FUZZY INSURANCE BY JEAN LEMAIRE Wharton School University of Pennsylvanta, U S.A. ABSTRACT Fuzzy set theory IS a recently developed field of mathematics, that introduces sets of objects whose boundaries are not sharply defined. Whereas in ordinary Boolean algebra an element is either contained or not contained m a given set, in fuzzy set theory the transition between membership and non-membership is gradual The theory aims at modehzmg situations described in vague or ~mpreclse terms, or situations that are too complex or all-defined to be analysed by conventional methods This paper alms at plesentlng the basic concepts of the theory In an insurance framework. First the basic defimtlons of fuzzy logic are presented, and applied to provide a flexible definmon of a "preferred policyholder" in life insurance. Next, fuzzy decision-making procedures are dlustrated by a reinsurance apphcation, and the theory of fuzzy numbers is extended to define fuzzy insurance premmms. KEYWORDS Fuzzy set theory; preferred pohcyholders in life insurance, optimal XL- retentions; net single premiums for pure endowment insurance l INTRODUCTION In 1965, ZADEtl published a paper entitled "Fuzzy Sets" in a httle known journal, Information and Control, introducing for the first time sets of objects whose boundaries are not sharply defined. This paper gave rise to an enormous interest among researchers, and mltiated the fulgurant growth of a new &sciphne of mathematics, fuzzy set theory. The number of papers related to the field exploded from 240 in 1975 (ZADEH et al.), to 760 m 1977 (GOPTA et al.), 2500 in 1980 (CHEN et al ), and 5000 m 1987 (ZIMMERMAN). Today, there are many more researchers in fuzzy set theory than in actuarial science, and they form a much more international group, with important contributions from China, Japan, and the Soviet Union. Two monthy scientific Journals publish new theoretical developments and applications, that are to be found m linguistics, risk analysis, artificial intelhgence (approxnnate reasoning, expert systems), pattern analysis and classification (pattern recognmon, clustering, image processing, computer vision), reformation processing, and declslon- making. In this paper we wdl explore some possible apphcations of fuzzy set theory to Insurance. ASTIN BULLETIN, Vol 20, No I
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`Liberty Mutual Exhibit 1026
`Liberty Mutual v. Progressive
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`34 JEAN LEMAIRE In ordinary Boolean algebra, an element is either contained or not contained m a given set. the transition from membership to non-membership ~s abrupt. Fuzzy sets, on the other hand, describe sets of elements or variables whose limits are ill-defined or imprecise The transition between membership and non-membership is gradual: an element can "more or less" belong to a set Consider for instance the set of "young drivers". In Boolean algebra, it is assumed that any indlwdual either belongs or does not belong to the set of young drivers. This xmphes that the individual will move from the category of "young drivers" to the complementary set of "not young drivers" overnight Fuzzy set theory allows for grades of membership. Depending on the specific application, one might for instance decide that drivers under 20 are dei]nitely young, that drivers over 30 are definitely not young, and that a 23-year-old driver is "more or less" young, or is young with a grade membership of 0.7, on a scale from 0 to 1 Fuzzy set theory thus alms at modehzmg imprecise, vague, fuzzy informa- tion, which abound in real world situations. Indeed, many practical problems are extremely complex and all-designed, hence difficult to modehze with precision To quote ZADEH, "as the complexity of a system increases, our ability to make precise and yet significant statements about its behavlour diminishes until a threshold is reached beyond which precision and slgmficance become almost exclusive characteristics" Computers cannot adequately handle such problems, because machine mtelhgence still employs sequential (Boolean) logic. The superiority of the human brain results from its capacity of handling fuzzy statements and decisions, by adding to logic