A New Approach to Solve Type-2 Fuzzy Linear Programming Problem Using Possibility, Necessity, and Credibility Measures

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1 New pproach to Solve Type- Fuzzy near Programmng Problem sng Possblty Necessty and Credblty Measures.Srnvasan G.Geetharaman Department of Mathematcs Bharathdasan Insttute of Technology (BIT) Campus-nna nversty Truchrappall-60 04Tamlnadu Inda. BSTRCT In ths paper a type - fuzzy lnear programmng model based on the possblty necessty and credblty relaton s ntroduced. By usng the degree of measures the satsfacton of constrants can be measured. Wth ths rankng nde the bound of optmal(opt) soluton s obtaned at dfferent degree of possblty and necessty measures. To valdate the proposed method Optmal soluton s obtaned for type - fuzzy lnear programmng problem at dfferent degree of satsfacton by usng smple method wth the help of MTB. Fnally the optmal soluton procedure s llustrated wth numercal eample. Keywords: Fuzzy near programmng Type- Fuzzy Set Interval Type- Fuzzy Set Interval Type- Fuzzy Number Perfectly normal. INTRODCTION Management Scence ncludes all ratonal approaches to management decson-makng that are based on an applcaton of scentfc and systematc procedures. The analyss process employed by the decson maker may take two basc forms: qualtatve and quanttatve. The qualtatve analyss approach s based prmarly upon the manager's udgment and eperence. In the quanttatve approach to the problem an analyst wll concentrate on the quanttatve facts or data assocated wth the problem and develop mathematcal epressons that descrbe the obectves constrants and relatonshps that est n the problem. There are several mportant tools or technques that have been found useful n the quanttatve analyss phase of the decson makng process. mong them near programmng s an unsurpassed technque to acheve the best outcome. The quanttatve analyss contans fve phases:. problem defnton.. Model development.. Data preparaton 4. Model soluton and 5. Report generaton. In model development phase the models are representatons of real obects. These representatons can be presented n varous forms alke Iconc models nalog models and Mathematcal models. The mathematcal models represent the real stuaton by a system of symbols and mathematcal relatonshps or epressons. The success of the mathematcal model and quanttatve approach wll depend heavly upon how accurately the obectve and constrants can be epressed n terms of mathematcal equatons or relatonshps. Such envronmental factors whch can affect both the obectve functon and constrants are referred to as the ncontrollable nput to the model. The nput whch are controlled or determned by the decson maker are referred to as the controllable nputs to the model thus are referred to as the decson varables of the model. Once all controllable and uncontrollable nputs are specfed the obectve functon and constrants can be evaluated and the output of the model determned []. In ths sense the output of the model s smply the proecton of what would happen f the partcular envronmental factors and decsons occurred n the real stuaton. The uncontrollable nputs can ether be known eactly or be uncertan and subect to varaton. If all uncontrollable nputs to a model are known and cannot vary the model s referred to as a determnstc model. If any of the uncontrollable nputs are uncertan and subectve to varaton uncertantes can be categorzed as probablstc or stochastc uncertanty and fuzzness as ponted out by Zmmerman stochastc uncertanty can be modeled and solved by stochastc mathematcal programmng technques and problems wth fuzzness can be modeled and solved by fuzzy mathematcal programmng technques. For the uncontrollable nput analyss gve an eample concernng a mathematcal model. In the producton model the number of man-hours requred per unt of producton could vary from to 6 hours dependng upon the qualty of the raw materal the model would have been stochastc. Fuzzy set provdes a rudmentary mathematcal framework for dealng wth ncomplete uncertan nformaton. It has long been proposed by Zadeh[] as an etenson of the classcal theory of crsp sets. Bellman and Zadeh [] prmarly make known to the noton of fuzzy decson makng whch was etensvely developed later by researchers. Fuzzy lnear programmng wth fuzzy coeffcents has been formulated by Negota and Stan [4] and followed by Zmmermann [5] and Tanaka and sa [6]. Snce then work on Fuzzy lnear programmng grew contnuously ts applcaton. ke Wu [7] presented possblty and necessty measures fuzzy optmzaton problems based on the embeddng theorem. Xu and Volume 5 Issue 4 prl 06 Page 96

