# How to Model Uncertain Service Life and Durability of Components in Life Cycle Cost Analysis Applications? The Stochastic Approach to the Factor Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Background

- ISO 15686-1:2000, Building and constructed assets—Service Life planning—Part 1: General principles [32];
- ISO 15686-2:2001, Building and constructed assets—Service Life planning—Part 2: Service Life prediction procedures [33];
- ISO 15686-7:2006, Building and constructed assets—Service Life planning—Part 7: Performance Evaluation for Feed-back of Service Life data from practice [34];
- ISO 15686-8:2008, Building and constructed assets—Service Life planning Part 8: Reference Service Life and Service Life estimation [35];
- UNI 11156-3: 2006, Valutazione della durabilità dei componenti edilizi. Metodo per la valutazione della durata (vita utile) [36].

- Expert opinions and experiences, and knowledge of component behavior (in similar conditions);
- Scientific research, technological information by producers, laboratory tests, and statistical analysis, etc.

- Deterministic approaches, for example, the “simple” FM, mathematically simple but not very affordable;
- Probabilistic approaches, very affordable and detailed, but very expensive in terms of input data required and calculation;
- Engineering approaches, for example, the engineering of the FM, able to maintain the simplicity of the method but reinforcing its affordability.

- Building component description and context variability through the factors of the FM;
- Relevant factors individuation, through scientific literature or experimental tests.

## 3. Methodological Background

_{GEnEc}= C

_{I}+ C

_{EE}+ C

_{EC}+ ∑ (C

_{m}+ C

_{r})/(1 + r)

^{t}+ (C

_{dm}+ C

_{dp}− V

_{r})/(1 + r)

^{N}

_{GEnEc}is the life cycle cost including environmental and economic indicators; C

_{I}the investment costs; C

_{EE}the costs related to embodied energy; C

_{EC}the costs related to the embodied carbon; C

_{m}the maintenance cost, C

_{r}the replacement cost; C

_{dm}the dismantling cost and C

_{dp}the disposal cost; V

_{r}the residual value; t the year in which the cost occurred and N the number of years of the entire period considered for the analysis; r the discount rate. A deterministic sensitivity analysis concludes the work.

_{r}is the residual value; t is the year in which the cost occurred; N is the number of years of the entire period considered for the analysis (representing the time required to renew and retrofit the building according to the regulations on energy consumptions and to the evolving functional requirements); and $\widehat{\mathrm{r}}$ is the stochastic discount rate. Notice that the residual value V

_{r}is due to the difference between the entire period of the analysis and the specific service life of components object of the study; simplifying, the residual value is considered as a deterministic input. An empirical modality is adopted by defining three different lifespan scenarios, representing the possible temporal variability of the components. As a consequence, three different residual values are obtained. Notice that the determination of possible temporal variability of components is the most crucial aspect of the analysis, and its rational quantification will be the focus of the present work.

- Determination of the estimated service life through the stochastic approach to the Factor Method
- −
- Step 1: Reference service life assumption. In this first step the RSL of the component is defined, through estimates based on empirical laboratory tests, generally developed by the manufacturers;
- −
- Step 2: Individuation of the factors for FM application, on the basis of the literature on the topic and on a hypothesis based on data deducted by the literature; preliminary hypothesis for factor values determination (according to Equation (3)) and individuation of alternative scenarios;
- −
- Step 3: individuation of the distribution type and PDFs calculation through Monte Carlo Method (MCM). This step is developed through iteration and sampling, according to the numerical methods;
- −
- Step 4: Stochastic estimated service life calculation, through the MCM (according to Equation (4)), on the basis of the stochastic factors defined above;
- −
- Step 5: Best-fit distribution calculation to obtain a PDF related to the E$\widehat{\mathrm{S}}$L. The preferable distribution function (best fit) is deducted by a ranking of distributions based on the results of statistic measures calculation (Chi-squared, Kolmogorov-Smirnov, Anderson-Darling, Root-Mean Squared Error), used for testing how the distribution fits the input data and to test the confidence on the distribution functions representativeness;

- Introduction of stochastic service lives input data in life cycle cost analysis
- −
- Step 6: Recalculation of the results of LCCA using the PDF of the E$\widehat{\mathrm{S}}$L as input data for the resolution of the Equation (5);

- Calculation of LCCA results and final considerations
- −
- Step 7: Defining the best-fitting distribution function for the output values (following the Step 5 procedure) calculated in Step 6 and results interpretation.

