FUZZY SET THEORY APPLIED TO MAKE MEDICAL PROGNOSES FOR CANCER PATIENTS834279/... · 2015-06-30 ·...

FUZZY SET THEORY APPLIED TO MAKE MEDICAL PROGNOSES FOR CANCER PATIENTS

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Hang Zettervall

Hang Z

ettervall

Blekinge Institute of Technology

Doctoral Dissertation Series No. 2014:01

Department of Mathematics and Natural Sciences2014:01

ISSN: 1653-2090

ISBN: 978-91-7295-271-3

ABSTRACTAs we all know the classical set theory has a deep-rooted influence in the traditional mathe-matics. Nevertheless, a feeling of imprecision in the two-valued logic, being the main tool for esta-blishing crisp sets, does not exist. With the rapid development of science and technology, a sub-stantial number of scientists have gradually app-reciated the vital importance of the multi-valued logic. The assumptions of a new theory of fuzzy sets, based on multi-valued logic and proposed for the first time in 1965, have given rise to model mathematically the real world’s occurrences in spite of their vague or incomplete nature.

This study aims at applying some classical and ex-tensional methods of fuzzy set theory in life ex-pectancy and treatment prognoses for cancer pa-tients. The research is based on real-life problems encountered in clinical works by physicians. From the introductory items of the fuzzy set theory to the medical applications, a collection of detailed analysis of fuzzy set theory and its extensions are discussed in the thesis. Concretely speaking, the

Mamdani fuzzy control systems and the Sugeno controller have been applied to prognosticate the survival length of gastric cancer patients. In or-der to make a surgery decision concerning can-cer patients, the fuzzy c-means clustering analysis has been adopted to investigate the possibilities for operation contra none operation. Furthermo-re, the approach of point set approximation has been proved to estimate the operation possibi-lities against to none operation for an arbitrary gastric cancer patient. In addition, in the domain of multi-expert decision-making, the probabilistic model, the model of 2-tuple linguistic represen-tations and the hesitant fuzzy linguistic term sets (HFLTS) have been utilized to select the most consensual treatment scheme(s) for two separate prostate cancer patients.

The obtained results have supplied the physicians with the reliable and helpful information. Therefo-re, the research work can be seen as the mathe-matical complement to the physicians’ queries.

2014:01

Fuzzy Set Theory Applied to Make Medical Prognoses

for Cancer Patients

Hang Zettervall

Fuzzy Set Theory Applied to Make Medical Prognoses

for Cancer Patients

Hang Zettervall

Doctoral Dissertation in Mathematics and its Applications

Blekinge Institute of Technology doctoral dissertation seriesNo 2014:01


SWEDEN

Department of Mathematics and Natural Sciences

2014 Hang Zettervall

Publisher: Blekinge Institute of Technology,SE-371 79 Karlskrona, SwedenPrinted by Lenanders Grafiska, Kalmar, 2014ISBN 978-91-7295-271-3 ISSN 1653-2090 urn:nbn:se:bth-00574

Department of Mathematics and Natural Sciences

Acknowledgements

This work has been carried out under the supervision of Professor ElisabethRakus-Andersson at the Department of Mathematics and Natural Sciences,Blekinge Institute of Technology, Karlskrona, Sweden.

My dream pursuing a Ph.D. degree could not become true without thehelp of a number of people. My particular thanks go to my main supervisor,Professor Elisabeth Rakus-Andersson. Thanks for leading me into the worldof academica and patiently keeping me on the track during these years. Alsothanks for spending countless hours to review my papers in order to keephigh research quality. It is the most deeply appreciated.

I also would like to express my gratitude to Associate Professor Hen-rik Forssell from Blekinge Competence Center and Dr. Janusz Frey fromBlekinge County Hospital in Karlskrona for the pleasant cooperation, theinspirational discussions and the valuable comments. The medical data sup-plied by Henrik Forssell and the physicians at the Department of Urology ofBlekinge County Hospital in Karlskrona in Sweden have been very valuablefor this research.

My thanks even go to the colleagues at the Department of Mathematicsand Natural Sciences at Blekinge Institute of Technology. In particular, Dr.Robert Nyqvist was generous with his time in helping me with LATEX type-setting. Also, I thank Dr. Claes Jogreus for taking time to proofread theparts of the thesis.

I am very grateful to Yao Yong and Muhammad Shahid for their gen-erosity when sharing the knowledge in MATLAB and LATEX with me.

Last but not least, I thank my beloved family for all that they have doneto support me during these years. I could not have done it without you!

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Abstract

As we all know the classical set theory has a deep-rooted influence in thetraditional mathematics. According to the two-valued logic, an element canbelong to a set or cannot. In the former case, the element’s membershipdegree will be assigned to one, whereas in the latter case it takes the zerovalue. With other words, a feeling of imprecision or fuzziness in the two-valued logic does not exist. With the rapid development of science andtechnology, more and more scientists have gradually come to realize thevital importance of the multi-valued logic. Thus, in 1965, Professor LotfiA. Zadeh from Berkeley University put forward the concept of a fuzzy set.In less than 60 years, people became more and more familiar with fuzzy settheory. The theory of fuzzy sets has been turned to be a favor applied tomany fields.

The study aims to apply some classical and extensional methods of fuzzyset theory in life expectancy and treatment prognoses for cancer patients.The research is based on real-life problems encountered in clinical works byphysicians. From the introductory items of the fuzzy set theory to the med-ical applications, a collection of detailed analysis of fuzzy set theory and itsextensions are presented in the thesis. Concretely speaking, the Mamdanifuzzy control systems and the Sugeno controller have been applied to predictthe survival length of gastric cancer patients. In order to keep the gastriccancer patients, already examined, away from the unnecessary suffering fromsurgical operation, the fuzzy c-means clustering analysis has been adopted toinvestigate the possibilities for operation contra to nonoperation. Further-more, the approach of point set approximation has been adopted to estimatethe operation possibilities against to nonoperation for an arbitrary gastriccancer patient. In addition, in the domain of multi-expert decision-making,the probabilistic model, the model of 2-tuple linguistic representations andthe hesitant fuzzy linguistic term sets (HFLTS) have been utilized to selectthe most consensual treatment scheme(s) for two separate prostate cancerpatients.

The obtained results have supplied the physicians with reliable and help-

vii

ful information. Therefore, the research work can be seen as the mathemat-ical complements to the physicians’ queries.

Keywords:

Fuzzy set theory, the Mamdani fuzzy control system, the Sugeno controller,fuzzy c-means clustering analysis, point set approximation, linguistic models,the 2-tuple linguistic representations, the hesitant fuzzy linguistic term sets

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Preface

“Even the longest journey begins with a single step”. Six years ago, I startedmy journey to pursue a Ph.D. degree with a focus on the applications ofclassical methods in fuzzy set theory to estimate the life expectancy forgastric cancer patients. My supervisor Professor Elisabeth Rakus-Anderssonwelcame me as a new member belonging to a research team consisted ofmathematicians and physicians.

The collaboration between mathematicians and physicians in BlekingeRegion in Sweden started this research. The project entitled “Fuzzy Sets,Rough Sets and Fuzzy Statistics in Treatment of Gastric Cancer Patients”pinpoints the research focus upon the estimation of the survival length andthe operation contra to nonoperation possibilities for gastric cancer patients.The later research subject turns to the discussion of multi-experts decisionmaking issues. The selection of the most consensual treatment scheme(s)for prostate cancer patients becomes the focal point. The objective of theresearch project therefore can be formulated as mathematical complementsto the physicians’ queries.

This dissertation is organized into three parts:

1. The first part contains the introductory items of fuzzy set theory andthe study of three different kinds of continuous fuzzy numbers.

2. The survival length prediction and the operation contra to nonoper-ation possibility estimation for gastric cancer patients constitute thesecond part.

3. The selection of the most consensual treatment scheme(s) for prostatecancer patients becomes the focus in the third part.

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Part I

Maybe a brief recall of some introductory items of fuzzy set theory can beuseful and helpful to understand better the later applications of the researchworks. Chapter 1 and Chapter 2 are included in the first part. Within thesection dealing with the fundamental items, some topological laws and theo-rems concerning fuzzy sets have been studied and proved to demonstrate thesimilarities and dissimilarities between the fuzzy set and the conventional(crisp) set. In Chapter 2, three special representations of continuous fuzzynumbers have been studied. These fuzzy numbers are named as fuzzy num-bers in the L-R form [89], the interval form [89] and the α−cut form [10],respectively. We demonstrate the arithmetic operations on these fuzzy num-bers and investigate the correlation between them. Moreover, the construc-tion of the π−membership functions and the s-membership functions [74]are discussed.

Part II

The second part consists of four chapters and focuses on the survival lengthprediction and the operation possibility estimations contra to the nonop-eration possibility evaluations. Such approaches as Mamdani fuzzy controltechnique [55], Sugeno fuzzy controller [101], fuzzy c-means clustering anal-ysis [11] and point set approximation [72] have been considered in Chapter3, Chapter 4, Chapter 5 and Chapter 6, respectively.

Once the diagnosis, “gastric cancer”, has been established, it is importantfor the physicians to estimate the life expectancy for the patients, since it notonly affects the life quality, but it also improves the quality of patient care.For keeping older patients with short survival length away from sufferingthe post-operative pain and the side effects, the surgical operation normallyis not recommended. Whereas, for younger patients with relatively longersurvival periods, some surgical measures will be suggested in order to keepthe patients enjoying the remaining lifetime. Rigorous analytic formulas arethe tools usually derived for describing the formal relationships between thesurvival length and the biological markers. If the mathematical formulasmay not exist or may be difficult to model, then the Mamdani fuzzy controlsystem can be seen as a good alternative.

The Mamdani control system arose in the 1970s. It is a fuzzy rule-basedsystem, which has the advantage dealing with information having a vaguecharacter. Experience-based knowledge plays an important role, since ithelps to generate a set of fuzzy control rules which constitute the core partof a fuzzy control system. The rules govern the relationship between thesurvival length and the biological parameters by using linguistic terms inthe “IF . . . , THEN . . . ” syntax.

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Therefore, in Chapter 3, the Mamdani fuzzy control technique has beenadopted to estimate the survival length for gastric cancer patients. The resultobtained by the Mamdani fuzzy control system is verified by the Sugenocontroller in Chapter 4.

So far, the surgical operation has been regarded as one of the most ef-ficient treatment therapies for gastric cancer. The possibilities of operationvary from person to person. In order to avoid the patients’ suffering frompost-operative pain and alleviate the side effects, therefore, designing a sys-tem that assists doctors in medical decision-making becomes imperative. Asthe synonym of pattern recognition, data mining, data analysis, etc., fuzzyc-means clustering analysis becomes quite outstanding within the generalframework of clustering analysis. The fuzzy clustering technique can help usin getting the interesting potential structure of a data set. In Chapter 5, thefuzzy c-means clustering technique is applied to partition a clinical data setconsisting of 25 gastric cancer patients in two clusters. One of them containspatients with operation possibilities whereas the other collects the patientsdeclared for nonoperation.

In Chapter 6, we use the joined truncated π−functions or s-functions toapproximate irregular point sets to estimate the possibilities for operationcontra to nonoperation for arbitrary gastric cancer patients selected fromthe data set of 25 gastric cancer patients.

Part III

Prostate cancer is one of the most common oncological diseases in the world.It is a frequent cause of death in male populations in the age groups olderthan 65 years. A new survey pointed out that the prostate cancer was aserious medical and an economical problem in Nordic countries.

Treatment modalities of prostate cancer can be divided, according to theEuropean Association of Urology (EAU) in three major groups named asdeferred treatment, curative treatment as well as hormonal therapy. The in-homogenous growth of the prostate cancer, various forms and types lead tomultifactorial choice of treatment strategies. Like other oncological diseases,the cooperations of health professionals are required to make the treatmentdecision. Therefore, the Department of Urology in Blekinge County Hospitalis currently using on a regular basis a multidisciplinary team meeting (MDT).It is the collaborative platform, consisting of urologists, radiologists, oncol-ogists as well as oncology and urology nurses, involved in making relevanttreatment decisions and developing treatment plans for cancer patients.

However, sometimes the physicians may have divergence opinions, some-times the patients are not interested in the treatment plan propsoed by theexperts. Therefore, the physicians need to find a model which can rangethe effectiveness of the treatment strategies. Hence, in the third part, multi-

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expert decision-making issues have been considered. Linguistic approacheshave been adopted in different models for selecting the most consensual treat-ment scheme(s) for prostate cancer patients. This part contains three chap-ters.

In Chapter 7, the probabilistic model with linguistic judgments is adoptedto select the most consensual alternative(s). In Chapter 8, the model of2−tuple linguistic representations has been utilized to range the effectivenessof treatment therapies in a decreased sequence. Finally, in Chapter 9, we usethe hesitant fuzzy linguistic term sets to express the medical professionals’preferences differentiating treatment strategies. Furthermore, the coordi-nates of fuzzy numbers’ centroid points will range the treatment schemesfrom the most recommended to contraindicated.

Summary

All in all, flexibilities, high reliabilities and preventing information loss can beshortly summerized as some of the advantages of classical approaches in thefuzzy set theory and its extensions. The applications of fuzzy set theory havebeen touched in many domains in medical community. The applications withpromising results have been gained. The expert knowledge-system has beenadopted to keep patient’s blood pressure on a balance level. It also has beenused to help patients in critical care environment with the transition fromcontrolled ventilation to total independance. Furthermore, the combinationof artificial intelligence system and fuzzy set theory has even been used inbreast cancer diagnosis for classifying the properties of the tumor. Moreapplication examples of fuzzy set theory used in the domain of artificialintelligence system in medicine can be found in [2, 97, 99].

In this dissertation, the applications of fuzzy set theory and its extensionsin medical prognoses for cancer patients have been discussed. The estimationof the survival length for gastric cancer patients, and the selection of the mostrecommended treatment therapy for prostate cancer patients by means ofadapted models constitute the main contributions of the thesis. The obtainedresults supplied the physicians with reliable mathematical complements. Weemphasize that the applications presented in this research work constituteits novel mathematical approach to medical prognoses. The research workconstitutes just a tiny part of the domain. Fuzzy set theory still has a largepotential to explore. Hopefully, the study can make readers get an interestinginsight of the subject.

Finally, my gratitude goes to Professor Elisabeth Rakus-Andersson forthe guidance and valuable comments. Also, I would like to give my thanksto medical experts, Henrik Forssell and Janusz Frey for the inspirationaldiscussions. My thanks go even to anonymous reviewers for the valuablefeedbacks.

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Publications

The following list contains the publications, in which I have made contribu-tions during my Ph.D. studies:

• H. Zettervall, E. Rakus-Andersson, J. Frey: Making Medication Prog-noses for Prostate Cancer Patients by the Application of LinguisticApproaches. The International Journal On Advances in Life Sciences,vol. 5, no. 3 & 4, December 2013, pp. 147–159.

• H. Zettervall, E. Rakus-Andersson, J. Frey: Solvning Multi-ExpertDecision-Making Issue by Linguistic Models for Prostate Cancer Pa-tients. In Proceedings of the Fifth International Conference on Bioin-formatics, Biocomputational Systems, and Biotechnologies, Lisbon, Por-tugal, March 2013, pp. 71–76.

• H. Zettervall, E. Rakus-Andersson, H. Forssell: Fuzzy C-Means Clus-ter Analysis and Approximated Data Strings in Operation Prognosisfor Gastric Cancer Patients. In: New Trends in Fuzzy Sets, Intuition-istic Fuzzy Sets, Generalized Nets and Related Topics. Volume II:Applications. Eds.: K.T. Atanassow, W. Homenda, O. Hryniewicz, J.Kacprzyk, M. Krawczak, Z. Nahorski, E. Szmidt, S. Zadrozny. IBSPAN-SRI PAS, Warsaw, 2013, pp. 181–200.

• H. Zettervall, E. Rakus-Andersson, H. Forssell: Fuzzy C-means Clus-tering Applied to Operation Evaluation for Gastric Cancer Patients.In Proceedings of the Fifth International Conference on eHealth, Tele-medicine and Social Medicine, Nice, France, 2013, pp. 228–233.

• H. Zettervall, E. Rakus-Andersson, H. Forssell: Sugeno Controller inPrediction of Survival Length in Elderly Patients with Gastric Can-cer. Licentiate Dissertation, Blekinge Institution of Technology, Karl-skrona, Sweden 2011.

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• H. Zettervall, E. Rakus-Andersson, H. Forssell: Mamdani Controllerin Prediction of Survival Length in Elderly Patients with Gastric Can-cer. In Proceedings of the International Conferences on Bioinformatics,Rome, Italy, 2011, pp. 283–286.

• E. Rakus-Andersson, H. Zettervall, H. Forssell: Fuzzy Controllers inEvaluation of Survival Length in Cancer Patients. Recent Advancesin Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and RelatedTopics. Volume II: Applications Polish Academy of Sciences, SystemResearch Institute, Warsaw, 2011, pp. 203–222.

• E. Rakus-Andersson, H. Zettervall, M. Erman: Prioritisation of Weight-ed Strategies in Multiplayer Games with Fuzzy Entries of the Pay-Off Matrix, International Journal of General Systems, (ISI list), DOI:10.1080/03081070903552882, 2010, Vol. 39, Issue 3, pp. 291–304.

• E. Rakus-Andersson, M. Salomonsson, H. Zettervall: Ranking of Weight-ed Strategies in the Two-Player Games with Fuzzy Entries of thePay-Off Matrix. In Proceedings of the 8-th International Conferenceon Hybrid Intelligent Systems, Eds: Fatos Xhafa, Francisco Herrera,Ajith Abraham et al., CDR by University Polytecnica de Catalunya,Barcelona, 2008.

• E. Rakus-Andersson, M. Salmonsson, H. Zettervall: Two-player Gameswith Fuzzy Entries of the Pay-Off Matrix. Computational Intelligencein Decision and Control - Proceedings of FLINS 2008, Madrid 2008,World Scientific, pp. 593–598.

• E. Rakus-Andersson, H. Zettervall: Approaches to Operations on Con-tinuous Fuzzy Numbers. In Developments of Fuzzy Sets, IntitionisticFuzzy Sets and Generalized Nets, Related Topics, Foundations, vol.I, Eds: Krassimir Atanassov, Panagiotis Chountas, Janusz Kacprzyk,Pedro Melo-Pinto et al., EXIT - The Publishing House of the PolishAcademy of Sciences, 2008, pp. 294–318.

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Contents

Contents xv

List of Figures xix

List of Tables xxi

1 Introductory Items of Fuzzy Set Theory 11.1 Preliminaries of Fuzzy Set Theory . . . . . . . . . . . . . . . 11.2 Basic Operations on Fuzzy Sets . . . . . . . . . . . . . . . . . 31.3 The Concepts of s-class Functions and Fuzzy Numbers . . . . 7

2 Different Approaches to Operations on Continuous FuzzyNumbers 112.1 Arithmetic Operations on Continuous Fuzzy Numbers in the L−

R Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Addition of ALR and BLR . . . . . . . . . . . . . . . . 122.1.2 Subtraction of ALR and BLR . . . . . . . . . . . . . . 142.1.3 Multiplication of ALR and BLR . . . . . . . . . . . . . 142.1.4 Division of ALR and BLR . . . . . . . . . . . . . . . . 15

2.2 Computations with Fuzzy Numbers in the Interval Form . . . 162.2.1 Addition of Aint and Bint . . . . . . . . . . . . . . . . 172.2.2 Subtraction of Aint and Bint . . . . . . . . . . . . . . . 172.2.3 Multiplication Aint and Bint . . . . . . . . . . . . . . . 182.2.4 Division of Aint and Bint . . . . . . . . . . . . . . . . 18

2.3 Arithmetic Operations in the Set of Fuzzy Numbers Convertedto α−cut Form . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Addition of A[α] and B[α] . . . . . . . . . . . . . . . . 212.3.2 Subtraction of A[α] and B[α] . . . . . . . . . . . . . . 222.3.3 Multiplication of A[α] and B[α] . . . . . . . . . . . . . 222.3.4 Division of A[α] and B[α] . . . . . . . . . . . . . . . . 23

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3 The Mamdani Controller in Prediction to the Survival Lengthin Elderly Gastric Cancer Patients 273.1 The Introduction to a Control System . . . . . . . . . . . . . 273.2 Fuzzification of Input and Output Variable Entries in Survival

Length Estimation . . . . . . . . . . . . . . . . . . . . . . . . 293.3 The Rule Based Processing Part of Surviving Length Model . 333.4 Defuzzification of the Output Variable . . . . . . . . . . . . . 353.5 The Survival Length Prognosis for a Selected Patient . . . . . 35

4 Verification of Survival Length Results by Means of SugenoController 414.1 Adaptation of the Processing Part of the Fuzzy Controller to

Sugeno-made Assumptions . . . . . . . . . . . . . . . . . . . . 414.2 Applications of the Sugeno Fuzzy Controller to Estimation of

the Survival Length in Gastric Cancer Patients . . . . . . . . 434.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Fuzzy C-means Clustering Applied to Operation Evaluationfor Gastric Cancer Patients 475.1 Description of Fuzzy C-Means Clustering Algorithm . . . . . 485.2 Determination of the Initial Membership Degrees in the Par-

tition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Evaluation of Operation Possibilities for Gastric Cancer Pa-tients by Means of Point Set Approximation 616.1 Determination of the Characteristic Values f c

xkfor Gastric

Cancer Patient xk by Adopting Data Code Vectors . . . . . . 626.2 Creation of the Truncated π-Functions to Approximate the

Point Set P c′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Determination of the Characteristic Values fxk

for GastricCancer Patient xk by the Clinical Data . . . . . . . . . . . . . 67

6.4 The Membership Function Approximation of Point Set P ′ byAdopting the Clinical Data . . . . . . . . . . . . . . . . . . . 68

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Solution of Multi-Expert Decision-Making Problem by theProbabilistic Model 717.1 Description of the Multidisciplinary Team Conference (MDT) 717.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.3 A Practical Study . . . . . . . . . . . . . . . . . . . . . . . . . 757.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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8 Selection of the Most Consensual Treatment Therapy for aProstate Cancer Patient by the 2-Tuple Linguistic Method 818.1 Description of the Model of 2-Tuple Linguistic Representation 818.2 A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9 Solution of Multi-Expert Decision-Making Problem by Hes-itant Fuzzy Linguistic Term Sets 879.1 The Preliminary Items of the Hesitant Fuzzy Linguistic Term

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.1.1 The Aggregation Phase . . . . . . . . . . . . . . . . . 899.1.2 The Exploitation Phase . . . . . . . . . . . . . . . . . 90

9.2 A Practical Study . . . . . . . . . . . . . . . . . . . . . . . . 939.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

10 Conclusions 10310.1 Research Summary . . . . . . . . . . . . . . . . . . . . . . . . 10310.2 Motivations of the Choice of Fuzzy Sets and Linguistic Ap-

proaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.2.1 The Mamdani Fuzzy Control . . . . . . . . . . . . . . 10410.2.2 Fuzzy C-Menas Clustering Analysis . . . . . . . . . . . 10510.2.3 Point Set Approximation by Clock-Shaped Member-

ship Functions . . . . . . . . . . . . . . . . . . . . . . 10610.2.4 Probabilistic Model with Linguistic Judgments . . . . 10710.2.5 The Model of 2-Tuple Linguistic Representations . . . 10710.2.6 The Hesitant Fuzzy Linguistic Term Sets . . . . . . . 108

10.3 Medical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 10810.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10910.5 Possible Future Directions . . . . . . . . . . . . . . . . . . . . 110

Appendix A 111

Appendix B 147

Appendix C 151

Appendix D 155

Bibliography 159

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List of Figures

1.1 The function s(x, 25, 37.5, 50) . . . . . . . . . . . . . . . . . . 81.2 The 0.5-level of A from Ex. 1.7 . . . . . . . . . . . . . . . . . 8

2.1 The membership function of A = (6, 2, 3)LR . . . . . . . . . 132.2 A+B =

(3, 1, 2

)LR

+(6, 2, 4

)LR

. . . . . . . . . . . . . . 132.3 The multiplication of A[α] = [1 + α, 4 − 2α] and B[α] =

[3 + 2α, 7− 2α] in the α-cut forms . . . . . . . . . . . . . . . 232.4 The division of A[α] = [α, 2−α] and B[α] = [1+α, 3−α] in

the α-cut forms . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 The membership functions for the “age” . . . . . . . . . . . . 323.2 The membership functions for the “CRP-value” . . . . . . . . 323.3 The membership functions for the “survival length” . . . . . . 333.4 The fuzzy subset of consequences constructed due to R(77, 16):1 373.5 The fuzzy subset of consequences constructed due to R(77, 16):2 373.6 The fuzzy subset of consequences constructed for R(77, 16):3 . 383.7 The fuzzy subset of consequences constructed in accord to

R(77, 16):4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 The total consequence set conseq(77, 16) . . . . . . . . . . . . 39

4.1 The linear functional dependency between age = X3 = “old ”, CRP= Y1 = “ low ” and survival length Z . . . . . . . . . . . . . . . 42

5.1 The collection of all the membership functions of operationpossibilities, L0 − L5 . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 The final cluster membership degrees of the operation and theno operation possibilities for 25 gastric cancer patients . . . . 58

6.1 The data points {(f c′xk, μc′(xk)

)} collected in the rearrangedpoint set P c′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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6.2 The membership functions of operation contra none operationpossibility for the data point set P c′ . . . . . . . . . . . . . . 67

6.3 The data points {(fxk, μ(xk)

)} aggregated in the rearrangedpoint set P ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4 The membership functions of the operation contra none oper-ation possibility for the point set P ′ . . . . . . . . . . . . . . 70

7.1 The s-parametric membership functions for linguistic fuzzysets, s0, . . . , s6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.1 The 2-tuple linguistic representation of βa2 = 1.75 . . . . . . . 84

9.1 The family of hesitant membership functions, s0 − s5 . . . . . 949.2 Conceivable effectiveness for the first treatment alternative a1 959.3 The membership function of Wa1 and Wa5 . . . . . . . . . . . 979.4 The membership function of Wa2 . . . . . . . . . . . . . . . . 989.5 The membership function of Wa3 and Wa4 . . . . . . . . . . . 999.6 The membership function of Wa6 and Wa7 . . . . . . . . . . . 100

C.1 The linguistic judgment for the first prostate cancer patientsupplied by expert one . . . . . . . . . . . . . . . . . . . . . . 151

C.2 The linguistic judgment for the first prostate cancer patientsupplied by expert two . . . . . . . . . . . . . . . . . . . . . . 152

C.3 The linguistic judgment for the first prostate cancer patientsupplied by expert three . . . . . . . . . . . . . . . . . . . . . 153

C.4 The linguistic judgment for the first prostate cancer patientsupplied by expert four . . . . . . . . . . . . . . . . . . . . . . 154

D.1 The hesitant preference supplied by expert one . . . . . . . . 155D.2 The hesitant preference supplied by expert two . . . . . . . . 156D.3 The hesitant preference supplied by expert three . . . . . . . 156D.4 The hesitant preference supplied by expert four . . . . . . . . 157D.5 The hesitant preference supplied by expert five . . . . . . . . 157

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List of Tables

3.1 Rule base of fuzzy controller estimating “survival length” . . . 34

4.1 The functional rule base table for combinations of X- and Y-levels in estimations of survival length . . . . . . . . . . . . . 43

5.1 The excerpted medical data set of 25 gastric cancer patients . 525.2 The data set with the initial membership values of patient xk be-

longing to cluster Si . . . . . . . . . . . . . . . . . . . . . . . 555.3 The excerpt data set including the first updated membership

values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1 The importance table of age, CRP -value and body weight . . 63

7.1 The judgment table in the probabilistic model . . . . . . . . . 737.2 The collection of the therapy judgments in the probabilistic

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.3 The aggregation of random preferences in the probabilistic

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.4 The collection of choice values in the probabilistic model . . . 80

8.1 The judgment table in which each judgment is expressed bysemantic word . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.2 The judgments collection of the treatment therapy expressedby the physicians . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3 The judgment table of the 2-tuple linguistic representations . 828.4 The judgments expressed in the 2-tuples representation model 858.5 The arithmetic means of six treatment therapies in the model

of 2-tuple linguistic representations . . . . . . . . . . . . . . . 86

9.1 The hesitant judgment table . . . . . . . . . . . . . . . . . . . 899.2 The hesitant preference table . . . . . . . . . . . . . . . . . . 909.3 The HFLTS’s subset of S . . . . . . . . . . . . . . . . . . . . 90

xxi

9.4 The Hesitant judgment table of practical study . . . . . . . . 94

A.1 The original medical data of 25 gastric cancer patients . . . . 112A.2 The first updated operation and nonoperation membership

degrees of 25 gastric cancer patients . . . . . . . . . . . . . . 113A.3 The updated operation and nonoperation membership degrees

of 25 gastric cancer patients . . . . . . . . . . . . . . . . . . . 114A.4 The operation and nonoperation possibilities of 25 gastric can-

cer patients after the 1st itreration . . . . . . . . . . . . . . . 115A.5 The operation and nonoperation possibilities of 25 gastric can-

cer patients after the 2nd iteration . . . . . . . . . . . . . . . 116A.6 The operation and nonoperation possibilities of 25 gastric can-

cer patients after the 3rd iteration . . . . . . . . . . . . . . . . 117A.7 The operation and nonoperation possibilities of 25 gastric can-

cer patients after the 4th iteration . . . . . . . . . . . . . . . . 118A.8 The operation and nonoperation possibilities of 25 gastric can-






cer patients after the 10th iteration . . . . . . . . . . . . . . . 124A.14 The operation and nonoperation possibilities of 25 gastric can-








cer patients after the 18th iteration . . . . . . . . . . . . . . . 132

xxii

A.22 The operation and nonoperation possibilities of 25 gastric can-cer patients after the 19th iteration . . . . . . . . . . . . . . . 133


A.24 The operation and nonoperation possibilities of 25 gastric can-cer patients after the 21st iteration . . . . . . . . . . . . . . . 135

A.25 The operation and nonoperation possibilities of 25 gastric can-cer patients after the 22nd iteration . . . . . . . . . . . . . . . 136

A.26 The operation and nonoperation possibilities of 25 gastric can-cer patients after the 23rd iteration . . . . . . . . . . . . . . . 137








A.34 The operation and nonoperation possibilities of 25 gastric can-cer patients after the 31st iteration . . . . . . . . . . . . . . . 145

B.1 The point set P c = {(f cxk, μSi(xk))}, of which the characteris-

tic values fxck

are determined by the code vectors, (ac, crpc, bwc)148B.2 The point set P = {(fxk

, μSi(xk))}, whose characteristic val-

ues fxkare calculated by the original clinical data, (a, crp, bw)149

xxiii

CHAPTER1Introductory Items of FuzzySet Theory

Before discussing the essence of the medical applications, tackling with thetreatment of gastric and prostate cancer patients, we wish to introduce somebasic formulations of fuzzy set theory.

1.1 Preliminaries of Fuzzy Set Theory

Definition 1.1. Fuzzy setIf X is a finite universe set, X = {xi, i = 1, · · · , n}, then a fuzzy set A in Xis defined as a set of ordered pairs A = {(xi, μA(xi)}, i = 1, . . . , n, in whichthe value of μA(xi) is called the degree of membership of xi in fuzzy set Aand μA : X → [0, 1] is called the membership function mapping X into theunit interval [0, 1] [10, 12, 48, 53, 60, 67, 68, 74, 96, 109, 114].

The support of the fuzzy set A, S(A) is defined as a non-fuzzy set ofthese elements, which have the membership degrees greater than zero.

There are four types of notations used for presenting a fuzzy set A [114].

1. A is formed as an ordered set of pairs, A = {(xi, μA(xi)}, i = 1, . . . , n,where xi denotes the element from the set X and μA(xi) denotes thedegree of the membership of xi in the fuzzy set A.