parallel and simultaneous information sources and thinking processes, and by filtering and selecting only those that are useful and relevant to its purposes. The human brain has many more thinking processes available and has developed a far greater filtering capacity than the machine A group of individuals is able to resolve the command "tall people m the back, short people m the front", a machine is not Fuzzy set theory explicitly introduces vagueness m the reasoning, hoping to provide decision-making procedures that are closer to the way the human brain performs. A clear distinction has to be made between fuzzy sets and probability theory. Uncertainty should not be confused with imprecision Probablhties are pri- marily intended to represent a degree of knowledge about real entities, while the degrees of membership defining the strength of participation of an entity m a class are the representation of the degree by which a proposition is partially true Probability concepts are derived from considerations about the uncer- tainty of propositions about the real world Fuzzy concepts are closely related to the multlvalued logic treatments of issues of imprecision m the definition of entities Hence, fuzzy set theory provides a better framework than probablhty theory for modelling problems that have some inherent imprecision The traditional approach to risk analysis, for instance, IS based on the premise that probablhty theory provides the necessary and sufficient tools for dealing with the uncertainty and imprecision which underline the concept of risk in decision analysis The theory of fuzzy sets calls Into question the valzdlty of this
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`FUZZY INSURANCE 35 premise. It does not equate imprecision with randomness It suggests that much of the uncertainty which ~s mtrinsm m risk analysis is rooted m the fuzziness of the reformation which is resident in the data base and m the imprecision of the underlying probabilities. Classical probability theory has its effectiveness limited when dealing with problems in which some of the prmc,pal sources of uncertainty are non-statistical in nature In the sequel we will present the basra principles of fuzzy logic, fuzzy decision-making, and fuzzy arithmetics, while developing three lnSul'ance examples We will show that fuzzy set theory could provide demsmn procedures that are much more flexible than those originating from conventional set theory Indeed, insurance executives and actuarms, much better trained to deal with uncerta,nty than with vagueness, have often transformed m~preclsc statements into "all-or-nothing" rules. For instance, Belgian insurers have used the fuzzy statistical evidence "Young dr,vers provoke more automobile accidents" to set up the a posteriorl i'atmg rule "Dr,vers under 23 years of age will pay a $150 deductible if they provoke an accident". Hence '" young" was equated with "under 23", a definite &storslon of the initial statement As another example, Belgmn regulatory authorlt,es define, for statistical purposes, a "severely wounded person" as "any person, wounded m an automobile accident, whose condmon requires a hospital stay longer than 24 hours", a very arguable "de-fuzzlficatlon" of a fuzzy health condmon In Section 2 we will present the bamc definitions of fuzzy Iogm and apply them to provide a more flexible defimtlon of a "preferred policyholder'" than the one currently used by some American life insurers Section 3 Introduces the main concepts of fuzzy decision-making, and uses them to select an optimal Excess of Loss retentmn. Fuzzy anthmetms are presented m Section 4, and applied to compute the fuzzy prem,um of a pure endowment policy First, let us introduce our three examples. Problem 1 Deftmt,on of a preferred pohcyhoMer h7 hfe insurance Heavy competition between Amerman hfe insurers has resulted m a greater subdlvlson of policyholders than in Europe U.S. insurers first began, in the mid 1960s, to award substantial discounts to nonsmokers purchasing a term or a whole life insurance. Then the "' preferred policyholder" category was further refined, and more discounts were granted to apphcants who met very stringent health reqmrements, such as a cholesterol level not exceeding 200, a blood pressure not exceeding 130/80, . For instance, one company offers a non- smoker bonus of 65 % more insurance coverage with no increase m premium if the apphcant has not smoked for 12 months prior to application A bonus of 100% is offered if the applicant:
