Generalized Grey Target Decision Method Based on Decision Makers Indifference Attribute Value Preferences

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1 Ameican Jounal of ata ining and Knowledge iscovey 27; 2(4): doi:.648/.admkd Genealized Gey Taget ecision ethod Based on ecision akes Indiffeence Attibute Value Pefeences a Jinshan School of Enegy Science and Engineeing, Henan Polytechnic Univesity, Jiaozuo, China addess: mscumttf@63.com To cite this aticle: a Jinshan. Genealized Gey Taget ecision ethod Based on ecision akes Indiffeence Attibute Value Pefeences. Ameican Jounal of ata ining and Knowledge iscovey. Vol. 2, o. 4, 27, pp doi:.648/.admkd Received: Octobe 8, 27; Accepted: Octobe 3, 27; Published: ecembe 5, 27 Abstact: In multi-attibute gey taget decision making, decision makes may have indiffeence pefeences towads some attibute values such that some supeio index values ae no diffeent within a small scope o some infeio index values ae indiffeence within a small ange. So the taget cente domain consisting of some supeio values and the taget edge domain compising some infeio values unde some attibute wee poposed based on the gey taget decision theoy. Based on the two domains, the Hamming distance of each index value to its taget cente domain can be calculated. Following this, the oiginal Hamming distances can be nomalized in a linea method individually. Then the decision can be made by the integated taget cente distances consideing each attibute s weight. A case study indicated that the genealized gey taget decision method impoved easily and combined with othe theoies can addess the decision makes indiffeence attibute value pefeences with its concise and simple technique compaed with the conventional gey taget method, which is supeio in handling many feasible altenatives with little diffeence of supeio values o infeio values. Keywods: Genealized Gey Taget ecision ethod, ecision akes, Indiffeence Attibute Value Pefeences, Taget Cente omain, Taget Edge omain. Intoduction The gey taget decision method poposed by eng has been used in a wide ange of fields []. ow, some scholas have made contibutions to the gey taget decision method. The incontinency poblem of eng s gey tansfomation was tested by Chen and Xie using the simulated method [2]. Song et al. impoved the calculation opeatos of gey taget [3-5]. Zhu et al. studied the weight detemination [6-8]. And the gey taget decision method fo mixed attibutes was studied by Luo et al. [9-4]. a and Ji poposed a genealized gey taget decision method [4, 5]. Besides, some othe theoies and methods wee also combined with the gey taget decision method [6-9]. These eseaches advanced the gey taget decision method. Howeve, the taget cente whethe expessed as a set fo multiple indices o singula index discussed in above eseach is a point which may be deteminacy fo indices of eal numbes o uncetainty fo those of fuzzy numbes. Futhemoe, the conventional taget cente epesents the best selection of all altenatives, which is to diffeentiate all indices absolutely. In pactice, decision makes may have indiffeence attibute pefeences towads supeio values o infeio values. Some supeio index values ae egaded as no diffeence when they ae in a small scope. Similaly, some infeio index values ae also thought as indiffeence while they ae in a small ange. But this issue has not been noticed by othe scholas. Thus the genealized gey taget decision method is poposed to cope with decision makes indiffeence attibute value pefeences. The conventional taget cente index is expanded to a domain that thee is no diffeence in it fo some supeio values with the poposed appoach. Similaly, the taget edge index can also be expanded to a domain that thee is indiffeence in it fo some infeio values. The poposed appoach, substituting the taget cente domain and the taget edge domain fo the taget cente and the taget edge espectively can in advance detemine some indices contibuting little to decision making. Futhemoe, it has the potential function of ewading the good and punishing the bad towads some indices. The poposed appoach need not nomalize the indices befoehand, and only nomalize the singula index taget cente distance fo compaison when aggegating integated taget cente distances [5], as can eflect the accuacy distances of all

