A novel gray forecasting model based on the box plot for small manufacturing data sets

Applied Mathematics and Computation 265 (2015) 400–408

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A novel gray forecasting model based on the box plot for small manufacturing data sets Che-Jung Chang a, Der-Chiang Li b,∗, Yi-Hsiang Huang b, Chien-Chih Chen b a Department of Management Science and Engineering, Business School, Ningbo University, No. 818, Fenghua Road, Ningbo City, Zhejiang Province 315211, China b Department of Industrial and Information Management, National Chen Kung University, No. 1, University Road, Tainan City 70101, Taiwan, ROC

a r t i c l e

i n f o

Keyword: Forecasting Small data set Gray model

a b s t r a c t Efficiently controlling the early stages of a manufacturing system is an important issue for enterprises. However, the number of samples collected at this point is usually limited due to time and cost issues, making it difficult to understand the real situation in the production process. One of the ways to solve this problem is to use a small data set forecasting tool, such as the various gray approaches. The gray model is a popular forecasting technique for use with small data sets, and while it has been successfully adopted in various fields, it can still be further improved. This paper thus uses a box plot to analyze data features and proposes a new formula for the background values in the gray model to improve forecasting accuracy. The new forecasting model is called BGM(1,1). In the experimental study, one public dataset and one real case are used to confirm the effectiveness of the proposed model, and the experimental results show that it is an appropriate tool for small data set forecasting. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Globalization has caused the manufacturing conditions that many firms operate under to change dramatically. It is thus very important for manufacturing firms to efficiently control their manufacturing systems [1]. The early stages of a manufacturing system are especially critical for managers [2]. If problems can be identified at an early stage and appropriate actions are taken to overcome them, the business will become more effective and efficient. However, the most commonly used forecasting techniques are restricted to big sample sizes, and are not suitable for use in the early stages of manufacturing systems. Therefore, it is of considerable interest to develop an appropriate forecasting model for use with small data sets [3], and this is the aim of the current study. Due to the limited information available from small data sets, only part of the overall data structure can be realized. Deng [4] thus proposed the gray system theory to overcome this problem. Unlike statistical approaches, this theory indirectly deals with original data through an accumulating generation operator (AGO), and tries to find the inherent structure in a system. In addition, assumptions concerning the statistical distribution are not required when using this approach, and the resulting flexibility means that it has been successfully applied in various fields [5–8].

∗

Corresponding author. Tel.: +886 6 2757575x53134; fax: +886 6 2362162. E-mail addresses: [email protected] (C.-J. Chang), [email protected], [email protected] (D.-C. Li), [email protected] (Y.-H. Huang), [email protected] (C.-C. Chen). http://dx.doi.org/10.1016/j.amc.2015.05.006 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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The conventional gray model, GM(1,1), is the main approach in the gray system theory, which was developed for time series forecasting with small samples and has good forecasting ability. Due to its convenience, it has become a popular forecasting technique, and produced many satisfactory results [9–12]. Although many studies have proposed methods to improve the forecasting accuracy of the GM(1,1) model [13–17], there is still some room for further improvement. This research analyzes the data profile of small data set by the box plot statistical approach to form a membership function and thus derive an improved modeling procedure for the gray forecasting model. This new method, called BGM(1,1), can establish a suitable model according to the features of the data, and has better forecasting accuracy than two popular gray models. This study employs the Synthetic Control Chart Time Series (SCCTS) dataset from the Knowledge Discovery Database, as well as the thermal copper pillar bump manufacturing dataset, to carry out the experimental analysis to examine the validity of the BGM(1,1) model, and compares the forecasting results with those of two other gray models. The experimental results show that the proposed method significantly improves the model’s accuracy. The remainder of this paper is organized as follows: Section 2 introduces the construction of the GM(1,1), the concept of the box plot, and details of the improved method. Next, Section 3 presents an application of the BGM(1,1). Finally, the conclusions of this work are presented in Section 4. 2. Methodology It is difficult to forecast data trends by using statistical approaches and data mining tool with a small data set, due to the lack of information, and it is this issue that gray system theory seeks to address. In this work, we develop a new modeling procedure, named BGM(1,1), to improve the forecasting performance of the gray approach. This section will present the concept of the BGM(1,1) and its implementation process. 2.1. The original model: GM(1,1) The conventional gray model, GM(1,1), is a small-data-set model for time series data. It employs an accumulating generation operator to obtain a smooth and increasing data series to establish a forecasting model. The GM(1,1) is the most popular gray theory method, as it is relatively simple to use. Therefore, this study will propose an improved building procedure based on the conventional gray model. The construction of the GM(1,1) model is as follows: 1. Given the original time series data set X (0 ) = {x(0 ) (1 ), x(0 ) (2 ), . . . , x(0 ) (n )}, n ≥ 4. 2. Convert the chaotic series X (0 ) into a monotonically increasing series X (1 ) by the accumulating generation operator.

