# The Use of Fuzzy Linear Regression and ANFIS Methods to Predict the Compressive Strength of Cement

^{*}

## Abstract

**:**

## 1. Introduction

_{3}S), which is responsible for the early strength and early characteristics of cement, dicalcium silicate (C

_{2}S), which contributes to cement’s long-term compressive strength, tricalcium aluminate (C

_{3}A), which in a low amount provides resistance to sulfates, tetracalcium aluminoferrite (C

_{4}AF) that works as a flux material and sometimes trioxide sulfate (SO

_{3}) [3], which is a gypsum that is added to the clinker after its cooling for minimizing the amount of grinding energy needed. Depending on its application, cement contains over 50–70% C

_{3}S, 15–30% C

_{2}S, 5–10% C

_{3}A, 5–15% C

_{4}AF [4] and less than 4% SO

_{3}[3]. Cement hydration also depends on the Blaine fineness of the cement particles, which ranges in value between 2000 and 5000 cm

^{2}/g, and by the alkali content in the cement composition.

_{2}) into the atmosphere, causing climate change and greenhouse gas emissions. On the other hand, non-conventional cementitious materials [5,6,7,8,9] are being developed in order to reduce environmental pollution and replace ordinary Portland cement with sustainable materials like self-healing concretes or alternative geopolymer-based concretes, which seem to be promising in concrete construction. Non-conventional cementitious materials also improve the characteristics of the cement and increase its compressive strength. Τhe ordinary cement compressive strength can also be improved by incorporating nanoparticles [10] in its composition to increase the density of the concrete. The prediction of the compressive strength of these types of concrete is a very important process, as it provides an option to modify the mix proportion in circumstances where the mandatory design strength is not attained, in order to avoid construction failures and substitute successfully the stability offered by Portland cement. As indicated, analytical models that include the effects of each of these factors on the compressive strength may be very complicated. Consequently, the use of fuzzy regression models, adaptive neuro-fuzzy inference systems (ANFISes) [11,12,13], as well as artificial neural networks (ANNs) [11,14,15] and fuzzy logic, seems to be a promising approach to the strength prediction problem, providing useful tools in the concrete industry.

_{3}S, SO

_{3}, Blaine fineness and alkali, were first normalized to ranges from 0.1 to 0.9, using the equation

_{i}and Xmax

_{i}are the lowest and highest values of each input, respectively, and X

_{i}is the value of each input of the ith node.

_{i}is the input parameter from the ith neuron and v

_{ij}is the weight from the previous level i in the next level j. Afterwards, the aforementioned values were transferred to the hidden layer. For determining the activation level, the sigmoid transfer function was used with the following form:

_{i}is the vector of the calculated output value, t

_{i}is the target output, P is the number of training patterns and p is the number of output neurons. The modification of the network weights was accomplished by using the following equation:

_{i}is the value of the output and μ(x

_{i}) is its membership value in the membership function.

## 2. Fuzzy Linear Regression (FLR)

_{i}are symmetric fuzzy numbers, which includes the inability to determine an exact association between the dependent and independent parameters. They are expressed as A = (r

_{i},c

_{i})

_{L}, where r

_{i}is the center in which the membership function is equal to 1 and c is the range of values. Thus, the membership function [22] is defined as

_{t}and F

_{t}are the observed and computed values, respectively, and n is the number of sets.

_{i}contributes to the fuzziness of the system. In order to find the effect of independent inputs on the calculation of the dependent variable Y, the degree of fuzziness was calculated separately for four different models. Each model was comprised of three input variables instead of four and each time a different variable was subtracted. The results are summarized in Table 2.

_{3}S was the most important factor in increasing the compressive strength at 28 days, as it had the highest effect on the system’s degree of fuzziness. It also constitutes 50–70% of cement’s composition, which defines it as the main component in the cement. However, all variables were important for developing a successful fuzzy model to predict the 28-day compressive strength of cement. More details on the fuzzy linear regression method can be retrieved from [23,24].

