Iformato Fuso Decso Makg uder Ucertaty Aleader Valshevsky, Arkady Borsov. Itroducto I ths aer a decso ad methodology based o f-graules s reseted. F-graules are at the kerel of the aroach, as they are used to descrbe alteratves. A mortat feature of f-graules s that t s ossble to use both fuzzy ad radom formato to descrbe roblem doma. F-graules used ca be cosdered to be robablty dstrbuto of fuzzy values. Dstctve feature of the aroach s that alteratves are evaluated based ot o ther erformace o crtera, as most methodologes, but o the lkelhood that erformace of alteratves o crtera wll be hgh. The lkelhood s reseted the form of terval robabltes, whch make rakg of alteratves a o-trval task ad allows for alteratves to be o-comarable. Thus, the geeral case oly a artal rakg of alteratves s obtaable. Whch s full coset wth tuto oe caot eect a methodology to gve a recse aswer to a etremely mrecse questo ad t relcates the stuato such decso ad methodologes as PROMETHEE or ELECTRE [, 2]. Usually t s ot ossble to obta a comlete rakg wthout loosg some formato, as the decso roblem ca be too hard for ay methodology to be able to make a smart trade-off betwee coflctg alteratves. Thus, t ca be advatageous to buld a artal rakg ad to delegate tradg-off to the decso maker. As wth ay decso ad methodology that s used for decso-makg uder ucertaty, our settg the oto of otmalty s vague. Thus, the ma task of the methodology s ot to etract a otmal alteratve, whatever way t s defed, but rather to use such tools as rakg ad sestvty aalyss to hel decso-maker to make a better ad more formatve decso. Ths methodology dffers from smlar decso ads several asects. The ma dfferece s that the roosed decso ad ca be used whe the avalable formato s very ucerta. It may be so ucerta that t wll ot be ossble to costruct a erformace table or to descrbe the roblem doma usg ay crs methodology or framework, cludg terval robabltes. Regardless of the fact that ths methodology uses terval robabltes, they are all derved wth the methodology descrbed [3] from fuzzy ad robablstc formato, rovded by the decso maker or a eert. I the et chater we gve a bref troducto to codtoal f-graules ad the we descrbe the methodology. 2. Codtoal fuzzy graules As was metoed above, codtoal fuzzy graules ca be vewed as robablty dstrbuto of fuzzy values, or to be more eact, they troduce robablstc costrats o ossblty dstrbuto of crtera. Detaled descrto of fuzzy graules ca be foud [3]. Fuzzy graules ca be rereseted the form of IF-THEN rules, where the remse art corresods to dfferet ossble o-deterstc outcomes. Outcomes that corresod to the same evet or heomeo are groued to evdece. The cosequece art cotas fuzzy value of a crtero, whch s a ossblstc costrat o ts value. To clarfy the oto of fuzzy evdece, let us cosder a smle eamle. Assume that we assess rsk of a costructo roect based o tye of sol, whch we are ucerta about. I
ths settg rsk s the crtero ad the tye of sol s a o-deterstc heomeo. After cosultg eerts ad eag samles of sol we may have to come to the followg evdece that cotas three graules: ) IF (sol s rocky) wth robablty 0.8 THEN rsk s VERY LOW 2) IF (sol s sady) wth robablty 0.5 THEN rsk s ABOVE AVERAGE 3) IF (sol s swamy) wth robablty 0.05 THEN rsk s VERY HIGH Each evdece s a robablty dstrbuto of fuzzy values. For eamle, we ca terret the aforemetoed evdece as the followg robablty dstrbuto: P(rsk s VERY LOW) = 0.8 P(rsk s ABOVE AVERAGE) = 0.5 P(rsk s VERY HIGH) = 0.05 Gve a set of such evdeces, whch descrbe crtero rsk ad obvously also other crtera, we ca detere what s the robablty that the rsk wll be LOW for a artcular costructo roect. Value of Prob(rsk=LOW) s a terval value. Detals o the calculato of the robablty bouds ca be foud [3]. Ths s the ma tool for descrbg alteratves ad for assessg ther