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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.17 No.1 pp.72-81

A Novel Grey Forecasting Model with Rolling Mechanism using Taguchi-Based Differential Evolution Algorithm to Optimize the Bicycle Industry in China

Xiao Han Liu, Seng Fat Wong*
College of Mechanical and Electric Engineering, Changchun University of Science and Technology, Changchun, China
Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau, China
Corresponding Author,
January 12, 2017 November 29, 2017 January 17, 2018


Sustainable manufacturing considers that efficiently controlling the early stages of a manufacturing system is an important issue for industry enterprise. However, the data collected at this point is usually limited due to cost and time issues, making it difficult to realize the real situation in the production process. One of the ways to solve this problem is to use a small data build prediction system, such as the various grey methods. Although the grey forecasting model has been successfully adopted in various fields and demonstrated promising results, it can still be further improved. Meanwhile, Chinese bicycles have had a tremendous effect on society. This paper thus combines Chinese bicycle industry proposes a new way to improve forecasting accuracy. The new prediction model is called Rolling-TDEGM(1, 1). The grey model contributes to forecasting the future demand, and the TDE method is applied to optimize the parameters of grey model based on the minimization of forecasting error. Also, the rolling mechanism can keep the real-time data. In this experimental study, the public dataset and a real case are used to confirm the effectiveness of the proposed model; the experiment indicates that the Rolling-TDE-GM(1, 1) can significantly improve prediction precision.



    Due to environmental and traffic problems, the bicycle has become a favorite transportation and fitness tool for many people around the world. Simultaneously, many bicycle manufacturing industries are facing the pressure to plan production projects that could satisfy the complicated and fickle market economy. However, in order to make the Chinese bicycle industry extremely competitive in the world, it is necessary to optimize the supply chain for controlling the production costs and market mediation costs because of the presence of demand uncertainty, precisely matching supply with the demand is quite difficult. So, the most important indicator of the supply chain is to obtain the high accuracy forecasting value. This paper proposes a novel forecasting model that can predict the future demand of the Chinese bicycle industry perfectly.

    Production planning is a core competency of global manufacturers. It determines how many items should be produced in each time over a given interval to meet the demands (Kim et al., 2015). Meanwhile, production forecasting is one of the crucial modules in the Enterprise Resource Planning system because a large quantity of raw material, new products, and inventory are all determined by it. Simultaneously, the supply chain extends from the final customer through varieties of distributors, wholesalers, and retailers, and returns to the manufacturers and their component and raw suppliers (Syntetos et al., 2016). Moreover, the forecast of future demand is the basis for all strategic and planning decisions in a supply chain of Enterprise Resource Planning system (Sepulveda-Rojas et al., 2015). With the development of technology and knowledge economy, artificial intelligence (AI) has exhibited significant progress in forecasting and non-linear modeling applications and in capturing the noise complexity in the dataset such as artificial neural network, evolutionary programming, and fuzzy logic (Eberhart et al., 1996). However, most of the traditional forecasting models such as regression analysis (Wang, 1998), time series (Mahmoud and Rice, 1990), neural network (Chiang et al., 1996), and exponential smoothing (Ramussen, 2004) have the limitation of requiring large data analysis with complicated calculations. Moreover, due to the complicated and fast-changing situation of the Chinese economy and the rapid innovation of bicycle manufacturing industry, it usually takes only a small amount of information from within a short time span to predict future changes and trends. To address this uncertain situation, a non-linear forecasting technique is used, the Grey System Theory proposed by Julong in 1982 (Julong, 1982). Grey system has become popular because it is highly complementary for tracking a wide variety of uncertain situations with samples and low-quality information. Because of the simple calculation, GM(1, 1) has been studied by many researchers in different ways. Akay and Atak (2007) improved grey prediction by applying the Rolling Mechanism.

    On the other hand, the approach of evolutionary computing technique has been applied to optimizing parameters of GM(1, 1) and has achieved great success. Several evolutionary algorithms, such as genetic algorithm (GA) and immune algorithm (IA) have been applied in the grey model. In 2013, Takeyasu et al. (2013) proposed a hybrid method to improve forecasting accuracy utilizing genetic algorithm.

