Fuzzy Set Theory
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Applications of Fuzzy Logic in Decision Making Amira Suliman Haron Ibrahim B.Ed.(Honor) in Mathematics and physics, University of Gezira (2011) Post graduate Diploma in Mathematical ,University of Gezira (2013) A Dissertation Submitted to the University of Gezira in Partial Fulfillment of the Requirements for the Award of the Degree of Master of Science in Mathematics Department of Mathematics Faculty of Mathematical and Computer Sciences February / 2015 1 2 Declaration This is to certify that I was completed my research by my self ,my effort and using references , without taking any information from others .This research was taken under supervision of my advisors. Name: Amira suliman Haron 3 iii DEDICATION To My father The light that illuminates the path of success. To My mother(Aziza) Which provide me with tenderness and love . To My brothers AND My sister. iv 4 ACKNOWLEDMENT First thank to Allah for giving me the ability to do this work .I would like to thank my main supervisor Dr. Abdulla Habila for his advise and did not stingy any information for me . Thank to the Co. supervisor Uz. Bakhit Osman Bakhit and all the staff in the faculty of Mathematical and computer sciences .I grateful to every one who helped me in preparing this study. Thanks to my family for their help and support. v 5 Applications of Fuzzy Logic in Decision Making Amira Suliman Haron Ibrahim Abstract Logic is a small part of the human thinking capability. It's can help us to organize words and make a clear sentences. From this perspective, it is clear that fuzzy logic is a method to formalize the human capability of imprecise thinking and it is a generalization of classical logic. It is considered a form multi-values logic derived from the fuzzy set theory. The fuzzy logic introduced in 1965 by the Iranian scholar Lotfi Zadeh, and it allows the definition of the intermediate values between the conventional evaluations like true/false. The fuzzy logic focuses on inferences through indefinite expressions and lingual pronunciations; therefore it solves the marginality problem. The scientific purpose of the fuzzy logic is to solve the problem of the representation of the approximate or indefinite information and to provide the required instrument to utilize knowledge and human expertise. The fuzzy logic importance is represented by its ability to deal with unclear information and controlling unstable systems which assists in finding solutions for many of the problems in engineering applications that cannot be solved by the classical logic. That will be done by attributing the variable degree from the real field [0,1] instead of {0,1}. The concept of the fuzzy set is considered one of the supportive principles in studying the fuzzy logic. Also, one of the important concepts is the fuzzy relations and the fuzzy rule base. It has many applications in our work lives, most of which we find in things like artificial intelligence or the advanced electronic devices. In this research it has been applied in medical diagnosis and the decision making process will be presented. As the concept of fuzzy rule base was utilized in the diagnosis of diseases because it contributes in organizing inferences when facts are not precise which assists doctors in determining the amount of the appropriate drug dose. Although the symptoms that the patients reports like high body temperature, high blood pressure cannot be taken by fuzzy values because the patient may exaggerate them in accordance with his/her feelings , however the analysis results are in most cases fuzzy. The values of investigation results are into logical equations to obtain fuzzy results. Fuzzy logic has also contributed in the decision making process by enabling individuals to take the optimal decision in issues characterized by ambiguity. It has effective results in linear programming. Two problems which cannot be directly solved by classical linear programming were solved. This shows its preference over the classical logic in engineering applications and in day-to-day life. It has other usages like whether identification, taking fair decision in competitions, in some electronic home appliances such as electric washing logic machines, ventilation, etc. The study recommends more concern with further studying fuzzy and striving for its development and expanding its usages. vi 6 تطبيقات المنطق الضبابي في صنع القرار أميرة سليمان هارون إبراهيم ملخص الدراسة المنطق هو جزء صغير من قدرة اﻹنسان على التفكير. إنه يمكن أن يساعدنا في تنظيم الكلمات لجعللا الجمللا ةا للضح. ةا لل مللن هلللا المنظللور المنطللق الضللءااي هللو ري للح ﻹ للفاء الطللاا الرسللمي علللى قللدرة اﻹنسللان علللى التفكيللر ديللر الللدقيق ةهللو تعملليم للمنطللق الك سلليكي. ةيعتءللر صورة لمنطق ال يم المتعددة المشت ح من نظريح المجموع الضءاايح. ةقد الدأ المنطلق الضلءااي فلي عللا 1965 مللن قءللا العللالم اﻹيرانللي لطفللي يادة ةأنلله يسللم اتعريلل ال لليم المتوسللطح اللين ال لليم الت ليديح مثا صضي /خا ئ. ةيركز المنطق الضءااي على اﻻستنتاج من خل التعلااير ةافلفلا اللغويح دير المضددة، لللك يعالج مشكلح الضديح، إن الغرض العلمي من المنطق الضءااي هلو للا مشللكلح تمثيللا المعلومللات الت ريءيللح أة ديللر المضللددة ةتللوفير اتليللح ال يمللح ﻻسللت دا المعللار ةال ءرات الءشريح. تتمثلا أهميتله فلي قدرتله عللى التعاملا مل المعلوملات الغامضلح ةاللتضكم فلي افنظمح دير المست رة، مما ساعد في إيجاد الضلو لكثير من المسائا في التطءي ات الهندسيح التي ﻻ يمكن للها االمنطق الك سيكي ةذلك اإسناد درجح المتغير من المجا الض ي لي [1,0 ]الدﻻ ملن {0,1} .