2 3 a Jinshan: Genealized Gey Taget ecision ethod Based on ecision akes Indiffeence Attibute Value Pefeences indices to taget cente indices. The genealized geat taget decision method can not only deal with decision makes indiffeence attibute value pefeences, simplify the calculation but also be easily impoved combined with othe theoies. The emainde of this pape is oganized as follows: Section 2 discusses the poposed method, Section 3 studies the impacts of the two domains on the index values of all altenatives unde some attibute, Section 4 pesents a case study, and Section 5 concludes this pape. A each o wose than cetain value ae egaded as infeio values with indiffeence, as is called indiffeence infeio attibute value pefeence. P efinition 5 Given the best value C unde attibute A, then the smalle scope of the value is called taget cente domain. Given the wost value C unde attibute A, then the smalle scope of the value is called taget edge domain. The taget cente domain and the taget edge domain can be pesented in Figue. 2. Genealized Gey Taget ecision odel 2.. Peliminaies efinition Let S ( S,S 2,,Sn ) ( ) = be an altenative set, A = A, A2,, Am be an attibute set, S ( i =,2,, n, =,2,, m) be the measue of altenative i S i unde attibute A, J +, J and J be benefit type attibute set, cost type attibute set and modeate type attibute set espectively[5], as ae the bases of multi-attibute decision making. P P P P efinition 2 Let C ( C, C2, C m ) = be a taget cente index set detemined by the altenative measue p S ( i =,2,, n, =,2,, m), and C satisfies i { i} { } + max S, Si P C = min Si, Si, i =,2,, n, =,2,, m (), Si whee is the standad o desiable value of modeate type index value, which can be given in advance [5]. efinition 3 Let C ( C, C2,Cm ) = be a taget edge index set detemined by the altenative measue S ( i =,2,, n, =,2,, m), and C satisfies i C + { Si} Si { Si} Si ( ) min, max, = { Si max Si, Si }, Si { Si max ( Si ), Si }, Si i =,2,, n, =,2,, m efinition 4 Some index values unde attibute A each o bette than cetain value ae thought as supeio values without diffeence, as is called indiffeence supeio attibute value pefeence. Similaly, some index values unde attibute (2) Figue. Taget cente domain and taget edge domain. In Figue, C,, R and R max ae the taget cente, the adius of the taget cente domain, the inne adius of the taget edge domain, and the taget edge The taget cente and the taget edge ae expanded to two domains, as ae shown by the shaded sections. While diand i ae the distances of Si to its taget cente and taget cente domain 2.2. Taget Cente omain and Taget Edge omain etemination Coefficient ethod () Taget cente domain detemination The taget cente domain detemined by coefficient is obtained with the best index value unde attibute A multiply the coefficientγ, whee γ (,.]. Because of the small scope of taget cente domain, it is bette fo γ no moe than., which is decided by the decision makes. Howeve, all the taget cente domains detemination fo all attibutes can employ the same coefficient o diffeent coefficients. The following discussion limits to singula index taget cente domain. E + =[a,b ]=[ x i (-γ),x i ] (3) E - =[c,d ]= [ x i,x i (+γ)] (4) E =[e,f ] =[ (-γ), (+γ)] (5)

3 Ameican Jounal of ata ining and Knowledge iscovey 27; 2(4): E + is the taget cente domain of benefit type index; a and b ae the lowe limits and uppe limits of the taget cente domain of benefit type index espectively; E- is the taget cente domain of cost type index; c and d ae the lowe limits and uppe limits of the taget cente domain of cost type index espectively; E is the taget cente domain of the modeate type index; e and f ae the lowe limits and uppe limits of the taget cente domain of modeate type index espectively; x i is the best value of all indices; is the desiable o standad value of modeate type index. (2) Taget edge domain detemination Let γ (,.] be the coefficient of the taget edge domain, then thee types of taget edge domains can be detemined as follows. E +=[a,b ]= [ x i,x i(+γ )] (6) E -=[c,d ]= [ x i(-γ ),x i] (7) E =[e d,f d] [e u,f u] =[x md,x md(+γ )] [x mu(-γ ),x mu] (8) E + is the taget edge domain of benefit type index; a and b ae the lowe limits and uppe limits of the taget edge domain of benefit type index espectively; E - is the taget edge domain of cost type index; c and d ae the lowe limits and uppe limits of the taget edge domain of cost type index espectively; E is the taget edge domain of the modeate type index; e d and f d ae the lowe limits and uppe limits of the downside of the taget edge domain of modeate type index espectively; e u and f u ae the lowe limits and uppe limits of the upside of the taget edge domain of modeate type index espectively; x i is the wost value of all indices; x md and x mu ae the wost values of the downside and the upside of the modeate value Adacent Value ethod () Taget cente domain detemination The taget cente