Towards Decisio Makig uder Iterval, Set-Valued, Fuzzy, ad Z-Number Ucertaity: A Fair Price Approach Joe Lorkowski, Rafik Aliev, ad Vladik Kreiovich Abstract I this paper, we explore oe of the possible ways to make decisios uder ucertaity: amely, we explai how to defie a fair price for a participatio i such a decisio, ad the select a alterative for which the correspodig fair price is the largest. This idea is explaied o the examples of iterval ucertaity, set-valued, fuzzy, ad Z-umber ucertaity. I. FORMULATION OF THE PROBLEM Need for decisio makig uder ucertaity. I may practical situatios, we eed to make a decisio i situatios whe we have a icomplete iformatio about cosequeces of differet decisios. Fair price approach: a idea. Whe we have a full iformatio about a object, we ca express our desirability of each possible situatio, e.g., by declarig a price that we are willig to pay to get ivolved i this situatio. Oce these prices are set, selectig the most preferable alterative is easy: we just select the alterative for which the participatio price is the highest sice this is clearly the most desirable alterative. I decisio makig uder ucertaity, the situatio is ot so clear, sice it is ot easy to come up with a fair price. A atural idea is to develop techiques for producig such fair prices these prices ca the be used i decisio makig, to select a appropriate alterative. Commet. A alterative approach to decisio makig uder ucertaity based o the extesio of the otio of utility to iterval, fuzzy, ad Z-umber ucertaity is described i [1]. II. CASE OF INTERVAL UNCERTAINTY Iterval ucertaity: descriptio. Let us start with a simple case of ucertaity, i which, istead of kowig the exact gai u of selectig a alterative, we oly kow the lower boud u ad the upper boud u o this gai ad we have o iformatio o which values from the correspodig iterval [u, u] are more probable or less probable. This situatio, i which the oly iformatio that we have about the gai u is that this gai belogs to the iterval [u, u], is called iterval ucertaity. Joe Lorkowski ad Vladik Kreiovich are with the Departmet of Computer Sciece, Uiversity of Texas at El Paso, El Paso, TX 79968, USA (emails: lorkowski@computer.org, vladik@utep.edu); Rafik Aliev is with the Azerbaija State Oil Academy, Azadlig Ave. 2, AZ11, Baki, Azerbaija (email: raliev@asoa.edu.az). This work was supported i part by the Natioal Sciece Foudatio grats HRD-734825 ad HRD-1242122 (Cyber-ShARE Ceter of Excellece) ad DUE-926721. Assigig fair price uder iterval ucertaity: descriptio of the problem. We wat to assig, to each iterval [u, u], a umber P ([u, u]) describig the fair price of this iterval. There are several reasoable requiremets that this fuctio must satisfy. First, sice i all cases, the gai is larger tha or equal to u ad is smaller tha or equal to u, it is reasoable to require that the price should also be larger tha or equal to u ad smaller tha or equal to u: u P ([u, u]) u. Secod, if we keep the lower edpoit u itact but icrease the upper boud, this meas that we are keepig all the previous possibilities, but we are allowig ew possibilities with a higher gai. I this case, it is reasoable to require that the correspodig price icreases (or at least that it does ot decrease). I other words, if u = v ad u < v, the P ([u, u]) P ([v, v]). Similar, if we dismiss some low-gai alteratives, this should icrease (or at least ot decrease) the fair price. So, i geeral, if u v ad u v, the we should have P ([u, u]) P ([v, v]). To describe the third requiremet, let us cosider the situatio whe we have two differet alteratives. If these alteratives are idepedet, i the sese that the overall gai u+v of selectig both alteratives is equal to the sum of the gais u ad v obtaied by selectig each of them, the it is reasoable to require that the fair price of the joit selectio should be equal to the sums of fair prices correspodig to both alteratives. Let us describe this requiremet for the case whe the cosequeces of each alterative are oly kow with iterval ucertaity. About the gai u from the first alterative, we oly kow that this (ukow) gai value belogs to the iterval [u, u]. About the gai v from the secod alterative, we oly kow that this gai belogs to the iterval [v, v]. The overall gai u + v ca thus take ay value from the iterval [u, u] + [v, v] = [u + v, u + v]. Thus, the above requiremet takes the form P ([u + v, u + v]) = P ([u, u]) + P ([v, v]). Thus, we arrive at the followig defiitio. Defiitio 1. By a fair price uder iterval ucertaity, we mea a fuctio P ([u, u]) that assigs, to every iterval, a real umber, ad which satisfies the followig properties: u P ([u, u]) u for all u (coservativeess); if u = v ad u < v, the P ([u, u]) P ([v, v]) (mootoicity); for all u, u, v, ad v, we have P ([u + v, u + v]) = P ([u, u]) + P ([v, v])