## 4. Case Study

- Initial investment cost (€) for timber frame 363,027.50, and aluminum frame 272,852.50;
- Annual running and replacement costs (€) for timber frame 1,593,370.21, and aluminum frame 853,198.40;
- Disposal costs (€) for timber frame 2,405.67, and aluminum frame 987.58;
- Embodied energy (MJ) for timber frame 2,333,539.78, and for aluminum frame 5,881,721.38;
- Embodied carbon (kg CO
_{2}eq—100 years) for timber frame 665,485, and for aluminum frame 1,100,860.

## 5. Application and Results

#### 5.1. Determination of the Estimated Service Life through the Stochastic Approach to the Factor Method

- Factors related to inherent quality characteristics (quality of components, design level, and work execution level): During the design and installation phase no relevant differences are detectable as respect the manufacturer’s indications. In other terms, no significant deviation is expected as respect the RSL (Factor values 1);
- Factors related to environment (indoor environment and outdoor environment): A reduction in respect to the RSL is expected (Factor values less than 1), due to more severe environmental conditions in respect to the RSL ones (outdoor conditions worse than indoor conditions). Notice that the building is located in Turin’s suburban area, devoted to commercial-tertiary use, and for these reasons it is reasonable to foresee a reduction in RSL. Furthermore, a reduction in relation to the Factor E value is foreseen more significant for timber frames than aluminum frames, assuming that external pollution has more effect on timber than on the aluminum;
- Factors related to operating conditions (in use conditions and maintenance level): An increase in respect to the RSL is expected (Factor values higher than 1), due to the foreseen presence of better operating conditions in respect to the RSL ones. In fact, a high level of quality of building and functions to be inserted are prefigured by the project.

- Low-impact factor scenarios: a minor deviation from RSL is hypothesized, in reduction and increase (minor impact of environmental factors produces a reduction, and operative an increment);
- High-impact factor scenarios: a greater deviation from RSL is hypothesized, in reduction and increase (higher impact of environmental factors—reduction and operative—increment).

- The factors can be expressed through a specific PDF. According to the literature, the lognormal distribution is the most frequently adopted and, in our case, the lognormal distribution is assumed for representing the factor’s values;
- The lognormal distribution reflects the greatest probability that the component could have a service life lower than the RSL. Distribution curves are of the lognormal type, are skewed to the left, with maximum probability given by peaks of the curves. The right side of the distribution reveals also a low probability of service life values higher than RSL.

#### 5.2. Introduction of Stochastic Service Lives Input Data in Life Cycle Cost Analysis

_{r}. The results of the calculation (by MCM simulation) are reported in following sub-section.

#### 5.3. Calculation of LCCA Results and Final Considerations

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Case study: (

**a**) cross section of the building; (

**b**) timber window frame; (

**c**) aluminum window frame.

**Figure 2.**Estimated service life, probability density function, low-impact factor scenarios timber/aluminum frames. Monte Carlo simulation output.

**Figure 3.**Estimated service life, probability density function, high-impact factor scenarios timber/aluminum frames. Monte Carlo simulation output.

**Figure 4.**Stochastic estimated service life, probability density function curve fitting, low-impact factor scenarios timber/aluminum frames.

**Figure 5.**Stochastic estimated service life, probability density function curve fitting, high-impact factor scenarios timber/aluminum frames.

**Figure 6.**Output probability distribution function, probability density function. Low-impact factor scenarios for stochastic estimated service life, timber/aluminum frames. Monte Carlo simulation output.

**Figure 7.**Input ranked by effect on output mean, timber/aluminum frames, low/high-impact factor scenarios for estimated service life. Monte Carlo simulation output.

