IJSRD - Iteratoal Joural for Scetfc Research & Develomet Vol. 2, Issue 07, 204 ISSN (ole): 232-063 Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves Poovarasa.V Joh Robso P 2,2 ssstat Professor,2 Deartmet of Mathematcs Sr Vdya Madr rts & Scece College, Uthagara 2 Bsho Heber College, Truchraall bstract The am of ths aer s to vestgate the multle attrbute decso mag roblems th tutostc fuzzy formato, hch the formato about attrbute eghts are comletely o, ad the attrbute values tae the form of tutostc fuzzy umbers. I order to get the eght vector of the attrbute, e establsh a otmzato model based o the basc deal of tradtoal gray relatoal aalyss (GR) method, by hch the attrbute eghts ca be determed. For the secal stuatos here the formato about attrbute eghts are comletely uo, e establsh aother otmzato model. By solvg ths model, e get a smle ad exact formula, hch ca be used to determe the attrbute eghts. The, based o the tradtoal GR method, calculato stes for solvg a terval-valued tutostc fuzzy evromet ad develoed modfed GR method for terval-valued tutostc fuzzy multle attrbutes decso-mag th comletely o attrbute eght formato. Ths aer rovdes a e method for sestvty aalyss of MDM roblems so that by sg t ad chagg the eghts of attrbutes, oe ca determe chages the fal results for a decso mag roblem. Fally, a llustratve examle s gve to verfy the develoed aroach ad to demostrate ts ractcalty ad effectveess. Keyords: MDM, ttrbute eght, IVIFs, Gray relatoal aalyss (GR) method, sestvty aalyss, Rag methods I. INTRODUCTION The am of ths aer s to exted the cocet of gray relatoal aalyss (GR) to develo a methodology for solvg MDM roblems th a terval valued tutostc fuzzy evromet ad develoed modfed GR method for terval-valued tutostc fuzzy multle attrbute decso-mag th comletely o attrbute eght formato. I order to do so, the rest of ths aer s orgazed as follos: ext secto brefly troduce some basc cocets related to tutostc fuzzy sets. I Secto 3, e have exteded the above results to a terval-valued tutostc fuzzy evromet ad develoed modfed GR method for terval-valued tutostc fuzzy multle attrbutes decso-mag th comletely o attrbute eght formato.. I Secto 4, e offer a e method for sestvty aalyss of MDM roblems so that by usg t ad chagg the eghts of attrbutes, oe ca determe chages the results of a decso mag roblem. I Secto 5 e llustrate our roosed algorthmc method th a examle. Thus, The fal secto cocludes. II. PRELIMINRIES I the follog, e troduce some basc cocets related to tutost fuzzy sets.. Defto Let X be a uverse of dscourse, the a fuzzy set s defed as { x, ( x) x X} () Whch s characterzed by a membersh fucto : X [0,], here () x deotes the degree of membersh of the elemet x to the set exteded the fuzzy set to the IFS, sho as follos B. Defto 2 IFS X s gve by { x, ( x), ( x) x X} (2) here : X [0,] ad : X [0,], th the codto 0 ( x) ( x), x X The umbers () x ad () x rereset, resectvely, the membersh degree ad o-membersh degree of the elemet x to the set. C. Defto 3 For each IFS X, f ( x) ( x) ( x), x X (3) The () x s called the degree of determacy of x to. III. GR METHOD FOR MULTIPLE TTRIBUTE DECISION MKING PROBLEMS WITH INTERVL-VLUED INTUITIONISTIC FUZZY INFORMTION Further troduced the terval-valued