(additivity). Propositio 1 [6]. Each fair price uder iterval ucertaity has the form P ([u, u]) = α H u + (1 α H ) u for some real umber α H [, 1]. Commet. This expressio was first proposed by the Nobelist L. Hurwicz; it is kow as the Hurwicz optimism-pessimism criterio [3], [5]. Proof. 1. Let us first cosider the value α H def = P ([, 1]) correspodig to the simplest possible iterval [, 1]. Due to coservativeess, we have α H 1. 2. Let us ow compute the value P ([, m]) for positive iteger values m. The iterval [, m] ca be represeted as the sum of m itervals equal to [, 1]: [, m] = [, 1]+...+[, 1] (m times). Thus, due to additivity, we have P ([, m]) = P ([, 1])+...+ P ([, 1]) (m times) = α H +... + α H (m times) = α H m. 3. Now, let us compute the value z def = P positive iteger. ([, 1 ]) for a I this case, the iterval [, [ 1] ca be represeted as the sum of itervals equal to, 1 ] [ : [, 1] =, 1 ] +... + [, 1 ] ( times). Thus, due to additivity, we have α H = z +...+z ( times), i.e., α H = z ad hece, z = α H 1. 4. For [ every two positive itegers m > ad >, the iterval, m ] ca be represeted as the sum of m itervals [ equal to, 1 ] ([. Thus, P, m ]) ([ = m P, 1 ]) = ( m α H 1 ) = α H m. 5. We have proved that for ratioal values r = m, we have P ([, r]) = α H r. Let us prove that the same property P ([, x]) = α H x holds for every positive real value x. To prove this property, we use mootoicity. Each real umber x ca be approximated, with arbitrary accuracy, by two ratioal umbers r < x < r. Due to mootoicity, we have P ([, r]) P ([, x]) P ([, r ]). Due to Part 4 of this proof, we thus coclude that α H r P ([, x]) α H r. Whe r x ad r x, we get α H r α H x ad α H r α H x ad thus, P ([, x]) = α H x. 6. Now, we are ready to prove the propositio. For each u ad u, we have [u, u] = [u, u] + [, u u]. Thus, due to additivity, P ([u, u]) = P ([u, u]) + P ([, u u]). For the first term, due to coservativeess, we have u P ([u, u]) u ad thus, P ([u, u]) = u. For the secod term, due to Part 5 of this proof, we get P ([, u u]) = α H (u u). Thus, the above additivity formula leads to P ([u, u]) = u + α H (u u), which is exactly α H u + (1 α H ) u. The propositio is prove. III. CASE OF SET-VALUED UNCERTAINTY Descriptio of the case. I some cases, i additio to kowig that the actual gai belogs to the iterval [u, u], we also kow that some values from this iterval caot be possible values of this gai. For example, if we gamble o a situatio with a ukow probability, we either get the price or lose the moey but we caot gai ay value i betwee. I geeral, istead of a (bouded) iterval of possible values, we ca cosider a more geeral bouded set of possible values. It makes sese to cosider closed bouded sets, i.e., bouded sets S that cotai all their limits poits. Ideed, if x S for all ad x x, the, for ay give accuracy, x is udistiguishable from some possible value x thus, i effect, the value x itself is possible. Assigig fair price uder set-valued ucertaity: descriptio of the problem. We wat to assig, to each bouded closed set S, a umber P (S) describig the fair price of this set. There are several reasoable requiremets that this fuctio must satisfy. First, for the case whe the set S is a iterval, we must get the fair price as described by Propositio 1. Secod, if we have two idepedet alteratives described by sets S ad S, the we should have P (S + S ) = P (S) + P (S ), where S + S def = {x + x : x S & x S } is the set of all possible sums x + x. Thus, we arrive at the followig defiitio. Defiitio 2. By a fair price uder set-valued ucertaity, we mea a fuctio P (S) that assigs, to every bouded closed set A, a real umber, ad which satisfies the followig properties: a restrictio of this fuctio to itervals S = [u, u] is a fair price uder iterval ucertaity i the sese of Defiitio 1 (coservativeess); for every two sets S ad S, we have P (S + S ) = P (S) + P (S ) (additivity). Propositio 2. Each fair price uder iterval ucertaity has the form P ([u, u]) = α H sup S + (1 α H ) if S for some real umber α H [, 1]. Proof. Due to coservativeess, for itervals S = [u, u]), the fuctio P (S) is a fair price uder iterval ucertaity ad thus, due to Propositio 1, has the form P ([u, u]) = α H u + (1 α H ) u.