    In 1995, an optimization algorithm named differential evolution (DE) was developed as an evolutionary computation technique by Storm and Price (1995). However, DE must sometimes explore too many search points before locating the global optimum. A mutation of DE, a perturbed vector was selected randomly that was not robust enough. In order to improve the efficiency and effectiveness of the optimization of the DE algorithm, Taguchi method has been utilized to solve problems. Taguchi technique is a powerful tool for the design of high-quality systems that were developed by Taguchi and Konishi in 1950 (Taguchi et al., 1987). Ho et al. (2016) used Taguchi method to propose the Taguchi-differential evolution algorithm (TDE) to solve the problem (Ho et al. 2016). In 2011, Zheng and Chou (2011) applied the TDE method with a grey model in business.

    In this work, the Taguchi method with an orthogonal array is used to optimize the mutation operator in differential evolution algorithm. The TDE method is applied to estimate the parameters of a grey forecasting model based on the minimization of forecasting error. The GA method is also applied in the GM(1, 1) model to optimize the parameters. Then, in order to keep the real-time information and operate the chaotic and poor data, the Rolling Mechanism was applied in this study. Therefore, three new models GAGM( 1, 1), TDE-GM(1, 1) and Rolling-TDE-GM(1, 1) are built in this paper. The experimental results show that the Rolling-TDE-GM(1, 1) can significantly improve the prediction precision when compared to the traditional models.


    This research adopts data provided by CEIC database. This is Chinese bicycle data, which focuses on the output, exports, and export value from July 2014 to July 2015. Table 1 shows the collected data. Table 2, 3


    As mentioned above, the object of this study is to present a novel computer simulation Rolling-TDEGM( 1, 1) approach to forecasting the production, exports, and export value of Chinese Bicycles. In addition, this paper uses one traditional Grey Model, and two improved methods compared with the proposed Rolling-TDEGM( 1, 1) to find a more precise prediction method.

    3.1 Original Grey Model

    Grey system theory is a newly developed theory to deal with the problem of uncertainties when there is insufficient information. This model has widely been used in many fields (Julong, 1982). It is developed from grey-box and black-box. Using the depth of the color expresses the entire extent of system information. In the forecasting areas, the GM(1, 1) means first differential and one variable. GM(1, 1) forecasting model is most widely used to deal with systems with imperfect and uncertain data information (Julong, 1989). The GM(1, 1) produce is shown as follows:

    Supposing original data are in original series of raw data contains k entries as in:

    x ( 0 ) ( k ) = x ( 0 ) ( 1 ) , x ( 0 ) ( 2 ) , ... , x ( 0 ) ( n )

    Where x(0) stands for the non-negative original historical time series data.

    Structure x(1) by accumulated generating operation (AGO) method, which is:




    The production of AGO is similar to the first order linear differential equation.




    k is a time point, a is called the development coefficient, and b is called driving coefficients. Applying the leastsquare method, in which we can estimate the values of a and b.


    where B and N Y are defined as follows


    Here, both parameters a and b of GM(1, 1) model can be obtained by the minimum least square estimation. The original forecasting equation of GM(1, 1) is denoted as follow:


    3.2 Improved Grey Model with Taguchi-based Differential Evolution Algorithm

    3.2.1 Review of Differential Evolution Algorithm

    Differential evolution (DE) has been the subject of intensive performance evaluation since it was proposed by Storm and Price (1995). The DE algorithm was successfully applied to the optimization of some well-known nonlinear, non-differentiable, and non-convex functions using its simple, powerful, and straightforward features (Storm and Price, 1995). Different with the genetic algorithm (GA), DE algorithm is a population-based algorithm using the following operators: initialization, mutation, crossover, and selection.

    However, GAs and DEs will require significant time for computing in the process. The convergence speed of GA is rather slow. Although the DE can speed up convergence, it has more control variables. According to the above problems, in this research, a Taguchi method is applied in the DE algorithm to make a fast selection of variables.

    3.2.2 Review of Taguchi Method

    A robust design approach, Taguchi method was proposed by Dr. Genichi Taguchi in 1949 (Ross, 1988). From an engineering background, Dr. Taguchi converted his study of statistics and advanced mathematics into systemmerging statistical techniques and engineering expertise (Peace, 1993). The main concept of the Taguchi method is optimizing the design of system parameters, refining the qualities of the products. Here, is also an application of Taguchi method to deal with problems in the design of evolutionary algorithm.

