# Development of a Novel Fuzzy Hierarchical Location-Routing Optimization Model Considering Reliability

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- Designing a comprehensive model of FHLRP.
- Considering reliability in positioning-routing to increase customer satisfaction.
- Using the fuzzy programming method to control the demand and transportation cost parameters due to the lack of access to historical data.

## 3. Definition of the Problem

- Network levels include production centers, warehouses, and customers.
- All customer demands for different products must be met.
- The amount of demand and transportation costs are uncertain.
- The capacity of production centers and warehouses is known.
- It is a single period and single product model.
- Various types of vehicles are considered.
- The reliability percentage of each vehicle is known.

## 4. Solution Methods

#### 4.1. Designing the Initial Solution

- Part 1: Location

- Part 2: Vehicle routing between warehouse and customers

- Part 3: Vehicle routing between production centers and warehouses

- Part 4: Calculation of the objective function

#### 4.2. Parameter Tuning

#### 4.3. TOPSIS Method

- Step 1: Normalize the decision matrix

- Step 2: Calculate the weighted normalized decision matrix

- Step 3: Determine the positive ideal and negative ideal solutions

- Step 4: Distance from the positive and negative ideal solutions

- Step 5: Calculate the relative closeness degree of alternatives to the ideal solution

## 5. Analysis of the Results

#### 5.1. Validation of the Model

#### 5.2. Analyzing a Small Numerical Example with GA and PSO

#### 5.3. Analysis of Large Numerical Examples with GA and PSO

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Parameters | |

${\stackrel{~}{dem}}_{l}$ | Customer demand $l\in L$ |

${c}_{v}$ | Capacity of vehicle $v\in V$ |

${ca}_{d}$ | Capacity of warehouse $d\in D$ |

${ca}_{m}$ | Capacity of production center $m\in M$ |

${fm}_{m}$ | The fixed cost of locating a production center at node $m\in M$ |

${fd}_{d}$ | The fixed cost of locating a warehouse at node $d\in D$ |

${\stackrel{~}{c}}_{ijv}$ | Transportation cost from node $i$ to node $j$ by vehicle $v\in V$ |

${p}_{m}$ | Product production cost in production center $m\in M$ |

${fv}_{v}$ | The fixed cost of using the vehicle $v\in V$ |

$\mathsf{\lambda}$ | Vehicle reliability percentage $v\in V$ |

$m\lambda $ | Minimum network reliability |

Decision variables | |

${R}_{vdl}$ | value 1; if vehicle
$v\in V$ moves between arc $(i,j)\in {N}_{2}$. value 0; otherwise. |

${G}_{vl}$ | value 1; if vehicle
$v\in V$ visits customer
$l\in L$. value 0; otherwise. |

${B}_{dl}$ | value 1; if warehouse
$d\in D$ serves customer
$l\in L$. value 0; otherwise. |

${Y}_{d}$ | value 1; if the warehouse is located at node $d\in D$. value 0; otherwise. |

${X}_{vmd}$ | value 1; if vehicle
$v\in V$ moves between arc $(i,j)\in {N}_{1}$. value 0; otherwise. |

${Z}_{m}$ | value 1; if the production center is located at node $m\in M$. value 0; otherwise. |

${Q}_{vd}$ | value 1; if vehicle
$v\in V$ visits warehouse $d\in D$. value 0; otherwise. |

${E}_{lv}$ | Product quantity delivered to customer $l\in L$ by vehicle $v\in V$ |

${D}_{dv}$ | Product quantity delivered to warehouse $d\in D$ by vehicle $v\in V$ |