tutostc fuzzy set (IVIFS), hch s a geeralzato of the IFS. The fudametal characterstc of the IVIFS s that the values of ts membersh fucto ad o-membersh fucto are tervals rather tha exact umbers.. Defto 4 Let X s a uverse of dscourse. IVIFS over X s a obect havg the form { x, ( x), ( x) x X} (4) Where ( x) [0,] ad ( x) [0,] are terval umbers, ad 0 su( ( x)) su( ( x)), x X. For coveece, let ( x) [ a, b], ( x) [ c, d], so ([ a, b],[ c, d]), here [ a, b] [0,], [ c, d] [0,], bd ll rghts reserved by.srd.com 68
Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves B. Defto 5 Let a ([ a, b ],[ c, d]) ad a2 ([ a2, b2 ],[ c2, d2]) be to terval-valued tutostc fuzzy values, the the ormalzed Hammg dstace betee a ([ a, b ],[ c, d]) ad a2 ([ a2, b2 ],[ c2, d2]) s defed as follos: d( a, a2) a a2 b b2 c c2 d d2 (5) 4 m m Let, G, ad H be reseted as Secto 2. Suose that R ( r ) ([ a, b ],[ c, d ]) s the tervalvalued tutostc fuzzy decso matrx, here [ a, b ] dcates the degree that the alteratve satsfes the attrbute G gve by the decso maer, [ c, d ] dcates the degree that the alteratve does ot satsfy the attrbute G gve by the decso maer, [ a, b ] [0,],[ c, d ] [0,], b d,,2,..., m,,2,...,. I the follog, e aly GR method to solve terval-valued tutostc fuzzy MDM th comletely o eght formato. The method volves the follog stes: ) Ste Determe the ostve deal th terval-valued tutostc fuzzy formato. r ([ a, b ],[ c, d ],[ a, b ],[ c, d ],...,[ a, b ],[ c, d ]) (6) 2 2 2 2 Where r ([ a, b ],[ c, d ]) max a,max b, max c,max d,,2,..., 2) Ste 2 Calculate the gray relatoal coeffcet of each alteratve from PIS usg the follog equato: m m d( r, r ) max max d( r, r ) (7) m m d( r, r ) max m max d( r, r ),2,..., m,,2,..., here the detfcato coeffcet 0.5. 3) Ste 3 Calculatg the degree of gray relatoal coeffcet of each alteratve from PIS usg the follog equato:,,2,..., m The basc rcle of the GR method s that the chose alteratve should have the largest degree of gray relato from the ostve deal soluto. Obvously, for the eght vector gve, the larger, the better alteratve s. If the formato about attrbute eghts are comletely o, order to get the, frstly, e must calculate the eght formato. The gray relatoal coeffcet betee PIS ad tself are (,... ), so the comrehesve gray relatoal coeffcet devato sum s (8) ( ) ( ) d (9) So, e ca establsh the follog multle obectve otmzato models to calculate the eght formato: m d( ) ( ),,2,..., m (0) subect to : H Sce each alteratve s o-feror, so there exsts o referece relato o the all the alteratves. The, e may aggregate the above multle obectve otmzato models th equal eghts to the follog sgle obectve otmzato model: m d( ) d ( ) ( ) subect to: H m () By solvg the model (), e get the otmal soluto = (, 2,..., ), hch ca be used as the eght vector of attrbutes. The, e ca get (=,2,,m) by Eq. (8). If the formato about attrbute eghts are comletely uo, e ca establsh aother multle obectve rogrammg model as follos: 2 m d( ) ( ), subect to : (2) Smlarly, e may aggregate the above multle obectve otmzato models th equal eghts to the follog sgle obectve otmzato model: m m d( ) d( ) ( ) subect to : 2 (3) To solve ths model, e costruct the Lagrage fucto: 2 (, ) m ( ) 2 L (4) here s the Lagrage multler. Dfferetatg Eq. (4) th resect to ( =,2,...,) ad, ad settg these artal dervatves equal to zero, e get a smle ad exact formula for determg the attrbute eghts as follos: m m 2 2 ( ) ( ) (5) The, e ca get (=,2,,m) by Eq. (8). 4) Ste 4 Ra all the alteratves ( =, 2,...,m) ad select the bet oe(s) accordace th (=,2,,m). If ay ll rghts reserved by.srd.com 69
Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves alteratve has the hghest mortat alteratve. 5) Ste 5 Ed. value, the, t s the most IV. DEVELOPING NEW METHOD FOR SENSITIVITY NLYSIS OF MDM PROBLEMS Earler researches o the sestvty aalyss of MDM roblems ofte focused o determg the most sestve attrbute. They also focused o fdg the least value of the chage. Hoever, a e method for sestvty aalyss of MDM roblems s cosdered ths artcle that calculates the chagg the fal score of alteratves he a chage occurs the eght of oe attrbute. The effects of chage the eght of oe attrbute o the eght of other attrbutes are, The vector for eghts of attrbutes s t,,..., W 2 here eghts are ormalzed th a sum of, that s: () Wth these assumtos, f the eght of oe attrbute chages, the the eght of other attrbutes chage accordgly, ad the e vector of eghts trasformed to t W, 2,..., The ext theorem dects the chages the eght of attrbutes.. Theorem 4.. I the MDM model, f the eght of the P th attrbute, chages by, the the eght of other attrbutes chage by, here:. ;,2,...,, Proof If the e eght of the attrbute s e eght of the P th attrbute chages as: (6) ad the (7) The, the e eght of the other attrbutes ould chage as ;,2,...,, d because the sum of eghts must be the: (8) 0 (9) Therefore: (20) Where:. ;,2,...,, (2) Sce: (22). Ma result. I a MDM roblem, f the eght of the P th attrbute chages from to as: (23) The, the eght of other attrbutes ould chage as:.. (24) Sce, for,2,...,, e have:.... ; (26),2,...,, (25),2,...,, The, e vector for eghts of attrbutes ould be t W, 2,...,, that s:.,,2,..., (27) f f,2,...,, (28) The sum of e eghts of attrbutes that are obtaed (23) s,because:. (29) ll rghts reserved by.srd.com 70
Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves V. NUMERICL ILLUSTRTION Let us suose there s a vestmet comay, hch ats to vest a sum of moey the best oto. There s a ael th fve ossble alteratves to vest the moey: s a car comay; 2 s a food comay; 3 s a comuter comay; 4 s a arms comay; 5 s a TV comay. The vestmet comay must tae a decso accordg to the follog four attrbutes: G s the rs aalyss; G 2 s the groth aalyss; G 3 s the socal-oltcal mact aalyss; G 4 s the evrometal mact aalyss. The fve ossble alteratves ( =,2,3,4,5) are to be evaluated usg the terval-valued tutostc fuzzy formato by the decso maer uder the above four attrbutes, as lsted the follog matrx. ([0.4,0.5],[0.3,0.4]) ([0.4,0.6],[0.2,0.4]) ([0.5,0.6],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) R ([0.3,0.5],[0.3,0.4]) ([0.,0.3],[0.5,0.6]) ([0.2,0.5],[0.3,0.4]) ([0.4,0.7],[0.,0.2]) ([0.3,0.4],[0.,0.3]) ([0.7,0. 