    3.2.3 Overview of the Rolling-TDE-GM(1, 1)

    As opposed to the ordinary least square method, the selection of the two parameters a and b is fundamental to the forecasting accuracy of GM(1, 1). The traditional GM(1, 1) usually uses the least-square method to estimate the parameters of a and b that will cause certain limitations. So, here is proposed an improved method, Rolling- TDE-GM(1, 1), to optimize the parameters a and b that will make the GM(1, 1) forecasting model more accurate and robust. The benefits are shown in Figure 1, using TMDE instead of the ordinary least square method.

    3.2.4 The Principle of Rolling-TDE-GM(1, 1)

    This study now addresses the problem of reducing the forecasting error and answering the question in context. The solution of the grey forecasting model using the TDE technique can be summarized as follows:

    Before optimizing a model via TDE algorithm, the fitness function should be confirmed. Mean absolute percentage error (MAPE) is calculated to evaluate the forecasting performance:


    where yt is the actual value, and ŷt is the predicted value.

    The algorithm is designed by initialization, mutation with difference vectors by Taguchi method, crossover, and selection.

    Step 1: Initialization

    The DE algorithm is started by generating a population of real-valued D-dimensional vectors.


    where X i G denotes the ith population vector in the G generation i = 1, 2, …, NP, Xmax and Xmin are upper and lower bound, respectively, and β is a random number, where β∈[0,1].

    Step 2: Mutation with difference vectors using Taguchi method

    In order to identify the best evolutionary direction, the advanced method uses Taguchi method to determine the best parent combination according to experiment results and consequently enhances the differential evolutional algorithm. The Taguchi method is based on orthogonal array experiments that provide many reduced variances and with the optimal setting of control parameters. Thus, the parameters a and b can be estimated from the TDE system.

    This updated function of the mutated individual can help us to obtain a best-perturbed vector V i G+1 then balance the evolution and improve the search capability. The mutation function in this research is as follows:


    The scaling factor F is a positive control parameter for scaling the difference vectors. XBest is the best individual vector with the best fitness in the population at generation G. The indices r1, r2, r3, and r4 are mutually exclusive integers randomly chosen from the range p [1, Np], and all are different from the index i. In the DE algorithm, these indices are randomly generated anew for each donor vector. Here these factors are adopted using Taguchi method.

    In these mutation rules, it has four factors, and each factor has two levels. So, the orthogonal array L8 (27) is enough in this research. In this set of mutation rules there are only have four factors, so the former four columns are selected.

    In order to understand the influence of each factor r1, r2, r3, r4, here add the performance value to evaluate the performance of each experiment using singlenoise ratio (SNR). The transformation of higher-the-better type has been applied in this research, and the S/N ratio is as follows:


    The homologous table of the factor to level is shown as in Table 4.

    In Table 4, rf ,l represents parental index corresponding to the level l of each factor rf . In this research, the roulette wheel approach is used, which selects the bestperforming individual.

    Step 3: Crossover

    Crossover is a mechanism designed for mutation. The crossover operation helps increase the diversity of mutation solution vectors. By setting of crossover rate, it causes particle vectors not to be converged prematurely or too quickly, resulting in reduced incomplete searches. This is used to produce a trial vector U ji G+1 by replacing the original population vector X i G from the perturbed vector V i G+1 . The new general solution vector is generated by the equation (9)


    where j denotes the dimension of particle and Cr ∈[0,1] is predefined crossover rate.

    Step 4: Selection

    Selection is also called mother-child competition. As its name implies, mother X i G and child U ji G+1 compete with each other to survive in the next generation. Briefly, the principle of selection is a retained better offspring. Mother-child competition is expressed mathematically as follows:


    Figure 2 shows the procedure of TDE-GM(1, 1). The process is that the loop continues to step 2 until the stopping criterion is satisfied.

    3.2.5 TDE-GM(1, 1) with Rolling Mechanism

    In the real world, the information data may show different trends or characteristics at different times. Addressing these different situations requires the use of the Rolling Mechanism. The Rolling Mechanism (RM) is an efficient technique that is used to increase forecasting accuracy (Storm et al., 1995).

