# Adaptation of a Cost Overrun Risk Prediction Model to the Type of Construction Facility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Concept of a Cost Overrun Risk Prediction Model

#### 3.1. Main Assumptions of the Model

- exceeding the time and cost of construction investments [34],
- occupational risks on construction sites [37],
- level of safety of construction workers [38],
- technological, financial, political, environmental, and legal risk factors in the life cycle of buildings [39],
- technological risk factors for old buildings [40].

#### 3.2. Block of Fuzzification

_{1}, X

_{2}, and X

_{3}. The domain (range of arguments) of the universes was determined as a percentage within the interval [0; 100%] for each input variable, with the model using the decimal notation corresponding to the interval [0; 1]. In defining the X consideration spaces, for all variables described by the linguistic terms “high”, “average”, and “low”, it was assumed that the adjacent fuzzy sets (representing consecutive linguistic terms) would overlap. According to Hovde and Moser [41], only this modelling of the linguistic terms for the input variables gives a favorable effect in the inference process.

_{2}) and L(X

_{3}), that is, for the input variables WC and PC. For the description of linguistic terms, membership functions with line graphs were used (triangular functions and classes Γ and L). The qualitative definition of fuzzy sets was based on the selection of appropriate types of membership functions. The quantitative definition was performed on the basis of the selection of the values of parameters characterizing the functional curves, which made it possible to precisely determine the degrees of membership of individual fuzzy sets. Degrees of membership for fuzzy sets are described in Table 1 (in the last column) by means of four numbers {α

_{1}, α

_{2}, α

_{3}, α

_{4}}. These parameters indicate, respectively, the intervals of achieving the value of membership degree 1.0 {α

_{2}, α

_{3}} and the left or right width of the distribution of the membership function to the value of the membership degree 0.0 {α

_{1}, α

_{4}}. It was assumed that linguistic values for both input variables (WC and PC) would remain unchanged regardless of the type of the building object.

- single-family residential buildings,
- multi-family residential buildings,
- office buildings,
- highways and expressways,
- sports fields.

- “high”—description of the variable would relate to the value “about or above quartile Q3”,
- “average”—description of the variable would relate to the value “about median”,
- “low”—description of the variable would relate to the value “about or below quartile Q1”.

#### 3.3. Block of Inference

_{1}), μ(x

_{2}), and μ(x

_{3}) for individual fuzzy sets of linguistic values. The resulting function often has a complex shape and its calculation is done by the so-called inference (inference process). The inference block consists of two basic elements, namely the rule base and the inference mechanism, the operation of which is based on the three following consecutive mathematical operations: aggregation of simple premises, implications of fuzzy inference rules, and aggregation of conclusions of all rules.

_{R}). This value was determined by applying the Mamdani fuzzy implication rule (T-norm), calculated according to the following formula:

_{1}, α

_{2}, α

_{3}, α

_{4}}.

#### 3.4. Block of Defuzzification

- are not able to implement the assumption adopted for the purposes of building the rule base, that with the increase in the share of element costs in the building costs (SE), predicted changes in the number of works (WC), and expected changes in the unit price (PC), the value of the risk level of exceeding the costs of a given element of the construction investment (R) will naturally and smoothly increase,
- result in sharp values, which will not in every case adequately represent the output fuzzy set, which is caused by the impact on the sharp result of only the most activated fuzzy set of the output variable.

## 4. Discussion

- diagrams of the result area for the output variable (R) due to the influence of the input variables PC and SE in the cross-section, when WC = 0.5, and WC and SE in the cross-section, when PC = 0.5,
- diagrams of the result area for the output variable (R) taking into account the set of input variables PC and WC in the cross-section, when SE = 0.5,
- flat diagrams of the resultant curves for the output variable (R) due to the influence of PC input variables in the cross-section, when WC = SE = 0.5, WC in the cross-section, when PC = SE = 0.5, and SE in the cross-section, when PC = WC = 0.5.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Result area in terms of input variables PC and SE (