${H}_{m}$ | The amount of product produced in the production center $m\in M$ |

## References

- Moonsri, K.; Sethanan, K.; Worasan, K.; Nitisiri, K. A hybrid and self-adaptive differential evolution algorithm for the multi-depot vehicle routing problem in egg distribution. Appl. Sci.
**2022**, 12, 35. [Google Scholar] [CrossRef] - Hemmelmayr, V.C.; Cordeau, J.F.; Crainic, T.G. An adaptive large neighborhood search heuristic for two-echelon vehicle routing problems arising in city logistics. Comput. Oper. Res.
**2012**, 39, 3215228. [Google Scholar] [CrossRef] - Stodola, P. Hybrid ant colony optimization algorithm applied to the multi-depot vehicle routing problem. Nat. Comput.
**2020**, 19, 46475. [Google Scholar] [CrossRef] - Dumez, D.; Tilk, C.; Irnich, S.; Lehuédé, F.; Olkis, K.; Péton, O. A matheuristic for a 2-echelon vehicle routing problem with capacitated satellites and reverse flows. Eur. J. Oper. Res.
**2023**, 305, 64–84. [Google Scholar] [CrossRef] - Eitzen, H.; Lopez-Pires, F.; Baran, B.; Sandoya, F.; Chicaiza, J.L. A multi-objective two-echelon vehicle routing problem. An urban goods movement approach for smart city logistics. In Proceedings of the 2017 XLIII Latin American Computer Conference (CLEI), Córdoba, Argentina, 4–8 September 2017; pp. 1–10. [Google Scholar]
- Zhou, L.; Baldacci, R.; Vigo, D.; Wang, X. A multi-depot two-echelon vehicle routing problem with delivery options arising in the last mile distribution. Eur. J. Oper. Res.
**2018**, 265, 765–778. [Google Scholar] [CrossRef] - Zhang, S.; Lee, C.K.M.; Choy, K.L.; Ho, W.; Ip, W.H. Design and development of a hybrid artificial bee colony algorithm for the environmental vehicle routing problem. Transp. Res. Part D Transp. Environ.
**2014**, 31, 85–99. [Google Scholar] [CrossRef] - Ghahremani-Nahr, J.; Ghaderi, A.; Kian, R. A food bank network design examining food nutritional value and freshness: A multi objective robust fuzzy model. Expert Syst. Appl.
**2023**, 215, 119272. [Google Scholar] [CrossRef] - Alinaghian, M.; Shokouhi, N. Multi-depot multi-compartment vehicle routing problem, solved by a hybrid adaptive large neighborhood search. Omega
**2018**, 76, 85–99. [Google Scholar] [CrossRef] - Brandão, J. Iterated local search algorithm with ejection chains for the open vehicle routing problem with time windows. Comput. Ind. Eng.
**2018**, 120, 146–159. [Google Scholar] [CrossRef] - Babaee Tirkolaee, E.; Hadian, S.; Golpira, H. A novel multi-objective model for two-echelon green routing problem of perishable products with intermediate depots. J. Ind. Eng. Manag. Stud.
**2019**, 6, 196–213. [Google Scholar] - Breunig, U.; Baldacci, R.; Hartl, R.F.; Vidal, T. The electric two-echelon vehicle routing problem. Comput. Oper. Res.
**2019**, 103, 198–210. [Google Scholar] [CrossRef] - Yan, X.; Huang, H.; Hao, Z.; Wang, J. A graph-based fuzzy evolutionary algorithm for solving two-echelon vehicle routing problems. IEEE Trans. Evol. Comput.