    In Rolling-TDE-GM(1, 1), x(0) (k +1) is predicted by applying GM(1, 1) to follow the formula:


    Where k < n. After the result is found, the procedure is repeated, but this time the newly predicted entry x(0) (k +1) and parameters a and b are added to the new model. Simultaneously, the old data and parameters are removed. The accuracy will be improved for looping this cycle, the proposed method’s efficiency, and the generalization performance.


    4.1 Bicycle Production Prediction Experiment Results

    With the low-quality information data, an advanced forecasting model Rolling-TDE-GM(1, 1) is proposed in this paper. This part of the research uses the output data of Chinese bicycle industry. The combination model involves a forecasting model and parameter estimate model. The parameter estimating model is built by Taguchi-based differential evolution algorithm. It is used to estimate the parameters a and b. Then, the parameters will be applied in the GM(1,1) to forecast the demanded data, which is the task that is common to all four models. Using the data from No. 1 to No. 8 as a training data. The testing data is from No. 9 to No. 11. The data are shown in Table 1. The result of this forecasting is also compared to the single forecasting method GM(1,1) and combination forecasting model GA-GM(1,1) and TDE-GM(1, 1).

    4.1.1 Experiment setup and comparisons of the parameters from differential models

    1) Original GM(1, 1)

    Using the data from No. 1 to No. 5 build the GM(1, 1) and then predict the data No. 9 to No. 11. The parameters a and b of GM(1, 1) are estimated by the least square method to be 0.00145 and 5325.340, respectively. The GM(1, 1) model can be expressed as:


    2) GA-GM(1, 1)

    Though a genetic algorithm is appropriate to search for a global optimization solution, it is also based on heuristics. Therefore, for each new problem to be solved, a GA requires different parameters to be set for each specific situation. These parameters include crossover rate: 0.5, mutation rate: 0.02, population size: 50, and solution method: recipe. Wang and Hsu (2008) allow for 30 generations to converge towards an optimal result. The GAGM( 1, 1) showed that the parameters of a and b of GAGM( 1, 1) estimated by genetic algorithm are 0.01157 and 5787.845, respectively. The GA-GM(1, 1) is listed as follows:


    3) TDE-GM(1, 1)

    For TDE-GM(1, 1), the best value of the parameter should be confirmed before the prediction model is constructed. The optimized parameters a and b are calculated by minimizing MAPE with the TDE algorithm. This case’s choice parameters include scaling factor F: 0.9, crossover rate CR: 0.5, and number of generation NP: 50. Using these factors to calculate parameters a and b, then replace the parameter of equation (9). The parameter of a and b of TDE-GM(1,1) are 0.00289 and 5318.408, respectively.


    4) Rolling-TDE-GM(1, 1)

    Owing to the complicated situation of the Chinese economy and the rapid development of the Chinese manufacturing industry, the data on Chinese bicycles exhibits substantial variability among many factors. Therefore, it is necessary to apply these data with the Rolling Mechanism. With the Rolling Mechanism, every step will make a new predicted data, use the new data to instead of the old one sequentially, then get the new parameters, after then build a new model again. The parameters a and b of TDEGM( 1,1) with Rolling mechanism are shown in Table 5.

    4.1.2 Comparisons of the Forecasting Results from Different Models

    The predicted results and Percentage Error (PE) are shown in Table 6 and Figure 3 by GM(1, 1) model, GAGM( 1, 1), TDE-GM(1, 1) and Rolling-TDE-GM(1, 1). The accuracy of different forecasting models are used by Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE) as shown in Table 7 and Figure 4.

    The MAPE values of these three models are 3.788% of GM(1, 1), 2.577% of GA-GM(1, 1), 2.252% of TDEGM( 1, 1) and 1.542% of Rolling-TDE-GM(1, 1). As shown in Table 7, the prediction accuracy has been improved significantly using Rolling-TDE-GM(1, 1). Simultaneously, from the Figure 3, it is made clear that only the method of Rolling-TDE-GM(1, 1) forecasting model is similar to the real data. Meanwhile, this novel method can minimize the error when compared with other methods. That means to optimize the parameters of a and b using Taguchi-based Differential Evolution Algo-rithm with rolling mechanism is the better way to im-prove the forecasting level of GM(1, 1) model. Table 6

    Accurate production forecasting can reduce the risk of resources waste (overproduction) within a manu-facturing plan.

