Abstract
The recent increase in the road transportation necessitates scheduling to reduce the adverse impacts of the road transportation and evaluate the effectiveness of previous actions taken in this context. However, it is impossible to undertake the scheduling and evaluation tasks unless previous information are available to predict the future. The grey model requires a limited volume of data for estimating the behavior of an unknown system. It provides high-accuracy predictions based on few data points. Various grey prediction models have been proposed so far, in which three different approaches are followed to increase the accuracy: (1) data preprocessing, (2) improved equation models, and (3) error improvement or error balancing. In this paper, firstly, a theorem is proposed and proved to recognize the parameters affecting two grey models, namely GM(1, 1) and FGM(1, 1). Then, the effective parameters are adjusted through particle swarm optimization (PSO) to formulate two adjusted models, namely IGM(1, 1) and IFGM(1, 1). According to the simulation results of the proposed models, accuracy of the modeling improved by a minimum of 14.24% and a maximum of 82.39%. Finally, the number of users of a public road transportation system was predicted using the proposed models. The results showed enhanced accuracy (by 7.7%) of the proposed models for predicting the number of users of the public road transportation system.
1 Introduction
The development of transportation systems dates back to the emergence of humans as they needed to transfer products from one point to another; it defines an inherent long-lasting system developed by humans. As one of the earliest forms of business in the world, the trade-in-goods requires product transportation. No government or organization can register its entity without transportation and communication. In the context of transportation, the scheduling refers to the procedure or operation that defines actions required for guiding a transportation system or set towards a desired state. The most significant challenge encountered in the scheduling is the providence, which is not feasible without prediction and estimation of the future state.
Various prediction techniques and tools have been proposed so far. Examples include artificial neural networks, machine learning, support vector machine, random jungle, fuzzy logic, cause-and-effect methods, linear regression, time series and Markov series [1]. Most of the proposed methods require sufficient data for satisfying the respective constraints. The cause-and-effect methods require a large volume of prior data to analyze the relationship between variables. The linear regression methods assume that the associated factors are independent of the normal distribution in the prediction procedure. The time series-based methods require stable trends for predicting a future event [2]. The probability of possible changes among different state serves as a requirement for prediction using the Markov model.
The grey systems theory, which includes the grey system analysis, modeling, prediction, control and decision-making, was first proposed by Dang in early 1980s [3]. According to this theory, known, unknown, and semi-known information of the system are represented in white, black and grey, respectively. The term semi-known here refers to the shortage or unreliability of information. The grey prediction models are widely used in the contexts of economy, education, agriculture, transportation, meteorology and military assessments [4, 5]. Although these models are popular for their efficiency in time-series prediction with few samples, most of them are basically linear, which limits their applicability in practice.Maet al. proposed a multi-variate nonlinear kernel-based grey model (GM) called KGM (1, n) [6]. In recent years, novel techniques have been used to optimize the grey model to achieve maximum possible prediction efficiency. Fractional computations can more accurately describe the real-order models as opposed to the integer-order models. Studies in this context can be divided to two broad categories: (1) discrete GM fractional accumulative models [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and (2) continuous fractional order GM [20, 21]. HE et al. [22] employed Tikhonov regularization (TR) and truncated singular value decomposition (TSVD) to resolve an ill-posed problem. The grey neural network prediction was developed to predict transportation failure and its effect on the market demand [23]. In their study, Bezuglov et al. considered the short-term traffic prediction along freeways using grey models [24]. Predicting the growth of demand for trip in the aerial industry [25], predicting the return flow of the end-of-life vehicles (ELVs) [26], and predicting short-term traffic in urban roads [27] are examples of other applications of the grey prediction models in the transportation industry.