Example 1.1.Let X = “ integers between 1 and 10 ” = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},then a fuzzy set A, “ integers which are close to 5 ” can be defined as

A = {(1, 0), (2, 0.3), (3, 0.5), (4, 0.7), (5, 1), (6, 0.9), (7, 0.8), (9, 0.3),(10, 0)}.

1

2 Chapter 1

2. A corresponds to “ the Zadeh notation ” A =∑n

i=1 μA(xi)/xi. Ele-ments with zero membership degrees are normally omitted. Note: thesigma-sign has just the symbolic function to connect the set elements.

Example 1.2.Let X = “ integers between 1 and 10 ” = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},Hence A = “ integers which are close to 6 ” can be defined as

A = 0.3/2 + 0.5/3 + 0.7/4 + 0.9/5 + 1/6 + 0.9/7 + 0.7/8 + 0.3/9.

3. A is characterized by an n-dimensional vector of membership degreesA =

(μA(xi), i = 1, . . . , n

). Elements with zero membership degrees

cannot be omitted.

Example 1.3.Let X = “ integers between 1 and 10 ” = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},A= “ close to 6 ” is thus assisted by

A = (0, 0.3, 0.5, 0.7, 0.9, 1, 0.9, 0.7, 0.3, 0).

4. A is listed as a collection of its elements and assigned to them degreesof membership in the form of A={μA(xi)/xi, i = 1, . . . , n}.Example 1.4.If X = “ integers between 1 and 10 ” = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},then A = “ close to 6 ” is defined by

A = {0/1, 0.3/2, 0.5/3, 0.7/4, 0.9/5, 1/6, 0.9/7, 0.7/8, 0.3/9, 0/10} .

Now we intend to demonstrate some useful definitions concerning fuzzysets.

Definition 1.2. Empty fuzzy setA fuzzy set A is said to be empty if all membership degrees assisting xi, i =1, . . . , n, are equal to zero in X. That is, μA(xi) = 0, xi ∈ X, i =1, . . . , n =⇒ A is empty.

Definition 1.3. Equal fuzzy setsTwo fuzzy sets A and B are equal, if and only if the membership degree inA is equal to the membership degree in B for each element xi ∈ X. Thisproperty can be expressed as

A = B ⇐⇒ μA(xi) = μB(xi), xi ∈ X, i = 1, . . . , n.

Definition 1.4. Complement of a fuzzy setThe complement of a fuzzy set A, denoted by A′, is defined by the followingformula

μA′(xi) = 1− μA(xi), xi ∈ X, i = 1, . . . , n.

Chapter 1 3

Definition 1.5. Fuzzy subsetA fuzzy set A is said to be a subset of another fuzzy set B if and only if

the membership degree in A is equal or less than the membership degree inB for each element xi ∈ X, which is equivalent to

A ⊆ B ⇐⇒ μA(xi) ≤ μB(xi), xi ∈ X, i = 1, . . . , n.

1.2 Basic Operations on Fuzzy Sets

To be able to perform some topological operations on fuzzy sets we determinetheir union, their intersection and their complements.

Definition 1.6. The topological or hard union of two fuzzy sets Aand BLet X be a common universe for fuzzy sets A and B. If A = {(xi, μA(xi))}and B = {(xi, μB(xi))}, i = 1, . . . , n, then the union of A and B is denotedby A ∪B and determined by a membership function

μA∪B(xi) = max[μA(xi), μB(xi)] = μA(xi) ∨ μB(xi), xi ∈ X, i = 1, . . . , n.

Lemma 1.1.The union of fuzzy sets A and B, due to definition 1.6, is the smallest fuzzyset which contains both A and B.

Proof.Suppose that μA∪B(xi) = max[μA(xi), μB(xi)], xi ∈ X, i = 1, . . . , n. Then,

max[μA(xi), μB(xi] = μA(xi)

ormax[μA(xi), μB(xi] = μB(xi).

Let D be an arbitrary fuzzy set containing both A and B. This implies that

μD(xi) ≥ μA(xi)

andμD(xi) ≥ μB(xi).

Hence, μD(xi) ≥ μA∪B(xi), xi ∈ X, i = 1, . . . , n.

Definition 1.7. The topological or hard intersection of two fuzzysets A and BThe intersection of two fuzzy sets A and B, A = {(xi, μA(xi))} and B ={(xi, μB(xi)), }, xi ∈ X, i = 1, . . . , n, is recognized by the notation A ∩ Band expressed by the membership function

μA∩B(xi) = min[μA(xi), μB(xi)] = μA(xi) ∧ μB(xi), xi ∈ X, i = 1, . . . , n.

4 Chapter 1

Lemma 1.2.The intersection of A and B is the largest fuzzy set, which is contained in

both A and B.

Proof. By definition 1.7 we get

μA∩B(xi) = min[μA(xi), μB(xi)] = μA(xi) ∧ μB(xi), xi ∈ X, i = 1, . . . , n.

It means thatmin[μA(xi), μB(xi)] = μA(xi)

ormin[μA(xi), μB(xi)] = μB(xi).

Suppose that M is an arbitrary fuzzy set which is contained in both Aand B. This implies that μA(xi) ≥ μM (xi) and μB(xi) ≥ μM (xi). Hence,μA∩B(xi) ≥ μM (xi), xi ∈ X, i = 1, . . . , n.

Some of the basic identities, such as the commutative laws, the distribu-tive laws and the associative laws, which hold for the topological union andintersection performed on the classical (crisp) sets, can be also encounteredin relations among fuzzy sets. We formulate and prove the following connec-tions involving fuzzy sets.

Theorem 1.1. Commutative lawsIf A and B are two fuzzy sets then the commutative laws A ∩ B = B ∩ A

and A ∪B = B ∪A hold.

Proof.

μA∩B(xi) = min[μA(xi), μB(xi)

]= min

[μB(xi), μA(xi)

]= μB∩A(xi), xi ∈ X, i = 1, . . . , n

For

μA∪B(xi) = max[μA(xi), μB(xi)

]= max

[μB(xi), μA(xi)

]= μB∪A(xi), xi ∈ X, i = 1, . . . , n.

Theorem 1.2. Distributive lawsFor three fuzzy sets, A, B and C we formulate the distributive laws C ∩(A ∪B) = (C ∩A) ∪ (C ∩B) and C ∪ (A ∩B) = (C ∪A) ∩ (C ∪B).

Chapter 1 5

Proof. The existence of the first distributive law can be verified by means ofthe topological operation performance on membership degrees of the fuzzysets A, B and C. Since the distributive laws are valid for the maximum andthe minimum operators then

μC∩(A∪B)(xi) = μC(xi) ∧ μA∪B(xi)

= μC(xi) ∧[μA(xi) ∨ μB(xi)

]=[μC(xi) ∧ μA(xi)

] ∨ [μC(xi) ∧ μB(xi)]

= μC∩A(xi) ∨ μC∩B(xi)= μ(C∩A)∪(C∩B)(xi).

For the second distributive law

μC∪(A∩B)(xi) = μC(xi) ∨ μA∩B(xi)

= μC(xi) ∨[μA(xi) ∧ μB(xi)

]=[μC(xi) ∨ μA(xi)

] ∧ [μC(xi) ∨ μB(xi)]

= μC∪A(xi) ∧ μC∪B(xi)= μ(C∪A)∩(C∪B)(xi).

We intend to formulate and to prove the existence of relations known asde Morgan’s laws for the topological operations on fuzzy sets.

Theorem 1.3. De Morgan’s lawsIf A = {(xi, μA(xi)), i = 1, . . . , n} and B = {(xi, μB(xi)), i = 1, . . . , n}, then{

(A ∪B)′ = A′ ∩B′

(A ∩B)′ = A′ ∪B′

Proof. We verify the first identity by means of the membership functions ofxi in A and B for each xi ∈ X. For

μ(A∪B)′(xi) = 1− μ(A∪B)(xi)

= 1−max[μA(xi), μB(xi)

]= 1− [

μA(xi) ∨ μB(xi)]

=[1− μA(xi)

] ∧ [1− μB(xi)]

= μA′(xi) ∧ μB′(xi) = min[μA′(xi), μB′(xi)

]= μA′∩B′(xi).

The second identity can also be verified by referring to the definitions ofthe topological operations. Hence

6 Chapter 1

μ(A∩B)′(xi) = 1− μ(A∩B)(xi)

= 1−min[μA(xi), μB(xi)

]= 1− [

μA(xi) ∧ μB(xi)]

=[1− μA(xi)

] ∨ [1− μB(xi)]

= μA′(xi) ∨ μB′(xi)

= max[μA′(xi), μB′(xi)

]= μA′∪B′(xi).

If we determine other types of the intersection and the union for twofuzzy sets, e.g., the operations algebraically defined on membership degreesof A and B as μA·B(xi) = μA(xi) ·μB(xi) and μA+B(xi) = μA(xi)+μB(xi)−μA(xi) · μB(xi), xi ∈ X, i = 1, . . . , n, then we can also prove the validity ofde Morgan’s laws.

A simple example confirms the validity of de Morgan’s law for topologicalunion and intersection of two fuzzy sets.

Example 1.5. Let X = {1, 2, 3, 4, 5, 6, 7, 8}. We define A = “ large integerin X ” = 0.25/5 + 0.5/6 + 0.8/7 + 1/8 and B = “ small integer in X ” =1/1 + 0.7/2 + 0.5/3 + 0.3/4 + 0.1/5. ThenA ∪B = 1/1 + 0.7/2 + 0.5/3 + 0.3/4 + 0.25/5 + 0.5/6 + 0.8/7 + 1/8,(A ∪B)′ = 0.3/2 + 0.5/3 + 0.7/4 + 0.75/5 + 0.5/6 + 0.2/7,A′ = 1/1 + 1/2 + 1/3 + 1/4 + 0.75/5 + 0.5/6 + 0.2/7,B′ = 0.3/2 + 0.5/3 + 0.7/4 + 0.9/5 + 1/6 + 1/7 + 1/8and A′ ∩B′ = 0.3/2 + 0.5/3 + 0.7/4 + 0.75/5 + 0.5/6 + 0.2/7.Hence, (A ∪B)′ = A′ ∩B′.

Theorem 1.4.For two fuzzy sets A = {(xi, μA(xi))} and B = {(xi, μB(xi))}, xi ∈ X, i =1, . . . , n, the law (A + B)′ = A′ · B′ is referred to the operations of thealgebraic (soft) intersection and the algebraic (soft) union yielded by themembership functions μA·B(xi) = μA(xi) ·μB(xi) and μA+B(xi) = μA(xi)+μB(xi)− μA(xi) · μB(xi).

Proof.We perform the set operations on membership degrees assisting A and B.Thus, μ(A+B)′(xi) = 1− (μA(xi)+μB(xi)−μA(xi) ·μB(xi)

)= 1−μA(xi)−

μB(xi)+μA(xi) ·μB(xi) = (1−μA(xi)) · (1−μB(xi)) = μA′(xi) ·μB′(xi).

So far, most of the identities which hold for the crisp set also hold for thefuzzy set. Except two axioms. These two axioms are known as the “axiom of

Chapter 1 7

contradiction”, A∩A′ �= ∅ and the “axiom of excluded middle”, A∪A′ �= X,[94].

Both the minimum operator and the multiplication operator, proposedin the intersection operations on membership degrees of fuzzy sets, fulfillthe properties of the function called a t-norm. The t-norm satisfies theproperties:

1. t(0, 0) = 0; t(μA(xi), 1

)= t

(1, μA(xi)

)= μA(xi), xi ∈ X,

2. t(μA(xi), μB(xi)

) ≤ t(μC(xi), μD(xi)

)if μA(xi) ≤ μC(xi) and μB(xi) ≤

μD(xi),

3. t(μA(xi), μB(xi)

)= t

(μB(xi), μA(xi)

),

4. t

(μA(xi), t

(μB(xi), μC(xi)

))= t

(t((μA(xi), μB(xi)

), μC(xi)

).

Generally, any function t, which has the features listed in points 1-4 canbe adopted as an operator of intersection between two fuzzy sets.

To define different approaches to the union of two fuzzy sets the charac-teristics of the function s, named the s-norm, are stated as

1. s(1, 1) = 1; s(μA(xi), 0

)=s(0, μA(xi)

)= μA(xi), xi ∈ X,

2. s(μA(xi), μB(xi)

) ≤ s(μC(xi), μD(xi)

)if μA(xi) ≤ μC(xi) and μB(xi) ≤

μD(xi),

3. s(μA(xi), μB(xi)

)= s

(μB(xi), μA(xi)

),

4. s

(μA(xi), s

(μB(xi), μC(xi)

))= s

(s(μA(xi), μB(xi)

), μC(xi)

).

It is easy to check that the maximum operator assisting the topologicalunion as well as the definition of the algebraically created membership func-tion for the algebraic union possess the properties 1-4 of the last connection.

All definitions and properties of fuzzy sets, discussed so far, can be trans-ferred into the continuous universe of discourse X = {x}, x ∈ X, in which afuzzy set A is denoted as a collection A =

∑x∈X μA(x)/x.

1.3 The Concepts of s-class Functions and FuzzyNumbers

Instead of predetermined discrete sets of membership degrees, correspondingto the set elements, continuous membership functions will be stretched overdomains of fuzzy sets. The functions constitute the sets’ restrictions in theuniverse X and, thus, they should take the values in the interval [0, 1]. Oneof the types of membership functions, commonly utilized in applications, isan s-class function defined below.

8 Chapter 1

Definition 1.8. s-class functionAs the membership function of the fuzzy set A we demonstrate the s-classfunction s(x, α, β, γ) with the parameters α, β and γ that are included inthe formula [1, 10, 48, 64, 72]

y = μA(x) = s(x, α, β, γ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 for x ≤ α,

2

(x− α

γ − α

)2

for α < x ≤ β,

1− 2

(x− γ

γ − α

)2

for β < x ≤ γ,

1 for x > γ,

(1.1)

where β = α+γ2 .

Example 1.6. The function s(x, 25, 37.5, 50) is plotted in Fig. 1.1.

50,5.37,25,xs

Figure 1.1: The function s(x, 25, 37.5, 50)

75 62.5 50 37.5 25

1

0.75

0.5

0.25

0

y y A(x)

5.0A

Figure 1.2: The 0.5-level of A from Ex. 1.7

Chapter 1 9

Definition 1.9. α-level of a fuzzy setFor a continuous fuzzy set A = {(x, μA(x)} , x ∈ X, we determine a none-fuzzy set Aα = {x : μA(x) ≥ α} called the α-level of A.

Example 1.7. For A given by the membership function [74]

μA(x) =

{s(x, 30, 40, 50) for x ≤ 50,

1− s(x, 50, 60, 70) for x ≥ 50,(1.2)

we state, e.g., A0.5 = [40, 60] in accordance with Fig. 1.2.

Definition 1.10.A fuzzy number A of a universe X = {x} is a normal fuzzy set possessingthe ascending left function and the descending right function around this xin which the membership degree equals one. A fuzzy set is recognized asnormal if at least one of its elements x has the degree of membership equalto one.

We intend to discuss the character and the properties of continuous fuzzynumbers in the next chapter.

10 Chapter 1

CHAPTER2Different Approaches toOperations on ContinuousFuzzy Numbers

This chapter is particularly addressed to the readers as our own study inperforming operations on continuous fuzzy numbers [89]. We discuss threedifferent approaches to operations on fuzzy numbers to make comparisons ofresults in the aspect of their advantages and disadvantages.

The theory of fuzzy number arithmetic constitutes an exciting part offuzzy set theory, therefore we have made a separate study of their nature.In this dissertation fuzzy numbers will be utilized in the chapters devotedto medical applications and crucial items of further investigations in devel-opments of algorithms. By demonstrating different possibilities of makingcalculations on fuzzy numbers we intend to select the proper approach tothe operations in the further research.

Fuzzy numbers, being normal fuzzy sets characterized by particularlydesigned membership functions, constitute important components of numer-ical operations involved in fuzzy calculus. The different approaches to thearithmetic over the space of fuzzy numbers have been suggested in manyworks [10, 15, 18, 23, 24, 29, 31, 32, 39, 40, 44, 49, 54, 57, 68, 100].

We often adopt three methods of performing the operations on continuousfuzzy numbers dependently on their forms, namely, we recognize the L −R form, the interval form and the α−cut form.

11

12 Chapter 2

2.1 Arithmetic Operations on Continuous Fuzzy Num-bers in the L−R Form

Definition 2.1. Fuzzy numbers in the L−R representationA continuous fuzzy number A can be represented by the L − R form ifthere exist reference functions L (for left) and R (for right) and scalars α >0 and β > 0, then the membership function μA(x) of A is defined as

y = μA(x) =

⎧⎪⎪⎨⎪⎪⎩L

(m− x

α

)for m− α ≤ x ≤ m

R

(x−m

β

)for m ≤ x ≤ m+ β

(2.1)

where m is called the mean value of A, m ∈ R, α and β are called the leftand the right spreads, respectively.

The fuzzy number A is represented by a triplet A = (m,α, β)LR, m ∈R. The support of A, S(A) = {x ∈ X : μA(x) > 0} is thus stated as aset S(A) = [m− α,m+ β]. The function L(x) and R(x) can have differentdefinitions. One of them introduces a very comfortable appearance of L(x) =R(x) = 1 − x [89] created for linear branches of A’s membership functions.Due to this formula we restrict μA(x) in the form of

y = μA(x) =

⎧⎪⎪⎨⎪⎪⎩L

(m− x

α

)= 1− m− x

αfor m− α ≤ x ≤ m

R

(x−m

β

)= 1− x−m

βfor m ≤ x ≤ m+ β

(2.2)

Example 2.1. If A = (6, 3, 2)LR, then

y = μA(x) =

⎧⎪⎪⎨⎪⎪⎩L

(6− x

3

)= 1− 6− x

3=

x− 3

3for 3 ≤ x ≤ 6

R

(x− 6

2

)= 1− x− 6

2=

8− x

2for 6 ≤ x ≤ 8

A = (6, 3, 2)LR has a triangular shape with the peak in 6. We plot themembership function of A in Fig. 2.1.

Let A = (mA, αA, βA)LR and B = (mB, αB, βB)LR be two continuousfuzzy numbers in the L − R form. We define the arithmetic operations onfuzzy numbers in the following way [23, 24].

2.1.1 Addition of ALR and BLR

A+B =(mA, αA, βA

)LR

+(mB, αB, βB

)LR

=(mA +mB, αA + αB, βA + βB

)LR

.(2.3)

Chapter 2 13

x

y

Figure 2.1: The membership function of A = (6, 2, 3)LR

Example 2.2. If A =(mA, αA, βA

)LR

=(3, 1, 2

)LR

and B =(mB, αB, βB

)LR

=(6, 2, 4

)LR

, then A+B =(3, 1, 2

)LR

+(6, 2, 4

)LR

=(9, 3, 6

)LR

.

After adopting (2.2) we plot the membership functions of A and B andthe result of A+B in Fig. 2.2.

0 2 4 6 8 10 12 14 16 18 200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x

y

Figure 2.2: A+B =(3, 1, 2

)LR

+(6, 2, 4

)LR

All membership functions are linear. If we wish to change the shape ofthe membership functions then we should design L(x) and R(x) in anotherform.

14 Chapter 2

2.1.2 Subtraction of ALR and BLR

If we determine −B as −B =(−mB, βB, αB

)LR

, then

A−B = A+(−B

)=(mA, αA, βA

)LR

+(−mB, βB, αB

)LR

=(mA −mB, αA + βB, βA + αB

)LR

.

(2.4)

Example 2.3.For A =

(5, 2, 3

)LR

and B =(6, 2, 1

)LR

, −B =(−6, 1, 2

)LR

, we get

A−B = A+ (−B) =(5, 2, 3

)LR

+(−6, 1, 2

)LR

=(5− 6, 2 + 1, 3 + 2

)LR

=(−1, 3, 5

)LR

2.1.3 Multiplication of ALR and BLR

We differentiate results dependently on the relative values of mA and mB.

Case 1

If mA > 0 and mB > 0, then

A ·B =(mA, αA, βA

)LR

· (mB, αB, βB)LR

=(mA ·mB, mAαB +mBαA, mAβB +mBβA

)LR

.(2.5)

Example 2.4. For A =(3, 1, 2

)LR

and B =(5, 4, 1

)LR

, we obtain afuzzy number by multiplication

A ·B =(3, 1, 2

)LR

· (5, 4, 1)LR

=(3 · 5, 3 · 4 + 5 · 1, 3 · 1 + 5 · 2)

LR

=(15, 17, 13

)LR

.

Case 2

If mA < 0 and mB > 0, then we design


)LR

· (mB, αB, βB)LR

=(mA ·mB, mBαA −mAαB, mBβA −mAβB

)LR

.(2.6)

Example 2.5. We state A =(−3, 2, 1

)LR

and B =(4, 3, 2

)LR

to expecta result

A ·B =(−3, 2, 1

)LR

· (4, 3, 2)LR

=(−3 · 4, 4 · 2− (−3) · 3, 4 · 1− (−3) · 2)

LR

=(−12, 17, 10

)LR

.

Chapter 2 15

Case 3

In the case of mA < 0 and mB < 0, the multiplication provides us witha fuzzy number


)LR

· (mB, αB, βB)LR

=(mA ·mB, −mAβB −mBβA, −mAαB −mBαA

)LR

.(2.7)

Example 2.6. We accept A =(−3, 2, 1

)LR

and B =(−1, 3, 2

)LR

tocompute

A ·B =(−3, 2, 1

)LR

· (−1, 3, 2)LR

=(−3 · (−1), 3 · 2− (−1) · 1, 3 · 3− (−1) · 2)

LR

=(3, 7, 11

)LR

.

Let us notice that for A =(mA, αA, βA

)LR

, the inverse number A−1 =(1

mA,

αA

m2A

,βAm2

A

)LR

.

Example 2.7.

A =(mA, αA, βA

)LR

=(2, 3, 2

)LR

=⇒ A−1 =

(1

2,3

4,1

2

)LR

.

2.1.4 Division of ALR and BLR

We recognize the cases of different interpretations of results in conformitywith the relative values of mA and mB as follows:Case 1For mA > 0 and mB > 0, we derive a formula

A

B=(mA, αA, βA

)LR

:(mB, αB, βB

)LR

=

(mA

mB,

(mAβB +mBαA

m2B

),

(mAαB +mBβA

m2B

))LR

.(2.8)

Example 2.8. The fuzzy numbers A =(6, 2, 1

)LR

and B =(3, 1, 2

)LR

,inserted as arguments of the division operation, lead to a result shown below:

A

B=(6, 2, 1

)LR

:(3, 1, 2

)LR

=

(6

3,

(6 · 2 + 3 · 2

32

),

(6 · 1 + 3 · 1

32

))LR

=(2, 2, 1

)LR

16 Chapter 2

Case 2We refer to mA > 0 and mB < 0 to create a result

A

B=(mA, αA, βA

)LR

:(mB, αB, βB

)LR

=

(mA

mB,

(mAβB −mBαA

m2B

),

(mAαB −mBβA

m2B

))LR

.(2.9)

Example 2.9. We state A =(2, 3, 1

)LR

and B =(−1, 2, 1

)LR

to expecta result shown below:

A

B=(2, 3, 1

)LR

:(−1, 2, 1

)LR

=

(2

−1,

(2 · 1 + 1 · 3

(−1)2

),

(2 · 2 + 1 · 1

(−1)2

))LR

=(−2, 5, 5

)LR

.

(2.10)

Case 3In the case of mA < 0 and mB < 0, the division provides us with a fuzzynumber

A

B=(mA, αA, βA

)LR

:(mB, αB, βB

)LR

=

(mA

mB,

(−mAαB −mBβAm2

B

),

(−mAβB −mBαA

m2B

))LR

.(2.11)

Example 2.10. We accept A =(−2, 3, 1

)LR

and B =(−1, 2, 3

)LR

tocompute

A

B=(−2, 3, 1

)LR

:(−1, 2, 3

)LR

=

(−2

−1,

(2 · 2 + 1 · 1

(−1)2

),

(2 · 3 + 1 · 3

(−1)2

))LR

=(2, 5, 9

)LR

.

Let us notice that the formulas, derived for arithmetical operations onfuzzy numbers in the L−R form, provide us only with supports of results. Toprognosticate shapes of the result membership functions in accordance withour expectations we have to model the reference functions L(x) and R(x) inadvance.

2.2 Computations with Fuzzy Numbers in the In-terval Form

The L−R form of a fuzzy number can be sometimes confusing when regardingthe order of parameters listed as the mean, the left spread and the right

Chapter 2 17

spread. It seems to be natural to experience the fuzzy number to appear asa triplet: the left border of the support, the mean and the right border ofthe support. Let us suggest a concept of the fuzzy number in the intervalform in conformity with the following definition.

Definition 2.2. Fuzzy numbers in the interval formIf A =

(mA, αA, βA

)LR

, then the interval form of A can be written as Aint =[aA, mA, bA], where mA is the mean value, aA and bA are the left and rightborders respectively. We determine aA = mA − αA and bA = mA + βA.

Example 2.11. Let A =(6, 3, 2

)LR

. Then A = [6 − 3, 6, 6 + 2]int =A = [3, 6, 8]int.

Suppose that Aint = [aA, mA, bA] and Bint = [aB, mB, bB] are triangu-lar fuzzy numbers in the interval form. We define the arithmetic operationson Aint and Bint by following the pattern below [10, 31].

2.2.1 Addition of Aint and Bint

Let us suppose that A = [aA, mA, bA]int and B = [aB, mB, bB]int indicatetwo fuzzy numbers in the interval form. The addition of A+B can be derivedfrom the following formula

[A+B]int = [aA + aB,mA +mB, bA + bB]. (2.12)

Example 2.12. Assume that A =(3, 2, 1

)LR

. Then Aint = [3− 2, 3, 3 +

1] = [1, 3, 4]. Also, we set B =(4, 1, 2

)LR

≡ Bint = [4 − 1, 4, 4 + 2] =[3, 4, 6]. The effect of the addition has been performed as [A + B]int =[1 + 3, 3 + 4, 4 + 6] = [4, 7, 10] ≡ (

7, 3, 3)LR

.

ConclusionIf we calculate the addition A+B in the L−R form, we will obtain A+B =(3 + 4, 2 + 1, 1 + 2

)LR

=(7, 3, 3

)LR

, which confirms the full similarity ofresults when proving different approaches to addition of fuzzy numbers.

2.2.2 Subtraction of Aint and Bint

Let A = [aA, mA, bA]int and B = [aB, mB, bB]int indicate two fuzzynumbers in the interval form. The subtraction of A and B can be given byformula

[A−B]int = [aA − bB,mA −mB, bA − aB]. (2.13)

Example 2.13. The setting of Aint = [1, 3, 4] ≡ (3, 2, 1

)LR

and Bint =

[3, 6, 8] ≡ (6, 3, 2

)LR

in (2.13) yields [A−B]int = [1− 8, 3− 6, 4− 3] =

[−7, −3, −1] ≡ (−3, 4, 4)LR

.

18 Chapter 2

Conclusion We execute the operation of subtraction for the same numbersAint = (3, 2, 1)LR, Bint = (6, 3, 2)LR and −Bint = (−6, 2, 3)LR in theL− R representation to be furnished with a number A− B = A+ (−B) =(3, 2, 1)LR + (−6, 3, 2)LR = (−3, 4, 4)LR.

This confirms the convergence of subtraction answers even if the compu-tations procedures have been conducted according to different approaches.Further, we intend to compare the result reports for multiplication and di-vision of two fuzzy numbers when performing the operations in the intervaland L−R forms. Thus we ought to determine the rules of multiplying anddividing for two fuzzy numbers in the interval performance to be capable ofmaking the mentioned confrontations.

2.2.3 Multiplication Aint and Bint

We suggest Aint = [aA, mA, bA] and Bint = [aB, mB, bB] as the data inthe interval multiplication. Let ε = min

(aAaB, aAbB, bAaB, bAbB

)and

φ = max(aAaB, aAbB, bAaB, bAbB

). We define the multiplication of two

fuzzy numbers [10] in the interval forms by

[aA, mA, bA] · [aB, mB, bB] = [ε, mA ·mB, φ]. (2.14)

Example 2.14. If A =(3, 1, 2

)LR

≡ Aint = [2, 3, 5] and B =(4, 3, 1

)LR

≡Bint = [1, 4, 5], then, for ε = min

(2 ·1, 2 ·5, 5 ·1, 5 ·5) = min

(2, 10, 5, 25

)= 2 and φ = max

(2 · 1, 2 · 5, 5 · 1, 5 · 5) = min

(2, 10, 5, 25

)= 25, we will

perform the operation of multiplication to obtain a number

[A ·B]int = [2, 3 · 4, 25]

= [2, 12, 5]

≡ (12, 10, 13

)LR

.

ConclusionWe set A =

(3, 2, 1

)LR

and B =(4, 3, 1

)LR

, due to (2.5), in A · B =(3, 2, 1

)LR

· (4, 3, 1)LR

=(12, 13, 11

)LR

to discover the divergent answerof multiplication when comparing forms of exploiting the definitions createdfor the interval and the L−R fuzzy numbers.

2.2.4 Division of Aint and Bint

We introduce the division of A and B in the interval form as

[A : B]int = [aA, mA, bA] · [1/bB, 1/mB, 1/aB]

=[ε,

mA

mB, φ

],

(2.15)

in whichε = min

(aAbB

,aAaB

,bAbB

,bAaB

)

Chapter 2 19

and

φ = max

(aAbB

,aAaB

,bAbB

,bAaB

).

Example 2.15. Suppose A =(6, 2, 1

)LR

and B =(2, 1, 2

)LR

. We will cal-culate A : B in the interval and the L−R form by utilizing (2.8) and (2.15) re-spectively, to check if we can produce the same answer in both cases. Welet mA > 0, mB > 0.

We take A =(6, 2, 1

)LR

≡ Aint = [4, 6, 7] and B =(2, 1, 2

)LR

≡Bint = [1, 2, 4]. We start with the calculation of [A : B]int = [4, 6, 7] ·[1/4, 1/2, 1/1]. After evaluating ε = min

(4

4,

4

1,

7

4,

7

1

)= 1 and φ =

max

(4

4,4

1,7

4,7

1

)= 7, we compute the division of A and B in the interval

form by (2.15) as:

[A : B]int =[ε,

mA

mB, φ

]=[1,

6

2, 7]

=[1, 3, 7

]≡ (

3, 2, 4)LR

.

We also calculate A : B in the L − R form as a number(A : B

)LR

forthe same fuzzy numbers A =

(6, 2, 1

)LR

and B =(2, 1, 2

)LR

to create theresult

(A : B

)LR

=(6, 2, 1

)LR

:(2, 1, 2

)LR

=

(6

2,

(6 · 2 + 2 · 2)

22,

(6 · 1 + 2 · 1)

22

)LR

=(3, 4, 2

)LR

.

We get different division results when comparing the interval and the L−R forms.

Let us emphasize that even the conceptions of arithmetic operations per-formed on fuzzy numbers in the interval forms provide us with supportsof results only. To attach the membership functions to these supports weshould combine the point

(a, 0

)corresponding to the left border with the

peak(m, 1

)to design the left branch of the membership function as well as

to tie(m, 1

)to(b, 0

)to produce the right part of the function. The shapes

are styled due to the users’ expectations.

20 Chapter 2

2.3 Arithmetic Operations in the Set of Fuzzy Num-bers Converted to α−cut Form

If we wish to find a membership curve fitted for results of arithmetical oper-ations without making some special selection arrangements, then we shouldapply the operation formulas already discussed for interval forms of data,provided that the data is interpreted by means of the α−cut.

Each α−cut of A constitutes an interval stretched along the abscissa.The α−cut form of a fuzzy number A, based on the determination of Aα anddenoted by A

(α), is a non fuzzy set, dependent on α, defined as an inter-

val A[α] = [a1(α), a2(α)], 0 < α < 1, where a1(α), a2(α) are the left and theright reference functions [10, 54].

Example 2.16. Let Aint = [4, 8, 10]. We will expand the general formulaof α−cuts of A, A[α] = [a1(α), a2(α)] for 0 < α < 1. For Aint = [4, 8, 10] weextract aA = 4, mA = 8 and bA = 10.