erformace, whch s actually the robablty that a artcular alteratve wll erform well. 3. Overvew of the methodology I ths chater we gve a overvew of the methodology that uses f-graules to descrbe avalable alteratves. The use of f-graules gves several advatages. It allows oe to use both fuzzy ad robablstc formato to descrbe alteratves. Ths ca be see as a trade-off betwee crs decso aalyss methodologes such as descrbed [, 2, 4, 5], whch do ot allow to use fuzzy descrtos of alteratves, ad rgorous aroaches that make etesve usage of fuzzy formato, such as [6], but whch requre etesve comutatoal ower. Recetly ew methods started to aear that deal wth multle sources of ucertaty decso-makg tasks. For eamle, [2] a aggregato rocedure s reseted, that eables oe to rocess smultaeously robablstc ad fuzzy formato. However, ths method s based o covertg all sorts of ucertaty to the robablty desty fuctos smly by ormalzato. Ths s agast tuto, sce fuzzess ad radomess are two dfferet tyes of ucertaty, whch are to be rocessed dstctvely. Thus, t s evdet that there s a eed for a decso-makg methodology that ca effectvely rocess dfferet tyes of ucertaty. The methodology descrbed ths aer ca be dvded to a umber of stes, whch ca be rereseted as a workflow, whch s show Fgure. Let us cosder the stes volved greater detals. 3.. Ste : Detere set of alteratves ad crtera Frst t s ecessary to detere what alteratves are there ad whch crtera wll be used to evaluate these alteratves. Nature of the roosed decso ad methodology requres the set of alteratves to be fte ad each alteratve to be dscrete ad defte.
Lke other aroaches, dfferet alteratves ad crtera ca be detered durg brastorg sessos or by careful aalyss of decso roblem.. Detere set of alteratves ad crtera 2. Descrbe crtera wth the hel of f-graules 3. Detere robabltes of crtera takg the desred values Perform rsk aalyss? Yes 4. Perform rsk aalyss No 5. Perform referece modellg Yes artal 6.. Perform artal rakg Perform artal or comlete rakg? comlete 6.2. Perform comlete rakg 7. Sestvty aalyss 8. Is further aalyss ecessary? No Preset aalyss results Fgure. Workflow of the f-graule based decso ad methodology 3.2. Ste 2: Descrbe crtera wth the hel of f-graules After havg detered a set of alteratves ad crtera t s ecessary to costruct evdeces that cosst of f-graules, as descrbed earler the aer. Each evdece cotas graules that descrbe ossble outcomes of some o-deterstc evet ad cosequece of these outcomes o the value of a crtero. Usually descrto of a alteratve cotas a set of evdeces, whch descrbe erformace of the alteratve o crtera, defed the frst ste. However, as wll be see later, alteratves are comared based o the robabltes that ther erformace o crtera wll be hgh. Iformato about oe crtero ca be dstrbuted amog several evdeces. However, each evdece should descrbe oly oe crtero. Further formato about the ossbltes
of usg fuzzy graules decso suort models ca be foud [7]. Whe a set of evdeces s defed ad agreed uo, f there s a grou of decso makers, we ca etract formato from these evdeces by deterg the lkelhood that the crtero wll be equal to some value. 3.3. Ste 3: Detere robabltes that the crtera wll be equal to the desred values Gve a set of evdeces that descrbe some crtera, we ca detere the lkelhood that the crtero wll be equal to a artcular fuzzy value. E.g., cosderg the eamle reseted above, we ca detere the robablty that rsk wll be LOW. The obtaed robablty s terval. It s coset wth our tuto, as the descrto of alteratves s ucerta wth radomess ad fuzzess reset so we caot eect the method to gve a recse aswer. Framework for aalysg fuzzy graules ad calculatg uer ad lower bouds of the robablty s reseted [3]. Oe ca use t as follows: frst t s ecessary to detere whch values of crtera are desrable ad the t s ecessary to calculate the robabltes that alteratves wll erform that well o these crtera,.e. the robabltes that evaluatos of crtera for these alteratves wll be equal to the desrable values. These are mzato decso attrbutes,.e. we wat to mze the robablty that crtera wll take desrable values. If there s a eed to avod some values, the t s ecessary to calculate the robabltes of crtera takg udesrable values, whch are the mzato decso attrbutes. 