developing defuzzifying method of fuzzy time-variant
TRANSCRIPT
DEVELOPING DEFUZZIFYING METHOD OF FUZZY TIME-VARIANT
SERIES FOR FORECASTING PRODUCT DEMAND
SAEED AKBARNATAJBISHEH
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Engineering (Industrial Engineering)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
JUNE 2013
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2 DEDICATION
It is my honor to dedicate this thesis to my kind parents who have always been
my motivator and supporter in each step of my life. I also dedicate this work
respectfully to my elder brother, Abed, and my dear sister, Yeganeh.
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3 ACKNOWLEDGEMENT
All praises are to Allah Al-Mighty for his grace and blessing and bestowing
me with health and opportunity to gain this treasure of knowledge. Apart from the
efforts of author, the success of any research project depends largely on the
encouragement and guidelines of many others. I take this opportunity to officially
express my gratitude to the people who have been instrumental in the successful
completion of this thesis.
Foremost, I would like to express my sincere gratitude to my supervisor, Dr.
Syed Ahmad Helmi Bin Syed Hassan for the continuous support of my Master study
and research, for his patience, motivation, enthusiasm, and immense knowledge. His
guidance helped me in all the time of research and writing of the thesis. Besides my
supervisor, I would like to appreciate one of my best friends, Seyed Navid Seyedi for
his assistance and comments on some parts of this thesis.
Last but not the least; I would like to thank parents for their selfless love and
their supports throughout my life.
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4 ABSTRACT
This thesis is about proposing a method of defuzzifying fuzzy time-variant
series on forecasting demand by developing time-variant model considering Mean
Absolute Error (MAE) and trend of primary forecasting (MAE&Trend). The
previous method of defuzzifying was based on Artificial Neural Network (ANN).
Defuzzifying by ANN needs model identification which is time consuming. To
justify the robustness of the proposed method, it is compared with Song‘s and Chen‘s
fuzzy time-invariant methods as well as Autoregressive Integrated Moving Average
(ARIMA) time series method. Advantage of fuzzy method is its robustness in
covering large range of data including linguistic data as well as numerical and its
ability to forecast by small amount of data. Accuracy of the methods is achieved by
comparing Mean Absolute Percentage Error (MAPE). The MAPE results are
modified by omitting outranged data to have more logical evaluations. Based on the
evaluations, the proposed model of MAE&Trend shows better accuracy in
forecasting demand.
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5 ABSTRAK
Kajian ini mengkaji kaedah defuzzifying siri masa fuzzy varian untuk
meramal permintaan dengan menghasilkan model siri masa yang mempertimbangkan
purata ralat mutlak (MAE) dan dan corak ramalan utama (MAE&Trend). Kaedah
sebelum ini menggunakan artificial neural network (ANN). Namun begitu,
defuzzifying menggunakan ANN memerlukan proses pengenal-pastian model dan ini
memakan masa. Bagi mengenalpasti keteguhan model yang dikaji, ia dibandingkan
dengan siri masa fuzzy invariant oleh Song dan Chen dan Autoregressive Integrated
Moving Average (ARIMA) model. Kelebihan kaedah fuzzy ialah ia mampu meramal
untuk pelbagai jenis data termasuk data nombor dan data bertulis dan ia mampu
meramal walaupun data input adalah sedikit. Ketepatan kaedah yang dikaji dicapai
dengan membandingkan purata peratusan ralat mutlak (MAPE). Keputusan MAPE
diubahsui dengan mengecualikan data di luar lingkungan untuk memperoleh
penilaian yang lebih logical. Berdasarkan keputusan MAPE, model yang dikaji, iaitu
(MAE& Trend) menunjukkan ketepatan yang lebih baik dalam meramal permintaan.