For each bouded set S, its ifimum s def = if S ad its supremum s def = sup S are fiite. By defiitio, if S is a lower boud (it is the greatest lower boud) ad sup S is a upper boud (it is the least upper boud); thus, we have S [s, s]. Both if S ad sup S are limits of poits from the set S; sice the set S is closed, it cotais these limits: {s, s} S [s, s]. Let us prove that [s, s]+s = [2s, 2s]. Ideed, by defiitio of set additio, if S S, the S + S S + S. I particular, {s, s} S [s, s] implies that [s, s] + {s, s} [s, s] + S [s, s] + [s, s]. Here, [s, s] + {s, s} = [2s, 2s] ad similarly [s, s] + [s, s] = [2s, 2s]. Thus, [2s, 2s] [s, s] + S [2s, 2s] ad so, ideed, [s, s] + S = [2s, 2s]. Now, additivity implies that P (S) = P ([2s, 2s]) P ([s, s]). Substitutig the expressio P ([u, u]) = α H u + (1 α H ) u for the fair price of itervals ito this formula, we get the desired expressio for P (S). The propositio is prove. IV. (CRISP) Z-NUMBERS, Z-INTERVALS, AND Z-SETS: CASES WHEN THE PROBABILITIES ARE CRISP Descriptio of the case. I the previous sectios, we assumed that we are 1% certai that the actual gai is cotaied i the give iterval (or set). I reality, mistakes are possible, so usually, we are oly certai that u belogs to the correspodig iterval or set with some probability < p < 1. I such situatios, to fully describe our kowledge, we eed to describe both the iterval (or set) ad this probability p. I the geeral cotext, after supplemetig the iformatio about a quatity with the iformatio of how certai we are about this piece of iformatio, we get what L. Zadeh calls a Z-umber [11]. Because of this: we will call a pair cosistig of a (crisp) umber ad a (crisp) probability a crisp Z-umber; we will call a pair cosistig of a iterval ad a probability a Z-iterval; ad we will call a pair cosistig of a set ad a probability a Z-set. I this sectio, we will describe fair prices for crisp Z- umbers, Z-itervals, ad Z-sets for situatios whe the probability p is kow exactly. Operatios o the correspodig pairs. Whe we have two idepedet sequetial decisios, ad we are 1% sure that the first decisio leads to gai u ad the secod decisio leads to gai v, the, as we have metioed earlier, the user s total gai is equal to the sum u + v. I this sectio, we cosider the situatio i which: for the first decisio, our degree of cofidece i the gai estimate u is described by some probability p; for the secod decisio, our degree of cofidece i the gai estimate v is described by some probability q. The estimate u + v is valid oly if both gai estimates are correct. Sice these estimates are idepedet, the probability that they are both correct is equal to the product p q of the correspodig probabilities. Thus: for crisp Z-umbers (u, p) ad (v, q), the sum is equal to (u + v, p q); for Z-itervals ([u, u], p) ad [v, v], q), the sum is equal to ([u + v, u + v], p q); fially, for Z-sets (S, p) ad (S, q), the sum is equal to (S + S, p q). Let us aalyze these cases oe by oe. Case of crisp Z-umbers. Sice the probability p is usually kow with some ucertaity, it makes sese to require that the fair price of a crisp Z-umber (u, p) cotiuously deped o p, so that small chages i p lead to small chage i the fair price ad the closer our estimate to the actual value of the probability, the closer the estimated fair price should be to the actual fair price. Thus, we arrive at the followig defiitios. Defiitio 3. By a crisp Z-umber, we mea a pair (u, p) of two real umbers such that < p 1. Defiitio 4. By a fair price uder crisp Z-umber ucertaity, we mea a fuctio P (u, p) that assigs, to every crisp Z-umber, a real umber, ad which satisfies the followig properties: P (u, 1) = u for all u (coservativeess); for all u, v, p, ad q, we have P (u+v, p q) = P (u, p)+ P (v, q) (additivity); the fuctio P (u, p) is cotiuous i p (cotiuity). Propositio 3. Each fair price uder crisp Z-umber ucertaity has the form P (u, p) = u k l(p) for some real umber k. Proof. 1. By additivity, we have P (u, p) = P (u, 1) + P (, p). By coservativeess, we have P (u, q) = u; thus, P (u, p) = u + P (, p). So, to fid the geeral expressio for the fair price fuctio P (u, p), it is sufficiet to fid the values P (, p) correspodig to u =. 2. For the values P (, p), additivity implies that P (, p q) = P (, p) + P (, q). 3. Let us first cosider the value p = e 1 which correspods to l(p) = 1. The correspodig value of P (, p) will be deoted by k def = P (, e 1 ). The, for p = e 1, we have the desired expressio P (, p) = k l(p). 4. Let us ow cosider the values P (, e m ) for positive iteger values m. The probability e m ca be represeted as a product of m values e 1 : e m = e 1... e 1 (m times). Thus, due to additivity, we have P (, e m ) = P (, e 1 )+...+P (, e 1 )(m times) = m k.