The present paper aims to improve two grey models, namely FGM (1, 1) and GM (1, 1) [28], using meta-heuristic methods. For this purpose, beginning with the GM (1, 1) model, the parameters affecting the growth factor and the grey control parameter are identified. Next, particle swarm optimization (PSO) is used to adjust the identified parameters and calculate the resultant error. Since FGM (1, 1) is an extension to GM (1, 1), the proposed approach is followed to propose new models including Improved Grey Model (IGM (1, 1)) and Improved First-entry Grey Model (IFGM (1, 1)).
The rest of this paper is organized as follows. Section 2 describes the GM (1, 1) and FGM (1, 1) models and analyzes the parameters affecting the accuracy of the two grey models. The proposed method is presented in Section 3, while Section 4 provides for the analysis and evaluation of the proposed method using benchmark data. Next, accuracy of the proposed models is evaluated by implementing them to predict the number of users of a public transportation fleet. Finally, Section 5 draws some conclusions and provides suggestions for future works.
2 Grey prediction models and analysis of the parameters affecting the accuracy of the model
Grey models investigate the sequence of primarily discrete data and convert grey difference equations to grey differential equations to extract potential rules. These models create a continuous and dynamic differential equation from discrete sequences to realize the prediction of time series. Each grey model is described as GM (H, N) where N is the order of the differential equation and H is the number of variables. The grey models require a limited volume of data for estimating the behavior of an unknown system. The following subsections explain the grey prediction model in more detail.
2.1 GM (1, 1) model
First introduced by Dang in 1982 [3], GM (l, 1) is the most widely used time-series prediction model for a sequence of positive numbers with a minimum of 4 samples. It applies the accumulative generation operation (AGO) to the primary data and proceeds to solve the resultant differential equation. Finally, performing aninverse AGO, predicted values of the primary data are calculated. Consider a single input-single output system and let the time series
Assume an initial sequence of x0s, as follows
where x0 is a sequence of nonnegative numbers and n is the number of samples. First-order AGO of x0 is obtained using Eq. (2).
The mean sequence of Z1is defined as follows.
Eq. (4) expresses basic form of the GM (1, 1) model.
with the whitening equation defined as Eq. (5).
where a is the growth factor and b is the grey control parameter. These can be calculated using Eq. (6).
The matrices B and Y are determined using Eqs. (7) and (8), respectively.
According to Eq. (5), final value of the whitening equation is calculated as follows:
By applying IAGO, the predicted values
GM (l, l) provides a higher accuracy for short-term prediction on nonnegative sequences with few training data points, as compared to other methods. However, it fails to provide adequate accuracy for non-uniform, cyclic and random sequences.
2.2 FGM (1, 1) model
This model is created based on the GM (1, 1) model by employing the first value of the primary sequence of the predictions [28]. The primary sequence is defined as follows:
Where the accumulated series X1 is calculated as follows:
Where the mean sequence is given in Eq. (13).
Therefore, according to GM (1, 1), we have:
Finally, prediction values for the initial data are calculated using Eq. (17).
Many of the researchers have proposed improvements to the grey models. The improvement methods can be divided into three general classes. The first class focuses on data-preprocessing methods such as exponential smoothing to increase the accuracy of the model [29]. The second class includes improved equation models [30]. And the third class considers the error balancing methods like Fourier error improvement method [2]. The proposed method in this research combines the advantages of the second and third classes to formulate a new approach to reduced prediction error.
2.3 Analyzing the parameters affecting the accuracy of the grey model
According to Eqs. (10) and (17), accuracy of the prediction models depends on the growth factor a and grey control parameter b (Eq. (6)). In order to identify the factors affecting these parameters, Eqs. (3) and (8) can be rewritten as follows:
Theorem
The values of α and β affect the values of a and b directly.
Proof. Denoting the adjacency matrix [31]
Considering the results obtained from Eq. (20), the value of a is calculated based on the value of Z. According to Eq. (17), the value of Z is in turn affected by α. In addition, the value of b is affected by not only β but also α. Accordingly, accuracy of the GM (1, 1) and FGM (1, 1) has contributions from α and β (the second column of matrix B in Eq. (19) and the coefficient in Eq. (18)). Upon identifying the parameters affecting the accuracy of the grey models, the parameters must be adjustment to minimize the prediction error. The parameterization can be defined as an optimization problem that can be addressed through a meta-heuristic algorithm, as detailed in the next section.