domain can be detemined accoding to the best value and the second best value unde attibute A with the following equations. E + =[a,b ]=[ x i, x i ] (9) E - =[c,d ]= [ x i,x i ] () E =[e,f ]= [x md,x mu ] () x i is the second best value of all indices; x md and x mu ae the second best values of the downside and the upside of the modeate value (2) Taget edge domain detemination The taget edge domain can be detemined by the wost value and the second wost value unde attibute A use the following equations. E +=[a,b ]=[ x i, x i] (2) E -=[c,d ]= [x i, x i] (3) E =[e d,f d] [e u,f u] =[x md,x md] [x mu,x mu] (4) x i is the second wost value of all indices; x md and x mu ae the second wost values of the downside and the upside of the modeate value Compehensive ethod Coefficient method and adacent method can be combined with each othe to detemine the taget cente domain o the taget edge domain by the decision makes intention. Futhemoe, the two methods can also be used individually to detemine the two domains. With espect to the two domains, decision makes can only detemine the taget cente domain o conside both of them fo special puposes Singula Index Taget Cente istance etemination Based on the theoy of gey taget decision, howeve the method of pocedue and technique is diffeent fom the classical one is efeed to as a genealized gey taget method. Compaed with the conventional model, the genealized gey taget method has two diffeences: no need to nomalize the index values S ( i =,2,, n, =,2,, m) and the i diffeence of taget cente distance calculation [4, 5]. iffeent fom the conventional gey taget decision method, the poposed appoach does not nomalize index values befoehand etemine Taget Cente istance Consideing Only Taget Cente omain Assume x is an index value unde attibute A, and then the distance of x to its taget cente distance can be obtained. () Taget cente distance fo benefit type index, x [ a, b] + = a x, x [ a, b] (5) whee E + =[a,b ] is a taget cente domain fo benefit type index, + is the distance of x to E +. (2) Taget cente distance fo cost type index, x [ c, d] = x d, x [ c, d] (6) whee E - =[c, d ] is a taget cente domain fo cost type index, is the distance of x to E -. (3) Taget cente distance fo the modeate type index

4 5 a Jinshan: Genealized Gey Taget ecision ethod Based on ecision akes Indiffeence Attibute Value Pefeences, x [ e, f] = e x, x < e x f, x > f (7) whee E =[e,f ] is a taget cente domain fo modeate type index, is the distance of x to E etemine Taget Cente istance Consideing Taget Edge omain The taget cente distance discussed above does not conside the taget edge domain. Consideing both the two domains, the taget cente distances of indices outside the taget edge domain ae still calculated by the above equations, while the taget cente distances of indices in the taget edge domain can be unified with the distance of the wost index value to the taget cente domain. Equations (8) to (2) can be used to solve the poblem. +max =a -x min (8) whee +max is the distance of the wost index value of benefit type to its taget cente domain, x min is the wost value. -max =x max -d (9) whee -max is the distance of the wost index value of cost type to its taget cente domain, x max is the wost value. maxd =e -x min o maxu =x max -f (2) whee maxd and maxu ae the downside and upside of the distances of the wost index values of modeate type to its taget cente domain espectively, x min and x max ae the wost value of the downside and the upside of the modeate value etemine Taget Cente istance Without Consideing Two omains The taget cente distance discussed above consides only the taget cente domain o both the two domains. Howeve, Equation (2) can be used to calculate the taget cente distance without consideing two domains. = x x (2) whee is the taget cente distance without consideing two domains, x and x ae the taget cente index and an index value unde attibute A 2.4. Taget Cente istance omalization Evey oiginal singula index taget cente distance can be nomalized using (22). i z =, = m i n i i= (22) whee i is the distance of S i to its taget cente domain unde attibute A, z i is the nomalized taget cente distance Weights etemination Thee ae thee types of weight detemination of attibutes: obective weights, subective weights and compehensive weights. Since weight detemination has been advanced by many scholas, this pape does not epeat it, and the inteested eades can see the liteatue [6-8] ecision aking The integated taget cente distances fo all altenatives can be calculated using (23). m wi = ω zi, i = n (23) = Thus, the decision can be made by the value