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`has not smoked for the past 12 months, and -- has a resting pulse of 72 or below, and -- has a blood pressure that does not exceed 134/80, and -- has a total cholesterol reading not exceeding 200, and
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`36 JEAN LEMA|RE does not engage in hazardous sports, and -- rigorously follows a 3-tlmes-a-week exercise program of at least 20 minutes, and
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`is within specified height and weight hmlts, and -- has no more than one death in immediate family prior to 60 years of age due to kidney or heart disease, stroke or diabetes Again this is a dlstors~on, or a least a very strict interpretation, of the medical statement "People who exercise, who do not smoke, who have a low level of cholesterol, low blood pressure, who are neither overweight nor severely underweight .... have a h~gher hie expectancy". Insurers demand all conditions to be strictly met, the shghtest infringement leads to automatic rejection of the preferred category For instance, a cholesterol level of 201 implies that the preferred rates won't apply, even ~f the applicant meets all other requirements. A cholesterol level of 200 is accepted, a level of 201 is not! We will show that fury set theory can be used to provide a more flexible definition of a preferred policyholder, that allows for some form of compensation between the selected criteria. Problem 2. Selection of an optmval excess of loss retentton Imprecise statements seem to be pervasive m reinsurance practlve, where vague recommendations and rules abound. "As a rule of thumb, an excess of loss (XL) retention should approximatively equal 1% of the premium income", "Our long-term relationship with our present reinsurer should in principle be maintained", "We could accept those conditions prowdlng substantial retro- cessions are offered ...., A ball-park figure for the cost of this reinsurance program is $10 nulllon", are fuzzy sentences frequently heard in practice. To illustrate fuzzy decision-making procedures, we shall consider the problem of the selection of the optimal retention of a pure XL treaty, given the four following fuzzy goals and constraints. Goal 1: The ruin probability should be substantially decreased, Ideally down to be neighbourhood of 10-5 Goal 2: The coefficient of varlatxon of the retained portfolio should be reduced; ~f possible it should not exceed 3 Constraint 1. The reinsurance premium should not exceed 2 5 % of the line's premium income by much. Constraint 2. As a rule of thumb, the retenuon should approxlmatlvely be equal to 1% of the line's premium income Problem 3 Computation oJ the fuzzy premmm of a pure endowment pohcy Forecasting interest rates is undoubtedly one of the most complex modelling problems. Money market interest rates seem to fluctuate according to monthly U.S. unemployment and trade deficit figures, vague statements made by Mr Kohl or Mr Greenspan, the markets' perceptxon of Mr Bush's wllhngness to tackle the deficit problem, the mood of the participants to an OPEC
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`2.1. Basic definitions
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`FUZZY INSURANCE 37 meeting, etc. To compute insurance premiums over a 40-year span with a fixed Interest rate of 4.75 % then seems to be an exercise in futdlty. We will show that the Introduction of fuzzy interest rates (and fuzzy survival probabilities) at least allows us to obtain a partial measure of our ignorance. As illustrated by our examples, fuzzy set theory attempts to modehze imprecise expressions like "more or less young", "neither overweight nor underweight", "in the nelghbourhood of", "in principle". In retreating from precision in the face of overpowering complexity, the theory explores the use of what might be called linguistic variables, that is, variables whose values are not numbers but words or sentences. In summary, fuzzy set theory endorses Bertrand Russell's opinion that "All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence" and reJects Yves Le Dantec's aphorism "That only is science which deals with the measurable". 2 FUZZY LOGIC AND FUZZY PREFERRED POLICYHOLDERS