2 Zhou [8] dscussed possblty necessty and credblty measures for fuzzy optmzaton. Fgueroa[9] presents some defntons about of Interval Type- Fuzzy Constrants regardng Interval Type- Fuzzy near Programmng models whch can be solved by classcal algorthms. In [0] he shows the use of nterval optmzaton models to solve lnear programmng problems wth Interval Type- fuzzy constrants and the concept of -cut of an Interval Type- fuzzy set s used to fnd optmal solutons to uncertan optmzaton problems. In [] he presents a general model for near Programmng where ts technologcal coeffcents are assumed as Interval Type- fuzzy sets and t s solved through an -cuts approach. nd [] he shows a method for solvng lnear programmng problems that ncludes Interval Type- fuzzy constrants and he proposed method fnds an optmal soluton n these condtons usng conve optmzaton technques. Jndong Qn and Xnwang u [] nvestgates an approach to multple attrbute group decson-makng problems n whch the ndvdual assessments are n the form of trangle nterval type- fuzzy numbers. Wth our best knowledge however none of them ntroduced Type- fuzzy lnear programmng model based on possblty necessty and credblty measures on nterval type- fuzzy set for upper and lower membershp functons. The Possblty necessty and credblty measures have a sgnfcant role n fuzzy and fuzzy optmzaton. The possblty measure s much sutable for the optmstc decson maker. If the decson-maker s pessmstc he may use the necessty measure as a tool to make decson and credblty measure as the average of possblty measure and necessty measure. In ths paper The uncontrollable nput due to envronmental factors or nputs that cannot be specfed by decson maker uncertanty and mprecson nvolved n decson maker knowledge can be well addressed usng type- fuzzy set. We consder the both membershp functon of perfectly normal Interval type-fuzzy set are trapezodal membershp functon and normal and conve. The advantages of the Perfectly normal Interval Type- fuzzy sets n near programmng nstead of Type- fuzzy sets allows us to handle hgher uncertanty levels whch come from typcal scenaros where the problem s beng defned by decson makers and they are not n agreement of usng a sngle fuzzy set for representng ther perceptons about the problem. The rest of ths paper s organzed as follows. In Secton and we recall some prelmnary knowledge about fuzzy and ts arthmetc operaton. In secton 4 the representaton for PnITTrFN ts propertes and some arthmetc operatons of PnITTrFN based on type fuzzy number are presented. Secton 5 has provded possblty necessty and credblty measures of PnITTrFN.In Secton 6 we have proposed type- fuzzy lnear programmng models based on possblty necessty and credblty measures. The soluton methodology of the proposed models usng possblty necessty and credblty measures has been dscussed n Secton 7.In Secton 8a numercal eample s presented to valdate the proposed method. The numercal and graphcal results at dfferent possblty and necessty levels of the gven problems have also been dscussed here. Secton 9 summarzes the paper and also dscusses about the scope of future work.. PREIMINRIES In ths secton some basc defntons of fuzzy set theory are revewed [4]. Defnton. et X be a non-empty set. fuzzy set n X s characterzed by ts membershp functon : X 0 and s nterpreted as the degree of membershp of element n fuzzy set for each X. It s clear that s completely determned by the set of tuples X. The famly of all fuzzy (sub) sets n X s denoted by F( X ). Fuzzy subsets of the real lne are called fuzzy quanttes. Defnton. et be a fuzzy subset of X : the support of denoted Supp s the crsp subset of X whose elements all have non-zero membershp grades n. Supp X Defnton. fuzzy subset of a classcal set X s called normal f there ests an X such that. Otherwse s subnormal. Defnton.4 n level set (or cut 0. ) of a fuzzy set of X s a non-fuzzy set denoted by and defned by Volume 5 Issue 4 prl 06 Page 97

3 X f 0 cl Supp f 0 Where cl Supp denotes the closure of the support of. Defnton.5 fuzzy set of X s called conve f Defnton.6 s a conve subset of X for all 0 The complement of a fuzzy set s defned as c Defnton.7[5]. et B be fuzzy sets wth the membershp functon 0 0 B Pos * B sup mn y * y y B B Nec * B nf ma y * y y respectvely Pos and Nec represent possblty of membershp and necessty of membershp functon respectvely. * s any of the relatons ± and. The dual relatonshp of possblty and necessty gves Pos B Nec * B * where * B represents complement of the event * B. Defnton.8[6] et be a fuzzy set. Then the fuzzy measures of for membershp functon s Me Pos Nec where Me represent measures of membershp functons and 0 s the optmstc-pessmstc parameter to determne the combned atttude of a decson maker. If then Me Pos ; t means the decson maker s optmstc and mamum chance of holds. If 0 then Me Nec ; t means the decson maker s pessmstc and mamum chance of holds.\newlne If 0.5 then Me Cre ; where Cre s the credblty measure; t means the decson maker takes compromse atttude. Defnton.9[4][7] fuzzy number s a fuzzy set of the real lne wth a normal(fuzzy)conve and contnuous membershp functon of bounded support. lternatvely the fuzzy subset of satsfed:. s normal.e. there est an such that s called a fuzzy number f the followng condtons are ;. The membershp functon s quas-concave\\.e. for all 0 ;. The membershp functon for all 0 ; s upper sem contnuous.e. 4. The 0 level set 0 s compact (closed and bounded n ) mn : s a closed subset of Volume 5 Issue 4 prl 06 Page 98