**left**) and PC and WC (

**right**)—last of maxima defuzzification method.

**Figure 9.**The result area for the output variable (R) in terms of the PC and WC variables in the cross-section, when SE = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable PC in the cross-section, when WC = SE = 0.5 (

**right**diagram).

**Figure 10.**The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (

**right**diagram)—single-family residential buildings.

**Figure 11.**The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (

**right**diagram)—multi-family residential buildings.

**Figure 12.**The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (

**right**diagram)—office buildings.

**Figure 13.**The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (

**right**diagram)—highways and expressways.

**Figure 14.**The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (

**left**diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (

**right**diagram)—sports.

**Figure 15.**The diagrams of the dependence between the output variable (R) and the input variable SE for all five types of construction objects.

**Table 1.**Fuzzy interpretations of the linguistic input variables “predicted changes in the number of works” (WC) or expected changes in the unit price (PC).

Fuzzy Set of Linguistic Values for WC or PC | Description of the Variables x_{2} or x_{3} | Fuzzy Evaluation of Membership μ(x_{2}) or μ(x_{3}) | |
---|---|---|---|

High | Hi | About or above 75.0% | (0.5; 0.75; 1.0; 1.0) |

Average | Av | About 50.0% | (0.25; 0.5; 0.5; 075) |

Low | Lo | About or below 25.0% | (0.0; 0.0; 0.25; 0.5) |

Type of Building | Cost Elements |
---|---|

Cubature facilities (single- and multi-family residential buildings, office buildings) | Earthworks, foundations (including walls and insulation of the ground floor of the building), ground walls, ceilings, stairs, partition walls, roof (construction and covering), sleepers and canals inside the building, insulation of the ground, plaster and interior cladding, windows and doors, painting work, floors (with layer), facades with works outside the building, water and sewage installations, central heating installations and electrical installations. |

Highways and expressways | Preparatory works, earthworks, drainage of road body, substructures, surfaces, finishing works, traffic safety equipment, street and road elements and other works. |

Sports fields | Site preparation and earthworks, substructures, sports surfaces, landscaping and equipment. |

Type of Building | Quartile Q1 | Median | Quartile Q3 |
---|---|---|---|

Single-family residential buildings | 3 | 6 | 9 |

Multi-family residential buildings | 3 | 5 | 8 |

Office buildings | 2 | 5 | 8 |

Highways and expressways | 2 | 6 | 21 |

Sports fields | 6 | 18 | 51 |

**Table 4.**Fuzzy interpretation of the linguistic input variable share of element costs in the building costs (SE).

Fuzzy Set of Linguistic Values for SE | Description of the Variable x _{1} | Fuzzy Evaluation of Membership μ(x _{1}) | |
---|---|---|---|

Single-family residential buildings | |||

High | Hi | About or above 9.0% | (0.06; 0.09; 1.0; 1.0) |

Average | Av | About 6.0% | (0.0; 0.06; 0.06; 0.09) |

Low | Lo | About or below 3.0% | (0.0; 0.0; 0.03; 0.06) |

Multi-family residential buildings | |||

High | Hi | About or above 8.0% | (0.05; 0.08; 1.0; 1.0) |

Average | Av | About 5.0% | (0.0; 0.05; 0.05; 0.08) |

Low | Lo | About or below 3.0% | (0.0; 0.0; 0.03; 0.05) |

Office buildings | |||

High | Hi | About or above 8.0% | (0.05; 0.08; 1.0; 1.0) |

Average | Av | About 5.0% | (0.0; 0.05; 0.05; 0.08) |

Low | Lo | About or below 2.0% | (0.0; 0.0; 0.02; 0.05) |

Highways and expressways | |||

High | Hi | About or above 21.0% | (0.06; 0.21; 1.0; 1.0) |

Average | Av | About 6.0% | (0.0; 0.06; 0.06; 0.21) |

Low | Lo | About or below 2.0% | (0.0; 0.0; 0.02; 0.06) |

Sports fields | |||

High | Hi | About or above 51.0% | (0.08; 0.51; 1.0; 1.0) |

Average | Av | About 8.0% | (0.0; 0.08; 0.08; 0.51) |

Low | Lo | About or below 6.0% | (0.0; 0.0; 0.06; 0.08) |

Rule No. | If (SE) | And (WC) | And (PC) | Then (R) | ||||
---|---|---|---|---|---|---|---|---|