`A fuzzy set is a class of objects in which there is no sharp boundary between those objects that belong to the class and those that do not. More precisely, let X = {x} denote a collection of objects denoted generically by x A fuzzy set A in X is a set of ordered pairs A = {x, UA (x)}, x ~ X where UA(x) is termed the grade of membership ofx in A, and UA:X ~ M is a function from X to a space M, called the membership space Hence a fuzzy set A on a referential set X can be viewed as a mapping UA from X to M. (Examples of membership functions are presented in all figures). For our purposes It IS sufficient to assume that M is the interval [0, 1], wlth 0 and 1 representing, respectively, the lowest and highest grade of membership The degree of membership of x in A corresponds to a "truth value" of the statement "x is a member of A ". When M only contains the two points 0 and 1, A is nonfuzzy. Problem 1 Let X be a set of prospective policyholders, x =.,~ (it, t2, t3, t4). For simplicity, assume that the requirements for the status of" preferred policyholder" will be based on the values taken by 4 variables t~, the total level of cholesterol in the blood, in mg/dl, t2, the systolic blood pressure, in mm of Hg
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`38 JEAN LEMAIRE t3, the ratio (in %) of the effective weight to the recommended weight, as a function of height and build &, the average consumption of cigarettes per day Using a classical approach, an insurance company would for instance define a preferred policyholder as a nonsmoker with a cholesterol level that does not exceed 200, and a blood pressure that does not exceed 130, and a weight that is comprised between 85% and 110% of his recommended weight. If a fuzzy set approach is to be used, membership functions have to be defined for all cnterm. National Institutes of Health nowadays recommend a level of less than 200 mg of cholesterol per deciliter of blood Levels between 200 and 240 mg/dl are considered to be borderhne high The fuzzy set A of the people with a low level of cholesterol can then UA (x, t~)
`1-2(---- UA (x; tt) = 2 / 240-t~ 40 .0 be defined by the membership function l I ~ 200 tl- 200 )2 40 200 < t, ~ 220
`220 < t~ ~240 240 < t~ The normal systolic blood pressure is about 130 mm of mercury. People with a blood pressure greater than 170 are five times more likely to suffer from coronary heart &sease than indwiduals with normal blood pressures Hence the fuzzy set B of the people with an acceptable blood pressure can be defined by the membership funcuon U~(x, t2)
`40 UB(x, t,) = 2( 170- tz ) 2 4O 150 < t2 =< 170 .0 170 < t 2 Overweight and underweight people have a shorter life expectancy, skinniness being less primordial than obesity. This is reflected in the asymmetric member- ship function Uc(x, t3) that characterizes the fuzzy set C of the people with adequate weight
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`FUZZY INSURANCE 39 0 2 (t3-60) 2
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`25- (85-t3) 2 I-2 - 25-- Uc(X, t3) = I 1-2( t3- 110 )20 2(130-t3) 220 0 13 ~ 60 60 < 13 ~ 72.5 72.5 < t 3 =< 85 85 < t 3 =< 110 110 < t 3 ~ 120 120 < t 3 ~ 130
`Even hght smokers are more prone to suffer from cancer and car&ovascular dxseases than nonsmokers Hence they cannot be considered as "preferred" and the set D of the nonsmokers is nonfuzzy 1 t4=O U o(X, /4) = 0 I4 > 0. The four selected membersh.p functions are represented m Figure 1. Admit- tedly, there ~s some arbitrariness m the defimton of these membership functmns, but fuzzy set theory contends that this is better than membership functions that abruptly jump from 1 to 0, m the classmal approach A fuzzy set is said to be normal lff Sup, UA(x) = 1 Subnormal fuzzy sets can be normalized by dwldmg each UA (x) by the factor Sup, UA (x) .,4 is said to be the complement of A iff U,i(x) = I-UA(x) Vx. A fuzzy set Is contained m or is a subset of a fuzzy set B (,4 c B) iff UA(x) ~_ UB(X) V x. The umon of A and B, denoted A U B, is defined as the smallest fuzzy set contalmng both A and B Its membership function is gwen by UAuB(X) = max [UA(X), UB(X)] x~X The intersection of A and B, denoted A f'l B, is defined as the largest fuzzy set contained m both A and B. Its membership functton ~s given by UAnR(X) = mm [gA(x), gB(x)] xeX The nouon of intersection bears a close relation to the notion of the connective "and", just as the umon of A and B bears a close relatmn to the connectwe "or". It can be shown that these definmons of fuzzy union and intersectmn are