يعللد مفهللو المجموعللح العائمللح مللن افسلل الداعمللح فللي دراسللح المنطللق الضللءااي ةمللن المفاهيم المهمح أيضا الع قلات الضلءاايح ةأسلال ال اعلدة الضلءاايح. لله العديلد ملن التطءي لات فلي لياتنللا العمليللح يالتللي ادلءهللا نجللدء فللي افكللياء مثللا اللللكاء اﻻصللطناعي أة افجهللزة اﻹلكترةنيللح المت دمح .في هلا الءضث ُءق في التش يص الطءي ةعمليح ات اذ ال رار يليث تم است دا مفهلو أسال ال اعدة الضلءاايح فلي تشل يص افملراض ةذللك فنله يسلهم فلي تنظليم اﻻسلتنتاجات عنلدما تكون الض ائق دير مضءو ح مما يساعد الطءيب في تضديلد م لدار جرعلح اللدةاء المناسلءح الالردم ملن أن افعلراض التلي يللكرها الملريل مثلا درجلح للرارة الجسلم. لغي اللد ﻻ يمكلن أخللها اال يم الضءاايح فن المريل قد يءالغ فيها لسب كعورء إﻻ أن نتائج التضاليا هلي أليانلا لءاايح. ةيتم إدخا قيم الفضوصات من معادﻻت منط يح للضصو على نتائج ءاايح. ةقلد سلاهم أيضلا في عمليح ات اذ ال رار اتمكين اففراد من ات اذ الضا افمثا في المسلائا التلي يعتريهلا الغملوض. ةله نتائج فعالح في الءرمجح ال طيح ةتم لا مسألتين ﻻ يمكن للهما االءرمجح ال طيلح الك سليكيح مءاكرة، مما أ هر أفضليته على المنطق العادي في التطءي ات الهندسيح ةفلي الضيلاة اليوميلح. ةلله اسلت دامات أخللرث مثللا تضديللد الط لل ي ات للاذ ال للرار العللاد فللي المسللاا اتي فللي اعللل افجهللزة اﻹلكترةنيللح المنزليللح مثللا الغسللاﻻت الكهرايللح ةالتكييلل .. الللة. توصللي الدراسللح ااﻻهتمللا أكثللر ادراسته ةالسعي في تطويرء ةتوسي است داماته . vi 7 Table of Contents Chapter One Fuzzy set 1.1 Introduction…………………………………………..…1 1.2 Classical set Theory ………………………………….....1 1.3 Fuzzy Set Theory…………………………………….….4 1.4 Properties of fuzzy set operation……………….……...10 Chapter Two Fuzzy Logic 2.1 Classical Logic…………………………………….……19 2.2 Fuzzy Logic……………………………………..……….25 2.3 Fuzzy Logic and Approximate reasoning………….…...29 2.4 Fuzzy Relation…………………………………….…….33 Chapter Three Fuzzy Rule Base 3.1 Rules…………………………………………………...39 3.2 Inference and Knowledge Representation…………....41 3.3 Structure of Fuzzy Rule Base………………….……..48 Chapter Four Some applications of Fuzzy Logic 4.1 Introduction………………………………….............54 4.2 Health Monitoring Fuzzy Diagnostic System….........54 4.3 Fuzzy Decision Making………………………..…….60 Chapter Five Conclusion and Recommendations…………...…………..73 vii 8 List of Tables Chapter One Fuzzy set Table 1.1 Properties of Classical Set Operation …..……..... 2 Chapter Two Fuzzy Logic Table 2.1 Boolean Operations……………………………..21 Table 2.2 Properties of the Boolean Algebra…………...….22 Table 2.3 A Three – valued logic…………………………...24 Chapter Three Fuzzy Rule Base Table 3.1 Interpretation of Fuzzy IF-THEN Rules………...50 viii 9 List of Figures Chapter One Figure 1.1 Fuzzy sets representing" young" and "very young"……………....6 Figure 1.2 A membership function for a positive and large real number….7 Figure 1.3 Subset…………………………..……………………………...…12 Figure 1.4 Two membership functions……………………………………...13 Figure 1.5 The -cut of the membership function (x) Sx …………………....14 Figure 1.6 The resulting membership function of Example 1.4…..…….15 Figure 1.7 Two membership functions for example 1.6…………………..17 Figure 1.8 The resulting membership function of Example 1.6…………..18 Chapter Two Fig 2.1 Linguistic variable "Age"…………………………………………28 Chapter Four Figure 4.1Definitions and membership functions of body temperatures, blood pressure and heart rate………………………………………....60 Figure 4.2 Fuzzy goal and constraint………………………………...64 10 ix Chapter One Fuzzy Set Theory 1.1 Introduction The classical set theory is built on the fundamental concept of “set” of which an individual is either a member or not a member. A sharp, crisp, and unambiguous distinction exists between a member and a nonmember for any well-defined “set” of entities in this theory, and there is a very precise and clear boundary to indicate if an entity belongs to the set. In other words, when one asks the question “Is this entity a member of that set?” The answer is either “yes” or “no.” This is true for both the deterministic and the stochastic cases. Many real-world application problems cannot be described and handled by the classical set theory. In order to introduce the concept of fuzzy sets, we first review the elementary set theory of classical mathematics. 1.2 Classical Set Theory, 1.2.1 Fundamental Concepts Let S be a nonempty set, called the universe set below, consisting of all the possible elements of concern in a particular context. Each of these elements is called a member, or an element, of S.