3 Improved First-entry Grey Model IFGM (1, 1) and Improved Grey Model IGM (1, 1)
Conventional optimization methods (e.g. linear programming, nonlinear programming and dynamic programming) try to find an optimal solution in the neighborhood of the starting point. However, as complexity of the problems and the number of local optima increase, these methods lose their efficiency in locating the global optimum. This has motivated researchers to propose new solution methods inspired by the nature. Various meta-heuristic methods have been proposed based on swarm intelligence. This class of algorithm makes use of the swarm intelligence and communication methods to achieve a particular goal. Usually, such behaviors result in extraordinary and unbelievable regularity. As a meta-heuristic method, PSO was developed by a social psychiatrist and an electric engineer, namely Kennedy and Eberhart [32, 33]. The PSO considers a number of particles distributed across the search space of the function to be optimized. Each particle evaluates
the optimality function at its current position and then compares the result to that at the best position across the population to select a movement orientation. In this way, each and any particle selects a movement orientation and a new generation of solution developed upon the movements. This procedure is iterated until an intent solution is obtained. Figure 1 shows the steps followed in the PSO algorithm (for more details, please see [34]).
Main steps of the PSO algorithm.
As proved in the previous section, the GM (1, 1) and FGM (1, 1) models have their accuracies dependent on the values of α and β. Therefore, in order to calculate optimal values of α and β, the following fitness function was defined for the PSO:
where
Adjusting the parameters of the PSO algorithm,
Calculating predicted values for the initial sequence,
Finding optimal values of α and β, and
Calculating predicted values for the initial sequence using the optimal values of α and β.
The proposed algorithm is demonstrated in Figure 2.
Pseudo-code of the proposed algorithm.
4 Simulation and experiments
In this section, the capabilities of the proposed methods (IGM (1, 1) and IFGM (1, 1)) are examined on the two ascending and descending series introduced in Ref. [7]. Then, application of the proposed models to model and predict the number of users of a public road transportation system is investigated. Here we set the maximum number of iterations of the PSO algorithm to 300. Efficiency of the proposed methods was evaluated by comparing the results to those of not only the conventional methods, but also a neural network with 12 neurons in the hidden layer and Levenberg-marquadt training algorithm [35]. Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) were used to evaluate the accuracy of the modeling. Let the real
Following with this section, results of the GM, FGM, ANN, IGM and IFGM are presented in Tables 1 and 2 for the ascending and descending series introduced in Ref. [7], respectively. According to the obtained values of MAPE in Table 1, the proposed methods improved the results of GM, FGM, and ANN by 48.47%, 28.79% and 14.24%, respectively. Figure 3 demonstrates the matching of the modeled values and Figure 4 shows the prediction errors produced using each method.
The results on the ascending sequence.