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`FIGURE I Membership funcuons Problem I 200 240 220 1:30 150 170 UB( x, t 2) 1 t 2 > z F ~ > m %(x, t 3) 60 70 80 90 100 110 120 130 UA(X,t 1 ) 1 t 3 10 20 t 4
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`FUZZY INSURANCE 41 the only ones that naturally extend the corresponding standard set theory notions, by satisfying all the usual requirements of assoclatlwty, commutatlv- ity, ~dempotency and dlstributw~ty. Problem 1 The fuzzy set E of the nonsmoklng individuals with low cholesterol, acceptable blood pressure and adequate wexght ts the intersection of the 3 fuzzy sets A, B, C, and the nonfuzzy set D. Its membership function is gwen by UE(X; tl , t2, t3, /4) = mm [UA (x; tl), UB(x, t2), Uc(x; /3), UD(X; 14)] So an individual can only be a full member of E if he doesn't smoke, has a cholesterol level not exceeding 200, a blood pressure not above 130, and a weight no less than 85 % and no more than 110 % of his recommended or ideal weight. Thts corresponds to the classical approach. A nonsmoker x = x(210, 145, 112, 0) with a cholesterol level of 210, a blood pressure of 145, and who ts overwetght by 12% ts a member of E wtth a grade of membership UE(x, 210, 145, 112, 0) = mm (0.875, 0.71875, 0 98, 1) = 0.71875. In other words, the "N" operation assigns a grade of membership that corresponds to the most severe of the infringements to "perfection", m this case blood pressure. Cumulative effects and interactions between the criteria are ignored, which Js not realistic. Obwously, the health consequences of high blood pressure are worse when there Is also an excess of weight and cholesterol. Also, since only the most severe cond~tton ~s considered, tt ~s tmposs~ble to introduce compensations or trade-offs m decision rules. A mild excess of weight cannot be compensated by ideal cholesterol and blood pressure
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`2.2. Other definitions of the intersection
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`The minimum operator that charactenzes the intersection corresponds to the "logical and" Other definitions of the intersection have been suggested, they correspond to "softer", more flexible interpretations of the connectwe "and ". They all amount to exactly the same in the conventional case of degrees of membership restricted to 0 and 1. The selection of a specific operator wdl depend on ~ts posslbd|tJes to allow for cumulative effects, interactions, and compensations between the criteria. We wish the following properties to be satisfied. Property 1 (cumulative effects): Two infringements are worse than one. UAns(X) < mln[UA(x),Us(x)] If UA(x) < 1 and Us(x) < 1. Property 2 (interactions between criteria). Assume UA (x) < Us(x) < 1. Then the effect of a decrease of UA(x) on UAnn(x) may depend on Ue(x)
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`42 JEAN LEMAIRE Property 3 (compensations between crlterm): If UA(r) and Uo(x) < 1, the effect of a decrease of Ua(x) on UAnb(X) can be erased by an increase of Us(x) (unless, of course, Us(x) reaches 1). The algebraic product F of A and B is denoted AB and Is defined by gA~(x) = UA (x)" UB(x) The bounded difference G of A and B is denoted A O B and is defined by UAes(x) = max [0, UA(x)+ UB(x)- 1] The Hamacher operator H defines the intersection of two fuzzy sets A and B by UA (x)" U~(x) UP(x) = o < p < l p + (I - p) [ UA (x) + UB (x) - u~ (x) UB (x)] The Yager operator Y defines the mtersectmn of two fuzzy sets A and B by U~(x) = I--mm{I,[(I--UA(x))P+(I--UB(x))P] I1p} p ~ I Problem 1 The generalized