An Improvement in Centroid Point Method for Ranking of Fuzzy Numbers

Transcription

1 From the SeletedWorks of Saeid asand n Improvement in Centroid Point Method for anking of Fuzz Numers Saeid asand T Hajjari vailale at:

2 J Si I U (JSIU), Vol, No 78/, Winter 9 n Improvement in Centroid Point Method for anking of Fuzz Numers S asand * Mathematis Department, Siene and esearh Branh, Islami zad Universit, Tehran, Iran T Hajjari Mathematis Department, Firuz Kuh Branh, Islami zad Universit, Firuz Kuh, Iran strat Introdution: In man appliations, ranking of fuzz numers is an important omponent of the deision proess Man authors have investigated the use of fuzz sets in ranking alternatives and the have studied different methods of raking fuzz sets Partiularl, the ranking of fuzz numers In a paper Cheng [ new approah for ranking fuzz numers distane method, Fuzz Sets and Sstems 95 (998) 7-7], a entroid-ased distane method was suggested for ranking fuzz numers, oth normal and non-normal The method utilizes the Eulidean distanes from the origin to the entroid point of eah fuzz numers to ompare and rank the fuzz numers It is found that the mentioned method ould not rank fuzz numers orretl For eample, it annot rank fuzz numers when the have the same entroid point Some other researhes suh as Chu and Tsao's, Wang and ee and Deng et al tried to overome the shortoming of the inonsisten of Cheng's method ut their methods still have drawak im: In this paper, we want to indiate these prolems of Cheng's distane, Chu and Tsao's area and Wang and ee's revised method and then we will propose an improvement method, whih an avoid these prolems for ranking fuzz numers Materials and Methods: In point of our view, ever fuzz numer ma lead to zero, positive or negative real numer To overome with this sujet, we first introdue the sign funtion B onneting the sign funtion, orreted entroid point formulas and Cheng s distane method, the improvement method will e presented esults: The proposed method inludes all situations of fuzz numers to rank them orretl Therefore, it improves the distane method of Cheng as well Conlusion: In order to overomes the shortoming in Cheng's distane method, Chu and Tsao's formulae and new revised method Wang and ee, a sign funtion was introdued and omposed an improvement strateg of Cheng s distane In this work those method are onsidered, whih utilized the entroid points The improved method an effetivel rank various fuzz numers and their images Thus, the method is superior to Cheng' distane, Chu and Tsao's area and Wang and ee's revised method Kewords: anking, Normal fuzz numers, Non-normal fuzz numers, Centroid point * Corresponding uthor

3 n Improvement in Centroid Point asand and Hajjari Introdution In man appliations, ranking of fuzz numers is an important omponent of the deision proess Man authors have investigated the use of fuzz sets in ranking alternatives and the have studied different methods of raking fuzz sets Partiularl, the ranking of fuzz numers [-5] In a paper Cheng, [6] a entroid-ased distane method was suggested for ranking fuzz numers The method utilizes the Eulidean distanes from the origin to the entroid point of eah fuzz numers to ompare and rank the fuzz numers Chu and Tsao [7] found that the distane method ould not rank fuzz numers orretl if the are negative and therefore, suggested using the area etween entroid point and the origin to rank fuzz numers Deng et al [8] utilized the entroid point of a fuzz numer and presented a new area method to rank fuzz numers with the radius of gration (OG) points to overome the drawak of the Cheng's distane method and Tsao's area method when some fuzz numers have the same entroid point However, OG method annot rank negative fuzz numers asand and sad [4] found that Tsao's area method ould sometimes lead to ounterintuitive ranking and hene suggested a sign distane More reentl, Wang et al [9] pointed out that the entroid point formulas for fuzz numers provided Cheng [6] are inorret and have led to some misappliation suh as Chu and Tsao, [6] [, ] Pan and Yeh and Deng et al [8] The presented the orret entroid formulae for fuzz numers and justified them from the viewpoint of analtial geometr Nevertheless, the main prolem, aout ranking fuzz numers aove methods, whih used the entroid point, is reminded In 8 Wang and ee [] revised Chu and Tsao's method and suggested a new approah for ranking fuzz numers ased on Chu and Tsao's method in awa to similar original point However, there is a shortoming in some situations It will e illustrated in eample 4 In the present paper, we disuss the prolem of aove methods In other words, Cheng's method has still drawak, ie, it annot rank fuzz numers in some situations To overome the shortoming in mentioned methods, we improve Cheng ased-distane method introduing a sign funtion The rest of the paper is organized as follows Setion ontains the asi definitions and notations use in the remaining parts of the paper In Setion, we introdue our idea to improve Cheng's method for ranking fuzz numers Setion 4 demonstrates several numerial eamples the fat that the entroid-ased method, the area method and Wang and ee revised method an signifiantl alter the result of the ranking proedure and lead to a wrong ranking order The paper is onluded in Setion 5 Materials and Methods In this setion, we introdue the asi onepts of fuzz numers and the entroid points of a fuzz numer Then we riefl review the entroid point presented heng [6] and Chu and Tsao [7] In addition the orreted formulae Wang et al [9] fuzz numer is a onve fuzz suset like of the real line is ompletel defined its memership funtion et e a fuzz numer, whose memership funtion f () an generall e defined as [] f f f a,, d, othrwise, ()