Sice for p = e m, we have l(p) = m, we thus have P (, p) = k l(p) for these values p. 5. Now, let us estimate the value P (, p) for p = e 1/, for a positive iteger. I this case, the value e 1 ca be represeted as a product of probabilities equal to e 1/ : e 1 = e 1/... e 1/ ( times). Thus, due to additivity, we have k = P (, e 1 ) = P (, e 1/ ) +... + P (, e 1/ ) ( times), i.e., k = P (, e 1/ ) ad hece, P (, e 1/ ) = k. Therefore, for p = e 1/, we also have P (, p) = k l(p). 6. For every two positive umbers m > ad >, the probability e m/ ca be represeted as the product of m probabilities equal to e 1/. Thus, due to additivity, we have P (, e m/ ) = m P (, e 1/ ) = k m. Hece, for the values p = e m/ for which the logarithm l(p) is a ratioal umber, we have P (, p) = k l(p). 7. Every real umber l def = l(p) ca be approximated, with arbitrary accuracy, by ratioal umbers l l for which def p = e l e l = p. For these ratioal umbers, we have P (, p ) = k l(p ). Thus, whe ad p p, by cotiuity, we have P (, p) = k l(p). From Part 1, we kow that P (u, p) = u + P (, p); thus, ideed, P (u, p) = u k l(p). The propositio is prove. Cases of Z-itervals ad Z-sets. Similar results hold for Z-itervals ad Z-sets; i both results, we will use the fact that we already kow how to set a fair price for the case whe p = 1. Defiitio 5. By a Z-iterval, we mea a pair ([u, u], p) cosistig of a iterval [u, u] ad a real umbers p such that < p 1. Defiitio 6. By a fair price uder Z-iterval ucertaity, we mea a fuctio P ([u, u], p) that assigs, to every Z-iterval, a real umber, ad which satisfies the followig properties: for some α H [, 1] ad for all u u, we have P ([u, u], 1) = α H u + (1 α H ) u (coservativeess); for all u, u, v, v, p, ad q, we have P ([u + v, u + v], p q) = P ([u, u], p) + P ([v, v], q) (additivity). Propositio 4. Each fair price uder Z-iterval ucertaity has the form P ([u, u], p) = α H u + (1 α H ) u k l(p) for some real umbers α H [, 1] ad k. Proof. By additivity, we have P ([u, u], p) = P ([u, u], 1) + P (, p). By coservativeess, we have P ([u, u], 1) = α H u + (1 α H ) u. For P (, p), similarly to the proof of Propositio 3, we coclude that P (, p) = k l(p) for some real umber k. The propositio is prove. Defiitio 7. By a Z-set, we mea a pair (S, p) cosistig of a closed bouded set S ad a real umbers p such that < p 1. Defiitio 8. By a fair price uder Z-set-valued ucertaity, we mea a fuctio P (S, p) that assigs, to every Z-iterval, a real umber, ad which satisfies the followig properties: for some α H [, 1] ad for all sets S, we have P (S, 1) = α H sup S + (1 α H ) if S (coservativeess); for all S, S, p, ad q, we have P (S + S, p q) = P (S, p) + P (S, q) (additivity). Propositio 5. Each fair price uder Z-set-valued ucertaity has the form P (S, p) = α H sup S + (1 α H ) if S k l(p) for some real umbers α H [, 1] ad k. Proof. By additivity, we have P (S, p) = P (S, 1)+P ({}, p). By coservativeess, we have P (S, 1) = α H sup S + (1 α H ) if S. For P ({}, p), similarly to the proof of Propositio 3, we coclude that P ({}, p) = k l(p) for some real umber k. The propositio is prove. V. (CRISP) Z-NUMBERS, Z-INTERVALS, AND Z-SETS: CASES WHEN PROBABILITIES ARE KNOWN WITH INTERVAL OR SET-VALUED UNCERTAINTY Motivatios. Whe we kow the exact probabilities p ad q that the correspodig estimates are correct, the the probability that both estimates are correct is equal to the product p q. Similarly to the fact that we ofte do ot kow the exact gai, we ofte do ot kow the exact probability p. Istead, we may oly kow the iterval [p, p] of possible values of p, or, more geerally, a set P of possible values of p. If we kow p ad q with such ucertaity, what ca we the coclude about the product p q? For positive values p ad q, the fuctio p q is icreasig as a fuctio of both variables: if we icrease p ad/or icrease q, the product icreases. Thus, if the oly iformatio that we have the probability p is that this probability belogs to the iterval [p, p], ad the oly iformatio that we have the probability q is that this probability belogs to the iterval [q, q], the: the smallest possible of p q is equal to the product p q of the smallest values; the largest possible of p q is equal to the product p q f the largest values; ad the set of all possible values p q is the iterval [p q, p q]. For sets P ad Q, the set of possible values p q is the set P Q def = {p q : p P ad q Q}. Let us fid the fair price uder such ucertaity. Case of crisp Z-umbers. Let us start with the case of crisp Z-umbers uder such ucertaity.