Let us suppose that

a1(α)= k1α+ c1 (2.16a)

a2(α)= k2α+ c2 (2.16b)

where k1, k2, c1 and c2 are constants. Equations (2.16a) and (2.16b) mustsatisfy conditions (2.17) and (2.18). For

α = 0 =⇒{a1(α)= aA

a2(α)= bA

(2.17)

and forα = 1 =⇒ a1

(α)= a2

(α)= mA (2.18)

Value α = 0 implies a1(α)= aA and a2

(α)= bA leading to k1 · 0 + c1 =

4 and k2 · 0 + c2 = 10 which provides us with c1 = 4 and c2 = 10.

If α = 1, we will expect that the system{a1(α)= mA

a2(α)= mA

equivalent to {k1 · 1 + c1 = 8

k2 · 1 + c2 = 8

reveals, after inserting c1 = 4 and c2 = 10, the rest of coefficients k1 = 4 andk2 = −2.

Due to (2.17) and (2.18), we obtain a1(α)= 4+4α and a2

(α)= 10−2α.

Hence, A[α] = [a1(α), a2(α)] = [4 + 4α, 10− 2α], 0 ≤ α ≤ 1.

Chapter 2 21

The α−cuts of fuzzy numbers are always closed and bounded intervals forevery α ∈ [0, 1]. Let us emerge two fuzzy numbers A and B in the α−cutforms as A[α] =

[a1(α), a2

(α)]

and B[α] =[b1(α), b2(α)]

, 0 ≤ α ≤ 1.On the basis of the arithmetic principles valid for interval forms of fuzzynumbers we thus generate the operations on fuzzy numbers in α−cut formsas follows.

2.3.1 Addition of A[α] and B[α]

The addition of A[α] and B[α] can be presented by

A[α] +B[α] =[a1(α)+ b1

(α), a2

(α)+ b2

(α)], 0 ≤ α ≤ 1. (2.19)

Example 2.17. Let Aint = [1, 3, 4] and Bint = [2, 5, 7] be two triangularfuzzy numbers. We wish to develop A[α] +B[α], 0 ≤ α ≤ 1.

After converting the interval forms to the α−cut forms, we get

Aint = [1, 3, 4] =⇒ A[α] = [1 + 2α, 4− α]

and

Bint = [2, 5, 7] =⇒ B[α] = [2 + 3α, 7− 2α]

to accomplish the addition as

A[α] +B[α] =[a1(α)+ b1

(α), a2

(α)+ b2

(α)]

=[(1 + 2α

)+(2 + 3α

),(4− α

)+(7− 2α

)]=[3 + 5α, 11− 3α

].

If α = 0, we obtain S(A+B

)=[3, 11

]. If α = 1, we obtain mA+B = 8.

Let us now demonstrate a method of employing the α−cut form of the ad-dition result to generate its membership function consisting of two unknownbranches.

Example 2.18. Since α ∈ [0, 1] constitutes the μA+B(x) value, then weset x = 3 + 5α for 3 ≤ x ≤ 8 and x = 11− 3α for 8 ≤ x ≤ 11. We solve theequations with respect to α to obtain α =

x− 3

5and α =

11− x

3i.e.,

y = μA+B(x) =

⎧⎪⎨⎪⎩x− 3

5for 3 ≤ x ≤ 8,

11− x

5for 8 ≤ x ≤ 11.

22 Chapter 2

2.3.2 Subtraction of A[α] and B[α]

The subtraction of A[α] and B[α] is given by

(A−B

)[α] =

[a1(α)− b2

(α), a2

(α)− b1

(α)], 0 ≤ α ≤ 1. (2.20)

Example 2.19. Let Aint = [1, 3, 4] and Bint = [3, 6, 8]. We concate-nate them to get

(A − B

)[α], 0 ≤ α ≤ 1. After interpreting Aint =

[1, 3, 4] as A[α] = [1 + 2α, 4 − α] and Bint = [3, 6, 8] as B[α] =[3 + 3α, 8− 2α], we are provided with the result(

A−B)[α] =

[(1 + 2α

)− (8− 2α

),(4− α

)− (3 + 3α

)]=[−7 + 4α, 1− 4α

],

which possesses S(A−B

)=[−7, 1

] (α = 0

)and mA−B = −3,

(α = 1

).

By extracting α from the equation x = −7 + 4α and x = 1 − 4α weexpand the membership function of

(A−B

)[α] in the form

y = μA−B(x) =

⎧⎪⎨⎪⎩x+ 7

4for −7 ≤ x ≤ −3,

1− x

4for −3 ≤ x ≤ 1.

2.3.3 Multiplication of A[α] and B[α]

Formula (2.21) presents the multiplication of A[α] and B[α] as

(A ·B)[α] = [

a1(α) · b1(α), a2

(α) · b2(α)], 0 ≤ α ≤ 1. (2.21)

Example 2.20. For Aint = [1, 2, 4] and Bint = [3, 5, 7] we will find theproduct of

(A ·B)[α], 0 ≤ α ≤ 1.

We first convert the proposed fuzzy numbers in the interval form tothe α−cut forms as follows:

Aint = [1, 2, 4] =⇒ A[α] = [1 + α, 4− 2α]

andBint = [3, 5, 7] =⇒ B[α] = [3 + 2α, 7− 2α]

to estimate(A ·B)[α] = [(

1 + α) · (3 + 2α

),(4− 2α

) · (7− 2α)]

=[2α2 + 5α+ 3, 4α2 − 22α+ 28

]as the number with S

(A ·B) = [3, 28] and mA·B = 10.

Chapter 2 23

To find the membership functions over the intervals [3, 10] and [10, 28],we let x = 2α2 + 5α + 3 and x = 4α2 − 22α + 28, respectively. Aftersolving x = 2α2+5α+3 with respect to α, we obtain two roots α1 = μ1(x) =−5 +

√1 + 8x

4and α2 = μ2(x) =

−5−√1 + 8x

4. To make a decision about

which part should be adopted as a left branch of the A · B’s membershipfunction, we suppose that the function value in the left border x = 3 is equalto zero. Because of μ1(3) = 0 contra μ2(3) �= 0, we select μ1 as the leftpart of the membership function of the product. We can even confirm theselection of μ1 by analyzing another property of μ1(10) = 1 typical of themean of a fuzzy number versus μ2(10) �= 1.

We repeat the same procedure for the equation 4α2−22α+28 = x to be

furnished with the appropriate membership function μ(x) =22−√

36 + 16x

8playing role of the right membership function of A ·B over x ∈ [10, 28].

Finally, we submit the membership function of A ·B as a split definition

y = μA·B(x)=

⎧⎪⎨⎪⎩−5 +

√1 + 8x

4for 3 ≤ x ≤ 10,

22−√36 + 16x

8for 10 ≤ x ≤ 28,

with a graph depicted in Fig. 2.3.

y

x

Figure 2.3: The multiplication of A[α] = [1 + α, 4 − 2α] and B[α] = [3 +2α, 7− 2α] in the α-cut forms

2.3.4 Division of A[α] and B[α]

The following formulation represents the division of A[α] and B[α]

24 Chapter 2

(A : B

)[α] =

[a1(α), a2

(α)] ·

[1

b2(α) , 1

b1(α)]

=

[a1(α)

b2(α) , a2

(α)

b1(α)], 0 ≤ α ≤ 1.

(2.22)

Example 2.21. Let A[α] = [α, 2−α] and B[α] = [1+α, 3−α], 0 ≤ α ≤ 1.Hence

(A : B

)[α] =

[α, 2− α

] ·[

1

3− α,

1

1 + α

]=

[a

3− α,2− α

1 + α

]

with S(A : B

)= [0, 2] and mA:B = 1/2. The left membership function,

that constitutes the solution ofα

3− α= x with respect to α, is found as

μleft(x) =3x

1 + x, 0 ≤ x ≤ 1/2. For

2− α

1 + α= x we seek μright(x) =

2− x

1 + x,

1/2 ≤ x ≤ 2. Both functions are depicted in Fig. 2.4.

x

y

Figure 2.4: The division of A[α] = [α, 2 − α] and B[α] = [1 + α, 3 − α] inthe α-cut forms

To demonstrate the utilization of the lastly introduced operations, letus apply them in the prediction of the survival length, expected as a fuzzynumber Y and affected by the biological marker X = CRP -value (C-reactiveproteins). The CRP is also estimated as a fuzzy number and consideredby a physician as the most crucial biological recognizer of the presence ofgastric cancer. We thus seek an equation of the fuzzy regression line, whichbinds X and Y .

Chapter 2 25

Example 2.22. Suppose that the physician has reported some data con-cerning the heightened CRP -value presence in a group of patients who diedafter suffering from gastric cancer. As the values of CRP for almost thesame post-operative survivals do not differ very much from each other, thenwe will decide to use fuzzy numbers in their mathematical representation.We sample the close CRP -values assisting the close survival periods in theconnected groups to treat the medians of the respective groups as meanvalues of fuzzy numbers being their representatives. The groups’ minimalvalues play roles of left borders of the numbers while the groups’ maximalquantities will constitute the right borders in fuzzy numbers.

Suppose that we have stated “CRP close to 16” = X1 = [13, 16, 18]int= [13 + 3α, 18 − 2α] corresponding to “survival length close to 2 years” =Y1 = [1, 2, 3]int = [1 + α, 3 − α] and “CRP very close to 30” = X2 =[28, 30, 32]int = [28+ 2α, 32− 2α] characteristic of “survival length close to1 year” = Y2 = [0, 1, 2]int = [α, 2− α].

We want to find the equation Y = AX+B which is particularly satisfiedfor the pairs (X1, Y1) and (X2, Y2). By solving the system of equations

{AX1 +B = Y1

AX2 +B = Y2

with respect to A and B, we evaluate A =Y2 − Y1X2 −X1

and B = Y2− Y2 − Y1X2 −X1

X2.

Thus

A =

[α, 2− α

]− [1 + α, 3− α

][28 + 2α, 32− 2α

]− [13 + 3α, 18− 2α

] =

[−3 + 2α

19− 5α,

1− 2α

10 + 4α

]

and

B =[α, 2− α

]−[−3 + 2α

19− 5α,

1− 2α

10 + 4α

][28 + 2α, 32− 2α

]

=

[−32 + 76α

10 + 4α,122− 79α+ α2

19− 5α

].

The regression line is built as

[Y1[α], Y2

[α]]

=

[−3 + 2α

19− 5α,

1− 2α

10 + 4α

]·[X1

[α], X2

[α]]

+

[−32 + 76α

10 + 4α,122− 79α+ α2

19− 5α

].

26 Chapter 2

For, e.g., X = [8, 9, 10]int = [8 + α, 10 − α], we estimate the survivallength as fuzzy number “close to 5 years” in spite of its borders and the mean,which are computed as fractions.

Even if the fuzzy regression constitutes a useful tool of inference we intendto test other methods to find more accurate evaluations of survival length.These can be affected by more independent variables than only one. In thenext chapters we will prove the action of fuzzy controllers.

To sum up the current chapter we can mention that all operations onfuzzy numbers, independently of their forms, are grounded on the extensionprinciple [23, 24, 114] that substantially affects obtained results.

The utilization of the α-cut forms of results facilitates a procedure ofgenerating their membership functions fitted best for the results’ occurrences,which should be regarded as an advantage of the method promoting the α-cutforms involved in calculations.

In the following chapters, the applications of medication prognoses will beconcerned. Chapter 3, the technique of Mamdani fuzzy controller is adoptedto evaluate the life expectancy to older gastric cancer patients.

CHAPTER3The Mamdani Controller inPrediction to the SurvivalLength in Elderly GastricCancer Patients

In the current chapter we will study the control actions to improve the es-timates of the survival length. We wish to treat the survival length as adependent variable, which is affected by some biological parameters. Strictanalytic formulas are the tools usually derived for determining the formal re-lationships between a sample of independent variables and a variable whichthey affect. If we cannot formalize the function tying the independent anddependent variables then we will utilize some control actions.

3.1 The Introduction to a Control System

Apart from crisp version of control we often adopt its fuzzy variant developedby Mamdani and Assilian [55]. Fuzzy control algorithm is furnished withsofter mechanisms when comparing it to classical control, which constitutesits advantage in the processing of vague or imprecise information.

The algorithm is particularly adaptable to support medical systems, of-ten handling uncertain premises and conclusions. From the medical point ofview it would be desirable to prognosticate the survival length for patientssuffering from gastric cancer. We thus formulate the objective of the chap-ter as the utilization of fuzzy control action for the purpose of making thesurvival prognoses [80, 90, 111].

Fuzzy set theory allows us to describe complex systems by using ourknowledge and experience in transparent English-like rules. It does not need

27

28 Chapter 3

complex mathematical equations and system modeling that governs the re-lation between inputs and outputs. Fuzzy rules are very appealing to use bynon-experts also.

Expert-knowledge designs together with assumptions of fuzzy set theoryhave given rise to the creation of fuzzy control, see e.g., [3, 5, 55, 62, 103].Experience-based rules constitute the crucial part of fuzzy controllers, whichhave found many adherents to apply them in order to support solutions ofcomplex systems not characterized by formally stated mathematical formu-las. Typical applications of fuzzy control have mostly concerned technicalprocesses. The applications range from cameras to model cars and trains[114].

Due to the possibility of making input and output variables verballyexpressed, which constitutes an advantage in the process of preparing theimprecise data strings, fuzzy control has also been tested in medicine. Thereduction of post-operative pain [33] or the blood pressure regulation [13,43] constitute some examples of medical experiments accomplished by fuzzycontrollers. We can learn about how adaptive mechanisms coupled with fuzzycontrollers regulate the mean arterial pressure and how the fuzzy controlsolutions deprive a patient of the postoperative pain.

The evaluation of survival length was already accomplished by statisticalmethods.

In the first trials of survival approximation a survival curve from censoreddata was introduced [47]. The model was used in cancer patient examinationsto estimate the length of living [61]. The Cox regression [19] of life lengthprediction was developed in such studies as logistic Cox regression [98]. Thestatistics-based models predicting the survival were compared by Everitt andRabe-Hesketh [27] who found such model disadvantages as the lack of normaldistribution or missing values among survival times.

A typical fuzzy control system normally consists of three parts. The firstpart refers to the fuzzification process of input and output variables. Theseare first linguistically differentiated in levels. The levels are listed as names,which are demonstrated by fuzzy numbers with assigned to them appropriatemembership functions.

The second part consists of processing procedures. We rely on own expe-rience when we prepare rules to link the linguistic terms of input variables tothe control output state. The rules employed in the model are constructed asIF-THEN statements. On the basis of the rules verbally formulated and ac-tual for individual data values we estimate their mathematical consequencesexpressed as a collection of fuzzy sets. The creation of a sampling of allconsequence sets results in one final consequence fuzzy set and terminatesthe procedures of the second step of fuzzy control. The last step of the fuzzycontrol system is to defuzzify the final fuzzy output set being the result ofthe second stage. We adopt the centre of gravity (COG) method [114] toconvert the fuzzy set into a crisp value corresponding to the initial crisp

Chapter 3 29

input data.We thus intend to prove fuzzy control to make some prognoses concerning

the survival length in patients whose disease has been diagnosed as gastriccancer. To make the conclusions reliable we select two clinical markers “age”and “CRP-value” due to the physicians’ expertise. The choice of CRP andage, as representative markers of post-surgical survival in cancer diseases,has been made due to the latest investigations revealing associations of theseindices with the progression of disease in many cancer types [20, 50].

3.2 Fuzzification of Input and Output Variable En-tries in Survival Length Estimation

Fuzzy control model is applied in research to some relationships betweena collection of independent variables and the dependent of them variablewhen we cannot mathematically formalize the functional connection amongthem. We are expected to evaluate the survival length in patients withdiagnosis “gastric cancer ”. The period of survival is affected by two biologicalparameters X = “age” and Y = “CRP-value”, which are selected as themost essential markers of making the prognosis. We cannot formally derivea function, which relates the independent variables X = “age” and Y =“CRP-value” to the dependent variable Z = “survival length”, therefore wewill adapt such fuzzy controller, which supports estimation of dependentvalues in spite of the lack of a formula concerning z = f(x, y), x ∈ X,y ∈ Y , z ∈ Z.

All variables are now differentiated into levels, which are expressed bylists of terms. The terms from the lists are represented by fuzzy numbers,restricted by the parametric s-functions lying over the variable domains[xmin, xmax], [ymin, ymax] and [zmin, zmax] respectively.

In conformity with the physician’s suggestions we introduce five levelsof X, Y and Z as the collections X = “age” =

{X0 = very young, X1 =

young, X2 = middle-aged, X3 = old, X4 = very old}, Y = “CRP-value” ={

Y0 = very low, Y1 = low, Y2 = medium, Y3 = high, Y4 = very high}

and Z= “survival length” =

{Z0 = very short, Z1 = short, Z2 = middle-long, Z3 =

long, Z4 = very long}.

To obtain a family of membership functions of fuzzy numbers standingfor the terms of the respective lists we will modify the parametric s-classfunctions [74, 114]. For Xi, i = 0, . . . , 4, we design [76, 83, 86, 87, 88]

μXi(x) =

{leftμXi(x),

rightμXi(x),(3.1)

30 Chapter 3

where

leftμXi(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

2

(x− ((xmin − hX) + hX · i)

hX

)2

for(xmin − hX

)+ hX · i ≤ x ≤ (

xmin − hX2

)+ hX · i,

1− 2

(x− (

xmin + hX · i)hX

)2

for(xmin − hX

2

)+ hX · i ≤ x ≤ (xmin) + hX · i,

(3.2)

rightμXi(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1− 2

(x− (xmin + hX · i)

hX

)2

for(xmin

)+ hX · i ≤ x ≤ (

xmin + hX2

)+ hX · i,

2

(x− ((xmin + hX) + hX · i)

hX

)2

for(xmin + hX

2

)+ hX · i ≤ x ≤ (xmin + hX) + hX · i.

(3.3)Formulas (3.2) and (3.3) depend on the minimal value xmin, which starts

the X-variable domain. The structures (3.2) and (3.3) are also affectedby the value of a parameter hX , which estimates the length between thebeginnings of membership functions constructed for two adjacent terms ofX. The hX quantity is adjusted to the number of functions in the X-list andto the distance between the minimal and the maximal value of the X-variabledomain.

The membership functions of Yj , j = 0, . . . , 4, constructed for theaccommodated values of parameters hY and ymin to the conditions of Y , areyielded by

μYj (y) =

{leftμYj (y),

rightμYj (y),(3.4)

for

leftμYj (y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

2

(y − ((ymin − hY ) + hY · j)

hY

)2

for(ymin − hY

)+ hY · j ≤ y ≤ (

ymin − hY2

)+ hY · j,

1− 2

(y − (

ymin + hY · j)hY

)2

for(ymin − hY

2

)+ hY · j ≤ y ≤ (ymin) + hY · j,

(3.5)and

Chapter 3 31

rightμYj (y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1− 2

(y − (ymin + hY · j)

hY

)2

for(ymin

)+ hY · j ≤ y ≤ (

ymin + hY2

)+ hY · j,

2

(y − ((ymin + hY ) + hY · j)

hY

)2

for(ymin + hY

2

)+ hY · j ≤ y ≤ (ymin + hY ) + hY · j.

(3.6)Finally, the Zk’s functions, k = 0, . . . , 4, are derived as

μZk(z) =

⎧⎪⎨⎪⎩leftμZk

(z)

middleμZk(z)

rightμZk(z)

(3.7)

with

leftμZk(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − (zmin − hZ

2 + hZ · k)hZ2

)2

for zmin − hZ2 + hZ · k ≤ z ≤ zmin − hZ

4 + hZ · k,1− 2

(z − (zmin + hZ · k)

hZ2

)2

for(zmin − hZ

4

)+ hZ · k ≤ z ≤ (zmin + hZ) · k,

(3.8)

the central part

middleμZk(z) = 1 for zmin + hZ · k ≤ z ≤ zmin +

hZ2

+ hZ · k (3.9)

and

rightμZk(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

1− 2

(z − (zmin + hZ

2 + hZ · k)hZ2

)2

for zmin + hZ2 + hZ · k ≤ z ≤ zmin + 3hZ

4 + hZ · k,2

(z − (zmin + hz + hZ · k)

hZ2

)2

for(zmin + 3hZ

4

)+ hZ · k ≤ z ≤ (zmin + hZ) + hZ · k.

(3.10)The parameter hZ allows designing five functions of fuzzy numbers from

Z over [zmin, zmax]. We return to the variable X = “age”, which is differ-entiated in five levels X = “age”=

{X0 = very young, X1 = young, X2 =

middle-aged, X3 = old, X4 = very old}

and restricted over the interval[xmin, xmax] = [0, 100], to state xmin = 0, hZ = 25 and i = 0, . . . , 4. For

32 Chapter 3

xiX ''''

0X 1X 2X 3X 4Xvery young young middle-aged old very old

x

Figure 3.1: The membership functions for the “age”

the terms of “age” from the list above we will obtain by (3.2) and (3.3) afamily of the membership functions sketched in Fig. 3.1.

By inserting new parameters of ymin and hY in (3.5) and (3.6) we generatethe membership functions for “CRP-value”. We have differentiated Y =“CRP-value” =

{Y0 = very low, Y1 = low, Y2 = medium, Y3 = high, Y4 =

very high}

over the interval [0, 50]. The design of installing five functionswith the peak of Y0 in (0, 1) and the peak of Y4 in (50, 1) demands theselection of hY = 15. We plot the Yj functions in Fig. 3.2 by setting in turnthe serial numbers of j in (3.5) and (3.6), where j = 0, . . . , 4.

yjY ''''

y

0Y 1Y 2Y 3Y 4Yvery low low medium high very high

Figure 3.2: The membership functions for the “CRP-value”

The output variable Z = “survival length” ={Z0 = very short, Z1 =

Chapter 3 33

short, Z2 = middle-long, Z3 = long, Z4 = very long}

takes the values in theinterval [0, 5] years. We determine hZ = 1 and set k = 0, . . . , 4 in (3.8),(3.9) and (3.10) to initialize the functions depicted in Fig. 3.3.

zkZ ''''

0Z 1Z 2Z 3Z 4Zvery short short middle-long long very long

z

Figure 3.3: The membership functions for the “survival length”

We emphasize the importance of the parametric design of functions. In-stead of implementing fifteen formulas of the similar nature we have sampledall functions in three generic groups. Any time we can involve the desiredfunction in necessary computations by setting its number in the proper for-mula concerning X, Y or Z. Moreover, the mathematical scenario of mem-bership functions is established in the formal and elegant designs, which canbe segments of a computer program for the reason of their nature letting thecreation of loops.

3.3 The Rule Based Processing Part of SurvivingLength Model

After the fuzzification procedure we are able to create the rule bases, whichlink the states of the two input variables to the state of the output variable.We thus design a table in which the entries are filled with terms of “survivallength”. To express the states of the survival length as logically as possible,we have studied the behavior of variables on the basis of biological datasamplings. The cells of the table are characterized by subintervals of domainsof X and Y .

We first estimate the survival length median in the samplings of thedata corresponding to considered cells. The median value was set as z inthe membership functions of all sets listed in the Z-space. We select thisfuzzy number Zk as a representative of the cell, in which the membership

34 Chapter 3

degree of the median was largest. The technique of combining the humanexperience with data sets obtained for discrete samples to make conclusionsfor continuous samples is a modern branch of so-called “integration systems”.The estimations of survival length are collected in Table 3.1.

Table 3.1: Rule base of fuzzy controller estimating “survival length”Xi � Yj very low low medium high very highvery youngyoungmiddle-age middle-longold middle-long short short short very shortvery old short short very short very short very short

In the first row of Table 3.1, the “CRP-values” are placed from “very low”to “very high”. In the first column, the “age” is presented from “very young”to “very old”. Some entries in the table are empty, since the essential datawere lacking for younger people. It rarely happens to find young patientswith diagnosis “gastric cancer”, therefore we could not make any reliableconclusions concerning survival in this age group.

Suppose that we would like to make the survival prognosis for (x, y),x ∈ X, y ∈ Y - with other words we want to evaluate z = f(x, y) whenassuming that the f -formula is not developed.

Furthermore, x belongs to the different fuzzy numbers Xi, i = 0, . . . , 4,being the fuzzy subsets of X, with different membership degrees equallingμXi(x). Element y associated to x is a member of the fuzzy numbers Yj ,j = 0, . . . , 4, constituting the fuzzy subsets of Y with the membershipdegrees μYj (y) . By means of the IF-THEN statements grounded on the basisof Table 3.1, we can determine the contents of rules by attaching the pair ofinput variable levels to a level of the output variable according to (3.11):

Rule R(x, y):l : If x is Xi:l and y is Yj:l, then z is Zk:l, (3.11)

where l is the rule number. The expressions Xi:l, Yj:l and Zk:l denote thefuzzy numbers Xi, Yj and Zk assisting rule number l, which has been foundfor actual x and y. To evaluate the influence of the input variables on theoutput consequences we need an estimate α(x, y):l found by performing theminimum operation

α(x, y):l = min(μXi:l(x), μYj :l(y)

), (3.12)

for each Xi:l and Yj:l concerning the choice of (x, y).We use α(x, y):l and the minimum operator to determine consequences of

all rules R(x, y):l for the output. Fuzzy sets Rconseq(x, y):l, stated in the output

space Z, will have the membership functions presented in

Chapter 3 35

μconseqR(x, y):l

(z) = min(α(x, y):l, μZk:l(z)

). (3.13)

In the last step of the algorithm we aggregate the consequence setsRconseq

(x, y):l in one common set conseq(x, y) allocated in Z over a continuousinterval [z0, zn]. To derive the membership function of conseq(x, y) we provethe action of the maximum operator in the form of (3.14):

μconseq(x, y)(z) = max

l

(μconseq(x, y):l(z)

)(3.14)

3.4 Defuzzification of the Output Variable

In order to assign a crisp value Z to the selected pair (x, y), we defuzzifythe consequence fuzzy set expressed by (3.14) in Z. We will thus indicatethe expected value of the survival length for a gastric cancer patient whose“age” x and “CRP-value” y have been examined.

One can find different kinds of defuzzification methods in literature [114].We are appealed by properties of the centre of gravity method (COG) as amodel of computing which is easy to perform and clearly interpretable. Weexpand COG as (3.15):

z = f(x, y)

=

zn∫z0

z · μconseq(z) dz

zn∫z0

μconseq(z) dz

=

z1∫z0

z · μconseq(z) dz + · · ·+zn∫

zn−1

z · μconseq(z) dz

z1∫z0

μconseq(z) dz + · · ·+zn∫

zn−1

μconseq(z) dz

(3.15)

with the inner borders z1, . . . , zn−1 being either z-coordinates of intersectionpoints between adjacent branches of the conseq(x, y) membership functionsor characteristic support values of fuzzy numbers included in conseq(x, y).

3.5 The Survival Length Prognosis for a SelectedPatient

Suppose that we examine a 77-year-old patient, whose “CRP-value” is 16.His diagnosis is determined by a physician as “gastric cancer”. We wish toestimate theoretically the expected value of his survival length by proving the

36 Chapter 3

algorithm sketched in the previous sections. The information is confidentialand used only by the physician.

Let x = 77 and y = 16. Age 77 belongs to the fuzzy number X3 =old. Therefore, for i = 3, hX = 25 and xmin = 0, we allocate x = 77 inthe interval xmin + hX · i ≤ x ≤ xmin + hX

2 + hX · i ⇐⇒ 0 + 25 · 3 ≤ x ≤0 + 25

2 + 25 · 3 ⇐⇒ 75 ≤ x ≤ 87.5 with the membership degree μX3(77) =

μold (77) = 1− 2(77−(xmin+hX ·i)

hX

)2= 1− 2

(77−(0+25·3)

25

)2= 0.9872.

The same x = 77 is a member of another fuzzy number X4 = very oldwith the membership degree of μX4(77) = μvery old(77) = 0.0128. The “CRP-value” 16 belongs to the fuzzy number Y1 = low with the membership degreeμY1(16) = μlow(16) = 0.991 and Y2 = medium with the membership degreeμY2(16) = μmedium(16) = 0.009.

In accordance with (3.11) the rules which connect the states of the inputvariables to the output variable levels are established as:

• R(77, 16):1: If “age” is old and the “CRP-value” is low, then “survivallength” will be short.

• R(77, 16):2: If “age” is old and the “CRP-value” is medium, then“survivallength” will be short.

• R(77, 16):3: If “age” is very old and the “CRP-value” is low, then“survivallength” will be short.

• R(77, 16):4: If “age” is very old and the “CRP-value” is medium, then“survival length” will be very short.

To evaluate the influences of the input variables on the output conse-quences due to (3.12), we estimate α(77, 16):l, l = 1, . . . , 4 as four quantities:

• α(77, 16):1 = min(μX3:1(77), μY1:1(16)

)= min

(μold(77), μlow(16)

)=

min(0.9872, 0.991)

)= 0.9872,

• α(77, 16):2 = min(μX3:2(77), μY2:2(16)

)= min

(μold(77), μmedium(16)

)=

min(0.9872, 0.009)

)= 0.009,

• α(77, 16):3 = min(μX4:3(77), μY1:3(16)

)=min

(μvery old(77), μlow(16)

)=

min(0.0128, 0.991)

)= 0.0128

and

• α(77, 16):4 = min(μX4:4(77), μY2:4(16)

)=min

(μvery old(77), μmedium(16)

)= min

(0.0128, 0.009)

)= 0.009.

In conformity with formula (3.13) we obtain the fuzzy subsets of theconsequences. The first set Rconseq

(77,16):1 has a membership function given by

Chapter 3 37

μconseqR(77, 16):1

(z) = min(α(77, 16):1, μZ1:1(z)

)= min

(0.9872, μ“short′′(z)

)and depicted in Fig. 3.4.

shortZ 1z

kZ ""

z

9872.01:16,77

Figure 3.4: The fuzzy subset of consequences constructed due to R(77, 16):1

The next set Rconseq(77,16):2 is characterized by a constraint

μconseqR(77, 16):2

(z) = min(α(77, 16):2, μZ1:2(z)

)= min

(0.009, μ“short′′(z)

)drawn in Fig. 3.5.

zkZ ''''

shortZ1

z009.02:16,77

Figure 3.5: The fuzzy subset of consequences constructed due to R(77, 16):2

The third set Rconseq(77,16):3 possesses a function presented by

38 Chapter 3

μconseqR(77, 16):3

(z) = min(α(77, 16):3, μZ1:3(z)

)= min

(0.0128, μ“short′′(z)

)whose graph is revealed in Fig. 3.6.

zkZ ''''

shortZ1

0128.03:16,77 z

Figure 3.6: The fuzzy subset of consequences constructed for R(77, 16):3

Finally, the last set Rconseq(77,16):4 is restricted by a function

μconseqR(77, 16):4

(z) = min(α(77, 16):4, μZ0:4(z)

)= min

(0.009, μ“veryshort′′(z)

)seen in Fig. 3.7.

zkZ ''''

z009.04:16,77

shortveryZ0

Figure 3.7: The fuzzy subset of consequences constructed in accord toR(77, 16):4

When applying formula (3.14) we concatenate all μconseqR(x, y):l

(z), l = 1, . . . , 4,in order to determine a common consequence of rules (3.11) fitted for the

Chapter 3 39

pair (77, 16). The fuzzy subset of the universe Z will be thus yielded by itsmembership function shown below:

μconseq(77, 16)(z) = max

1≤l≤4

(μconseq(77, 16):l(z)

)The fuzzy set conseq(77, 16) is aggregated in Fig. 3.8.