w i, the smalle value of it, the bette of the altenative Steps of the Genealized Gey Taget ecision ethod Step etemine evey attibute s taget cente index and taget edge index. Use Equations () and (2) to calculate the taget cente indices and the taget edge indices. Step 2 etemine evey attibute s taget cente domain o meanwhile detemine the taget edge domain. Use Equations (3) to (5) o (9) to () to detemine the taget cente domain. Use Equations (6) to (8) o (2) to (4) to detemine the taget edge domains. Step 3 Calculate the Hamming distance of evey index to its taget cente domain use Equations (5) to (2). Without the two domains, Equation (2) can be used to calculate the taget cente distance. Step 4 omalize evey index s taget cente distance using (22). Step 5 etemine the weights of all attibutes. Step 6 Aggegate evey nomalized taget cente distance unde all attibutes using (23), then the altenative anking can be made accoding to the integated taget cente distances with the smalle value the bette. 3. The Impacts of Two omains on Altenatives Genealized gey taget decision method consideing eithe of the taget cente domain and the taget edge domain has the function of ewading good and punishing bad towads some indices of the altenatives, as is simila to the pevious study in the aticle [5]. In [5], the singula index taget cente detemined by the selection pefeences can educe the diffeence of supeio indices and enlage the diffeence of othe indices; howeve, the singula index taget cente detemined by the desiable pefeences can educe the diffeence of the indices. Hee, the impacts of both the two domains on the indices ae discussed.

5 Ameican Jounal of ata ining and Knowledge iscovey 27; 2(4): Assume Si( i =,2,, n, =,2,, m), S, and S e ae the index value, the taget cente value and the taget edge value unde attibute A Let S ( i + ) and S be any two indices, and d ( i + ), the taget cente Also, let the distances of ( i ) i d ae thei distances to i domain Suppose that ( i + ) ( i + ) and be S to thei taget cente d ( i + ) > d and >, the esults will not change. The discussion only limits to unde attibute A, thus the subscipt fo distances will be omitted. In Figue, the meaning of the paametes C,, and R max ae as above. Let be the nomalized diffeence of S ( i + ) and S to thei taget cente. So Equation (22) can be obtained i without consideing the two domains. d d d d = = i + i i + i n n n d d d i i i i= i= i= d i is the distance of S i to its taget cente. (24) Suppose that thee ae p index values in the taget cente domain, q index values in the taget edge domain, while othe (n-p-q) index values outside the two domains, if the two domains ae both consideed. Thus Equation (25) can be obtained. = = i + p q n pq + + i k i= = k= i p q n pq + + i k i= = k= i + i p q n pq + + i k i= = k= (25) Seen fom Figue, the distance of index value to its taget cente domain will be shote than that to its oiginal taget cente, thus Equation (26) can be obtained. = d, = d (26) i + i + i i The following discussions only concen thee conditions: both the two indices in the taget cente domain, both of them in the taget edge domain and none of them in the two domains. () If ( i ) S i ae both included in the taget cente domain, then we can achieve Equation (27) using Equation (5). i + = i = (27) So Equation (28) can be obtained with the compaison of Equations (24) and (25). > = (28) Equation (28) means that the two index taget distances become smalle when consideing taget cente domain, namely both of them ae supeio index values. (2) If ( i ) S fall within the taget edge domain, i then Equation (29) can be obtained using (8). i + = = R (29) i Compae with Equations (24) and (25), the following equation can be obtained: > = (3) Equation (3) indicates that the diffeence of them becomes smalle such that both of them ae infeio index values consideing taget edge domain. (3) If ( i ) S fall outside the two domains, then i Equation (3) can be obtained by Equations (8), (26) and (25). = p i= = = d d i + i p q n pq + + ( d ) i k i= = k= i + i n pq i + i n pq q( R ) + d ( n q p) max k k = d qr + d ( n p) d d d max k k = q (3) i =, = q( Rmax ), as calculated by = Equations (4) and (5). Compae with (24) and (3), thei numeatos ae equal. So only the denominatos ae consideed to detemine the elationship of and. Howeve the elationship of n pq max k k = n di and i= ( qr + d ( n p) ) is unclea, as they ae detemined by d i, R max, R,, n, p and q. Theefoe, the elationship of and is uncetain. The pupose of this wok is to seek fo the impacts of the two domains on indices of all altenatives unde some attibute.