8],[0.,0.2]) ([0.3,0.4],[0.4,0.5]) ([0.5,0.6],[0.,0.3]) ([0.5,0.6],[0.3,0.4]) ([0.4,0.7],[0.,0.2]) ([0.2,0.5],[0.4,0.5]) ([0.2,0.3],[0.4,0.6]) ([0.4,0.5],[0.3,0.5]) ([0.5,0.8],[0.,0.2]) ([0.5,0.6], [0.2,0.4]) ([0.6,0.7],[0.,0.2]) I such case, e ca utlze the roosed rocedure II to get the most desrable alteratve (s).. Case The formato about the attrbute eghts are artly o ad the o eght formato s gve as follos: H 0.23 0.26, 0.5 0.8, 0.30 2 0.35, 0.25 0.28, 0, 3 4 4,2,3,4, Ste. Determe the ostve deal r [([0.5,0.6],[0.,0.3]),([0.7,0.8],[0.,0.2]), ([0.5,0.6],[0.2,0.4]),([0.6,0.8],[0.,0.2])] Ste 2. Calculate the gray relatoal coeffcet of each alteratve from PIS 0.25 0.2 0.75 0. 0.025 0. 0.025 0.075 d( r, r ) 0.5 0.475 0.75 0.4 0.75 0. 0. 0.025 0. 0 0 0.025 0.556 0.5429 0.3636 0.75.0000 0.7037 0.7778 0.882 5 4 0.6774 0.3333 0.5758 0.48 0.4286 0.7037 0.4667.0000 0.4667.0000.0000 0.8889 Ste 3. Utlze the model () to establsh the follog sgle obectve rogrammg model: m ( ).877.764.86.3 Subect to : H 2 3 4 Solve ths model; e get the eght vector of attrbutes: = (0.23, 0.5, 0.35, 0.27) The, e ca get the degree of gray relatoal coeffcet of each alteratve from PIS 0.5389, 0.8287, 0.585, 2 3 0.6375, 0.8473, 4 5 Ste 4. ccordg to the relatve relatoal degree, the rag order of the fve alteratves are: 5 2 4, ad thus the most desrable 3 alteratve s 5. Case 2: If the formato about the attrbute eghts are comletely chaged, e utlze the e method for Sestvty alyss develoed to get the most desrable alteratve(s). No e assume that the eght of the 2 d attrbute creased by 0.500 ad be 2 2 2 0.500 0.500 0.3000. The by equato (24), the eght of other attrbutes chage as (30):. ;,2,3 4 4 T (0.894, 0.3000, 0.2882, 0.2224). (30) The, e ca get the degree of gray relatoal coeffcet of each alteratve from PIS 0.5397, 0.8066, 0.4858, 2 3 0.6492, 0.8743, 4 5 ccordg to the relatve relatoal degree, the rag order of the fve alteratves are: 5 2 4, ad 3 thus the most desrable alteratve s also 5. From the roceedg fgures, t ca be observed that eve though there are much varato the attrbute eghts comuted from the to dfferet models, there are less varato observed the coeffcets calculated from the to models, hece resultg same rag of the alteratves from the to models. Hece the roosed MDM model th coeffcet of IVIFs s a effectve model because of ts ucomromsg rag of the alteratves eve he the attrbute eghts are artally comletely uo ad comletely chaged. ll rghts reserved by.srd.com 7
Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves Model- Model-2 0.4 0.3 0.2 0. 0 2 3 4 Fg. : Comarso of Weghts W W2 W3 W4 Model- 0.2300 0.500 0.3500 0.2700 Model-2 0.894 0.3000 0.2882 0.2224 Fg. 2: Comarso of Coeffcet 2 3 4 5 0.5389 0.8287 0.585 0.6375 0.8473 0.5397 0.8066 0.4858 0.6492 0.8743 VI. CONCLUSION Decso mag s the tegral art of huma lfe. Regardless of the varety of decso mag roblems, e ca categorze them to to categores, mult obectve decso mag roblems that decso maer must desg a aroach that has the most utlty by cosderg lmted resources ad mult-attrbute decso mag roblems that decso maer must select oe alteratve from amog avalable alteratves so that has the most utlty. Naturally, for selectg a alteratve e must cosder several ad ofte coflctg attrbutes. I ths aer, e have vestgated the roblem of calculatg IVIF method for acqurg best alteratve of attrbute eght. Whe the attrbute eghts of alteratves are chaged through Sestve aalyss method ad he t has bee calculated the result hch e acqured the above method that s IVIF method