0z 1z 2z4z 5z

6z3z

zconseq

z

Figure 3.8: The total consequence set conseq(77, 16)

Formula (3.15) constitutes a basis of an estimation of the survival lengthexpected when assuming “age” = 77 and “CRP −value” = 16. Over interval[z0, z6] = [0, 2], which contains characteristic points z0 = 0, z1 = 0.533,z2 = 0.75, z3 = 0.96, z4 = 1.54, z5 = 1.75 and z6 = 2, we compute thez-prognosis as follows:

z = f(77, 16)

=

0.533∫0

0.009 · z dz + · · ·+2∫

1.75

2(z−20.5

)2 · z dz0.533∫0

0.009 dz + · · ·+2∫

1.75

2(z−20.5

)2dz

= 1.05

For the patient who is 77 years old and has the CRP-value equal to 16, thetheoretical estimated survival length is about 1 year. The result convergeswith the physician’s own judgment made on the basis of his medical reports.For each pair (x, y) we can arrange new computations due to the fuzzycontrol algorithm to estimate the patient’s period of surviving in the case ofsuffering from gastric cancer.

40 Chapter 3

The Mamdani fuzzy control system is a powerful method, which mostlyis applied to technologies controlling complex processes by means of humanexperience. In this work we have proved that the expected values of patients’survival lengths can be estimated even if the mathematical formalizationinvolving independent and dependent variables is unknown. For each x andeach y belonging to continuous spaces X and Y respectively, we are ableto repeat the control algorithm in order to cover the space of pairs over theCartesian product of X and Y with a continuous surface. This will constitutea part of our future work.

In the next chapter we wish to confirm the magnitude of survival lengthapproximation by testing the Sugeno controller.

CHAPTER4Verification of Survival LengthResults by Means of SugenoController

In the rules, constituting the crucial part of control processing, the levelsof the independent variables have been tied to a selected level representingthe dependent parameter. All levels have been further replaced by fuzzynumbers. The operations recommended by the Mamdani controller havebeen performed on membership functions of these fuzzy representatives oflevels.

To shorten the action of the processing part in the Mamdani controller,Sugeno [101, 102] proposed another approach to the creation of rules, inwhich the dependent variable level will be determined by a functional con-nection of independent variables.

4.1 Adaptation of the Processing Part of the FuzzyController to Sugeno-made Assumptions

We still wish to evaluate the survival length in gastric cancer patients dueto information about their age and CRP-value. In this new version of afuzzy controller, called the Sugeno controller, we preserve the former resultsof the fuzzification of independent variables, i.e., we still keep alive the levelsof variables X = “age” and Y = “CRP-value” with assisting membershipfunctions (3.1), (3.2) and (3.3) for X, as well as (3.4), (3.5) and (3.6) for Y .

The dependent variable Z = “survival length” is not differentiated intolevels anymore. Instead, for each combination of X- and Y -levels we derivea linear function of the general shape f(x, y) = ax+ by+ c. This procedurecan only work in the case of possessing some data triples (x, y, z), which

41

42 Chapter 4

come from the examinations carried out on patients belonging to the desiredcombinations of levels. We support our experience by engaging discretepoint sets to predict the information, which can be obtained for continuousintervals of X and Y .

Example 4.1. The triples (“age”, “CRP-value”, “survival length”) = (x, y, z)belong to the set {(77, 18, 0.5), . . . , (81, 21, 0.9)}, in which x-values cor-respond to level X3 = “old ” and y-values are typical of Y1 = “ low ”. Thedependent variable z = f(x, y) has taken values between 0.1 and 0.8. Thedata is withdrawn from the patients’ reports. Thus, by the adoption oflinear regression, for the couple of levels X3 and Y1 we find the functionaldependency z = f(x, y) = 0.13057x + 3.4256 · 10−3y − 9.4426 shown inFig. 4.1.

Age

80 10040

0

1

2Survival length

5

80

3

4

60

CRP-value6020

0 20 400

Figure 4.1: The linear functional dependency between age = X3 =“old ”, CRP = Y1 = “ low ” and survival length Z

In the IF . . . THEN . . . rule (3.11) the Z-level indicates the character ofdependency between levels of X and Y . In the Sugeno IF . . . THEN . . . rulethe function z = f(x, y) is not assimilated to any level of variable Z fromChapter 3. We formulate a new pattern of (3.11) as

Rule R(x, y):l : If x is Xi:l and y is Yj:l, then z(x, y):l is zk = fk(x, y) (4.1)

where l is the rule number, Xi:l and Yi:l represent the fuzzy numbers Xi andYj , which are associated with the rule number l for the actual pair (x, y).

The formula z(x, y):l = zk = fk(x, y) = akx+bky+ck is a control functionfound for levels Xi:l and Yj:l. The quantities ak, bk and ck are constants. Thecontrol functions have been constructed by the Maple computer program.

As we want to find the functional evaluations for all possible connectionsof levels, selected for independent variables due to Table 3.1, then the number

Chapter 4 43

of functions will equal 11. Hence, the index k, k = 1, . . . , 11, constitutesthe function number with accordance to the next rule base table, introducedas Table 4.1.

Table 4.1: The functional rule base table for combinations of X- and Y-levelsin estimations of survival length

Xi�Yj very low low medium high very highvery young

youngmiddle-age z1

old z2 z3 z4 z5 z6very old z7 z8 z9 z10 z11

As before, we have left some empty cells in the table because of the lackof data for younger patients with diagnosis “gastric cancer”.

Instead of the sophisticated procedure of looking for the final consequenceset, characteristic of the Mamdani controller, we adopt the control func-tion [101, 102, 114].

z = fSugeno(x, y) =

∑l α(x, y):l · (z(x, y):l = fk(x, y))∑

l α(x, y):l(4.2)

which directly delivers a crisp control value of the output variable “survivallength”.

To obtain the value α(x, y):l, we need to perform the minimum operationfor the membership degrees of μXi:l(x) and μYj :l(y) according to (3.12).

4.2 Applications of the Sugeno Fuzzy Controller toEstimation of the Survival Length in GastricCancer Patients

We return to the case of the patient already presented in section 3.5 inChapter 3. By testing the Sugeno controller let us now evaluate the expectedvalue of the survival length for a 77-year-old patient, whose CRP-value is16.

We let x = 77 and y = 16. We have previously stated that age 77 belongsto the fuzzy number X3 = “old ” with membership degree μX3(77) = 0.9872.Element x = 77 also is a member of X4 = “very old ” with the membershipdegree μX4(77) = 0.0128.

The CRP-value y = 16 is found in fuzzy number Y1 = “ low ”, where itsmembership degree equals μY1(16) = 0.991. The same y = 16 takes placein Y2 = “medium”, but the membership degree is determined as μY2(16) =0.009.

44 Chapter 4

The rules IF . . . THEN, which associate the input variables with theoutput variable are determined according to formula (4.1) and Table 4.1 as:

• R(77, 16):1: IF x is X3 and y is Y1 , THEN z(77, 16):1=z3 = f3(x, y)=−4.3948 · 10−2x− 9.6411 · 10−3y + 4.3938,

• R(77, 16):2: IF x is X3 and y is Y2 , THEN z(77, 16):2=z4 = f4(x, y)=−1.0077 · 10−2x− 2.7872 · 10−2y + 2.0658,

• R(77, 16):3: IF x is X4 and y is Y1 , THEN z(77, 16):3=z8 = f8(x, y)=−6.2637 · 10−2x+ 0.26035y + 4.1361

and

• R(77, 16):4: IF x is X4 and y is Y2 , THEN z(77, 16):4=z9 = f9(x, y)=−9.7739 · 10−3x+ 3.3194 · 10−4y + 1.1162.

To estimate the value of α(x, y):l, we perform the minimum operationon each pair of values μXi:l(x) and μYj :l(y). We refer to previously knownresults obtained in Chapter 3.

• α(77, 16):1 = min(μX3:1(77), μY1:1(16)

)= min

(μ“old′′(77), μ“low′′(16)

)= min

(0.9872, 0.991)

)= 0.9872,

• α(77, 16):2 = min(μX3:2(77), μY2:2(16)

)= min

(μ“old′′(77), μ“medium′′(16)

)= min

(0.9872, 0.009)

)= 0.009,

• α(77, 16):3 = min(μX4:3(77), μY1:3(16)

)=min

(μ“very old′′(77), μ“low′′(16)

)= min

(0.0128, 0.991)

)= 0.0128

and

• α(77, 16):4 = min(μX4:4(77), μY2:4(16)

)=min

(μ“very old′′(77), μ“medium′′(16)

)= min

(0.0128, 0.009)

)= 0.009.

After substituting the values of α(x, y):l, l = 1, . . . , 4 and correspondingto them z(x, y):l = fk(x, y), k = 3, 4, 8 and 9, in (4.2) we get

z = fSugeno(77, 16) =

=α(77, 16):1 · f3(77, 16) + α(77, 16):2 · f4(77, 16) + . . .+ α(77, 16):4 · f9(77, 16)

α(77, 16):1 + α(77, 16):2 + . . .+ α(77, 16):4

=0.9872 · 0.8555 + 0.009 · 0.84392 + . . .+ 0.009 · 0.36892

0.9872 + 0.009 + . . .+ 0.009≈ 0.88.

When comparing the results yielded by two controllers we make the con-clusion about their convergence to the approximated survival value aboutone year. The deviation between quantities, related to survival length of

Chapter 4 45

the 77-year-old patient with CRP equaling 16, can be an effect of usingthe planar surface instead of an irregular one in the approximation of pointsets in the Sugeno controller. We wish to formulate the following conclusionsumming up the comparison of both controllers.

4.3 Summary

The algorithm of the Mamdani controller demands a large number of oper-ations in the processing phase, but we can always construct logical rules IF. . . THEN . . . based on own experience in how to cope with different com-binations of levels. Even if the data from point sets are lacking, it is stillpossible to make a trial of designing membership functions for all levels ofvariables by relying on the human expertise.

The Sugeno controller does not need so many operations in the processingstage. Nevertheless, its use is impossible in practice when we cannot befurnished with discrete data sets to accomplish the design of functions fk.The choice of the method is thus dependent on the access to data.

The controllers encounter results coming from statistical experiments,and they do not need special assumptions like normal distributions of thedependent variables.

In the end we emphasize that, unlike the traditional control methods,fuzzy control is the methodology, which deals with many real-life problemssuccessfully. As the conventional control methods often are based on ad-vanced mathematical models, such as differential equations sometimes im-possible to solve, the method of fuzzy control is much more convenient toapply.

In next chapter, the approach of fuzzy c-means clustering analysis will bestudied for exploring the internal structure of a dataset consisting of 25 gas-tric cancer patients. The operation possibility and none operation possibilitywill be analyzed.

46 Chapter 4

CHAPTER5Fuzzy C-means ClusteringApplied to OperationEvaluation for Gastric CancerPatients

Nowadays we are living in an era of the rapid development of informationtechnology. A large number of information is sent and received every day.Therefore, finding some data processing methods to discover the partialstructure in a data set and to utilize useful information to solve efficientlydaily issues becomes of vital importance.

A lot of approaches such as data analysis, pattern recognition and datamining have been put forward. Cluster analysis is one of them. This methodinvolves the task of dividing data points into homogeneous classes or clustersso that items in the same class are as similar as possible and items in differentclasses are as dissimilar as possible [108]. Among clustering approaches thefuzzy c-means clustering (FCM) is regarded as well-known and efficient [11,26, 70, 93].

In hard clustering, data points are divided into crisp clusters, where eachdata point belongs to exactly one cluster [4]. In many situations, boundarydata points can be difficult to be allocated. Therefore, the realistic pictureof the data structure may not be correctly presented by the crisp clustering.However, fuzzy partition can make up the flaw, due to the advantage thatdata points are allowed to belong to more than one cluster.

Making medication prognoses concerning the survival length predictionfor gastric cancer patients presented in Chapter 3 and 4 were the first inter-esting medication applications in the research work. The application of fuzzyc-means cluster with respect to the operation possibility evaluation gives thesecond challenge. In accordance with the physician’s clinical experience, the

47

48 Chapter 5

surgical operation is one of the most effective treatment modalities for gas-tric cancer patients. Three biological parameters, such as the CRP-value(C-reactive proteins), the age and the body weight, play a key roll in thedetermination of operation possibility. Like consequence of any other sur-gical measures, the operation leads to the post-operation pain, side affectsand even risks. In order to keep gastric cancer patients away from unneces-sary suffering, therefore, we attempt to utilize the FCM algorithms to dividea clinical data set of 25 gastric cancer patients into two clusters, whereone presents the positive prognosis for “operation” and the other samplespatients-vectors classified for “no operation”.

5.1 Description of Fuzzy C-Means Clustering Algo-rithm

Let us suppose that X = {x1, . . . , xn} is a finite set of data points. Eachdata point xk = (xk1 , . . . , xkp), k = 1, . . . , n, is a pattern vector in Rp. Fuzzyc-means algorithm tries to partition X in a collection of Si subsets, 2 ≤ i ≤ c,called fuzzy clusters. By running the algorithm repeatedly, a list of vi clustercenters and a partition matrix U are returned.

The fuzzy c-means algorithm is based on minimizing the objective func-tion J with respect to the membership values μsi(xk) and the distanced(vi, xk) [11] in

J =

n∑k=1

c∑i=2

(μsi(xk)

)m · d(vi, xk), (5.1)

n is the number of data points and c is the number of clusters. The valueof μsi(xk) or μik represents the value of membership degree of xk in clusterSi. Moreover, the sum of the membership degrees for each xk sample in allclusters is equal to one. The notation of d(vi, xk) indicates the Euclideandistance between the cluster center vi and xk. The constant m > 1 is calledweighting exponent, which determines the grade of fuzziness of the resultingclusters.

A linguistic description of the FCM algorithm is presented by the follow-ing steps:

1. Select the number of clusters c, initialize the value of fuzzy parameterm (2 ≤ m < ∞) and the termination tolerance ε.

2. Set l = 0.

3. Determine the initial values of membership degrees in partition matrixU l.

Chapter 5 49

4. Calculate cluster centers vli, i = 1, . . . , c, due to [11], as

vli =

∑nk=1

((μlik

)m · xk)

∑lk=1

(μlik

)m . (5.2)

5. Calculate the updated partition matrix U l+1 by using vli in formula

μl+1ik =

(1

d(xk,vli)

)1/m−1

∑cj=1

(1

d(xk,vlj)

)1/m−1. (5.3)

6. If∥∥U l+1 − U l

∥∥ ≥ ε, then set l = l+1, and go to step 4. If∥∥U l+1 − U l

∥∥ ≤ε, then stop the procedure. Matrix U l+1 is the most optimal distribu-tion of membership degrees of xk in clusters Si. Here,

∥∥U l+1 − U l∥∥

=√∑

(μl+1ik − μl

ik)2 represents the Euclidean distance between the

partition matrices U l+1 and U l, in which μl+1ik and μl

ik describes themembership degrees of patient xk belong to cluster Si after the l+1-threspective the l-th iteration, i = 2, . . . , c and k = 1, . . . , n.

The prior determination of the membership degrees of xk in Si plays acrucial role in this algorithm, as their choice not only can affect the con-vergence speed, but also may have a direct impact on the results of theclassification [107].

The initial cluster centers are just prototypes and unstable. Therefore,they need to be iteratively updated. Every iteration guarantees an improve-ment of the coordinates of clustering centers. The updating procedure con-tinues until two adjacent membership matrices cease to change. Later on,we wish to demonstrate how FCM algorithm has been applied to operationevaluations for gastric cancer patients.

Furthermore, the calculation of clustering centers depends on the valuesof initial membership degrees in the partition matrix. To avoid inaccuracy infinal results we will use the technique of calculation of membership degreespresented in [78, 79, 80, 81] to avoid guessing at their values intuitively.

5.2 Determination of the Initial Membership De-grees in the Partition Matrix

The accurate evaluation of the membership of xk in Si can improve theiteration time and the convergence speed. The s-class membership functionis adopted for the further calculations due to [78, 79, 80, 81]. We recall theformula of the s-function as

50 Chapter 5

s(z, α, β, γ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ α,

2

(z − α

γ − α

)2

for α ≤ z ≤ β,

1− 2

(z − γ

γ − α

)2

for β ≤ z ≤ γ,

1 for z ≥ γ.

(5.4)

The curve, implemented as a graph of (5.4), starts with point (0, α) andends with (γ, 1), whereas β is the arithmetic mean value of α and γ.

By referring to the most decisive medical factors, such as the patient’sage, body weight and CRP-value, operation prognoses usually can be ex-pressed by “operation” and “no operation”. The possibilities of the decisionevaluation can be described by some linguistic terms.

We intend now to utilize the technique of buiding three families of fuzzysets over the common reference space, expanded in [78, 79, 80, 81]. Letus suppose that L = {L1, . . . , Lω} is a linguistic list consisting of ω words,where ω is a positive odd integer. Each word is associated with a fuzzy set.Furthermore, let E be the length of a common reference set R, designed forall restrictions characterizing the fuzzy sets from L, provided that z ∈ R.We now wish to divide the linguistic terms into three groups recognized asa left group, a middle group and a right group.

The membership functions assigned to the leftmost terms are parametricfunctions, which are presented by (5.5) as

μLt(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1

for z ≤ E(ω − 1)

2(ω + 1)δ(t),

1− 2

(z − E(ω−1)

2(ω+1) δ(t)

E(ω−1)ω(ω+1) δ(t)

)2

for E(ω−1)2(ω+1) δ(t) ≤ z ≤ E(ω−1)

2ω δ(t),

2

(z − E(ω−1)(ω+2)

2ω(ω+1) δ(t)

E(ω−1)ω(ω+1) δ(t)

)2

for E(ω−1)2ω δ(t) ≤ z ≤ E(ω−1)(ω+2)

2ω(ω+2) δ(t),

0

for z ≥ E(ω−1)(ω+2)2ω(ω+1) δ(t),

(5.5)

where δ(t) = 2tω−1 , t = 1, . . . , ω−1

2 is a parametric function depending onleft function number t. When t is equal to 1, the formula implies the firstleftmost membership function. If t takes the value of ω−1

2 , then we willobtain the last left membership function [80].

Chapter 5 51

The membership function in the middle has the form of a clock. It isgiven by (5.6) in the form of

μLω+12

(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ E(ω − 2)

2(ω),

2

(z − E(ω−2)

2ωEω

)2

for E(ω−2)2ω ≤ z ≤ E(ω−1)

2ω ,

1− 2

(z − E

2Eω

)2

for E(ω−1)2ω ≤ z ≤ E

2 ,

1− 2

(z − E

2Eω

)2

for E2 ≤ z ≤ E(ω+1)

2ω ,

2

(z − E(ω+2)

2ωEω

)2

for E(ω+1)2ω ≤ z ≤ E(ω+2)

2ω ,

0 for z ≥ E(ω+2)2ω .

(5.6)

Finally, the membership functions on the right-hand side can be ex-pressed by as

μLω+32 +t−1

(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0

for z ≤ E − E(ω − 1)(ω + 2)

2ω(ω + 1)· ε(t),

2

(z − (E − E(ω−1)(ω+2)

2ω(ω+1) · ε(t))E(ω−1)ω(ω+1) · ε(t)

)2

for E − E(ω−1)(ω+2)2ω(ω+1) · ε(t) ≤ z ≤ E − E(ω−1)

2ω · ε(t),

1− 2

(z − (E − E(ω−1)

2(ω+1) · ε(t))E(ω−1)ω(ω+1) · ε(t)

)2

for E − E(ω−1)2ω · ε(t) ≤ z ≤ E − E(ω−1)

2(ω+1) · ε(t),1

for z ≥ E − E(ω−1)2(ω+1) · ε(t),

(5.7)A new function ε(t) = 1 − 2(t−1)

ω−1 , t = 1, . . . , ω−12 allows generating all

rightmost functions one by one when setting t-values in (5.7) [80].

5.3 A Case Study

To make a decision “operate” contra “do not operate”, concerning an individ-ual patient in accordance with his/her biological markers’ values, we haveto involve the medical experience in the decision process. To facilitate a

52 Chapter 5

conversation with a physician we have prepared a linguistic list named “Theoperation possibility”. We denote the linguistic list by L = {L1, . . . , L5}, inwhich L1 = “none”, L2 = “little”, L3 = “medium”, L4 = “large” and L5 = “to-tal”. The list for “no operation” becomes the complement of the “operation”list.

The excerpt of the data set, shown in Table 5.1, consists of the patients’clinical records and primary judgments of “operation” respective “no oper-ation” possibilities made by the medical expert. The total medical reportcontains 25 gastric cancer patients randomly selected. The entire medicaldata set can be found in Table A.1 in Appendix A.

Table 5.1: The excerpted medical data set of 25 gastric cancer patientsPatient xk Attribute vectors and operation possibilities

Attribute vectors Operation No Operation(age, weight, crp) Cluster S1 Cluster S2

x1 (71, 85, 1) Total Littlex2 (81, 70, 9) Medium Largex3 (50, 67, 4) Large Medium. . . . . . . . . . . .x25 (54, 49, 36) None Large

Two surgery states “operate” and “do not operate” assist two clustersS1 and S2 respectively. By selecting words from the lists the experiencedsurgeon makes the primary graded decision about possibilities of operatingor not operating on the patient.

Each verbal expression, being the term of L, is associated with a fuzzyset. L1 and L2 represent two left fuzzy sets, L3 is the fuzzy set in the mid-dle, whereas L4 and L5 constitute two rightmost fuzzy sets. Unfortunately,these linguistic items do not provide us with any information about mem-bership degrees expected in matrix U0 as primary recommendation states of“operate” or “do not operate”. Therefore we adopted the following technique[78, 79, 80, 81] to assign numerical substitutes to verbal expressions from thelist.

By inserting E = 100, the length of the reference set R = [0, 100] –typical of density measures in medical investigations, ω = 5 and t = 1, 2in (5.5), we obtain the membership functions of the first two fuzzy sets,namely, L1 = “none” given as:

Chapter 5 53

μL1(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 for z ≤ 16.7,

1− 2

(z − 16.7

6.6

)2

for 16.7 ≤ z ≤ 20,

2

(z − 23.3

6.6

)2

for 20 ≤ z ≤ 23.3,

0 for z ≥ 23.3

(5.8)

and L2 = “little” prepared as:

μL2(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 for z ≤ 33.3,

1− 2

(z − 33.3

13.4

)2

for 33.3 ≤ z ≤ 40,

2

(z − 46.7

13.4

)2

for 40 ≤ z ≤ 46.7,

0 for z ≥ 46.7.

(5.9)

By substituting E = 100 and ω = 5 in (5.6), the membership function ofL3 =“middle” is given as the structure

μL3(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ 30,

2

(z − 30

20

)2

for 30 ≤ z ≤ 40,

1− 2

(z − 50

20

)2

for 40 ≤ z ≤ 50,

1− 2

(z − 50

20

)2

for 50 ≤ z ≤ 60,

2

(z − 70

20

)2

for 60 ≤ z ≤ 70,

0 for z ≥ 70.

(5.10)

Finally, for E = 100, ω = 5 and t = 1, 2 inserted in (5.7), we get themembership functions of L4 =“large” in the form of

μL4(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ 53.3,

2

(z − 53.3

13.4

)2

for 53.3 ≤ z ≤ 60,

1− 2

(z − 66.7

13.4

)2

for 60 ≤ z ≤ 66.7,

1 for z ≥ 66.7

(5.11)

54 Chapter 5

and L5 = “total” as

μL5(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ 76.7,

2

(z − 76.7

6.6

)2

for 76.7 ≤ z ≤ 80,

1− 2

(z − 83.3

6.6

)2

for 80 ≤ z ≤ 83.3,

1 for z ≥ 83.3.

(5.12)

When substituting α = 0, β = 50 and γ = 100 in a new s-functionimpacted over set R we determine

s(z, 0, 50, 100) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 for z ≤ 0,

2

(z

100

)2

for 0 ≤ z ≤ 50,

1− 2

(z − 100

100

)2

for 50 ≤ z ≤ 100,

1 for z ≥ 100.

(5.13)

After sampling all membership functions (5.8)-(5.13) in Fig. 5.1, we aimat evaluating the membership degrees taking place in the first partition ma-trix U0.

none1L little2L middle3L large4L total5L

16.7 33.3 66.7 83.3

z = possibilities

The

mem

bers

hip

degr

eeso

fpos

sibi

litie

s

Figure 5.1: The collection of all the membership functions of operation pos-sibilities, L0 − L5

Chapter 5 55

In the interval [0, 16.7], the membership degree of L1 = “none” equals1, which means that the possibility of none operation is the highest in thisregion. As the membership degrees decrease from 1 to 0 over (16.7, 23.3],then z = 16.7 will become a natural border for sure members in L1. In (5.13),z = 16.7 ∈ [0, 50]. From the formula of membership function (5.13), whichis lying over the interval [0, 100], we choose the segment 2( z

100)2 in which we

set z = 16.7 to obtain μR(16.7) = 0.056. This represents numerically L1 inTable 5.2 which is a mathematical adaptation of Table 5.1.

We apply the procedure to the second fuzzy set L2 = “little”, where weselect z = 33.3 for calculating its membership degree by employing (5.13)to get μR(33.3) = 0.22. For the third fuzzy set L3 =“middle” represent-ing the membership value is specified to be μR(50) = 0.50. The last fuzzysets L4=“large” and L5=“total” are represented by μR(66.7) = 0.78 andμR(88.3) = 0.944, respectively. After the data arrangement, the linguis-tic words in Table 5.1 are replaced by numerical values put in Table 5.2.The whole initial membership degrees of the 25 gastric cancer patients arepresented in Table A.2 in Appendix A.

Table 5.2: The data set with the initial membership values of patient xk be-longing to cluster Si

Patient xk Attribute vectors and operation possibilitiesAttribute vectors Operation No Operation(age, weight, crp) Cluster S1 Cluster S2

x1 (71, 85, 1) 0.944 0.220

x2 (81, 70, 9) 0.5 0.78

x3 (50, 67, 4) 0.78 0.5

. . . . . . . . . . . .x25 (54, 49, 36) 0.056 0.78

It is assumed that the sum of membership grades in clusters S1 and S2

should be equal to 1 for each xk, k = 1, . . . , 25. It can happen that thedistinct sums differ from 1. In such cases some adjustments need to bemade, therefore the following techniques presented in [113] are applied.Case 1: μS1(xk) + μS2(xk) > 1.If the sum is greater than 1, we calculate a quotient q1, designed as

q1 =μS1(xk) + μS2(xk)− 1

2. (5.14)

Hence, two adjusted membership degrees are given by the following for-mulations:

μ′S1(xk) = μS1(xk)− q1 (5.15a)

μ′S2(xk) = μS2(xk)− q1 (5.15b)

56 Chapter 5

Proof.

μ′S1(xk) + μ

′S2(xk) = μS1(xk)− q1 + μS2(xk)− q1

= μS1(xk) + μS2(xk)− 2q1

= μS1(xk) + μS2(xk)− 2 · μS1(xk) + μS2(xk)− 1

2= μS1(xk) + μS2(xk)− μS1(xk)− μS2(xk) + 1

= 1.

(5.16)

In contrast with Case 1, Case 2 handles the situation that the sum isless than 1.

Case 2: μS1(xk) + μS2(xk) < 1.

We now have to derive another fraction q2, given by:

q2 =1− μS1(xk)− μS2(xk)

2. (5.17)

Membership values typical of Case 2 are verified by:

μ′S1(xk) = μS1(xk) + q2 (5.18a)

μ′S2(xk) = μS2(xk) + q2 (5.18b)

Proof.

μ′S1(xk) + μ

′S2(xk) = μS1(xk) + q2 + μS2(xk) + q2

= μS1(xk) + μS2(xk) + 2q2

= μS1(xk) + μS2(xk) + 2 · 1− μS1(xk)− μS2(xk)

2= μS1(xk) + μS2(xk) + 1− μS1(xk)− μS2(xk)

= 1.

(5.19)

After revising the membership degrees due to Case 1 or Case 2 werearrange the last two columns of Table 5.2 to renew it as Table 5.3. Thecomplete data set containing the first updated membership grades is givenin Table A.3 in Appendix A.

The entries of the initial partition matrix U0 consist of the values comingfrom the last two columns in Table 5.3. U0 is a 2× 25 matrix given by:

Chapter 5 57

Table 5.3: The excerpt data set including the first updated membershipvaluesPatient xk Attribute vectors and operation possibilities

Attribute vectors Operation No Operation(age, weight, crp) Cluster S

′1 Cluster S

′2

x1 (71, 85, 1) 0.862 0.138

x2 (81, 70, 9) 0.36 0.64

x3 (50, 67, 4) 0.64 0.36

. . . . . . . . . . . .x25 (54, 49, 36) 0.138 0.862

U0 =

[0.862 0.36 0.64 · · · 0.1380.138 0.64 0.36 · · · 0.862

]2×25

The numerical values in the first row in matrix U0 propose membershipdegrees for patients xk, k = 1, . . . , 25, in cluster S1. And the second rowsuggests the membership values for patients xk, k = 1, . . . , 25, in cluster S2.The sum of membership degrees in each column is equal to one.

If we go back to the FCM algorithm and let l = 0, m = 3 and ε =10−8 then, by using Matlab after 31 iterations, the cluster partition matricesbecome stable and do not change their coordinates, due to ‖U31 − U30‖ =9.93692 × 10−9 < 10−8. The last two partition matrices and the optimalcluster centers are listed in the patterns:

U30 =

[0.74366062 · · · 0.394472290.25633938 · · · 0.60552771

]2×25

and

U31 =

[0.74366062 · · · 0.394472290.25633938 · · · 0.60552771

]2×25

as well as

v311 =(65.9704, 74.2257, 6.50373

)and

v312 =(70.4353, 69.735, 35.8068

).

The final membership degrees for 25 gastric cancer patients, classifiedin S1 and S2, are depicted in Figure 5.2. In this manner the primary op-eration hypotheses, formulated by verbal structures, have been secondarilyconfirmed or denied by the strength of corresponding membership degreesin both clusters. Readers, who are interested in the membership degrees inthe partition matrix and the coordinates of respective cluster centres, arerefered to Table A.4 - A.34 aggregated in Appendix A.

58 Chapter 5

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pat ient Index

MembershipDegree

OperationNonoperation

Figure 5.2: The final cluster membership degrees of the operation and theno operation possibilities for 25 gastric cancer patients

5.4 Summary

In this study we have adopted fuzzy 2-means clustering analysis to partitiona patient data set, containing clinical records of 25 gastric cancer patients, intwo fuzzy clusters. These reveal the numerical decision of states ”operation”and ”no operation” by the values of membership degrees due to the rule: thehigher the degree is the more certain decision should be made with respectto the cluster considered.

We notice that the patients’ original clinical marker quantities lead tohigher membership degrees in the initial partition matrix when comparingthem to the lower values in the final matrix. This phenomenon can beexplained by the fact that the decision for an individual patient has beenmade by the assistance of all data filling the data set. This means that themedical knowledge provided in the form of the collective information, resetnumerically, could decide ”softer” decisions, which have not deprived thepatient of a chance for surgery. We have engaged a new form of experienceperformed as computerized experience constituting a database [112, 113].

The obtained results converge to cautious expertise made by physicianswho want the patient to survive as well as possible without any unnecessaryrisks. Therefore, fuzzy c-means cluster analysis can be seen as one of the ap-proaches that would assist medical operation diagnosis. The method can beapplied for a large number of patients. Lastly, we wish to emphasize that theadaptation of membership function families to the purpose of determiningthe initial membership degrees in the partition matrix has been an efficienttool in the algorithm. The functions, furnished with parameters, allow con-structing arbitrary linguistic lists containing many verbal judgments. Themathematical translation of words to numbers has been done systematically

Chapter 5 59

without predetermining any casual values. This has improved definitely theconvergence speed of the algorithm.

60 Chapter 5

CHAPTER6Evaluation of OperationPossibilities for Gastric CancerPatients by Means of Point SetApproximation

In this chapter the objective focuses on the design of two-dimensional contin-uous functions approximating irregular point sets, whose shapes are clock-like. Each point in the point sets consists of two elements, in which the firstelement symbolizes the characteristic value [81, 112] for each gastric cancerpatient, whereas the second one denotes the membership degree of operationpossibility obtained from the fuzzy c-means clustering analysis in Chapter5. The characteristic values are computed by means of code vectors [81]representing clinical data or real medical data, respectively. The point setapproximating function of none operation possibilities will be delivered ascomplement to the operation.

