Set pair analysis of lattice order decision-making model and application

Transcription

1 Avalable onlne Journal of Checal and Pharaceutcal Research 0 6(:5-58 Research Artcle IN : CODEN(A : JCPRC5 et par analyss of lattce order decson-akng odel and applcaton Je un Lhong L Yan L and Baoang Lu* College of cences Hebe nted nversty 6 Xnhua Road Chna ABTRACT When the theores and ethods of set par analyss are appled to the lattce order decson- akng a new decson-akng odel-- set par analyss of lattce order decson akng odel s created. Based on the portray of the lattce order of the set par utlty functon the best soluton of the decson odel s found. Key words: lattce order decson-akng set par utlty functon connecton degree set par analyss INTRODCTION Professor Guo Yaohuang usng algebra lattce theory prooted the total order depct of Von Neuann-Morgenstern s ratonal behavor ao syste to the lattce ordered depct and then he proposed a new lattce order decson-akng theory. In ths way Professor Guo buld a lattce order decson-akng[] syste of ratonal behavor. Connectvty ao and transtvty of the tradtonal VNM ratonal behavor ao syste requre that all decson consequences should be coparable wth the preference relaton and should he transtve as well. Ths actually akes preference relatonshp nto a total order relatonshp whch sees to be unreasonable. Ths condton s too strong to far apart wth the actual decson stuatons. Therefore under the condton of keepng the ratonal decson-akng behavor followng the ndependence ao lattce order decson akng theory weakens the tradtonal connectedness ao to connectvty ao of lattce. But connectvty ao of lattce does not requre decson schee be coparable n pars. On the one hand t ensures that there s a certan relatonshp between the varous solutons on the other hand t avods the decson proble whch has no coparatve nforaton. The creaton of lattce order decson akng theory flled the blank of lattce theory appled n the decson scence and opened a research drecton n the sae feld. In recent years people for the lattce order decson odel have carred out n-depth research and tred to gve out a practcal ethod of Lattce order decson-akng. At present the aor proble s how to establsh and perfect the lattce order decson akng odel of the gvng set par utlty functon. Ths essay applyng the theory and ethod of set par utlty functon to the lattce order decson akng analyzes lattce order structure characterstcs of the sets of utlty functon then establshes lattce order decson akng odel of the set par analyss and at last gves soe applcaton eaple.. ET PAIR ANALYI AND ET PAIR TILITY FNCTION. THE BAIC THEORY OF ET PAIR ANALYI et par s a par whch has a certan connecton of two sets. The core dea of set par analyss s to analyze the certanty and uncertanty of the obects as a certan-uncertan syste []. In soe specfc contet the characterstcs of the two target set are analyzed fro such perspectves as ther slartes ther dfferences. Meanwhle a quanttatve analyss s ade. In ths way the connecton degree epresson of the two sets s got. Based on these analyses such aspects as contact decson-akng forecast control sulaton evaluaton evoluton and utaton[] are the target of the further analyss. 5