4 J Si I U (JSIU), Vol, No 78/, Winter is a onstant, [ ] [ ] and [ ] [ ] where < f : a,, f :, d, are two stritl monotoni and ontinuous mapping from to losed interval[,] If, then is normal fuzz numer; otherwise, it is a trapezoidal fuzz numer and is usuall denoted ( a,,, d; ) or ( a,,, d) if In Partiular, when, the trapezoidal fuzz numer is redued to a triangular fuzz numer denoted ( a,, d; ) or ( a,, d) if So triangular fuzz numers are speial ases of trapezoidal fuzz numers Sine f () and f () are oth stritl monotoni and ontinuous funtions, their inverse funtions eist and should e ontinuous and stritl monotoni et g : [, ] [ a, ] and g : [, ] [, d] e the inverse funtions of f () and f (), respetivel Then g () and g () should e integrale on the lose interval[,] In other words, oth g ( ) d and g d ( ) should eist In the ase of trapezoidal fuzz numer, the inverse funtion g () and () an e analtiall epressed as g g g a ( ) ( ) a + ( a) / d ( d ) / () In order to determine the entroid point ( ), ( )) provided the following entroid formulae of a fuzz numer, Cheng [6] ( f d + d + a d f d + d + a g + ( ) d g ( ) d + g g d f f ( ) d ( ) d d d () Normal fuzz numers an e seen as speial ases of non-normal fuzz numers with In Chu and Tsao, [7] the entroid point formulae were given as f a d + f d + d + a ( ) g + d g ( ) d + d + d f f d d g ( ) d g ( ) d d (4) Pan and Yeh [, ] also adopted the aove formulae However, it is found that the formulas () and (4) are inorret despite the fat that formulas of in relations () and (4) are onsistent

5 n Improvement in Centroid Point asand and Hajjari + with Yager's ranking inde [4, 5] F( ) g( ) f d / f d with the weighted funtion g and and the same as Murakami et al's ranking inde [6] for normal + + fuzz sets Z ( ) f d / f d Wang et al [9] found from the point of view of analtial geometr and showed the orreted entroid point as follows: + f a d + f d + d + a g ( ) d g ( ) d d + d f f d d g ( ) d g ( ) d d (5) For non-normal trapezoidal fuzz numer ( a,, d; ) formulas (5) lead to following results respetivel d a a d ( ) ( ) d + a + + ( ) ( ) d + a + (6) Sine non-normal triangular fuzz numers are speial ases of normal trapezoidal fuzz numers with, formulas (6) an e simplified as [ a + + d] (7) Oviousl, for normal trapezoidal (triangular) fuzz numers, the formulas of in relations (6) and (7) an e simplified as ( and ) ( ), + ( d + ) ( a + )