3.4. Ste 4: Calculate formato value of each alteratve Throughout the aer t was metoed that codtoal f-graules allow oe to erform decso aalyss uder ucertaty, whe the formato avalable s very lmted. But how much formato s lackg? How much formato s out there? Whch alteratve s more formatve? To aswer these ad other questos a formato value calculato methodology has bee develoed. The method s descrbed [8]. The dea of the method s to measure the amout of lackg formato by calculatg etroy value for each alteratve. Etroy value shows how much formato s lackg, so t ca be cosdered as a measure of ucertaty for a alteratve. I order to calculate etroy t s ecessary to reset decso roblem formato theoretc terms. Iterval evaluatos are used throughout the decso ad methodology, so t s ecessary to geeralze Shao s etroy to the case of terval robabltes. Such geeralzato s reseted Chater 4. Iformato obtaed ths ste ca be used to check whether formato avalablty for the alteratves s more or less equal. I case there s much less formato avalable about oe alteratve, tha about aother, t would be advsable to sed some tme collectg addtoal formato, f t s reasoable. 3.5. Ste 5: Perform referece modellg We roose to use weghtg scheme order to model refereces. There are a umber of weghtg schemes readly avalable for use. For eamle, t s ossble to adat a value-tree-based aroach used AHP [5]. Moreover, we ca use such methods for weght evaluato as SMART, SWING or ther geeralzato to terval case [9]. The basc dea of the latter aroach s to f oe so-called referece attrbute, to assg t some value ad the to detere relatve mortace of other attrbutes by assgg them evaluatos relatve to
that of the referece attrbute. Weghts are the detered by ormalzg the sum of these evaluatos to oe. I ts tur, AHP cossts of costructg a herarchy of crtera, sub-crtera. Weghts are assged based o ar-wse comarsos of crtera that have a commo aret. 3.6. Ste 6: Perform rakg The cosdered decso ad methodology s teded for solvg mult-crtera decso aalyss tasks uder ucertaty. Thus, t s etremely dffcult, f ot mossble to defe a otmalty crtero ths settg. The ma tools that hel a decso maker to make a better decso are rakg ad sestvty aalyss. Comarso of alteratves s based o terval robabltes, ad a geeral case t s ossble to costruct a artal rakg oly. Let us troduce artal rakg that s smlar to that used the PROMETHEE method [], ad s accomlshed by evaluatg outgog ad cog flows. Frst let us troduce several deftos. Let us assume that we have a set of alteratves A = {a, a 2,, a }, where s the umber of alteratves. Set of crtera that s used to evaluate alteratves s C = {c, c 2,, c m }, where m s the umber of crtera. Let us assume that usg a weghtg scheme the followg weghts have bee assged to crtera: w, w 2,, w m. Each alteratve s evaluated o m crtera ad the evaluatos are terval-valued robabltes. For alteratve A we have the followg evaluatos: E, E 2,, E m. Evaluatos are terval, so: E ( e, e ) =. We defe outgog ad cog flows as follows: φ + φ m ( A ) = k= m ( A ) = k= w w k k e k e k Partal rakg s erformed by the followg reorder, where a ad b are two alteratves from set A, whle P s outrakg relato, I s dfferece relato ad R s comarablty relato: + ( a) > φ ( b), + + ( a) = φ ( b) ad φ ( a) = φ ( b) apb f φ aib f φ, arb otherwse. If we reset some foldg of terval evaluatos, the a comlete rakg would be ossble. But, as oted elsewhere the lterature, comlete rakg s ossble oly f some formato s hdde or gored. A atural way to fold terval robabltes s the followg: e + e e = 2 Now let us defe erformace of a alteratve a as follows: The the comlete rakg s defed accordg to the followg reorder, where a ad a are two alteratves from set A: apa ( a outraks a ) f > aia ( a s dfferet to a ) f = = m k= w e k k