| Row | Real data | GM(1,1) | FGM(1,1) | ANN | IGM(1, 1) | IFGM(1, 1) |
|---|---|---|---|---|---|---|
| 1 | 7 | 7.0000 | 0.00 | 10.09 | 7.00 | 0.00 |
| 2 | 9.4 | 9.4000 | 9.97 | 10.88 | 9.40 | 9.40 |
| 3 | 12.5 | 12.5840 | 11.74 | 12.76 | 11.25 | 11.25 |
| 4 | 14.0 | 14.0920 | 13.82 | 14.50 | 13.47 | 13.47 |
| 5 | 15.9 | 16.3332 | 16.26 | 16.68 | 16.12 | 16.12 |
| 6 | 19.3 | 18.9309 | 19.14 | 19.53 | 19.30 | 19.30 |
| 7 | 24.1 | 21.9416 | 22.53 | 24.02 | 23.10 | 23.10 |
| 8 | 25.8 | 25.4312 | 26.52 | 24.57 | 27.65 | 27.66 |
| 9 | 28.7 | 29.4758 | 31.21 | 25.06 | 33.10 | 33.11 |
| 10 | 39.6 | 34.1636 | 36.74 | 39.60 | 39.63 | 39.63 |
| 11 | 42.2 | 39.5969 | 43.24 | 48.09 | 47.43 | 47.44 |
| 12 | 58.3 | 45.8944 | 50.90 | 58.30 | 56.78 | 56.79 |
| 13 | 77.5 | 53.1934 | 59.91 | 77.50 | 67.97 | 67.98 |
| 14 | 89.6 | 61.6533 | 70.51 | 89.60 | 81.36 | 81.37 |
| 15 | 98.0 | 71.4586 | 82.99 | 98.00 | 97.39 | 97.41 |
| 16 | 106.4 | 82.8233 | 97.69 | 106.40 | 116.58 | 116.60 |
| MAPE | 10.75 | 7.78 | 6.46 | 5.536 | 5.540 | |
| MAE | 7.49 | 4.91 | 1.07 | 2.78 | 2.78 | |
The results on the descending sequence.
| Row | Real data | GM(1,1) | FGM(1,1) | ANN | IGM(1, 1) | IFGM(1, 1) |
|---|---|---|---|---|---|---|
| 1 | 7 | 7 | 0 | 7.00 | 7.00 | 0.00 |
| 2 | 1931 | 1931 | 1933.34 | 1999.86 | 1931.04 | 1931.04 |
| 3 | 1724 | 1720.23 | 1713.41 | 1724.00 | 1716.33 | 1716.33 |
| 4 | 1517 | 1519.19 | 1518.49 | 1607.56 | 1525.50 | 1525.50 |
| 5 | 1345 | 1341.65 | 1345.74 | 1345.00 | 1355.88 | 1355.88 |
| 6 | 1207 | 1184.86 | 1192.65 | 1213.85 | 1205.13 | 1205.13 |
| 7 | 1069 | 1046.39 | 1056.97 | 1069.00 | 1071.13 | 1071.13 |
| 8 | 952 | 924.10 | 936.73 | 952.00 | 952.04 | 952.04 |
| 9 | 848 | 816.10 | 830.17 | 815.15 | 846.18 | 846.18 |
| 10 | 745 | 720.73 | 735.73 | 745.00 | 752.10 | 752.10 |
| 11 | 669 | 636.50 | 632.03 | 669.00 | 668.48 | 668.47 |
| MAPE | 1.76 | 1.24 | 1.27 | 0.310 | 0.311 | |
| MAE | 15.51 | 10.99 | 18.10 | 3.69 | 3.69 | |
Matching of the modeled values with the corresponding real data in the ascending sequence.
Modeling error.
An investigating into Table 2 and Figure 5 shows higher efficiency of the proposed methods, as compared to other methods for modeling the descending sequence. The obtained values of MAPE show the improved efficiency of the proposed method by a minimum of 75% and a maximum of 82.39%. Figure 6 signifies smaller error of the proposed methods compared to other methods (seventh data point of the first series is ignored).
Matching of the modeled values with the corresponding real data in the descending sequence.
Modeling error.
Now, efficiency of the proposed methods (IGM (1, 1) and IFGM (1, 1)) for modeling the number of users a public road transportation system is presented. The required data were provided by the Iran Road Maintenance and Transportation Organization [36]. In this respect, the data collected during 2007-2015 was used for modeling and predictions were made to forecast the data collected during 2016-2017.
According to Table 3, the proposed models could predict the data collected during 2016-2017 at higher accuracy (by 7.7%) upon using the calculated values of α and β.