**Figure 8.**Spearman correlation coefficients, timber/aluminum frames, low/high-impact factor scenarios for stochastic estimated service life. Monte Carlo simulation output.

**Figure 9.**Spider graphs, timber/aluminum frames, low/high-impact factor scenarios for stochastic estimated service life. Monte Carlo simulation output.

**Figure 10.**Stochastic Gc

_{EnEc}for timber/aluminum frames: Probability density function curve fitting, low-impact factor scenarios for stochastic estimated service life, timber/aluminum frames.

**Figure 11.**Stochastic Gc

_{EnEc}for timber/aluminum frames: Probability density function curve fitting, high-impact factor scenarios for stochastic estimated service life, timber/aluminum frames.

Low-Impact Factor Scenarios | High-Impact Factor Scenarios | ||||
---|---|---|---|---|---|

Factor | Timber Factor Value | Aluminum Factor Value | Timber Factor Value | Aluminum Factor Value | |

A | Quality of component | 1 | 1 | 1 | 1 |

B | Design level | 1 | 1 | 1 | 1 |

C | Work execution level | 1 | 1 | 1 | 1 |

D | Indoor environment | 0.9 | 0.9 | 0.8 | 0.8 |

E | Outdoor environment | 0.8 | 0.9 | 0.7 | 0.8 |

F | In use conditions | 1.1 | 1.1 | 1.2 | 1.2 |

G | Maintenance level | 1.2 | 1.2 | 1.3 | 1.3 |

**Table 2.**Factor values and probability distribution, low-impact factor scenarios. Monte Carlo simulation output.

Factor | Distribution | Graph | Min | Mean | Max | 5% | 95% | Std Dev |
---|---|---|---|---|---|---|---|---|

Quality of component | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 3.85 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 3.44 | 0.92 | 1.18 | 0.1 | |

Design level | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 5.19 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 3.45 | 0.92 | 1.18 | 0.1 | |

Work execution level: | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 3.43 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 3.73 | 0.92 | 1.18 | 0.1 | |

Indoor environment | ||||||||

Timber Weighting value | Lognormal | 0.81 | 0.9 | 4.7 | 0.83 | 1.06 | 0.09 | |

Aluminum Weighting value | Lognormal | 0.81 | 0.9 | 3.89 | 0.83 | 1.06 | 0.09 | |

Outdoor environment | ||||||||

Timber Weighting value | Lognormal | 0.72 | 0.8 | 2.84 | 0.73 | 0.94 | 0.08 | |

Aluminum Weighting value | Lognormal | 0.81 | 0.9 | 3.53 | 0.83 | 1.06 | 0.09 | |

In use conditions | ||||||||

Timber Weighting value | Lognormal | 0.99 | 1.1 | 3.85 | 1.01 | 1.3 | 0.11 | |

Aluminum Weighting value | Lognormal | 0.99 | 1.1 | 3.75 | 1.01 | 1.3 | 0.11 | |

Maintenance level | ||||||||

Timber Weighting value | Lognormal | 1.08 | 1.2 | 4.3 | 1.1 | 1.41 | 0.12 | |

Aluminum Weighting value | Lognormal | 1.08 | 1.2 | 4.97 | 1.1 | 1.41 | 0.12 |

**Table 3.**Factor values and probability distribution, high-impact factor scenarios. Monte Carlo simulation output.

Factor | Distribution | Graph | Min | Mean | Max | 5% | 95% | Std Dev |
---|---|---|---|---|---|---|---|---|

Quality of component | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 4.24 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 3.44 | 0.92 | 1.18 | 0.1 | |

Design level | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 3.98 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 4.19 | 0.92 | 1.18 | 0.1 | |

Work execution level: | ||||||||

Timber Weighting value | Lognormal | 0.9 | 1 | 3.49 | 0.92 | 1.18 | 0.1 | |

Aluminum Weighting value | Lognormal | 0.9 | 1 | 3.54 | 0.92 | 1.18 | 0.1 | |

Indoor environment | ||||||||

Timber Weighting value | Lognormal | 0.64 | 0.8 | 5.26 | 0.67 | 1.08 | 0.16 | |

Aluminum Weighting value | Lognormal | 0.64 | 0.8 | 4.99 | 0.67 | 1.08 | 0.16 | |