6 J Si I U (JSIU), Vol, No 78/, Winter respetivel In this ase, normal triangular fuzz numers an e ompared or ranked diretl in terms of their entroid oordinates on horizontal ais For more detail, we refer the reader to [9] Cheng [6] formulated his idea as follows: ( ) + (8) ( ) ( ) Chu and Tsao's [7] omputed the area etween the entroid and original points to rank fuzz numers as: S ) ( ) ( ) (9) ( Wang and ee [] in 8 proposed a revised method for ranking fuzz numers The revised method is ased on the Chu and Tsao's method [7] In short the ranked the fuzz numers ased on their s values if the are different In the ase that the are equal, the further ompare their s values to form their ranks Further, for two fuzz numers and B if ( ) ( B) ased on ( ) ( B), then B To emphasis, we note that in this work trapezoidal fuzz numers are onsidered n Improvement in Cheng s Distane-Based Method s we alread mentioned, in 6 Wang et al [9] pointed out that the entroid point formulas for fuzz numers provided Cheng [6] are inorret In this setion, we deide to utilize Chen's distane method with orreted formulae and onnet it to a sign funtion, whih will e introdued to improve the mentioned method Sine in Cheng's method none of fuzz numer is respeted to zero Therefore, it would e a drawak In point of our view, ever fuzz numer ma lead to zero, positive or negative real numer To overome with this sujet, we first introdue the sign funtion as follows: Definition et E stands the set of non-normal fuzz numers, e a onstant provided that < γ : E,, e a funtion that is defined as: and { } ( g g ), E : γ ( ) sign + d ie γ ( ) ( g + g ) ( g + g ) ( g + g ) d, d, d () It is lear for normal fuzz numers

7 4 n Improvement in Centroid Point asand and Hajjari emark If inf (supp ()) thenγ ( ) emark If sup (supp ()) < then γ ( ) emark If ( a,, d; ) e a smmetri trapezoidal fuzz numer suh that + then γ ( ) Consider fuzz numer, its entroid point is denoted as ( ), ( )), where ( ) ( and ( ) an e alulated orreted entroid point (6) B onneting the sign funtion and relations (8), the improved method denoted I I () instead of () (Cheng [6] proposed), it will e presented as: In other words I( ) γ ( ) ( ) I ( ) γ ( ) + () ( ) ( ) Here, the improved method I () is utilized to rank fuzz numers Therefore, for an two fuzz numers and, I ( ) > I( ) if and onl if f I ( ) < I( ) if and onl if p I ( ) I( ) if and onl if Then we formulate the order and as if and onl if f or, if and onl if p or We onsider the following reasonale properties for the ordering approahes, see [6] For an aritrar finite suset Γ of E and, Γ For an aritrar finite suset Γ of E and (, ) Γ, and, we should have For an aritrar finite suset Γ of E and (,, ) Γ, and, we should have 4 For an aritrar finite suset Γ of E and (, ) Γ, inf{supp( )}> sup{supp( ), we should have 5 For an aritrar finite suset Γ of E and (, ) Γ, inf{supp( )}> sup{supp( )}, we should have f 6 et,, + and + e elements of E If, then + + emark 4 The funtion I () has the properties i, ii, iii,, vi Numerial Eamples Eample Consider two fuzz numers (,,) and, B (,484,968) see Fig

8 J Si I U (JSIU), Vol, No 78/, Winter 5 Fig - Fuzz numers (-,, ) and B (, 484, 968, 9) Intuitivel the ranking order is p B Tale shows the results otained improvement entroid method, Chu and Tsao's formula, revised entroid and Cheng's method, respetivel It is ver lear that Cheng's method lead to an inorret ranking order f B,, whih is ontrar to the ranking order p B otained using our proposed method Moreover, some other methods that utilize the entroid points This shows the fat Cheng's method an lead to wrong ranking orders Tale - Comparative results of Eample Fuzz Numers Centroid Point, Improvement Centroid I ( ) γ () + Chu and Tsao s Method S ( ) Wang and ee s evised Cheng s Distane ( ) + B, 484, esults p B p B p B f B 5 9 Eample Consider two fuzz numers (,,,) and B (,, ), whih are demonstrated in Fig Intuitivel, oth the numers are negative and the ranking order is p B However, Cheng's method, the ranking order is f B, whih unreasonale result is On the other hand 5 9 the images of these to fuzz numers are - (,,, ) and B (,, ), respetivel B our method produing ranking order f B Clearl, similar to Chu and Tesao's area method [7] as ou see in Tale On the ontrar, the result of Cheng's distane [6] and revised entroid [] is p B, whih is unreasonale The proposed method an also overome the shortoming of Wang and ee's method [] in ranking fuzz numers and their images