operators provide a more realistic way of modelling this specific problem because they explicitly allow for compensations and interac- tions between the selected criteria First consider the algebraic product. The grade of membership of individual x(210, 145,112,0) in the fuzzy set F = ABCD is UF(X; 210, 145, 112, 0) = (0 875) (0.71875) (0.98) (1) = 0 6163 The effect of high blood pressure is here amphfied by the presence of a shght obesity and a cholesterol level mildly above normal This operator satisfies all three properties. The grade of membership of the same mdwldual m the fuzzy set G = A O B O C O D corresponding to the bounded difference operation is Ua(x; 210, 145, 112, 0) = max [0, 0 875+0.71875+0 98+ I -3] = 0 57375 Hence the effects of the cnterm are addltwe; no interactions are introduced, since the consequences of cholesterol are the same whatever the blood pressure and the weight. This operator satisfies properties 1 and 3, but not property 2 The minimum and algebrmc product operators model two extreme situa- tions. The minimum operator does not satisfy any property Compensations and interactions cannot be introduced. The algebraic product allows for compensation and maximum interaction, since the effect of one criterion fully impacts the others. The Hamacher and Yager operators model mtermedmte situations, wtth flexlblhty provided by the parameter p. The Hamacher operator reduces to the algebraic product when p = 1. For p < 1, the denominator Is less than 1 and UH(X) > UF(X): the producl~
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`FUZZY INSURANCE 43 operator is "softened"; this operator models weaker interactions It reduces the effect of combined infringements The reduction effect is greater for severe Infringements. Also, the lower the selected p, the greater the reduction effect Hence this operator can be used if it ~s considered that the combined effect of high cholesterol and high blood pressure is somewhat less than multlphcatlve Selecting p = 0.5 for our example, we obtain successwely UffZ(x, 210, 145) - (0 875) (0 ...... 71875) = 0.6402 0 5+(1-0 5)[0.875+0.71875-(0.875)(0.71875)] Uff2(x, 210, 145, 112, 0) = uffZ(x, 210, 145, 112) (0 6402) (0 98) - -- 0 6296 0.5+(1 -0 5)[0 6402+0 98-(0 6402) (0.98)] This operator satisfies all three properties. The Yager operator reduces to the bounded difference operator when p = 1, and to the mlmmum operator when p --, ~. UPr(x) is an Increasing funcnon of p. Hence all intermediate SltUatmns can be modelled, from the strongest to the weakest "and" Selecting p = 2, we obtain U~,(x) = 1 -ram {1, [(1 -0.875)2+(I -0 71875)2+(I -0 98)2] I/2} = 0.69157 This operator satisfies all three properties, except in the case p = ~.
`If A is a fuzzy subset of X, ItS a-cut A~ is defined as the nonfuzzy subset such that A~ = {x[U~(x) > a} for 0 < a ~ I An a-cut can be interpreted as an error interval whose truth value is a. Problem 1 The notion of a-cut provides a flexible way of defining preferred policyholders. The "classical" approach corresponds to l-cuts such as E~ or F~. Lower values of a provide generahzatlons of this defimtlon For instance preferrred customers could be defined as the members of E075 or F 06o- E075 IS the set of pohcyholders for whmh the grade of membership attains at least 0 75 for each of the selected criteria (for our specific membership functions, t~ < 214, tz ~_ 144, 76.2 < t 3 < 117.1, t4 = 0). Hence this amounts to relaxing all criteria in a uniform way F06o is the set of policyholders for which the product of the four grades of membership attains at least 0 60 The latter definmon is more realistic because it allows for interactions and compensatmns An excess of blood pressure can for instance be compensated by normal or near-normal weight and cholesterol
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`2.3. Selection of a decision rule
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`44 JEAN LEMAIRE levels Policyholder x(210, 145, 112, 0) is accepted as preferred using the second criterion. He is not accepted if the first criterion is used Similar decision rules can be constructed using the other operators, ~f medical considerations hint that they provide a better model of the problem.