Oce aga we would lke to ot out that the comlete rakg mght be decetve, as t hdes away features of alteratves that may make them comarable. Ths cocludes the art about rakg, whch s oe of the most mortat tools for mult-crtera decso-makg uder ucertaty. Net chater cosders how t s ossble to erform sestvty aalyss. 3.7. Ste 7: Perform sestvty aalyss Sestvty aalyss s a mortat ad owerful tool of a decso maker, as t hels to detere how sestve a decso model s to chages ts arameters. Lke other aroaches, the roosed decso ad methodology t s ossble to chage values of weghts ad to follow alteratos rakg of alteratves. Should t be ecessary, defto of alteratves or desred crtera values ca be chaged as well. These chages requre that robabltes be re-calculated (ste 3). There s aother aroach that ca ad erforg sestvty aalyss. It s based o deterg whch features of alteratves are desrable. It ca be cosdered as a sort of reverse egeerg: frst we defe whch values of crtera are desrable ad the we detere a dream alteratve, whch s close to satsfyg these desres. Results of such a aalyss ca be used for otg crtcal features of alteratves, whch we should ay our atteto to. Ths kd of aalyss s based o the adatve etwork ANGIE (Adatve Network for Graular Iformato ad Evdece rocessg) that s descrbed [0]. Descrto of the ANGIE-based sestvty aalyss ca be foud []. 3.8. Ste 8: Aalysg results At ths ot t should be decded whether further aalyss s requred. Decso aalyss s ecessarly a teratve task, so oe should ot eglect data adustmet ad revso. Several reasos for revsg data are: rsk aalyss shows that there s too lttle formato avalable about a alteratve; there are crtcal features that have ot bee roerly eaed; further sestvty aalyss s requred; ew data s avalable ad so o. 4. Geeralzato of Shao s Etroy to the Case of Iterval Probabltes I ths chater the geeralzato roblem s reseted as a otmzato task wth equalty costrats ad the atteto s focused o solvg ths task. We gve a formal statemet of the roblem ad reset ts aalytc. The roblem s solved usg method of Lagrage multlers [3], as t has both equalty ad equalty costrats. The tradtoal sgle-valued etroy s descrbed [4]. 4.. Geeral Problem Statemet Let us assume that a system ca be states, ad the robablty that the system s the -th state s terval ad s equal to [, ]. I case whe robabltes are sglevalued rather tha terval-valued, t s requred that robabltes sum to,.e. =. () If robabltes are terval-valued, the () ca be rewrtte as (2):. (2) It s easy to show that () s a secal case of (2) whe =. :
The etroy s terval as well: H = [ H, H ]. Geeral roblem statemet for the calculato of the terval-valued etroy ca be stated the followg way: Let [, ], [ 2, 2 ],, [, ] be terval-valued robabltes, the lower ad uer boudary of etroy ca be calculated as follows: subect to H : l ad H : l, =, =., = = Natural logarthms are used, as t wll smlfy calculatos that wll follow, but t does ot chage the essece of the roblem. 4.2. Aalytcal Soluto of the Problem We use method of Lagrage multlers [3] order to solve roblem (3). It s ecessary to covert all equalty costrats to by multlyg by. Moreover, t s ecessary to covert a mzato roblem to mzato roblem. Frst let us cosder soluto of the mzato roblem (4). H : = l subect to,, =,. = = = (3) (4) The Lagraga for rogram (4) s the followg: L,...,, λ, μ, μ, μ, μ,..., μ, μ, = ( ) 2 2 22 ( μ ( ) + μ2 ( ) l + λ + = = = Partal dervatves of the Lagraga (5) are: L = l + λ + μ μ2, =,. Thus, t s ecessary to solve the followg roblem: 2 (5) l + λ + μ μ2 = 0, =,, (6) = =, (7) ( ) = 0, μ ( ) = 0, μ, (8) 2 =,, μ 0, =,, =,2, (9)