4 Fgure Trapezodal Fuzzy Number We denote by F the set of all fuzzy numbers. If s a fuzzy number then from Zadeh[]. level set s a conve set from condton (). Combnng ths fact wth condton () the level set s a compact and conve set for all 0 ( snce 0 s bounded t says that 0 s also bounded for all (0] ). Therefore we can wrte a a. Defnton.0 The trapezodal fuzzy number s fully determned by quadruples a a of crsp numbers such that a a 0 0 whose membershp functon can be denoted by when a a / a a a a a / a a 0 otherwse a the trapezodal fuzzy number becomes a trangular fuzzy number. If the trapezodal fuzzy number becomes a symmetrcal trapezodal fuzzy number. a a s the core of and 0 0 are the lefthand and rght-hand spreads(see Fgure \ref{fg:fg}).it can easly be shown that a a a a a a The support of s a a. rthmetc operatons a In ths sub secton addton subtracton and scalar multplcaton operaton of trapezodal fuzzy numbers are revewed [4][8] a a B b b be two trapezodal fuzzy numbers then et and a a and B b b B a b a b B a b a b a a 0 a a 0. Volume 5 Issue 4 prl 06 Page 99

5 INTERV TYPE- FZZY SETS n Interval type- fuzzy set (ITFS) s a specal case of type- fuzzy set whch play an mportant role n management and engneerng applcatons. These fuzzy sets are characterzed by ther footprnts of uncertanty. Defnton.[9] u Type- fuzzy set n the unverse of dscourse X can be represented by a type- membershp functon as follows: where 0 u u X u J 0 0 u J s the prmary membershp functon at and uj u / u ndcates the second membershp at. For dscreet stuatons s replaced by. Defnton.[0] et be a type- fuzzy set n the unverse of dscourse X represented by a type- membershp functon u If all u. then s called an nterval type- fuzzy set. n nterval type- fuzzy set can be regarded as a specal case of the type- fuzzy set whch s defned as where s the prmary varable 0 / u s the secondary membershp functon at. uj / u / u / X uj X uj J s the prmary membershp of u s the secondary varable and It s obvous that the nterval type- fuzzy set defned on X s completely determned by the prmary membershp whch s called the footprnt of uncertanty and the footprnt of uncertanty can be epressed as follows: Defnton.[][] 0 FO J u u J X X et be an nterval type- fuzzy set uncertanty n the prmary membershp of a type- fuzzy set conssts of a bounded regon called the footprnt of uncertanty whch s the unon of all prmary membershps. Footprnt of uncertanty s characterzed by upper membershp functon and lower membershp functon. Both of the membershp functons are type- fuzzy sets. pper membershp functon s denoted by 0 and lower membershp functon s denoted by 0 respectvely. Defnton.4[] n nterval type- fuzzy number s called trapezodal nterval type- fuzzy number where the upper membershp functon and lower membershp functon are both trapezodal fuzzy numbers.e. a a a a ; H H a a a a ; H H 4 4 where H and respectvely. H denote membershp values of the correspondng elements a and a Volume 5 Issue 4 prl 06 Page 00