Defiitio 9. By a crisp Z-umber uder iterval p- ucertaity, we mea a pair (u, [p, p]) cosistig of a real umber u ad a iterval [p, p] (, 1]. Defiitio 1. By a fair price uder crisp Z-umber p-iterval ucertaity, we mea a fuctio P (u, [p, p]) that assigs, to every crisp Z-umber uder iterval p-ucertaity, a real umber, ad which satisfies the followig properties: for some real umber k, we have P (u, [p, p]) = u k l(p) for all u ad p (coservativeess); for all u, v, p, p, q, ad q, we have P (u + v, [p q, p, q]]) = P (u, [p, p]) + P (v, [q, q]) (additivity); the fuctio P (u, [p, p]) is cotiuous i p ad p (cotiuity). Propositio 6. Each fair price uder crisp Z-umber p- iterval ucertaity has the form P (u, [p, p]) = u (k β) l(p) β l(p) for some real umber β [, 1]. Proof. 1. By additivity, we have P (u, [p, p]) = P (u, p) + P (, [p, 1]), where p def = p/p. By coservativeess, we kow that P (u, p) = u k l(p). Thus, P (u, p) = u k l(p) + P (, [p, 1]). So, to fid the geeral expressio for the fair price fuctio P (u, [p, p]), it is sufficiet to fid the values P (, [p, 1]) correspodig to u = ad p = 1. 2. For the values P (, [p, 1]), additivity implies that P (, [p q, 1]) = P (, [p, 1]) + P (, [q, 1]). 3. Let us first cosider the value p = e 1 which correspods to l(p) = 1. The correspodig value of P (, [e 1, 1]) will be deoted by β=p (, [e 1, 1]). The, for p = e 1, we have the expressio P (, [p, 1]) = β l(p). 4. Let us ow cosider the values P (, [e m, 1]) for positive iteger values m. The probability e m ca be represeted as a product of m values e 1 : e m = e 1... e 1 (m times). Thus, due to additivity, we have P (, [e m, 1]) = P (, [e 1, 1]) +... + P (, [e 1, 1]) (m times) = m β. Sice for p = e m, we have l(p) = m, we thus have P (, [p, 1]) = β l(p) for these values p. 5. Now, let us estimate the value P (, [p, 1]) for p = e 1/, for a positive iteger. I this case, the value e 1 ca be represeted as a product of probabilities equal to e 1/ : e 1 = e 1/... e 1/ ( times) Thus, due to additivity, we have β = P (, [e 1, 1]) = P (, [e 1/, 1]) +... + P (, [e 1/, 1]) ( times), i.e., β = P (, [e 1/, 1]) ad hece, P (, [e 1/, 1]) = β. Therefore, for p = e 1/, we also have P (, [p, 1]) = β l(p). 6. For every two positive umbers m > ad >, the probability e m/ ca be represeted as the product of m probabilities equal to e 1/. Thus, due to additivity, we have P (, [e m/, 1]) = m P (, [e 1/, 1]) = β m. Hece, for the values p = e m/ for which the logarithm l(p) is a ratioal umber, we have P (, [p, 1]) = k l(p). 7. Every real umber l def = l(p) ca be approximated, with arbitrary accuracy, by ratioal umbers l l for which def p = e l e l = p. For these ratioal umbers, we have P (, [p, 1]) = β l(p ). Thus, whe ad p p, by cotiuity, we have P (, [p, 1]) = β l(p). From Part 1, we kow that thus, P (u, [p, p]) = u k l(p) + P (, [p, 1]); P (u, [p, p]) = u k l(p) β l(p). Substitutig p = p/p ito this formula ad takig ito accout that l(p) = l(p) l(p), we get the desired formula. Defiitio 11. By a crisp Z-umber uder set-valued p- ucertaity, we mea a pair (u, P) cosistig of a real umber u ad a bouded closed set P (, 1] for which if P >. Defiitio 12. By a fair price uder crisp Z-umber p-setvalued ucertaity, we mea a fuctio P (u, P) that assigs, to every crisp Z-umber uder set-valued p-ucertaity, a real umber, ad which satisfies the followig properties: for some real umbers k ad β, we have P (u, [p, p]) = u (k β) l(p) β l(p) for all u, p, ad p (coservativeess); for all u, v, P, ad Q, we have P (u + v, P Q) = P (u, P) + P (v, Q) (additivity). Propositio 7. Each fair price uder crisp Z-umber p-setvalued ucertaity has the form P (u, P) = u (k β) l(sup P) β l(if P) for some real umber β [, 1]. Proof. By additivity, we have P (u, P) = P (u, 1) + P (, P), i.e., due to coservativeess, P (u, P) = u + P (, P). So, to fid the expressio for P (u, P), it is sufficiet to fid the values P (, P). Similarly to prove of Propositio 2, we ca prove that P [if P, sup P] = [(if P) 2, (sup P) 2 ]. Due to additivity, this implies that P (, [(if P) 2, (sup P) 2 ] = P (, P) + P (, [if P, sup P]),