Outdoor environment | ||||||||

Timber Weighting value | Lognormal | 0.56 | 0.7 | 4.33 | 0.59 | 0.95 | 0.14 | |

Aluminum Weighting value | Lognormal | 0.64 | 0.8 | 4.63 | 0.67 | 1.08 | 0.16 | |

In use conditions | ||||||||

Timber Weighting value | Lognormal | 0.96 | 1.2 | 8.22 | 1 | 1.63 | 0.24 | |

Aluminum Weighting value | Lognormal | 0.96 | 1.2 | 8.94 | 1 | 1.63 | 0.24 | |

Maintenance level | ||||||||

Timber Weighting value | Lognormal | 1.04 | 1.3 | 7.71 | 1.09 | 1.76 | 0.26 | |

Aluminum Weighting value | Lognormal | 1.05 | 1.3 | 8.23 | 1.09 | 1.76 | 0.26 |

Timber Frame | Aluminum Frame | ||||||
---|---|---|---|---|---|---|---|

Input Data | Unit | Low Range | Point Estimate | High Range | Low Range | Point Estimate | High Range |

Initial investment costs (elements costs) | €/m^{2} | 218.06 | 229.53 | 252.49 | 157.99 | 166.31 | 182.94 |

Annual running and replacement costs: | |||||||

- Inspection | € per year | 6220 | 6547 | 7202 | 2253 | 2372 | 2609 |

- Preemptive maintenance | € per year | 15,550 | 16,369 | 18,005 | 11,267 | 11,860 | 13,046 |

- Maintenance work (light) | € every 5 years | 62,201 | 65,474 | 72,022 | 45,067 | 47,439 | 52,183 |

- Maintenance work (main) | € every 10 years | 117,854 | 130,949 | 157,138 | 74,717 | 83,019 | 99,622 |

- Replacement | € | 339,561 | 377,290 | 452,748 | 258,404 | 287,115 | 344,538 |

Dismantling cost | €/m^{2} | 29.7 | 33 | 39.6 | 29.7 | 33 | 39.6 |

Disposal cost | €/ton | 49.5 | 55 | 66 | −640 | −800 | −880 |

Discount rate | % | 1.25 | 1.39 | 2.50 | 1.25 | 1.39 | 2.50 |

Embodied Energy | €/Kwh | 0.145 | 0.153 | 0.168 | 0.145 | 0.153 | 0.168 |

Embodied Carbon | €/ton | 13.5 | 22.25 | 33 | 13.5 | 22.25 | 33 |

**Table 5.**Input data and probability distribution values. Low-impact factor scenarios for estimated service life. Monte Carlo simulation output.

Input data | Distribution | Graph | Min | Mean | Max | 5% | 95% | Std Dev |
---|---|---|---|---|---|---|---|---|

Disposal cost_glass | Triangular | 72.03 | 82.67 | 95.96 | 75.1 | 91.62 | 4.99 | |

Disposal cost_timber | Triangular | 49.52 | 56.83 | 65.96 | 51.63 | 62.99 | 3.43 | |

Disposal cost_aluminum | Triangular | 640.47 | 773.33 | 879.57 | 683.82 | 849.02 | 49.89 | |

Dismantling cost | Triangular | 29.7 | 34.1 | 39.58 | 30.98 | 37.79 | 2.06 | |

Discount rate | Triangular | 1.25% | 1.71% | 2.5% | 1.34% | 2.24% | 0.28% | |

Embodied Energy_cost of electricity | Triangular | 0.15 | 0.16 | 0.17 | 0.15 | 0.16 | 0.01 | |

Embodied Carbon_Carbon Tax mean EU | Triangular | 13.53 | 22.92 | 32.99 | 16.42 | 29.76 | 3.99 | |