6 Defnton.5[] The upper membershp functon and lower membershp functon of an nterval type- fuzzy set are type- membershp functon respectvely. Defnton.6[4] nterval type- fuzzy set s sad to be perfectly normal f both ts upper and lower membershp functon are normal.e. sup sup. 4 PERFECTY NORM ITTRFN In ths secton the concepts of Perfectly normal nterval type- trapezodal fuzzy number (PnITTrFN) have been dscussed. It s the etenson work of Chao[5][6]. Defnton 4. PnITTrFN a a a a of crsp numbers such that a a a a 0 and 0. a a s the core of and 0 0 are the lefthand and rght-hand spreads and a a s the core of and 0 0 are the left-hand and rghthand spreads such that the membershp functon are as follows(see Fgure ).: a / a a a a a / a a 0 otherwse. a / a a a a a / a a 0 otherwse. Obvously If a a a a the perfectly normal nterval type- trapezodal fuzzy number reduce to the perfectly normal nterval type- trangular fuzzy number. If fuzzy number becomes a type- trapezodal fuzzy number[][7]. then the perfectly normal nterval type- trapezodal Fgure The upper Trapezodal membershp functon ITFS. Defnton 4. (Prmary cut of an PnITFS) The prmary cut u J u 0 of an PnITFS s and the lower trapezodal membershp functon whch s bounded by two regons of the Volume 5 Issue 4 prl 06 Page 0

7 0 and 0. Defnton 4. (Crsp bounds of PnITTrFN) The crsp bounds of the prmary cut of the PnITTrFN a a a a s closed nterval shall be obtaned as follows (0] thus are defned as the upper and lower membershp functons. The and are the lower and upper nterval valued bounds of. lso let the bounds of defned as the boundares of the cuts of each nterval type- fuzzy set as follows: nf sup al a u and be nf sup al a u R R nf u sup u al a l au a u a a a a whch s equvalent to say cb R R a a a a. cb Evdently from the Fgure a a a a R R. Defnton 4.4 fuzzy number 0 0. Defnton 4.5 Fgure Crsp bounds of PnITTrFN a a a a s sad to be non-negatve PnITTrFN f Volume 5 Issue 4 prl 06 Page 0

8 fuzzy number 0 0. Defnton 4.6 a a a a s sad to be non-postve PnITTrFN f fuzzy number a a a a s sad to be zero PnITTrFN f a b 0 a b 0 0 and 0. Defnton 4.7 PnITTrFNs and B B B b b b b are s sad to be dentcally equal B f and only f a b a b and. rthmetc Operatons on PnITTrFN Defnton 4.8 If and B are PnITTrFNs then C B s also a PnITTrFN and defned by C a b a b a b a b Defnton 4.9 If and B are PnITTrFNs then C B s also a PnITTrFN and defned by C a b a b a b a b Defnton 4.0 et R. If s also a PnITTrFN and s gven by a a a a ; f 0 C a a a a ; f 0 5. POSSIBIITY NECESSITY ND CREDIBIITY MESRES OF PNITTRFN Defnton 5. et and B be two PnITTrFNs wth lower and upper membershp functon and s the set of real numbers. Then the possblty degrees of lower and upper membershp functons are defned as follows: R R R ma a b 0 ma a b 0ma a b 0ma a b 0 a b Pos B ma ma 0 0 () R R R R a b a b a b a b b b a a Pos Defnton 5. R R R ma a b 0 ma a b 0 ma a b 0 ma a b 0 a b ma ma 0 0 R R R R a b a b a b a b b b a a B () The dual relatonshp of possblty and necessty gves Nec B Pos B () Nec B Pos B (4) et ba a type- fuzzy number. Then the fuzzy measures of for membershp functon s Me Pos Nec (5) Me Pos Nec (6) Volume 5 Issue 4 prl 06 Page 0

9 where Me and Me are represent measures of lower and upper membershp functons and 0 s the optmstc-pessmstc parameter to determne the combned atttude of a decson maker. If then Me Pos Me Pos ; t means the decson maker s optmstc and mamum chance of holds. If 0 then Me Nec Me Nec ; t means the decson maker s pessmstc and mamum chance of holds. If 0.5 then Me Cre Me Cre ; where Cre s the credblty measure; t means the decson maker takes compromse atttude. Defnton 5.[4][7][8] et and B be two PnITTrFNsFrom Defnton5. the possblty of lower and upper membershp functons are as follows: a b b a Pos B a b a b (7) 0 b a a b b a u Pos B a b a b (8) 0 b a The possblty of the event ± B for lower and upper membershp functon are as follows: a b a b Pos ± B a b a b 0 a b (9) a b a b Pos ± B a b a b 0 a b (0) Defnton 5.4 et and B be two PnITTrFNs From Defnton5. the necessty of lower and upper membershp functons are as follows: 0 b a b a Nec B Pos B b a a b () b a. Volume 5 Issue 4 prl 06 Page 04