**2019**, 24, 129–141. [Google Scholar] [CrossRef] - Ji, Y.; Du, J.; Han, X.; Wu, X.; Huang, R.; Wang, S.; Liu, Z. A mixed integer robust programming model for two-echelon inventory routing problem of perishable products. Phys. A Stat. Mech. Its Appl.
**2020**, 548, 124481. [Google Scholar] [CrossRef] - Dellaert, N.; Van Woensel, T.; Crainic, T.G.; Saridarq, F.D. A multi-commodity two-Echelon capacitated vehicle routing problem with time windows: Model formulations and solution approach. Comput. Oper. Res.
**2021**, 127, 105154. [Google Scholar] [CrossRef] - Du, J.; Wang, X.; Wu, X.; Zhou, F.; Zhou, L. Multi-objective optimization for two-echelon joint delivery location routing problem considering carbon emission under online shopping. Transp. Lett.
**2022**, 1–19, in press. [Google Scholar] [CrossRef] - Huang, H.; Yang, S.; Li, X.; Hao, Z. An Embedded Hamiltonian Graph-Guided Heuristic Algorithm for Two-Echelon Vehicle Routing Problem. IEEE Trans. Cybern.
**2021**, 52, 5695–5707. [Google Scholar] [CrossRef] [PubMed] - Zhou, H.; Qin, H.; Zhang, Z.; Li, J. Two-echelon vehicle routing problem with time windows and simultaneous pickup and delivery. Soft Comput.
**2022**, 26, 3345360. [Google Scholar] [CrossRef] - Goli, A.; Golmohammadi, A.M.; Verdegay, J.L. Two-echelon electric vehicle routing problem with a developed moth-flame meta-heuristic algorithm. Oper. Manag. Res.
**2022**, 15, 891–912. [Google Scholar] [CrossRef] - Nozari, H.; Tavakkoli-Moghaddam, R.; Gharemani-Nahr, J. A neutrosophic fuzzy programming method to solve a multi-depot vehicle routing model under uncertainty during the covid-19 pandemic. Int. J. Eng.
**2022**, 35, 36071. [Google Scholar] - Hajghani, M.; Forghani, M.A.; Heidari, A.; Khalilzadeh, M.; Kebriyaii, O. A two-echelon location routing problem considering sustainability and hybrid open and closed routes under uncertainty. Heliyon
**2023**, 9, e14258. [Google Scholar] [CrossRef] - Zhou, H.; Qin, H.; Cheng, C.; Rousseau, L.M. An exact algorithm for the two-echelon vehicle routing problem with drones. Transp. Res. Part B Methodol.
**2023**, 168, 124–150. [Google Scholar] [CrossRef] - Jia, S.; Deng, L.; Zhao, Q.; Chen, Y. An adaptive large neighborhood search heuristic for multi-commodity two-echelon vehicle routing problem with satellite synchronization. J. Ind. Manag. Optim.
**2023**, 19, 1187–1210. [Google Scholar] [CrossRef] - Du, J.; Wang, X.; Ma, B.; Zhou, F. Two-echelon joint delivery capacitated vehicle routing problem considering carbon emissions of online shopping. Int. J. Shipp. Transp. Logist.
**2023**, 16, 37298. [Google Scholar] [CrossRef] - Sluijk, N.; Florio, A.M.; Kinable, J.; Dellaert, N.; Van Woensel, T. A chance-constrained two-echelon vehicle routing problem with stochastic demands. Transp. Sci.
**2023**, 57, 252–272. [Google Scholar] [CrossRef] - Jiménez, M.; Arenas, M.; Bilbao, A.; Rodrı, M.V. Linear programming with fuzzy parameters: An interactive method resolution. Eur. J. Oper. Res.
**2007**, 177, 1599–1609. [Google Scholar] [CrossRef]