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6 TABLE OF CONTENT
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENT vii
LIST OF TABLES xi
LIST OF FIGURES xiii
LIST OF ABBREVIATIONS xiv
LIST OF APPENDICES xv
1 INTRODUCTION 1
1.1 Introduction 1
1.2 Background 1
1.3 Problem Statement 2
1.4 Objective 3
1.5 Scope 3
1.6 Summary of Literature 4
1.7 Conceptual Framework 4
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1.8 Significance of the Study 5
1.9 Overview and Conclusion 6
1.10 Conclusion 7
2 LITERATURE REVIEW 8
2.1 Introduction 8
2.2 Fuzzy Time series forecasting 8
2.2.1 Fuzzy logic 9
2.2.2 Fuzzy Set 9
2.2.3 Fuzzy Time Series Concept 10
2.2.4 Modeling the Fuzzy Time Series 16
2.2.5 Fuzzy Time-invariant Method 17
2.2.6 Fuzzy Time-variant Method 22
2.3 Artificial Neural Network (ANN) 27
2.3.1 Biological Pattern of ANN 27
2.3.2 Neural Network Model 28
2.3.3 Multilayer Neural Network 30
2.3.4 Back Propagation Networks 31
2.3.5 Advantages and Disadvantages of ANN 32
2.4 ARIMA 33
2.4.1 ARIMA Model 33
2.4.2 Steps of Applying ARIMA Model 35
3 RESEARCH METHODOLOGY 38
3.1 Introduction 38
3.2 Research Framework 38
3.3 Methodology 40
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3.3.1 Error Estimation Methods 40
3.3.2 Defuzzifying Fuzzy Time-variant Series by
MAE&Trend 41
3.3.3 Proposed Defuzzification Method 43
3.3.4 Method Validation 44
3.3.5 Comparing Defuzzifying of Proposed Method
and Neural Network 48
3.4 Conclusion 49
4 DATA GATHERING AND ANALYSIS 50
4.1 Introduction 50
4.2 Summary of case study 51
4.3 Data Collection 51
4.4 Data Analysis for Fuzzy Time series 52
4.5 Conclusion 56
5 RESULTS AND CONCLUSION 57
5.1 Introduction 57
5.2 Fuzzy Time-invariant Method 57
5.2.1 Song‘s Method Results 58
5.2.2 Chen‘s Method Results 60
5.3 Fuzzy Time-variant Method 62
5.3.1 Results of Defuzzifying by Proposed Model of
MAE&Trend 62
5.3.2 Results of Defuzzifying by Neural Network 65
5.4 Results of ARIMA Model 69
5.5 Comparison of Applied Methods 72
5.5.1 Comparison of MAPE 72
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5.5.2 Graphs of Actual Demand and Forecasted
Demand for All Methods 74
5.6 Conclusion, Significance of the Study and
Recommendations 77
REFERENCES 79
APPENDIX A 82
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7 LIST OF TABLES
TABLE TITLE PAGE
2.1 Vectors of memberships of the intervals of fuzzy sets 14
2.2 Enrollments and memberships of (Song and Chissom 1993) 15
2.3 Defuzzified values of fuzzy time-invariant model (Song and
Chissom,1993) 19
2.4 Fuzzy logical transitions group (Chen 1996) 21
2.5 Output results for time-invariant model (Song and Chissom,1994) 24
3.1 MAPE of time-invariant for 42
3.2 Primary forecasting of fuzzy time-variant model 43
3.3 Proposed time-variant forecasting 44
3.4 Results of Assumption (a) 45
3.5 Results of Assumption (b) 46
3.6 Results of Assumption (c) 47
3.7 Comparing MAPE of proposed model and neural network results 48
4.1: Bina Paint Company's Products 51
4.2 Actual demand from 2009 to 2012 52
4.3 Membership vectors of fuzzy time-invariant series 55
4.4 Fuzzified Demand from 2009 to2011 56
5.1 Song‘s fuzzy time-invariant method‘s forecasting 59
5.2 Group of transitions 60
5.3 Calculation of forecasting for each group of Chen‘s method 60
5.4 Chen‘s method forecasting results 61
5.5 Fuzzy membership outputs and primary forecasting 63
5.6 Calculation of MAE 64
5.7 Defuzzification by proposed model 65
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5.8 Training data for the network 66
5.9 Output of defuzzifying by neural network 68
5.10 ARIMA forecasting results 72
5.11 MAPE comparison 73
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8 LIST OF FIGURES
FIGURE TITLE PAGE
1.1 Conceptual Framework 5
2.1 Neuron scheme (Jain, Mao et al. 1996) 28
2.2 Neuron model (Negnevitsky 2005) 28
2.3 Graph of three different Activation functions (Negnevitsky 2005) 30
2.4 Multilayer perceptron with two hidden layers (Negnevitsky 2005) 30
2.5 Three-layer back-propagation neural network (Negnevitsky 2005) 32
3.1 Research framework 39
4.1 Membership function plot 54
5.1 Neural network model 67
5.2 Autocorrelation of error 67
5.3 Correlation between input and error 68
5.4 ACF for actual demand 70
5.5 PACF for actual demand 70
5.6 Residual plots for ARIMA model 71
5.7 Time-invariant methods‘ plot 74
5.8 Time-variant and ARIMA plot 75
5.9 ARIMA results 75
5.10 Proposed MAE&Trend method plot 76
5.11 Neural network‘s output 76
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9 LIST OF ABBREVIATIONS
MAE Mean Absolute Error
MAPE Mean Absolute Percentage Error
ANN Artificial Neural Network
ARIMA Autoregressive Integrated Moving Average
ACF Autocorrelation Function
PACF Partial Autocorrelation Function
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10 LIST OF APPENDICES
APPENDIX TITLE PAGE
Appendix A Work Report Evaluation Form for Bina Paint Manufacturing
Company 82
CHAPTER 1
1 INTRODUCTION
.