10 0 b a b a Nec B Pos B b a a b () b a. The necessty of the event ± B for lower and upper membershp functon are as follows: 0 a b a b Nec ± B Pos B a b a b a b. () 0 a b a b Nec ± B Pos B a b a b a b (4) Defnton 5.5 et and B be two PnITTrFNs By Defnton 5. measures of the event upper and lower membershp functons are as follows: 0 b a Me B Pos B Nec B b a b a b a b a b a b a b a a b b a 0 b a Me B Pos B Nec B b a b a b a b a b a b a b a a b b a The measures of event ± B for lower and upper membershp functon are as follows: (5) (6) Volume 5 Issue 4 prl 06 Page 05

11 ± ± ± Me B Pos B Nec B b a a b b a b a b a b a a b a a a b Me B Pos B Nec B 0 b a ± ± ± b a a b b a b a (8) b a b a a b a a a b 0 b a For b a b a b a a b Cre B b a b a (9) a b b a a b b a (7) b a 0 b a b a a b Cre B b a b a (0) a b b a a b b a Volume 5 Issue 4 prl 06 Page 06

12 Cre b a a b b a a b ± B b a b a () a b b a a b 0 b a Cre b a a b b a a b ± B b a b a () a b b a a b 0 b a Theorem 5.6 et and B be two PnITTrFNs and Pos B p Proof : If p (0] Pos B p f and only f b a p. p then from Pos B f and only f b a p one can get that b a and vce versa. If 0 p then b a and b a Pos B b a p f and only f p that s b a p. Smlarly for Pos B p f and only f b a p. Theorem 5.7 et and B be two PnITTrFNs and Nec B p Proof: If b a p (0] Nec B p f and only f p then from Nec B and.e. b a p f and only f. b a p we can get that b a and vce versa. If 0 p then b a b Nec B p f and only f b a p Smlarly for. Nec B pf and only f b a p. Theorem 5.8 If and B be two PnITTrFNs a p (0] Cre B p f and only f b a p b a p. p Volume 5 Issue 4 prl 06 Page 07

13 Proof: et us consder Cre B p Now from equaton 9 b a b a Cre B p p p. Remark : a a Cre p p p. 6. TYPE- FZZY INER PROGRMMING MODES In ths secton we propose a type- fuzzy lnear programmng models based on Chance-Constraned Programmng Models(CCM)[9] wth Type- fuzzy parameter. We can use the chance operator (possblty or necessty or credblty measure) to transform the type- fuzzy model n to crsp lnear programmng model. general sngle obectve lnear programmng model wth Type- fuzzy parameter should have the followng form: Opt f c () Subect to g a b 0 where a b b b... b m mn T c c c... cn and... m;... n... T and a b c are an PnITFN n. s a Type- fuzzy partal order. Defnton 6. Consder a set of rght-hand-sde(resources) parameters of a Fuzzy lnear programmng problem defne as an PnITFS b defned on the closed nterval b R R b b b b and n. The membershp functon whch represents the fuzzy space Supp b s b / u / b n Jb 0. b u Jb Here b s bounded by both lower and upper prmary membershp functon namely b u b b wth parameter b & b R and b u b b wth parameter b b R and. Defnton 6. Consder a technologcal coeffcent of an fuzzy lnear programmng problem defne as an PnITFS closed nterval nf sup R R a a u a u a a a a a a a defned on the a a n m Volume 5 Issue 4 prl 06 Page 08

14 The membershp functon whch represents the fuzzy space Suppa s Here a / u / a n m J a 0. a uj a a s bounded by both lower and upper prmary membershp functon namely a u a a wth parameter a & R a and a u a a R wth parameter a & a. Defnton 6. Consder a proft or cost coeffcent of an fuzzy lnear programmng problem defne as an PnITFS c defned on the closed nterval R R c c c c c and Supp c s n. The membershp functon whch represents the fuzzy space c / u / c n J c 0. c uj c Here c s bounded by both lower and upper prmary membershp functon namely c u c c wth parameter c & c R and c u c c wth parameter c c R and. 6. Type- Fuzzy near programmng model wth PnITFN The general type- fuzzy lnear programmng models based on CCM wth PnITTrFN for model equaton () s as follows: Volume 5 Issue 4 prl 06 Page 09