Solution Method | Factor | L1 | L2 | L3 | Optimum Value |
---|---|---|---|---|---|

GA | Max it | 100 | 150 | 200 | 200 |

N pop | 100 | 150 | 200 | 200 | |

Pc | 0.7 | 0.8 | 0.9 | 0.7 | |

Pm | 0.05 | 0.06 | 0.07 | 0.06 | |

PSO | Max it | 100 | 150 | 200 | 200 |

N particle | 100 | 150 | 200 | 200 | |

C1 | 1 | 1.5 | 2 | 1.5 | |

C2 | 1 | 1.5 | 2 | 2 | |

w | 0.5 | 0.6 | 0.8 | 0.5 |

Parameter | Value | Parameter | Value |
---|---|---|---|

${dem}_{l}^{o}$ | $~U\left(100,150\right)$ | ${c}_{vij}^{o}$ | $~U\left(30,40\right)$ |

${dem}_{l}^{m}$ | $~U\left(150,200\right)$ | ${c}_{vij}^{m}$ | $~U\left(40,50\right)$ |

${dem}_{l}^{p}$ | $~U\left(200,250\right)$ | ${c}_{vij}^{p}$ | $~U\left(50,60\right)$ |

${c}_{v}$ | $~U\left(400,500\right)$ | ${p}_{m}$ | $~U\left(2,5\right)$ |

${ca}_{d}$ | $~U\left(600,800\right)$ | ${fv}_{v}$ | $~U\left(1000,1500\right)$ |

${ca}_{m}$ | $~U\left(700,1000\right)$ | ${fd}_{d}$ | $~U\left(\mathrm{10,000},\mathrm{12,000}\right)$ |

${fm}_{m}$ | $~U\left(\mathrm{10,000},\mathrm{12,000}\right)$ |

$\mathit{\alpha}$ | Total Cost (USD) | Changes (%) |

0.1 | 39,527.61 | −13.74 |

0.2 | 41,688.30 | −9.02 |

0.3 | 43,685.58 | −4.67 |

0.4 | 44,124.15 | −3.71 |

0.5 | 45,823.83 | 0 |

0.6 | 46,937.18 | +2.43 |

0.7 | 48,536.33 | +5.92 |

0.8 | 52,348.66 | +14.26 |

0.9 | 54,366.47 | +18.64 |

${\mathit{c}}_{\mathit{v}}\text{}\left(\mathbf{\%}\right)$ | Total Cost (USD) | Vehicle Number |

−30 | 46,238.44 | 5 |

−20 | 46,238.44 | 5 |

−10 | 45,823.83 | 4 |

0 | 45,823.83 | 4 |

+10 | 45,823.83 | 4 |

+20 | 44,259.67 | 3 |

+30 | 43,211.28 | 3 |

${\mathit{c}\mathit{a}}_{\mathit{d}}-{\mathit{c}\mathit{a}}_{\mathit{m}}\text{}\left(\mathbf{\%}\right)$ | Total Cost (USD) | Total Centers |

−30 | 58,947.22 | 4 |

−20 | 58,647.66 | 4 |

−10 | 45,823.83 | 3 |

0 | 45,823.83 | 3 |

+20 | 45,823.83 | 3 |

+30 | 31,246.91 | 2 |

Solution Methods | Production Centers | Warehouse | Customer | Optimum Routing | Vehicle Number |
---|---|---|---|---|---|

Baron | 1 | 2-4 | - | ${M}_{1}\to {D}_{2}\to {D}_{4}\to {M}_{1}$ | 2 |

- | 2 | 1-5 | ${D}_{2}\to {L}_{5}\to {L}_{1}\to {D}_{2}$ | 3 | |

- | 2 | 2 | ${D}_{2}\to {L}_{2}\to {D}_{2}$ | 1 | |

- | 4 | 3-4-6 | ${D}_{4}\to {L}_{4}\to {L}_{6}\to {L}_{3}\to {D}_{4}$ | 4 | |

GA | 1 | 2-4 | - | ${M}_{1}\to {D}_{2}\to {D}_{4}\to {M}_{1}$ | 3 |

- | 2 | 1-2 | ${D}_{2}\to {L}_{2}\to {L}_{1}\to {D}_{2}$ | 2 | |

- | 2 | 3 | ${D}_{2}\to {L}_{3}\to {D}_{2}$ | 1 | |

- | 4 | 4-5-6 | ${D}_{4}\to {L}_{4}\to {L}_{6}\to {L}_{5}\to {D}_{4}$ | 4 | |

PSO | 2 | 1-4 | - | ${M}_{1}\to {D}_{1}\to {D}_{4}\to {M}_{1}$ | 1 |

- | 1 | 4-6 | ${D}_{1}\to {L}_{4}\to {L}_{6}\to {D}_{1}$ | 4 | |

- | 1 | 3 | ${D}_{1}\to {L}_{3}\to {D}_{1}$ | 2 | |

- | 4 | 1-2-3 | ${D}_{4}\to {L}_{2}\to {L}_{1}\to {L}_{3}\to {D}_{4}$ | 3 |

Sample Problem | Production Center | Warehouse | Customers | Vehicles |
---|---|---|---|---|