## 3. Adaptive Neuro-Fuzzy Modeling (ANFIS)

_{Ai}(x

_{1}), μ

_{Bi}(x

_{2}), μ

_{Ci}(x

_{3}) and μ

_{Di}(x

_{4}) were the trapezoidal membership functions in this study and i = 1,2,3. At each node, the incoming signals were multiplied and this information was sent to the output, using the min or prod operator. The prod method was used in this study with the following form:

_{i}is the weight parameter from the ith neuron. In the third layer, the degree of membership function of a rule from each node resulted, using the equation

## 4. Model Application

_{3}S, SO

_{3}, Blaine fineness and alkali are the inputs, strength is the output, FLR (left), FLR (center) and FLR (right) are the values of the fuzzy linear regression in the boundaries and in the center of regression, μ

_{A}(y

_{i}) is the membership grade of the FLR method and ANFIS presents the results obtained from the ANFIS algorithm.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- European Committee for Standardization (CEN). Cement Part 1: Composition, specifications and conformity criteria for common cements. Br. Stand. Inst.
**2000**, BS EN 197-1:2000, 12–15. [Google Scholar] - Klieger, P.; Lamond, J.F. Significance of Tests and Properties of Concrete and Concrete-Making Materials, 4rd ed.; ASTM International: Philadelphia, PA, USA, 1994; pp. 464–465. [Google Scholar]
- Mohammed, S.; Safiullah, O. Optimization of the SO
_{3}content of an Algerian Portland cement: Study on the effect of various amounts of gypsum on cement properties. Constr. Build. Mater.**2018**, 164, 362–370. [Google Scholar] [CrossRef] - Taylor, H.F.W. Cement Chemistry, 2nd ed.; Thomas Telford: London, UK, 1997; pp. 1–2. [Google Scholar]
- Gliozzi, A.S.; Scalerandi, M.; Anglani, G.; Antonaci, P.; Salini, L. Correlation of elastic and mechanical properties of consolidated granular media during microstructure evolution induced by damage and repair. Phys. Rev. Mater.
**2018**, 2, 013601. [Google Scholar] [CrossRef] - Ouarabi, M.A.; Antonaci, P.; Boubenider, F.; Gliozzi, A.S.; Scalerandi, M. Ultrasonic monitoring of the interaction between cement matrix and alkaline silicate solution in self-healing systems. Materials
**2017**, 10, 46. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Palmero, P.; Formia, A.; Tulliani, J.M.; Antonaci, P. Valorisation of alumino-silicate stone muds: From wastes to source materials for innovative alkali-activated materials. Cem. Concr. Compos.
**2017**, 83, 251–262. [Google Scholar] [CrossRef] - Humad, A.M.; Habermehl-Cwirzen, K.; Cwirzen, A. Effects of Fineness and Chemical Composition of Blast Furnace Slag on Properties of Alkali-Activated Binder. Materials
**2019**, 12, 3447. [Google Scholar] [CrossRef] [Green Version] - Cui, Y.; Gao, K.; Zhang, P. Experimental and Statistical Study on Mechanical Characteristics of Geopolymer Concrete. Materials
**2020**, 13, 1651. [Google