Our purpose is to evaluate the operation possibility contra none operationpossibility by entering the characteristic values into suitable expressions ofthe membership functions. By utilizing the approximation function, we canmake the predictions for any arbitrary patient whose data is placed inside theset of points known already. To accomplish the approximation, the truncatedπ-functions designed by Rakus-Andersson in [72, 74, 82, 84, 85, 112] havebeen taken into account.

61

62 Chapter 6

6.1 Determination of the Characteristic Values f cxk

for Gastric Cancer Patient xk by Adopting DataCode Vectors

The patients’ ages, the CRP -values and the body weights play a significantrole in the operation diagnosis. Let us denote the space of ages [0, 100] byA, the space of CRP -values [0, 85] by CRP and express the body weightby BW = [40, 120]. The characteristic value equation [81] will be used tocalculate the characteristic values f c

xkfor the patient xk already known from

Chapter 5, k = 1, . . . , 25, in accord with

f cxk(ac, crpc, bwc) = w1 · ac + w2 · crpc + w3 · bwc, (6.1)

where (ac, crpc, bwc) is the code vector consisting of codes representing theindices of subintervals which make partitions of age A, CRP -value CRP andbody weight BW. The coefficients w1, w2 and w3 indicate the importanceweights [71, 73, 74, 75, 81, 88, 112] which stress the dominance of the medicalparameters in operation decision for gastric cancer patients.

The technique, introduced by Saaty in [95], determining the values ofw1, w2 and w3 will be put in use. Let us suppose that sg, g = 1, . . . , 3 rep-resents medical parameters. Also, we need to compare pairwise all medicalparameters. The value of Vgh yields the range when comparing sg with sh.Similarly, the value of Vhg presents the scale of comparison of sh with sg.Especially, Vgh ·Vhg = 1 is one of the properties concerning these two factors.The other properties presented in references [71, 73, 74, 75, 81, 88] are givenby the following statements:

1. If sg and sh have the equal importance, then Vgh = 1.

2. If sg shows weak importance over sh, then Vgh = 3.

3. If sg reveals strong importance over sh, then Vgh = 5.

The pairwise comparisons of 3 medical parameters generate an 3× 3 im-portance matrix W [81]. The entries in matrix W concern the values of Vgh

and Vhg, g, h = 1, . . . , 3. The quantities of the importance weights w1, w2

and w3 will be determined as the components of the eigenvector correspond-ing to the largest in magnitude eigenvalue of matrix W . In the light of thesuggestions, which were given by an expert in gastric cancer diagnosis, theimportances of the age, the CRP -value and the body weight are ranked asCRP -value > age > body weight, assuming that “>” denotes higher impor-tance. This means that the CRP -value has weak importance over the age butstrong importance over the body weight. Moreover, the age presents weakimportance over the body weight. Thus, we state Table 6.1 to illustrate theimportance strength among three medical parameters mentioned.

Chapter 6 63

Table 6.1: The importance table of age, CRP -value and body weightAge CRP -value Body weight

Age 1 1/3 3

CRP -value 3 1 5

Body weight 1/3 1/5 1

Based on Table 6.1, the matrix W is given as follows:

W =

⎡⎣ 1 1/3 3

3 1 51/3 1/5 1

⎤⎦

By using Matlab, we obtain the largest eigenvalue of the matrix W equalto λ = 3.0385, and the corresponding eigenvector is determined as V =(0.37, 0.92, 0.15). Thereby, w1 = 0.37, w2 = 0.92 and w3 = 0.15. Accordingto equation (6.1) the characteristic function can be presented by

f cxk(ac, crpc, bwc) = 0.37 · ac + 0.92 · crpc + 0.15 · bwc. (6.2)

When looking through the original data set, we assign the age intervalas A = [0, 100], denote the space for the CRP -value by CRP = [0, 85]and define the region of body weight as BW = [40, 120] as it has beenmentioned. We divide the interval of A into five subintervals, which arepresented by A0 = [0, 20], A1 = [20, 40], A2 = [40, 60], A3 = [60, 80] andA4 = [80, 100]. The collection of CRP is divided into five subcollections,CRP0 = [0, 17], CRP1 = [17, 34], CRP2 = [34, 51], CRP3 = [51, 68] andCRP4 = [68, 85]. In the end, the sample of BW will be separated into fivesubintervals too, such as, BW0 = [40, 56], BW1 = [56, 72], BW2 = [72, 88],BW3 = [88, 104] and BW4 = [104, 120]. To each subinterval the relevantcode is assigned by the index of the respective subinterval.

Example 6.1. Let us assume that the clinical data of patient xk are givenby the vector (83, 78, 67.5), where 83 indicates the patient’s age, 78 repre-sents the CRP -value and 67.5 symbolizes the body weight. The code vector(ac, crpc, bwc) can be expressed by (4, 4, 1), since 83 ∈ A4, 78 ∈ CRP4

and 67.5 ∈ BW1. We note that A4 has the code 4, CRP4 has the code 4and BW1 has the code 1. By substituting the values of ac = 4, crpc = 4and bwc = 1 into the equation (6.2), we estimate the patient’s characteristicvalue by equation

f cxk(ac, crpc, bwc) = 0.37 · 4 + 0.92 · 4 + 0.15 · 1 = 5.31. (6.3)

64 Chapter 6

6.2 Creation of the Truncated π-Functions to Ap-proximate the Point Set P c′

After substituting the code values of the age, the CRP -value and the bodyweight in the code vector (ac, crpc, bwc) put into equation (6.2), we obtainthe characteristic values of the gastric cancer patients. These characteris-tic values together with the membership degrees of operation possibilitiesobtained from the fuzzy c-means clustering analysis consistitute a pointset P c =

{(f c

xk, μU31

S1(xk))

}, where k = 1, · · · , 25 [112]. The whole point

set data are presented in Table B.1 in Appendix B. In a two-dimensionalcoordinate system, the characteristic values, arranged in the ascending or-der, will be assigned to the abscissa. The membership degrees will be allo-cated on the ordinate. After the rearrangment of the characteristic values,P c′ =

{(f c′

xk, μc′

xk)}

, k = 1, · · · , 25 indicates the point set which we wish toapproximate. The points are depicted in Fig. 6.1 in the two-dimensionalcoordinate system.

Figure 6.1: The data points {(f c′xk, μc′(xk)

)} collected in the rearranged pointset P c′

To be able to perform the point set approximation by means of the trun-cated π-functions, we need three important points chosen from the pointset P c′ . The start point (f c′1 , μc′1) = (min(fxk

), μ(xk)) possesses the min-imal characteristic value. The second point (f c′2 , μc′2) = (fxk

,max(μ(xk)))has the maximal membership degree. And the terminate point (f c′3 , μc′3) =(max(fxk

), μ(xk)) has the maximal characteristic value , k = 1, · · · , 25.The detailed descriptions about the construction of truncated π-membership

functions have been carefully elaborated in [72, 74, 82, 84, 85]. Readers who

Chapter 6 65

are interested in more detailed information about that are referred to thecontributions. According to the references mentioned above, the generalmembership function approximating the point set P c′ , created for 25 gastriccancer patients for operation possibility, is given by

μ′′op′′(fc′xk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

μc′2

⎛⎝1− 2

(f c′xk

− f c′2

f c′2 − α

)2⎞⎠ for f c′1 ≤ f c′

xk≤ f c′2 ,

μc′2

⎛⎝1− 2

(f c′xk

− f c′2

γ − f c′2

)2⎞⎠ for f c′2 ≤ f c′

xk≤ β,

2μc′2

(f c′xk

− γ

γ − f c′2

)2

for β ≤ f c′xk

≤ f c′3 ,

(6.4)

where α = f c′2 − fc′2−fc′1√μc′2−μ

c′12μ

c′2

, γ =fc′3−fc′2

√μc′3

2μc′2

1−√

μc′3

2μc′2

and β = fc′2+γ2 .

From the rearranged data set of characteristic values in compliance withtheir ascending order, we choose (f c′1 , μc′1) = (0.67, 0.562), (f c′2 , μc′2) =(1.26, 0.824) and (f c′3 , μc′3) = (5.31, 0.374). The substitution of the data inequation (6.4) implies the membership function of the operation possibilitygiven as

μ′′op′′(fc′xk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0.824

⎛⎝1− 2

(f c′xk

− 1.26

1.26 + 0.22

)2⎞⎠

for 0.67 ≤ f c′xk

≤ 1.26,

0.824

⎛⎝1− 2

(f c′xk

− 1.26

8.99− 1.26

)2⎞⎠


≤ 5.125,

2 · 0.824(

f c′xk

− 8.99

8.99− 1.26

)2

for 5.125 ≤ f c′xk

≤ 5.31.

(6.5)

The none operation possibility will be treated as the complement to the

66 Chapter 6

operation case. The membership function is presented by the equation

μ“none op′′(fc′xk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1− μc′2

⎛⎝1− 2

(f c′xk

− f c′2

f c′2 − α

)2⎞⎠

for f c′1 ≤ f c′xk

≤ f c′2 ,

1− μc′2

⎛⎝1− 2

(f c′xk

− f c′2

γ − f c′2

)2⎞⎠

for f c′2 ≤ f c′xk

≤ β,

1− 2μc′2

(f c′xk

− γ

γ − f c′2

)2

for β ≤ f c′xk

≤ f c′3 .

(6.6)

The insertion of the point data in equation (6.6) yields the membershipfunction of none operation possibility given by (6.7). Figure 6.2 illustratesthe shapes of the π-membership functions approximating the point sets ofoperation possibiblity and none operation possibility.

μ′′none op′′(fc′xk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1− 0.824

⎛⎝1− 2

(f c′xk

− 1.26

1.26 + 0.22

)2⎞⎠


≤ 1.26,

1− 0.824

⎛⎝1− 2

(f c′xk

− 1.26

8.99− 1.26

)2⎞⎠


≤ 5.125,

1− 2 · 0.824(

f c′xk

− 8.99

8.99− 1.26

)2

for 5.125 ≤ f c′xk

≤ 5.31.

(6.7)

Example 6.2. From Ex. 6.1, the obtained characteristic value of the pa-tient xk is 5.31. Since 5.125 ≤ 5.31 ≤ 5.31, we substitute f c′

xk= 5.31 in the

third equation of the equation (6.5). The approximated membership degreeof operation possibility is equal to 0.374. The value of 0.374 indicates alower operation possibility, which agrees with the physician’s criteria. Forolder and underweighted patients, whose CRP -value are high, the operationis actually not recommended.

We utilized the code vectors in the previous section to determine thecharacteristic values of biological markers measured for gastric cancer pa-tients. During the calculating process, we noticed that some of the patients

Chapter 6 67

'cx k

f

'cx k

f

'

''''c

xop kf

'

''''c

xopnone kf

Figure 6.2: The membership functions of operation contra none operationpossibility for the data point set P c′

get the same characteristic values but different membership degrees of oper-ation possibilities. Therefore, we wish to use the original data of the medicalparameters to settle the characteristic values. Hopefully, each patient willbe assigned to only one characteristic value.

6.3 Determination of the Characteristic Values fxk

for Gastric Cancer Patient xk by the ClinicalData

We recall the characteristic function (6.2). Instead of the code vector, wewill employ the vector consisting of original medical values of the age, theCRP-value and the body weight. Therefore, the characteristic function ispresented by the equation (6.8)

fxk(a, crp, bw) = 0.37 · a+ 0.92 · crp+ 0.15 · bw, (6.8)

where a ∈ A, crp ∈ CRP and bw ∈ BW .

Example 6.3. Vector (71, 1, 85) supplies us with the information that thereis a seventy-one year-old patient xk, whose crp is 1 and the body weight is85 kg. By substituting the vector coordinates in equation (6.8), we obtainfxk

(71, 1, 85) = 0.37 · 71 + 0.92 · 1 + 0.15 · 85 = 39.94. The value of 39.94indicates the characteristic value of the patient xk.

By the same procedure, all the patients’ characteristic values fxkare

calculated and placed on the abscissa in an ascending order. The earlier

68 Chapter 6

obtained membership degrees of the operation possibilities μk will be put onthe ordinate. Consequently, P ′ = {(fxk

, μ(xk))} , k = 1, . . . , 25, becomes thepoint set which we attempt to approximate. The entire point set data arecollected in Table B.2 in Appendix B. In Fig. 6.3 the points

(fxk

, μ(xk))

aredepicted.

Figure 6.3: The data points {(fxk, μ(xk)

)} aggregated in the rearrangedpoint set P ′

6.4 The Membership Function Approximation ofPoint Set P ′ by Adopting the Clinical Data

The point set approximation will be accomplished by using the truncatedπ-functions again. Thus, three crucial points characterized by the earliersimilar properties are selected. The start point is presented by (f1, μ1) =(min(fxk

), μ(xk)). The middle point possesses the maximal membershipdegree and is decided as (f2, μ2) = (fxk

,max(μ(xk))). And the end point(f3, μ3) = (max(fxk

), μ(xk)) has the maximal characteristic value, k =1, · · · , 25. From the real data set, coming from Chapter 5, we choose (f1, μ1)= (32.23, 0.682), (f2, μ2) =(36.428, 0.824) and (f3, μ3) = (112.60, 0.374)to substitute them in equations (6.4) and (6.6). We obtain the followingmembership functions for operation possibility and none operation possibil-ity, which are presented in (6.9) and (6.10) as

Chapter 6 69

μ′′op′′(fxk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0.824

(1− 2

(fxk

− 36.428

36.428− 22.127

)2)

for 32.23 ≤ fxk≤ 36.428,

0.824

(1− 2

(fxk

− 36.428

181.9− 36.428

)2)

for 36.428 ≤ fxk≤ 109.164,

2 · 0.824((

fxk− 181.9

181.9− 36.428

)2)

for 109.164 ≤ fxk≤ 112.6,

(6.9)

and

μ′′none op′′(fxk) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1− 0.824

(1− 2

(fxk

− 36.428

36.428− 22.127

)2)

for 32.23 ≤ fxk≤ 36.428,

1− 0.824

(1− 2

(fxk

− 36.428

181.9− 36.428

)2)

for 36.428 ≤ fxk≤ 109.164,

1− 2 · 0.824(

fxk− 181.9

181.9− 36.428

)2

for 109.164 ≤ fxk≤ 112.6.

(6.10)

We aggregate and plot both the membership functions in one coordinatesystem. Figure 6.4 provides us with an image of the point set P ′ and itscomplement.

Example 6.4. A twenty-seven year-old patient has 11 as the CRP -value andhis body weight is 87 kg. By inserting these values into equation (6.8), theobtained characteristic value is equal to 33.16. If we substitute fxk

= 33.16in the first equation of (6.9), we obtain μ′′op′′(33.16) = 0.738. The value of0.738 confirms that the operation is recommended.

Example 6.5. Suppose that (83, 78, 67.5) represents the clinical data forthe cancer patient xk from Ex.6.1. 83 indicates the age, the CRP -value is78 and the body weight is 67.5 kg. By substituting a = 83, crp = 78 andbw = 67.5 in equation (6.8), the characteristic value is equal to 112.6. Ifwe now substitute fxk

= 112.6 in the third equation of (6.9), we obtainμ′′op′′(112.6) = 0.374, which is indentical with the earlier result obtainedfrom Ex.6.2.

70 Chapter 6

kxf

kxf

kxop f''''

kxopnone f''''

Figure 6.4: The membership functions of the operation contra none operationpossibility for the point set P ′

6.5 Summary

By applying the code vectors transformed from clinical data and the realmedical data, we obtained the characteristic values for 25 randomly selectedcancer patients. These characteristic values together with the membershipdegrees of operation possibility obtained from fuzzy c-means clustering analy-sis generated two irregular clock-like point sets. The truncated π−functionshave been applied to approximate the point sets for operation possibility.The possibility of none operation has been treated as the complement to thecase of operation. If we insert the characteristic value of an arbitrary cancerpatient, belonging to the domain of the point set, in an appropriate equation,then the operation or the none operation possibility can be evaluated by thestrength of membership degrees of the π-function.

Old patients, who possess lower CRP -values and normal weight, willobtain lower characteristic values but higher operation possibilities. Forvery old patients, who have higher CRP -values and are underweighted, theoperation possibility degrees show contraditions in the decision to operateon the patient. In the end, young patients with lower CRP -values obtainlower characteristic values and higher operation possibilities. The results ofthe point set approximation logically agree with the physician’s criteria ofgastric cancer prognosis.

In Chapter 7, 8 and 9, multi-expert decision-making issues constitute thefocuses of the second part of the thesis. In Chapter 7, the probabilistic modelwith linguistic judgments has been considered to range the effectiveness oftreatment modalities for prostate cancer.

CHAPTER7Solution of Multi-ExpertDecision-Making Problem bythe Probabilistic Model

In this chapter, a probabilistic model is discussed and applied to solve amulti-expert decision making issue (MEDM). At the stage of data samplingin the considered MEDM problem, we involve a group of physicians whoindependently assess the effectiveness of a set of treatment therapies for aprostate cancer patient. The objective of the chapter is to find the mostoptimal treatment by means of a model presented in [41, 42], in which prob-abilities play a crucial role. We also use linguistic terms to express the ex-perts opinions. Later, to each linguistic term a fuzzy set with a parametricfunction of the s-type is assigned.

7.1 Description of the Multidisciplinary Team Con-ference (MDT)

Multidisciplinary team conferences or multidisciplinary cancer conferencesplay a very important role in decision-making process in modern treatmentof cancer patients. At the Urology Department of Blekinge County Hos-pital, Karlskrona, the MDT is a forum of health care providers includingmedical oncologists, urologists, urology sub-specialized nurses, radiologistand pathologists. The aim of the conference is to establish the treatmentdecisions for particular prostate cancer patients with a spectrum of prob-lematic urological conditions that cannot be easily solved by means of avail-able resources. The long term aim of the conferences is to discuss the bestand available treatment modalities of all newly diagnosed cases of prostatecancer. Quite often the decision making process is very clear and straight

71

72 Chapter 7

forward, but some cases lay outside the frames of guidelines and recommen-dations. Obviously the final choice of the treatment is also on discretion ofthe patient. This approach has two pitfalls. One of them is when there is adiscrepancy between forum members and the other one is when the patientis not interested in the treatment modality chosen by the panel. The bestsolution is to obtain a method for solving discrepancy and simultaneously tofind a method that ranges the degrees of treatment recommendation fromstrongly recommended to contraindicated. Such approach should be veryhelpful in diseases such as a prostate cancer, which has a broad spectrumof treatment methods that can be tailored to the particular patient’s needsand requirements.

In actual life, we often are in such situations that we need to evaluatesome information by means of numerical values. But when the numericalvalues are no longer available, then the linguistic approach [110] can be seenas a good alternative. Especially, in medical community, the informationoften is characterized vaguely and imprecisely. Thus it is hard to evaluatethe information by numberical values. For example, the expressions such as“very painful”, “slightly painful”, “medium” and “not very painful” are justsome examples of the linguistic evaluations of postoperative pain degrees.Also, in group decision making cases, when the experts assess the effective-ness of treatment therapies for prostate cancer, the semantic terms suchas “contraindicated”, “doubtful”, “acceptable”, “possible”, “suitable”, “recom-mended” and “strongly recommended” can be used. When comparing withthe quantitative factor, the linguistic approach is regarded by [21, 34, 37] asa more realistic, intuitionistic and natural method. Due to the advantages ofthe linguistic approach, an extensive application has been presented in thereferences [14, 45, 46, 63].

By involvning the probabilistic model in [41, 42] in the multi-expert de-cision making (MEDM) problem, we wish to select the most consensualtreatment for a prostate cancer patient. The entire process will be definedin the linguistic framework.

7.2 Preliminaries

The following four steps are involved in the probabilistic model:

1. Collection of all the assessments in an information table.

2. Calculating the random preference value Xai for each alternative ai, i =1, . . . , n.

3. Computing the choice value V (ai), for each alternative ai, i = 1, . . . , n,by the choice function.

Chapter 7 73

4. Ranking the choice values obtained in the previous step to select theoptimal one by formula aoptimal = maxai∈A(V (ai)).

In references [41, 42], a general property of a MEDM problem is consid-ered as an introduction of a finite set of experts denoted by E = {e1, . . . , ep},who are asked to select assessments from another finite set of alternativesA = {a1, . . . , an}. The assessments/evaluations are expressed by semanticwords in an order structured linguistic term set S = {s0, . . . , sg}, in whichsuch that sk < sl if and only if k < l, k, l = 0, . . . , g. An example of theorder structured linguistic term set S is given in Example 7.1.

Example 7.1. Suppose S = {s0, s1, s2, s3, s4, s5, s6} represents a linguisticterm set, consisting of seven semantic words expressing the physicians’ judg-ments of the effect of some treatment schemes for prostate cancer patients.Here s0 = “contraindicated” = “C”, s1 = “doubtful” = “D”, s2 = “acceptable”= “A”, s3 = “possible” = “P”, s4 = “suitable” = “S”, s5 = “recommended” =“R” and s6 = “strongly recommended” = “SR”. In the linguistic term set theelements are arranged in an ascending order such as s0 < s1 < s2 < s3 < s4< s5 < s6.

We are now going to describe in details the model, in which probabilitiesplay a crucial role. In the first step, all the assessments are collected in ajudgment table as shown in Table 7.1. Here each judgment Lij , i = 1, . . . , n

Table 7.1: The judgment table in the probabilistic modelAlternatives Experts

e1 · · · epa1 L11 · · · L1p

a2 L21 · · · L2p

· · · · · · · · · · · ·an Ln1 · · · Lnp

and j = 1, . . . , p, is expressed by the linguistic term selected from the linguis-tic term set S. We should emphasize that each linguistic term is associatedwith a general s-parametric membership function [83, 88, 111] given by (7.1)

74 Chapter 7

as

μsl(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − (

(zmin − hz) + hz · l)

hz

)2

for(zmin − hz

)+ hz · l ≤ z ≤ (

zmin − hz2

)+ hz · l,

1− 2

(z − (zmin + hz · l)

hz

)2

for(zmin − hz

2

)+hz · l ≤ z ≤ zmin + hz · l,

1− 2

(z − (zmin + hz · l)

hz

)2

for zmin + hz · l ≤ z ≤(zmin + hz

2

)+ hz · l,

2

(z − (

(zmin + hz) + hz · l)

hz

)2

for(zmin + hz

2

)+ hz · l ≤ z ≤ (

zmin + hz)+ hz · l.

(7.1)

where z ∈ [0, 1] is a symbolic reference set of effectiveness, zmin = 0, and hzis defined as the distance of the peaks between two adjacent fuzzy sets. Ifwe set zmin och hz as fixed values when choosing l = 0, . . . , g, then we willobtain the membership functions for s0, . . . , sg.

Xai in the second step is assumed as a random preference value for eachalternative ai, i = 1, . . . , n, with associated probability distribution P definedby [41, 42] as

P (Xai = sl) = PE

({ej ∈ E | Lij = sl}

). (7.2)

It is worth highlighting that the statement of random preference Xai

is a crucial procedure in the approach of probability. Since each Xai isstochastically independent of each other, it makes the comparisons of anytwo random preferences be possibly performed.

The choice value Vai , defined in step 3, for each alternative ai, i =1, . . . , n, is computed by the choice function presented by (7.3) as

V (ai) =∑i �=j

P(Xai ≥ Xaj

)

=∑i �=j

∑sl∈S

[P (Xai = sl)

∑Lij∈Ssl≥Lij

P (Xai = Lij)

],

(7.3)

where the quantity P (Xai ≥ Xaj ) could be interpreted as the probability of“the performance of ai is as least as good as that of aj .”

Chapter 7 75

Finally, by ranking the choice values obtained in the former step, we canselect the optimal one by (7.4) as

aoptimal = maxai∈A

(V (ai)

). (7.4)

In order to show how this model is applied in practice, we would like todemonstrate an example based on the real medical application.

7.3 A Practical Study

In this section, a practical study in medical group decision making task isdemonstrated. The members of a physician group are asked for providingthe opinions on the effect of some treatment schemes for a prostate cancerpatient. The probabilistic model is applied and the result is presented.

Let us suppose that E = {e1, e2, e3, e4} denotes a collection consistingof four physicians. And another set A = {a1, a2, a3, a4, a5, a6} contains sixtypes of treatment schemes for a prostate cancer patient, where a1 = “waitand see”, a2 = “active monitoring”, a3 = “symptom based treatment”, a4 =“brachytherapy”, a5 = “external beam radiation therapy” and a6 = “radicalprostatectomy”. Also, L = {s0, s1, s2, s3, s4, s5, s6} includes seven linguisticterms, in which s0 = “contraindicated”, s1 = “doubtful”, s2 = “acceptable”,s3 = “possible”, s4 = “suitable”, s5 = “recommended” and s6 = “stronglyrecommended”

By instering zmin = 0, hz = 0.167 and l = 0 in (7.1), we obtain thefunction for s0 = “contraindicated” expanded by

μs0(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z + 0.167

0.167

)2

for −0.167 ≤ z ≤ −0.0835,

1− 2

(z

0.167

)2

for −0.0835 ≤ z ≤ 0,

1− 2

(z

0.167

)2

for 0 ≤ z ≤ 0.0835,

2

(z − 0.167

0.167

)2

for 0.0835 ≤ z ≤ 0.167.

(7.5)

By following the same procedure for l = 1, 2, 3, 4, 5 and 6, we generatemembership functions from (7.6) to (7.11).

76 Chapter 7

μs1(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z

0.167

)2

for 0 ≤ z ≤ 0.0835,

1− 2

(z − 0.167

0.167

)2

for 0.0835 ≤ z ≤ 0.167,

1− 2

(z − 0.167

0.167

)2

for 0.167 ≤ z ≤ 0.2505,

2

(z − 0.334

0.167

)2

for 0.2505 ≤ z ≤ 0.334,

(7.6)

μs2(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − 0.167

0.167

)2

for 0.167 ≤ z ≤ 0.2505,

1− 2

(z − 0.334

0.167

)2

for 0.2505 ≤ z ≤ 0.334,

1− 2

(z − 0.334

0.167

)2

for 0.334 ≤ z ≤ 0.4175,

2

(z − 0.501

0.167

)2

for 0.4175 ≤ z ≤ 0.501,

(7.7)

μs3(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − 0.334

0.167

)2

for 0.334 ≤ z ≤ 0.4175,

1− 2

(z − 0.501

0.167

)2

for 0.4175 ≤ z ≤ 0.501,

1− 2

(z − 0.501

0.167

)2

for 0.501 ≤ z ≤ 0.5845,

2

(z − 0.668

0.167

)2

for 0.5845 ≤ z ≤ 0.668,

(7.8)

μs4(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − 0.501

0.167

)2

for 0.501 ≤ z ≤ 0.5845,

1− 2

(z − 0.668

0.167

)2

for 0.5845 ≤ z ≤ 0.668,

1− 2

(z − 0.668

0.167

)2

for 0.668 ≤ z ≤ 0.7515,

2

(z − 0.835

0.167

)2

for 0.7515 ≤ z ≤ 0.835,

(7.9)

Chapter 7 77

μs5(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − 0.668

0.167

)2

for 0.668 ≤ z ≤ 0.7515,

1− 2

(z − 0.835

0.167

)2

for 0.7515 ≤ z ≤ 0.835,

1− 2

(z − 0.835

0.167

)2

for 0.835 ≤ z ≤ 0.9185,

2

(z − 1.002

0.167

)2

for 0.9185 ≤ z ≤ 1.002

(7.10)

and

μs6(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − 0.835

0.167

)2

for 0.835 ≤ z ≤ 0.9185,

1− 2

(z − 1.002

0.167

)2

for 0.9185 ≤ z ≤ 1.002,

1− 2

(z − 1.002

0.167

)2

for 1.002 ≤ z ≤ 1.0855,

2

(z − 1.169

0.167

)2

for 1.0855 ≤ z ≤ 1.169.

(7.11)

We sample all functions from (7.5) to (7.11) in a family of fuzzy setrestrictions, which are plotted in Fig. 7.1.

By using the probabilistic model, we collect all the experts’ judgments inTable 7.2, whereas the random preference values of each judgment is givenin Table 7.3.

Table 7.2: The collection of the therapy judgments in the probabilistic modelAlternatives Experts

e1 e2 e3 e4a1 s0 s0 s0 s0a2 s6 s6 s5 s5a3 s0 s0 s0 s0a4 s3 s2 s4 s4a5 s3 s1 s3 s4a6 s4 s5 s4 s5

78 Chapter 7

z = The effectivness of the treatment schemes

0s 1s 2s 3s 4s 5s 6sz

Figure 7.1: The s-parametric membership functions for linguistic fuzzy sets,s0, . . . , s6

Table 7.3: The aggregation of random preferences in the probabilistic modelRandom Preferences

s0 s1 s2 s3 s4 s5 s6Xa1 1 0 0 0 0 0 0

Xa2 0 0 0 0 0 0.5 0.5

Xa3 1 0 0 0 0 0 0

Xa4 0 0 0.25 0.25 0.5 0 0

Xa5 0 0.25 0 0.5 0.25 0 0

Xa6 0 0 0 0 0.5 0.5 0

The choice value of a1, V (a1) is computed by (7.12) as

V (a1) =∑1 �=j

P(Xa1 ≥ Xaj

)

=∑1 �=j

∑sl∈S

[P (Xa1 = sl)

∑L1j∈Ssl≥L1j

P (Xaj = L1j)

]

= P (Xa1 ≥ Xa2) + . . .+ P (Xa1 ≥ Xa6)

= 0 + 1 + 0 + 0 + 0 = 1,

(7.12)

where

Chapter 7 79

P (Xa1 ≥ Xa2) = 1 · 0 + 0(2 · 0) + 0(3 · 0) + 0(4 · 0) + 0(5 · 0)= 0 + 0 + 0 + 0 + 0

= 0,

P (Xa1 ≥ Xa3) = 1 · 1 + 0(0 + 1) + 0(2 · 0 + 1) + 0(3 · 0 + 1)

+ 0(4 · 0 + 1) + 0(5 · 0 + 1) + 0(6 · 0 + 1)

= 1 + 0 + 0 + 0 + 0 + 0 + 0

= 1,

P (Xa1 ≥ Xa4) = 1 · 0 + 0(2 · 0) + 0(3 · 0 + 2 · 0.25 + 0.5)

+ 0(4 · 0 + 2 · 0.25 + 0.5)

= 0 + 0 + 0 + 0 + 0

= 0,

P (Xa1 ≥ Xa5) = 1 · 0 + 0(2 · 0 + 0.25) + 0(3 · 0 + 2 · 0.25 + 0.5)

+ 0(4 · 0 + 2 · 0.25 + 0.5)

= 0 + 0 + 0 + 0

= 0

and

P (Xa1 ≥ Xa6) = 1 · 0 + 0(2 · 0) + 0(3 · 0) + 0(4 · 0)+ 0(5 · 0 + 2 · 0.5)= 0 + 0 + 0 + 0 + 0

= 0.

For other ai, i = 2, 3, 4, 5, 6, V (ai) are calculated in the similar way as

V (a2) =∑2 �=j

P(Xa2 ≥ Xaj

)

=∑2 �=j

∑sl∈S

[P (Xa2 = sl)

∑L2j∈Ssl≥L2j

P (Xaj = L2j)

]

= P (Xa2 ≥ Xa1) + . . .+ P (Xa2 ≥ Xa6)

= 1 + 1 + 1 + 1 + 1 = 5.

(7.13)

V (a3) = 1 + 0 + 0 + 0 + 0 = 1,

80 Chapter 7

V (a4) = 1 + 0 + 1 + 0.75 + 0.25 = 3,

V (a5) = 1 + 0 + 1 + 0.5 + 0.125 = 2.625 and

V (a6) = 1 + 0.25 + 1 + 1 + 1 = 4.25.

The collection of choice values for ai, i = 1, . . . , 6, is aggregated in Ta-ble 7.4.