1 | 5 | 6 | 7 | 6 |

2 | 5 | 6 | 8 | 8 |

3 | 8 | 6 | 10 | 10 |

4 | 8 | 8 | 12 | 12 |

5 | 10 | 8 | 15 | 14 |

6 | 10 | 8 | 18 | 16 |

7 | 12 | 10 | 20 | 18 |

8 | 12 | 10 | 23 | 20 |

9 | 15 | 10 | 28 | 22 |

10 | 15 | 12 | 32 | 24 |

11 | 18 | 12 | 35 | 26 |

12 | 18 | 15 | 40 | 28 |

13 | 20 | 15 | 45 | 30 |

14 | 25 | 18 | 50 | 32 |

15 | 30 | 20 | 60 | 35 |

Sample Problem | Total Cost (USD) | CPU-Time (s) | ||||
---|---|---|---|---|---|---|

Baron | GA | PSO | GAMS | GA | PSO | |

1 | 69,745.28 | 70,391.12 | 70,370.21 | 128.49 | 21.72 | 18.66 |

2 | 88,672.24 | 89,589.11 | 89,730.15 | 331.97 | 23.63 | 20.37 |

3 | 95,748.66 | 96,822.96 | 96,837.32 | 597.29 | 27.77 | 23.94 |

4 | 112,343.28 | 114,075.63 | 113,857.73 | 972.66 | 32.91 | 28.37 |

5 | - | 127,890.05 | 125,566.02 | <1000 | 40.32 | 34.76 |

6 | - | 141,782.57 | 142,023.64 | <1000 | 52.40 | 45.17 |

7 | - | 161,659.52 | 162,441.07 | <1000 | 69.59 | 59.99 |

8 | - | 172,868.33 | 176,139.36 | <1000 | 89.09 | 76.80 |

9 | - | 192,507.95 | 194,335.84 | <1000 | 111.87 | 96.44 |

10 | - | 209,109.05 | 208,593.57 | <1000 | 144.69 | 124.73 |

11 | - | 221,201.84 | 222,350.19 | <1000 | 193.12 | 166.48 |

12 | - | 235,330.92 | 235,916.97 | <1000 | 249.01 | 214.66 |

13 | - | 257,636.53 | 260,863.93 | <1000 | 313.00 | 269.83 |

14 | - | 283,291.13 | 285,776.63 | <1000 | 400.05 | 344.87 |

15 | - | 298,740.32 | 299,020.68 | <1000 | 488.21 | 429.47 |

Solution Method | Mean of Objective Function | Mean of CPU-Time |
---|---|---|

GA | 178,193.1 | 150.49 |

PSO | 178,921.6 | 130.30 |

Weight | 0.5 | 0.5 |

Type | Negative | Negative |

Normalized | ||

GA | 0.706 | 0.756 |

PSO | 0.709 | 0.655 |

Solution Method | Mean of Objective Function | Mean of CPU-Time |
---|---|---|

GA | 0.353 | 0.354 |

PSO | 0.327 | 0.378 |

Distance to positive ideal | Distance to positive negative | |

GA | 0.051 | 0.001 |

PSO | 0.001 | 0.051 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ghahremani-Nahr, J.; Nozari, H.; Rahmaty, M.; Zeraati Foukolaei, P.; Sherejsharifi, A.
Development of a Novel Fuzzy Hierarchical Location-Routing Optimization Model Considering Reliability. *Logistics* **2023**, *7*, 64.
https://doi.org/10.3390/logistics7030064

**AMA Style**

Ghahremani-Nahr J, Nozari H, Rahmaty M, Zeraati Foukolaei P, Sherejsharifi A.
Development of a Novel Fuzzy Hierarchical Location-Routing Optimization Model Considering Reliability. *Logistics*. 2023; 7(3):64.
https://doi.org/10.3390/logistics7030064

**Chicago/Turabian Style**

Ghahremani-Nahr, Javid, Hamed Nozari, Maryam Rahmaty, Parvaneh Zeraati Foukolaei, and Azita Sherejsharifi.
2023. "Development of a Novel Fuzzy Hierarchical Location-Routing Optimization Model Considering Reliability" *Logistics* 7, no. 3: 64.
https://doi.org/10.3390/logistics7030064