# Healthcare Facility Location-Allocation Optimization for China’s Developing Cities Utilizing a Multi-Objective Decision Support Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Healthcare Facility Location Problems

#### 2.2. Multiple Objective Decision Making (MODM) Methods

#### 2.3. Uncertainty Analysis

#### 2.4. Meta-Heuristic Algorithm

## 3. Problems Description and Framework

#### 3.1. Challenge Description

#### 3.2. Research Framework

## 4. Materials and Methods

#### 4.1. MODM Programming

_{1}: S and F

_{2}: L represent the social and environmental benefits, which are established from the perspective of customers. The objectives of G

_{1}: S and G

_{2}: C based on the suppliers’ angle pursue social and economic benefits respectively. The detailed description of each function is stated below.

#### 4.1.1. Upper-Level Programming: Objective Functions

#### 4.1.2. Upper-Level Programming: Constraints

#### 4.1.3. Lower-Level Programming: Objective Functions

#### 4.1.4. Lower-Level Programming: Constraints

#### 4.2. Particle Swarm Optimization (PSO) Algorithm for Healthcare Facility Location Problems (HCFLPs)

#### 4.2.1. Bi-Level Multi-Objective Particle Swarm Optimization (BLMOPSO)

#### 4.2.2. Overall Procedure of the Proposed Algorithm

- Set the parameters in the upper-level programming, including swarm size, particle position and velocity, iterations, inertia weights, acceleration coefficients and random variables $\tilde{\xi}$.
- Update the control parameters and compute the fitness values of two upper-level objectives.
- Estimate and replace the upper-level pareto solutions.
- Obtain the $pbes{t}_{s}$, $gbes{t}_{s}$, $lbes{t}_{s}$, $nbes{t}_{s}$ through the aforementioned approach.
- Set the similar type of parameters as step 1 on the lower level, and generate fuzzy variables $\tilde{\eta}$ based on confidence levels $\alpha $.
- Renewal the correlative parameters on the lower level.
- Compute the fitness values by incorporating solutions from upper level.
- Obtain the $pbes{t}_{s}{}^{\u2019}$, $gbes{t}_{s}{}^{\u2019}$, $lbes{t}_{s}{}^{\u2019}$, $nbes{t}_{s}{}^{\u2019}$ on the lower level.
- Check the lower level termination: if the algorithm acquires the best solution or met the maximum iteration, stop the lower level program. Otherwise, go back to Step 6.
- Check the BLMOPSO termination: if the algorithm gains the appropriate Pareto solutions or met the maximum iteration, then stop the BLMOPSO procedure. Otherwise, go back to Step 2.

#### 4.2.3. Solution Representation

#### 4.2.4. Parameter Setting

#### 4.2.5. Particle Evaluation

#### 4.2.6. Particle Updating

## 5. Case Study

#### 5.1. Study Area

#### 5.2. Date Acquisition and Processing

#### 5.3. Case Solution

#### 5.4. Analytic Results

#### 5.5. Comparative Analysis

#### 5.6. Stability Analysis

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Framework of healthcare facility location-allocation optimization for developing cities in China. BLMOPSO, bi-level multi-objective particle swarm optimization.

**Figure 4.**Flow chart of the bi-level multi-objective particle swarm optimization (BLMOPSO) algorithm.

Authors | Major Approach | Problem Type |
---|---|---|

Karatas et al. [6], etc. | Multi-objective optimization | Facility location |

Czerwiński et al. [16], etc. | Mixed-integer linear programming | Healthcare location-allocation |