2 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58 Defnton.: two sets A and B are gven and H ( A B s a set par ade up wth the two sets. In soe specfc contetw set par H has N features aong whch features are utual of A and B. They are opposte on P features. They are nether opposte nor slar n the rest F features F N P. We defne the rato as follow: N s the dentty degree of A and B under background W shortened as dentty degree F N s the dscrepancy degree of A and B under background W shortened as dscrepancy degree P N s the contrary degree of A and B under background W shortened as contrary degree F P All these can be represented by the forula ( W. (W s the degree contact of set A and N N N set B. For splcty f we let a N b F N c P N then t can be recorded as the followng: u a b c. s the ark of dfference degree and. a b c. 0 a b c and s the ark of contrary degree and. Obvously. THE RELATED KNOWLEDGE OF THE ET TILITY FNCTION.. TILITY FNCTION In decson theory utlty s a concept whch represents the results of the schee satsfes and acheves the decson aker s preference degree at the sae te t s value whch can be tested wth soe specfc ethods and can be used as the bass of decson analyss. Assue that each feasble schee of the decson proble ay be results n dfferent ways each result can be of dfferent value and effects to ther decson-akers accordng to ther subectve desre and value-orentaton each value of the result of decson akers have dfferent value and effect and reacton result value for decson akers. Thus utlty should be soe volue of value and effect that perfor to the decson akers[9]. The utlty functon u s a knd of relatve easureent generally rangng between 0 and ( 0 u whch s proft or onetary value and the utlty functon s a ncreasng functon of... ET PAIR TILITY FNCTION In the decson akng process decson akers tend to show the hestant and perpleed psychologcal whch can be agreeent obecton or neutralty whose counterparts are respectvely dentty degree dfference degree and contrary degree. Therefore t s necessary to buld a set par utlty functon[8] and to depct postve utlty negatve utlty and uncertanty utlty n order to ake the decson-akng ore obectve and feasble. Decson proble can be epressed n a forat whch s called a decson table also called decson atr. For splcty assue that there are certan knds of possble states whch s ncopatble arks as V V V V V (naed as set of states and the probablty of occurrence of varous natural states s n represented by P. At the sae te assue that there are certan knds of possble actons whch ake up an acton set represented as. Decson-akers ust select only one of these actons. If the consequences of. (actons and V (real state arked as Table.: the general for of the decson table V ( P V ( P V n ( P n n n n n then we can get the decson table shown n table 5

3 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58 Based on the decson forat contet above to structure a set par utlty functon the procedure s as follows[8]: tep : In the forat contet of decson probles select the best value ake the nu value and then ake 0 tep : eek for a sutable value sq between the best value and the nu value to ake 0.5 fg fg sq tep : Choose the type of utlty curve. Generally we present paraeters as A B C to the three functon values above we can get the forula for the utlty functon: tep : Construct the set par utlty functon: f a b c( nder the condton of the probablty P to take the bggest. ln. Accordng a s the postve utlty a P c s the negatve utlty c P b under the condton of the probablty P to take the nu s the utlty of uncertanty b a b P P. In ths way the set par utlty functon can be shown as follows: P P P P f tep 5: The set of set par utlty functon s got: n P f. ANALYI DEVICE OF LATTICE ORDER DECIION OF THE ET OF ET PAIR TILITY FNCTION. THE PARTIAL ORDER RELATION OF THE ET OF ET PAIR TILITY FNCTION can be sply epressed as Generally the set of set par utlty functon a b c a b c a b c a b c. Defnton.[]: If A s a set relaton R arked as arb If a b R R A A R s for a bnary relaton n A. If a b R A and b have a a and b have no relatonshp lke R arked as ar ' b or non- arb. Defnton.: uppose as a set of set par utlty functon n soe contet and a b c a b c then: ( If a a b b c c and are equvalence arked as (If a a a b a b takes prorty to arked as (If a a a b a b takes absolute prorty to arked as Defnton. shows that precedence relaton of the set of set par utlty functon s has the followng characters: Character.: reflevty: for any then Character.: antsyetry: f Character.: transtvty: f then then Obvously set under the precedence relatonshp consttutes a poset. In a poset ( suppose A and B are poset eleents n If or otherwse and are not coparable arked as. and s coparable 5