1.1 Introduction
In this chapter firstly background of study is explained which is mostly about
anticipating future demand with historical data. Secondly, potential problems are
stated which is generally about limitations of using neural network for defuzzifying
fuzzy time-variant method. In addition, reaching new way of defuzzifying fuzzy
time-variant method with mean absolute error (MAE) and trend of primary
forecasting with better accuracy is stated for the objective. After defining scope of
this thesis, a brief summary of literature as well as conceptual framework and
significance of study is explained. Finally, in thesis organization, the overview of
each chapter is given.
1.2 Background
Predicting future demand is one of the most crucial concerns of managements
in a company. Moreover, accuracy of the forecast and adapting the prediction to the
real world play a vital role in forecasting. To have more accurate forecast, more
historical data should be gathered. Since data ,in this case, are categorized into
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linguistic and numerical data the most useful approach to deal with data is fuzzy
logic (Song and Chissom, 1993b) which is able to use both types of data as an input
and produce fuzzy data as output. Fuzzy time series are classified into two
categories, fuzzy time-invariant and fuzzy time-variant series(Song and Chissom,
1993b). Defuzzifying of fuzzy time-invariant series is applied by two methods,
Song‘s method(Song and Chissom, 1993a) and Chen‘s method(Chen, 1996); but the
only defuzzification method for fuzzy time-variant series is by neural network (Song
and Chissom, 1994). Neural network uses fuzzy memberships as input as well as
actual data for target values. The major disadvantage of neural network is the process
of model identification. It means that you may not reach the best result in the first
train. Hence, it is proposed a new way of defuzzification for fuzzy time-variant series
to cope with this problem.
Forecasting by various methods gives us the chance to evaluate that which
method has better accuracy. ARIMA method is selected to be compared with the
proposed method because of its robustness and dealing with demand fluctuation over
time (Wang, 2011). It is also able to cover time series with seasonality and
nonseasonality. Making the best decision among results could be achieved by
suitable comparison between methods. Hence, the results of methods would be
compared by MAPE (mean absolute percentage error). Output of MAPE is
percentage of error which is difference between actual data and forecasted data.
1.3 Problem Statement
Defuzzification of time-variant model was by applying neural network so far.
One of the neural network disadvantages in defuzzifying is that it needs model
identification. Identifying the best model in neural network is time consuming. In
addition, MAPE of defuzzifying by ANN in time-variant series is higher than time-
invariant series.
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In this paper, reaching a method to defuzzifying time-variant series is
targeted that lead to less error in comparison with ANN. Dealing with this problem it
is proposed new defuzzification method based on MAE and trend of forecasting.
Considering MAE as well as trend of time series would affect positively on accuracy
that means reaching less error.
1.4 Objective
In this project, it is proposed a new way of defuzzifying fuzzy time-variant
method which firstly proposed by Song and Chissom (1994). The previous method
has utilized neural network for defuzzifying that has some limitations and
disadvantageous such as model identification which would be explained in further
sections. Proposed defuzzification method of fuzzy time-variant method is developed
by consideration of MAE and trend of primary forecasting.
Comparison of proposed model by ARIMA and previous method of
defuzification would be applied by MAPE to determine the accuracy of proposed
model.
1.5 Scope
The scope of this project is as follow:
a) Data has been collected in paint manufacturing company ―BINA
Integrated‖.
b) Fuzzy time series and ARIMA model are applied on historical data.
c) MATLAB and MINITAB software is used for getting results.