9 6 n Improvement in Centroid Point asand and Hajjari Fig - Fuzz numers (-5/,-/,, ) and B (-9/4,-/4, /4) Tale - Comparative results of Eample Fuzz Numers Centroid Point, Improvement Centroid I ( ) γ () + Chu and Tsao s Method S ( ) Wang and ee s evised Cheng s Distane ( ) + B -565,75-58, esults p B p B p B f B - -B 565,75 58, esults f B f B p B f B Eample Consider three triangular fuzz numers (,,), B (,,5) and C (,,) (See Fig ), whose memership funtions are respetivel defined as f ( + ) p,, p, othrwise,

10 J Si I U (JSIU), Vol, No 78/, Winter 7 f B + ( 5) 7 p,, p 5, othrwise, f C ( + ) 4 + p, p, othrwise, The orreted entroid point formulas (5) for mentioned numers ield the same results, ie, ( ) and ( ) Then using Cheng's entroid - ased distane method ( ) ( B) ( C) On the other hand appling Tsao's method S ( ) S( B) S( C) Therefore, oth the distane method and area method produing the ranking order B C It shows the fat that the mentioned methods an lead to wrong ranking order ording to formulae () λ( ), γ ( B) and γ ( C) + lso from the formulae () the results will e otained as I ( ), I( B) and I ( C) Hene, the ranking order is p B p C Therefore, using our improved method an effiientl deal with the fuzz ranking prolems when different generalized fuzz numers have the same entroid point Fig - Fuzz numers (-,,), B(-,,5) and C(-,,) Eample 4 The two triangular fuzz numers (,,), B (,,9 ) shown in Figure 4 are ranked our method It will e otain I ( ), I( B) 474 To ompare with other method we refer the reader to Tale

11 8 n Improvement in Centroid Point asand and Hajjari Fig 4- Fuzz numers (-,, ) and B (-,, 9) Tale - Comparative results of Eample 4 Fuzz Numers Modified Centroid Point Sign Distane P Sign Distane P Chu and Tsao Cheng s Distane B esults f B f B f B B B Conlusion: In this paper, we improved Cheng distane method and presented an improvement method The improvement method overomes the shortoming in Cheng's distane method, Chu and Tsao's formulae and new revised method Wang and ee In this work those method are onsidered, whih utilized the entroid point The improved method an effetivel rank various fuzz numers and their images Thus, the method is superior to Cheng' distane, Chu and Tsao' area and Wang and ee's revised method The proposed method inludes all situations of mentioned method, so it improves the distane method of Cheng as well eferenes: asand, S, uas, C, and sad, B, Quarterl J Si, Teaher Training Universit,, 8 () asand, S, and sad, B, ppl Math Comput, 56, 8 (4) asand, S, Otadi, M, and Mosleh, M, Math Sientifi J Islami zad Universit of rak,, (6-7) 4 asand, S, and sad, B, Inform Si, 76, 45 (6) 5 asand, S, and Hajjari, T, Comput Math ppl, 57, 4(9) 6 Cheng, CH, Fuzz Sets and Sst, 95, 7(998)

12 J Si I U (JSIU), Vol, No 78/, Winter 9 7 Chu, T, and Tsao, C, Comput Math ppl, 4, () 8 Deng, Y, Zhenfu, Z, and Qi,, Comput Math ppl, 5, 7 (6) 9 Wang, YM, Yang, JB, Xu, D, and Chin, KS, Fuzz Sets and Sst, 57, 99 (6) Pan, H, and Yeh, CH, Pro th IEEE Internat Conf On Fuzz Sstems, 755 () Pan, H, and Yeh, CH, Metaheuristi pproah to Fuzz Projet Sheduling, in: V, Palade, J, Howlett, C, Jain (Eds), Knowledge-ased Intelligent Information and Engineering Sstems, Springer, Berlin () Wang, YJ, and ee, HSh, Comput Math ppl, 55, (8) Duios, D, and Prade, H, Internat J Sstem Si, 9, 6 (978) 4 Yager,, Pro 7th IEEE Conf on Cerneti and Soiet, 9 (978) 5 Yager,, Inform Si, 4, 4 (98) 6Wang, X, and Kerri, EE, Fuzz Sets and Sst, 8, 75 ()