Ye et al. [25], etc | GIS integration | |

Schuldt et al [30], etc. | Multilevel programming | Hospital network planning |

Schuldt et al. [27] | Conjoint analysis | |

Mestre et al. [8] | Uncertainty modelling | |

Syam and Côté [26], etc. | Integer programming |

Authors | Factors Type | Factors Name | Total Cite |
---|---|---|---|

Güneş et al. [5], etc. | social | travel distance/time | 11 |

Schuldt et al. [27] | service quality | 4 | |

Zhang et al. [36], etc. | expected waiting time | 2 | |

Vidyarthi and Jayaswal [3] | traffic congestion | 1 | |

Current et al. [7], etc. | economic | facility cost | 7 |

Jia et al. [4], etc. | capacity | 6 | |

Güneş and Nickel [9], etc. | travel cost | 3 | |

Ye and Kim [25], etc. | facility amount | 2 | |

Syam at al. [26] | operate cost | 1 | |

Brimberg et al. [18] | service costs | 1 | |

Jia et al. [4], etc. | environmental | geographic accessibility | 3 |

Zarrinpoor et al. [31] | disruption risk | 1 |

Authors | Area of Modification | Detail Description |
---|---|---|

Ratnaweera et al. [50] | Linear varying inertia weight | Control the individual velocity |

Naka et al. [51] | Nonlinear inertia weight | Ensure the velocity toward the lowest dynamic range |

Clerc and Kennedy [52] | Constriction Factor | Adjust the updating of the whole velocity |

Xing and Xiao [53] | Acceleration Coefficients | Generate stochastic influence on velocity of different groups |

Wang et al. [49] | Topologies | Exchange the cooperative information amongst each particle |

Li et al. [54], Niknam et al. [55], Mandloi and Bhatia [56] | Hybrid Technique | Integrate others intelligent algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA) and Ant Colony Optimization (ACO) |

Environmental Type | |||

${\alpha}_{1}$ | ${\alpha}_{2}$ | ||

0.34 | 0.66 | ||

Transportation Advantage Area | |||

${\varphi}_{1}$ | ${\varphi}_{2}$ | ${\varphi}_{3}$ | ${\varphi}_{4}$ |

0.13 | 0.18 | 0.37 | 0.32 |

Environmentally Tranquil Area | |||

${\phi}_{1}$ | ${\phi}_{2}$ | ${\phi}_{3}$ | ${\phi}_{4}$ |

0.11 | 0.31 | 0.36 | 0.22 |

No. | N | E | Patient Area | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |||

1 | 31°41′35.05″ | 103°51′28.77″ | 2 | 4 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

2 | 31°41′37.09″ | 103°51′23.34″ | 2 | 4 | 2 | 1 | 1 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

3 | 31°41′40.18″ | 103°51′33.00″ | 2 | 4 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

4 | 31°41′39.32″ | 103°51′37.63″ | 2 | 4 | 2 | 1 | 1 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

5 | 31°40′59.34″ | 103°51′26.53″ | 2 | 4 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

6 | 31°41′53.40″ | 103°51′44.01″ | 2 | 4 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

7 | 31°41′35.52″ | 103°51′35.94″ | 2 | 4 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 5 |

No. | Number of Sickbed | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |||||||

Original | - | 200 | 120 | 107 | 60 | 40 | ||||||

1 | 289 | +289 | 394 | +194 | 319 | +199 | 34 | −73 | 102 | +42 | 318 | +278 |

2 | 133 | +133 | 500 | +300 | 212 | +92 | 144 | +37 | 402 | +342 | 105 | +65 |

3 | 433 | +433 | 341 | +141 | 447 | +327 | 261 | +154 | 417 | +357 | 396 | +356 |

4 | 309 | +309 | 432 | +232 | 429 | +309 | 329 | +222 | 384 | +324 | 246 | +206 |

5 | 174 | +174 | 305 | +105 | 264 | +144 | 255 | +148 | 214 | +154 | 233 | +193 |

6 | 377 | +377 | 401 | +201 | 438 | +318 | 305 | +198 | 393 | +333 | 306 | +266 |

7 | 56 | +56 | 489 | +289 | 335 | +215 | 349 | +242 | 242 | +182 | 58 | +18 |

8 | 190 | +190 | 394 | +194 | 383 | +263 | 313 | +206 | 302 | +242 | 229 | +189 |

9 | 32 | +32 | 365 | +165 | 370 | +250 | 396 | +289 | 69 | +9 | 66 | +26 |

Iteration | The Average Distance | The Distribution | The Extent | The Set Convergence | The Solution Amount |
---|---|---|---|---|---|