15 Opt f f where Ch ± Subect to f c ± f p Ch f c f p Ch g a b p Ch g a b p 0.. m;... n. a a a a a a a b b b b b b b and (4) c c c c c c c are PnITTrFNs are the decson varable. The abbrevatons Ch and Ch represent chance operator (.e. possblty or necessty or credblty measure for) lower and upper membershp functons s predetermned confdence levels such that 0 p for.. m;... n. 6.. Type- fuzzy lnear programmng model based on possblty measure. The Type- fuzzy lnear programmng model wth PnITTrFN (4) based on CCM and possblty measure s as follows: Opt f f Subect to Pos f c ± f p Pos f c ± f p Pos g a b p Pos g a b p 0... m;... n. Where p s the predetermned confdence level such that 0 p for... m;... n. Defnton 6.4 * soluton Pos g a b p and (5) of the problem equaton (5) satsfes Pos g a b p... m;... n. s called a feasble soluton at p cut possblty level. Defnton 6.5 * feasble soluton at p cut possblty level s sad to be p cut effcent soluton for problem (5) f and only f there ests no other feasble soluton at p cut possblty level such that Pos f c f p and * * wth f f f Pos f c f p Volume 5 Issue 4 prl 06 Page 0

16 6.. Type- fuzzy lnear programmng model based on necessty measure. The Type- fuzzy lnear programmng model wth PnITTrFN (4) based on CCM and necessty measure s as follows: Opt f f Subect to Nec f c ± f p Nec f c ± f p Nec g a b p Nec g a b p 0... m;... n. where p s the predetermned confdence level such that 0 p for... m;... n. Defnton 6.6 * soluton Nec g a b p Nec g a b p of the problem equaton (6) satsfes and... m;... n. s called a feasble soluton at p cut necessty level Defnton 6.7 * feasble soluton at p cut necessty level s sad to be p cut effcent soluton for problem (6) f and only f there ests no other feasble soluton at p cut necessty level such that Nec f c f and * * Nec f c f wth f f f 6.. Type- fuzzy lnear programmng model based on credblty measure. The Type- fuzzy lnear programmng model wth PnITTrFN (4) based on CCM and credblty measure s as follows: Opt f f Subect to Cre f c ± f p Cre f c ± f p Cre g a b p Cre g a b p 0... m;... n. Where $ p $ s the predetermned confdence level such that 0 p for... m;... n. Defnton 6.8 * soluton Cre g a b p and (7) (6) of the problem equaton (7) satsfes Cre g a b p... m;... n. s called a feasble soluton at p cut credblty level. Defnton 6.9 * feasble soluton at p cut credblty level s sad to be p cut effcent soluton for problem (7) f and only f there ests no other feasble soluton at p cut credblty level such that Cre f c f p and * * wth f f f Cre f c f p Volume 5 Issue 4 prl 06 Page

17 7 PROPOSED METHOD TO SOVE TYPE- FZZY INER PROGRMMING MODES: To solve type fuzzy lnear programmng model based on possblty or necessty or credblty measures we propose the followng method. Step. pply chance operator possblty/necessty/credblty n type - fuzzy lnear programmng model () can be converted nto followng form. Opt f f Subect to Pos f c f p Pos f c f p or Nec f c f p Nec f c f p or Cre f c f p Cre f c f p Pos g a b p Pos g a b p or Nec g a b p Nec g a b p or Cre g a b p Cre g a b p 00 p (0) for =...m;=...n.where p s the predefned confdence level. Step. sng Theorems and/or Theorem 5.8 the above problem n Step can also be wrtten as Opt f f Subect to f f Z () (9) -(0) where Z s obtaned by applyng Theorems and/or Theorem 5.8 n (8) Step. The above model s equvalent to Opt Z 9 0 for... m;... n. Step 4. Crsp programmng model obtaned n step can be solved usng smple method to get the optmal soluton 8 NMERIC ISTRTION Subect to farmer has about " acres of cultvable land and he wanted to grow multple vegetable crops vz. Brnal ades fnger Btter guard and Tomato n a season. Out of hs eperence he stated that labour work tme avalable wth hm s 0 hours and avalablty of water s 5 acre-nches. The proft coeffcents (lakh rupees) requred work tme and water for each crop for one acre of land are provded n the Table-. How many acres he has to consder for each crop n order to get guaranteed net returns out of volatlty among proft coeffcents? (9) () (8) Volume 5 Issue 4 prl 06 Page