X ( 1 ) = {x ( 1 ) ( 1 ) , x ( 1 ) ( 2 ) , . . . , x ( 1 ) ( n ) }, x ( 1 ) ( 1 ) = x ( 0 ) ( 1 ) ; k x(0) (i ), k = 2, 3, . . . , n x (1 ) ( k ) = i=1

(1)

3. Determine the background values z(1 ) (k ).

z ( 1 ) ( k ) = ( 1 − α )x ( 1 ) ( k − 1 ) + α x ( 1 ) ( k ),

α ∈ (0, 1 ), k = 2, 3, . . . , n

(2)

4. Use Eq. (3) to estimate the developing coefficient a and the gray input b by the least-squares method, and establish the first order gray differential equation from Eq. (4)

x(0) (k ) + az(1) (k ) = b

(3)

dx(1) + ax(1) = b· dt

(4)

The estimated coefficients, [a, b]T , can be evaluated by Eq. (5).

[a, b]T = (BT B )−1 BT Y where

⎡ ⎢ ⎣

B=⎢

−z(1) (2 ) −z(1) (3 ) .. . −z(1) (n )

1 1

(5)

⎤

⎥ ⎥, Y = [x(0) (2 ), x(0) (3 ), . . . , x(0) (n )]T . ⎦ 1 1

5. Use the estimated coefficients a and b together with the initial condition x(0 ) (1 ) = x(1 ) (1 ) to solve Eq. (3), and then the desired forecasting output at step k + 1 can be extracted by Eq. (6).



⎧ ⎨ xˆ(1) (k + 1 ) = x(0) (1 ) − b e−ak + b a a ⎩ (0 ) ( 1) ( 1) xˆ (k + 1 ) = xˆ (k + 1 ) − xˆ (k )

(6)

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C.-J. Chang et al. / Applied Mathematics and Computation 265 (2015) 400–408

Outlier

*

Upper Limit (UL): Q3+1.5(Q3− Q1) Maximum observation Third quartile, Q3

Interquartile range (IQR): Q3− Q1

Median, Q2

First quartile, Q1 Minimum observation Lower limit (LL): Q1−1.5(Q3−Q1) Outlier

* Fig. 1. Box plot.

2.2. The box plot The box plot is a useful graphic approach for displaying some important data features such as the central tendency, dispersion, skewness and potential outliers through three quartiles and the minimum and maximum observations [18]. In a box plot (shown in Fig. 1), the box surrounds the interquartile range (IQR), and its lower and upper bounds are the first and third quartile, respectively. There are two limits in a box plot, the lower limit (LL) and upper limit (UL). The lower limit is defined as 1.5 × IQR lower than Q1 , and the upper limit is defined as 1.5 × IQR higher than Q3 . Any observations outside of these are called outliers [19]. 2.3. The proposed method: an improved gray forecasting model based on the box plot, BGM(1,1) This subsection will explain the main concept of the proposed approach and its building procedure. 2.3.1. A new computing formula for the background value The purpose of the background value in the conventional gray model is to mitigate the fluctuation in data, and it is the most important factor that influences the model’s prediction accuracy. Researchers generally set a fix value α = 0.5 to simplify the evolution procedure. However, this will ignore the valuable information of the data trend, and lead to greater forecasting errors. This study thus further analyzes the influence of α and rewrites Eq. (2) as z(1 ) (k ) = x(1 ) (k − 1 ) + αk x(0 ) (k ), αk ∈ (0, 1 ), k = 2, 3, . . . , n. We can clearly see that the influence of α on the background value is significantly affected by the newest datum. We thus need a proper tool to analyze data features in order to determine the α coefficient. 2.3.2. The building procedure of membership function based on the box plot This paper considers the information obtained from samples taken from the overall data, with the box plot being used to understand the data profile. Based on the concept of the box plot, the reasonable range within which most values are located should be within the two limits, LL and UL. However, the minimum or maximum observations may still fall outside of these two limits, and the estimated limits are thus extended here as