Scholar] [CrossRef] [Green Version] - Sheikh, N.A.; Ching, D.L.C.; Khan, I.; Ahmad, A.; Ammad, S. Concrete Based Jeffrey Nanofluid Containing Zinc Oxide Nanostructures: Application in Cement Industry. Symmetry
**2020**, 12, 1037. [Google Scholar] [CrossRef] - Khademi, F.; Akbari, M.; Jamal, S.M.; Nikoo, Μ. Multiple linear regression, artificial neural network, and fuzzy logic prediction of 28 days compressive strength of concrete. Front. Struct. Civ. Eng.
**2017**, 1, 90–99. [Google Scholar] [CrossRef] - Ahmad, S.S.S.; Othman, Z.; Kasmin, F.; Borah, S. Modeling of Concrete Strength Prediction using Fuzzy Type-2 Techniques. J. Theor. Appl. Inf. Technol.
**2018**, 96, 7973–7983. [Google Scholar] - Behfarnia, Κ.; Khademi, F. A comprehensive study on the concrete compressive strength estimation using artificial neural network and adaptive neuro-fuzzy inference system. Int. J. Optim. Civ. Eng.
**2017**, 7, 71–80. [Google Scholar] - Asteris, P.G.; Kolovos, K.G.; Douvika, M.G.; Roinos, K. Prediction of self-compacting concrete strength using artificial neural networks. Eur. J. Environ. Civil Eng.
**2016**, 20, 102–122. [Google Scholar] [CrossRef] - Naderpour, H.; Kheyroddin, A.; Godrati-Amiri, G. Prediction of FRP-confined compressive strength of concrete using artificial neural networks. Compos. Struct.
**2010**, 92, 2817–2829. [Google Scholar] [CrossRef] - Akkurt, S.; Tayfur, G.; Can, S. Fuzzy logic model for the prediction of cement compressive strength. Cem. Concr. Res.
**2004**, 34, 1429–1433. [Google Scholar] [CrossRef] [Green Version] - Akkurt, S.; Ozdemir, S.; Tayfur, G.; Akyol, B. The use of GA-ANNs in the modelling of compressive strength of cement mortar. Cem. Concr. Res.
**2003**, 33, 973–979. [Google Scholar] [CrossRef] [Green Version] - Asteris, P.G.; Armaghani, D.J.; Hatzigeorgiou, G.D.; Karayannis, C.G.; Pilakoutas, K. Predicting the shear strength of reinforced concrete beams using Artificial Neural Networks. Comput. Concr.
**2019**, 24, 469–488. [Google Scholar] [CrossRef] - Haykin, S. Neural Networks A Comprehensive Foundation, 3rd ed.; Prentice-Hall: Upper Saddle River, NJ, USA, 2007. [Google Scholar]
- Jantzen, J. Design of Fuzzy Controllers; Technical Report (98-E864); Department of Automation. Technical University of Denmark: Lyngby, Denmark, 1998. [Google Scholar]
- Papadopoulos, B.K.; Sirpi, M.A. Similarities in Fuzzy Regression Models. J. Optim. Theory Appl.
**1999**, 102, 373–383. [Google Scholar] [CrossRef] - Tanaka, H.; Uejima, S.; Asai, Κ. Linear Regression Analysis with Fuzzy Model. IEEE Trans. Syst. Man Cybern.
**1982**, 12, 903–907. [Google Scholar] - Terano, Τ.; Asai, Κ.; Sugeno, Ν. Fuzzy Systems Theory and Its Applications; Academic Press: London, UK, 1992; pp. 69–80. [Google Scholar]
- Klir, G.; Yuan, B. Fuzzy Sets and Fuzzy Logic: Theory and Applications; Prentice-Hall: Upper Saddle River, NJ, USA, 1995; pp. 454–459. [Google Scholar]