18 Table : Proft coeffcents labour requrement and water for entre duraton of crop Brna ades Btter Tomato l fnger guard Proft coeffcents (lakh rupees)(about) abour requrement per acre(' hours)(about) Water requrement per acre (acre-nch)(about) Here we llustrate soluton of the problem by the workng procedure provded n the secton-\ref{methodology}. et for 4. be the number of acres to be consdered for Brnal ades fnger Btter guard and Tomato respectvely and the undertaken problem s to solve. Ma Z c c c c 44 S. t a a a a 44 b a a a a 44 b a a a a b where\\ c ; c ; a ; a ; a a ; a ; a ; a a ; c ; a ; c ; a ; a ; a b b and () b TYPE- FZZY INER PROGRMMING PROBEM BSED ON CCM ND POSSIBIITY MESRE. Now by usng Step of the method eplaned n Secton 7 and theorem 5.6f we apply the possblty measure n type- fuzzy lnear programmng problem s converted nto the followng crsp programmng problems: Z MaZ MaZ MaZ ( p) ( p) ( p) ( p) S. t (.4 0. p) 4 4 (.5 0. p) ( p) (.6 0. p) (.6 0. p) p 4 p p p p 4 ( ) ( ) ( ) ( ) 80 0 p 0 4. (4) Volume 5 Issue 4 prl 06 Page

19 MaZ ( p) ( p) ( p) ( p) S. t ( p) 4 4 ( p) ( p) ( p) (. 0.5 p) p 4 p p p p 4 ( ) ( ) ( ) ( ) 0 50 p 0 4. Solvng the above crsp problem for effcent levels ( and ) and dfferent possblty levels we get dfferent optmal solutons. Optmal soluton of 4 and 5 at dfferent possblty levels are presented n Table (5) Table : Optmal soluton of 4 and 5 at dfferent possblty levels. Optmal soluton and Optmal Value Optmal soluton and Optmal Value Z Z = = = = = = Z TYPE- FZZY INER PROGRMMING PROBEM BSED ON CCM ND NECESSITY MESRE. Now by usng Step of the method eplaned n Secton 7 and theorem 5.7f we apply the necessty measure n type- fuzzy lnear programmng problem s converted nto the followng crsp programmng problems: Z MaZ MaZ MaZ (0.80) (0.60) () (0.7) 4 4 S. t ( p) (.8 0. p) (. 0. p) ( p) ( p) p 4 ( p) ( p) ( p) ( p) 60 0 p MaZ (0.80) (0.60) () (0.7) 4 S. t ( p) ( p) (. 0.5 p) ( p) ( p) p 4 ( p) ( p) ( p) ( p) p Table : Optmal soluton of 6 and 7 at dfferent necessty levels. (6) (7) Optmal soluton and Optmal Value Z = = = Optmal soluton and Optmal Value Z = Volume 5 Issue 4 prl 06 Page 4 = = Solvng the above crsp problem for effcent levels ( and ) and dfferent necessty levels we get dfferent optmal solutons. Optmal soluton of 6 and 7 at dfferent necessty levels are presented n Table. 9. Concluson In ths paper we have developed the possblty necessty and credblty measures on type- fuzzy set. We have also developed the theoretcal calculaton on possblty necessty and credblty measures for defuzzfy type- fuzzy lnear programmng model usng chance operators. To valdate the proposed method we have dscussed three dfferent Z