hece P (, P) = P (, [(if P) 2, (sup P) 2 ] P (, [if P, sup P]). Due to coservativeess, we kow the values i the righthad side of this equality. Substitutig these values, we get the desired formula. Case of Z-itervals ad Z-sets. Let us exted the above results to Z-sets (ad to their particular case: Z-itervals). Defiitio 13. By a Z-set uder set-valued p-ucertaity, we mea a pair (S, P) cosistig of a bouded closed set S ad a bouded closed set P (, 1] for which if P >. Defiitio 14. By a fair price uder Z-set p-set-valued ucertaity, we mea a fuctio P (S, P) that assigs, to every Z-set uder set-valued p-ucertaity, a real umber, ad which satisfies the followig properties: for some real umber α H [, 1], we have P (S, 1) = α H sup S + (1 α H ) if S for all S (coservativeess); for some real umbers k ad β, we have P (u, P) = u (k β) l(sup P) β l(if P) for all u ad P (coservativeess); for all S, S, P, ad Q, we have P (S + S, P Q) = P (S, P) + P (Q, Q) (additivity). Propositio 8. Each fair price uder Z-set p-set-valued ucertaity has the form P (S, P) = α H sup S + (1 α H ) if S (k β) l(p) β l(p). VI. CASE OF FUZZY AND Z-NUMBER UNCERTAINTY Fuzzy umbers: remider. I the above text, we first cosidered situatios whe about each value of gai u, the expert is either absolutely sure that this value is possible or absolutely sure that this value is ot possible. The, we took ito accout the possibility that the expert is ot 1% certai about that but we assumed that the expert either kows the exact probability p describig his/her degree of certaity, or that the expert is absolutely sure which probabilities ca describe his/her ucertaity ad which caot. I reality, a expert is ofte ucertai about the possible values, ad ucertai about possible degrees of ucertaity. To take this ucertaity ito accout, L. Zadeh itroduced the otio of a fuzzy set [4], [9], [1], where, to each possible value of u, we assig a degree µ(u) [, 1] to which this value u is possible. Similarly, a fuzzy set µ p : [, 1] [, 1] ca describe the degrees to which differet probability values are possible. I this paper, we restrict ourselves to fuzzy umbers s, i.e., fuzzy sets for which the membership fuctio is differet from oly o a bouded set, where it first mootoically icreases util it reaches a poit s at which µ(s) = 1, ad the mootoically decreases from 1 to. Operatios o fuzzy umbers. Operatios o fuzzy umbers are usually described i terms of Zadeh s extesio priciple: if two quatities u ad v are described by membership fuctios µ 1 (u) ad µ 2 (v), the their sum w = u + v is described by the membership fuctio µ(w) = max mi(µ 1(u), µ 2 (v)), ad their product u,v: u+v=w w = u v is described by the membership fuctio µ(w) = max u,v: u v=w mi(µ 1(u), µ 2 (v)). It is kow that these operatios ca be equivaletly described i terms of the α-cuts. A α-cut of a fuzzy umber µ(u) is defied as a iterval u(α) = [u (α), u + (α)], where u (α) def = if{u : µ(u) α} ad u + (α) def = sup{u : µ(u) α}. The α-cuts correspodig to w = u + v ca be described, for every α, as [w (α), w + (α)] = [u (α), u + (α)] + [v (α), v + (α)], or, equivaletly, as [w (α), w + (α)] = [u (α) + v (α), u + (α) + v + (α)]. Similarly, whe both fuzzy umbers u ad v are limited to the iterval [, 1], the α-cuts correspodig to the product w = u v ca be described as [w (α), w + (α)] = [u (α), u + (α)] [v (α), v + (α)], or, equivaletly, as [w (α), w + (α)] = [u (α) v (α), u + (α) v + (α)]. Fair price of fuzzy umbers. Let us start with describig the fair price of fuzzy umbers. Similarly to the iterval case, a atural requiremet is mootoicity: if for all α, we have s (α) t (α) ad s + (α) t + (α), the the fair price of t should be larger tha or equal to the fair price of s. It is also reasoable to require cotiuity: that small chages i µ(u) should lead to small chages i the fair price. Defiitio 15. By a fair price uder fuzzy ucertaity, we mea a fuctio P (s) that assigs, to every fuzzy umber s, a real umber, ad which satisfies the followig properties: if a fuzzy umber s is located betwee u ad u, the u P (s) u (coservativeess); if a fuzzy umber w is the sum of fuzzy umbers u ad v, the we have P (w) = P (u) + P (v) (additivity); if for all α, we have s (α) t (α) ad s + (α) t + (α), the we have P (s) P (t) (mootoicity); if a sequece of membership fuctios µ uiformly coverges to µ, the we should have P (µ ) P (µ) (cotiuity).
