Taylor Series Approximation to Solve Neutrosophic Multiobjective Programming Problem

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1 Taylor Seres Approxmato to Solve Neutrosophc Multobectve Programmg Problem Abstract. ths paper Taylor seres s used to solve eutrosophc mult-obectve programmg problem (NMOPP. the proposed approach the truth membershp determacy membershp falsty membershp fuctos assocated wth each obectve of mult-obectve programmg problems are trasformed to a sgle obectve lear programmg problem by usg a frst order Taylor polyomal seres. Fally to llustrate the effcecy of the proposed method a umercal expermet for suppler selecto s gve as a applcato of Taylor seres method for solvg eutrosophc mult-obectve programmg problem at ed of ths paper. Keywords: Taylor seres; Neutrosophc optmzato; Multobectve programmg problem. 1 troducto 1995 startg from phlosophy (whe [8] fretted to dstgush betwee absolute truth ad relatve truth or betwee absolute falsehood ad relatve falsehood logcs ad respectvely betwee absolute membershp ad relatve membershp or absolute o-membershp ad relatve o-membershp set theory [12] bega to use the o-stadard aalyss. Also spred from the sport games (wg defeatg or te scores from votes (pro cotra ull/black votes from postve/egatve/zero umbers from yes/o/na from decso makg ad cotrol theory (makg a decso ot makg or hestatg from accepted/reected/pedg etc. ad guded by the fact that the law of excluded mddle dd ot work ay loger the moder logcs. [12] combed the o-stadard aalyss wth a tr-compoet logc/set/probablty theory ad wth phlosophy.how to deal wth all of them at oce s t possble to uty them?[12]. Netrosophc theory meas Neutrosophy appled may felds order to solve problems related to determacy. Neutrosophy s a ew brach of phlosophy that studes the org ature ad scope of eutraltes as well as ther teractos wth dfferet deatoal spectra. Ths theory cosders every etty <A> together wth ts opposte or egato <ata> ad wth ther spectrum of eutraltes <euta> betwee them (.e. ettes supportg ether <A> or<ata>. The <euta> ad <ata> deas together are referred to as <oa>. Neutrosophy s a geeralzato of Hegel's dalectcs (the last oe s based o <A> ad <ata> oly. Accordg to ths theory every etty <A> teds to be eutralzed ad balaced by <ata> ad <oa> ettes - as a state of equlbrum. a classcal way <A> <euta> <ata> are dsot two by two. But sce may cases the borders betwee otos are vague mprecse Sortes t s possble that <A> <euta> <ata> (ad <oa> of course have commo parts two by two or eve all three of them as well. Hece oe had the Neutrosophc Theory s based o the trad <A> <euta> ad <ata>. the other had Neutrosophc Theory studes the determacy labeled as wth = for 1 ad m + = (m+ eutrosophc structures developed algebra geometry topology etc. The most developed felds of Netrosophc theory are Neutrosophc Set Neutrosophc ogc Neutrosophc Probablty ad Neutrosophc Statstcs - that started 1995 ad recetly Neutrosophc Precalculus ad Neutrosophc Calculus together wth ther applcatos practce. Neutrosophc Set ad Neutrosophc ogc are geeralzatos of the fuzzy set ad respectvely fuzzy logc (especally of tutostc fuzzy set ad respectvely tutostc fuzzy logc. eutrosophc logc a proposto has a degree of truth (T a degree of determacy ( ad a degree of falsty (F where TF are stadard or o-stadard subsets of ] [. Mult-obectve lear programmg problem (MOPP a promet tool for solvg may real decso makg problems lke game theory vetory problems agrculture based maagemet systems facal ad corporate plag producto plag marketg ad meda selecto uversty plag ad studet admsso health care ad hosptal plag ar force mateace uts bak braches etc. Our obectve ths paper s to propose a algorthm to the soluto of eutrosophc mult-obectve