LL =

 UL =

Q1 − 1.5 × IQR min

if Q1 − 1.5 × IQR ≤ min otherwise

(7)

Q3 + 1.5 × IQR max

if Q3 + 1.5 × IQR ≥ max otherwise

(8)

where min and max are the minimum and maximum observations, respectively. To describe the data distribution, fuzzy theory is employed to build an asymmetric triangular membership function (MF), and this is then used to revise the conventional gray model. The detailed building process for the MF is as follows: 1. Compute the three quartiles of the given dataset, Q1 , Q2 , and Q3 , respectively. 2. Determine the upper limit (UL) and lower limit (LL) through Eq. (7) and Eq. (8), respectively. 3. Employ Q2 , LL, and UL to form a triangular membership function (MF). Fig. 2 is a simple example of an MF.

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MF(x)

1 LL

Q2

UL

x

Fig. 2. The asymmetric triangular membership function.

4. Calculate the MF values of the existing observations. Here, we set the MF value of Q2 to be 1, and use the ratio rule of the triangle to find the MF values of the others observations. Consequently, if the given datum is x, then its MF value can be obtained by Eq. (9).

MF(x ) =

⎧ ⎪ ⎪ ⎪ ⎨

1 x − LL Q2 − LL

⎪ ⎪ ⎪ ⎩ x − UL

Q2 − UL

if x = Q2 if x < Q2

(9)

if x > Q2

2.3.3. Determine the background values by the box-plot membership function In gray system theory, the main intention of using the background values is to smooth data for unrandomness. In addition, based on the analysis in Subsection 2.3.1, we know that the background values are closely related to the newest datum, and thus the newest datum is important for the whole data. Therefore, we need a smaller α coefficient if the data series has a higher fluctuation, otherwise a larger α coefficient would be selected to emphasize the importance of the newest datum. The box-plot membership function (BMF) generates MF values according to the data characteristics, and these values represent their degree of typicality in the dataset. A bigger MF value reveals that it is closer to the most of the data, so this datum only has a less influence for whole fluctuation, and thus it is suitable to choose a larger α coefficient, and the reverse is true for a smaller MF value. In summary, the correlation between the α coefficient and the MF value is positive, and their possible values lie in the same interval between 0 and 1. It is thus possible to replace the α coefficient with the MF value to form a revised gray model considering data dynamic information, known as BGM(1,1). The procedure is as follows: 1. 2. 3. 4. 5. 6.

Given n time series data {x(0 ) (1 ), x(0 ) (2 ), . . . , x(0 ) (n )}. Calculate the MF values of these existing data by the BMF as {MFk } = {MF1 , MF2 , . . . , MFn }, k = 1, 2, . . . , n. Apply Eq. (1), AGO, to transform the original series into an increasing series X (1 ) = {x(1 ) (1 ), x(1 ) (2 ), . . . , x(1 ) (n )}. Compute the background values z(1 ) (k ) = x(1 ) (k − 1 ) + MFk × x(0 ) (k ), k = 2, 3, . . . , n. Build the gray differential equation and estimate the gray coefficients a and b using Eqs. (3) to (5). Form the gray model according to the estimated gray coefficients a and b, and substitute k = 2, 3, . . . , n into the model to obtain the desired forecasting outputs.

3. Experimental analyses This research selects one public database and one actual database to verify the effectiveness of the proposed model based on the box plot, and these are the Synthetic Control Chart Time Series (SCCTS) dataset, and the thermal copper pillar bump (TCPB) manufacturing data. The SCCTS dataset contains six different data types that can help to determine what kind the BGM(1,1) model is best suited to. In addition, the other data set is used to further confirm the practicality of using the proposed approach in manufacturing industries. The detailed experimental process is described below. 3.1. Synthetic control chart time series data The SCCTS dataset is a public database obtained from the Knowledge Discovery Database (http://kdd.ics.uci.edu/ databases/synthetic_control/synthetic_control.html), and it contains the most common data types from various manufacturing environments. The SCCTS dataset has a total of 600 observations of synthetic control charts, and it is separated into six different categories, namely normal, cyclic, increasing trend, decreasing trend, upward shift and downward shift. There are 100 examples in each category, and every example has 60-stage data. All 600 examples are used for the experimental analysis, and one part of this dataset is in Table 1. To clearly grasp the data profile, we use descriptive statistics to implement the pre-test, and the results are shown in Table 2. We find that the normal type owns the minimum standard deviation, and the cyclic type has the maximum one. The big amplitude implies that the cyclic data is relatively unstable among all the data types, and this may impact the forecasting performance. Because the gray approach is specifically designed for data analyses with very small data sets, we thus suppose that each example has only four data, {x1 , x2 , x3 , x4 }, and the task is thus to forecast the next observation, x5 .