6 7 a Jinshan: Genealized Gey Taget ecision ethod Based on ecision akes Indiffeence Attibute Value Pefeences Fom the discussion above, we can daw the conclusion that the supeio index values and infeio index values may contibute little to make a decision, and the othe index values inteacted with each othe may act as the main oles to detemine the esults. 4. Case Study 4.. Backgound To evaluate coal mines safety pefomance, conside eight indices, including seam dip ( ), methane emission ate (m 3 /t), wate inflow (m 3 /h), spontaneous combustion peiod (month), ventilating stuctues qualified ate (%), equivalent oifice (m 2 ), motality pe million tons (peson/ 6 t), and accident economic loss ( 5 Yuan) [5, 2] denoted by A to A, and the altenatives ae denoted by S to S. The data is shown in Table, the benefit type attibutes ae A 4 to A 6, and the othes ae cost type attibutes. ecision makes have indiffeence supeio value pefeences towads all attibutes, while have indiffeence infeio value pefeences towads attibutes A 2 and A 8. The taget cente domain of attibute A 5 is detemined by coefficient method with the value.5, and the othes ae detemined by the adacent value method. And the taget edge domains of attibutes A 2 and A 8 ae detemined by the adacent value method. Table. Safety data fo coal mines. S i A A 2 A 3 A 4 A 5 A 6 A 7 A 8 S S S S S S S S S S Pocess to ecision aking () Calculate taget cente indices P The taget cente indices ae C =(, 3.7, 2, 2,, 3.6,, 3) calculated by Equation (), and the taget edge indices ae C =(4, 4, 6, 6, 86,.2,.34, 5.6) calculated by Equation (2). (2) etemine the taget cente domain and the taget edge domain Use Equations (3), (9) and (), taget cente domains can be obtained as E P =([, 2], [3.7, 4], [2, 3], [, 2], [95, ], [2.7, 3.6], [,.8], [3, 38]). And use Equations (2) and (3), the taget edge domains of attibutes A 2 and A 8 ae [2.8, 4] and [964.7, 5.6] (3) Calculate oiginal index taget cente distances Use Equations (5), 6), (8) and (9), all index taget cente distances can be calculated as shown in Table 2. Table 2. All index taget cente distances. i A A 2 A 3 A 4 A 5 A 6 A 7 A * * 8 28 * * ote: the value with the mak * means the value is obtained consideing the taget edge domain. (4) omalize the oiginal taget cente distances Use Equation (22), the nomalized taget cente distances can be obtained shown in Table 3. Table 3. Evey index taget cente distance. Z i A A 2 A 3 A 4 A 5 A 6 A 7 A 8 Z Z Z Z Z Z Z Z Z Z (5) ecision making If the attibute weights given by the expets ae ω=(.6,.5,.3,.8,.2,.3,.27,.6), then all the integated taget cente distances can be calculated as w =(.4336,.2397,.49643,.44664,.7823,.3353,.95989,.75255,.373,.767)using (23). So the altenative anking is S 6 S S 4 S S 5 S 2 S 9 S 3 S 8 S 7. (6) iscussion Table 4 is pesented to show the compaison between the two methods of consideing taget cente domain and taget edge domain o not. If the weights given by the expets ae ω =(.6,.5,.3,.8,.2,.3,.27,.6), then use Equations fom (2) to (23), all the integated taget cente distances without consideing two domains ae w =(.46647,.9429,.5459,.49688,.8932,.9647,

7 Ameican Jounal of ata ining and Knowledge iscovey 27; 2(4): ,.7589,.2966,.767,.68827), So the altenative anking is S 6 S S 4 S S 5 S 2 S 9 S 3 S 7 S 8. Table 4. Results compaison between the two methods. S i o domains anking omains Ranking Changing S S S S S S S S S S Seen fom Table 4, except fo the altenatives S 7 and S 8, the bette altenatives anking emains steady whethe consideing the domains o not. Though compaing the esults, we daw the conclusion that some supeio index values o infeio index values contibuting little to decision making. Thus, some supeio index values can be thought as indiffeences and the same with some infeio index values, as may not affect the decision making that seeking fo some bette altenatives. eanwhile, the above esults also indicate that the altenative may not be an excellent altenative with only patially bette index values. 