The number of users of the public road transportation system: available data and the modeling results.
| Year | Number of passengers | GM(1, 1) | FGM(1, 1) | IGM(1, 1) | IFGM(1, 1) |
|---|---|---|---|---|---|
| (thousand) | α = 0.99990, | α = 0.99999 | |||
| β = 1.01071 | β = 1.01079 | ||||
| 2007 | 1881 | 1881 | 0 | 1881 | 0 |
| 2008 | 2438 | 2736 | 2696 | 2696 | 2696 |
| 2009 | 2664 | 2705 | 2665 | 2664 | 2664 |
| 2010 | 2754 | 2674 | 2635 | 2632 | 2633 |
| 2011 | 2934 | 2644 | 2605 | 2601 | 2601 |
| 2012 | 2949 | 2614 | 2575 | 2571 | 2571 |
| 2013 | 2560 | 2585 | 2546 | 2540 | 2540 |
| 2014 | 2452 | 2555 | 2517 | 2510 | 2510 |
| 2015 | 2287 | 2526 | 2488 | 2480 | 2480 |
| 2016 | 2080 | 2498 | 2460 | 2451 | 2451 |
| 2017 | 2261 | 2469 | 2432 | 2422 | 2422 |
| MAPE | 14.64 | 12.92 | 12.48 | 12.48 | |
5 Conclusion
In this paper, a new design of the grey prediction system was presented and implemented to predict the number of users of a public road transportation system. The new design included the analysis of the grey models to identify the parameters affecting the accuracy of the model followed by optimizing the parameters using PSO. According to the simulation results, the proposed approach enhanced the accuracy of modeling the number of users of the considered public road transportation system during 2007-2015 by 14.75% and 3.41% compared to GM (1, 1) and FGM (1, 1), respectively. In addition, 7.7% improvement in accuracy was obtained for predicting the data during 2016-2017. Such accuracy enhancement can reduce the management risks and increase the scheduling performance. A possible application of the proposed approach is the prediction of the number of road accidents. In addition, the theorem and the algorithm presented in this paper can be used to identify the parameters affecting other grey prediction models and develop them accordingly. As a future trend, online grey prediction model is focused on real-time tuning of the model parameters as new data become available, so as to develop a multi-step prediction.
References
[1] L.-C. Hsu, "Using improved grey forecasting models to forecast the output of opto-electronics industry," Expert Systems with Applications vol. 38, no. 11, pp. 13879-13885, 2011.10.1016/j.eswa.2011.04.192Search in Google Scholar
[2] E. Kayacan, B. Ulutas, and O. Kaynak, "Grey system theory-based models in time series prediction," Expert Systems with Applications, vol. 37, no. 2, pp. 1784-1789, 2010.10.1016/j.eswa.2009.07.064Search in Google Scholar
[3] D. Ju-Long, "Control problems of grey systems," Systems & Control Letters vol. 1, no. 5, pp. 288-294, 1982.10.1016/S0167-6911(82)80025-XSearch in Google Scholar
[4] C.-F. Tsai, "The Application of Grey Theory to Taiwan Pollution Prediction," in World Congress on Engineering 2012, vol. Vol II.Search in Google Scholar
[5] P. H. Hsin and C. I. Chen, "Forecasting of Optimal Selecting-stock Strategy by Rolling Grey Model RGM (1, 1): Evidence from Taiwan 50 Index Stocks," in Second International Conference on Innovative Computing, Informatio and Control (ICICIC 2007) 2007, pp. 198-198.10.1109/ICICIC.2007.315Search in Google Scholar
[6] X. Ma and Z.-b. Liu, "The kernel-based nonlinear multivariate grey model," Applied Mathematical Modelling vol. 56, pp. 217-238, 2018.10.1016/j.apm.2017.12.010Search in Google Scholar