Now t s ecessary to cosder 2 2 cases deedg o values of μ comlemetary slackess codtos (8), amely, deedg whether a artcular μ s or s ot equal to zero. ) If μ = 0, =,, =, 2 ad each terval robablty cotas value, the the mum etroy s reached at ths ot. Otherwse roceed to the et ste. 2) Check each cofgurato whch a umber of μ s are ot equal to zero for some ad. It s useful to ote that f μ or 2 or ( ) must be equal to zero accordg to (8). Thus, f μ or μ 2 s ot equal to zero, the, accordgly, = or =. μ s ot equal to zero, the ( ) Hece, frst for all ostve μ or μ 2 detere values of robabltes. Values of all other robabltes must be equal, whch follows from (6) ad from the fact that μ = 0 ad μ 2 = 0 for all that caot be detered. Values of ca be detered usg (7) ad the fact that all ukow robabltes are equal. At ths ot values of all robabltes for a artcular cofgurato have bee detered. Now t s ecessary to check whether robabltes sum to ad whether all μ s are oegatve accordg to (9). I order to detere the latter t s ecessary to calculate a system of lear equatos derved from (6). If all codtos are satsfed, calculate value of etroy at the corresodg ot. After all the cofguratos have bee rocessed choose the bggest value, whch s the sought uer boudary of etroy H. H : Soluto of the mzato roblem (0) s very smlar to that of (4). = l subect to,, =,. = = (0) I order to solve (0) t s ecessary to covert t to mzato roblem by multlyg the target fucto by. After ths oerato the rogram (0) looks as follows: H : = l stated that,, =,. = = Soluto of rogram () s comletely aalogous wth that of (4), ecet for the last ste where t s ecessary to choose the smallest value, whch wll be the lower boudary of the terval etroy H. Moreover, Lagraga ad artal dervatves do ot look the same, amely: ()
The Lagraga for () s: L,...,, λ, μ, μ, μ, μ (,..., μ, μ,) 2 2 22 ( μ ( ) + μ2 ( ) l + λ + = = = Ad the artal dervatves of (2) are: L = l + + λ + μ μ2, =,. 2 = (2) 4.3. Addtvty I order to rove that the terval etroy calculated above s addtve we ca use the followg reasog. Accordg to [4], f two systems ad y are deedet, the H (, y) = H ( ) + H (y). If the etroy values are terval H ( ) = [, ] ad H ( y) = [, ] H, the H H order to detere etroy value of a ot system wth mum ossble etroy value, t s ecessary to sum the lower boudares of corresodg tervals. If there s a system wth smaller value, the t meas that ether systems ad y are ot deedet or the boudares H `H `H y or are chose correctly. Assume that there ests some + `H + H y H y, the ether `H H or `H y, but sce ad are mum ossble etroy values for dvdual systems, the: `H = H =. Proof for the uer boudary s aalogous. [ H, H Hece, we have rove that f H ( ) = ], ( y) [, ] ad y are deedet, the: (, y) [ H + H, H H ] H = +. y ad H = ad systems Ths fshes the geeralzato of Shao s Etroy to the case of terval robabltes. 5. Cocluso I ths aer a decso ad methodology s roosed. F-graules are at the core of ths methodology, whch eables oe to use both fuzzy ad robablstc formato to descrbe alteratves. Ths aroach has several smlar ad dstctve features to other methodologes. It s smlar referece modellg, as almost arbtrary weghtg scheme ca be aled to the descrbed methodology. Moreover, rakg s smlar to that aled other aroaches, for eamle []. However, such decso aalyss tools as rsk ad sestvty aalyss are ot smlar to those used elsewhere. Refereces to aers cotag descrtos of both tyes of aalyss are cluded. Moreover, the method descrbed does ot requre the decso maker to costruct a erformace table. Besdes that, the geeralzato of Shao s etroy to the case of terval robabltes s reseted. y
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