ad sestve aalyss both rereseted the same desrable alteratve. Fally, a llustratve examle s gve to verfy the develoed aroach ad to demostrate ts ractcalty ad effectveess. REFERENCES [] taassov, K. Itutostc fuzzy sets. Fuzzy Sets ad Systems, 20, (986), 87 96. [2] taassov, K. More o tutostc fuzzy sets. Fuzzy Sets ad Systems, 33, (989), 37 46. [3] taassov, K. Oerators over terval-valued tutostc fuzzy sets. Fuzzy Sets ad Systems, 64(2), (994), 59 74. [4] taassov, K. To theorems for tutostc fuzzy sets. Fuzzy Sets ad Systems, 0, (2000), 267 269. [5] taassov, K., & Gargov, G. Iterval-valued tutostc fuzzy sets. Fuzzy Sets ad Systems, 3, (989), 343 349. [6] Bora, F. E., Geç, S., Kurt, M., & ay, D. multcrtera tutostc fuzzy grou decso mag for suler selecto th TOPSIS method. Exert Systems th lcatos, 36(8), (2009), 363 368. [7] Bustce, H. Costructo of tutostc fuzzy relatos th redetermed roertes. Fuzzy Sets ad Systems, 09, (2000), 379 403. [8] Bustce, H., & Burllo, P. Correlato of tervalvalued tutostc fuzzy sets. Fuzzy Sets ad Systems, 74, (995), 237 244. [9] Che, S. M., & Ta, J. M. Hadlg mult crtera fuzzy decso-mag roblems based o vague set theory. Fuzzy Sets Systems, 67, (994), 63 72. [0] Deg, J. L. Itroducto to grey system. The Joural of Grey System (UK), (), (989), 24. [] Hog, D. H. ote o correlato of terval-valued tutostc fuzzy sets. Fuzzy Sets ad Systems, 95, (998), 3 7. [2] Hog, D. H., & Cho, C. H. Mult crtera fuzzy decso-mag roblems based o vague set theory. Fuzzy Sets Systems, 4, (2000), 03 3. [3] Km, S. H., & h, B. S. Iteractve grou decso mag rocedure uder comlete formato. Euroea Joural of Oeratoal Research, 6, (999), 498 507. [4] Km, S. H., Cho, S. H., & Km, J. K. teractve rocedure for multle attrbute grou decso mag th comlete formato: Rage based aroach. Euroea Joural of Oeratoal Research, 8, (999), 39 52. [5] L, L., Yua, X. H., & Xa, Z. Q. Mult crtera fuzzy decso-mag methods based o tutostc fuzzy sets. Joural of Comuter ad System Sceces, 73, (2007), 84 88. [6] Lu, H. W. Mult-crtera decso-mag methods based o tutostc fuzzy sets. Euroea Joural of Oeratoal Research, 79(), (2007), 220 233. [7] L, D. F., Wag, Y. C., Lu, S., & Sha, F. Fractoal rogrammg methodology for mult-attrbute grou decso mag usg IFS. led Soft Comutg Joural, 9, (2009), 29 225. [8] L, G. D., Yamaguch, D., & Naga, M. grey-based decso-mag aroach to the suler selecto ll rghts reserved by.srd.com 72
Sestvty alyss of GR Method for Iterval Valued Itutost Fuzzy MDM: The Results of Chage the Weght of Oe ttrbute o the Fal Rag of lteratves roblem. Mathematcal ad Comuter Modelg, 46(3 4), (2007), 573 58. [9] G.W. We, method for multle attrbute grou decso mag based o the ET-WG ad ET- OWG oerators th 2-tule lgustc formato, Exert Systems th lcatos, vol. 37, o. 2,. (200), 7895-7900. [20] Gu-Wu-We, Gray relatoal aalyss method for tutostc fuzzy multle attrbute decso mag. Exert Systems th lcatos 38 (20) 67-677. [2] Barro.H, Schmdt.C. P, Sestvty alyss of addtve mult attrbutes values models. Oeratos Research, 46, (2002), 22-27. [22] Eshlaghy., Paydar. N. R, Joda. K, Paydar. N.R, Sestvty aalyss for crtera values decso mag matrx of SW method. Iteratoal Joural of Idustral Mathematcs,, (2009), 69-75.x ll rghts reserved by.srd.com 73