# A Model for Shovel Capital Cost Estimation, Using a Hybrid Model of Multivariate Regression and Neural Networks

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- small shovels (0.5–2 m
^{3}bucket size); - medium shovels (2–5 m
^{3}bucket size); - large-size shovels (5–25 m
^{3}bucket size); - very large-size shovels (with a bucket larger than 25 m
^{3}).

^{2}), Normalized Mean Square Error (NMSE), and Mean Absolute Percentage Error (MAPE), and the best model is identified according to calculation results.

## 2. A Multivariate Regression Model

_{1}+ b

_{2}BC + b

_{3}BL + b

_{4}W + b

_{5}HP + e,

_{1}is a constant term; b

_{2}, b

_{3}, b

_{4}, and b

_{5}are the regression coefficients; and, e is an error term.

## 3. An Artificial Neural Network

_{i}(I = 1, 2, …, n); and, the output of the neuron j is assigned by y

_{j}, which can be mathematically computed by Equation (4):

## 4. The Hybrid Methodology

_{t}and L

_{t}indicate the nonlinear and linear components, respectively. Therefore, the main idea is to employ, in the first place, the MVR model, and, next, to apply the ANN to formulate the residuals of the linear structure. A schematic diagram of the proposed model is shown in Figure 3.

## 5. Model Performance

^{2}), and Mean Absolute Percentage Error (MAPE), to analyse the performance of each equation. These indicators can be calculated by as follows:

_{i}denotes the predicted values, A

_{i}denotes the observed values, N is the number of the datasets, and ${\overline{A}}_{i}$ is the average of the observed set.

^{2}shows to what extent the independent variable(s) can explain the variability in the dependent variable. R

^{2}belongs to the closed interval zero and one and can only take a positive value [53]. The value of close to zero for R

^{2}shows a poor fit of the predictive model, while the value of close one to presents a good fit. The R

^{2}value of 0.9 and higher is considered to be very satisfactory, while the values of 0.8–0.9 represent a relatively acceptable formula, and those below 0.8 are taken into account to be unacceptable [54].

## 6. The Data

## 7. The Implementation of the Proposed Model

^{1−4−1}). The correlation between the actual values and the output of the proposed model for testing data is presented in Figure 4.

## 8. The Comparison of the Developed Model with Other Models

^{5−8−1}). The results of different models used for testing data are presented in Table 2.

^{2}value for the proposed model is 0.9965, which is bigger than those that are yielded by MVR and ANN and making 0.9941 and 0.9924, respectively. The experimental results presented in Table 2 show that the hybrid models are more accurate. This conclusion can be derived because the hybrid models integrate linear and nonlinear information for predicting, while the individual model uses only linear or nonlinear information for modeling.

## 9. Sensitivity Analysis

_{ij}) between the shovel capital cost and the considered independent component [56]. The larger the value of CAM, the higher its impact on the capital cost. If the shovel capital cost is not related to the independent variable, then, the CAM value is zero. The independent variable plays a positive role the shovel capital cost where the CAM value is non-negative and plays a negative role in the shovel capital cost where the CAM value is non-positive.

_{1}, x

_{2}, …, x

_{n}}, while each of its elements, x

_{i}, in the data array X, is itself a vector of length m, and can be expressed as:

_{ij}, results from a pairwise comparison of two data samples. The strength of the relationship between the data samples, x

_{i}and x

_{j}, is given by the membership value, expressing this strength:

## 10. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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Machine Type | Operator | Bucket Capacity (BC) | Boom Length (BL) | Weight (W) | Horsepower (HP) | Capital Cost (CC) |
---|---|---|---|---|---|---|

Cable | Min | 7 | 35 | 556,000 | 685 | 2,800,000 |

Max | 80 | 64 | 4,500,000 | 7000 | 17,900,000 | |

Mean | 31.29 | 50.57 | 1,517,071 | 2923.85 | 7,871,429 | |

Standard deviation | 21.27 | 9.04 | 1,084,127.97 | 1989.71 | 4,443,392.42 | |

Hydraulic | Min | 3 | 15.4 | 95,900 | 250 | 605,000 |

Max | 44 | 39.3 | 1,399,000 | 3000 | 6,900,000 | |

Mean | 12.74 | 28.74 | 385,318.8 | 873.56 | 2,068,313 | |

Standard deviation | 9.97 | 6.96 | 334,189.25 | 700.14 | 1,781,125.17 |

Case | Actual | MVR | ANN | Proposed Model |
---|---|---|---|---|

1 | 680,000 | 120,755.6 | 815,732.3 | 716,660.3 |

2 | 1,263,000 | 120,2621 | 1,299,821 | 1,115,828 |

3 | 1,850,000 | 2,338,051 | 2,145,405 | 2,293,947 |

4 | 4,400,000 | 3,903,419 | 5,124,051 | 3,802,985 |

5 | 3,200,000 | 3,528,523 | 3,719,688 | 3,470,780 |

6 | 8,500,000 | 7,865,014 | 9,298,305 | 8,150,003 |

7 | 3,100,000 | 3,431,278 | 4,803,671 | 3,378,503 |

8 | 17,900,000 | 16,661,424 | 18,551,802 | 17,801,455 |

Model | R^{2} | NMSE | MAPE |
---|---|---|---|

Proposed model | 0.9965 | 0.0035 | 9.59% |

ANN | 0.9924 | 0.0076 | 17.44% |

MVR | 0.9941 | 0.0059 | 20% |

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**MDPI and ACS Style**

Yazdani-Chamzini, A.; Zavadskas, E.K.; Antucheviciene, J.; Bausys, R.
A Model for Shovel Capital Cost Estimation, Using a Hybrid Model of Multivariate Regression and Neural Networks. *Symmetry* **2017**, *9*, 298.
https://doi.org/10.3390/sym9120298

**AMA Style**

Yazdani-Chamzini A, Zavadskas EK, Antucheviciene J, Bausys R.
A Model for Shovel Capital Cost Estimation, Using a Hybrid Model of Multivariate Regression and Neural Networks. *Symmetry*. 2017; 9(12):298.
https://doi.org/10.3390/sym9120298

**Chicago/Turabian Style**

Yazdani-Chamzini, Abdolreza, Edmundas Kazimieras Zavadskas, Jurgita Antucheviciene, and Romualdas Bausys.
2017. "A Model for Shovel Capital Cost Estimation, Using a Hybrid Model of Multivariate Regression and Neural Networks" *Symmetry* 9, no. 12: 298.
https://doi.org/10.3390/sym9120298