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C.-J. Chang et al. / Applied Mathematics and Computation 265 (2015) 400–408 Table 1 Part of the SCCTS data. No.

1 2 3 4 5 6 7 8 9 . . . 598 599 600

Time stage 1

2

3

4

5

28.7812 24.8923 31.3987 25.7740 27.1798 25.5067 28.6989 30.9493 35.2538

34.4632 25.7410 30.6316 30.5262 29.2498 29.7929 29.2101 34.3170 34.6402

31.3381 27.5532 26.3983 35.4209 33.6928 28.0765 30.9291 35.5674 35.7584 . . . 34.1911 28.2814 31.9529

31.2834 32.8217 24.2905 25.6033 25.6264 34.4812 34.6229 34.8829 28.5510

28.9207 27.8789 27.8613 27.9700 24.6555 33.8000 31.4138 30.6691 25.6518

35.8990 24.5383 34.3354

26.6719 24.2802 30.9375

...

...

.. 35.8270 27.1316 31.1457

25.1009 26.6623 24.5189

.

...

60 25.8717 26.6910 29.3430 25.3069 31.0179 35.4907 26.4637 34.5230 32.3833 . . . 17.4747 17.4599 10.1521

Table 2 The descriptive statistics of the SCCTS data. Data types

Average

Standard deviation

Normal Cyclic Increasing trend Decreasing trend Upward shift Downward shift

30.01 30.38 40.25 19.53 37.17 23.21

3.48 9.54 7.41 7.67 8.16 7.90

3.2. Thermal copper pillar bump manufacturing data Thermal copper pillar bumps (TCPB) are advanced thermoelectric devices which are embedded on thin-film thermoelectric materials in flip chips for use in electronics and optoelectronic packaging. Unlike conventional solder bumps, which provide an electrical path and a mechanical connection to the package, a thermal bump acts as solid-state heat pump and adds thermal management functionality locally on the surface of a chip. A thermal bump can directly use the thermoelectric effect to convert temperature differences to electric voltage and vice versa. This effect can be used to generate electricity, to measure temperature, to cool objects, or to heat them. Thermal bumps mean that transistors, resistors and capacitors can be integrated in conventional circuit designs, which is an important process in the packaging industry. The TCPB process relies on the combination of many different materials, and the physical properties of the TCPB will be affected by the rigidity, the thermal expansion coefficient, and the temperature variation of materials used. Solder balls and copper pillars are the key mediators in TCPB process which enable signal transduction, heat transmission, and absorption of dilatant difference between components. The reliability of these two elements thus has a major impact on the packaging process, and so they are very important and need to be effectively controlled to maintain a good yield. In addition, solder balls are used for surface mounting in general, because of their low melting point. However, they can suffer from both creeping and plastic behavior in the mounting step under common operation conditions, and this can cause the height of the solder balls to be unstable, thus impairing the functions of the final products. Therefore, how to effectively assess the critical dimension of solder balls and the critical dimension of copper pillar is a key focus of this study. The real observations in this study were collected from a leading packaging manufacturer in Taiwan in September of 2011. The case company wanted to better understand the new TCPB process by carrying out post-installation testing, but had only very limited data to work with. Fifteen lots of experimental data were obtained in this pilot run, and the details are listed in Table 3. In this sheet, CD1 is the critical dimension of the solder balls and CD2 is the critical dimension of the copper pillars, and the unit of measurement for each of these is micrometers (μm). Both CD1 and CD2 needed to be inspected on five different points, so a total of 150 observations were collected. Table 4 is the descriptive statistics of the TCPB manufacturing data, where we find that the TCPB data is a steady data because of its smaller standard deviation, and this feature results in small forecasting errors. The aim of this study is to develop an appropriate forecasting model to figure out the variations in this process over time. It is thus assumed that each example has only four data, and the forecasting model then uses these to predict the next datum.

