The Mean Area Measurement Method to. Multiple Attribute Decision Making within. Triangular Fuzzy Numbers 1

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1 Apped Mathematca Sceces, Vo. 9, 015, o. 8, HIKARI Ltd, The Mea Area Measuremet Method to Mutpe Attrbute Decso Makg wth Traguar Fuzzy Numbers 1 Tachao Wag, Yg Meg, Ru Ma, Jyag Che, Da Gao, Qgja Zhou ad Ja Jao Coege of Scece, Daa Natoates Uversty Daa, , Cha Copyrght 01 Tachao Wag et a. Ths s a ope access artce dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orga work s propery cted. Abstract I ths paper, the Mea Area s adopted to express the correspodg traguar fuzzy umber. So the Mea Area Measuremet Method s used to sove mutpe attrbute decso makg wth attrbute vaues wth traguar fuzzy umbers. Keywords: Mea area measuremet method, Mutpe attrbute, Decso makg, Traguar fuzzy umber 1 Itroducto Mutpe attrbute decso makg s characterzed by a decso maker, who s caed to rak a the ateratves as we as seect the best. I may stuatos, attrbute vaues are gve the form of traguar fuzzy umber. So far a ot of research has bee doe to t, see [1-]. I ths paper, frsty we use Mea Area to express the correspodg traguar fuzzy umbers. Secody the Mea Area 1 Ths work s supported by the Educato Departmet foudato of Laog Provce (L0159), the Doctor Support Foudato of DLNU(110090), Su Brd Fouds of DLNU(ty015,0139,015), Coege Studet s Iovatve Etrepreeura Trag Program of Laog Provce ( ) ad of DLNU(x010918) Correspodg author

2 39 Tachao Wag et a. Measuremet Method s used to sove mutpe attrbute decso makg wth attrbute vaues wth traguar fuzzy umbers. Decso makg Method Defto 1 M [, m, u] s caed a traguar fuzzy umber, ad ts membershp fucto s the foowg: x, x [, m]; m u x ( x), x [ m, u]; M u m 0, others. Here, 0 m u are rea umbers. Specay, whe m u, M s degraded to a rea umber, as [1,3,]. Traguar fuzzy umbers ca express ucerta formato very we. For some [0,1], we deote cut set of M as u M [ m ( ), m ( )], So u m ( ) m ( ) mm ( ) s the mea of cut set of M h M Defto s( M ) m( M ) d s caed the Mea Area of M. 0 here, h M s the heght of M. For ay traguar fuzzy umber M [, m, u], accordg to the defto of the Mea Area of M, we ca get ( m u) sm ( ) Defto 3 et WWA : R R, f WWA (,,, ), 1 j j j1 where =( 1,,, ) T s the weghted vector, j [0,1], ad j =1, j1 R s the set of a rea umbers, the WWA s caed ad weghted averagg operator.

3 The mea area measuremet method 395 So we ca use Mea Area to express the correspodg traguar fuzzy umbers. Secody ths paper the Mea Area Measuremet Method s used to sove mutpe attrbute decso makg wth attrbute vaues wth traguar fuzzy umbers accordg towwa agorthm. 3 Decso Makg Steps Step 1 Repace each traguar fuzzy umber M [, m, u] of the orga decso matrx wth ts Mea Area sm ( ), so the orga matrx s trasformed to the matrx composed by ther Mea Area vaues A= ( a), here a s( M% ), Step Let I1, I represet the subscrpt sets of the beeft type ad the cost beeft attrbutes. Stadard A= ( a)to stadardzato matrx R ( r ) as: a r =, 1, L j ; I1 (1) max( a ) m( a ) r =, 1, L ; j I () a Step 3 Obta =( 1, L, ) accordg to the mportace of each attrbute, ad here we adopt WWA agorthm to get overa vaue for each ateratve m 1 m j j1 z ( ) WWA ( r, r, L, r ) (3) Step Rak the ateratves ad seect the best by z ( ). Iustratve Exampe Cosder the foowg exampe. We w evauate four departmets x ( 1,,3, ) oe uversty agast four attrbutes: u : Teachg eve, 1 u : Research eve, u : Operatg cost, 3 u : Satsfacto degree of studets Ad the evauatg resuts are show Tabe 1 wth attrbuto vaues the form of traguar fuzzy umbers, where Teachg eve, Research eve ad Satsfacto degree of studets are beeft attrbutes, ad Operatg cost s cost attrbute. Whch departmet s the best. []

4 396 Tachao Wag et a. TABLE 1. Decso Makg Matrx Tabe u1 u u3 u x 1 [7,8,10] [5,6,8] [7,8,9] [5,7,8] x [5,8,10] [,5,7] [,6,7] [6,8,10] x [6,7,9] [6,7,8] [7,8,9] [6,8,9] 3 x [6,8,9] [6,7,8] [6,8,10] [5,7,9] Step 1 Repace each attrbute vaue of traguar fuzzy umber of M [ a, am, au ] the orga decso matrx Tabe 1 wth ts Mea Area ad get the matrx composed by the Mea Area, A=( a )= Step Cacuate the stadardzato matrx by (1) ad () R( r ), Step 3 Here we set =(0.3,0.3,0.,0.) based o the mportace of the four attrbutes ad cacuate z ( ) by (3) z ( )= , z ( )= , z ( )= 0.901, z ( )= , 1 3 Step Utze z ( ) to rak the ateratves: x f x3 f x f x1, ad thus the best ateratve s x, so the best departmet s the th departmet. Refereces [1] D. F. L, Fuzzy mutobjectve may-perso decso makg ad games. Beg: Natoa Defece Idustry Press, 003. [] Z. S. Xu, Ucerta mutpe attrbuto decso makg: methods ad appcatos. Beg: Tsghua Uversty Press, 00.

5 The mea area measuremet method 397 [3] Q. J. Zhou, J. Jao ad D. L. Yag, The formato etropy based agorthm to mutpe attrbute decso makg wth attrbute vaues the form of traguar fuzzy umbers, ICIC Express Letters Part B: Appcatos, (01,5(5):179-18). [] Q. J. Zhou ad J. Jao. The Ordered Weghted Averagg Agorthm to Mutpe Attrbute Decso Makg wth Traguar Fuzzy Numbers, Apped Mathematca Sceces, Vo. 8, 01, o. 56, Receved: December 1, 01; Pubshed: Jauary 6, 015