Table 7.4: The collection of choice values in the probabilistic modelThe collection of choice values for each alternative aiV (a1) V (a2) V (a3) V (a4) V (a5) V (a6)

1 5 1 3 2.625 4.25

We choose the optimal therapy alternative by means of equation (7.4).We obtain

aoptimal = maxai∈A

(V (ai)

)= max {1, 5, 1, 3, 2.625, 4.25}= 5 = V (a2)

(7.14)

The value of 5 indicates the choice value of a2 to be maximal. This meansthat the second therapy alternative is the most efficacious.

7.4 Summary

In this chapter, the probabilistic model has been applied in a MEDM prob-lem to select the most consensual treatment scheme for a prostate cancerpatient. Moreover, the independent assumed preferences of each alternativehave made the computation of choice values V (ai) be easily performed. Theuse of s-parametric membership functions makes the fuzzy sets intuitionis-tically. In the next chapter, we wish to adopt the model of 2-tuple fuzzylinguistic representations [36] to verify the high reliability of adopting thelinguistic approach in solving group decision making problems.

CHAPTER8Selection of the MostConsensual Treatment Therapyfor a Prostate Cancer Patientby the 2-Tuple LinguisticMethod

In the previous chapter, we adopted the technique of probability to seekthe moste optimal treatment therapy for a prostate cancer patient. In thepresent chapter, we wish to utilize the model of the 2-tuple linguistic repre-sentation [36] to confirm the result obtained by the probablistic model andsimultaneously verify the high reliability of adopting the linguistic approachin solving group decision-making problems.

8.1 Description of the Model of 2-Tuple LinguisticRepresentation

Let us assume that there is a finite set of expert denoted by E = {e1, . . . , ep},who are asked to select assessments from another finite set of alternativesA = {a1, . . . , an}. The assessments/evaluations are expressed by semanticwords in an order structured linguistic term set S = {s0, . . . , sg}, in whichsuch that sk < sl if and only if k < l.

In the present model, the physicians’ judgments of the treatments effectare represented by 2-tuples in the form of (sl, α), where sl ∈ S is representedby a semantic term, to which a fuzzy set is assigned. α ∈ [−0.5, 0.5

)is

defined as a numerical value.

81

82 Chapter 8

Each judgment, which is expressed by a semantic word, collected in Ta-ble 8.1 will be transformed into 2-tuple linguistic representation such as(sl, α

). If sl ∈ S, then

(sl, 0

)reflects sl. Next, xai = {(sl, α)}, is de-

fined as a finite set that consists of judgments of the 2-tuple fuzzy linguisticrepresentations for each alternative ai, i = 1, . . . , n.

Table 8.1: The judgment table in which each judgment is expressed by se-mantic word

Alternatives Expertse1 · · · ep

a1 L11 · · · L1p

a2 L21 · · · L2p

· · · · · · · · · · · ·an Ln1 · · · Lnp

Example 8.1. Let us suppose the judgment of an alternative, a2, is ex-pressed by s3 = “acceptable” = A, then (A, 0) is the 2-tuple fuzzy linguisticrepresentation of “acceptable”.

The experts’ judgments about the effectiveness of each treatment therapyare gathered in Table 8.2.

Table 8.2: The judgments collection of the treatment therapy expressed bythe physicians

Alternatives Expertse1 e2 e3 e4

a1 s0 s1 s2 s3a2 s2 s0 s1 s4a3 s3 s4 s5 s1a4 s2 s1 s2 s0

By the technique of 2-tuple linguistic representation, the judgments aregiven in Table 8.3

Table 8.3: The judgment table of the 2-tuple linguistic representationsExperts Alternatives

a1 a2 a3 a4e1 (C, 0) (A, 0) (P, 0) (A, 0)

e2 (D, 0) (C, 0) (S, 0) (D, 0)

e3 (A, 0) (D, 0) (R, 0) (A, 0)

e4 (P, 0) (S, 0) (D, 0) (C, 0)

Chapter 8 83

Two transformations also are involved in this model. The first one mapsa 2-tuple representation (sl, α) which belongs to the space of S× [−0.5, 0.5)into a numerical value β

ejai ∈ [0, g], i = 1, . . . , n, j = 1, . . . , p, in which β

ejai

= l + α. The action of Δ1 is formalized by

Δ1 : S × [−0.5, 0.5) → [0, g]

(sl, α) → βejai = l + α.

(8.1)

We explicate the performance of Δ1 by the following example.

Example 8.2. Let S = {s0, . . . , s6} and βejai ∈ [0, 6]. In Table 8.2, the

assessment of α1, given by expert e3, is expressed by the semantic term s2= “acceptable” = A. By the model of 2-tuple representation we can employthe judgment by (A, 0) presented in Table 8.3 for s2 = “acceptable” = Aand α = 0. The 2-tuple linguistic representation for other judgments areaggregated in Table 8.3.

Due to the first transformation, the 2-tuple representation of (A, 0) canbe performed as a numerical value βe3

a1 = l+α = 2+0 = 2, which belongs tothe interval [0, 6]. Furthermore, xa1 = {(C, 0), (D, 0), (A, 0), (P, 0)} con-sists of the judgments of the 2-tuple linguistic representations for alternativea1.

In addition, we use the notation, βai , to represent the arithmetic meanof the sum of βej

ai , in which i = 1, . . . , n and j = 1, . . . , p. The computationof βai is given by

βai =1

p

p∑j=1

βejai , (8.2)

where i = 1, . . . , n and j = 1, . . . , p.

Example 8.3. From Table 8.3 we obtain xa1 = {(C, 0), (D, 0), (A, 0), (P, 0)}= {(s0, 0), (s1, 0), (s2, 0), (s3, 0)}, which leads to βe1

a1 = 0 + 0 = 0, βe2a1

= 1 + 0 = 1, βe3a1 = 2 + 0 = 2 and βe4

a1 = 3 + 0 = 3. According to (8.2), the

arithmetic mean of βa1 is equal to0 + 1 + 2 + 3

4= 1.5.

The second transformation Δ2 can be regarded as an inverse of the firstone, i.e., it maps a numerical value βai ∈ R into a 2-tuple (sl, α) by (8.3)

Δ2 : βai → S × [−0.5, 0.5)

βai → (sl, α).(8.3)

Here sl has the closest index label to βai , the interval of [0, g] representsthe space consisting of the semantic label indices in the linguistic term setS = {sl}, l = 0, . . . , g.

84 Chapter 8

Example 8.4. Let S = {s0, . . . , s6}. According to (8.2), βa2 =2 + 0 + 1 + 4

4= 1.75. Since 1.75 is closer to s2 than to s1, then we choose s2 as the se-mantic word. The difference between 1.75 and 2 is 0.25, and 1.75 lies to theleft of 2. Therefore, we choose −0.25 to be the value of α. By means of thesecond transformation, Δ2(1.75) = (s2,−0.25), which is depicted in Fig. 8.1.

0s 1s 2s 3s 4s 5s 6s

0 1 2 3 4 5 6

S

25.0,75.1 2s

Figure 8.1: The 2-tuple linguistic representation of βa2 = 1.75

The third step contains the computation of the arithmetic mean of xeaiof 2-tuples for each alternative ai, i = 1, . . . , n. This is formalized by

xeai = Δ2

( g∑l=0

1

nΔ1(sl, α)

). (8.4)

Since the arithmetic means, supplied from the previous step, are pre-sented by 2-tuples, a computational technique to compare the arithmeticmean for each alternative proposed in [36] is given as follows.

Let (sk, α1) and (sl, α2) be two 2-tuples linguistic representations, witheach one representing a counting of information as follows:

1. If k < l, then (sk, α1) is smaller than (sl, α2).

2. If k = l, then the following conditions will be checked:If α1 = α2, (sk, α1) and (sl, α2) represent the same information.If α1 < α2, (sk, α1) is smaller than (sl, α2).If α1 > α2, (sk, α1) is greater than (sl, α2).

At last, by comparing the arithmetic means with each other and rankingthe alternatives, the optimal alternative(s) will be obtained.

8.2 A Case Study

In this section, by utilizing the model of 2-tuple linguistic representation, wewant to verify the result obtained in the previous chapter.

We recall the practical study presented in Chapter 7. Let us supposethat E ={e1, e2, e3, e4} denotes a collection consisting of four physicians.Another set denoted by A= {a1, a2, a3, a4, a5, a6} contains six types of treat-ment schemes for a prostate cancer patient, where a1 = “wait and see”, a2 =

Chapter 8 85

“active monitoring”, a3 = “symptom based treatment”, a4 = “brachytherapy”,a5 = “external beam radiation therapy” and a6 = “radical prostatectomy”.Also, L = {s0, s1, s2, s3, s4, s5, s6} includes seven linguistic terms, in whichs0 = “contraindicated” = “C”, s1 = “doubtful” = “D”, s2 = “acceptable” =“A”, s3 = “possible” = “P”, s4 = “suitable” = “S”, s5 = “recommended” = “R”and s6 = “strongly recommended” = “SR”.

According to the algorithm for the model of 2-tuple representation, thejudgment which is transformed into 2-tuples is given in Tab. 8.4

Table 8.4: The judgments expressed in the 2-tuples representation modelExperts Alternatives

a1 a2 a3 a4 a5 a6e1 (C, 0) (SR, 0) (C, 0) (P, 0) (P, 0) (S, 0)

e2 (C, 0) (SR, 0) (C, 0) (A, 0) (H, 0) (R, 0)

e3 (C, 0) (R, 0) (C, 0) (S, 0) (P, 0) (S, 0)

e4 (C, 0) (R, 0) (C, 0) (S, 0) (S, 0) (R, 0)

We calculate the arithmetic mean for the first alternative a1 by meansof (8.4). xa1 = {(C, 0), (C, 0), (C, 0), (C, 0)} is a finite set consisting offour 2-tuple linguistic representations for alternative a1. By adopting (8.4),the arithmetic mean value for a1 is calculated as:

xea1 = Δ2(14(0 + 0 + 0 + 0)

)= Δ2(0) = (s0, 0).

For the second alternative the arithmetic mean is given as follows:

xea2 = Δ2(14(6 + 6 + 5 + 5)

)= Δ2(5.5) = (s5, 0.5).

And by the same reasoning, when setting i = 3, 4, 5, 6 in (8.4), weobtain:

xea3 = Δ2(14(0 + 0 + 0 + 0)

)= Δ2(0) = (s0, 0),

xea4 = Δ2(14(3 + 2 + 4 + 1)

)= Δ2(2.5) = (s2, 0.5),

xea5 = Δ2(14(3 + 1 + 3 + 4)

)= Δ2(2.75) = (s3, −0.25) and

xea6 = Δ2(14(4 + 5 + 4 + 5)

)= Δ2(4.5) = (s4, 0.5).

We present the collection of the arithmetic means for all alternatives inTable 8.5

According to the computational technique presented earlier, we comparethe above 2-tuples which represent the arithmetic means for all the alter-natives. We start the comparisons of a1 and a2. (s0, 0) represents thearithmetic mean of a1 and (s5, 0.5) denotes the arithmetic mean of a2.

86 Chapter 8

Table 8.5: The arithmetic means of six treatment therapies in the model of2-tuple linguistic representations

The Collection of the Arithmetic Meansxea1 xea2 xea3 xea4 xea5 xea6

(s0, 0) (s5, 0.5) (s0, 0) (s2, 0.5) (s3, −0.25) (s4, 0.5)

Since 0 < 5 → (s0, 0) < (s5, 0.5) → a1 < a2. By the same procedure, weget the final result ranged as a2 > a6 > a5 > a4 > a1 = a3, in which a2indicates the most optimal treatment scheme. This result converges to theanswer obtained from Chapter 7.

8.3 Summary

In this chapter, the model of 2-tuple linguistic representation has been ap-plied in a multipel-expert decision-making problem for a prostate cancerpatient. Also we have verified the result agrees with the answer obtainedin Chapter 7. The convergence results from both of the approaches demon-strate the high reliability of adopting the linguistic approach in solving groupdecision-making problems. Especially, the use of the model of 2-tuple lin-guistic representation provents the information loss and make the result moreprecis.

In next chapter, Chapter 9, the hesitate fuzzy linguistic term sets willbe applied to make the medication prognoses for another prostate cancerpatient. We intend to explore if it is possible to rank the effectiveness oftreatment therapies from the most recommended to the contraindicated,even if the judgments are expressed hesitantly by physicians.

CHAPTER9Solution of Multi-ExpertDecision-Making Problem byHesitant Fuzzy Linguistic TermSets

The probabilistic model [42] and the model of 2-tuple linguistic representa-tions [36] have been described in Chapter 7 and 8 as two examples of lin-guistic approaches dealing with group decision-making problems. Now, thehesitant fuzzy linguistic term sets (HFLTS) [92] can be regarded as a newtechnique tackling with decision making issues under imprecise conditions,in which experts hesitate among several preferences to assess an alternative.By the inspiration of the concept of HFLTS, the aggregated preferences pro-vided by experts are expressed by comparative linguistic terms rather thansingle words. In addition, the L−R form of a fuzzy number has been utilizedto represent the assessments. The algorithm for computing the union of twofuzzy numbers in the L−R form are applied. By comparing the coordinatesof the centroid points of fuzzy numbers, we rank the effectiveness of treat-ment alternatives from the most recommended to the contraindicated for aprostate cancer patient.

9.1 The Preliminary Items of the Hesitant FuzzyLinguistic Term Sets

Linguistic approaches [109] and its extensions [6, 25, 58, 59, 106] solvinggroup decision-making problems sucessfully have been put forward. Actu-ally, in a real-world decision-making environment, it is quite difficult fordecision-makers to assess an alternative by using only simple words. Espe-

87

88 Chapter 9

cially in medical community, before the final diagnosis is established, cautioushealth professionals with rich clinical experience often provide patients withinformation in a broader spectrum. Therefore, the hesitant fuzzy linguisticterm sets (HFLTS) can be seen as a good alternative when experts hesitatebetween several preferences to assess an alternative [92].

In this section, we need shortly review some of the definitions of HFLTSintroduced by Rodriguez et al. in [92] and the techniques ranking fuzzynumbers in [104] for better understanding of the later medication application.

Definition 9.1. A Hesitant Fuzzy Linguistic Term SetLet S = {s0, . . . , sg} be a linguistic term set. A hesitant fuzzy linguistic termset (HFLTS), denoted by HS , is an ordered finite subset of the consecutivelinguistic terms of S.

The empty HFLTS and the full HFLTS for elements s ∈ S are definedas follows:

1. empty HFLTS: HS(s) = ∅,

2. full HFLTS: HS(s) = S.

Example 9.1.Let us assume that S = {s0, s1, s2, s3, s4, s5} is a linguistic term setdescribing the effectiveness of some treatment schemes for prostate cancerpatients. If s0 = “contraindicated” = “C”, s1 = “acceptable” = “A”, s2 = “pos-sible” = “P”, s3 = “suitable” = “S”, s4 = “recommended” = “R”, s5 = “stronglyrecommended” = “SR”, then a HFLTS might be HS(s) = {s2, s3, s4} ={possible, suitable, recommended} = between “P” and “R”.

Differing from the probabilistic model and the model of 2-tuple linguis-tic representation, in which the experts’ preferences are expressed by singlewords, in the HFLTS, the judgment expressions based on comparative termslike, e.g., between . . . and . . . , greater than . . . , or less than . . . , [92] will beused to supply the preferences. The comparative terms refer to expressionssuch as “between acceptable and possible”, “less than possible”, “greater thansuitable” and so on. Symbolically, we denote “between acceptable and possi-ble” as [A, P], “less than possible” as ≤ P and “greater than suitable” as ≥ S.Single words such as “contraindicated” are abbreviated as “C”.

By studying research reports and literatures concerning group decision-making issues, we may find that there are two phases, known as an aggrega-tion phase and an exploitation phase [42] included in a general resolution pat-tern in solving conventional decision-making issues. The assessments givenby individual experts in the first phase will be arranged in the second phase.Under a given criterion, the most consensual alternative or alternatives canbe selected.

Chapter 9 89

9.1.1 The Aggregation Phase

Let us suppose that A = “Alternatives” = {ai}, i = 1, · · · , n, representsa set including n types of alternatives. We assume that E = “Experts” ={ej}, j = 1, · · · , p, denotes a collection of p experts and, S = “Linguisticterm set” = {sk}, k = 0, . . . , g consists of g + 1 linguistic assessments. Weuse the combination of comparative terms and the words selected from S toexpress the judgments Pij (the judgments of ej referring to treatment ai).Especially, S contains the elements ordered in such a way that sq ≤ sr if andonly if q ≤ r, q, r = 0, · · · , g [42]. It is worth highlighting that each sk isrepresented by continuous fuzzy numbers in the Left-Right form, L−R form[89]. The aggregated preferences from individual experts are presented inTable 9.1.

Table 9.1: The hesitant judgment tableAlternatives Experts

e1 e2 · · · ep

a1 P11 P12 · · · P1p

a2 P21 P22 · · · P2p

· · · · · · · · · · · · · · ·an Pn1 Pn2 · · · Pnp

Sets H ijS contains these elements of S which consider the judgments Pij .

H ijS ⊆ S, i = 1, · · · , n, j = 1, · · · , p. By utilizing the operation of union on

sets H ijS on each row, the new generated HFLTS, Uai , i = 1, · · · , n, becomes

a subset of the linguistic term set S and obtains all conceivable effectivenessassessments. Subsequently, the union of all the elements in Uai yields theeffectiveness of each alternative denoted by Eff(ai) = Wai . We illustrate thisby Ex. 9.2.

Example 9.2. Consider three alternatives {a1, a2, a3} ⊂ A which representthree different kinds of treatment schemes. Three experts {e1, e2, e3} ⊂ Eexpress their preferences about these treatment alternatives by combiningcomparative terms and words selected from the linguistic term set S ={s0, s1, s2, s3, s4, s5}, in which s0 = “contraindicated” = “C”, s1 = “accept-able” = “A”, s2 = “possible” = “P”, s3 = “suitable” = “S”, s4 = “recommended”= “R”, s5 = “strongly rekommended” = “SR”. Table 9.2 displays the collectionof preferences. Table. 9.3 shows the HFLTS’s subset of S.

We use the operation of union on H ijS on each row, denoted by Uai =⋃

H ijS , i, j = 1, · · · , 3, to obtain all conceivable effectiveness assessments for

each alternative.

90 Chapter 9

Table 9.2: The hesitant preference tableAlternatives Experts

e1 e2 e3

a1 ≤ A [A, P] Ra2 [A, S] [P, S] Ca3 [S, R] [C, P] [P, S]

Table 9.3: The HFLTS’s subset of SAlternatives Experts

e1 e2 e3

a1 H11S ={s0, s1} H12

S ={s1, s2} H13S ={s4}

a2 H21S ={s1, s2, s3} H22

S ={s2, s3} H23S ={s0}

a3 H31S ={s3, s4} H32

S ={s0, s1, s2} H33S ={s2, s3}

Ua1 = {C, A} ∪ {A, P} ∪ {R}= {C, A, P, R}= {s0, s1, s2, s4},

Ua2 = {A, P, S} ∪ {P, S} ∪ {C}= {C, A, P, S}= {s0, s1, s2, s3},

Ua3 = {S, R} ∪ {C, A, P} ∪ {P, S}= {C, A, P, S, R}= {s0, s1, s2, s3, s4}.

Hence, the effectiveness assessments of a1, a2 and a3 can be given as follows:

Eff(a1) = Wa1 = s0 + s1 + s2 + s4,Eff(a2) = Wa2 = s0 + s1 + s2 + s3,Eff(a3) = Wa3 = s0 + s1 + s2 + s3 + s4.

We show how the collected linguistic assessments can be transformed intonumerical values in the exploitation phase, and later on how the optimalalternative or alternatives are obtained in next section.

9.1.2 The Exploitation Phase

After obtaining the HFLTS containing all conceivable effectiveness assess-ments, we would like at first to utilize the algorithm for calculating the sum

Chapter 9 91

of fuzzy numbers in the L−R form, and later on to transform the L−R forminto the interval form [89]. Finally, by adopting the technique of rankingfuzzy numbers in [104], we hopefully can select the most consensual alterna-tive or alternatives.

We recall the information about fuzzy numbers expressed in the L − Rform. We suppose that sq and sr are two fuzzy numbers in the L−R form,in which q, r = 0, . . . , g. We describe sq = (msq , αsq , βsq)LR and sr =(msr , αsr , βsr)LR, in which msq and msr are called the mean values, αsq andαsr are defined as the left spreads, βsq and βsr are known as right spreads,respectively. The union of sq and sr is calculated by

sq + sr = (msq +msr , αsq + αsr , βsq + βsr). (9.1)

Being able to rank the fuzzy numbers obtained from (9.1), we need firsttransfer them into interval forms.

We review the fuzzy number tranformation from the L − R form intointerval form [89]. Assume Wai = (mWai

, αWai, βWai

)LR is a fuzzy numberin the L−R form. The interval form of Wai is given by (9.2)

Wai = [b−Wai, mWai

, b+Wai]int, (9.2)

in which mWaiis the mean value, b−Wai

= mWai− αWai

and b+Wai= mWai

+

βWaiare defined as the left and the right border, respectively. The member-

ship function associated with the fuzzy number Wai = [b−Wai, mWai

, b+Wai]int

can be given by the following s-functions [83, 88, 111]:

y = μWai(z)

{Left(μWai

(z)) for z ≤ mWai,

Right(μWai(z)) for z ≥ mWai

,(9.3)

in which

Left(μWai(z)) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2

( z − b−Wai

mWai− b−Wai

)2

for b−Wai≤ z ≤ c1Wai

,

1− 2

(z −mWai

mWai− b−Wai

)2

for c1Wai≤ z ≤ mWai

(9.4)

and

Right(μWai(z)) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1− 2

(z −mWai

b+Wai−mWai

)2

for mWai≤ z ≤ c2Wai

,

2

( z − b+Wai

b+Wai−mWai

)2

for c2Wai≤ z ≤ b+Wai

,

(9.5)

92 Chapter 9

where c1Wai=

b−Wai+mWai

2and c2Wai

=b+Wai

+mWai

2are arithmetic mean

values.Ranking fuzzy numbers in a decision-making environment is a very im-

portant and complex procedure. So far, the approaches ranking fuzzy num-bers have been proposed in [9, 14, 16, 17, 22, 30, 51, 91]. Some of them aredifficult to perform and others lead to different outcomes for a same prob-lem. Therefore, a revised approach, based on [17], was explicated by Wangand Lee in [104]. In [104], the authors argued that “multipying the valueon the horizontal axis with the value on the vertical axis often degrades theimportance of the value on horizontal axis in ranking fuzzy numbers”. In-stead, Wang and Lee proposed a technique to overcome the shortcomings.The revised method is given by the following criteria:

• If z(Wai) > z(Waj ), then Wai > Waj .

• If z(Wai) < z(Waj ), then Wai < Waj .

• If z(Wai) = z(Waj ), then Wai = Waj . Thereby, we check the followingconditions:

• If μWai(z) > μWaj

(z), then Wai > Waj .

• If μWai(z) < μWaj

(z), then Wai < Waj .

• If μWai(z) = μWaj

(z), then Wai = Waj ,

in which

z(Wai) =

∫mWai

b−Wai

zLeft(μWai(z)) dz +

∫ b+WaimWai

zRight(μWai(z)) dz

∫mWai

b−Wai

Left(μWai(z)) dz +

∫ b+WaimWai

Right(μWai(z)) dz

(9.6)

and

μWai(z) =

∫ 10 μLeft(μWai

(z))−1 dμ+∫ 10 μRight(μWai

(z))−1 dμ∫ 10 Left(μWai

(z))−1 dμ+∫ 10 Right(μWai

(z))−1 dμ. (9.7)

Here,(z(Wai), μ(Wai)

)is the centroid point of the fuzzy number of Wai ,

Left(μWai

(z))

and Right(μWai

(z))

are called the left and the right mem-bership functions of Wai , Left(μWai

(z))−1 and Right(μWai(z))−1 are known

as the inverse functions of Left(μWai(z)) and Right(μWai

(z)), respectively.

Chapter 9 93

9.2 A Practical Study

In this section, a practical study based on HFLTS in a multi-expert decision-making issue is concerned. Five medical professionals at the Urology De-partment of Blekinge County Hospital in Karlskrona in Sweden suppliedus with a new multip-expert decision-making case in prostate cancer. Thistime, a wider spectrum of the judgments are provided by using the compar-ative terms and single words. Also, the s−parametric membership functionshave been designed to represent the judgments and transform the linguisticassessments into numerical factors. The union operation on fuzzy sets inthe L−R form are used to obtain assessments as comprehensive as possiblewithout any information loss.

Let us assume that E = {e1, e2, e3, e4, e5} denotes a group consist-ing of five medical experts, A = {a1, a2, a3, a4, a5, a6, a7} symbolizesseven treatment alternatives, in which a1 = “active expectance”, a2 = “ac-tive monitoring”, a3 = “symptom based treatment”, a4 = “brachytherapy”,a5 = “external beam radiation therapy”, a6 = “adjuvant hormonal therapy”and a7 = “radical prostatectony”.

Furthermore, a linguistic term set S = {s0, s1, s2, s3, s4, s5} includessix linguistic terms describing the effectiveness of the treatment schemes, inwhich s0 = “contraindicated” = “C”, s1 = “acceptable” = “A”, s2 = “possible”= “P”, s3 = “suitable” = “S”, s4 = “recommended” = “R” and s5 = “stronglyrecommended” = “SR”. Each linguistic term is associated with general s-parametric membership function [83, 88, 111] given by

μsk(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

(z − (

(zmin − hz) + hz · k)

hz

)2

for(zmin − hz

)+ hz · k ≤ z ≤ (

zmin − hz2

)+ hz · k,

1− 2

(z − (zmin + hz · k)

hz

)2

for(zmin − hz

2

)+hz · k ≤ z ≤ zmin + hz · k,

1− 2

(z − (zmin + hz · k)

hz

)2

for zmin + hz · k ≤ z ≤(zmin + hz

2

)+ hz · k,

2

(z − (

(zmin + hz) + hz · k)

hz

)2

for(zmin + hz

2

)+ hz · k ≤ z ≤ (

zmin + hz)+ hz · k.

(9.8)

where zmin ∈ Z, Z = [0, 1] is a reference interval, hz is defined as the distanceof the peaks between two adjacent fuzzy numbers. If we hold zmin and hzas fixed values, by choosing k = 0, · · · , 5, we will obtain a family of six

94 Chapter 9

membership functions that map the effectiveness of the treatment therapies.Functions sk are presented in Fig. 9.1.

0s 1s 2s 3s 4s 5s

z

zks

Figure 9.1: The family of hesitant membership functions, s0 − s5

Here, s0 = “contraindicated” = “C” = (0, 0.2, 0.2)LR, s1 = “accept-able” = “A” = (0.2, 0.2, 0.2)LR, s2 = “possible” = “P” = (0.4, 0.2, 0.2)LR,s3 = “suitable” = “S” = (0.6, 0.2, 0.2)LR, s4 = “recommended” = “R” =(0.8, 0.2, 0.2)LR and s5 = “strongly recommended” = “SR” = (1.0, 0.2, 0.2)LR,

By combining comparative terms with single words, the experts expressthe preferences of the treatment therapies in a broader spectrum. Theseassessments are aggregated in Table 9.4.

Table 9.4: The Hesitant judgment table of practical studyAlternatives Experts

e1 e2 e3 e4 e5

a1 [A, P] ≥ S C [A, P] ≤ Pa2 ≤ A C [A, P] ≤ A Ra3 [A, S] C C C ≤ Aa4 [A, S] [P, S] C [A, S] ≤ Sa5 [S, R] [P, S] ≥ S ≥ P ≤ Aa6 [A, S] [P, S] C [P, R] Ca7 [S, R] [C, P] [P, S] [P, R] [P, R]

The assessment “[A, P]” denotes a comparative term, which indicatesthe terms s ∈ S between “acceptable” and “possible”. It is also a hesitantfuzzy linguistic term set H11

S = {s1, s2}, in which {s1, s2} ⊂ S. “≥ S” canbe interpreted as “greater than suitable”, which symbolizes another hesitant

Chapter 9 95

fuzzy linguistic term set H12S = {s0, s1, s2, s3}. Furthermore, “≤ A” means

“less than acceptable”, which assigns H21S = {s0, s1}.

In order to obtain the assessments as comprehensive as possible andprevent the information loss, for individual alternative, we perform the op-eration of union on fuzzy sets to aggregate all possible preferences in one set.Therefore,

Ua1 = {A, P} ∪ {S, R, SR} ∪ {C} ∪ {A,P} ∪ {C,A, P}= {C, A, P, S, R, SR}= {s0, s1, s2, s3, s4, s5}.

Figure 9.2 illustrates the possible effectiveness of the first treatment alterna-tive “active expectance”.

0s 1s 2s 3s 4s 5s

z

zks

Figure 9.2: Conceivable effectiveness for the first treatment alternative a1

For the second alternative a2, we get

Ua2 = {C, A} ∪ {C} ∪ {A, P} ∪ {C, A} ∪ {R}= {C, A, P, R}= {s0, s1, s2, s4}.

In a similar way, we obtain the possible effectiveness for the remainedalternatives. For the third alternative a3, we get

Ua3 = {A, P, S} ∪ {C} ∪ {C} ∪ {C} ∪ {C, A}= {C, A, P, S}= {s0, s1, s2, s3}.

96 Chapter 9

For the fourth alternative a4, we obtain

Ua4 = {A, P, S} ∪ {P, S} ∪ {C} ∪ {A, P, S} ∪ {C, A, P, S}= {C, A, P, S}= {s0, s1, s2, s3}.

Both a3 and a4 have the same preferences.The treatment alternative a5 possesses the following possible assessments:

Ua5 = {S, R} ∪ {P, S} ∪ {S, R, SR} ∪ {P, S, R, SR} ∪ {C, A}= {C, A, P, S, R, SR}= {s0, s1, s2, s3, s4, s5}.

We noticed that a1 and a5 contain the identical effectiveness assessments.The last two alternatives a6 and s7 gain assessments shown as follows:

Ua6 = {A, P, S} ∪ {C} ∪ {P, S} ∪ {C} ∪ {P, S, R} ∪ {P, S, R}= {C, A, P, S, R}= {s0, s1, s2, s3, s4}

and

Ua7 = {S, R} ∪ {C, A, P} ∪ {P, S} ∪ {P, S, R} ∪ {P, S, R}= {C, A, P, S, R}= {s0, s1, s2, s3, s4}.

We recall that the union of two fuzzy numbers can be performed by (9.1).Thereby, the effectiveness of a1 and a5 can be calculated

Eff(a1) = Wa1 = s0 + s1 + s2 + s3 + s4 + s5

= (0, 0.2, 0.2)LR + . . .+ (1.0, 0.2, 0.2)LR

= (3, 1.2, 1.2)LR

≡ [1.8, 3.0, 4.2]int

and

Eff(a5) = Wa5 = s0 + s1 + s2 + s3 + s4 + s5

= (0, 0.2, 0.2)LR + . . .+ (1.0, 0.2, 0.2)LR

= (3, 1.2, 1.2)LR

≡ [1.8, 3.0, 4.2]int.

The membership functions for a1 and a5 are given by

μWa1(z) = μWa5

(z)

{Left(μWa1

(z)) for z ≤ 3.0,

Right(μWa1(z)) for z ≥ 3.0,

(9.9)

Chapter 9 97

in which

Left(μWa1(z)) = Left(μWa5

(z)) =

⎧⎪⎪⎨⎪⎪⎩2

(z − 1.8)

3.0− 1.8

)2

for 1.8 ≤ z ≤ 2.4,

1− 2

(z − 3.0

3.0− 1.8)

)2

for 2.4 ≤ z ≤ 3.0

(9.10)and

Right(μWa1(z)) = Right(μWa5

(z)) =

⎧⎪⎪⎨⎪⎪⎩1− 2

(z − 3.0

4.2− 3.0

)2

for 3.0 ≤ z ≤ 3.6,

2

(z − 4.2

4.2− 3.0

)2

for 3.6 ≤ z ≤ 4.2.

