# Formula for Determining the Construction Workers Productivity Including Environmental Factors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Factors Affecting Labour Productivity

#### 3.2. Parameterization of the Factors

- The membership functions of the following factors are captured as linguistic values: ergonomics of equipment and tools; wage; organization of work and work stations; stress; fatigue of the employee; and health. The membership function is:
- Very good (0.8; 0.9; 1.0; 1.0);
- Good (0.6; 0.7; 0.8; 0.9)
- Average (0.3; 0.4; 0.6; 0.7);
- Weak (0.1; 0.2; 0.3; 0.4)
- Bad (0.0; 0.0; 0.1; 0.2)

- For the factor: noise, the membership function is described in Equation (2).$${\mu}_{A}(x)=\{\begin{array}{l}1\text{}for\text{}x\le 52dB\\ \frac{85dB-x}{33dB}for\text{}52dBx85dB\\ 0\text{}for\text{}x\ge 85dB\end{array}$$
- For the factor: duration of work shift, the membership function is described in Equation (3).$${\mu}_{A}(x)=\{\begin{array}{l}0\text{}for\text{}x\le 6h\text{}and\text{}x\ge 12h\\ \frac{x-6h}{1.5h}for\text{}6hx7.5h\\ \frac{12h-x}{3h}for\text{}9hx12h\\ 1\text{}for\text{}7.5h\le x\le 9h\end{array}$$
- For the factor: regeneration of strength, the membership function is described in Equation (4).$${\mu}_{A}({T}_{fo/T})=\{\begin{array}{l}0\text{}for\text{}{T}_{fo/T}\le 6\%\text{}and\text{}{T}_{fo/T}\ge 30\%\\ \frac{{T}_{fo/T}-6\%}{8\%}for\text{}6\%{T}_{fo/T}14\%\\ \frac{30\%-{T}_{fo/T}}{10}for\text{}20\%{T}_{fo/T}30\%\\ 1\text{}for\text{}14\%\le {T}_{fo/T}\le 20\%\end{array}$$
_{fo}is time to rest; and ${T}_{fo/T}=\frac{{T}_{fo}}{T}100\%$ - For the factor: precipitation, the membership function is described in Equation (5).$${\mu}_{A}(x)=\{\begin{array}{l}1for\text{}x=0mm\\ \frac{10-x}{10}for\text{}0mmx10mm\\ 0\text{}for\text{}10mm\le x\end{array}$$
- For the factor: wind, the membership function is described in Equation (6).$${\mu}_{A}(x)=\{\begin{array}{l}1\text{}for\text{}x=0m/s\\ \frac{10-x}{10}for\text{}0m/sx10m/s\\ 0\text{}for\text{}10m/s\le x\end{array}$$
- For the factor: air temperature, the membership function is described in Equation (7).$${\mu}_{A}(x)=\{\begin{array}{l}0\text{}for\text{}x\le 4\xb0C\\ \frac{x-4\xb0C}{12\xb0C}for\text{}4\xb0Cx16\xb0C\\ 1\text{}for\text{}x=16\xb0C\\ \frac{28\xb0C-x}{12\xb0C}for\text{}16\xb0Cx28\xb0C\\ 0\text{}for\text{}28\xb0C\le x\end{array}$$
- For the factor ‘worker’s absence,’ the membership function is described in Equation (8).$${\mu}_{A}(x)=\{\begin{array}{l}0\text{}for\text{}x=0r-d\\ \frac{x}{5}for\text{}1r-dx5r-d\\ 1\text{}for\text{}x\ge 5r-d\end{array}$$
- For the factor: adaptation to new working conditions, the membership function is described in Equation (9).$${\mu}_{A}(x)=\{\begin{array}{l}0\text{}for\text{}x=1r-d\\ \frac{x-1}{15}for\text{}2r-d\le x\le 15r-d\\ 1\text{}for\text{}x\ge 16r-d\end{array}$$

- For the factor ‘time spent with family,’ the membership function is described in Equation (10).$${\mu}_{A}(x)=\{\begin{array}{l}1\text{}for\text{}x\le 5r-d/week\\ 0\text{}for\text{}x\ge 6r-d/week\end{array}$$
- For the factor ‘day of the week,’ the membership function is described in Equation (11).$${\mu}_{A}(x)=\{\begin{array}{l}0.38\text{}for\text{}x=\u201cMonday\u201d\\ 0.87\text{}for\text{}x=\u201cTuesday\u201d\\ 1\text{}for\text{}x=\u201cWednesday\u201d\\ 0.88\text{}for\text{}x=\u201cThursday\u201d\\ 0.84\text{}for\text{}x=\u201cFriday\u201d\\ 0\text{}for\text{}x=\u201cSaturday\u201d\end{array}$$