20 approaches to defuzzfy the type- fuzzy relatons usng possblty necessty and credblty measures. sng chance operator we can convert a problem under mprecse models to correspondng crsp models. t dfferent levels of possblty necessty and credblty we have acheved dfferent optmal soluton. References []. Davd Ray nderson and Thomas Wllams. n Introducton to Management Scence Quanttatve pproach to Decson Makng. West Publshng Company 979. []... Zadeh. Fuzzy sets. Informaton and Control 8():8 pp []. R. E. Bellman and.. Zadeh. Decson-makng n a fuzzy envronment. Management Scence 7(4): [4]. S. Mnou C.V. Negota and E. Stan. On consderng n precson n dynamc lnear programmng. The Centre of Economc Computaton and Economc Cybernetcs Bucharest Romana (): [5]. H.-J. Zmmermann. Fuzzy mathematcal programmng. Computers & Operatons Research 0(4): [6]. H. Tanaka and K. sa. Fuzzy lnear programmng problems wth fuzzy numbers. Fuzzy Sets and Systems (): [7]. Hsen-Chung Wu. Fuzzy optmzaton problems based on the embeddng theorem and Possblty and necessty measures. Mathematcal and Computer Modellng 40(4): [8]. Jupng Xu and Xaoyang Zhou. Fuzzy-ke Multple Obectve Decson Makng. 0. [9]. Juan Carlos Fgueroa Garca and German Hernandez. near programmng wth nterval type- fuzzy constrants. In Constrant Programmng and Decson Makng pages [0]. Juan Carlos Fgueroa Garca and German Hernandez. Solvng lnear programmng problems wth nterval type- fuzzy constrants usng nterval optmzaton. In Jont IFS World Congress and NFIPS nnual Meetng IFS/NFIPS 0 Edmonton lberta Canada June pages []. J.C. Fgueroa Garca. general model for lnear programmng wth nterval type- fuzzy technologcal coeffcents. In Fuzzy Informaton Processng Socety (NFIPS) 0 nnual Meetng of the North mercan pages -4 ug 0. []. Juan Carlos Fgueroa-Garca and German Hernandez. method for solvng lnear programmng models wth Interval Type- fuzzy constrants. Pesqusa Operatonal 4: []. Jndong Qn and Xnwang u. Frank aggregaton operators for trangular nterval type- fuzzy set and ts applcaton n multple attrbute group decson makng. Journal of ppled Mathematcs 04:4 04. [4]. Robert Fuller. Fuzzy Reasonng and Fuzzy Optmzaton. Turku Centre for Computer Scence 998. [5]. Jaroslav Ramk. Dualty n fuzzy lnear programmng wth possblty and necessty relatons. Fuzzy Sets and Systems 57(0): [6]. Dpak Kumar Jana Dpankar Chakraborty and Tapan Kumar Roy. new approach to solve ntutonstc fuzzy optmzaton problem usng possblty necessty and credblty measures. Internatonal Journal of Engneerng Mathematcs 04: 04. [7]. Hsen-Chung Wu. Dualty theory n fuzzy lnear programmng problems wth fuzzy coeffcents. Fuzzy Optmzaton and Decson Makng (): [8]. Honga and Zengta Gong. Fuzzy lnear programmng wth possblty and necessty relaton. In Bng-yuan Cao Guo-un Wang S-zong Guo and Shu-l Chen edtors Fuzzy Informaton and Engneerng 00 volume 78 of dvances n Intellgent and Soft Computng pages 05-. Sprnger Berln Hedelberg 00. [9]. Behrooz Safarneadan Parsa Ghane and Hossen Monrvaghef. Fault detecton n non-lnear systems based on type- fuzzy logc. Internatonal Journal of Systems Scence 46(): [0]. Zhmng Zhang and Shouhua Zhang. novel approach to mult attrbute group decson makng based on trapezodal nterval type- fuzzy soft sets. ppled Mathematcal Modellng 7(7): []. Yanbng Gong. Fuzzy mult-attrbute group decson makng method based on nterval type- fuzzy sets and applcatons to global suppler selecton. Internatonal Journal of Fuzzy Systems 5(4):9-400 December 0. []. Junhua Hu Yan Zhang Xaohong Chen and Yongme u. Mult-crtera decson makng method based on possblty degree of nterval type- fuzzy number. Knowledge- Based Systems 4:-9 0. []. Shy-Mng Chen and -We ee. Fuzzy multple attrbutes group decson-makng based on the rankng values and the arthmetc operatons of nterval type- fuzzy sets. Epert Systems wth pplcatons 7(): [4]. D.S. Dnagar and. nbalagan. new type- fuzzy number arthmetc usng etenson prncple. In dvances n Engneerng Scence and Management (ICESM) 0 Internatonal Conference on pages -8 March 0. [5]. Kuo-Png Chao. Multple crtera group decson makng wth trangular nterval type- fuzzy sets. In Fuzzy Systems (FZZ) 0 IEEE Internatonal Conference on pages June 0. [6]. Kuo-Png Chao. Trapezodal nterval type- fuzzy set etenson of analytc herarchy process. In Fuzzy Systems (FZZ-IEEE) 0 IEEE Internatonal Conference on pages -8 June 0. Volume 5 Issue 4 prl 06 Page 5

21 [7]. Xnwang u. Measurng the satsfacton of constrants n fuzzy lnear programmng. Fuzzy Sets and Systems (): [8]. Dder Dubos and Henr Prade. Rankng fuzzy numbers n the settng of possblty theory. Informaton Scences 0(): [9]. ng Yang. Fuzzy chance-constraned programmng wth lnear combnaton of possblty measure and necessty measure. ppled Mathematcal Scences (46): Volume 5 Issue 4 prl 06 Page 6