LV | Weight | LV | Weight | LV | Weight | Product | Concl. | |

1 | Lo | 1 | Lo | 1 | Lo | 1 | 1 | Vl |

2 | Lo | 1 | Lo | 1 | Av | 2 | 2 | Vl |

3 | Lo | 1 | Lo | 1 | Hi | 3 | 3 | Ql |

4 | Lo | 1 | Av | 2 | Lo | 1 | 2 | Vl |

5 | Lo | 1 | Av | 2 | Av | 2 | 4 | Ql |

6 | Lo | 1 | Av | 2 | Hi | 3 | 6 | Av |

7 | Lo | 1 | Hi | 3 | Lo | 1 | 3 | Ql |

8 | Lo | 1 | Hi | 3 | Av | 2 | 6 | Av |

9 | Lo | 1 | Hi | 3 | Hi | 3 | 9 | Qh |

10 | Av | 2 | Lo | 1 | Lo | 1 | 2 | Vl |

11 | Av | 2 | Lo | 1 | Av | 2 | 4 | Ql |

12 | Av | 2 | Lo | 1 | Hi | 3 | 6 | Av |

13 | Av | 2 | Av | 2 | Lo | 1 | 4 | Ql |

14 | Av | 2 | Av | 2 | Av | 2 | 8 | Av |

15 | Av | 2 | Av | 2 | Hi | 3 | 12 | Qh |

16 | Av | 2 | Hi | 3 | Lo | 1 | 6 | Av |

17 | Av | 2 | Hi | 3 | Av | 2 | 12 | Qh |

18 | Av | 2 | Hi | 3 | Hi | 3 | 18 | Vh |

19 | Hi | 3 | Lo | 1 | Lo | 1 | 3 | Ql |

20 | Hi | 3 | Lo | 1 | Av | 2 | 6 | Av |

21 | Hi | 3 | Lo | 1 | Hi | 3 | 9 | Qh |

22 | Hi | 3 | Av | 2 | Lo | 1 | 6 | Av |

23 | Hi | 3 | Av | 2 | Av | 2 | 12 | Qh |

24 | Hi | 3 | Av | 2 | Hi | 3 | 18 | Vh |

25 | Hi | 3 | Hi | 3 | Lo | 1 | 9 | Qh |

26 | Hi | 3 | Hi | 3 | Av | 2 | 18 | Vh |

27 | Hi | 3 | Hi | 3 | Hi | 3 | 27 | Vh |

Fuzzy Set of Linguistic Values for R | Description of the Variable y | Fuzzy Evaluation of Membership μ(y) | |
---|---|---|---|

Very high | Vh | About or above 0.9 | (0.7; 0.9; 1.0; 1.0) |

Quite high | Qh | About 0.7 | (0.5; 0.7; 0.7; 0.9) |

Average | Av | About 0.5 | (0.3; 0.5; 0.5; 0.7) |

Quite low | Ql | About 0.3 | (0.1; 0.3; 0.3; 0.5) |

Very low | Vl | About or below 0.1 | (0.0; 0.0; 0.1; 0.3) |

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**MDPI and ACS Style**

Plebankiewicz, E.; Wieczorek, D.
Adaptation of a Cost Overrun Risk Prediction Model to the Type of Construction Facility. *Symmetry* **2020**, *12*, 1739.
https://doi.org/10.3390/sym12101739

**AMA Style**

Plebankiewicz E, Wieczorek D.
Adaptation of a Cost Overrun Risk Prediction Model to the Type of Construction Facility. *Symmetry*. 2020; 12(10):1739.
https://doi.org/10.3390/sym12101739

**Chicago/Turabian Style**

Plebankiewicz, Edyta, and Damian Wieczorek.
2020. "Adaptation of a Cost Overrun Risk Prediction Model to the Type of Construction Facility" *Symmetry* 12, no. 10: 1739.
https://doi.org/10.3390/sym12101739