(9.11)Figure 9.3 depicts the membership function of Wa1 and Wa5 .

zzaa WW 51

z

y

Figure 9.3: The membership function of Wa1 and Wa5

The effectivness of a2 is given by

Eff(a2) = Wa2 = s0 + s1 + s2 + s4

= (1.4, 0.8, 0.8)LR

≡ [0.6, 1.4, 2.2]int.

(9.12)

The membership functions of Eff(a2) is given by (9.13) and (9.14) as:

Left(μWa2(z)) =

⎧⎪⎪⎨⎪⎪⎩2

(z − 0.6

1.4− 0.6

)2

for 0.6 ≤ z ≤ 1,

1− 2

(z − 1.4

1.4− 0.6)

)2

for 1 ≤ z ≤ 1.4

(9.13)

98 Chapter 9

and

Right(μWa2(z)) =

⎧⎪⎪⎨⎪⎪⎩1− 2

(z − 1.4

2.2− 1.4

)2

for 1.4 ≤ z ≤ 1.8,

2

(z − 2.2

2.2− 1.4

)2

for 1.8 ≤ z ≤ 2.2.

(9.14)

The membership function of Wa2 is plotted in Fig. 9.4.

zaW 2

z

y

Figure 9.4: The membership function of Wa2

For a3 and a4, the result is shown by

Eff(a3) = Wa3 = s0 + s1 + s2 + s3

= (0, 0.2, 0.2)LR + . . .+ (0.6, 0.2, 0.2)LR

= (1.2, 0.8, 0.8)LR

≡ [0.4, 1.2, 2.0]int

and by

Eff(a4) = Wa4 = s0 + s1 + s2 + s3

= (0, 0.2, 0.2)LR + . . .+ (0.6, 0.2, 0.2)LR

= (1.2, 0.8, 0.8)LR

≡ [0.4, 1.2, 2.0]int.

The membership functions of Wa3 and Wa4 are presented by (9.15) and

Chapter 9 99

(9.16) as


(z)) =

⎧⎪⎪⎨⎪⎪⎩2

(z − 0.4

1.2− 0.4

)2

for 0.4 ≤ z ≤ 0.8,

1− 2

(z − 1.2

1.2− 0.4)

)2

for 0.8 ≤ z ≤ 1.2

(9.15)and


(z)) =

⎧⎪⎪⎨⎪⎪⎩1− 2

(z − 1.2

2.0− 1.2

)2

for 1.2 ≤ z ≤ 1.6,

2

(z − 2.0

2.0− 1.2

)2

for 1.6 ≤ z ≤ 2.0.

(9.16)Figure 9.5 represents the membership function of Wa3 and Wa4 as follows:

zzaa WW 43

z

y


And finally, the effectivness of a6 and a7 is yielded by

Wa6 = Wa7 = s0 + s1 + s2 + s3 + s4

= (0, 0.2, 0.2)LR + . . .+ (0.8, 0.2, 0.2)LR

= (2.0, 1.0, 1.0)LR

≡ [1.0, 2.0, 3.0]int.

(9.17)

The membership functions of Wa6 and Wa7 are given by (9.18) and (9.19)

100 Chapter 9

as


(z)) =

⎧⎪⎪⎨⎪⎪⎩2

(z − 1.0

2.0− 1.0

)2

for 1.0 ≤ z ≤ 1.5,

1− 2

(z − 2.0

2.0− 1.0)

)2

for 1.5 ≤ z ≤ 2.0

(9.18)

and


(z)) =

⎧⎪⎪⎨⎪⎪⎩1− 2

(z − 2.0

3.0− 2.0

)2

for 2.0 ≤ z ≤ 2.5,

2

(z − 3.0

3.0− 2.0

)2

for 2.5 ≤ z ≤ 3.0.

(9.19)

Figure 9.6 depicts the membership functions of Wa6 and Wa7 .

zzaa WW 76

z

y


We first use (9.6) to calculate the horizontal coordinate of the cen-troid point of each fuzzy number Wai . If there exists identical horizon-tal coordinates, then (9.7) will be use to compute the vertical coordinate.By the insertion of the left respective the right membership functions andthe borders of Wa1 and Wa5 in (9.6), we obtain the horizontal coordinateof Wa1 and Wa5 presented as follows:

Chapter 9 101

z(Wa1) = z(Wa5) =

∫ 3.01.8 zLeft(μWa1

(z)) dz +∫ 4.23.0 zRight(μWa1

(z)) dz∫ 3.01.8 Left(μWa1

(z)) dz +∫ 4.23.0 Right(μWa1

(z)) dz

=3.6

1.2= 3.0.

(9.20)

By the same procedure, we obtain the horizontal coordinates for theremained alternatives presented below:

z(Wa2) =




(z)) dz +∫ 2.21.4 Right(μWa2

(z)) dz

=1.12

0.8= 1.4,

(9.21)

z(Wa3) = z(Wa4) =




(z)) dz +∫ 2.01.2 Right(μWa3

(z)) dz

=2.073

0.8≈ 2.5913,

(9.22)

z(Wa6) = z(Wa7) =




(z)) dz +∫ 3.02.0 Right(μWa6

(z)) dz

=2

1= 2.

(9.23)

Since no identical horizontal coordinates are found, we do not need tocompute the values of the vertical coordinates. By means of the criteriaintroduced in [104], we obtain the following results:

3 > 2.5913 > 2 > 1.4, i.e. z(Wa1) = z(Wa5) > z(Wa3) = z(Wa4) >z(Wa6) = z(Wa7) > z(Wa2). The first a1 = “active expectance” and the fifthealternative a5 = “external beam radiation therapy” have the most optimaleffectivness according to our computation.

9.3 Summary

According to the physicians’ requirement, a multi-expert decision-makingproblem involved the selection of the most consensual treatment therapy for

102 Chapter 9

a prostate cancer patient has been considered. The treatment strategies pro-posed by the MDT members have been ranged from the most recommendedto contraindicated. The first and the fifth alternatives named as “active ex-pectance” respective “external beam radiation therapy” are obtained as themost recommended.

Hesitant fuzzy linguistic term sets (HFLTS) have been utilized to presentthe preferences expressed by the physicians. In order to avoid the informationloss, we use the union performed on the continuous fuzzy sets to collectthe assessments as comprehensive as possible. Moreover, the s−membershipfunctions facilitate the transformation process from the linguistic preferencesto the numerical values.

The horizontal coordinate of the centroid point of each fuzzy number,which indicates the treatment effectivness are adopted for ranking the fuzzynumbers in a decision-making environment. The calculating process has itscomplexity but the technique is reliable.

CHAPTER10Conclusions

This dissertation principally focuses on the applications of fuzzy sets in med-ication prognoses for gastric cancer patients as well as for prostate cancerpatients. This chapter summarizes the research results, the contributions ofthe research work and possible future directions.

10.1 Research Summary

This research has its inception from the research project entitled “Fuzzy Sets,Rough Sets and Fuzzy Statistics in Treatment of Gastric Cancer Patients”. Ithas been a cooperation between mathematicians and physicians in BlekingeRegion in Sweden. The purpose is to provide the physicians with math-ematical complements in real-life problems concerning decisions based onclinical investigations. This project has resulted in four applications, whichare presented in Chapter 3, 4, 5 and 6, respectively.

Interesting questions which have been investigated in this project can beformulated as:

• Is it possible to predict the life expectancy for gastric cancer patientsby means of fuzzy sets theory?

• Is it possible to evaluate the operation respective none operation pos-sibility for gastric cancer patients by the technique of data analysis?

Relevant approaches for solving the problems are listed below:

• The Mamdani fuzzy controller.

• The Sugeno controller.

• Fuzzy c-means clustering analysis.

103

104 Chapter 10

• Approximation of irregular point sets by clock-formed continuous mem-bership functions.

Later on, the research topic focuses on multi-expert decision-makingproblems involving ranking the effectiveness of the treatment schemes fortwo separate prostate cancer patients. As the physicians’ requirement, werank the treatment schemes from the most recommended to the contraindi-cated. The following linguistic approaches have been taken into account tomeet the demand:

• The probabilistic model with linguistic judgments.

• The 2-tuple linguistic representations.

• The hesitant fuzzy linguistic term sets.

The results are presented in Chapter 7, 8 and 9. In the next section, themotivations, why the fuzzy sets and the linguistic approaches are adoptedin the applications of medication prognoses for cancer patients, are brieflysummarized.

10.2 Motivations of the Choice of Fuzzy Sets andLinguistic Approaches

The applications of fuzzy sets and linguistic approaches play an importantrole in this disseration, since these approaches possess advantages which areunavailable for classical analytical models or statistical techniques.

10.2.1 The Mamdani Fuzzy Control

Strict analytical models are often used to describe the relation between theindependent variables and the dependent variable. However, for the prob-lems derived from the project, the strict analytical models connecting theindependent variable (such as the patients’ age and the CRP -values) and thedependent variable (defined as the survival length) are unavailable. More-over, statistical technique is a frequently used alternative to approximatethe survival time for cancer patients [19, 47]. Once the observation data arenot complete, then the out-put step-function becomes discontinuous, thatmeans the survival length can not be estimated. This shortcomings can becompensated by fuzzy sets due to the overlappings between the membershipfunctions. Therefore, being able to answer the first research question, wechoose the control actions to complete the prediction.

A fuzzy controller includes three phases, the fuzzification phase, the con-troller procedure and the defuzzification process. Fuzzy sets assigned withs-parametric membership functions have been employed to fuzzify variables

Chapter 10 105

such as the age, the CRP -values and the survival period. The experience-based knowledge generates a set of control rules characterized by the “IF. . . , THEN . . . ” syntax, which connects the variables and facilitates thecontroller procedure. In the defuzzification phase, the method of the centerof gravity (COG) [114] belonging to the centroid strategy has been takeninto consideration to estimate the survival length, since it is the most trivialweighted average. This technique generates a concrete expected value.

10.2.2 Fuzzy C-Menas Clustering Analysis

According to the physicians’ experience in the clinical work, the operation isso far the most effective treatment scheme for gastric cancer. Being able topartition a data set consisting of 25 gastric cancer patients into two clusters,in which one contains individuals with operation prognosis and the otherincludes patients classified for no operation, the technique of fuzzy c-meansclustering analysis is adopted.

By comparing with the hard clustering, data points are divided into crispclusters, in which each data point belongs to exactly one cluster. Boundarypoints can be allocated with difficulty by the hard clustering. However, fuzzyclustering analysis makes the boundary points possible to be placed in morethan one cluster.

The purpose of the technique of fuzzy c-means clustering analysis in ourproject is to minimize the objective function J with respect to μSi(xk) andd(vi, xk), i = 1, 2 and k = 1, . . . , 25. Here μSi(xk) represents the value ofthe membership degree of xk in cluster Si. The notation of d(vi, xk) sym-bolizes the distance between the cluster center and the vector xk consistingof age, CRP -value and body weight.

The determination of the membership degrees of xk in cluster Si is ofimportance in the algorithm, since the membership degrees have an influenceon the convergence speed. It is rather difficult to determine the value of theinitial membership degrees. Random selection of the values is an availabletechnique. To be able to estimate the initial membership degrees as preciseas possible, the s-class membership functions [78, 79, 80, 81] are utilized. Byusing MATLAB, after 31 iterations, we obtain that ||U31−U30|| = 9.93692×10−9 < 10−8, which means that the membership degrees in the last twopartition matrices no longer alter the values. The partitioned clusters becomestable. Thereby, the action of clustering partitions have been successfullycompleted.

The control action and the technique of data analysis have provided thephysicians and the patients with promising information, which improves thequality of the patient care as well as the life quality for gastric cancer patients.

106 Chapter 10

10.2.3 Point Set Approximation by Clock-Shaped Member-ship Functions

The point sets approximation applied in Chapter 6 is an extended version ofthe fuzzy c-means clustering analysis. The idea is to design two-dimensionalcontinuous truncated π-functions to approximate the operation possibilityfor any arbitrary gastric cancer patient whose clinical data is placed withinthe domain of the point sets.

Two types of point sets have been created. The first one is indicated by{(f c

xk, μSi(xk))} and the second one is denoted by {(fxk

, μSi(xk))}. Here

f cxk

is the characteristic value of the patient xk, k = 1, . . . , 25, which isdetermined by the code vector (ac, crpc, bwc). The notation of fxk

indicatesthe characteristic value of the patient xk, which is obtained by the patients’original clinical data of age, CRP -value and body weight. Further, μSi(xk)symbolizes the membership degrees of operation possibility obtained in thelast partition matrix in Chapter 5.

In a two-dimensional coordinate system, the characteristic values of f cxk

,arranged in the ascending order, are assigned to the abscissa, whereas theoperation membership degrees obtain in the last partition matrix in Chapter5 are allocated on the ordinate. These data points

(f cxk, μSi(xk)

)are irreg-

ularly spread in the coordinate system. Three special points are selectedto design the π-functions. The start point (f c′1 , μc′1) =

(min(f c

xk), μSi(xk)

)possesses the minimal characteristic value. The second point (f c′2 , μc′2) =(f cxk,max(μSi(xk))

)has the maximal membership degree and the terminate

point (f c′3 , μc′3) =(max(f c

xk), μSi(xk)

)has the maximal characteristic value.

By the inspiration of the truncated π-functions designed by Rakus-Andersson[72, 74, 84, 85, 82, 112], the clock-shaped membership function approximat-ing the point set {(f c

xk, μSi(xk)

)} is accomplished.Since we noticed that, by the code vectors, some of the patients get the

same characteristic values, but different membership degrees of operationpossibilities. Therefore, the original clinical data of the age, the CRP -valueand the body weight have been put in use to be able to approximate thecharacteristic values of fxk

more precisly. In a similar way, the membershipfunctions for the data point set {(fxk

, μSi(xk))} is designed.Thereby, when we insert the data of any arbitrary gastric cancer patient

belonging to the point sets and choose the suitable membership function, theestimation of the operation possibility is obtained. The approximation of nooperation possibility is treated as the complement of operation possibility.

Linguistic approach and fuzzy sets supply with each other in this disser-tation. The use of the linguistic terms makes the communication and thejudgments easier and naturally performed, whereas the fuzzy sets exchangethe linguistic terms to numerical values. This makes the further calcula-tions be possibly completed. Therefore, besides the applications of fuzzysets, the linguistic approaches have been adopted in multi-expert decision-

Chapter 10 107

making problems concerning ranking the effectiveness of treatment therapiesfor two separate prostate cancer patients. By studying a large amount of re-search reports, three linguistic models, stated as the probablistic model withlinguistic judgments, the model of 2-tuple linguistic representations and thehesitant fuzzy linguistic term sets, have been selected to investigate the effec-tiveness of treatment therapies of prostate cancer. The first two mentionedmodels are employed to select the medication for the first patient while thethird technique is used to determine the treatment scheme for the secondpatient.

10.2.4 Probabilistic Model with Linguistic Judgments

In the probabilistic model, four physicians, denoted as e1, e2, e3 and e4 inprostate cancer have been independently asked to assess the effectiveness ofa set of six treatment therapies for the first prostate cancer patient. The setof six treatment therapies is given by A = {ai}, i = 1, . . . , 6, in which a1 =“wait and see”, a2 = “active monitoring”, a3 = “symptom based treatment”, a4= “brachytherapy”, a5 = “external beam radiation therapy” and a6 = “radicalprostatectomy”. Moreover, seven linguistic terms describing the effectivenessare formulated as s0 = “contraindicated”, s1 = “doubtful”, s2 = “acceptable”,s3 = “possible”, s4 = “suitable”, s5 = “recommended” and s6 = “strongly rec-ommended”. The judgment table collecting all the linguistic judgments playsan important role. From this table, the preference value of each ai, denotedas P

(Xai = sl

), i = 1, . . . , 6, l = 0, . . . , 6, is determined. Subsequently,

the choice value of ai, V (ai), is computed. By employing the maximumoperator on all the choice values, the most recommended treatment ther-apy is obtained. For the first prostate cancer patient, the effectiveness ofthe treatment alternatives are ranked as a2 > a6 > a4 > a5 > a1 = a3.The result indicates that the second alternative stated as “a2 = active mon-itoring” is the most recommended, whereas the first alternative “a1 = waitand see” and the third alternative “a3 = symptom based treatment” are thecontraindicated.

10.2.5 The Model of 2-Tuple Linguistic Representations

The model of 2-tuple linguistic representations is another form of the lin-guistic approach. We use the model to verify the result obtained from thecase study in the probablistic model. Instead of expressing the judgmentsby single linguistic words, the 2-tuples denoted by (sl, α), l = 0, . . . , 6 andα ∈ [−0.5, 0.5

), are used. Also, two transformations have been involved

in the model. The first transformation maps a 2-tuple representation into anumerical factor β

ejai , i = 1, . . . , 6, j = 1, . . . , 4, whereas the second trans-

formation exchanges the arithmetic mean value of βejai to a 2-tuple linguistic

representation. By comparing the arithmetic mean value of each treatment

108 Chapter 10

alternative given in the form of 2-tuple representation, the second treatmenttherapy a2 suggested as “active monitoring” obtains the maximal arithmeticmean value. This suggestion agrees with the result obtained from the prob-abilistic model. The convergence results demonstrate the high reliability ofadopting the linguistic approach in the solution of group decision-makingproblems. Furthermore, the use of the model of 2-tuple linguistic represen-tation prevents the loss of information and makes the result more precis.

10.2.6 The Hesitant Fuzzy Linguistic Term Sets

The technique of hesitant fuzzy linguistic term sets is the third challenge con-cerning ranking the effectiveness of treatment schemes of prostate cancer. Forthe second prostate cancer patient, five experts, {e1, . . . , e5} are involvedto express the preferences of seven treatment alternatives {a1, . . . , a7}, inwhich a1 = “active expectance”, a2 = “active monitoring”, a3 = “symptombased treatment”, a4 = “brachytherapy”, a5 = “external beam radiation ther-apy”, a6 = “adjuvant hormonal therapy” and a7 = “radical prostatectony”.Neither single linguistic terms nor 2-tuple linguistic representations are uti-lized. Instead, a wider spectrum of the judgments is supplied by the hesitantfuzzy linguistic term sets, [92] i.e., the combinations of six linguistic wordsand comparative terms are employed to express the judgments concerningthe seven treatment alternatives. Here, s0 = “contraindicated”, s1 = “ac-ceptable”, s2 = “possible”, s3 = “suitable”, s4 = “recommended” and s5 =“strongly recommended”. The comparative terms refer to words such as“less than . . . ”, “greater than . . . ”, “between . . . and . . . ” etc. Furthermore,s0, . . . , s5 are fuzzy sets assigned with s-parametric membership functions.Since the judgments are expressed by the hesitant fuzzy linguistic term sets,which constitute the union of the fuzzy sets or parts of the fuzzy sets inthe L-R form, we choose the addition operation on the L-R fuzzy numbersto determine the effectiveness of each treatment alternative. The horizon-tal coordinate of the centroid point [104] of each treatment alternative iscalculated to choose the most recommended treatment alternative(s). Theobtained results are presented as a1, “active expectance”, and a5, “externalbeam radiation therapy”.

From the clinical point of view, the results obtained from the linguisticapproaches have supplied the physicians with reliable information. The lin-guistic technique can be seen as a trustworthy basis when the disagreementsoccur. Of course, it is up to the patients if they agree to the suggestions ornot.

10.3 Medical Remarks

There is a need to develop some methods that could facilitate concludingin such a multivariate analysis as a medical or surgical treatment decision

Chapter 10 109

making. As far as we are concerned there is not any mathematical processavailable now, coping with such problems, apart from statistics. The sta-tistical approach is the well known and widely accepted way to deal withmedical problems, but there are situations in which the way general statis-tics is not sufficient. For example the statistical approach cannot be appliedwhen the patients groups are too small or the data are missing. Also theexperience-based knowledge, which is commonly used when the informationis vague or incomplete, cannot be used by statistical processes either.

In such cases there is a huge discrepancy in the medical background andthe supportive information for decision-makers. When the good quality dataare present, the decisions are more easy to take than in cases where almost allmedical information is based on descriptive studies of single cases or on theopinion of an expert or an organization (the lowest grade in the frequentlyused Oxford Evidence-Based Medicine scale), that is not supported by highquality studies.

To fill this significant gap we have intended to apply fuzzy mathematicsto test its reliability in two different diseases: gastric cancer and prostatecancer.

In summary we can conclude that from our medical consultants’ point ofview the described methods are very flexible and reliable. One of the mostimportant features is to include physician experiences to warrant the morevaluable results. The usage of linguistic terms also increases the flexibilityand simplicity in the clinical terms of use, as well as in terms of communi-cation between mathematicians, physicians and patients.

Results obtained, when using the methods mentioned above, seem to bereliable and easy to be communicated to the patient. We are aware of thatfuzzy mathematics has restrictions and cannot replace statistical techniquesand clinical studies, but we think that it is a complementary method thatcan refine decision-making process and it probably should be tested in thereal-life conditions on the more numerous groups of patients / physicians.

10.4 Contributions

This research work has generated several contributions. The extensive liter-ature investigations lead to the choices of suitable approaches. For the firsttime, the adapted fuzzy models have been applied to the clinical problemsconcerning the medication prognoses for gastric cancer patients as well asfor prostate cancer patients. The adaptation of models has demanded owncontributions providing with constructions of specially designed membershipfunctions and other complements. This constitutes the unique contributionin this dissertation. Moreover, the life expectancy and the operation possi-bilities have been investigated and estimated for the gastric cancer patientsin this research work. The thesis also has demonstrated the high reliabil-

110 Chapter 10

ity of linguistic approaches in the solutions of multi-expert decision-makingproblems involving medical prognoses for the prostate cancer patients. Thefinal results not only provide the physicians with reliable mathematical com-plements, but also improve the life quality and the medical care for thepatients.

10.5 Possible Future Directions

The research presented in this dissertation has provided a number of inter-esting mathematical complements to real-life questions arise in clinical in-vestigations. The obtained answers assisted the physicians in the prognosesfor gastric cancer patients as well as for prostate cancer patients. Well, newchallenges and ideas arise while the research progresses. In the followingtext, there are a few suggestions for possible future research directions:

This research work shows the indications that it probably may be testedon a larger samples of patients to implement dense surfaces of points insteadof singletons. It would be desirable to prove controllers with a larger numberof biological variables than two. Also the fuzzy sets of type 2 may be apossible alternative suggestion in the control action. In the domain of multi-expert decision making, it may be interesting to rank fuzzy numbers witharea technique using radius of gyration.

APPENDIXA

In this chapter, the original clinical data concerning the operation and thenone operation possibilities for 25 gastric cancer patients are presented inTable A.1. The original membership degrees of the operation and the nooperation are collected in Table A.2. The membership degrees in U0 aregiven in Table A.3. Finally, the operation contra none operation possibilitiesobtained from MATLAB are presented in the Tables A.4 – A.34.

111

112 Appendix A

Table A.1: The original medical data of 25 gastric cancer patientsAge Body weight CRP-value Operation Nonoperation

p1 71 85 1 total littlep2 81 70 9 middle largep3 50 67 4 large middlep4 64 84 13 large littlep5 41 95 4 large littlep6 67 72 32 none largep7 56 77 0.9 total nonep8 27 87 11 middle middlep9 83 67.5 78 none totalp10 71 66.7 0.9 total nonep11 71 50 17 little middlep12 77 74 11 large middlep13 68 56.5 54 none totalp14 66 107.5 2.9 large middlep15 92 85.7 8 none totalp16 80 67 30 none largep17 71 81.9 41 none largep18 74 59 4 middle middlep19 80 70 52 none totalp20 55 63 4.7 middle littlep21 85 70 7 little middlep22 58 73 3.5 large littlep23 67 69 1.4 total nonep24 76 74 4 large middlep25 54 49 36 none large

Appendix A 113

Table A.2: The first updated operation and nonoperation membership de-grees of 25 gastric cancer patients

Age Body weight CRP-value Operation Nonoperationp1 71 85 1 0.944 0.220p2 81 70 9 0.5 0.78p3 50 67 4 0.78 0.5p4 64 84 13 0.78 0.22p5 41 95 4 0.78 0.22p6 67 72 32 0.056 0.78p7 56 77 0.9 0.944 0.056p8 27 87 11 0.5 0.5p9 83 67.5 78 0.056 0.944p10 71 66.7 0.9 0.944 0.056p11 71 50 17 0.22 0.5p12 77 74 11 0.78 0.5p13 68 56.5 54 0.056 0.944p14 66 107.5 2.9 0.78 0.5p15 92 85.7 8 0.056 0.944p16 80 67 30 0.056 0.78p17 71 81.9 41 0.056 0.78p18 74 59 4 0.5 0.5p19 80 70 52 0.056 0.944p20 55 63 4.7 0.5 0.22p21 85 70 7 0.22 0.5p22 58 73 3.5 0.78 0.22p23 67 69 1.4 0.944 0.056p24 76 74 4 0.78 0.5p25 54 49 36 0.056 0.78

114 Appendix A

Table A.3: The updated operation and nonoperation membership degreesof 25 gastric cancer patients

Age Body weight CRP-value Operation Nonoperationp1 71 85 1 0.862 0.138p2 81 70 9 0.36 0.64p3 50 67 4 0.64 0.36p4 64 84 13 0.78 0.22p5 41 95 4 0.78 0.22p6 67 72 32 0.138 0.862p7 56 77 0.9 0.944 0.056p8 27 87 11 0.5 0.5p9 83 67.5 78 0.056 0.944p10 71 66.7 0.9 0.944 0.056p11 71 50 17 0.36 0.64p12 77 74 11 0.722 0.278p13 68 56.5 54 0.056 0.944p14 66 107.5 2.9 0.722 0.278p15 92 85.7 8 0.056 0.944p16 80 67 30 0.138 0.862p17 71 81.9 41 0.138 0.862p18 74 59 4 0.5 0.5p19 80 70 52 0.056 0.944p20 55 63 4.7 0.64 0.36p21 85 70 7 0.36 0.64p22 58 73 3.5 0.78 0.22p23 67 69 1.4 0.944 0.056p24 76 74 4 0.64 0.36p25 54 49 36 0.138 0.862

Appendix A 115

Table A.4: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 1st itreration

Operation Nonoperationp1 0.77141255 0.22858745p2 0.58728384 0.41271616p3 0.71194117 0.28805883p4 0.72219995 0.27780005p5 0.65279521 0.34720479p6 0.24394788 0.75605212p7 0.84204421 0.15795579p8 0.59881503 0.40118497p9 0.35410700 0.64589300p10 0.72874725 0.27125275p11 0.46671149 0.53328851p12 0.62211916 0.37788084p13 0.28998740 0.71001260p14 0.62946832 0.37053168p15 0.55241778 0.44758222p16 0.21111264 0.78888736p17 0.27411622 0.72588378p18 0.61688816 0.38311184p19 0.24317751 0.75682249p20 0.69959150 0.30040850p21 0.57478575 0.42521425p22 0.85160295 0.14839705p23 0.79505855 0.20494145p24 0.71317966 0.28682034p25 0.39366951 0.60633049v1 = (63.0724, 76.9922, 3.89619),v2 = (74.3811, 68.5914, 36.5088),||U1 − U0|| = 1.26692 > 10−8.

116 Appendix A

Table A.5: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 2nd iteration


Appendix A 117

Table A.6: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 3rd iteration


118 Appendix A

Table A.7: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 4th iteration


Appendix A 119



120 Appendix A



Appendix A 121



122 Appendix A



Appendix A 123



124 Appendix A



Appendix A 125


Operation Nonoperationp1 0.74363699 0.25636301p2 0.64547969 0.35452031p3 0.68173993 0.31826007p4 0.69927011 0.30072989p5 0.60638037 0.39361963p6 0.17970019 0.82029981p7 0.76591619 0.23408381p8 0.56201659 0.43798341p9 0.37391207 0.62608793p10 0.76685723 0.23314277p11 0.50352113 0.49647887p12 0.68564295 0.31435705p13 0.30843722 0.69156278p14 0.60043067 0.39956933p15 0.57552481 0.42447519p16 0.28902938 0.71097062p17 0.27057575 0.72942425p18 0.65982170 0.34017830p19 0.28232914 0.71767086p20 0.69135199 0.30864801p21 0.62322341 0.37677659p22 0.80183372 0.19816628p23 0.82414175 0.17585825p24 0.75881279 0.24118721p25 0.39450879 0.60549121v1 = (65.958, 74.229, 6.50248),v2 = (70.4447, 69.7321, 35.8054),

||U11 − U10|| = 0.000745362 > 10−8.

126 Appendix A



||U12 − U11|| = 0.000423947 > 10−8.

Appendix A 127



||U13 − U12|| = 0.000241189 > 10−8.

128 Appendix A



||U14 − U13|| = 0.000137244 > 10−8.

Appendix A 129



||U15 − U14|| = 7.81123× 10−5 > 10−8.

130 Appendix A



||U16 − U15|| = 4.44671× 10−5 > 10−8.

Appendix A 131



||U17 − U16|| = 2.53197× 10−5 > 10−8.

132 Appendix A



||U18 − U17|| = 1.44207× 10−5 > 10−8.

Appendix A 133



||U19 − U18|| = 8.21546× 10−6 > 10−8.

134 Appendix A



||U20 − U19|| = 4.68175× 10−6 > 10−8.

Appendix A 135

Table A.24: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 21st iteration


||U21 − U20|| = 2.66887× 10−6 > 10−8.

136 Appendix A

Table A.25: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 22nd iteration


||U22 − U21|| = 1.52197× 10−6 > 10−8.

Appendix A 137

Table A.26: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 23rd iteration


||U23 − U22|| = 8.68271× 10−7 > 10−8.

138 Appendix A



||U24 − U23|| = 4.95559× 10−7 > 10−8.

Appendix A 139



||U25 − U24|| = 2.82973× 10−7 > 10−8.

140 Appendix A



||U26 − U25|| = 1.61668× 10−7 > 10−8.

Appendix A 141



||U27 − U26|| = 9.24168× 10−8 > 10−8.

142 Appendix A



||U28 − U27|| = 5.28631× 10−8 > 10−8.

Appendix A 143



||U29 − U28|| = 3.02589× 10−8 > 10−8.

144 Appendix A



||U30 − U29|| = 1.73331× 10−8 > 10−8.

Appendix A 145

Table A.34: The operation and nonoperation possibilities of 25 gastric cancerpatients after the 31st iteration


||U31 − U30|| = 9.93692e× 10−9 < 10−8.

146 Appendix A

APPENDIXB

Appendix B contains the point sets, P c = {(f cxk, μSi(xk))} and P = {(fxk

, μSi(xk))}.

In P c, the characteristic values f cxk

of 25 gastric cancer patients determinedby the code vectors, (ac, crpc, bwc), are presented in Table B.1, whereas inP , the characteristic values fxk

of 25 gastric cancer patients calculated bythe orignial clinical data, (a, crp, bw), are presented in Table B.2.