Element lifespan: | ||||||||

Timber | Pearson | 10.72 | 19.01 | 139.71 | 13.51 | 28.43 | 5.13 | |

Aluminum | Pearson | 12.02 | 21.38 | 147.44 | 15.21 | 31.94 | 5.74 | |

Fixture elements cost: | ||||||||

Timber | Triangular | 218.11 | 233.36 | 252.41 | 222.5 | 246.2 | 7.16 | |

Aluminum | Triangular | 158.03 | 169.08 | 182.88 | 161.21 | 178.38 | 5.19 | |

Inspection: | ||||||||

Timber | Triangular | 6221.15 | 6656.55 | 7200.04 | 6346.84 | 7022.85 | 204.15 | |

Aluminum | Triangular | 2254.01 | 2411.5 | 2608.33 | 2299.3 | 2544.2 | 73.96 | |

Maintenance work (light): | ||||||||

Timber | Triangular | 62,211.8 | 66,565 | 72,018 | 63,468 | 70,229 | 2042 | |

Aluminum | Triangular | 45,079 | 48,230 | 52,170 | 45,986 | 50,884 | 1479 | |

Maintenance work (main): | ||||||||

Timber | Triangular | 117,923 | 135,314 | 157,064 | 122,925 | 149,965 | 8166 | |

Aluminum | Triangular | 74,747 | 85,786 | 99,570 | 77,932 | 95,075 | 5177 | |

Preemptive maintenance: | ||||||||

Timber | Triangular | 15,554 | 16,641 | 18,001 | 15,867 | 17,557 | 510.4 | |

Aluminum | Triangular | 11,268 | 12,057 | 13,042 | 11,496 | 12,721 | 369.8 | |

Replacement: | ||||||||

Timber | Triangular | 339,684 | 389,866 | 452,648 | 354,173 | 432,081 | 23,528 | |

Aluminum | Triangular | 258,553 | 296,685 | 344,467 | 269,523 | 328,811 | 17,905 |

**Table 6.**Output values of stochastic Gc

_{EnEc}for timber/aluminum frames: probability distribution function and statistics. Low-impact factor scenarios for stochastic estimated service life. Monte. Carlo simulation output.

Output | Graph | Min | Mean | Max | 5% | 95% | Std Dev |
---|---|---|---|---|---|---|---|

GcEnEc Timber | €1,601,748 | €1,993,344 | €2,483,597 | €1,802,961 | €2,184,788 | €115,848 | |

GcEnEc Aluminum | €1,059,654 | €1,313,428 | €1,595,593 | €1,197,441 | €1430861 | €71,045 |

**Table 7.**Output values of stochastic Gc

_{EnEc}for timber/aluminum frames: probability distribution function and statistics. High-impact factor scenarios for stochastic estimated service life. Monte. Carlo simulation output.

Output | Graph | Min | Mean | Max | 5% | 95% | Std Dev |
---|---|---|---|---|---|---|---|

GcEnEc Timber | €1,504,770 | €2,077,165 | €2,861,121 | €1,811,771 | €2,384,404 | €174,995 | |

GcEnEc Aluminum | €989,123 | €1,366,702 | €1,888,321 | €1,202,502 | €1,562,951 | €110,546 |

© 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**

Fregonara, E.; Ferrando, D.G. How to Model Uncertain Service Life and Durability of Components in Life Cycle Cost Analysis Applications? The Stochastic Approach to the Factor Method. *Sustainability* **2018**, *10*, 3642.
https://doi.org/10.3390/su10103642

**AMA Style**

Fregonara E, Ferrando DG. How to Model Uncertain Service Life and Durability of Components in Life Cycle Cost Analysis Applications? The Stochastic Approach to the Factor Method. *Sustainability*. 2018; 10(10):3642.
https://doi.org/10.3390/su10103642

**Chicago/Turabian Style**

Fregonara, Elena, and Diego Giuseppe Ferrando. 2018. "How to Model Uncertain Service Life and Durability of Components in Life Cycle Cost Analysis Applications? The Stochastic Approach to the Factor Method" *Sustainability* 10, no. 10: 3642.
https://doi.org/10.3390/su10103642