Variable | Estimate R_{i} | Estimate C_{i} | SE | T | P-Value |
---|---|---|---|---|---|

A_{0} | 13.624 | 0.000 | 12.268 | 1.111 | 0.273 |

A_{1} | 0.411 | 0.056 | 0.135 | 3.047 | 0.004 |

A_{2} | 6.633 | 0.000 | 1.482 | 4.475 | 0.000 |

A_{3} | −0.002 | 0.000 | 0.002 | −0.802 | 0.427 |

A_{4} | 4.192 | 0.000 | 3.813 | 1.099 | 0.277 |

Model with Included Inputs | F(M^{h}) |
---|---|

SO_{3}, Blaine, alkali | 1.355 |

C_{3}S, Blaine, alkali | 0.639 |

C_{3}S, SO_{3}, alkali | 0.577 |

C_{3}S, SO_{3}, Blaine | 1.283 |

**Table 3.**Comprehensive strength of cement at 28 days as determined by the fuzzy linear regression (FLR) and ANFIS methods across 50 sets.

C_{3}S | SO_{3} | Blaine | Alkali | Strength | FLR (Left) | FLR (Center) | FLR (Right) | μ_{A}(y_{i}) | ANFIS |
---|---|---|---|---|---|---|---|---|---|

54 | 3 | 3530 | 1.1 | 53.9 | 50.7 | 53.7 | 56.8 | 0.9 | 53.9 |

54.8 | 2.9 | 3680 | 0.9 | 51.9 | 49.2 | 52.3 | 55.3 | 0.9 | 51.9 |

57.3 | 2.8 | 3560 | 1 | 53.9 | 50.1 | 53.3 | 56.5 | 0.8 | 53.9 |

64.6 | 2.6 | 3850 | 1 | 50.8 | 50.8 | 54.4 | 58 | 0 | 50.8 |

56.9 | 2.7 | 3580 | 0.8 | 54.5 | 48.4 | 51.6 | 54.8 | 0.1 | 54.5 |

61.3 | 2.3 | 3780 | 0.9 | 50.4 | 47.3 | 50.8 | 54.2 | 0.9 | 50.5 |

62.3 | 2.8 | 3640 | 0.9 | 55.4 | 51.3 | 54.8 | 58.3 | 0.8 | 55.4 |

62.4 | 2.8 | 3590 | 0.9 | 58.4 | 51.4 | 54.9 | 58.4 | 0 | 58.4 |

64.6 | 2.5 | 4090 | 0.8 | 54.8 | 48.8 | 52.5 | 56.1 | 0.4 | 54.7 |

59.3 | 2.8 | 3500 | 1.1 | 51.8 | 51.3 | 54.6 | 58 | 0.1 | 51.8 |

61.8 | 2.7 | 3630 | 1.1 | 51.3 | 51.3 | 54.8 | 58.2 | 0 | 51.3 |

61.3 | 3 | 3580 | 1 | 54.7 | 52.8 | 56.2 | 59.7 | 0.6 | 54.7 |

60.4 | 2.6 | 3680 | 1 | 54.1 | 49.6 | 53 | 56.4 | 0.7 | 54.1 |

55.6 | 3.1 | 3510 | 1 | 54.5 | 51.6 | 54.7 | 57.8 | 0.9 | 54.6 |

62.4 | 2.5 | 3590 | 1.1 | 51.5 | 50.3 | 53.8 | 57.2 | 0.4 | 51.5 |

63.1 | 2.6 | 3540 | 0.9 | 52.1 | 50.4 | 54 | 57.5 | 0.5 | 52.1 |

61.2 | 2.7 | 3610 | 0.9 | 51.7 | 50.3 | 53.7 | 57.1 | 0.4 | 51.7 |

55.6 | 2.7 | 3620 | 0.9 | 54.2 | 48.3 | 51.4 | 54.5 | 0.1 | 54.2 |

67.3 | 2.6 | 4020 | 0.8 | 53.8 | 50.6 | 54.4 | 58.1 | 0.8 | 53.8 |

58.7 | 3 | 3550 | 0.9 | 51.5 | 51.5 | 54.8 | 58.1 | 0 | 51.5 |

65.4 | 2.3 | 3730 | 0.9 | 48.9 | 48.9 | 52.6 | 56.2 | 0 | 48.9 |

58 | 2.7 | 3420 | 1 | 53.2 | 49.9 | 53.2 | 56.4 | 0.99 | 53.3 |

65 | 2.5 | 4070 | 0.8 | 54.7 | 49 | 52.7 | 56.3 | 0.4 | 54.7 |

62 | 2.9 | 3720 | 1 | 54.3 | 52.1 | 55.6 | 59.1 | 0.6 | 54.3 |

61.4 | 2.7 | 3840 | 0.9 | 52.5 | 49.9 | 53.4 | 56.8 | 0.7 | 52.5 |

63.5 | 2.5 | 3540 | 1 | 51.3 | 50.3 | 53.9 | 57.4 | 0.3 | 51.3 |