1 | 0.0568 | 0.3333 | 3.8649 | 0.3333 | 3 |

2 | 0.0547 | 0.6000 | 5.6127 | 0.6000 | 5 |

3 | 0.0547 | 0.6000 | 5.6127 | 1.0000 | 5 |

4 | 0.0409 | 0.5000 | 5.8634 | 0.7500 | 4 |

5 | 0.0409 | 0.5000 | 5.8634 | 1.0000 | 4 |

6 | 0.0762 | 0.6000 | 5.8634 | 0.8000 | 5 |

7 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

8 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

10 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

12 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

15 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

18 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

20 | 0.0762 | 0.6000 | 5.8634 | 1.0000 | 5 |

22 | 0.0922 | 0.6667 | 5.8634 | 0.6667 | 6 |

23 | 0.0922 | 0.6667 | 5.8634 | 1.0000 | 6 |

24 | 0.0922 | 0.6667 | 5.8634 | 1.0000 | 6 |

25 | 0.0922 | 0.6667 | 5.8634 | 1.0000 | 6 |

26 | 0.0425 | 0.7143 | 5.8634 | 0.8571 | 7 |

27 | 0.0425 | 0.7143 | 5.8634 | 1.0000 | 7 |

28 | 0.0425 | 0.7143 | 5.8634 | 1.0000 | 7 |

29 | 0.0425 | 0.7143 | 5.8634 | 1.0000 | 7 |

30 | 0.0425 | 0.7143 | 5.8634 | 1.0000 | 7 |

Algorithm Type | Iteration | The average Distance | The Distribution | The Extent | The set Convergence | The Solution Amount |
---|---|---|---|---|---|---|

BLMOPSO | 30 | 0.0425 | 0.7143 | 5.8634 | 1.0000 | 7 |

Basic PSO | 30 | 0.1712 | 0.5000 | 5.3036 | 1.0000 | 4 |

Solution Amount | Occurrence Amount | Percentage |
---|---|---|

7 | 12 | 33.33% |

8 | 4 | 16.67% |

6 | 3 | 16.67% |

5 | 3 | 13.33% |

10 | 2 | 6.67% |

others | 4 | 13.33% |

total | 30 | 100.00% |

No. | Solution Amount | The Extent | Error Rate | |
---|---|---|---|---|

Original | 7 | 5.8634 | - | |

1 | 7 | 5.7702 | −0.0932 | −1.59% |

2 | 7 | 5.9289 | 0.0655 | 1.12% |

3 | 7 | 5.9289 | 0.0655 | 1.12% |

4 | 7 | 5.7494 | −0.1140 | −1.94% |

5 | 7 | 5.4991 | −0.3643 | −6.21% |

6 | 7 | 5.9435 | 0.0801 | 1.37% |

7 | 7 | 6.1374 | 0.2740 | 4.67% |

8 | 7 | 5.5974 | −0.2660 | −4.54% |

9 | 7 | −0.1889 | −0.1889 | −3.22% |

10 | 7 | 5.6943 | −0.1691 | −2.88% |

11 | 7 | 6.2093 | 0.3459 | 5.90% |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Shi, H.; Gan, L.
Healthcare Facility Location-Allocation Optimization for China’s Developing Cities Utilizing a Multi-Objective Decision Support Approach. *Sustainability* **2018**, *10*, 4580.
https://doi.org/10.3390/su10124580

**AMA Style**

Wang L, Shi H, Gan L.
Healthcare Facility Location-Allocation Optimization for China’s Developing Cities Utilizing a Multi-Objective Decision Support Approach. *Sustainability*. 2018; 10(12):4580.
https://doi.org/10.3390/su10124580

**Chicago/Turabian Style**

Wang, Li, Huan Shi, and Lu Gan.
2018. "Healthcare Facility Location-Allocation Optimization for China’s Developing Cities Utilizing a Multi-Objective Decision Support Approach" *Sustainability* 10, no. 12: 4580.
https://doi.org/10.3390/su10124580