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`2.4. Fuzzy operations
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`UDILCA)(X) =
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`The concept of grades of membership allows to define the following operations that have no counterpart m ordinary set theory; they are uniquely fuzzy. Concentratton: A fuzzy set as concentrated by reducing the grade of member- ship of all elements that are only partly in the set, m such a way that the less an element as in the set, the more its grade of membershap as reduced The concentration of a fuzzy set A as denoted CON (A) and defined by UCON~A)(X) = U](x) a > 1 Ddatton: Dilation as the opposate of concentration A fuzzy set is dilated or stretched by increasing the grade of membership of all elements that are partly m the set. The dilation of a fuzzy set A as denoted DIL(A) and defined by
`U~(x) a < I a is called the power of the operation. Intensification: A fuzzy set can be antensafied by increasing the grade of membership of all the elements that are at least half m the set and decreasing the grade of membership of the elements that are less than half m the set The intensification of a fuzzy set as denoted INT (A) and is defined by {2U~(x) 0 <U(x)__<05 Uir~'r~A)(x) = I-2[l-UA(x)] 2 05 < U(x)< 1 Fuzztfication. A fuzzy set can be fuzztfied or de-mtensffed by increasing the extent of its fuzziness. There are several ways of achaeving this. Problem 1 The operations of concentrataon and dilation roughly approximate the effect of the llngmstic mo&fiers "very" and "more or less". They are used whenever the different crlterm have to be weaghted. The presentataon of problem 1 so far implicitly assumes that each criterion has the same importance. If for me&cal reasons this is not desirable, fuzzy operahons can be used. Suppose that cholesterol level is the better predictor of future heart problems, while the importance of blood pressure has to be downgraded. This can be reflected by
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`FUZZY INSURANCE 45 assigning powers of 2 and 0.5 to the two criteria. The modified fuzzy set E, corresponding to the minimum operator, is characterized by UR(x; tl, t2, /'3, /'4) = min [U2A (x, /'1), UsI/2(x;/'2), Uc(X; 13), UD(X;/'4)] The modified fuzzy set FP, corresponding to the algebraic product, has the membership function Up(X; /'1, /'2, /'3, /'4) = U2A( X, /'1) UBI/2(X; 12) Uc(x; 13) UD(X; /'4) Prospective pohcyholder x (210, 145, 112, 0) has a grade of membership of mm [(0.875) 2, (0.71875) t/2, 0 98, 1] = 0.7656 in ~7, and of (0.875)2-(0 71875) ~/2 (0.98) (1) = 0 6361 in F. He is now accepted as a preferred customer under each of the two crlterm of Section 2 3, since x(210, 145, 112,0) is included in both E07s and if060. 3 DECISION-MAKING WITH FUZZY GOALS AND CONSTRAINTS AND FUZZY REINSURANCE In the classical approach to decision-making, the principal ingredients of a decision problem are (a) a set of alternatives, (b) a set of constraints on the choice between different alternatwes, and (c) an objective function which associates with each alternative its evaluation. There is however an intrinsic similarity between objective functions and constraints, a similarity that becomes apparent when for instance Lagrangian multipliers are introduced This slmdanty is made explicit m the formulation of a decision problem m a fuzzy environment Let X = {x} be a given set of alternatives. A fuzzy goal G m X, or simply a goal G, is expressed and identified with a given fuzzy set G in X In other words, a fuzzy goal is an objectwe which can be characterized as a fuzzy set m the space of alternatives In the classical approach, the objectwe function serves to define a linear ordering on the set of alternatives. Clearly the membership function Uc(x) of a fuzzy goal serves the same purpose, and may even be derived from a given objective function by normalization, which leaves the hnear ordering unaltered Such normalization provides a common denom- inator for the various goals and constraints and makes ~t possible to treat