62.8 | 2.3 | 3580 | 0.9 | 51.1 | 48.3 | 51.8 | 55.3 | 0.8 | 51.1 |

56.4 | 3 | 3370 | 1.1 | 52.5 | 51.9 | 55 | 58.2 | 0.2 | 52.5 |

62.8 | 3 | 3750 | 1.1 | 54.1 | 53.4 | 56.9 | 60.5 | 0.2 | 54.1 |

58.9 | 3 | 3540 | 1 | 53.5 | 52 | 55.3 | 58.6 | 0.5 | 53.5 |

62.3 | 2.5 | 3910 | 0.9 | 53.6 | 48.8 | 52.3 | 55.8 | 0.6 | 53.6 |

57.7 | 2.7 | 3480 | 1 | 55.4 | 49.7 | 52.9 | 56.2 | 0.2 | 55.5 |

55.8 | 3.1 | 3420 | 0.9 | 53.7 | 51.4 | 54.5 | 57.6 | 0.7 | 53.7 |

55.9 | 2.8 | 3620 | 1 | 55.6 | 49.5 | 52.6 | 55.7 | 0.04 | 55.6 |

60.7 | 2.8 | 3740 | 1.1 | 55.2 | 51.4 | 54.8 | 58.2 | 0.9 | 55.2 |

50.3 | 2.5 | 3750 | 1.1 | 55.5 | 48.9 | 52.2 | 55.5 | 0 | 55.5 |

60.8 | 2.2 | 3520 | 1.1 | 49.8 | 47.8 | 51.2 | 54.6 | 0.6 | 49.8 |

60.7 | 3 | 3840 | 0.9 | 55.6 | 51.7 | 55.1 | 58.5 | 0.8 | 55.6 |

63.2 | 2.5 | 4010 | 0.9 | 52.1 | 48.9 | 52.5 | 56 | 0.9 | 52.1 |

59.3 | 2.6 | 3450 | 1 | 51.6 | 49.7 | 53 | 56.3 | 0.6 | 51.6 |

65.8 | 2.6 | 4050 | 0.9 | 53 | 50.4 | 54.1 | 57.8 | 0.7 | 53 |

57.4 | 2.5 | 3390 | 1.1 | 50.5 | 48.9 | 52.1 | 55.3 | 0.5 | 50.7 |

62 | 2.4 | 3490 | 1 | 54 | 49.2 | 52.7 | 56.2 | 0.6 | 54 |

59.7 | 2.2 | 3890 | 1 | 52.1 | 46.3 | 49.7 | 53 | 0.3 | 52.1 |

56.8 | 2.7 | 3620 | 1 | 53.8 | 49.1 | 52.3 | 55.5 | 0.5 | 53.8 |

61.7 | 2.4 | 3630 | 0.9 | 53.6 | 48.4 | 51.9 | 55.3 | 0.5 | 53.6 |

63.9 | 2.8 | 3680 | 0.9 | 53 | 51.7 | 55.2 | 58.8 | 0.4 | 53 |

61.6 | 2.8 | 3630 | 1.1 | 53.5 | 51.9 | 55.3 | 58.8 | 0.5 | 53.5 |

64.9 | 2.4 | 3900 | 1 | 49.9 | 49.5 | 53.1 | 56.8 | 0.1 | 49.9 |

61 | 2.8 | 3700 | 0.9 | 54.2 | 50.7 | 54.1 | 57.5 | 0.98 | 54.2 |

**Table 4.**The root mean square error (RMSE) and mean absolute percentage error (MAPE) results of all of the methods.

Metric | ANN | Fuzzy | Fuzzy Linear Regression | ANFIS |
---|---|---|---|---|

RMSE | 1.70 | 1.84 | 2.04 | 0.04 |

MAPE | 2.41% | 2.69% | 3.28% | 0.0270% |

© 2020 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**

Gkountakou, F.; Papadopoulos, B.
The Use of Fuzzy Linear Regression and ANFIS Methods to Predict the Compressive Strength of Cement. *Symmetry* **2020**, *12*, 1295.
https://doi.org/10.3390/sym12081295

**AMA Style**

Gkountakou F, Papadopoulos B.
The Use of Fuzzy Linear Regression and ANFIS Methods to Predict the Compressive Strength of Cement. *Symmetry*. 2020; 12(8):1295.
https://doi.org/10.3390/sym12081295

**Chicago/Turabian Style**

Gkountakou, Fani, and Basil Papadopoulos.
2020. "The Use of Fuzzy Linear Regression and ANFIS Methods to Predict the Compressive Strength of Cement" *Symmetry* 12, no. 8: 1295.
https://doi.org/10.3390/sym12081295