)}_{min}$ | $c{\left(E\right)}_{max}$ | $c{\left(E\right)}_{min}$ | $c{\left(O\right)}_{max}$ |
---|---|---|---|

7,383,873,820 | 10,410,177 | 218,552 | 15,235,138,900 |

Exemplification | Total Cost in the Most Efficient Supply Chains (RMB) | Total Cost in the Most Robust Supply Chains (RMB) | Total Cost of the Optimal Objective Functions (RMB) |
---|---|---|---|

Disrupted distribution center | 7,394,283,997 | 15,235,357,452 | 7,386,193,817 |

Link Disruption Probability | BJ | LN | SH | HUB | GD | SC | SHX |
---|---|---|---|---|---|---|---|

Beijing | 0.041 | 0.045 | 0.021 | 0.036 | 0.0295 | 0.0685 | 0.031 |

Shanghai | 0.021 | 0.025 | 0.001 | 0.016 | 0.0095 | 0.0485 | 0.011 |

**Table 9.**Boundary values of $c\left(O\right)$ and $c\left(E\right)$ in Link disruption between manufacturing center and distribution center.

$c{\left(O\right)}_{min}$ | $c{\left(E\right)}_{max}$ | $c{\left(E\right)}_{min}$ | $c{\left(O\right)}_{max}$ |
---|---|---|---|

7,383,873,820 | 15,091,028 | 1,147,609 | 1,523,5138,900 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shu, T.; Gao, X.; Chen, S.; Wang, S.; Lai, K.K.; Gan, L.
Weighing Efficiency-Robustness in Supply Chain Disruption by Multi-Objective Firefly Algorithm. *Sustainability* **2016**, *8*, 250.
https://doi.org/10.3390/su8030250

**AMA Style**

Shu T, Gao X, Chen S, Wang S, Lai KK, Gan L.
Weighing Efficiency-Robustness in Supply Chain Disruption by Multi-Objective Firefly Algorithm. *Sustainability*. 2016; 8(3):250.
https://doi.org/10.3390/su8030250

**Chicago/Turabian Style**

Shu, Tong, Xiaoqin Gao, Shou Chen, Shouyang Wang, Kin Keung Lai, and Lu Gan.
2016. "Weighing Efficiency-Robustness in Supply Chain Disruption by Multi-Objective Firefly Algorithm" *Sustainability* 8, no. 3: 250.
https://doi.org/10.3390/su8030250