    4.2 Exports Prediction Experiment

    China has a powerful manufacturing industry, es-pecially bicycle manufacture, consumption, and export. Since 2004, the annual production of bicycles is almost 80 million. The United States imports 18 million bicycles per year. Japan imports 8 million bicycles every year. And the more than 70 percent of bicycle importations of both of these two countries were from China. Since 2012, China has been exporting the bicycle, which was attractive in price and quality to 167 countries and re-gions. Therefore, it is necessary to control the exports and export values for the Chinese bicycle industry.

    4.2.1 Experiment Setup and Parameters Comparisons from Differential

    1) Original GM(1, 1)

    Using the data from No. 1 to No. 5 build the GM(1, 1) and then predict the data No. 9 to No. 11. The parameters a and b of GM(1, 1) are estimated using the least square method 0.058748 and 642.910, respectively.

    2) GA-GM(1, 1)

    The choice of F, CR, and NP depend on the specif-ic problem. These values directly influence the perfor-mance of the GAs. In this situation, these values are selected as follows: crossover rate (CR): 0.5, mutation rate (F): 0.02, population size (NP): 50. The GA-GM(1, 1) showed that the parameters of a and b of GA-GM(1, 1) estimated by genetic algorithm are 0.001202 and 601.428, respectively.

    3) TDE-GM(1, 1)

    For TDE-GM(1, 1), the best value of the parameter should be confirmed before the prediction model is constructed. The optimized parameters a and b are calculated by minimizing MAPE with the TDE algorithm.

    Though a DE is appropriate to use when searching for an optimized global solution, it is also based on heuristics. But when the number of the factors increases, the traditional factorial design may cost a large-scale computation and the result may not be as expected. In this study case, the update function (Equation 13) of mutated individuals can help to obtain the perturbed vector V i,1 G+1 as it only has four factors, and each fac-tor has two levels. Since the value of scaling factor F can impact the performance of DE directly, here are considered two values for the scaling factors F1, F2 to improve the performance. The actual setting of level case for VG+1 depends on the levels F1, F2 of scaling factor F as shown in the following Table 8.

    Where V i,1 G+1 denotes the vector V i G+1 obtained from function (13) with F= F1 and r1,1, r2,1, r3,1, r4,1. Simultaneously, the vector V i,2 G+1 Here, F1=0.7, F2= 0.9 are set as an assumption to select the proper value of F; meanwhile, obtain the best value of the perturbed vec-tor.

    In order to understand the influence of each factor, we add the performance value to evaluate the perfor-mance of each experiment and transfer the quality characteristic into the S/N ratio. The transformation type is higher is better (HB) in this research.


    Though the OA has eight experiments, each exper-iment has a corresponding performance value. Based on the performance value, the best combination is ob-tained. As is shown in Table 9, No. 1 is the best perfor-mance value. From Table 9, we obtain F= 0.7 as the scaling factor to apply in the mutation strategy.

    This case choice parameter includes scaling factor F: 0.7, crossover rate CR: 0.5, and number of generation NP: 50. Using these factors to calculate parameters a and b, then replace the parameter of Equation (9). The parameter of a and b of TDE-GM(1, 1) are 0.050313 and 633.120, respectively.

    4) Rolling-TDE-GM(1, 1)

    Using five data from No. 1 to No. 5 build the TDE grey model and then predict the new data. Use the ac-tual data from No. 2 to No. 5 and add the new forecast data No.6, then use the data column to estimate the parameters a and b. Utilize a new group of parameters to build a new forecasting model.

    The comparisons of the parameters a and b from different forecasting models are shown in Table 10.

    4.2.2 Forecasting Results of Different Models

    The exports of Chinese bicycle are shown in Table 11. The MAPE values of these three models are 22.915% of GM(1, 1), 22.045% of GA-GM(1,1), 17.919% of TDE-GM(1, 1), and 3.237% of Rolling-TDE-GM(1, 1). With the rolling mechanism, the errors have increased nearly 15 percent.