5. Conclusions The poposed gey taget decision method expanding the conventional taget cente to a domain and also consideing taget edge domain can effectively deal with decision makes supeio o infeio indiffeence attibute value pefeences. The appoach has its advantage to cope with multi-attibute altenatives with little diffeence of some index values especially fo so many altenatives. It can not only simplify the calculation but also keep the accuacy of esults, at least fo some excellent altenatives. oeove, decision makes can employ this method of decision making by thei pefeences with consideing the two domains o eithe of them. Acknowledgements The authos thank the suppot by the Key Reseach Poect of Science and Technology of Henan povince (Gant o. 3B6233) and octoal Fund of Henan Polytechnic Univesity (Gant o. B26-53). Refeences [] eng, J. L., Gey System Theoy. 22, Wuhan: Huazhong Univesity of Science and Technology Pess. [2] Chen, Y.. and H. Y. Xie, Test of the inconsistency poblem on eng s gey tansfomation by simulation. Systems Engineeing and Electonics, (8): p [3] Song, J., et al., Gey Risk Goup ecision Based on the aoant Opeato of "Rewading Good and Punishing Bad". Jounal of Gey System, 29. 2(4): p [4] Song, J., Y. G. ang and Z. X. Wang, ulti-attibute decision model of gey taget based on maoant opeato of ewading good and punishing bad. Systems Engineeing and Electonics, 2. 32(6). [5] ang, Y. G., et al., ulti-attibute decision model of gey taget consideing weights. Statistics and ecision, 24(3): p [6] Zhu, J. J. and K. W. Hipel, ultiple stages gey taget decision making method with incomplete weight based on multi-ganulaity linguistic label. Infomation Sciences, : p [7] Song, J., et al., The ecision-making odel of Haden Gey Taget Based on Inteval umbe with Pefeence Infomation on Altenatives. Jounal of Gey System, 29. 2(3): p [8] Wang, Z. X., Y. G. ang and H. Yang, Impovements on decision method of gey taget. Systems Engineeing and Electonics, 29. 3(): p [9] Luo,. and X. Wang, The multi-attibute gey taget decision method fo attibute value within thee-paamete inteval gey numbe. Applied athematical odelling, (5): p [] Luo,., ulti-obective gey taget decision model based on positive and negative clouts. Contol and ecision, (2): p [] ang, Y. G., S. F. Liu and B. Liu, Study on the ulti-attibute ecision odel of Gey Taget Based on Inteval umbe. Engineeing Science, 25. 7(8): p [2] Shen, C. G., Y. G. ang and L. L. Pei, Hybid multi-attibute decision model of gey taget. Statistics and ecision, 2(2): p [3] Song, J., et al., ew decision model of gey taget with both the positive clout and negative clout. Systems engineeing Theoy & pactice, 2. 3(): p [4] a, J. S. and C. S. Ji, Genealized Gey Taget ecision ethod fo ixed Attibutes Based on Connection umbe. Jounal of Applied athematics, : p. 8. [5] a, J. S., Gey Taget ecision ethod fo a Vaiable Taget Cente Based on the ecision ake's Pefeences. Jounal of Applied athematics, : p. 6. [6] Wang, H. H., J. Zhu and F. Z. Geng, Gey taget cluste decision method on linguistic evaluation case-based. Systems Engineeing Theoy & Pactice, (2): p [7] Zhu, J. J., et al., Gey taget decision method based on uncetain evidence aggegation unde conflict inteest paticipants. Contol and ecision, (7): p [8] Liu, Y., et al., ulti-obective gey taget decision-making based on pospect theoy. Contol and ecision, (3): p [9] Zeng, B., et al., Gey taget decision-making model of inteval gey numbe based on cobweb aea. Systems Engineeing and Electonics, (): p [2] Cao, S. G., et al., Compehensive Assessment of Secuity in Coal ines Based on Gey Relevance Analysis. Jounal of ining & Safety Engineeing, (2): p

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