Riema-Stieltjes itegral: remider. As we will see, the fair price of a fuzzy umber is described i terms of a Riema-Stieltjes itegral. This itegral is a atural geeralizatio of the usual (Riema) itegral. I geeral, a ituitive meaig of a Riema itegral b f(x) dx is that it is a area uder the curve y = f(x). a To compute this itegral, we select poits a = x 1 < x 2 <... < x 1 < x = b, ad approximate the curve by a piecewise costat fuctio f(x) = f(x i ) for x [x i, x i+1 ). The subgraph of this piece-wise costat fuctio is a uio of several rectagles, so its area is equal to the sum of the areas of these rectagles f(x i ) (x i+1 x i ). This sum is kow as the itegral sum for the itegral b f(x) dx. Riema s a itegral ca be formally defied as a limit of such itegral sums whe max(x i+1 x i ). A Riema-Stieltjes itegral b f(x) dg(x) is similarly a defied as the limit of the sums f(x i ) (g(x i+1 ) g(x i )) whe max(x i+1 x i ). Propositio 9. For a fuzzy umber s with a cotiuous membership fuctio µ(x), α-cuts [s (α), s + (α)] ad a poit s at which µ(s ) = 1, the fair price is equal to P (s) = s + k (α) ds (α) for appropriate fuctios k (α) ad k + (α). k + (α) ds + (α), Discussio. Whe the fuctio g(x) is differetiable, the itegral b a f(x) dg(x) is equal to the usual itegral b a f(x) g (x) dx, where g (x) deotes the derivative. Whe the fuctio f(x) is also differetiable, we ca use itegratio by part ad get yet aother equivalet form f(b) g(b) f(a) g(a) + b a F (x) g(x) dx, with F (x) = f (x). I geeral, a Stieltjes itegral ca be represeted i a similar form for some geeralized fuctio F (x) (see, e.g., [2]; geeralized fuctio are also kow as distributios; we do ot use this term to avoid cofusio with probability distributios). Thus, the above geeral formula ca be described as P (s) = K (α) s (α) dα + K + (α) s + (α) dα for appropriate geeralized fuctios K (α) ad K + (α). Coservativeess meas that for a crisp umber located at s, we should have P (s) = s. For the above formula, this meas that K (α) dα + K + (α) dα = 1. For a fuzzy umber which is equal to the iterval [u, u], the above formula leads to ( ) ( ) P (s) = K (α) dα u + K + (α) dα u. Thus, Hurwicz optimism-pessimism coefficiet α H is equal to K+ (α) dα. I this sese, the above formula is a geeralizatio of Hurwicz s formula to the fuzzy case. Proof. 1. For every two real umbers u ad γ [, 1], let us defie a fuzzy umber s γ,u (x) with the followig membership fuctio: µ γ,u () = 1, µ γ,u (x) = γ for x (, u], ad µ γ,u (x) = for all other x. For this fuzzy umbers, α-cuts have the followig form: s γ,u (α) = [, ] for α > γ, ad s γ,u (α) = [, u] for α γ. Based o the α-cuts, oe ca easily check that s γ,u+v = s γ,u + s γ,v. Thus, due to additivity, P (s γ,u+v ) = P (s γ,u ) + P (s γ,v ). Due to mootoicity, the value P (s γ,u ) mootoically depeds o u. Thus, similarly to the proof of Propositio 1, we ca coclude that P (s γ,u ) = k + (γ) u for some value k + (γ). By defiitio, the fuzzy umber s γ,u is located betwee ad u, so, due to coservativeess, we have P (s γ,u ) u for all u. This implies that k + (γ) 1. 2. Let us ow cosider a fuzzy umber s whose membership fuctio is equal to for x <, jumps to 1 for x =, ad the cotiuously decrease to. For this fuzzy umber, all α- cuts have the form [, s + (α)] for some s + (α). By defiitio of a α-cut, the value s + (α) decreases with α. For each sequece of values α = 1 < α 1 < α 2 <... < α 1 < α = 1, we ca defie a fuzzy umber s with the followig α-cuts: s (α) = for all α; ad whe α [α i, α i+1 ), the s + (α) = s + (α i ). Sice the membership fuctio of s is cotiuous, whe max(α i+1 α i ), we have s s, ad thus, P (s ) P (s). Oe ca check that the fuzzy umber s ca be represeted as a sum of fuzzy umbers s = s α 1,s + (α 1) + s α 2,s + (α 2) s + (α 1) +... + s α1,α 1 α 2. Thus, due to additivity, we have P (s ) = P (s α 1,s + (α 1 )+ P (s α 2,s + (α 2) s + (α 1)) +... + P (s α1,α 1 α 2 ). Substitutig the expressio for P (s γ,u ) from Part 1 of this proof, we coclude that P (s ) = k + (α 1 ) s + (α 1 )+ k + (α 2 ) (s + (α 2 ) s + (α 1 )) +... + k + (α 1 ) (α 1 α 2 ). The right-had side is mius the itegral sum for the Riema-Stieltjes itegral k+ (γ) ds + (γ). Sice we have P (s ) P (s), this meas that the itegral sums always coverges, the Riema-Stieltjes itegral is defied, ad the limit P (s) is equal to this itegral. 3. Similarly, for fuzzy umbers s whose membership fuctio µ(x) cotiuously icreases from to 1 as x icreases to ad is equal to for x >, the α-cuts are equal to