    4.3 Exports Value Prediction

    4.3.1 Experiment Setup and Parameters Comparisons from Differential

    All the models have used the data from No. 1 to No. 5 build the GM(1, 1) and then predict the data No.9 to No.11. For GA-GM(1, 1) the selection of these pa-rameters : crossover rate (CR): 0.5, mutation rate (F): 0.04, popu-lation size (NP): 50. For TDE-GM(1, 1), this case choice parameters include scaling factor F: 0.7, crossover rate CR: 0.5 and number of generation NP: 50. Then, it uses a rolling mechanism to enhance the optimization of the parameters. Meanwhile, it optimizes the whole forecast-ing model and increases the accuracy of the model. The comparisons of the parameters a and b from different forecasting models are shown in Table 12.

    4.3.2 Forecasting Results of Different Models

    China is a powerful global bicycle manufacturer, consumer, and exporter, and its factory output is seen as a key indicator of the whole manufacturing supply chain.

    The export value predicted results are shown in Ta-ble 13 by GM(1, 1) model, GA-GM(1, 1), TDE-GM(1, 1) and Rolling-TDE-GM(1, 1) in terms of mean average precision error in the table, where the significantly better results are verified. The MAPE values of Rolling-TDE-GM(1, 1) is 3.971%, which is better than the other three models.


    In this paper, the uncertain economic situation and the rapid innovation of manufacturing technology in China led to a situation where there is neither long-term historical data nor a smoothing of data available on the Chinese bicycle manufacturing industry. There are also very few forecasting models about the demand for bicycles. Simultaneously, hard facts on bicycle production, sales, imports, exports, and trade are difficult to come by and constantly changing, which means it is extremely hard to keep the high-quality information data and improve the forecasting accuracy level. Therefore, in this paper, we proposed a novel Rolling-TDE-GM(1, 1), which combines the grey forecasting model and parameter estimate model.

    From the above experimental results and comparisons, Rolling-TDE-GM(1, 1) forecasting model, which only requires a small amount of data samples and incomplete information, can get higher prediction accuracy than the traditional GM(1, 1) model and the improved GA-GM(1, 1) and TDE-GM(1, 1), which are more commonly used. Meanwhile, it can solve the forecasting problem under different situations. It is suitable for bicycle manufacturers to conduct a preliminary demand forecast and optimize the supply chain.

    By measuring three main aspects of Chinese bicycle industry, output, exports, and export value, each aspect has three forecast values with the test error; it is able to ensure that the actual production value falls in the ranges of the forecast, which can verify the forecast accuracy.

    From the above experimental results and comparison, it is shown that Rolling-TDE-GM(1, 1) has the best performance for this prediction problem. At the same time, from the prediction results by Rolling-TDE-GM(1, 1) model, it is not difficult to see that the output of Chinese bicycle is on the downward trend. Although there were some irregularity in the bicycle industry, like share bikes in 2016, it does not have a big impact on bicycle production industry. Chinese bicycles are mainly focused on the export of various types of bikes, like athletic cycling, mountain bikes, road bikes, sharing bikes as the lowest type of bicycle may have an impact on bicycle production. Therefore the forecasting trend by Rolling- TDE-GM(1, 1) is still functional.

    To sum up, Rolling-TDE-GM(1, 1) is considered a new, efficient, and effective forecasting model for us in evaluating the bicycle manufacturing industry.


    The authors acknowledge the funding support by University of Macau under research grant number MYRG082 (Y1-L2)-FST12-WSF.



    The principle diagram of the advanced forecasting model.


    The flow chart of TDE-GM(1, 1).


    Compare performance index of different forcasting models


    Forecasting results from diffrent models


    The data of Chinese bicycle

    MAPE criteria for model evaluation

    The orthogonal array with four factors

    The Factors to level

    The parameters of TDE-GM(1,1) with rolling mechanism

    Forecasting result and PE

    Compare performance index of different forecasting models

    The assignment of factors

    L8 (27) Two-level orthogonal array with evaluation

    Comparisons of the parameters from different models

    Forecasting result and PE

    Comparisons of the parameters from different models

    Forecasting result and PE


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