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1.6 Summary of Literature
Fuzzy time series was proposed in 1993 by Song et al. as ―a special dynamic
process with linguistic values‖ (Song and Chissom, 1993b) that is able to forecast
future by use of fuzzy sets and variables. Two main categories of fuzzy time series
are fuzzy time-invariant and fuzzy time-variant series: In fuzzy time-invariant series,
for example, forecasting month is done by use of data of , but in fuzzy time-
variant series sets of data is considered for forecasting month . In
addition defuzzifying methods of these two categories are different; two methods of
defuzzifying fuzzy time-invariant series, Song‘s method and Chen‘s method, are
defined in this project as well as one method of defuzzifying fuzzy time-variant
series which is defuzzification by neural network. Neural network is a system, in
which an input is presented to this system, and a corresponding desired response is
set at the output, an error is extracted from the difference of the desired response and
the real system output.
One of the issues in forecasting is comparison various methods to determine
that which one has the best accuracy or lower MAPE (mean absolute percentage
error); hence, another method which is considered to be compared with fuzzy time
series, is ARIMA (Autoregressive integrated moving average) model. This method is
proposed by Box and Jenkins (1970) and it is a linear combination of pervious points
(past observations or historical data) and random errors.(GE, 1970)
1.7 Conceptual Framework
The proposed method starts by defining parameter of time-variant method
( ); then fuzzy membership of fuzzy time-variant is extracted. Fuzzy memberships
are used to calculating primary forecasting ( by applying defuzzifying rules of
time-invariant (Song‘s method). Next, calculate mean absolute error ( ) of time-
invariant forecasting in period of . Trend ( ) of the primary forecasting is another
5
factor which is considered. Finally, the results are defuzzified values (forecasting) for
time-variant method by multiplying , and .
MAE
Trend
Defining time-variant
parameter (w)
Fuzzy time-
invariant
Primary
forecasting
[PF(i)]
Fuzzy
membership
Fuzzy time-
variant
Forecasting
Figure 1.1: Conceptual Framework
1.8 Significance of the Study
This study is mainly determined some peroperties of fuzzy time variant
method and proposed a new defuzzification method of fuzzy time-variant series. The
proposed model is based on mean absolute error (MAE) and trend of primary
forecasting while the previous method of defuzzifying fuzzy time-variant was by use
of neural network. In addition, it is found that there is a limitation of defining
parameter of time-variant method ( ), as well as the easier way of calculating
(matrix) which lead to shorten the time of calculation in fuzzy forecasting. For
calculating MAPE of each method, it is asserted that if outranged data be omitted,
the comparison is more logical; because by omitting outranged data, diverse effect of
external factors are reduced.
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1.9 Overview and Conclusion
This thesis is inclusive of 5 chapters; the first one is generally gives an
overview of project; in this chapter introduction and background of project are stated
then set the objective due to stated problem. The objective is proposing new method
of defuzzification for fuzzy time-variant series. To manage the project, some
methods and software for the scope of this thesis are considered. Significance of
study and theoretical framework are brought to give us the image of what have been
done in this thesis.
Chapter 2 is about reviewing the literatures for the topic; in this part,
background of project is explained more and definitions and application of each
method is clarified. Some advantages and disadvantages of each method are
discussed according to related papers.
In Chapter 3 the properties of the proposed model are explained. Then the
methodology of applying new way of defuzzification for fuzzy time-variant series is
clarified. Finally, the result of applying proposed model for enrollment for University
of Alabama are compared with neural network defuzzification method.
Chapter 4 is dedicated to data collection and analysis. Demand of paint
manufacturing company is gathered to justify the robustness of the proposed method.
Since this data is fluctuated, performance of any methods would be measured by
comparing the MAPE of results.
In the last chapter, results of each method are calculated. In addition, some
constraints of fuzzy time series such as determining (parameter of time-variant
method) are defined. To reduce the diverse effect of external factors outranged data
are omitted. The meaning of outranged data and corresponding reasons are stated.
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Finally results of methods are compared by MAPE (mean absolute percentage error)
in which the lower MAPE of method illustrates its better accuracy in forecasting.
1.10 Conclusion
The overall view of this thesis is mentioned in the first chapter. Defining
objective and problem statement are two important parts of this chapter. The
conceptual framework shows then map of proposed method of defuzzifying time-
variant series. In next chapter definitions of each method as well as literature review
will explain more to clarify application of methods.
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