#### 3.3. Impact of Identified Factors on Work Productivity of Construction Workers

_{3}, c

_{10}, c

_{14}, and c

_{15}. Their values are included in the range <3.09, …, 3.20>. Due to the small differences between the means for these factors, they were categorized into one group: those with the lowest influence on the construction workers’ productivity. The next group included the factors c

_{2}, c

_{6}, c

_{9}, c

_{11}, c

_{12}, c

_{13}, c

_{16}, and c

_{17}, whose means fit the range <3.38, …, 3.83>. This group comprises factors described as having an average influence on the productivity of construction workers. Yet another group was based on mean values of responses in the range <4.01, …, 4.27>, and included the factors c

_{1}, c

_{5}, c

_{7}, and c

_{8}. These were called factors that had a large degree of influence on construction workers’ productivity. The last factor, c

_{4}, was assigned to the group with a very high degree of influence on the productivity of construction workers, as the mean value of the responses was the highest. All the values were assigned on the basis of the assumptions of the Likert scale. The results of the survey were such that the set of factors having a very small degree of influence on workers’ productivity remained empty.

#### 3.4. Formula for Determining the Productivity of Construction Workers by Considering the Influencing Factors

_{1}, c

_{2}, …, c

_{17}was described using two variables. According to Table 2, each of the factors was assigned a function μ

_{A}(c), where c∈<c

_{1}, c

_{2}, …, c

_{17}> and the function value of μ

_{A}(c) ∈<0,1>. The other variable was the weight coefficient w, for which the values w(c)∈<0.25,0.5,0.75,1> are predefined in Table 4, where c∈<c

_{1}, c

_{2}, …, c

_{17}>. Summing up, all the factors were described in terms of pairs of numbers, which are the results of the previously specified functions and w(c).

_{A}(c

_{i}) refers to the value of the membership function of the set of high productivities for the i-th factor; w(c

_{i}) refers to the value of the function of the degree of the impact of the i-th factor on labour productivity, and the y–coefficient correcting the interval width of the possible occurrences of the Wp function values.

_{1}, …, z

_{2}>, depending on the value of the power of y, are presented in Table 4. Value z

_{1}is the minimum theoretical value of the function Wp, while z

_{2}is the maximum.

_{A}(c

_{i}) can exceed 700%, which is doubtful, if not impossible. Choosing the width of the productivity range depends on the nature of the worker’s activity and his or her sensitivity to the changing factors. Simple and repetitive tasks have less sensitivity to the variability of factors, while complicated and atypical tasks are less predictable. Therefore, determining the appropriate value of the exponent y is vital for obtaining reliable results. It is proposed that its value should be determined by using the standard deviation of the test sample. This assumption is based primarily on the lack of sensitivity of the model to the change of exponent y. The basic assumption is the alignment of the standard deviation of the actual results with the theoretical results. It is therefore possible to present the assumption using formula (13):

_{w}—standard deviation of the actual results; σ

_{t}—standard deviation of the theoretical results, y—expected exponent of formula (1).

#### 3.5. Example of Formula Usage

_{A}(c

_{i}), are presented in Table 5. The data in the column “measurement result” come from observations and membership functions were determined on the basis of graphs (Figure 1 and Figure 2).

_{i}), were used. All the necessary values for calculating Wp were determined. The calculations proceeded as follows:

^{2}/w-h. Therefore, taking into account the conditions determined in Table 5, the productivity of the worker should be 0.800 m

^{2}/w-h.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Graphic interpretation of the fuzzy value of the membership function to the high productivity sets for the worker’s age factor.