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Table 3 TCPB manufacturing data. Lot

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CD1

CD2

1

2

3

4

5

1

2

3

4

5

53.00 54.00 54.40 52.20 54.30 53.30 53.00 52.20 54.30 53.94 54.68 54.43 54.00 54.19 54.00

53.40 53.10 54.60 53.60 54.90 54.50 53.80 53.10 54.90 53.20 54.19 53.45 54.80 54.43 53.00

53.70 53.40 54.10 53.20 53.90 54.00 53.70 53.10 53.90 53.45 54.43 53.45 54.00 53.69 53.20

53.40 53.00 53.70 52.40 54.00 54.20 53.10 53.90 54.00 53.94 53.94 54.43 53.32 53.69 53.80

53.50 52.40 54.40 53.20 53.00 54.30 53.40 52.60 53.00 53.45 53.94 53.20 53.71 54.43 53.20

52.00 53.10 53.10 50.40 51.20 51.40 51.70 51.90 51.20 50.70 50.80 51.00 50.80 50.90 52.10

51.40 52.70 52.80 50.00 50.30 51.70 51.40 51.90 50.30 51.00 51.00 50.90 51.00 51.10 51.50

52.00 53.00 52.10 50.20 51.20 51.50 51.10 51.60 51.20 50.90 51.10 50.80 51.10 51.20 52.00

51.70 53.00 53.10 50.20 50.40 51.00 50.80 51.20 50.40 50.80 51.00 51.00 51.00 50.80 51.90

51.20 53.10 52.80 50.20 50.30 51.20 51.70 51.50 50.30 51.20 50.70 50.90 50.70 50.80 51.90

Table 4 The descriptive statistics of the TCPB manufacturing data.

CD1 CD2

Average

Standard deviation

53.69 51.32

0.62 0.79

Table 5 Computation of the BGM(1,1). Time stage

xk

MFk

x1(1)

zk(1)

xˆk

1 2 3 4 5

28.7812 34.4632 31.3381 31.2834 28.9207

1.0000 0.5000 1.0000 0.9904 –

28.7812 63.2444 94.5825 125.8659 –

– 46.0128 94.5825 125.5652 –

– 34.1621 32.7528 31.4016 30.1061

x(1) (k + 1 ) = (−828.0930 )e−0.0421k + 856.8742

3.3. Comparison methods To further verify the effectiveness of the proposed method, the results of using it are compared with those of two other gray approaches: one is the conventional gray model, GM(1,1), the most representative model in gray system theory, while the other is the discrete gray forecasting model (DGM) developed by Xie and Liu [20], which is one of the most accurate gray models. The same four data as were used with the BGM(1,1) model are used to create GM(1,1) and DGM models to predict the corresponding value xˆ5 , and the building parameter α in GM(1,1) is fixed at 0.5. 3.4. Modeling of the BGM(1,1) We use the first four observations of the first subset in the SCCTS dataset to create a BGM(1,1) model. The details of this process are described below, and the computations are listed in Table 5. 1. 2. 3. 4. 5.

The original data set is X (0 ) = {28.7812, 34.4632, 31.3381, 31.2834}. Use the BMF to calculate the MF values and obtain MF = {1, 0.5, 1, 0.9904}. Apply AGO to create a new series X (1 ) = {28.7812, 63.2444, 94.5825, 125.8659}. Find out the background values Z (1 ) = {46.0128, 94.5825, 125.5652}. Estimate the gray coefficients aˆ = (BT B )−1 BT Y = [0.0421, 36.0993]T , where



T

Y = [34.4632, 31.3381, 31.2834]

and

B=



−46.0128 1 −94.5825 1 . −125.5652 1

6. Use Eq. (6) to create the model x(1 ) (k + 1 ) = (−828.0930 )e−0.0421k + 856.8742 which is then used to predict the next datum, which is xˆ5 = 30.1061.

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C.-J. Chang et al. / Applied Mathematics and Computation 265 (2015) 400–408 Table 6 Performance of forecasting methods of SCCTS data. Methods

MSE

MAE

MAPE (%)

SCCTS BGM(1,1) GM(1,1) DGM

40.3199 46.6526 45.9792

4.8458 5.2965 5.2471

15.5639 16.9975 16.8108

TCPB BGM(1,1) GM(1,1) DGM

0.8707 1.1759 1.1717

0.7129 0.8416 0.8301

1.3543 1.6034 1.5811

Table 7 Improvement between BGM(1,1) and GM(1,1) in the SCCTS database. Methods

GM(1,1)

BGM(1,1)

Improvement (%)

