# A Hybrid Genetic Algorithm-Based Fuzzy Markovian Model for the Deterioration Modeling of Healthcare Facilities

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Fuzzy Deterioration Framework

_{A}(∙): X → [0,1] is the membership function of elements present in “A”, all belonging to universe X, and μ

_{A}(x) denotes the membership degree of element x within set A.

^{®}using the Fuzzy Logic Toolbox™ are illustrated in Figure 1.

#### 3.2. Markovian Deterioration Prediction Model

_{cal}representing the percentage of systems in every given final condition state. This allowed for the consequent calculation of a final condition rating for each of the systems being studied on a scale from 10–100 representing the observed condition rating of the systems at the final available inspection.

#### 3.3. Genetic Algorithm Optimization

_{Cal}or Cr

_{Act}that represent the overall “Calculated” or “Actual” condition ratings for oxygen gas systems (OGS) in hospital buildings; i is the counter for the condition states available evaluated on a scale from excellent with a value of 6 to critical with a value of 1; Csi is the percentage of the OGS in condition state i and it can be represented by the following parameters, either Csi

_{Cal}or Csi

_{Act}, according to the method of condition rating collection and retrieval; Csi

_{TR}is the respective value for state i in the transpose vector of condition ratings, ranging from excellent with a value of 100 to critical with a value of 10.

_{Cal}and Cr

_{Act}are the values obtained from the two previous equations and are representatives of a final evaluation of the overall OGS condition; T is the total number of condition rating observations filtered for model training; MAE is the mean absolute error between each set of two values as described previously, demonstrating the efficiency of the developed framework in relation to the actual datasets utilized for model training purposes.

#### 3.4. Model Performance Evaluation

_{i}is an indicator for the actual condition value observable for component i within the hospital inspection process; C

_{i}is an indicator for the predicted value as per the prediction methodology used, either the Weibull-based distribution function, or the proposed model for component i; N is the total number of observations available for condition values of the oxygen gas systems within different hospitals that were filtered and specified for model validation purposes.

## 4. Results

#### 4.1. Fuzzy-Based Model Sensitivity Analysis

#### 4.2. GA Optimization Results

#### 4.3. Overall Life Expectancy Results

#### 4.4. Model Validation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Mean absolute error (MAE) results of different values of age incrementation as compared to the base dataset with 0% incrementation.

**Figure 6.**MAE results of different values of condition incrementation as compared to the base dataset with 0% incrementation.

**Figure 10.**Fuzzy-based (initial) and optimum GA-derived (modified) transition probabilities for each zone in system’s life.

Excellent | Good | Acceptable | Marginal | Poor | Critical | Total | |
---|---|---|---|---|---|---|---|

No. of Systems | 15 | 12 | 10 | 8 | 3 | 2 | 50 |

% of Systems | 0.3 | 0.24 | 0.2 | 0.16 | 0.06 | 0.04 | 1 |

Fuzzy GA-Based Markov Model | Weibull Distribution | |
---|---|---|

Mean Absolute Error | 1.405 | 9.852 |

Root-Mean Squared Error | 1.853 | 11.429 |

Mean Absolute Percentage Error | 5.358% | 40.496% |

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**MDPI and ACS Style**

Ahmed, R.; Zayed, T.; Nasiri, F. A Hybrid Genetic Algorithm-Based Fuzzy Markovian Model for the Deterioration Modeling of Healthcare Facilities. *Algorithms* **2020**, *13*, 210.
https://doi.org/10.3390/a13090210

**AMA Style**

Ahmed R, Zayed T, Nasiri F. A Hybrid Genetic Algorithm-Based Fuzzy Markovian Model for the Deterioration Modeling of Healthcare Facilities. *Algorithms*. 2020; 13(9):210.
https://doi.org/10.3390/a13090210

**Chicago/Turabian Style**

Ahmed, Reem, Tarek Zayed, and Fuzhan Nasiri. 2020. "A Hybrid Genetic Algorithm-Based Fuzzy Markovian Model for the Deterioration Modeling of Healthcare Facilities" *Algorithms* 13, no. 9: 210.
https://doi.org/10.3390/a13090210