[s (α), ], ad P (s) = k (γ) ds (γ) for a appropriate fuctio k (γ). 4. A geeral fuzzy umber g, with α-cuts [g (α), g + (α)] ad a poit g at which µ(g ) = 1, ca be represeted as a sum of three fuzzy umbers: a crisp umber g ; a fuzzy umber whose α-cuts are equal to [, g + (α) g ]; ad a fuzzy umber whose α-cuts are equal to [g g (α), ]. By coservativeess, the fair price of the crisp umber is equal to g. The fair prices of the secod ad the their fuzzy umbers ca be obtaied by usig the formulas from Parts 2 ad 3 of this proof. By additivity, the fair price of the sum is equal to the sum of the prices. By takig ito accout that for every costat g, d(g(x) g ) = dg(x) ad thus, f(x) d(g(x) g ) = f(x) dg(x), we get the desired expressio. Case of Z-umber ucertaity. I this case, we have two fuzzy umbers: the fuzzy umber s which describes the values ad the fuzzy umber p which describes our degree of cofidece i the piece of iformatio described by s. Defiitio 16. By a fair price uder Z-umber ucertaity, we mea a fuctio P (s, p) that assigs, to every pair of two fuzzy umbers s ad p such that p is located o a iterval [p, 1] for some p >, a real umber, ad which satisfies the followig properties: if a fuzzy umber s is located betwee u ad u, the u P (s, 1) u (coservativeess); if w = u + v ad r = p q, the (additivity); if for all α, we have P (w, r) = P (u, p) + P (v, q) s (α) t (α) ad s + (α) t + (α), the we have P (s, 1) P (t, 1) (mootoicity); if s s ad p p, the P (s, p ) P (p, s) (cotiuity). Propositio 1. For a fuzzy umber s with α-cuts [s (α), s + (α)] ad a fuzzy umber p with α-cuts [p (α), p + (α)], we have P (s, p) = K (α) s (α) dα + L (α) l(p (α)) dα + K + (α) s + (α) dα+ L + (α) l(p + (α)) dα for appropriate geeralized fuctios K ± (α) ad L ± (α). Proof. Due to additivity, we have P (s, p) = P (s, 1) + P (, p). We already kow the expressio for P (s, 1); we thus eed to fid the expressio for P (, p). For logarithms, we have l(p q) = l(p)+l(q), so i terms of logarithms, additivity takes the usual form P (, l(p) + l(q)) = P (, l(p)) + P (, l(q)). Thus, similarly to the proof of Propositio 9, we coclude that P (, p) = L (α) l(p (α)) dα+ L + (α) l(p + (α)) dα. By applyig additivity to this expressio ad to the kow expressio for P (s, 1), we get the desired formula. VII. REMAINING PROBLEMS I this paper, we described how to defie fair price whe we have oe piece of iformatio: a fuzzy set S of gais with a fuzzy set P describig how cofidet we are i S. I practice, we may several such pieces of iformatio. It is desirable to come up with formulas which describe fair price uder such multiple pieces of iformatio formulas which are uiquely determied by additivity ad similar reasoable coditios. Aother ope questio is how to exted the above formulas for fair price to the case of iterval-valued ad, more geeral, type-2 fuzzy sets; see, e.g., [7], [8]. REFERENCES [1] R. A. Aliev, Fudametals of the Fuzzy Logic-Based Geeralized Theory of Decisios, Spriger Verlag, Berli, Heidelberg, 213. [2] I. M. Gelfad, G. E. Shilov, ad N. Ya. Vileki, Geeralized Fuctios, Academic Press, New York, 1964. [3] L. Hurwicz, Optimality Criteria for Decisio Makig Uder Igorace, Cowles Commissio Discussio Paper, Statistics, No. 37, 1951. [4] G. Klir ad B. Yua, Fuzzy Sets ad Fuzzy Logic, Pretice Hall, Upper Saddle River, New Jersey, 1995. [5] R. D. Luce ad R. Raiffa, Games ad Decisios: Itroductio ad Critical Survey, Dover, New York, 1989. [6] J. McKee, J. Lorkowski, ad T. Ngamsativog, Note o Fair Price uder Iterval Ucertaity, Joural of Ucertai Systems, 214, Vol. 8, to appear. [7] J. M. Medel, Ucertai Rule-Based Fuzzy Logic Systems: Itroductio ad New Directios, Pretice-Hall, Upper Saddle River, 21. [8] J. M. Medel ad D. Wu, Perceptual Computig: Aidig People i Makig Subjective Judgmets, IEEE Press ad Wiley, New York, 21. [9] H. T. Nguye ad E. A. Walker, A First Course i Fuzzy Logic, Chapma ad Hall/CRC, Boca Rato, Florida, 26. [1] L. A. Zadeh, Fuzzy sets, Iformatio ad Cotrol, 1965, Vol. 8, pp. 338 353. [11] L. A. Zadeh, A Note o Z-Numbers, Iformatio Scieces, 211, Vol. 181, pp. 2923 2932.