Normal (1–100) MSE 33.3475 MAE 4.6007 MAPE(%) 15.5389

29.6988 4.2305 14.3568

12.66 8.75 8.23

Cyclic (101–200) MSE 93.1094 MAE 7.7753 MAPE 20.1112

81.3966 7.0421 18.2356

14.39 10.41 10.29

Increasing trend (201–300) MSE 50.5916 MAE 5.7804 MAPE 19.1527

48.0004 5.5785 18.3827

5.40 3.62 4.19

Decreasing trend (301–400) MSE 40.4088 MAE 5.0819 MAPE 17.9345

32.5055 4.5699 16.1682

24.31 11.20 10.92

Upward shift (401–500) MSE 33.1443 MAE 4.4765 MAPE 15.2991

27.4199 4.0070 13.7520

20.88 11.72 11.25

Downward shift (501–600) MSE 29.2040 MAE 4.0639 MAPE 13.9488

22.8979 3.6468 12.4878

27.54 11.44 11.70

Average MSE MAE MAPE

40.3199 4.8458 15.5639

15.71 9.30 9.21

46.6526 5.2965 16.9975

3.5. Measurement of the forecasting performance Yokum and Armstrong [21] indicated that a model’s accuracy can help researches to determine whether or not it is an appropriate forecasting tool. Liu and Lin [22] pointed that when testing whether the gray model is an appropriate approach, it is necessary to use a forecasting error, with the mean absolute error being the most frequently used indicator for this. We thus select three common measuring indices to evaluate the performances of the forecasting methods used in this study, and these are the mean absolute percentage error (MAPE), the mean squared error (MSE), and the mean absolute error (MAE). Since MAPE is a relative percentage of errors corresponded to the number of real observations, it can help engineers to understand the risks associated with using the forecasting tool. The MSE and MAE are both show the average degrees of errors, with the former being very sensitive with regard to bigger errors, and thus it can obtain more information about the forecasting precision of a particular model. With yˆi and yi are the predictive output and the actual observation, respectively, the MAPE, MSE and MAE are defined below.





1 m  yˆi − yi  MAPE = × 100% i=1  m yi 

(10)

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407

MSE =

1 m (yˆi − yi )2 i=1 m

(11)

MAE =

 1 m  yˆi − yi  i=1 m

(12)

96

GM(1,1)

BGM(1,1)

84

72 60 48 36 24 12 0

Normal

Cyclic

Increasing Decreasing Upward Downward Average trend trend shift shift

Fig. 3. Bar chart of MSE values in the SCCTS database.

8

GM(1,1)

BGM(1,1)

7 6 5 4 3 2 1 0 Normal

Cyclic

Increasing Decreasing Upward Downward Average trend trend shift shift

Fig. 4. Bar chart of MAE values in the SCCTS database.

24

GM(1,1)

BGM(1,1)

18

12

6

0 Normal

Cyclic

Increasing Decreasing Upward Downward Average trend trend shift shift

Fig. 5. Bar chart of MAPE(%) values in the SCCTS database.

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3.6. Comparisons among the forecasting methods Table 6 shows a summary of the forecasting results. It indicates that the BGM(1,1) has the best forecasting performance among the three models in all tree measurements, clearly demonstrating that it is an effective forecasting method for small data sets. Table 7 shows that compared with the original GM(1,1) model, the improvement of BGM(1,1) with regard to MSE is over 15%, showing that the proposed model can enhance the conventional gray model, and achieve better forecasting outcomes by more accurately reflecting the data characteristics. The performance of BGM(1,1) was then analyzed using six different data types in the SCCTS database (Table 7), and the results showed that the new model is superior to GM(1,1) with all six types of data. Therefore, the BGM(1,1) model is considered an appropriate forecasting tool for use with small data sets (Figs. 3–5). 4. Conclusions and discussion A significant issue for managers of manufacturing firms is how to effectively control the early stages of a manufacturing system, as if any adverse phenomena are detected at this time, some preventive actions can be taken before many faulty products are produced. However, one problem that arises in this situation is the limited amount of data available which can be used to make predictions. To solve the small data set forecasting problem, this article presents a procedure to improve the performance of the conventional gray forecasting model. The proposed method, known as BGM(1,1), first applies a box plot to analyze the features of the collected observations, and then builds a calculating formula using in the background values. This approach can enhance the ability to obtain the data trend, and thus lead to more accurate forecasting results. The experimental results show that the BGM(1,1) model can produce satisfactory results that are better than those obtained with two widely used models, and thus is a useful tool for manufacturing firms. Future studies may apply this method to other real-world industrial cases to confirm its effectiveness. Acknowledgment This research is partially sponsored by the K.C. Wong Magna Fund in Ningbo University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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