programmg problem (NMOPP wth the help of the frst order Taylor s theorem. Thus eutrosophc eutrosophc mult-obectve lear programmg problem s reduced to a equvalet mult-obectve lear programmg problem. A algorthm s proposed to determe a global optmum to the problem a fte umber of steps. The feasble rego s a bouded set. the proposed approach we have attempted to reduce computatoal complexty the soluto of (NMOPP. The proposed algorthm s appled to suppler selecto problem. The rest of ths artcle s orgazed as follows. Secto 2 gves bref Some prelmares. Secto 3 descrbes the

2 Formato of The Problem. Secto 4 presets the mplemetato ad valdato of the algorthm wth practcal applcato. Fally Secto 6 presets the cocluso ad proposals for future work. 2 Some prelmares Defto 1. [1] A real fuzzy umber J % s a cotuous fuzzy subset from the real le R whose µ % J s defed by a tragular membershp fucto ( cotuous mappg from R to the closed terval [01] where (1 µ ( J = 0 J a J% for all ( 1] (2 µ J % ( J s strctly creasg o [ m ] (3 µ J% ( J = 1 for J = m (4 µ J % ( J s strctly decreasg o J [ m a2 ] (5 µ % ( J = 0 for all [ 2 J J 1 Ths wll be elcted by: J a1 a1 J m m a 1 a2 µ J% ( J = m J a2 (1 a2 m 0 otherwse. Fgure 1: Membershp Fucto of Fuzzy Number J. where m s a gve value a 1 ad a 2 deote the lower ad upper bouds. Sometmes t s more coveet to use the otato explctly hghlghtg the membershp fucto parameters. ths case we obta J a1 a2 J µ ( J ; a1 m a2 = Max M 0 (2 m a1 a2 m what follows the defto of the α-level set or α-cut of the fuzzy umber J % s troduced. Defto 2. [1] et X = {x 1 x 2 x } be a fxed oempty uverse a tutostc fuzzy set FS A X s defed as A = x µ x υ x x X (3 { A ( A ( } whch s characterzed by a membershp fucto µ A : X [ 01] ad a o-membershp fucto [ ] ( x ( x υa : X 01 wth the codto 0 µ A + υa 1 for all x X where µ A ad υa represet respectvely the degree of membershp ad o-membershp of the elemet x to the set A. addto for each FS A X π A ( x = 1 µ A ( x υa ( x for all x X s called the degree of hestato of the elemet x to the set A. Especally f π A ( x = 0 the the FS A s degraded to a fuzzy set. Defto 3. [4] The α-level set of the fuzzy parameters J % problem (1 s defed as the ordary α J % for whch the degree of membershp set ( fucto exceeds the level α α [ 01] where: ( = { J% ( } α J % J R µ J α (4 For certa values α to be the ut terval Defto 4. [10] et be a space of pots (obects ad. A eutrosophc set s defed by a truth-membershp fucto ( a determacymembershp fucto ( ad a falsty-membershp fucto (. t has bee show fgure 2. ( ( ad ( are real stadard or real ostadard subsets of ]0 1+[. That s ( : ]0 1+[ ( : ]0 1+[ ad F ( : ]0 1+[. There s ot restrcto o the sum of ( ( ad ( so 0 sup ( sup ( ( 3+. the followg we adopt the otatos μ ( σ ( ad ( stead of ( ( ad ( respectvely. Also we wrte SVN umbers stead of sgle valued eutrosophc umbers. Defto 5. [10] et be a uverse of dscourse. A sgle valued eutrosophc set over s a obect havg the form ={ μ ( σ ( ( : } where μ ( : [01] σ ( : [01] ad ( : [01] wth 0 μ ( + σ ( + ( 3 for all. The tervals μ ( σ ( ad ( deote the truth- membershp degree the determacy-

3 membershp degree ad the falsty membershp degree of to respectvely. For coveece a SVN umber s deoted by =( where [01] ad Defto 6 et J % be a eutrosophc umber the set of real umbers R the ts truth-membershp fucto s defed as J a1 a1 J a2 a2 a1 a2 TJ% ( J = a2 J a3 (5 a3 a2 0 otherwse. ts determacy-membershp fucto s defed as J b1 b1 J b2 b2 b1 b2 J ( J = b2 J b3 b3 b2 0 otherwse. % (6 ad ts falsty-membershp fucto s defed as J c1 c1 J c2 c2 c 1 c2 FJ ( J = c2 J c3 c3 c2 1 otherwse. % (7 Fgure 2: Neutrosophcato process [11] 3 Formato of The Problem The mult-obectve lear programmg problem ad the mult- obectve eutrosophc