147

148 Appendix B

Table B.1: The point set P c = {(f cxk, μSi(xk))}, of which the characteristic

values fxck

are determined by the code vectors, (ac, crpc, bwc)

f cxk

Operation Nonoperationp1 1.41 0.74366062 0.25633938p2 1.63 0.64570218 0.35429782p3 0.89 0.68159676 0.31840324p4 1.41 0.69917978 0.30082022p5 1.19 0.60626631 0.39373369p6 2.18 0.17955501 0.82044499p7 1.04 0.76572316 0.23427684p8 0.67 0.56190473 0.43809527p9 5.31 0.37393372 0.62606628p10 1.26 0.76699189 0.23300811p11 1.11 0.50360426 0.49639574p12 1.41 0.68588580 0.31411420p13 4.02 0.30844468 0.69155532p14 1.71 0.60039667 0.39960333p15 1.78 0.57564271 0.42435729p16 2.18 0.28925142 0.71074858p17 3.25 0.27054644 0.72945356p18 1.26 0.65995112 0.34004888p19 4.02 0.28238871 0.71761129p20 0.89 0.69125182 0.30874818p21 1.63 0.62341688 0.37658312p22 1.04 0.80161258 0.19838742p23 1.26 0.82420632 0.17579368p24 1.41 0.75903477 0.24096523p25 2.58 0.39447229 0.60552771

Appendix B 149

Table B.2: The point set P = {(fxk, μSi(xk)

)}, whose characteristic valuesfxk

are calculated by the original clinical data, (a, crp, bw)

f cxk

Operation Nonoperationp1 39.94 0.74366062 0.25633938p2 48.75 0.64570218 0.35429782p3 32.23 0.68159676 0.31840324p4 48.24 0.69917978 0.30082022p5 33.1 0.60626631 0.39373369p6 65.03 0.17955501 0.82044499p7 33.098 0.76572316 0.23427684p8 33.16 0.56190473 0.43809527p9 112.6 0.37393372 0.62606628p10 37.103 0.76699189 0.23300811p11 49, 41 0.50360426 0.49639574p12 49.71 0.68588580 0.31411420p13 83.315 0.30844468 0.69155532p14 43.213 0.60039667 0.39960333p15 54.255 0.57564271 0.42435729p16 67.25 0.28925142 0.71074858p17 76.275 0.27054644 0.72945356p18 39.91 0.65995112 0.34004888p19 87.94 0.28238871 0.71761129p20 34.124 0.69125182 0.30874818p21 48.39 0.62341688 0.37658312p22 35.63 0.80161258 0.19838742p23 36.428 0.82420632 0.17579368p24 42.9 0.75903477 0.24096523p25 60.45 0.39447229 0.60552771

150 Appendix B

APPENDIXCAppendix C consists of the judgment tables for the first prostate cancerpatient obtained from the physicians at the Urology Department of BlekingeCounty Hospital in Karlskrona, Sweden.

Patient Expert

Initial: Patient One Initial: Expert One

Age: 64 Date: 2012-10-28

Contraindicated Doubtful Acceptable Possible Suitable Recommended Strongly

recommended

Wait and see X

Active monitorering X

Symptom based treatment X

Brachytherapy X

External beam radiation therapy X

Radical prostatektomy X

Figure C.1: The linguistic judgment for the first prostate cancer patientsupplied by expert one

151

152 Appendix C

Patient Expert

Initial: Patient One Initial: Expert Two

Age: 64 Date: 2012-10-28


recommended

Wait and see X



Brachytherapy X



Figure C.2: The linguistic judgment for the first prostate cancer patientsupplied by expert two

Appendix C 153

Patient Expert

Initial: Patient One Initial: Expert Three

Age: 64 Date: 2012-10-28


recommended

Wait and see X



Brachytherapy X



Figure C.3: The linguistic judgment for the first prostate cancer patientsupplied by expert three

154 Appendix C

Patient Expert

Initial: Patient One Initial: Expert Four

Age: 64 Date: 2012-10-28


recommended

Wait and see X



Brachytherapy X



Figure C.4: The linguistic judgment for the first prostate cancer patientsupplied by expert four

APPENDIXDThe judgment tables consist of hesitant preferences for the second prostatecancer patient, supplied by the physicians at the Urology Department ofBlekinge County Hospital in Karlskrona, Sweden, are aggregated in Ap-pendix D.

Patient Expert

Initial: Patient Two Initial: Expert One

Age: 78 Date: 2013-06-17

Contraindicated Acceptable Possible Suitable Recommended Strongly

recommended

Active expectance

Active monitorering

Symptomatic therapy

Brachytherapy

Extern radical therapy

Adjuvant hormonal therapy

Radical prostatektomy

Figure D.1: The hesitant preference supplied by expert one

155

156 Appendix D

Patient Expert

Initial: Patient Two Initial: Expert Two

Age: 78 Date: 2013-06-17


recommended

Active expectance

Active monitorering

Symptomatic therapy

Brachytherapy




Figure D.2: The hesitant preference supplied by expert two

Patient Expert

Initial: Patient Two Initial: Expert Three

Age: 78 Date: 2013-06-17


recommended

Active expectance

Active monitorering

Symptomatic therapy

Brachytherapy




Figure D.3: The hesitant preference supplied by expert three

Appendix D 157

Patient Expert

Initial: Patient Two Initial: Expert Four

Age: 78 Date: 2013-06-17


recommended

Active expectance

Active monitorering

Symptomatic therapy

Brachytherapy




Figure D.4: The hesitant preference supplied by expert four

Patient Expert

Initial: Patient Two Initial: Expert Five

Age: 78 Date: 2013-06-17


recommended

Active expectance

Active monitorering

Symptomatic therapy

Brachytherapy




Figure D.5: The hesitant preference supplied by expert five

158 Appendix D

Bibliography

[1] K.-P. Adlassnig: “A Fuzzy Logical Model of Computer-assisted MedicalDiagnosis”. Methods of Information in Medicine vol. 19, issue 3 (1980)141-148

[2] K.-P. Adlassnig: “CADIAG-2: Computer-Assisted Medical DiagnosisUsing Fuzzy Subsets”. M. M. Gupta, E. Sanchez (Eds.): ApproximateReasoning in Decision Analysis. New York: North-Holland PublishingCompany, (1982) 219-242

[3] A. I. Al-Odienat, A. A. Al-Lawama: “The Advantages of PID FuzzyControllers over the Conventional Types”. American Journal of AppliedSciences, vol. 5, issue 6 (2008) 653–658

[4] S. Albayark, F. Aasyali: “Fuzzy C-Means Clustering on Medical Di-agnostic Systems”, Internationall XII Turkish Symposium on ArtificialIntelligence and Neural Networks, 2003

[5] N. Andrei: Modern Control Theory: “A Historical Perspective. Re-search Institute for Informatics”, Centre for Advanced Modelling andOptimization, Romania, http://www.ici.ro/camo/neculai/history.pdf(2005)

[6] K. Atanassov: “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems 20(1986) 87–96

[7] H. Becker: “Computing with words and machine learning in medicaldiagnosis”, Inf. Sci., vol.134 (2001) 53–69

[8] D. Ben-Arieh, Z. Chen: “Linguistic-Labels Aggregation and ConsensusMeasure for Autocratic Decision Making Using Group Recommenda-tions”. IEEE Transactions on Systems, Man, and Cybernetics-Part A:Systems and Humans 36, (2006) 558–568

159

160 Bibliography

[9] G. Bortolan, R. Degani: “A review of some methods for ranking fuzzynumbers”,Fuzzy Sets and Systems, 15 (1985) 1–19

[10] J. Buckley, E. Eslami: An Introdution to Fuzzy Logic and Fuzzy Sets.Advances in Soft Computing Series, Springer-verlag (2001) 95–124

[11] J. C. Bezdek: Pattern Recognition with Fuzzy Objective Function Al-gorithms, Plenum Press, New York, 1981

[12] C. Carlsson, R. Fuller: Fuzzy Reasoning in Decision Making and Op-timization. Studies in Fuzziness and Soft Computing Series, Springer-Verlag, Berlin Heidelberg New York (2001)

[13] Chin-Te Chen, Win-Li Lin, Te-Son Kuo, Cheng-Yi Wang: “Blood Pres-sure Regulation by Means of a Neuro-fuzzy Control System”. The 18thAnnual International Conference of the IEEE Engineering in Medicineand Biology Society, Amsterdam (1996) 1725–1726

[14] S.J. Chen, C.L. Hwang: Fuzzy Multiple Attribute Decision Making,Springer, New York, 1992

[15] Chich-Hui Chiu, Wen-June Wang: “A Simple Computation of MINand MAX Operations for Fuzzy Numbers”. Fuzzy Sets and Systems,126 (2002) 237–276

[16] F. Choobineh, H. Li: “An Index for Ordering Fuzzy Numbers”. FuzzySets and Systems 54 (1993) 287–294

[17] T. C. Chu, C. T. Tsao: “Ranking Fuzzy Numbers with an Area Betweenthe Centroid and the Original Points”. Computers and Mathematicswith Applications 43 (2002) 111–117.

[18] S. Coupland, R.I. John: “An Approach to Type-2 Fuzzy Arithmetic”.Proc. UK Workshop on Computational Intelligence (2003) 107–114

[19] D. Cox: “Regression Models and Life Tables”. J Roy Stat Soc B, 4(1972) 187–220

[20] J. de Mello, L. Struthers, R. Turner, E. H. Cooper, G. R. Giles: “Mul-tivariate Analyses as Aids to Diagnosis and Assessment of Prognosisin Gastrointestinal Cancer”. Br. J. Cancer, 48 (1983) 341–348

[21] M. Delgado, J.L. Verdegay, M.A. Vila: “Linguistic Decision MakingModels”. Int. J. Intell. Syst., vol. 7 (1992) 479–492

[22] G. Dias: “Ranking alternatives using fuzzy numbers”: A Computa-tional Approach, Fuzzy Sets and Systems 56 (1993) 247–252

Bibliography 161

[23] D. Dubois, H. Prade: “Fuzzy Real Algebra”. Fuzzy Sets and Systems,2 (1978a) 327–348

[24] D. Dubois, H. Prade: “Operations on Fuzzy Numbers”. Int. J. SystemsSci.,vol. 9, nr. 6 (1978b) 613–626

[25] D. Dubois, H. Prade: Fuzzy Sets and Systems: Theory and Applica-tions. New York: Kluwer, 1980.

[26] R. Duda, P. Hart, D. Stork: Pattern Classification, 2nd ed., JohnWiley and Sons, New York, USA, 2001

[27] B. Everitt, S. Rabe-Hesketh: Analyzing Medical Data Using S-PLUS,Springer, New York 2001

[28] G. Feng, Y. Chen, B. Wang, Y. Chen: “Rough Set Based Classifi-cation Rules Generation for SARS Patients”. Proceedings of the 2005IEEE, Engineering in Medicine and Biology 27th Annual Conference,Shanghai, China 2005

[29] D. P. Filev, R. R. Yager: “Operations on Fuzzy Numbers via FuzzyReasoning”. Fuzzy Sets and Systems, vol. 91, Issue 2 (1997) 137–142

[30] P. Fortemps, M. Roubens: “Ranking and Defuzzication Methods Basedon Area Compensation”, Fuzzy Sets and Systems 82 (1996) 319–330.

[31] R. Goetschel, W. Voxman: “Elementary Fuzzy Calculus”. Fuzzy Setsand Systems, vol. 18, Issue 1 (1986) 31–43

[32] S. Heilpern: “Representation and Application of Fuzzy Numbers”.Fuzzy Sets and Systems 91 (1997) 259–268

[33] C. Hernandez, A. Carollo, C. Tobar: “Fuzzy Control of PostoperativePain”. Proceedings of the Annual International Conference of the IEEE(1992) 2301–2303

[34] F. Herrera, E. Herrera-Viedma, J. L. Verdegay: “A Model of Consensusin Group Decision Making under Linguistic Assessments,” Fuzzy SetsSyst., vol. 79 (1996) 73–87

[35] F. Herrera, E. Herrera-Viedma, L. Martinez: “A Fuzzy LinguisticMethodology to Deal with Unbalanced Linguistic Term Sets”. IEEETransactions on Fuzzy Systems 16 (2008) 354–370

[36] F. Herrera, L. Martinez: “A 2-Tuple Linguistic Representation Modelfor Computing with Words”, IEEE Transactions on Fuzzy Systems,vol. 8, no. 6, (2000) 746–752

162 Bibliography

[37] F. Herrera, J. L. Verdegay: “A Linguistic Decision Process in GroupDecision Making,” Group Decision Negotiation, vol. 5, (1996) 165–176

[38] E. Herrera-Viedma, L. Martinez, F. Mata, F. Chiclana: “A Consen-sus Support System Model for Group Decision-Making Problems withMulti-Granular Linguistic Preference Relations”. IEEE Transactionson Fuzzy Systems 13, (2005) 644–658

[39] Dug Hun Hong: “Shape Preserving Multiplications of Fuzzy Numbers”.Fuzzy Sets and Systems, 123 (2001) 81–84

[40] Dug Hun Hong, Hae Young Do: “Fuzzy System Reliability Analysisby the Use of Tw (the weakest t-norm) on Fuzzy Number ArithmeticOperations”. Fuzzy Sets and Systems, vol. 90, Issue 3 (1997) 307–316

[41] V. N. Huynh, Y. Nakamori: “Multi-Expert Decision-Making with Lin-guistic Information: A Probabilistic-based Model”. Proceedings of the38th Hawaii International Coference on System Sciences, (2005) 1–9

[42] V. N. Huynh, Y. Nakamori: “A Satisfactory-Oriented Approach toMulti-Expert Decision-Making under Linguistic Assessments”, IEEETrans. Systems, Man, and Cybernetics, vol. SMC-35, April 2005, 184–196.

[43] S. Isaka, A. V. Sebald: “An Adaptive Fuzzy Controller for Blood Pres-sure Regulation”. IEEE Engineering in Medicine and Biology Society- The 11th Annual International Conference (1989) 1763–1764

[44] J. Kacprzyk: Fuzzy Sets in System Analysis, PWN, Warszawa (in Pol-ish), 1986

[45] J. Kacprzyk: “Group decision making with a fuzzy linguistic majority,”Fuzzy Sets Syst., vol. 18 (1986) 105-118

[46] J. Kacprzyk, M. Fedrizzi: Multiperson Decision Making Models Us-ing Fuzzy Sets and Possibility Theory, Dordrecht: Kluwer AcademicPublishers, 1990

[47] E. Kaplan, P. Meier: “Nonparametric Estimation from Incomplete Ob-servations”. Journal American Statistical Association 53 (1958) 457–481

[48] A. Kaufmann, M. M. Gupta: Introduction to Fuzzy Arithmetic Theoryand Application. Van Nostrand Reinhold, New York (1991)

[49] P. S. Kechagias, B. K. Papadopoulos: “Computational Method to Eval-uate Fuzzy Arithmetic Operations”. Applied Mathematics and Compu-tation, vol. 185, Issue 1 (2007) 169–177

Bibliography 163

[50] Do Kyong-Kim, Sung Yong Oh, Hyuk-Chan Kwon, Suee Lee, KyungA Kwon, Byung Geun Kim, Seong-Geun Kim, Sung-Hyun Kim, JinSeok Jang, Min Chan Kim, Kyeong Hee Kim, Jin-Yeong Han, Hyo-JinKim: “Clinical Significances of Preoperative Serum Interleukin-6 andC-reactive Protein Level in Operable Gastric Cancer”. BMC Cancer 9(2009) 155–156

[51] K. M. Lee, C. H. Cho, H. Lee-Kwang: “Ranking fuzzy values withsatisfaction”, Fuzzy Sets and Systems 64 (1994) 295–311

[52] D. F. Li, ”TOPSIS-Based Nonlinear-Programming Methodology forMultiattribute Decision Making with Interval-Valued IntuitionisticFuzzy Sets,” IEEE Trans. Fuzzy Syst., vol. 18, no.2, April, 2010, 299–311

[53] R. Lowen: Fuzzy Set Theory: Basic Concepts, Techniques and Bibli-ography. Kluwer Academic Publishers, Dordrecht 1996

[54] B. T. Luke: “Fuzzy Sets and Fuzzy logic”. ([email protected]), Learn-ingFromTheWeb.net (2006)

[55] E. H. Mamdani, S. Assilian: “An Experiment in Linguistic Synthesiswith a Fuzzy Logic Controller”. Int. J. Man-Machine Studies 7 (1973)1–13

[56] L. Martinez: “Sensory evaluation based on linguistic decision analysis”.Int. J. Approx. Reason., 44, no. 2, (2007) 148–164

[57] J. M. Mendel: Uncertain Rule-Based Fuzzy Logic Systems: Introduc-tion and New Directions. Prentice-Hall, Upper-Saddle River, NJ 2001

[58] S. Miyamoto: “Multisets and Fuzzy Multisets”, in Z.-Q. Liu, S.Miyamoto (Eds.), Soft Computing and Human-Centered Machines,Springer, Berlin, (2000) 9–33

[59] S. Miyamoto: “Remarks on Basics of Fuzzy Sets and Fuzzy Multisets”,Fuzzy Sets and Systems 156 (2005) 427–431

[60] J. N. Mordeson, S. M. Davender, Shih Chuang Cheng: Fuzzy Mathe-matics in Medicine. Studies in Fuzziness and Soft Computing Series,Springer-verlag, Berlin Heidelberg New York 2000

[61] R. C. Newland, O. F. Dent, M. N. Lyttle, P. H. Chapuis, E. L. Bokey:“Pathologic Determinants of Survival Associated with Colorectal Can-cer with Lymph Node Metastases”. A Multivariate Analysis of 579Patients. Cancer 73(8) (1994) 2076-2082

[62] H. T. Nguyen, N. R. Prasad, C. L. Walker, E. A. Walker: A FirstCourse in Fuzzy and Neural Control, Chapman and Hall/CRC 2002

164 Bibliography

[63] S. A. Orlovsky, “Decision Making with a Fuzzy Preference Relation,”Fuzzy Set Syst., vol. 1, (1978) 155-167

[64] S. K. Pal, P. Mitra: “Case Generation Using Rough Sets with FuzzyRepresentation”. IEEE Transactions on Knowledge and Data Engi-neering, vol 16, no. 3 (2004) 292-300

[65] Q. Pang, A. Hou, Q. Hua, H. Tang: “Mining Classification Rules ofCancer Patients for TCM Treatments: A Rough Set Based Approach”.Fifth International Conference on Fuzzy Systems and Knowledge Dis-covery, Haikou City, Hainan, China (2007)

[66] Z. Pawlak, J. Grzymala-Busse, J. Slowinski, W. Ziarko: “Rough set”.Communications of the ACM, vol. 38, no. 11 (1995).

[67] W. Pedrycz: Fuzzy Sets Engineering, CRC Press, Boca Raton, FL 1995

[68] W. Pedrycz, F. Gomide: An Introduction to Fuzzy Sets. Analysis andDesign. MIT Press 1998

[69] W. Pedrycz, S. Mingli: ”Analytic Hierarchy Process (AHP) in GroupDecision Making and its Optimization with an Allocation of Informa-tion Granularity,” IEEE Trans. Fuzzy Syst., vol. 19, no.3, Jun. 2011,527–539

[70] W. Pedrycz, A. Amato, V. Di Lecce, V. Piuri: “Fuzzy Clustering WithPartial Supervision in Organization and Classification of Digital Im-ages”, IEEE Transactions on fuzzy systems, vol. 16, no. 4 (2008) 1008–1026

[71] E. Rakus-Andersson: “A Choice of Optimal Medicines by Fuzzy De-cision Making Including Unequal Objectives”. Issues in the Represen-tation and Processing of Uncertain and Imprecise Information, EXIT– The Publishing House of the Polish Academy of Sciences, (2005)307–321

[72] E. Rakus-Andersson: “S-truncated Functions and Rough Sets in Ap-proximation and Classification of Polygons”. Proc. of Modeling De-cisions for Artificial Intelligence - MDAI 2005, Tsukuba, CD-ROM,paper nu 049, Consejo Superior de Investigaciones Cientificas (2005)

[73] E. Rakus-Andersson: “Minimization of Regret versus Unequal Multi-objective Fuzzy Decision Process in a Choice of Optimal Medicines”.Proceedings of the XIth International Conference IPMU–InformationProcessing and Management of Uncertainty in Knowledge-based Sys-tems, vol. 2, Edition EDK, Paris-France, (2006) 1181-1189

Bibliography 165

[74] E. Rakus-Andersson: Fuzzy and Rough Sets in Medical Diagnosis andMedication. Springer, Berlin Heidelberg 2007

[75] E. Rakus-Andersson: Decision-making techniques in ranking ofmedicine effectiveness. In: Sordo M., Vaidya S, Jain L. C., eds.,Advanced Computational Intelligence Paradigms in Healthcare 3,Springer-verlag, Berlin Heidelberg, (2008) 51-73

[76] E. Rakus-Andersson: “Approximate Reasoning in Surgical Decisions”.Proceedings of the International Fuzzy Systems Association WorldCongress - IFSA 2009, Lisbon, Instituto Superior Technico (2009) 225–230

[77] E. Rakus-Andersson: “Rough Set Theory in the Classification of Di-agnoses”. In Computers in Medical Activity, Eds: Kacki, E., Rudnicki,M., Stempczynska, J., Springer-Verlag, Berlin Heidelberg (2009) 42–51

[78] E. Rakus-Andersson: “Adjusted s-parametric Functions in the Cre-ation of Symmetric Constraints”, In Proceedings of the 10th Interna-tional Conference on Intelligent Systems Design and Applications -ISDA 2010, Cairo, Egypt, Nov. 2010, 451–456

[79] E. Rakus-Andersson: “Approximate Reasoning in Cancer Surgery”, InProceedings of International Conference on Fuzzy Computation Theoryand Applications 2011 - FCTA 2011, Paris, France, Oct. 2011, 466–469

[80] E. Rakus-Andersson: “The Mamdani Controller with Modeled Familiesof Constraints in Evaluation of Cancer Patient Survival Length”, InEmerging Paradigms in Machine Learning, Editors: Ramanna, S. andJain, L. and Howlett, R. Vol. 13, (2012) 359–378, Springer, BerlinHeidelberg, New York.

[81] E. Rakus-Andersson: “Selected Algorithms of Com-putational Intelligence in Surgery Decision making”Open Access book Gastroenterology in SCITECHhttp://www.intechopen.com/articles/show/title/selected-algorithms-of-computational-intelligence-in-cancer-surgery-decision-making,2012

[82] E. Rakus-Andersson: “Approximation and Rough Classification ofLetter-Like Polygon Shapes. In Rough Sets and Intelligent Systems–Professor Zdzislaw Pawlak in Memoriam, Skowron A. Suraj Z., eds.,ISRL 43, Springer, Berlin Heidelberg, (2013) 455-474

[83] E. Rakus-Andersson, L. Jain: “Computational Intelligence in MedicalDecisions Making”. In Recent Advances in Decision Making, Rakus-Andersson E., Yager R. R., Jain L., (Eds) in Series: Studies of Com-putational Intelligence, Berlin Heidelberg, Springer (2009) 145–159

166 Bibliography

[84] E. Rakus-Andersson, M. Salomonsson: “The truncated π-functions inApproximation of Multi-shaped Polygons”. deBeats B, de Caluwe R.,de Tre G, Fodor J., Kacprzyk J., Zadrozny S., eds., In: Current Issuesin Data and Knowledge Engineering, Polish Academy of Sciences - ThePublishing House ”Exit”, (2004) 444–452.

[85] E. Rakus-Andersson, M. Salomonsson: “π-truncated Functions andRough Sets in the Classification of Internet Protocols. In Proceedingsof Eleventh International Fuzzy Systems Association World Congress– IFSA 2005, Beijing, China, Tsinghua University Press – Springer,(2005) 1487–1492.

[86] E. Rakus-Andersson, M. Salomonsson, H. Zettervall: “Ranking ofWeighted Strategies in the Two-player Games with Fuzzy Entries of thePayoff Matrix”. In Proceedings of the 8th International Conference onHybrid Intelligent Systems, Barcelona 2008, Eds: Fatos Xhafa, Fran-cisco Herrera, Ajith Abraham et al., CDR by Universitat Polytecnicade Catalunya

[87] E. Rakus-Andersson, M. Salomonsson, H. Zettervall: “Two-playerGames with Fuzzy Entries of the Payoff Matrix”. Computational Intel-ligence in Decision and Control - Proceedings of FLINS 2008, Madrid2008, World Scientific 593–598

[88] E. Rakus-Andersson, H. Zettervall, M. Erman: “Prioritization ofWeighted Strategies in the Multi-player Games with Fuzzy Entries ofthe Payoff Matrix”. Int. J. of General Systems, Vol. 39, Issue 3 (2010)291–304

[89] E. Rakus-Andersson, H. Zettervall: “Different Approaches to Opera-tions on Continuous Fuzzy Numbers. Developments of Fuzzy Sets”. InIntuitionistic Fuzzy Sets and Generalized Nets, Related Topics, Foun-dations, vol.I, Eds: Atanassov, K., Chountas, P., Kacprzyk, J., Melo-Pinto, P. et al., EXIT - The Publishing House of The Polish Academyof Sciences (2008) 294–318

[90] E. Rakus-Andersson, H. Zettervall, H. Forssell: “Fuzzy Controllers inEvaluation of Survival Length in Cancer Patients”. In Recent Advancesin Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and RelatedTopics. Editors: Ramanna, S. and Jain, L. and Howlett, R. Vol. II,Applications Polish Academy of Sciences, System Research Institute,Warsaw (2011) 203–222

[91] I. Requena, M. Delgado, J. I. Verdagay: “Automatic Ranking of FuzzyNumbers with the Criterion of Decision-Maker learnt by an ArtificialNeural Network”, Fuzzy Sets and Systems. Vol. 64 (1994) 1–19.

Bibliography 167

[92] R. M. Rodriguez, L. Martinez, F. Herrera: “Hesitant Fuzzy LinguisticTerm Sets for Decision Making”. IEEE Trans. on Fuzzy Systems, vol.20, no. 1, February 2012, 109–119.

[93] K. Rose, E. Gurewitz, G. C. Fox: “Constrained Clustering as Op-timization Method”, IEEE Trans. on Pattern Analysis and MachineIntelligence, vol. 15, no. 8, (1993) 785–794

[94] T. J. Ross: Fuzzy Logic with Engineering Applications. John Wiley &Sons Ltd, England 2005

[95] T. L. Saaty: “Exploring the Interface between Hierarchies, MultipliedObjectives and Fuzzy Sets”. Fuzzy Sets and Systems 1, (1978) 57-58

[96] K. Sadegh-Zadeh: “The Fuzzy Revolution: Goodbye to the AristotelianWeltanschauung”. Artificial Intelligence in Medicine 21 (2000) 1-25

[97] E. Sanchez: “Medical Diagnosis and Composite Fuzzy Relations”. M.M. Gupta, R. K. Ragade, R. R. Yager (Eds.): Advances in Fuzzy SetTheory and Applications. Amsterdam: North-Holland, (1979) 437-444

[98] D. J. Sargent: “Comparison of Artificial Networks with Other Statis-tical Approaches”. Cancer, vol. 91 (2001) 1636–1942

[99] R. Seising, C. Schuh, K.-P. Adlassnig: “Medical Knowledge, Fuzzy Setsand Expert Systems”. Workshop on intelligent and adaptive systems inmedicine, Prague, (2003)

[100] Divyendu Sinha: “A General Theory of Fuzzy Arithmetic”. Fuzzy Setsand Systems, vol. 36, Issue 3 (1990) 339–363

[101] M. Sugeno: “An Introductory Survey of Fuzzy Control”. Inf. Sci.36 (1985) 59–83

[102] M. Sugeno, M. Nishida: “Fuzzy Control of Model Car”. Fuzzy Sets andSystems (1985) 103–113

[103] R. Sutton, D. R. Towill: “An Introduction to the Use of Fuzzy Setsin the Implementation of Control Algorithms”. IEEE Trans., UDC510.54:62–519:629.12.014.5 (1985) paper no. 2208/ACS39.

[104] Y. J. Wang, H. S. Lee: “The Revised Method of Ranking Fuzzy Num-bers with an Area Between the Centroid and Original Points”, Com-puters and Mathematics with Applications. vol. 55 (2008) 2033–2042.

[105] R. R. Yager: “A New Methodology for Ordinal Multi-Objective Deci-sions Based on Fuzzy Sets”, Decision Sci., vol. 12, (1981) 589-600

168 Bibliography

[106] R. R. Yager: “On the Theory of Bags”. International Journal of GeneralSystems 13 (1986) 23–37

[107] J. Yang, Y. Ning: ”Research on Initial Clustering Centers of FuzzyC-Means Algorithm and its Application to Instrusion Detection,” InProceedings of the International Conference on Environmental Scienceand Information Application Technology 2010, Wuhan, China, July2010, vol. 3, 161–163

[108] X. Yang, W. Wang: “GIS Based Fuzzy C-Means Clustering Analysis ofUrban Transit Network Service,” the Nanjing City Case Study, Roadand Transport Research China, (2001)

[109] L. A. Zadeh: “Fuzzy sets”. Inf. Control 8 (1965) 338–353

[110] L. A. Zadeh: “The Concept of a Linguistic Variable and Its Appli-cations to Approximate Reasoning,” Inform. Sci., pt. I, no. 8, (1975)199–249

[111] H. Zettervall, E. Rakus-Andersson, H. Forssell: “The Mamdani Con-troller in Prediction of the Survival Length in Elderly Gastric Patients”.In Proceedings of Bioinformatics 2011, Rome, (2011), 283–286

[112] H. Zettervall, E. Rakus-Andersson, H. Forssell: “Fuzzy C-Means Clus-ter Analysis and Approximated Data Strings in Operation Prognosisfor Gastric Cancer Patients”. In New Trends in Fuzzy Sets, Intuitionis-tic Fuzzy Sets, Generalized Nets and Related Topics. Volume II: Appli-cations, (K.T. Atanassow, W. Homenda, O. Hryniewicz, J. Kacprzyk,M. Krawczak, Z. Nahorski, E. Szmidt, S. Zadrozny, Eds.), IBS PAN-SRI PAS, Warsaw, (2013) 181-200

[113] H. Zettervall, E. Rakus-Andersson, H. Forssell: “Fuzzy C-meansClustering Applied to Operation Evaluation for Gastric Cancer Pa-tients”. In Proceedings of the Fifth International Conference on eHealth,Telemedicine, and Social Medicine, Nice, France, (2013) 228–233

[114] H. J. Zimmermann: Fuzzy Set Theory and Its Applications. KluwerAcademic Publishers 2001

FUZZY SET THEORY APPLIED TO MAKE MEDICAL PROGNOSES FOR CANCER PATIENTS

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Doctoral Dissertation Series No. 2014:01

Department of Mathematics and Natural Sciences2014:01

ISSN: 1653-2090

ISBN: 978-91-7295-271-3

ABSTRACTAs we all know the classical set theory has a deep-rooted influence in the traditional mathe-matics. Nevertheless, a feeling of imprecision in the two-valued logic, being the main tool for esta-blishing crisp sets, does not exist. With the rapid development of science and technology, a sub-stantial number of scientists have gradually app-reciated the vital importance of the multi-valued logic. The assumptions of a new theory of fuzzy sets, based on multi-valued logic and proposed for the first time in 1965, have given rise to model mathematically the real world’s occurrences in spite of their vague or incomplete nature.

This study aims at applying some classical and ex-tensional methods of fuzzy set theory in life ex-pectancy and treatment prognoses for cancer pa-tients. The research is based on real-life problems encountered in clinical works by physicians. From the introductory items of the fuzzy set theory to the medical applications, a collection of detailed analysis of fuzzy set theory and its extensions are discussed in the thesis. Concretely speaking, the

Mamdani fuzzy control systems and the Sugeno controller have been applied to prognosticate the survival length of gastric cancer patients. In or-der to make a surgery decision concerning can-cer patients, the fuzzy c-means clustering analysis has been adopted to investigate the possibilities for operation contra none operation. Furthermo-re, the approach of point set approximation has been proved to estimate the operation possibi-lities against to none operation for an arbitrary gastric cancer patient. In addition, in the domain of multi-expert decision-making, the probabilistic model, the model of 2-tuple linguistic represen-tations and the hesitant fuzzy linguistic term sets (HFLTS) have been utilized to select the most consensual treatment scheme(s) for two separate prostate cancer patients.

The obtained results have supplied the physicians with the reliable and helpful information. Therefo-re, the research work can be seen as the mathe-matical complement to the physicians’ queries.

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FUZZY SET THEORY APPLIED TO MAKE MEDICAL PROGNOSES FOR CANCER PATIENTS834279/... · 2015-06-30 ·...

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Transcript of FUZZY SET THEORY APPLIED TO MAKE MEDICAL PROGNOSES FOR CANCER PATIENTS834279/... · 2015-06-30 ·...