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

iii

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.

vii

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

viii

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

x

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

xii

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

xiii

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

xv

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

2

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.

3

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.

4

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.

6

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.

7

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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