13 مجله علوم پايه دانشگاه آزاد اسلامي (JSIU) سال شماره 78/ زمستان بهبودي در روش مركز ثقل جهت رتبه بندي اعداد فازي سعيد عباس * بندي گروه رياضي واحد علوم و تحقيقات دانشگاه آزاد اسلامي تهران ايران طيبه حجاري گروه رياضي واحد فيروزكوه دانشگاه آزاد اسلامي فيروزكوه ايران 88// تاريخ دريافت: 86//6 تاريخ پذيرش: چكيده مقدمه: در بسياري از مساي ل كاربردي رتبه بندي اعداد فازي يكي از مولفه هاي مهم مراحل تصميم گيري است لذا اين موضوع از اهميت ويژه اي برخوردار است محققين زيادي در اين زمينه تحقيقات وسيعي انجام داده و به روش هاي متفاوتي دست يافته اند چنگ در يكي از مقاله هاي خود (روشي جهت رتبه بندي اعداد فازي با استفاده ازروش فاصله مجموعه هاي فازي و سيستم ها جلد 998) 95 روشي به نام "فاصله ي مركز ثقل" را جهت رتبه بندي اعداد فازي نرمال و غير نرمال پيشنهاد داد در روش وي از فاصله اقليدسي بين مبدا مختصات و مركز ثقل استفاده شده است پس از مدتي معلوم شد كه روش فاصله مركزثقل چنگ داراي نقص مي باشد و قادر به رتبه بندي همه اعداد فازي نيست بعنوان مثال آن دسته از اعدادي كه داراي مركز ثقل يكسان هستند محققين ديگري بر آن شدند كه نقص روش چنگ را برطرف كنند و به اين ترتيب بر اساس مختصات مركزثقل روشهاي ديگري را اراي ه شد هدف: در اين مقاله ضمن مطرح كردن اشكالات روش هايي كه بر اساس مختصات مركز ثقل اراي ه شده روش بهبود يافته اي بر اساس روش فاصله چنگ پيشنهاد داده بطوريكه روش بهبود يافته پيشنهادي قادر به رتبه بندي اعداد گوناگون فازي باشد و مشكلات روش هاي قبلي را نداشته باشد روش بررسي: از نقطه نظر ما هر عدد فازي ممكن است صفر مثبت يا منفي باشد جهت فاي ق آمدن بر مشكلات موجود در روش هايي كه بر اساس مركز ثقل هستند ابتدا تابعي به نام تابع علامت معرفي مي كنيم با استفاده ازتابع علامت مختصات مركز ثقل تصحيح يافته و روش فاصله چنگ روش بهبود يافته اي بر اين مبنا ساخته مي شود * عهدهدار مكاتبات: aasand@ahooom تلفن: 956 فكس: 8-784

14 67 مجله علوم پايه دانشگاه آزاد اسلامي ( JSIU ) جلد 9 شماره 74/ زمستان 89 نتايج: روش پيشنهادي قادر به رتبه بندي كليه اعداد فازي در هر وضعيتي مي باشد لذا بدين ترتيب روش فاصله چنگ به خوبي بهبود يافته است نتيجه گيري: با استفاده از يك تابع علامت تركيب آن با روش فاصله چنگ و مختصات مركز ثقل تصحيح يافته مي توان به روش مناسبي دست يافت كه در واقع بهبود يافته روش چنگ است و قادر به رتبه بندي كليه اعداد فازي در هر وضعيت ميباشد و مشكلات روش هاي پيشين را ندارد واژههاي كليدي: رتبه بندي اعداد فازي نرمال اعداد فازي غير نرمال مركز ثقل