**Figure 2.**Shapes of the membership functions for factors: (

**a**) with the membership functions described as linguistic values; (

**b**) noise; (

**c**) duration of the work shift; (

**d**) regeneration of strength; (

**e**) precipitation; (

**f**) wind; (

**g**) air temperature; (

**h**) worker’s absence; (

**i**) adaptation to new working conditions.

Factor Group | Factors | Study |
---|---|---|

Time spent outside work | worker’s absence | Hsie (2009) [22]; Ahn et al. (2013) [24] |

time spent with the family (WLB) | Townsend (2012) [23] | |

Weather conditions | air temperature | Moselhi and Khan (2012) [5]; Lee et al. (2009) [25]; Zhao et al. (2009) [26] |

wind | ||

precipitation | ||

Psychophysical conditions | stress | Bowen et al. (2013) [29] |

fatigue | Bowen et al. (2013) [29] | |

health | Helmer (1996) [27] | |

age | Helmer (1996) [27] | |

recovery | Chan et al. (2012) [28] | |

Organization and management of the worker | ergonomics noise duration of work shift salary organization of work and workstations | Malara (2014) [30]; Plebankiewicz et al. (2015) [31] |

Remaining factors | day of the week adaptation to new operating conditions or a new technology | Malara (2014) [30] |

Factor Number | Factor Name | Average Value |
---|---|---|

c_{1} | Ergonomics | 4.01 |

c_{2} | Noise | 3.65 |

c_{3} | Duration of work shift | 3.20 |

c_{4} | Salary | 4.51 |

c_{5} | Organization of the workstations | 4.17 |

c_{6} | Stress | 3.38 |

c_{7} | Fatigue | 4.18 |

c_{8} | Health | 4.27 |

c_{9} | Age of the worker | 3.83 |

c_{10} | Recovery of strength | 3.09 |

c_{11} | Precipitation | 3.49 |

c_{12} | Air temperature | 3.49 |

c_{13} | Wind | 3.49 |

c_{14} | Time spent with the family | 3.16 |

c_{15} | Worker’s absence | 3.16 |

c_{16} | Day of the week | 3.65 |

c_{17} | Adaptation to new operating conditions | 3.73 |

Group Name—Influence on Work Performance of Construction Workers | Low | Average | High (Important) | Very High (Very Important) |
---|---|---|---|---|

Factors assigned | c_{3}, c_{10}, c_{14}, c_{15} | c_{2}, c_{6}, c_{9}, c_{11},c _{12}, c_{13}, c_{16}, c_{17} | c_{1}, c_{5}, c_{7}, c_{8} | c_{4} |

Weight coefficient | 0.25 | 0.5 | 0.75 | 1 |

y | 0 | 0.25 | 0.5 | 0.75 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|---|

z_{1} | 1 | 0.84 | 0.71 | 0.59 | 0.5 | 0.25 | 0.13 | 0.06 | 0.03 |

z_{2} | 1 | 1.11 | 1.22 | 1.36 | 1.5 | 2.25 | 3.38 | 5.06 | 7.59 |

Factor Symbol | Factor Name | Measurement Result | Function µ_{A}(c_{i}) Value |
---|---|---|---|

c_{1} | Ergonomics | good | 0.75 |

c_{2} | Noise | approx. 78 dB | 0.2 |

c_{3} | Duration of work shift | 9 h | 1 |

c_{4} | Salary | good | 0.8 |

c_{5} | Organization of the workstations | good | 0.8 |

c_{6} | Stress | low | 0.8 |

c_{7} | Fatigue | high | 0.25 |

c_{8} | Health | average | 0.5 |

c_{9} | Age of the worker | approx. 42 years | 1 |

c_{10} | Recovery of strength | 8% | 0.28 |

c_{11} | Precipitation | N/A | 1 |

c_{12} | Air temperature | 8 °C | 0.33 |

c_{13} | Wind | 5 m/s | 0.5 |

c_{14} | Time spent with the family | 2 days | 1 |

c_{15} | Worker’s absence | 1 workday | 0.2 |

c_{16} | Day of the week | Thursday | 0.88 |

c_{17} | Adaptation to new operating conditions | 2nd day | 0.07 |

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**MDPI and ACS Style**

Malara, J.; Plebankiewicz, E.; Juszczyk, M. Formula for Determining the Construction Workers Productivity Including Environmental Factors. *Buildings* **2019**, *9*, 240.
https://doi.org/10.3390/buildings9120240

**AMA Style**

Malara J, Plebankiewicz E, Juszczyk M. Formula for Determining the Construction Workers Productivity Including Environmental Factors. *Buildings*. 2019; 9(12):240.
https://doi.org/10.3390/buildings9120240

**Chicago/Turabian Style**

Malara, Jarosław, Edyta Plebankiewicz, and Michał Juszczyk. 2019. "Formula for Determining the Construction Workers Productivity Including Environmental Factors" *Buildings* 9, no. 12: 240.
https://doi.org/10.3390/buildings9120240