them alike. A fuzzy constraint C in X, or smaply a constraint C, is similarly defined to be a fuzzy set C m X. An important aspect of those definitions is thus that the notions of goal and constraint both are defined as fuzzy sets in the space of alternatives. Hence they can be treated identically m the decision process Since we want to satisfy (optimize) the objcctwe function as well as the constraints, a decision m a fuzzy environment is defined as the selection of activities which simultaneously satisfy objective functions and constraints. A decision can therefore be viewed as the intersection of fuzzy constraints and fuzzy objective function(s) The relationship between constraints and ob.lectwe functions in a fuzzy environment is therefore fully symmetric
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`46 JEAN LEMAIRE Assume we are given a fimte set of alternatives X = {xj, x2 ..... x,,}, a set of goals G~ .... Gp, characterized by their respectwe membership functions Uc,(x), ., Uo~(x), and a set of constraints Ci .... Cq, characterized by their respective membership functions Uc (x) ..... Uc (x). Finiteness is assumed for expository purposes only and can be easily rela~ed. A decision is a choice or a set of choices drawn from the available alternatives, satisfying the constraints and the goals. The constraints and goals combine to form a demsmn D, which ~s naturally defined as the mtersecUon of the fuzzy sets G's and C's. D = Gi {7 G2 f'l ... C1GpN Ci ["l C2 N ... ~ Cq Consequently a decision D is a fuzzy set in the space of alternatives whose membership function ~s Up(r) = mm[Ua,(x), ., Ueo(x), Uc.(.v), ., Uc~(X)] This decision membership funcuon can be interpreted as the degree to wh,ch each of the alternatives satisfies the goals and constraints As m example 1, concentrations and dilations can be performed to reflect unequal importances of the goals and constraints, and other intersection operators can be used. Let K be the (nonfuzzy) set consisting of all the alternatives for which Up(X) reaches its maxmaal value K is called the optimizing set, and any alternative in K is an optnnal decision. The decision-maker simply selects as best alternative the one that has the maxmmm value of membership m D This decision-making procedure is essentially a maxlmln technique, similar to the selectmn of an optimal strategy in noncooperative game theory. For each alternative the minimum possible grade of membership of all the goals and constraints is computed to obtain D Then the maximuln value over the alternatives m D is selected Problem 2 Gwen the formulation of the problem, a reinsurance program is characterized by its XL deductible, and evaluated by means of 4 different variables tl = probability of rum (x 104 ) tz = coefficient of variation of the retained portfolio reinsurance premium 13 = (in %) cedent's premium income deductible t4 = (m %) cedent's premium income Assume the reinsurer offers 10 different XL deductibles, arranged m increasing order (x = 1,2,. , 10). The values taken by the selected variables are pro- vlded m Table 1
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`FUZZY INSURANCE 47 TABLE 1 CIIARACIERIST[CS OF THE [0 XL REINSURANCE PROGRAMS Program I 2 3 4 5 6 7 8 9 10 61 tl 339 280 200 200 313 339 360 388 419 465 G2 t2 2 98 3 00 3 03 3 07 3 12 3 19 3 28 3 52 3 80 4 20 Ch t3 3 20 3 00 2 85 2 73 2 64 2 57 2 52 2 48 2 45 2 43 C2 t4 4 6 8 9 10 I I 12 14 16 18 The following membership funchons have been chosen They are represented m Figure 2 Goal 1 (probablhty of rum) U~, (x, tl) = "1 1-2(tl-'00002) 2.00008 (.0001-t~ )2 2 -- 00008 t l ~ .00002 .00002 < l I ~ .00006 00006 < t~ < .0001 .0 0001 < t~ Goal 2 (coefficient of varlahon) UG2(X, t2) = .l--t 2 Constraint 1 (reinsurance premium)
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`I-2 t3- 2.5 06 3061-t3 )2 2 Uc, (x, t3) =
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`.0 t2 < 3-1 = 3.1 <t2=<41 41<t2 13 ~ 2.5 2 5 < t 3 ~ 2.8 2.8 < t~ N 31 3.1 <t' 3
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`U (x, t 1) G 1 0001 t 1 U (x,t 4) C2 i ! 00002 00006