[7] L.-F. Wu, S.-F. Liu, and L.-G. Yao, "Discrete grey model based on fractional order accumulate," System Engineering Theory and Practice, vol. 34, no. 7, pp. 1822-1827, 2014.Search in Google Scholar
[8] S.-H. Mao, M.-Y. Gao, and X.-P. Xiao, "Fractional order accumulation time-lag GM(1,N,τ model and its application," System Engineering Theory and Practice vol. 35, no. 2, pp. 430-436, 2015.Search in Google Scholar
[9] B.-h. Yang and J.-s. Zhao, "Fractional order discrete grey GM(1, 1) power model and its application," Control Decis. vol. 30, pp. 1264-1268, 2015.Search in Google Scholar
[10] L. Wu, S. Liu, and J. Liu, "GM(1, 1) model based on fractional order accumulating method and its Stability," Control and Decision vol. 29, no. 5, pp. 919-924, 2014.Search in Google Scholar
[11] L. Wu, S. Liu, L. Yao, S. Yan, and D. Liu, "Grey system model with the fractional order accumulation," Communications in Nonlinear Science and Numerical Simulation vol. 18, no. 7, pp. 1775-1785, 2013.10.1016/j.cnsns.2012.11.017Search in Google Scholar
[12] X.-p. Xiao, H. Guo, and S.-h. Mao, "The modeling mechanism, extension and optimization of grey GM (1, 1) model," Applied Mathematical Modelling vol. 38, no. 5, pp. 1896-1910, 2014.10.1016/j.apm.2013.10.004Search in Google Scholar
[13] L.-F. Wu, S.-F. Liu, W. Cui, D.-L. Liu, and T.-X. Yao, "Non-homogenous discrete grey model with fractional-order accumulation," Neural Computing and Applications vol. 25, no. 5, pp. 1215-1221, 2014.10.1007/s00521-014-1605-1Search in Google Scholar
[14] Y. Shen and P. Qin, "Optimization of grey model with the fractional order accumulation," Journal of Grey System vol. 17, no. 3, pp. 127-132, 2014.Search in Google Scholar
[15] S. F. Su and X. N. Su, "The prediction of the doubling capacity of the farmers of Jiangsu province based on GMλ(1,1) model of fractional order accumulating generation," Math Pract theory vol. 44, no. 13, pp. 58-62, 2014.Search in Google Scholar
[16] L. Wu, S. Liu, Z. Fang, and H. Xu, "Properties of the GM(1,1) with fractional order accumulation," Applied Mathematics and Computation, vol. 252, pp. 287-293, 2015.10.1016/j.amc.2014.12.014Search in Google Scholar
[17] M. Wei, Z. Bo, L. Si-feng, and F. Zhi-geng, "Study on expansion and properties of grey accumulating generation operator," presented at the Grey systems and intelligent services (GSIS), 2015.Search in Google Scholar
[18] F. Shili, W. Lifeng, Y. Liang, and F. Zhigeng, "Using fractional GM(1,1) model to predict maintenance cost of weapon system," in Proceedings of 2013 IEEE International Conference on Grey systems and Intelligent Services (GSIS) 2013, pp. 177-181.10.1109/GSIS.2013.6714760Search in Google Scholar
[19] L. Wu, S. Liu, L. Yao, R. Xu, and X. Lei, "Using fractional order accumulation to reduce errors from inverse accumulated generating operator of grey model," Soft Computing journal article vol. 19, no. 2, pp. 483-488, 2015.10.1007/s00500-014-1268-ySearch in Google Scholar
[20] L. I. U. S.-f. Y. A. O. L.-g. Wu Li-feng, "Grey model with Caputo fractional order derivative," Systems Engineering - Theory & Practice vol. 35, no. 5, pp. 1311-1316, 2015.Search in Google Scholar
[21] S. Mao, M. Gao, X. Xiao, and M. Zhu, "A novel fractional grey system model and its application," Applied Mathematical Modelling vol. 40, no. 7, pp. 5063-5076, 2016.10.1016/j.apm.2015.12.014Search in Google Scholar