4 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58. LATTICE ORDERED TRCTRE OF THE ET OF ET PAIR TILITY FNCTION s a set of set par utlty functon let a b c and a b c suppose: ( upreu of the set s a connecton degree epresson of a b c as sup({ } then: a a ( a b a b a( a b a b / ( a b a b a( a b a / b a b c a b ( Infu of the set s a connecton degree epresson of a b c as sup({ } then: a n ( a b a b n( a b a b / ( a b a b n( a b a / b c n b a b. upreu and Infu of connecton degree can be calculated wth the followng rules: Idepotent rate: sup({ } nf({ } Coutaton rate: sup({ } sup({ } nf({ } nf({ } Cobne rate: sup({ }{ } sup({ }{ } nf({ }{ } nf({ }{ } Defnton.: ( s a poset of set par utlty functons f any of the two eleents have a supreu and an nfu on partal order " " consttutes a lattce of set par utlty functon and " "s a lattce order of set par utlty n. Theore.: In a poset ( s ( s ( s } addng ( eleents at ost then we can { consttute an etended poset akng a lattce ordered structure. Deonstraton: Consderng the worst case of poset naely the eleents n ( s ( s ( s } for a ant-chan. e. any two eleents are not coparable. These eleents { can ne regarded as a leaf nodes. sng the structure theory of coplete bnary tree we can structure superposton of two bnary tree and then we get a lattce. Accordng to the propertes of the bnary tree suppose the nuber of nodes of 0 s n the nuber of nodes of 00 s n then the relaton between the nuber of leaf nodes and the nuber of the nodes of 0 can be represented as n n. o n the ant-chan addng ( eleents wll result n a lattce structure. 0 Defnton.: In any lattce u u u a b c u a b c the dstance functon can be defned as d u u a a b b. In any lattce u u u dstance functon has the followng characters: Character.: d u u du u Character.5: d u u du u Character.6: d u u du u du u k. k k 55

5 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58 AMPLE ANALYI uppose decson probles wth four states and four proects the probablty of occurrence of each state and the revenue of each proect are shown n the followng table: V P V P V P V P revenue revenue revenue revenue Frst construct a set par utlty functon: tep : In the forat contet of decson probles select the best value 00 and ake tep : eek for a sutable value fg and the nu value 0 between the best value and the nu value to eet that 55 fg tep : Accordng to three pont ethod we can get A.7 B. C 0 and then the soluton to the.7.ln 0 utlty functon s tep : Construct the set par utlty functon: f c( u a b c tep 5: Output the set of set par utlty functon: 0. f f f f f 0 0. f f 5 0. f f 5 0. f f 65 0.f f 0 0. f f 5 0. f econdly construct lattce structure: ( Any two eleents n the set of set par utlty functon cannot be copared ( The supreu of ( and s R ( The supreu of and s R ( ( R ( ( R prorty than (5 The nfu of and ( ( (6 The nfu of and ( ( s T ( T s sq 56

6 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58 (7 The nfu of ( T and ( T s T Then partal order structure of the set of set par utlty functon Hasse dagra s fgure.. ( R of the ( R ( ( ( ( ( T T ( Fnally the analyss result s as followngs: ( R d R R d R ( T 0 Fgure.: Hasse dagra d takes prorty to ( d takes prorty to ( d R d R takes prorty to ( d R d R takes prorty to. (Note: by coparng the dstance between each node wth the nfu the optal soluton of the odel wll be found s the optal soluton of the odel. Through the analyss above CONCLION et par analyss s a better way to ake a general and practcal research n lattce order decson. It s a knd of developent and proveent for ratonal behavor decson-akng theory. Ths artcle apples set par analyss ethod to the lattce order decson akng theory akes a n-depth study on the lattce ordered characterstcs of the set of set par utlty functon. Then a lattce structure s establshed. At last ore ratonal and obectve l soluton of the odel s found. Acknowledgents upported by Natonal cence Fund of Chna (NO Natonal cence Fund of Hebe Provnce Chna (NO.A00900 REFERENCE [] Guo Yaohuang. hangha cence and Technology Press 00. [] Zhao Keqn. Zheang cence and Technology Press000. [] Guo Chunang Guo Yaohuang. ystes Engneerng-Theory Methodology Applcatons00( [] Jang Yunlang Xu Congfu. Coputer cence006( [5] Hong WeWu ChengzhenL Zhenyou. Journal of Fuan College of Forestry 9955( [6] Guo ChunXang Guo Yaohuang. ystes Engneerng 006(9 95. [7] Wang Guohua Lang Lang. nversty of cence and Technology of Chna press 006. [8] Lu Baoang Yang Yafeng L Yankun. Internatonal Conference on Engneerng and Busness Manageent

7 Baoang Lu et al J. Che. Phar. Res. 0 6(:5-58 [9] Hao Guang Mou Qfeng Zhang Danye Guo Yaohuang. Journal of outhwest Jaotong nversty 006( [0] Da Yu Zhou DeQun. Chnese Journal of Manageent cence0075(