lear programmg problem are descrbed ths secto. A. Mult-obectve Programmg Problem (MOPP ths paper the geeral mathematcal model of the MPP s as follows[6]: m/ max z1 x1... x z 2 x1... x... z p x1... x (8 subect to x S x 0 ( ( ( (9 S = x R AX = b X 0. B. Neutrosophc Mult-obectve Programmg Problem (NMOPP f a mprecse asprato level s troduced to each of the obectves of MOPP the these eutrosophc obectves are termed as eutrosophc goals. et z z z deote the mprecse lower ad upper bouds respectvely for the th eutrosophc obectve fucto. for maxmzg obectve fucto the truth membershp determacy membershp falsty membershp fuctos ca be expressed as follows:

4 1 f z z µ ( z = f z z 0 f z (10 1 f z z σ ( z = f z z 0 f z (11 0 f z z υ ( z = f z z 1 f z (12 for mmzg obectve fucto the truth membershp determacy membershp falsty membershp fuctos ca be expressed as follows: 1 f z z µ ( z = f z z 0 f z 1 f z z σ ( z = f z z 0 f z 0 f z z υ ( z = f z z 1 f z (13 (14 (15 4 Algorthm for Neutrosophc Mult-Obectve Programmg Problem The computatoal procedure ad proposed algorthm of preseted model s gve as follows: x = x 1 x 2... x that s used to Step 1. Determe ( maxmze or mmze the th truth membershp fucto µ ( X the determacy membershp σ ( X ad the falsty membershp fuctos υ ( X ad s the umber of varables.. =12..p Step 2. Trasform the truth membershp determacy membershp falsty membershp fuctos by usg frst-order Taylor polyomal seres ( x ( x + ( x x = 1 µ µ σ ( x σ ( x + ( x x = 1 ( x υ ( x + ( x x = 1 υ µ (16 σ (17 υ (18 Step 3. Fd satsfactory x ( x 1 x 2... x = by solvg the reduced problem to a sgle obectve for the truth membershp determacy membershp falsty membershp fuctos respectvely. p µ ( x p( x = µ ( x x + = 1 = 1 p q x x x x = 1 = 1 ( = σ ( + ( σ (19 p υ h( x = υ ( x x + = 1 = 1 Thus eutrosophc multobectve lear programmg problem s coverted to a ew mathematcal model ad s gve below: Maxmze or Mmze p(x Maxmze or Mmze q(x Maxmze or Mmze h(x Where µ ( X σ ( X ad υ ( X calculate usg equatos (10 (11 ad (12 or equatos (13 (14 ad (15 accordg to type fuctos maxmum or mmum respectvely. 4.1 llustratve Example A mult-crtera suppler selecto s selected from [2]. For supplyg a ew product to a market assume that three supplers should be maaged. The purchasg crtera are et prce qualty ad servce. The capacty costrats of supplers are also cosdered. t s assumed that the put data from supplers performace o these crtera are ot kow precsely. The eutrosophc values of ther cost qualty ad servce level are preseted Table 1. The mult-obectve lear formulato of umercal example s preseted as m z 1 max z 2 z 3 :

5 m z 1 = 5x 1 + 7x x 3 max z 2 = 0.80x x x 3 max z 3 = 0.90x x x 3 s. t. : x 1 + x 2 + x 3 = 800 x x x x 0 = Table 1: Supplers quattatve formato Z1 Cost Z2Qualty (% Z3 Servce (% Capacty Suppler Suppler Suppler The truth membershp determacy membershp falsty membershp fuctos were cosdered to be eutrosophc tragular. Whe they deped o three scalar parameters (a1ma2. z 1 depeds o eutrosophc asprato levels ( whe z2 depeds eutrosophc asprato levels ( ad z3 depeds eutrosophc asprato levels ( The truth membershp fuctos of the goals are obtaed as follows: 0 f z z1 f 3550 z µ 1 ( z1 = 4900 z1 f 4225 z f z f z z f z µ 2 ( z2 = z2 660 f 660 z f z f z3 700 z f z µ 3 ( z3 = z f z f z f 4225 ( 5x1 + 7x 2 + 4x ( 5x1 + 7x 2 + 4x 3 µ 1 ( z1 = max m ( 0.8 x1+ 0.9x x ( z2 = m(max( 21 ( 0.8x x x ( 0.9x x x ( z3 = m(max( ( 0.9x x x µ µ The µ 1 ( µ 2 ( µ 3 ( The truth membershp fuctos are trasformed by usg frst-order Taylor polyomal seres ( x = ( ( x 350 µ µ ( 1 µ ( ( µ µ ( x ( x µ 1 ( x x x x the smlar way we get µ 2 ( x x x x µ 3 ( x 0.042x x x The the p(x s p ( x = µ 1 ( x + µ 2 ( x + µ 3 ( x ( p x x x x s. t. : x 1 + x 2 + x 3 = 800 x x x x 0 = The lear programmg software NGO 15.0 s used to solve ths problem. The problem s solved ad the optmal soluto for the truth membershp model s obtaed s as follows: (x 1 x 2 x 3 = ( z 1 =3550 z 2 = z 3 = The truth membershp values are µ 1 = 1 µ 2 = µ 3 = The truth membershp fucto values show that both goals z 1 z 3 ad z 2 are satsfed wth 100% 11.63% ad 89.4% respectvely for the obtaed soluto whch s 1=350; 2= 0 x3=450. the smlar way we get σ ( X q(x Cosequetly we get the optmal soluto for the determacy membershp model s obtaed s as follows: (x 1 x 2 x 3 =( z 1 =3550 z 2 =662.5 z 3 =697.5

6 ad the determacy membershp values are µ 1 = 1 µ 2 = µ 3 = The determacy membershp fucto values show that both goals z 1 z 3 ad z 2 are satsfed wth 100% 11.63% ad 89.4% respectvely for the obtaed soluto whch s 1=350; 2= 0 x3=450. the smlar way we get υ ( X ad h(x Cosequetly we get the optmal soluto for the falsty membershp model s obtaed s as follows: (x 1 x 2 x 3 =( z 1 =3550 z 2 =662.5 z 3 =697.5 ad the falsty membershp values are µ 1 = 0 µ 2 = µ 3 = The falsty membershp fucto values show that both goals z 1 z 3 ad z 2 are satsfed wth 0% 88.37% ad 10.6% respectvely for the obtaed soluto whch s 1=350; 2= 0 x3= Coclusos ad Future Work ths paper we have proposed a soluto to Multobectve programmg problem (NMOPP. The truth membershp determacy membershp falsty membershp fuctos assocated wth each obectve of the problem are trasformed by usg the frst order Taylor polyomal seres. The eutrosophc multobectve programmg problem s reduced to a equvalet multobectve programmg problem by the proposed method. The soluto obtaed from ths method s very ear to the soluto of MOPP. Hece ths method gves a more accurate soluto as compare to other methods. Therefore the complexty solvg NMOPP has reduced to easy computato. the future studes the proposed algorthm ca be solved by metaheurstc algorthms. Referece [1] O.M. Saad B.M.A. Hassa ad.m. Hzam Optmzg the udergroud water cofed steady flow usg a fuzzy approach teratoal Joural of Mathematcal Archve (JMA SSN vol [2] X. X Y. E ad W. DA tutostc fuzzy teger programmg based o mproved partcle swarm optmzato [J] Joural of Computer Applcatos vol p [3] A. YüCel ad Al Fuat GüNer. "A weghted addtve fuzzy programmg approach for multcrtera suppler selecto." Expert Systems wth Applcatos 38 o. 5 (2011: [4].M. Hzam O. A. Raouf M. M. Hadhoud " Solvg Fractoal Programmg Problems sg Metaheurstc Algorthms der certaty " teratoal Joural of Advaced Computg vol p [5]. Y. We-m ad X. Xao-la tutostc Fuzzy Blevel Programmg by Partcle Swarm Optmzato Computatoal tellgece ad dustral Applcato PACA 08. Pacfc-Asa Workshop o EEE 2008 pp [6] Abdel-Baset Mohamed ad brahm M. Hezam. "A mproved Flower Pollato Algorthm for Ratos Optmzato Problems." Appled Mathematcs & formato Sceces etters A teratoal Joural 3 o. 2 (2015: [7] R. ree HepzbahR ad Vdhya tutostc Fuzzy Mult-Obectve lear Programmg Problem(FMOPP usg Taylor Seres teratoal Joural of Scetfc ad Egeerg Research (JSER Vol. 3 ssue 6 Jue [8] A. Amd S. H. Ghodsypour ad Ch O Bre. "Fuzzy multobectve lear model for suppler selecto a supply cha." teratoal Joural of Producto Ecoomcs 104 o. 2 (2006: [9] D. Ptu Tapa K. R. Mult-obectve olear programmg problem based o Neutrosophc Optmzato Techque ad ts applcato Rser Desg Problem Neutrosophc Sets ad Systems Vol : [10] R. Şah ad Muhammed Y. "A Mult-crtera eutrosophc group decso makg metod based TOPSS for suppler selecto." arxv لتعاريف (2014. arxv: preprt [11] S. Aggarwal Rat B. ad A. Q. Asar. "Neutrosophc modelg ad cotrol." Computer ad Commucato Techology (CCCT 2010 teratoal Coferece o pp EEE [12] Smaradache F. "A Geometrc terpretato of the Neutrosophc Set-A Geeralzato of the tutostc Fuzzy Set." arxv preprt math/ (2004.