[22] Z. He, Y. Shen, J. Li, and Y. Wang, "Regularized multivariable grey model for stable grey coefficients estimation," Expert Systems with Applications vol. 42, no. 4, pp. 1806-1815, 2015.10.1016/j.eswa.2014.09.058Search in Google Scholar
[23] C. Liu, T. Shu, S. Chen, S. Wang, K. K. Lai, and L. Gan, "An improved grey neural network model for predicting transportation disruptions," Expert Systems with Applications vol. 45, pp. 331-340, 2016.10.1016/j.eswa.2015.09.052Search in Google Scholar
[24] A. Bezuglov and G. Comert, "Short-term freeway traffic parameter prediction: Application of grey system theory models," Expert Systems with Applications vol. 62, pp. 284-292, 2016.10.1016/j.eswa.2016.06.032Search in Google Scholar
[25] R. B. Carmona Benítez, R. B. Carmona Paredes, G. Lodewijks, and J. L. Nabais, "Damp trend Grey Model forecasting method for airline industry," Expert Systems with Applications vol. 40, no. 12, pp. 4915-4921, 2013.10.1016/j.eswa.2013.02.014Search in Google Scholar
[26] S. Ene and N. Öztürk, "Grey modelling based forecasting system for return flow of end-of-life vehicles," Technological Forecasting and Social Change vol. 115, pp. 155-166, 2017.10.1016/j.techfore.2016.09.030Search in Google Scholar
[27] H. Guo, X. Xiao, and F. Jeffrey, "Urban Road Short-term Traffic Flow Forecasting Based on the Delay and Nonlinear Grey Model," Journal of Transportation Systems Engineering and Information Technology vol. 13, no. 6, pp. 60-66, 2013.10.1016/S1570-6672(13)60129-4Search in Google Scholar
[28] T.-L. Tien, "A new grey prediction model FGM(1, 1)," Mathematical and Computer Modelling vol. 49, no. 7, pp. 1416-1426, 2009.10.1016/j.mcm.2008.11.015Search in Google Scholar
[29] L. Qiu-hua, "A Forecast Model of Monthly Electric Power Demand Based on Season Exponent and Grey Forecast," Journal of Nanjing Institute of Technology(Natural Science Edition) vol. 1, 2006.Search in Google Scholar
[30] G. D. Li, S. Masuda, D. Yamaguchi, and M. Nagai, "A New Reliability Prediction Model in Manufacturing Systems," IEEE Transactions on Reliability vol. 59, no. 1, pp. 170-177, 2010.10.1109/TR.2009.2035795Search in Google Scholar
[31] K. M. Abadir and J. R. Magnus, Matrix Algebra Cambridge University, 2005.10.1017/CBO9780511810800Search in Google Scholar
[32] B. Qinghai, "Analysis of Particle Swarm Optimization Algorithm," Computer and Information Science vol. 3, no. 1, pp. 180-184, 2010.10.5539/cis.v3n1p180Search in Google Scholar
[33] S. Asghari and N. J. Navimipour, "Review and Comparison of Meta-Heuristic Algorithms for Service Composition in Cloud Computing," Majlesi Journal of Multimedia Processing vol. 4, no. 4, pp. 28-34, 2015.Search in Google Scholar
[34] M. Mariappan Kadarkarainadar and T. Prabaharan, "Solving realistic industrial scheduling problems using a multi-objective improved hybrid particle swarm optimisation algorithm," International Journal of Operational Research vol. 23, 2015.10.1504/IJOR.2015.068752Search in Google Scholar
[35] Y. Yang and D. Xue, "Continuous fractional-order grey model and electricity prediction research based on the observation error feedback," Energy vol. 115, pp. 722-733, 2016.10.1016/j.energy.2016.08.097Search in Google Scholar
[36] RMTO. (2018). Salname Amari Available: http://rmto.ir/Pages/SalnameAmari.aspxSearch in Google Scholar
© 2020 S. Balochian and H. Baloochian, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
